Deformation and Buckling of Isolated and Interacting Thin Shells in an Elastic Medium

DEFORMATION AND BUCKLING OF ISOLATED AND INTERACTING THIN SHELLS IN AN ELASTIC MEDIUM

A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Engineering and Physical Sciences

2016

By Maria Thorpe School of Mathematics Contents

Abstract 17

Declaration 18

Acknowledgements 20

1 Introduction 21 1.1 Introduction...... 21

2 Background 26 2.1 Tensor notation and coordinate systems ...... 26 2.1.1 Curvilinear coordinates: tensors and vectors ...... 27 2.1.2 Vectors ...... 28 2.1.3 Tensors ...... 28 2.1.4 Cartesiancoordinates ...... 29 2.1.5 Spherical polar coordinates ...... 30 2.2 ElasticityTheory...... 31 2.2.1 LinearElasticity ...... 32 2.2.2 Equations of linear elasticity in spherical coordinates ...... 34 2.2.3 Constitutive equations in spherical coordinates ...... 35 2.3 Nonlinear elasticity for shells ...... 36 2.3.1 The Boussinesq-Papkovich stress functions ...... 43 2.3.2 Multipolesolutions...... 45 2.4 Singlecavityproblems ...... 48 2.4.1 Steady state heat conduction in an inﬁnite medium containing a sphericalcavity...... 49 2.4.2 Spherical cavity in an elastic medium: A spherically symmetric problem...... 51 2.4.3 Spherical cavity in an elastic medium: An axisymmetric problem 54

2 2.5 Singleshellproblems...... 61 2.5.1 Thick spherical shell embedded in an elastic medium: Spherically symmetricproblem...... 61 2.5.2 Thin spherical shell embedded in an elastic medium: Spherically symmetricproblem-asecondapproach ...... 71 2.5.3 Spherical shell embedded in an elastic medium: Axisymmetric problem...... 77 2.5.4 Shellunderuniaxialpressure ...... 88

3 Single shell buckling problems 100 3.1 Bucklingtheory...... 100 3.1.1 Previouswork...... 100 3.1.2 Buckling methodology for an embedded spherical shell under ex- ternalpressure ...... 101 3.1.3 TheTreﬀtzCriterion...... 102 3.1.4 Rayleigh-Ritz and the critical buckling stress ...... 106 3.2 Buckling of a spherical shell in an elastic medium ...... 107 3.2.1 Hydrostatic pressure: The shell equilibrium solution and its approximations ...... 108 3.2.2 Uniaxial compression: The shell equilibrium solution and its approximations ...... 109 3.2.3 Criticalbucklingpressure ...... 110 3.2.4 TheEigenvalueproblem ...... 124 3.2.5 Results ...... 126

4 Cavities interacting in an elastic medium 141 4.1 Problemconﬁguration ...... 141 4.1.1 Conditionsuponthesystem...... 142 4.1.2 Nondimensionalising ...... 143 4.1.3 Application of the boundary conditions ...... 144 4.2 Displacementofthemedium ...... 144 4.2.1 Boussinesq-Papkovich stress function method ...... 145 4.2.2 Expansion of the displacement about cavity i ...... 146 4.3 Stresses upon cavity i ...... 149

4.3.1 The radial stress σriri ...... 149

4.3.2 The radial shear stress σriφi ...... 150 4.3.3 Applyingtheboundaryconditions ...... 150 4.4 Results...... 154 4.4.1 Agreement with single cavity solution ...... 154

3 4.4.2 Equalsizedcavities...... 154 4.4.3 Cavitieswithdiﬀeringradii ...... 159

5 Shells interacting in an elastic medium 162 5.1 Displacement and stresses throughout the composite ...... 164 5.1.1 Stresses and displacements within the host ...... 165 5.1.2 Stresses and displacements within shell i ...... 166 5.2 Boundary conditions upon shell i ...... 167 5.2.1 Solvingtheboundaryconditions ...... 172 5.3 Results...... 172 5.3.1 Interaction of shells ...... 172

6 Buckling of interacting shells 184 6.1 Bucklingmethodology ...... 184 6.1.1 TheTreﬀtzCriterion...... 184 6.1.2 Rayleigh-Ritz and the critical buckling stress ...... 187 6.2 EnergyIntegrals ...... 188 6.2.1 TheFirstEnergyIntegral ...... 188 6.2.2 TheSecondEnergyIntegral ...... 194 6.2.3 TheThirdEnergyIntegral ...... 194 6.3 TheEigenvalueProblem...... 202 6.4 Results...... 205 6.4.1 Agreement with single shell buckling - dilute limit ...... 206 6.4.2 Equalsizedshells...... 208 6.4.3 Unequalshells ...... 213

7 Summary and Further Work 216 7.1 Summary ...... 216 7.2 FurtherWork ...... 217

Appendices 220

A Solution methods in linear elasticity 220 A.1 Laplace’s equation and solutions ...... 221

B Expressing displacements at inﬁnity 224

C Thin shell approximation for radial stress 227

D Useful identities for Legendre polynomials 229

4 References 231

5 List of Tables

3.1 Critical buckling pressures and associated mode numbers derived using various methods for an embedded shell under hydrostatic pressure where

1 = 200, ν1 = 0.3, ν0 = 0.49985, h = 0.01 ...... 132 h 6.1 Critical buckling pressures for single shells of thickness ratio and R+ h equal small shells of radius R+ = 0.01 and thickness ratio de- R+ rived using the single shell buckling pressures for shells under hydrostatic pressure (3.112), and the two-shell system. Shell parameters are

1 = 200, ν1 = 0.3, host parameter ν0 = 0.49985...... 207

6 List of Figures

1.1 Rubberlike composite used by the underwater engineering sector: X- ray micrograph produced by Fonseca, McDonald and Withers (School of Materials, University of Manchester)...... 22 1.2 Schematic of rubberlike composite containing spherical gas-filled shells. . 23 1.3 Experimentally determined stress-strain curve associated with a Silicone RTV microsphere elastomer filled with various volume fractions of Ex- pancel microspheres under uniaxial tension. The solid curve is unfilled and others refer to increasing volume fractions of the microsphere phase (dashed 10%, dot-sashed 20%, dotted 30%, dash-dot-dot 40%) [23]. . . . 24

2.1 Cartesiancoordinatesystem...... 30 2.2 Sphericalpolarcoordinatesystem...... 30 2.3 A point P in an undeformed body Ω can be represented by position vector X. In the deformed body Ω′ the point P moves to position P ′ and is represented by position vector x. The displacement vector u is the change in position vector from the undeformed to deformed state. . 32

2.4 Shell surface geometry such that θ1 and θ2 are coordinates in the shell

mid-surface whilst θ3 describes movement away from this mid-surface. . 37

2.5 Spherical shell geometry (θ1,θ2,θ3) in relation to spherical coordinates (r, φ, θ)...... 39

2.6 Vector schematic of a point P with reference to two diﬀerent origins O1

and O2 ...... 46

2.7 Schematic of a point P with reference to two different origins O1 and O2 47 2.8 Spherical cavity in an infinite medium...... 49 2.9 Temperature distribution ψ throughout the medium, on y = 0, when ψ = T cos φ is applied on the surface of the cavity (for T = 8 and cavity radius R = 1) and ψ → 0 as radial distance r tends to infinity...... 50 2.10 Spherical cavity in an infinite elastic medium...... 51

7 2.11 Absolute value of the radial displacement in an infinite elastic medium a) with a cavity of unit radius (blue) and b) with no cavity (pink).r ¯ is the radial distance from the centre of the cavity and medium...... 54 2.12 Off center spherical cavity in an infinite elastic medium...... 54 2.13 Deformation of cavity in spherically symmetric (pink, dotted) and ax-

isymmetric (blue, solid) displacement ﬁelds, where ν0 = 0.49985, L = 100, p =2...... 60

2.14 Radial displacement throughout a composite, with a host medium (ν0 = 0.49985) under far ﬁeld hydrostatic pressure (p = 1), containing: a shell

(1 = 5,ν1 = 0.25) of thickness h = 0.1 with inner shell boundary

R0 = 0.9 (blue, solid); a cavity of radius 0.9(pink,dashed)...... 65 2.15 Radial displacement near the shell/cavity boundary in a composite (with

host medium ν0 = 0.49985) under far ﬁeld hydrostatic pressure, p = 1,

containing: a shell (ν1 = 0.25,1 = 5)of thickness h = 0.1 with inner

shell boundary R0 = 0.9 (blue, solid) ; a cavity of radius 0.9 (pink, dashed) 65 2.16 Radial displacement from undeformed shell on the shell-medium inter-

face, R1 = 1, as shell thickness, h, changes for shells in which ν0 = ν1 = 3 2 0.49985 for shear moduli 1 = 10 (blue, solid), 1 = 10 (pink, small

dashing), 1 = 10 (yellow, dot-dashed), 1 = 1 (green, large dashing), under hydrostatic pressure p =1...... 66 2.17 Radial displacement from undeformed shell on the shell-medium inter-

face R1 = 1 as shell thickness h changes for shells with shear modulus

equal to that for the host 1 = 1 with host Poisson ratio ν0 = 0.49985 ν0 ν0 and shell Poisson ratios: ν1 = 4 (blue, solid), ν1 = 2 (pink, small 3ν0 dashing), ν1 = 4 (yellow, dot-dashed), ν1 = ν0 (green, large dashing), under hydrostatic pressure p =1...... 66

2.18 Radial displacement on the outer boundary of a shell with ν1 = 0.25 and

1 = 1.05 within a medium in which ν0 = 0.49985 and 0 = 1, plotted against the shell thickness, h. This uses the full thick shell solution (2.85) (blue, solid) and the thin shell approximation (2.90) (pink, dashed). The medium is subject to hydrostatic pressure p =1...... 68 2.19 Relative change between the radial displacement of a shell (with material

parameters as in above graph) using the full thick shell solution ur and

the thin shell approximation, approx(ur), as the shell thickness h changes. 68

2.20 Radial displacement of a shell (ν1 = 0.25,1 = 1.05h) within an inﬁnite

medium (ν0 = 0.49985) under far-ﬁeld hydrostatic pressure p = 1, using

the full thick shell solution ur (blue, solid) and the thin shell approxi-

mation, approx(ur) (pink, dashed) as shell thickness h changes...... 69

8 2.21 Radial displacement on the boundary of a shell (ν1 = 0.25, η = 1.05)

within an infinite medium (ν0 = 0.49985) under far-field hydrostatic pressure p = 1, as the shell thickness, h, changes. Using the full thick shell solution (2.85) (blue, solid) and the stiff-thin shell approximation (2.92)(pink,dashed)...... 70

2.22 Relative change between the radial displacement of a shell (ν1 = 0.25, η =

1.05) within an inﬁnite medium (ν0 = 0.49985) under far-ﬁeld hydro-

static pressure p = 1, using the full thick shell solution ur and the stiﬀ-

thin shell approximation, approx(ur) as shell thickness, h, changes. . . . 70 2.23 Sphericalshellinaninﬁnitemedium...... 71 u − approx(u ) 2.24 Relative error, r r , in approximation of the full thick shell u r solution ur by the approximate solution, approx(ur), given by:(i) the Fok-Allwright solution (blue, solid),(ii) the thin shell solution (pink, dashed) and (iii) the stiﬀ-thin shell solution (yellow, dot-dashed), for

a shell (1 = 200, ν1 = 0.25) in a nearly incompressible medium,(ν0 = 0.49985) under far ﬁeld hydrostatic pressure p = 1, as shell thickness, h, changes...... 75 u − approx(u ) 2.25 Relative error, r r , in approximation of the full thick shell u r solution ur by the approximate solution, approx(ur), given by:(i) the Fok-Allwright solution (blue, solid),(ii) the thin shell solution (pink, dashed) and (iii) the stiﬀ-thin shell solution (yellow, dot-dashed), for

a shell (1 = 200, h = 0.01) in a nearly incompressible medium,(ν0 = 0.49985) under far ﬁeld hydrostatic pressure p = 1, as shell Poisson ratio,

ν1,changes...... 76 u − approx(u ) 2.26 Relative error, r r , in approximation of the full thick shell u r solution ur by the approximate solution, approx(ur), given by:(i) the Fok-Allwright solution (blue, solid),(ii) the thin shell solution (pink, dashed) and (iii) the stiﬀ-thin shell solution (yellow, dot-dashed), for

a shell (h = 0.01, ν1 = 0.25) in a nearly incompressible medium,(ν0 = 0.49985) under far ﬁeld hydrostatic pressure p = 1, as shell shear mod-

ulus, 1,changes ...... 77 2.27 Off center spherical shell in an infinite elastic medium...... 78 2.28 Off center spherical shell in an infinite elastic medium...... 89

9 2.29 Hoop stress σφφ on the shell-host interface for shells, in which 1 = 1750,

ν1 = 0.3 and ν0 = 0.49985, under pressure p = 0.1, with meridional angle φ, for shells of various thicknesses, h, such that: (i) h = 0.001 (blue, solid), (ii) h = 0.01 (pink, dashed), (iii) h = 0.03 (yellow, dot-dashed), (iv) h = 0.05(green,dotted)...... 94

2.30 Hoop stress σφφ on the shell-host interface for an embedded shell in

which h = 0.01, ν1 = 0.3 and ν0 = 0.49985, with meridional angle φ for

various shell shear moduli: (i) 1 = 2000 (blue, solid), (ii) 1 = 1000

(pink, dashed), (iii) 1 = 100 (yellow, dot-dashed), (iv) 1 = 10 (green,

dotted), (v) 1 =1(orange,longdashes)...... 95

2.31 Hoop stress σφφ on shell-host interface for shells in which h = 0.01,

1 = 1750 and ν0 = 0.49985, with meridional angle φ for varying values

of ν1: (i) ν1 = 0.49985 (blue, solid), (ii) ν1 = 0.4 (pink, dashed), (iii)

ν1 = 0.3 (yellow, dot-dashed), (iv) ν1 = 0.2 (green, dotted), (v) ν1 = 0.1 (orange,longdashes)...... 95

2.32 Hoop stress σφφ on shell-host interface with meridional angle φ for shells

of thickness h = 0.01, shear modulus 1 = 1750 and Poisson ratio ν0 =

0.3, within mediums of various Poisson ratio ν0, such that: (i) ν0 =

0.49985 (blue, solid), (ii) ν0 = 0.4 (pink, dashed), (iii) ν0 = 0.3 (yellow,

dot-dashed), (iv) ν0 = 0.2 (green, dotted), (v) ν0 = 0.1 (orange, long dashes)...... 96

2.33 Hoop stress σφφ on shell-host interface with meridional angle φ for shells

of thickness h = 0.01, shear modulus 1 = 200 and Poisson ratio

ν1 = 0.3, within mediums of Poisson ratio ν0 = 0.49985 and pressure p = 0.1 using various methods: (i) exact hoop stress (2.148) (blue, solid), (ii) stiﬀ-thin shell approximation (pink, dashed), (iii) thin shell approximation (yellow, dot-dashed)...... 99

2.34 Hoop stress σφφ on shell-host interface with meridional angle φ for shells

of thickness h = 0.01, shear modulus 1 = 1.1 and Poisson ratio ν1 = 0.3,

within mediums of Poisson ratio ν0 = 0.49985 and pressure p = 0.1 using various methods: (i) exact hoop stress (2.148) (blue, solid), (ii) stiﬀ- thin shell approximation (pink, dashed), (iii) thin shell approximation (yellow,dot-dashed)...... 99

10 3.1 Buckling pressure p at each buckling mode number n, nondimension- 2 4hˆ 1(1 + ν1) alised on classical buckling pressure p0 = as derived us- ˆ2 2 R 3(1 − ν1 ) ing Koiter’s buckling pressure (blue, solid) and using the unembedded shell limit, (3.108), of the full solution buckling pressure (pink, dashed). Note: lines overlay each other...... 128 3.2 Buckling pressure p at each buckling mode number n, nondimension-

alised on classical buckling pressure p0, for a thin glassy shell 1 =

200,ν1 = 0.3 and h = 0.01 within a nearly incompressible host ν0 = 0.49985, where the buckling pressure is derived via: (i) the full solution (blue, solid); (ii) the Jones et al. approximation (pink, dashed); (iii) the stiﬀ-thin shell approximation (yellow, dot dashed); (iv) the Fok- Allwright approximation (green, dotted). Note: blue and yellow curves overlay...... 130 3.3 Buckling pressure, p, at each buckling mode number, n, nondimension-

alised on classical buckling pressure, p0, for a thin shell, ν1 = 0.3 and

h = 0.01, within a nearly incompressible host, ν0 = 0.49985, when shell

and host medium have the same shear modulus 1 = 1, where the buckling pressure is derived via: (i) the full solution (blue, solid); (ii) the Jones et al. approximation (pink, dashed); (iii) the stiﬀ-thin shell approximation (yellow, dot dashed); (iv) the Fok-Allwright approximation (green, dotted). Note: blue, pink and yellow curves overlay...... 131

3.4 Critical buckling pressure pc, for a thin shell, ν1 = 0.3 and h = 0.01,

within a nearly incompressible host, ν0 = 0.49985, as the shell shear

modulus 1 changes, where the buckling pressure is derived via: (i) the full solution (blue, solid); (ii) the Jones et al. approximation (pink, dashed); (iii) the stiﬀ-thin shell approximation (yellow, dot dashed); (iv) the Fok-Allwright approximation (green, dotted). Note: blue and yellow curves overlay, and are partially overlaid by the green curve...... 133

3.5 Critical buckling pressure pc, for a thin stiﬀ shell ν1 = 0.3 and 1 = 200,

within a nearly incompressible host ν0 = 0.49985, as the shell thickness h changes, where the buckling pressure is derived via: (i) the full solution (blue, solid); (ii) the Jones et al. approximation (pink, dashed); (iii) the stiﬀ-thin shell approximation (yellow, dot dashed); (iv) the Fok- Allwright approximation (green, dotted).Note: blue and yellow curves overlay...... 134

11 3.6 Critical buckling pressure pc nondimensionalised on classical buckling

pressure p0, for a thin shell h = 0.01 and 1 = 200, within a nearly

incompressible host ν0 = 0.49985, as the shell Poisson ratio ν1 changes, where the buckling pressure is derived via: (i) the full solution (blue, solid); (ii) the Jones et al. approximation (pink, dashed); (iii) the stiﬀ- thin shell approximation (yellow, dot dashed); (iv) the Fok-Allwright approximation (green, dotted). Note: blue and yellow curves overlay. . . 135

3.7 Critical buckling pressure pc nondimensionalised on classical buckling

pressure p0, for an embedded thin stiﬀ shell h = 0.01,ν1 = 0.3 and

1 = 200 as the host Poisson ratio ν0 changes, where the buckling pressure is derived via: (i) the full solution (blue, solid); (ii) the Jones et al. approximation (pink, dashed); (iii) the stiﬀ-thin shell approximation (yellow, dot dashed); (iv) the Fok-Allwright approximation (green, dotted). Note: blue and yellow curves overlay...... 136

3.8 Critical uniaxial buckling pressure pc, for a thin shell h = 0.01, ν1 = 0.3

embedded in an elastic matrix ν0 = 0.49985 as a function of the shell

shear modulus 1, where the buckling pressure is derived via: (i) the full solution (antisymmetric buckling- blue, solid; symmetric buckling- green, dotted); (ii) the Jones et al. approximation (antisymmetric buckling- yel- low, solid; symmetric buckling- red, dashed); (iii) the stiﬀ-thin shell approximation (antisymmetric buckling- pink, solid; symmetric buckling- black, dotted). Note: blue and pink overlay, as do green and black similarlyyellowandred...... 138

3.9 Critical uniaxial buckling pressure pc, for a thin shell h = 0.01, ν1 = 0.3

embedded in an elastic matrix ν0 = 0.49985 as a function of the shell

3.10 Odd critical buckling for a thin stiﬀ shell h = 0.01,ν1 = 0.3 and 1 = 200

embedded in an elastic matrix ν0 = 0.49985, where the buckling pattern isfoundviavariousshellsolutions...... 140

4.1 Two vertically aligned spherical cavities (known as cavities 1 and 2) a distance Lˆ apart in an inﬁnite elastic medium with local coordinate

systems (ˆr1, φˆ1, θˆ) and (ˆr2, φˆ2, θˆ) respectively...... 142

12 4.2 Hoop stress on surface of cavity 1 for a nearly incompressible medium ν = 0.49985 under hydrostatic pressure p = 0.1, for cavities of equal radius r such that: r = 0.01 (dotted, blue), r = 0.1 (small dashes, pink), r = 0.2 (large dashes, yellow), r = 0.4 (dot-dashed, green), r = 0.45 (solid,black)...... 157

4.3 Percentage change between maximum absolute hoop stress (at φ1 = π) on surface of cavity 1 and isolated cavity hoop stress for a nearly incompressible medium ν = 0.49985 under hydrostatic pressure p = 0.1, for cavities of equal radius r...... 158

4.4 Percentage change between maximum absolute hoop stress (at φ1 = π) on the surface of cavity 1 and the isolated cavity hoop stress for a medium under hydrostatic pressure p = 0.1, for cavities of equal radius r, where the medium has Poisson ratio: ν = 0.49985 (blue, solid); ν = 0.4 (pink, dashed); ν = 0.25 (yellow, dotted); ν = 0.1 (green, dot-dashed). . . . . 159 4.5 Absolute percentage change of the hoop stress on the surface of cavity 1

at radius R1, from when cavity 1 has a near neighbour cavity of radius

R2 (varying with R2 axis) to when cavity 1 is isolated. The medium is nearly incompressible (ν = 0.49985) and experiences pressure p = 0.1,

various radii of R1 are given by: R1 = 0.01 (blue, dotted), R1 = 0.1

(pink, small dashes), R1 = 0.2 (yellow, medium dashes), R1 = 0.3 (green,

dot-dashed), R1 = 0.4 (black, large dashes), R1 = 0.45 (brown, solid). . 160 4.6 Absolute percentage change of the hoop stress on the surface of cavity

1 at radius R1 = 0.4, from when cavity 1 has a near neighbour cavity of

radius R2 to when cavity 1 is isolated. The medium experiences pressure p = 0.1 and is of various Poisson ratios given by: ν = 0.49985 (blue, dotted), ν = 0.4 (pink, dashes), ν = 0.25 (yellow, dot-dashed), ν = 0.1 (green,solid)...... 161

5.1 Two spherical shells in an inﬁnite elastic host medium...... 163

5.2 Hoop stress σφ1φ1 upon shell 1 for equal sized shells R1 = R2 with ﬁxed h h shell thickness to radius ratio, 1 = 2 = 0.01. The shells are stiﬀ R1 R2 and glassy (ν1,ν2 = 0.25,1,2 = 200) whilst the host medium is nearly

incompressible ν0 = 0.49985. Various radii are exhibited: R1 = 0.01

(blue, solid), R1 = 0.1 (pink, dashed), R1 = 0.2 (yellow, dot-dashed),

R1 = 0.3 (green, dotted), R1 = 0.4 (black, dashed), R1 = 0.45 (red, solid)...... 175

13 5.3 Minimum hoop stress, in magnitude, σφ1φ1 upon shell 1 for equal sized

shells R1 = R2 as radius changes, with ﬁxed shell thickness to radius h1 h2 ratio, = = 0.01. The shells are stiﬀ and glassy (ν1,ν2 = R1 R2 0.25,1,2 = 200) whilst the host medium is nearly incompressible

ν0 = 0.49985...... 176

5.4 Each figure shows the hoop stress σφ1φ1 upon shell 1 at the shell-matrix interface for variously sized shell 2, with fixed shell thickness to radius h1 h2 ratios, = = 0.01. The shells are stiff and glassy (ν1,ν2 = R1 R2 0.25,1,2 = 200) whilst the host medium is nearly incompressible

ν0 = 0.49985. Various radii are exhibited in each ﬁgure: R2 = 0.01

(blue, solid), R2 = 0.1 (pink, dashed), R2 = 0.2 (yellow, dot-dashed),

R2 = 0.3 (green, dotted), R2 = 0.4 (black, dashed), R2 = 0.45 (red, solid).177

5.5 Hoop stress σφ1φ1 upon shell 1 for equal sized shells R1 = R2 = 0.4. The

shells are stiﬀ and glassy (ν1,ν2 = 0.25,1,2 = 200) whilst the host

medium is nearly incompressible ν0 = 0.49985. Various shell thicknesses

are exhibited: h1, h2 = 0.005R1 (blue, solid), h1, h2 = 0.01R1 (pink,

dashed), h1, h2 = 0.02R1 (yellow, dot-dashed), h1, h2 = 0.03R1 (green,

dotted), h1, h2 = 0.04R1 (black, dashed), h1, h2 = 0.05R1 (red, solid). . 178

5.6 Diﬀerence between hoop stress σφ1φ1 upon shell 1 for equal sized shells

R1 = R2 = 0.4 and the hoop stress of an isolated shell when R1 = 0.4.

The shells are stiﬀ and glassy (ν1,ν2 = 0.25,1,2 = 200) whilst the host

medium is nearly incompressible ν0 = 0.49985. Various shell thicknesses

are exhibited: h1, h2 = 0.005R1 (blue, solid), h1, h2 = 0.01R1 (pink,

dashed), h1, h2 = 0.02R1 (yellow, dot-dashed), h1, h2 = 0.03R1 (green,

dotted), h1, h2 = 0.04R1 (black, dashed), h1, h2 = 0.05R1 (red, solid). . 179

5.7 Hoop stress σφ1φ1 on the shell-matrix interface of shell 1 for two identical

shells of radius R1 = R2 = 0.4, thickness h1 = h2 = 0.01R1 and shear

modulus 1 = 2 = 200 under hydrostatic pressure p = 0.1. The shells

have various Poisson ratios: ν1 = ν2 = 0.1 (blue, solid); ν1 = ν2 = 0.2

(pink, dotted); ν1 = ν2 = 0.3 (yellow, dot-dashed); ν1 = ν2 = 0.4 (green,

dotted); ν1 = ν2 = 0.49985 (black, dashed)...... 180

5.8 Hoop stress σφ1φ1 on the shell-matrix interface of shell 1 for shells of

radius R1 = R2 = 0.4, ν1 = ν2 = 0.25 in a nearly incompressible host

medium ν0 = 0.49985 for shells of various shear moduli: 1 = 2 = 1

(blue solid), 1 = 2 = 10 (pink, dashed), 1 = 2 = 50 (yellow, dot-

dashed), 1 = 2 = 150 (green, dotted), 1 = 2 = 500 (black, dashed). 181

14 5.9 Change in the hoop stress on the shell-matrix interface of shell 1 in a two- two iso shell system from an isolated shell, ∆|σφ1φ1 | = |σφ1φ1 |−|σφ1φ1 |, for shells of radius R1 = R2 = 0.4 and ν1 = ν2 = 0.25 in a nearly incompressible

host medium ν0 = 0.49985 for shells of various shear moduli: 1 = 2 = 1

(blue solid), 1 = 2 = 10 (pink, dashed), 1 = 2 = 50 (yellow, dot-

dashed), 1 = 2 = 150 (green, dotted), 1 = 2 = 500 (black, dashed). 181 5.10 Change in the hoop stress on the shell-matrix interface of shell 1 in a two- two iso shell system from an isolated shell, ∆|σφ1φ1 | = |σφ1φ1 |−|σφ1φ1 |, for shells of radius R1 = R2 = 0.4, ν1 = ν2 = 0.25 and 1 = 2 = 200 in hosts

of various Poisson ratio: ν1 = ν2 = 0.49985 (blue solid), ν1 = ν2 = 0.4

(pink, dashed), ν1 = ν2 = 0.3 (yellow, dot-dashed), ν1 = ν2 = 0.25

(green, dotted), ν1 = ν2 = 0.2 (black, dashed), ν1 = ν2 = 0.1 (red, solid). 182

5.11 Hoop stress σφ1φ1 on the shell-matrix interface of shell 1 for shells of

radius R1 = R2 = 0.4, ν1 = ν2 = 0.25 and 1 = 2 = 200 in hosts of

various Poisson ratio: ν1 = ν2 = 0.49985 (blue solid), ν1 = ν2 = 0.4

(pink, dashed), ν1 = ν2 = 0.3 (yellow, dot-dashed), ν1 = ν2 = 0.25

(green, dotted), ν1 = ν2 = 0.2 (black, dashed), ν1 = ν2 = 0.1 (red, solid). 183 h 6.1 Buckling patterns of two identical shells of = 0.01 and R+ = 0.01 R+ at critical buckling pressure pc = 4.604575. Amplitude of buckling is arbitrary. Shell 1 (pink) is situated a unit distance above shell 2 (blue). 207 h 6.2 Buckling patterns of two identical shells of = 0.03 and R+ = 0.01 R+ at critical buckling pressure pc = 0.993704. Amplitude of buckling is arbitrary. Shell 1 (pink) is situated a unit distance above shell 2 (blue). 208

6.3 Critical buckling pressure pc of interacting shells, each of outer radius + R , shell parameters ν1 = ν2 = 0.3, 1 = 2 = 200 and host material h h ν = 0.49985 for various shell thickness: = 0.01 (blue, solid), = 0 R+ R+ h 0.03 (pink, dashed), and = 0.05 (yellow, dot-dashed)...... 209 R+ 6.4 Critical buckling pressure pc of interacting shells, each of outer radius + R , shell parameters ν1 = ν2 = 0.3, 1 = 2 = 200 and host material h h ν = 0.49985 for various shell thickness: = 0.01 (blue), = 0.03 0 R+ R+ h (pink), and = 0.05(yellow)...... 210 R+ h 6.5 Buckling patterns of two identical shells of = 0.01 for various shell R+ radii at critical system buckling pressure pc . Amplitude of buckling is arbitrary. Shell 1 (pink) is situated a unit distance above shell 2 (blue). 211

15 h 6.6 Buckling patterns of two identical shells of = 0.05 for various shell R+ radii at critical system buckling pressure pc . Amplitude of buckling is arbitrary. Shell 1 (pink) is situated a unit distance above shell 2 (blue). 212

6.7 Critical buckling pressure pc of a system of interacting shells of outer

radii R1 and R2, for diﬀering R1 as R2 varies, with shell parameters

h1 = 0.01R1, h2 = 0.01R2, ν1 = ν2 = 0.3, and 1 = 2 = 200 embedded

in host material ν0 = 0.49985 for various radii R1: R1 = 0.01 (blue),

R1 = 0.1 (pink, dashed), R1 = 0.3 (yellow, dot-dashed), and R1 = 0.4 (green, dotted). Markers denote points where buckling patterns are shownbelow...... 214

6.8 Buckling patterns of a system of interacting shells of outer radii R1 and

R2, with shell parameters h1 = 0.01R1, h2 = 0.01R2, ν1 = ν2 = 0.3, and

1 = 2 = 200 embedded in host material ν0 = 0.49985 at the critical

buckling pressure pc denoted by the black circle in ﬁgure 6.7. Shell 1 (pink) is situated a unit distance directly above shell 2 (blue) ...... 214

6.9 buckling patterns of system of interacting shells of outer radii R1 and

R2, with shell parameters h1 = 0.01R1, h2 = 0.01R2, ν1 = ν2 = 0.3, and

1 = 2 = 200 embedded in host material ν0 = 0.49985 at the critical

buckling pressure pc denoted by the black square in ﬁgure 6.7. Shell 1 (pink) is situated a unit distance directly above shell 2 (blue)...... 215

6.10 buckling patterns of system of interacting shells of outer radii R1 and

R2, with shell parameters h1 = 0.01R1, h2 = 0.01R2, ν1 = ν2 = 0.3, and

1 = 2 = 200 embedded in host material ν0 = 0.49985 at the critical

buckling pressure pc denoted by the black triangles in ﬁgure 6.7. Shell 1 (pink) is situated a unit distance directly above shell 2 (blue)...... 215

B.1 Two vertically aligned spherical coordinate systems with origins a distance L apart...... 225

16 Abstract

This thesis, Deformation and buckling of isolated and interacting thin shells in an elastic medium, is submitted by Maria Thorpe to The University of Manchester for the degree of PhD in September 2015. This thesis aims to model the effects of interaction and buckling upon pairs of micro-shells embedded within an elastic medium under far field hydrostatic pressure. This analysis is motivated by the role shell buckling plays in the nonlinear nature of the pressure relative volume curve of elastomers containing micro-shells. Current models of the effective properties of these types of composites assume shells are in a dilute distribution within the host medium, and as such assume shells will buckle at the pressure of the associated isolated embedded shell model. For composites with a high volume fraction of micro-shells, or in poorly mixed composites, the dilute distribution model may provide a first approximation to the effective properties of the composite, however, interaction between shells must be considered to find a more accurate model. We begin the process of modelling the buckling of interacting embedded shells by considering the buckling of an isolated embedded thin spherical shell. For a host medium undergoing far field hydrostatic pressure we demonstrate the parameter ranges in which Jones et al. thin shell buckling theory [11] agrees with the thin shell buckling theory of Fok and Allwright [10]. We then use scalings to increase the range of validity of the thin shell approximation used in the Jones et al. theory to include composites with a high contrast between medium and shell materials. This enables more accurate predictions of buckling pressures of embedded shells under far field axially symmetric pressures to also be found, as is demonstrated for an embedded shell under far field axial compression. We model the linear elastic deformation of pairs of embedded micro-shells using the Boussinesq-Papkovich stress function method, before employing the thin shell linear analysis method developed in previous chapters, based upon the Jones et al. thin shell theory, to calculate the critical buckling pressure and buckling patterns of the pair of embedded shells.

17 Declaration

No portion of the work referred to in this thesis has been submitted in support of an application for another degree or qualiﬁcation of this or any other university or other institute of learning.

18 Copyright

i. The author of this thesis (including any appendices and/or schedules to this thesis) owns certain copyright or related rights in it (the “Copyright”) and s/he has given The University of Manchester certain rights to use such Copyright, including for administrative purposes. ii. Copies of this thesis, either in full or in extracts and whether in hard or electronic copy, may be made only in accordance with the Copyright, Designs and Patents Act 1988 (as amended) and regulations issued under it or, where appropriate, in accordance with licensing agreements which the University has from time to time. This page must form part of any such copies made. iii. The ownership of certain Copyright, patents, designs, trade marks and other intellectual property (the “Intellectual Property”) and any reproductions of copyright works in the thesis, for example graphs and tables (“Reproductions”), which may be described in this thesis, may not be owned by the author and may be owned by third parties. Such Intellectual Property and Reproductions cannot and must not be made available for use without the prior written permission of the owner(s) of the relevant Intellectual Property and/or Reproductions. iv. Further information on the conditions under which disclosure, publication and commercialisation of this thesis, the Copyright and any Intellectual Property and/or Reproductions described in it may take place is available in the Uni- versity IP Policy (see http://www.campus.manchester.ac.uk/medialibrary/ policies/intellectual-property.pdf), in any relevant Thesis restriction dec- larations deposited in the University Library, The University Library’s regulations (see http://www.manchester.ac.uk/library/aboutus/regulations) and in The University’s policy on presentation of Theses

19 Acknowledgements

I would like to thank Thales Underwater Systems and the EPSRC for joint funding of this thesis, my supervisors Will Parnell and David Abrahams and all the academics who have helped shape this work, and most of all my partner Liam Brown for his unending support during my PhD.

20 Chapter 1

Introduction

1.1 Introduction

Many materials today can be purpose engineered to suit the task they will perform. For example, concrete can be reinforced with metal rods for tensile strength, or resin combined with carbon nanotubes when high strength to weight ratios are needed [40]. These material mixtures are known as composites and are used in many forms by the manufacturing sector to help structures withstand extreme environmental conditions. This thesis will consider one type of composite, often known as microsphere composites, which typically consist of a host material containing spherical particles, such as Ex- pancel microspheres [8], each a few microns in diameter with thermoplastic shells and thickness to diameter ratios of the order of 0.01 [46]. Microsphere composites are used in many industrial applications, from low density fillers in automotive parts to blowing agents in printing inks, [19, 8, 28]. Whilst a major reason for using fillers is to reduce costs they are also used to reduce density, improve stability and impact strength, increase thermal insulation and compressibility and provide a smoother surface fin- ish. Of particular interest to the underwater industry is the sonar transparency that micro-shells can sometimes provide when the composite has optimal acoustic damping characteristics. In particular when the host is highly attenuating and so mode conversion from compressive to shear waves at the shell interfaces leads to attenuation due to scattering. As such, when designing hull cladding, the underwater engineering sector often uses microsphere composites such as the one imaged in figure 1.1. This shows a slice of an approximately 2mm by 2mm material sample. The main component of the sample is a grey coloured elastomeric host into which white flecked barytes and black gas-filled microspheres have been mixed. The barytes flecks, due to the high atomic number of barium in comparison to the host, are used to tune the density of the material whilst the stiff linearly elastic shells of the gas-filled spheres delay cavity collapse and as such protect the composite from a reduction in acoustic performance.

21 CHAPTER 1. INTRODUCTION 22

Figure 1.1: Rubberlike composite used by the underwater engineering sector: X-ray micrograph produced by Fonseca, McDonald and Withers (School of Materials, University of Manchester).

The behaviour of a composite such as this is shaped by both the static (mechanical) and dynamic (with or without pre-stress) response of the composite. Modelling the full static and dynamic constitutive behaviour of an inhomogeneous composite material such as this in its full form would, however, be very complex. Instead the problem is simplified to include only the most important structural characteristics. We neglect thermal, viscous or plastic effects, and retain only elasticity and potential hysteretic behaviour due to buckling, and model only the quasi-static constitutive behaviour of the material. With this simplified model it is possible to mathematically predict how environmental or structural changes affect the behaviour of the composite. When considering how to simplify the structure of the material in figure 1.1 we may, in this case: consider the host as homogeneous and nearly incompressible; neglect the inclusion of barytes, since the contrast in material properties between the microspheres and the host is much greater than between the host and the barytes; and consider the material to be infinite in all directions with pressure applied in the far field. With these simplifications we can create a schematic representation of the material, as in figure 1.2, which provides a base for our mathematical models. CHAPTER 1. INTRODUCTION 23

Figure 1.2: Schematic of rubberlike composite containing spherical gas-ﬁlled shells.

Such microsphere composites behave in a rather complex manner under pressure, as can be seen in figure 1.3, exhibiting nonlinearity (and hysteresis during unloading). Whilst hysteresis is associated with viscoelasticity, the nonlinearity of the pressure relative volume change curve of the composite is thought to arise due to the inherent nonlinearity of the elastomeric matrix but also, rather importantly, from the buckling of the microsphere shells, as was suggested by experimental work by Shorter et al. [23]. As such, models for the effective constitutive behaviour of the composite require accurate predictions of the shell buckling. Much work has been completed on the deformation of porous media, where micro- shells are replaced by gas-filled cavities. For an early model of the effective linear elastic properties see Mackenzie [24] or for rubber foams see, for example, Gent and Thomas [5], Gibson and Ashby [35] or Lakes et al. [3]. As such the deformation of these microvoided media is now relatively well understood. Modelling of materials containing micro-shells on the other hand has tended towards modelling only the effective linear elastic constitutive behaviour, for example see Parnell [47], with much focus on syntactic foams, which encase micro-shells in a polymer matrix and have both a low density and high damage threshold. Some static homogenization techniques used recently in this area are given by Porfiri and Gupta [36] and Tagliavia et al. [37]. De Pascalis et al. [46] have, however, considered how the buckling of micro-shells may affect the effective properties of the composite when it’s subject to hydrostatic pressure, though only when the shells are in a dilute distribution. Considering the shells within a composite to be in a dilute dispersion assumes each shell is far enough away from any other shell that they do not interact and as such allows each shell to be modelled as an isolated shell in an unbounded medium. Thus a dilute distribution allows the material to be broken into volumes containing only one microsphere. The deformation and buckling of this microsphere can then be modelled separately before homogenisation is used to create an CHAPTER 1. INTRODUCTION 24

Figure 1.3: Experimentally determined stress-strain curve associated with a Silicone RTV microsphere elastomer filled with various volume fractions of Expancel microspheres under uniaxial tension. The solid curve is unfilled and others refer to increasing volume fractions of the microsphere phase (dashed 10%, dot-sashed 20%, dotted 30%, dash-dot-dot 40%) [23]. effective medium with the same properties as the composite. The buckling of spherical shells, where pressure is applied directly to the shell surface, is well understood, see Koiter [50], or more recently Fu [51] and Ben-Amar and Goriely [1]. However there is little work on the buckling of spherical shells embedded within an elastic medium. The first work in this area, by Fok and Allwright [10], considered a thin shell embedded within an elastic matrix under far field hydrostatic pressure. It is the buckling criterion predicted by this model that is used in the De Pascalis [46] homogenization scheme for dilute dispersions. The work of Fok and Allwright was extended by Jones et al. [11] to include far field loading which is not spherically symmetric. For certain materials, where the shell is much stiffer than the host medium, the Jones et al. theory predicts very different buckling pressures to the Fok and Allwright work, for an embedded shell under hydrostatic pressure. Due to this disagreement in the results the Jones et al. model tends to be less frequently used. The dilute scheme is not always appropriate since composites used in manufacturing processes can have filler volume fractions of up to 50% and in composites with lower volume fractions aggregation of micro-shells may occur. In this case, as in figure 1.1, interaction between shells must also be considered. Although many micromechanical CHAPTER 1. INTRODUCTION 25 methods have considered approximate schemes to incorporate the effect of interaction on the effective linear elastic behaviour of inhomogeneous media, it is the objective here to start to consider interaction effects associated with buckling. In order to accurately model the effective material properties of a composite with a high volume fraction of micro-shells it is necessary to understand how not only isolated but near neighbour shells buckle. The objective of this thesis is therefore to predict the buckling pressure between two micro-shells embedded within an elastic matrix under far field hydrostatic pressure. In order to complete this it will also be necessary to resolve differences between the Fok and Allwright and Jones et al. buckling models, as the ability of the Jones et al. model to be applied to non-spherically symmetric problems will be needed for the interaction model. With this in mind we will provide the reader with information on the mathematical methods needed to formulate embedded shell deformation problems, and canonical examples of their uses, in chapter 2. In chapter 3 we discuss the work of Jones, Chapman and Allwright [17] in developing a method for predicting the buckling pressure of embedded shells which is applicable to media under non-spherically symmetric loadings. We demonstrate the links between the Jones et al. theory and work done by Fok and Allwright [10], for spherically symmetric loadings, and by Koiter [50], for unembedded shells, before increasing the range of validity of the Jones et al. method to include composites with a high contrast between medium and shell materials. The second half of this thesis concentrates on the effects of interaction between cavities and shells within a medium. We show, in chapters 4 and 5, the effects of near neighbour cavities or shells upon the linear elastic deformation of the medium. These linear deformations are then used in the novel application of calculating the buckling pressures and buckling patterns of near neighbour shells using our corrections to the Jones et al. theory, as developed in chapter 3. Chapter 2

Background

This chapter will review the elasticity theory and mathematical methods relevant to this thesis. It is assumed that the reader is familiar with continuum mechanics, though if not a good introduction to the subject can be found in Spencer [7] and Reddy [27]. A good introduction to nonlinear elasticity can be found in Green and Zerna [45], Ogden [39], and Fung [52]. As we are concerned with only elastic materials, as experiments have shown, for a ﬁrst approximation we can neglect any dissipative eﬀects. We will discuss the material within this background section in the same order as it is needed within the thesis. Therefore we will start with elasticity theory, covering both linear and nonlinear theory, before moving onto methods of solution for linear elastic problems such as the Boussinesq-Papkovich stress function method and multipole solutions. After considering canonical linear elasticity problems involving single cavities and shells we will discuss the application of buckling theory to these problems.

2.1 Tensor notation and coordinate systems

As we are going to use a variety of coordinate systems we will provide the reader with a basic understanding of vector and tensor relations in curvilinear coordinate systems before specialising to specific coordinate systems, such as spherical polar coordinates, which are used most often within this thesis. We will define the stress, strain and displacement relations that are used in linear elasticity, and their spherical coordinate form, before finally discussing the role of nonlinear elasticity in thin shell deformation. The notation is mainly consistent with the thesis of Jones [17], which itself is based upon Green and Zerna [45] and Koiter [50]. As such lower case Latin indices run over the numbers 1 to 3, whilst Greek indices vary over 1 to 2. The summation convention applies unless stated otherwise.

26 CHAPTER 2. BACKGROUND 27

2.1.1 Curvilinear coordinates: tensors and vectors

A curvilinear coordinate system comprises a set of coordinates (θ1,θ2,θ3) and a position vector

r = r(θ1,θ2,θ3).

Within this system covariant basis vectors are introduced as

gi = r,i , (2.1) where the notation ,i implies differentiation with respect to θi, and contravariant basis vectors gi are defined via j j gi g = δi . (2.2) j Here the notation implies the scalar product and δi is the mixed second-rank Kronecker delta tensor defined as

j 1 for i = j, δi = 0 for i = j.  The usual notation for the Kronecker delta δij is the covariant form of this tensor. The deﬁnition of these basis vectors allow the covariant and contravariant metric tensors, ij gij and g to be deﬁned as

ij i j gij = gi gj , g = g g .

The covariant and contravariant basis vectors are also related by the expressions:

j i ij gi = gijg , g = g gj . (2.3)

If the basis vectors are also orthogonal then

ij gij = g = 0 i = j, and 1 g = , ii gii i where summation over i is not applied. In general the basis vectors gi and g are not unit vectors (nor orthogonal), and as such the covariant and contravariant components of vectors and tensors referred to these basis vectors are not the physical components i of the coordinate system. To diﬀerentiate between basis vectors, gi and g , and unit i gi g basis vectors, and i we will use numerical indices 1, 2, 3 for vectors and tensors |gi| |g | referred to the basis vectors and algebraic indices, r, φ, θ for example, for vectors and CHAPTER 2. BACKGROUND 28 tensors referred to the unit base vectors, the physical components.

2.1.2 Vectors

Within a curvilinear system covariant and contravariant vectors form bases for vectors in R3. As such a vector v can be represented in either basis as

i i v = v gi , or v = vig , where covariant components are given by

vi = v gi , and contravariant components similarly,

vi = v gi .

Conversion between covariant and contravariant states for components work similarly to basis vectors, (2.3): j i ij vi = gijv , v = g vj . (2.4)

When diﬀerentiating vectors with respect to a coordinate we must remember that the basis vectors themselves are not necessarily independent of the coordinate, as such

i i v,j = vi,jg + vig,j i = vi|jg , where the notation vi|j deﬁnes the covariant derivative of vi given by

r vi|j = vi,j − Γijvr , (2.5) in which r r r Γij =Γji = −gi g,j , are the Christoﬀel symbols of the second kind.

2.1.3 Tensors

A second order tensor A is a linear operator which when acting upon a vector v generates another vector w, i.e. w = Av. CHAPTER 2. BACKGROUND 29

Components of a tensor can be deﬁned with respect to the covariant and contravariant basis vectors in a similar way to components of a vector, (2.4). The four component types

i i Aij = gi Agj, Aj = g Agj, j j ij i j Ai = gi Ag , A = g Ag , are known as the covariant, mixed (two varieties) and contravariant components respectively. The mixed components assume the order of the indices is important; when this is not important the notation is often omitted. Covariant diﬀerentiation of tensors follows a similar pattern to that of vector diﬀerentiation:

m m Aij|r = Aij,r − Γir Amj − ΓjrAim, ij ij i mj j im A |r = A,r − ΓrmA − ΓrmA . (2.6)

Tensors of higher order are also possible, and can be expressed as linear combinations of the tensor product, ⊗, of basis vectors, for example a second order tensor may be expressed as i j A = Aijg ⊗ g , whilst a fourth order tensor A may be expressed as

i j k l A = Aijklg ⊗ g ⊗ g ⊗ g , where the tensor product ⊗ of two vectors u and v is deﬁned via

(u ⊗ v)w = u(v w) = (v w)u.

2.1.4 Cartesian coordinates

We can specialise this generalised curvilinear system to speciﬁc coordinate systems. Within this thesis we will concentrate on Cartesian and spherical polar coordinates. We will also deﬁne the physical components of each coordinate system. CHAPTER 2. BACKGROUND 30

We will refer to Cartesian coordinates as ei-

ther (x,y,z) or (x1,x2,x3) as appropriate. z = x3 The two notations are related by x = x1,

y = x2, z = x3. The physical Cartesian ba- y = x sis vectors are deﬁned to be (e1, e2, e3). As 2 such the position vector r is deﬁned to be x = x1

r = xe1 + ye2 + ze3. Figure 2.1: Cartesian coordinate system

The covariant and contravariant basis vectors, via (2.1) and (2.2), are

1 2 3 g1 = g = e1, g2 = g = e2, g3 = g = e3, implying the basis vectors in Cartesian coordinates are unit vectors, and covariant and contravariant metric tensors are

ij 1 i = j, gij = g = 0 i = j. 

2.1.5 Spherical polar coordinates  In spherical polar coordinates the coordinate system is denoted (r, φ, θ), with position vector r = rer, in terms of unit basis x3 r vectors. This is related to the Cartesian φ system (x1,x2,x3) by

x1 = r sin φ cos θ, θ x2

x2 = r sin φ sin θ, x1

x3 = r cos φ, (2.7) Figure 2.2: Spherical polar coordinate system where r ∈ [0, ∞), φ ∈ [0,π] and θ ∈ [0, 2π).

In terms of Cartesian basis vectors (e1, e2, e3), the position vector r can then be written as

r = r sin φ cos θe1 + r sin φ sin θe2 + r cos φe3. CHAPTER 2. BACKGROUND 31

This deﬁnition of a position vector allows us to ﬁnd the covariant basis vectors via (2.1) and (2.3):

g1 = sin φ cos θe1 + sin φ sin θe2 + cos φe3,

g2 = r cos φ cos θe1 + r cos φ sin θe2 − r sin φe3,

g3 = −r sin φ sin θe1 + r sin φ cos θe2. (2.8)

It should be remembered that these are not unit vectors. The covariant metric tensors are thus

2 g11 = 1, g22 = r , 2 2 g33 = r sin φ, gij =0 if i = j, implying that coordinates are orthogonal. We can use the covariant basis vectors and metric tensors to ﬁnd the contravariant metric tensors:

1 g11 = 1, g22 = , r2 1 g33 = , gij =0 for i = j. r2 sin2 φ

Additionally the contravariant basis vectors are given by:

1 g = g1, 1 g2 = g , r2 2 1 g3 = g . (2.9) r2 sin2 φ 3

Our work with spherical coordinates will also require knowledge of the Christoﬀel symbols in spherical coordinates, the only nonzero elements of which are:

1 1 2 Γ22 = −r, Γ33 = −r sin φ, 1 Γ2 =Γ2 = , Γ2 = − sin φ cos φ, 12 21 r 33 1 Γ3 =Γ3 = , Γ3 =Γ3 = cot φ. (2.10) 13 31 r 23 32

2.2 Elasticity Theory

Elasticity is the study of the deformation of solid objects under external loading [45]. As depicted in figure 2.3, given an elastic body Ω, the position of any point P within the body can be represented by the position vector X. Under external loading the CHAPTER 2. BACKGROUND 32 body will deform, to a new configuration Ω′, as such every point within the body will move. The new position of point P , say P ′, can be denoted by the vector x. The change in position of a point, from P in the undeformed state to P ′ in the deformed state, then defines the displacement u within the medium, so that

u = x − X. (2.11)

The deformation of the body inherently implies the solid will experience strain. One

Ω′ Ω ′ u P P x3

x X

Figure 2.3: A point P in an undeformed body Ω can be represented by position vector X. In the deformed body Ω′ the point P moves to position P ′ and is represented by position vector x. The displacement vector u is the change in position vector from the undeformed to deformed state. measure of the strain within an elastic body is the Lagrange strain, with tensor notation E, the Cartesian components of which are given by [4]

∂u ∂u ∂u ∂u E = i + j + k k . (2.12) ij ∂x ∂x ∂x ∂x j i j i 2.2.1 Linear Elasticity

Whilst the Lagrange strain tensor is used in calculations where large shape changes are expected, if only small shape changes are expected it is possible to use an approximation to the Lagrange strain tensor to quantify strain within the body. The presumption of small displacement gradients is central to the theory of linear elasticity. In this case the nonlinear components of the Lagrangian strain tensor are negligible and this allows the strain within a linearly elastic body to be approximated by the Cauchy strain eij, [4], where 1 e = (u | + u | ). (2.13) ij 2 i j j i CHAPTER 2. BACKGROUND 33

If the material constitutive behaviour (which could be obtained experimentally) is linearly elastic the (ijth component of) the Cauchy stress σij within the material can be related to the strain through Hooke’s Law [4]:

ij ijkl σ = A ekl, (2.14) where Aijkl is the fourth order isotropic elasticity tensor, given by

Aijkl = λgijgkl + (gikgjl + gilgjk). (2.15)

In this gij are contravariant metric tensors previously deﬁned in (2.3). The constants and λ are material constants of the medium, as deﬁned below. Throughout this thesis we will denote material constants as follows:

ν Poisson ratio The negative ratio of transverse to axial strain Shear modulus The ratio of shear stress to shear strain E Youngs modulus The longitudinal stress-strain ratio for a slender rod k Bulk modulus The ratio of the inﬁnitesimal pressure increase to the resulting relative decrease in volume λ Lam´emodulus First elastic modulus.

See [18] for further information on material constants and their experimental deter- mination. These constants are related by

E 2ν E k = , λ = , and 2 = . 3(1 − 2ν) 1 − 2ν 1+ ν

The ﬁnal governing equation of linear elasticity relates the state of stress within the body directly to the displacement through balance of linear momentum. This is known as the equation of motion. When the body force per unit mass acting upon the medium is denoted f, then this equation can be written

ij i i σ |j + ρf = ρu¨ , where ρ is mass density, and ˙ represents diﬀerentiation with respect to time. For static deformations without body forces this simpliﬁes to the equation of equilibrium [4],

ij σ |j = 0. (2.16)

Using linear elastic theory the potential energy density of a medium can then be written

1 1 V = σije = Aijkle e . (2.17) 2 ij 2 ij kl CHAPTER 2. BACKGROUND 34

Potential energy stored in the elastic body Ω can therefore be written as

1 W = Aijkle e dV. (2.18) 2 ij kl Ω 2.2.2 Equations of linear elasticity in spherical coordinates

Using the covariant derivative (2.5) and the Christoﬀel symbols in spherical coordinates, (2.10), the strain-displacement relations in terms of spherical basis vectors can be derived as:

1 u e = u , e = (u + u ) − 2 , 11 1,1 12 2 1,2 2,1 r 1 u e = (u + u ) − 3 , e = u + ru , 13 2 1,3 3,1 r 22 2,2 1 1 e = (u + u ) − u cot φ, e = u + ru sin2 φ + u sin φ cos φ. (2.19) 23 2 2,3 3,2 3 33 3,3 1 2

It should be remembered that vector components with numerical indices refer to components in the non-unit basis vectors whilst algebraic indices refer to physical components, those defined with respect to the unit basis vectors. We can use (2.14) and (2.15) to determine the stress-strain relations, which are simplified by defining the dilation ∆ by

kl 1 1 ∆= g ekl = e11 + e22 + e33, r2 r2 sin2 φ to

λ 2 σ11 = λ∆ + 2e , σ22 = ∆+ e , 11 r2 r4 22 12 2 33 λ 2 σ = e12, σ = ∆+ e33, (2.20) r2 r2 sin2 φ r4 sin4 φ 13 2 23 2 σ = e13, σ = e23. r2 sin2 φ r4 sin2 φ

Spherical coordinates: physical components

i As the basis vectors g and gi are not necessarily unit vectors, it may sometimes be easier to use the physical components, in this case er, eφ and eθ, than the basis vectors. In spherical polar coordinates the unit basis vectors are related to the covariant basis vectors by

1 1 1 er = g1, eφ = g2, eθ = g3. |g1| |g2| |g3| CHAPTER 2. BACKGROUND 35

In terms of the spherical basis vectors (2.8), these can be written as

1 1 e = g = g1, e = g = rg2, e = g = r sin φg3. r 1 φ r 2 θ r sin φ 3

i Similarly the vector v = v gi can be written as

v = vrer + vφeφ + vθeθ, in which components are related by

v v v = v = v1, v = 2 = rv2, v = 3 = r sin φ v3. (2.21) r 1 φ r θ r sin φ

We can similarly ﬁnd the physical components of a tensor. For example two components of the stress tensor σ are given by:

σ = σ11 = λ∆ + 2e , σ = rσ12 = e , . (2.22) rr 11 rφ r 12

Equation of equilibrium: physical coordinates

We can use this process to write the equation of equilibrium,

ij σ |j = ∇ σ = 0, (2.23) in spherical coordinates as:

∂σ 1 ∂σ 1 ∂σ 1 rr + rφ + rθ + (2σ − σ − σ + σ cot φ) = 0, ∂r r ∂φ r sin φ ∂θ r rr φφ θθ rφ ∂σ 1 ∂σ 1 ∂σ 1 rφ + φφ + θφ + [(σ − σ ) cot φ + 3σ ] = 0, ∂r r ∂φ r sin φ ∂θ r φφ θθ rφ ∂σ 1 ∂σ 1 ∂σ 1 rθ + θφ + θθ + (2σ cot φ + 3σ ) = 0. ∂r r ∂φ r sin φ ∂θ r θφ rθ

2.2.3 Constitutive equations in spherical coordinates

The stress-strain relations (2.22) can be combined with the strain-displacement relations (2.19) to yield a relationship between stress and displacement. Under assumptions of axisymmetric displacement, where uθ = 0 we then ﬁnd that

∂u 2u 1 ∂u u ∆= r + r + φ + φ cot φ, (2.24) ∂r r r ∂φ r CHAPTER 2. BACKGROUND 36 and thus

∂u σ = λ∆ + 2u = λ∆ + 2 r rr 1,1 ∂r 2 ∂u 2u 1 ∂u u = (1 − ν) r + ν r + φ + cot φ φ , (2.25) (1 − 2ν) ∂r r r ∂φ r 2 1 u 1 ∂u 1 ∂(ru ) 2u σ = (u + u ) − 2 = r + φ − φ rφ r 2 1,2 2,1 r r ∂φ r ∂r r 1 ∂u ∂u u = r + φ − φ , (2.26) r ∂φ ∂r r 2 u ∂u 1 − ν ∂u ν σ = r + ν r + φ + cot φ u , (2.27) φφ 1 − 2ν r ∂r r ∂φ r φ 2 u (1 − ν) ∂u ν ∂u σ = r + cot φ u + ν r + φ , (2.28) θθ 1 − 2ν r r φ ∂r r ∂φ σrθ = σφθ = 0. (2.29) where (2.21) are used to relate components in the basis vectors to physical components.

2.3 Nonlinear elasticity for shells

Whilst for many applications within this thesis linear elasticity applies, when considering the stability of thin shells due to the possibility of large displacements we will need to employ nonlinear elasticity theory, and specifically thin shell theory. However, we assume the constitutive behaviour of the shell is still linear, as defined by Hooke’s law. A fully nonlinear theory of shells was developed by Koiter [50], and separately by Sanders [26], in the 1960s and has therefore come to be known as Sanders-Koiter thin shell theory. In the context of shells, various approximations of the full Lagrangian strain are taken in order to incorporate the dominant effects. As such strain (and therefore stress) are nonlinear functions of the displacement gradients.

Surface geometry

An elastic shell is a surface with a “thickness” dimension small in comparison to all other dimensions [16]. The shell mid-surface is parameterised by

r = r˜(θ1,θ2), (2.30) then a general r can be written as

r = r˜(θ1,θ2)+ θ3a3, (2.31) CHAPTER 2. BACKGROUND 37

with θ1,θ2 giving the point on the mid-surface closest to r and the unit vector a3

(which depends only upon θ1 and θ2), is normal to the mid-surface at that point.

θ3

θ2

θ1

Figure 2.4: Shell surface geometry such that θ1 and θ2 are coordinates in the shell mid-surface whilst θ3 describes movement away from this mid-surface.

Using the deﬁnition of covariant basis vectors, (2.1), we can thus obtain

gα = aα + θ3a3,α, g3 = a3, (2.32) where aα = r˜,α and α = 1, 2. Choosing the sign of a3 appropriately allows us to form a right handed basis (a1, a2, a3) to describe the surface. Covariant metric tensors for the surface are deﬁned by

aαβ = aα aβ, with contravariant basis vectors given by

β β aα a = δα.

With these deﬁnitions it is easy to show that

1 1 a1 = (a a − a a ), a2 = (a a − a a ), a 22 1 12 2 2 11 2 21 1 where a = a11a22 −a12a21 is the determinant of the covariant metric tensor aαβ. There- fore contravariant metric tensors, deﬁned by

aαβ = aα aβ, are explicitly a a a a11 = 22 , a = 11 , and a12 = a21 = − 12 , a 22 a a CHAPTER 2. BACKGROUND 38 and the covariant and contravariant basis vectors are thus related by

α αβ β a = a aβ, aα = aαβa . (2.33)

The covariant metric tensor aαβ is also known as the ﬁrst fundamental tensor of the surface. The second fundamental tensor of the surface, bαβ, allows us to describe curvature of the surface and is deﬁned by

bαβ = −aα a3,β = −aβ a3,α = a3 aα,β = a3 aβ,α.

Using the second fundamental tensor the basis vector, as deﬁned in (2.32), can be written as β gα = ζα aβ, (2.34) where β β β ζα = δα − θ3bα, (2.35)

β and the mixed metric tensor bα is given by,

β βλ βλ bα = a bαλ = aαλb . (2.36)

Using the second fundamental tensor also allows us to describe the mean curvature,

1 1 H = bα = aαβb , (2.37) 2 α 2 αβ and Gaussian curvature, b K = = b1b2 − b1b2, (2.38) a 1 2 2 1 where b is the determinant of bαβ. Within the surface geometry we also deﬁne covariant λ λ diﬀerentiation. For vector components vλ and v , in terms of basis vectors a and aλ,

¯ vλ|α = vλ,α − Γλαv, λ λ ¯λ v |α = v,α + Γαv ,

¯α where the Christoﬀel symbols Γβγ are:

¯α α α Γβγ = a aβ,γ = −aγ a,β. CHAPTER 2. BACKGROUND 39

Spherical surface geometry

Given a spherical shell of thickness h and mid-surface radius R, we deﬁne the shell coordinates (θ1,θ2,θ3) to be

θ1 = φ, θ2 = θ, θ3 = r − R, where (r, φ, θ) are as in (2.7).

θ3 r θ2 θ1

R φ x2

Figure 2.5: Spherical shell geometry (θ1,θ2,θ3) in relation to spherical coordinates (r, φ, θ).

This coordinate system allows the mid-shell surface to be described by θ3 = 0 and in-plane coordinates to be represented by θ1 and θ2. It is important to note that this deﬁnition of (θ1,θ2,θ3) is not the same as in section 2.1.1, where we considered spherical coordinates within a medium, rather than a shell. As such we need to deﬁne the covariant and contravariant basis vectors and matrices for this coordinate system. If any point in the medium is given by (2.31),

r = r˜(θ1,θ2)+ θ3a3, then points upon the shell mid-surface are described by r˜(θ1,θ2) where

r˜(φ, θ)= R sin φ cos θex + R sin φ sin θey + R cos φez. CHAPTER 2. BACKGROUND 40

This implies the basis vectors are given by

a1 a = = R cos φ cos θe + R cos φ sin θe − R sin φe , 1 R2 x y z a2 a2 = = −R sin φ sin θex + R sin φ cos θey, R2 sin2 φ 3 a3 = a = er = sin φ cos θex + sin φ sin θey + cos φez.

The covariant and contravariant metric tensors are thus

1 a = = R2, 11 a11 1 a = = R2 sin2 φ, 22 a22 12 21 a12 = a21 = a = a = 0, and the determinant a is given by R4 sin2 φ. Using this the second fundamental covariant metric tensor bαβ becomes

b11 = −R, 2 b22 = −R sin φ,

b12 = b21 = 0, and thus the mean and Gaussian curvatures, (2.37) and (2.38), are

1 1 H = − , K = . R R2

The only nonzero Christoﬀel symbols are:

¯1 ¯2 ¯3 ¯3 2 Γ22 = − sin φ cos φ, Γ12 = cot φ, Γ11 = −R, Γ22 = −R sin φ,

¯1 ¯2 though only Γ22 and Γ12 will be used later in covariant diﬀerentiation. In terms of spherical polar unit vectors (er, eφ, eθ), we already know er = a3 by deﬁnition. The in-plane unit vectors can also be written as

a1 = Reφ, a2 = R sin φ eθ.

Thus any shell displacement vector v can be written in terms of either basis or unit vectors α v = v aα + wa3 = vrer + vφeφ + vθeθ, CHAPTER 2. BACKGROUND 41 and the components can be related by

1 2 vr = w, vφ = Rv , vθ = R sin φ v . (2.39)

Stress and strain within shells

Given the position vector

r = r˜(θ1,θ2)+ θ3a3,

h h then any point within the shell lies in the range − 2 ≤ θ3 ≤ 2 , where h is the shell thickness. The points on the mid-surface of the shell are then described by the position vector r˜(θ1,θ2). To ﬁnd strains within the shell Koiter [50], considers two shell states. State I, prior to deformation, in which the position vector is given by r and state II, after deformation, (represented by ˆ notation), with position vector rˆ. The displacement vector of the shell can then be denoted

α 3 v = rˆ − r = vαa + wa , where vα describe in-surface displacements and w out of surface displacement. Funda- mental tensors can be formed for each state, such thata ˆαβ and ˆbαβ are the fundamental tensors in state II, and are used to deﬁne the middle surface strain tensor γαβ and tensor of changes of curvature ραβ:

1 γ = (ˆa − a ), (2.40) αβ 2 αβ αβ

ραβ = ˆbαβ − bαβ. (2.41)

Useful measures of stress in the shell are the stress resultants nαβ and stress couples mαβ, determined from the stress tensor σij and mean and Gaussian curvatures, H and K, [45]: h h αβ 2 αβ αβ 2 αβ n = τ dθ3, m = τ θ3 dθ3. (2.42) h h − 2 − 2 Within these deﬁnitions; αβ α βλ τ = η ζλ σ ,

α where ζλ is as in (2.35) and

3 η = 1 − 2θ3H + (θ3) K, combines the mean (2.37) and Gaussian (2.38) curvatures. CHAPTER 2. BACKGROUND 42

Shallow buckling modes approximation

If the radius of curvature of the shell is much greater than the characteristic wavelength of the deformation pattern we can simplify the equations for the stresses and strains within the shell by use of the Donnell shallow shell approximation, a simple nonlinear theory only requiring one nonlinear term. In the approximation appropriate to shallow buckling modes the mid-surface strain (2.40) and the tensor of changes of curvature (2.41) are given by [50] (originally Donnell [31])

1 γ = θ + w w , (2.43) αβ αβ 2 ,α ,β

ραβ = w|αβ , (2.44) where θαβ represents the linear part of the mid-surface strain tensor given by

1 θ = (v | + v | ) − b w. αβ 2 α β β α αβ

When the characteristic wavelength is unknown then this approximation should be used with care and the characteristic wavelength of solutions should be calculated to ensure that the assumption of shallow shell theory still holds. If results do not comply with the approximation then a better approximation should be employed.

Linear stress and strain measures

If the additional approximation of linear shell theory is employed, in which ραβ = w|αβ, as in thin shell theory but now with γαβ = θαβ, its linearised counterpart, the stress resultants and stress couples reduce to [45]:

αβ αβλκ n = hE θλκ, (2.45)

h3 mαβ = Eαβλκρ , (2.46) 12 λκ where 2ν Eαβλκ = (aαλaβκ + aακaβλ + aαβaλκ), (2.47) 1 − ν is the elasticity tensor for shells. Using this we can deﬁne the potential energy density within a shell to be

h h3 V = Eαβλκγ γ + Eαβλκρ ρ , (2.48) 2 αβ λκ 24 αβ λκ CHAPTER 2. BACKGROUND 43 and the stored potential energy within the shell is

W = V dS. Mid-shell surface h Using these approximations, for a linear thin shell in which ≪ 1, we can for example R write 1 2 n11 ∼ h lim τ dξ, h 11 1 →0 − 2 Rˆ where θ3 = hξ and

hξ 3 1 lim τ11 = lim 1+ τφφ , (2.49) h h R 2 R →0 R →0 (R + hξ) 1 = lim τ . (2.50) 2 h φφ R R →0

2.3.1 The Boussinesq-Papkovich stress functions

The equation of equilibrium in linear elasticity given in (2.16) can also be written as

∇ σ = 0 and can be expressed in terms of displacements u, as [2], as

∇(∇ u) + (1 − 2ν)∇2u = 0.

Using the Boussinesq-Papkovich stress functions ξ and ψ [2] we can represent the solution to this as 2u = ∇(ξ + x ψ) − 4(1 − ν)ψ, (2.51)

2 2 where ∇ ξ = ∇ ψi = 0and ,ν are the shear modulus and Poisson ratio of the medium. We can demonstrate that (2.51) satisﬁes the equation of equilibrium in the following way, though see [13] for a discussion on the proof. Whilst a tensor equation is true in all coordinate systems we need only prove it in one coordinate system, so for simplicity Cartesian coordinates are chosen. In Cartesian coordinates the equilibrium equation can be written as ∂2u ∂2u j + (1 − 2ν) i = 0, ∂xi∂xj ∂xj∂xj and the displacement (2.51), as

∂ 2ui = (ξ + xjψj) − 4(1 − ν)ψi, (2.52) ∂xi CHAPTER 2. BACKGROUND 44

where ξ,kk = ψi,kk = 0. We can expand (ξ + xjψj),i as

(ξ + xjψj),i = ψi + xjψj,i + ξ,i, and so substitute back into ui to get

2ui = xjψj,i + ξ,i + (4ν − 3)ψi. (2.53)

If we now diﬀerentiate (2.53), we ﬁnd that

2ui,k = ψk,i + xjψj,ik + ξ,ik + (4ν − 3)ψi,k, (2.54)

2ui,kk = 2ψk,ik + xjψj,ikk + ξ,ikk + (4ν − 3)ψi,kk. (2.55)

Since ξ and ψi satisfy Laplace’s equation (2.55) reduces to

ui,kk = ψk,ik, (2.56) and if we set i = k in (2.54) then by the same argument this reduces to

uk,k = (2ν − 1)ψk,k. (2.57)

If we substitute (2.56) and (2.57) into (2.52) we have

uj,ji + (1 − 2ν)ui,jj = (2ν − 1)ψj,ji + (1 − 2ν)ψj,ji = 0, showing that displacements in the form given by (2.51) satisfy the equation of equilibrium.

Axisymmetric problems

If we have an axisymmetric problem, symmetric about the x3 or z-axis say, then only two functions ξ and ψ3 = ψ are needed [2] and the Boussinesq-Papkovich stress function can be simpliﬁed further to

∞ 2(u − u )= ∇(ξ + zψ) − 4(1 − ν)ψez, (2.58) where the total displacement vector has been split into that experienced at inﬁnity due to forces in the medium, u∞, and the equilibrium solution upon the medium, u. Ina CHAPTER 2. BACKGROUND 45 spherical coordinate system the component displacements can then be expressed as

∂ξ ∂ψ 2(u − u∞)= + cos φ r + (4ν − 3)ψ , r r ∂r ∂r 1 ∂ξ ∂ψ 2(u − u∞)= + cos φ + ψ(3 − 4ν)sin φ. (2.59) φ φ r ∂φ ∂φ

Using the constitutive relations in section 2.2.3 we can also write the stresses σrr and σrφ in terms of the stress functions ψ and ξ. Substituting (2.59) into (2.25) we can write the radial stress as

∂2ξ 2ν ∂ξ ν ∂2ξ ν ∂ξ (1 − 2ν) (σ − σ∞) =(1 − ν) + + + cot φ rr rr ∂r2 r ∂r r2 ∂φ2 r2 ∂φ ∂2ψ ∂ψ ν ∂2ψ + cos φ (1 − ν)r + (2 − 4(1 − ν)2) + ∂r2 ∂r r ∂φ2 ν ∂ψ + (cot φ + (2 − 4ν) tan φ) . r ∂φ Here we note that since both ξ and ψ satisfy Laplace’s equation by construction we can use ∂2ξ 2 ∂ξ 1 ∂2ξ 1 ∂ξ + + + cot φ = 0 ∂r2 r ∂r r2 ∂φ2 r2 ∂φ and the similar expansion for ψ, to rearrange the radial stress into the form

∂2ξ ∂2ψ ∂ψ 2ν ∂ψ σ − σ∞ = + cos φ r − 2(1 − ν) + tan φ . (2.60) rr rr ∂r2 ∂r2 ∂r r ∂φ

Similarly we can ﬁnd σrφ throughout the medium. In terms of the displacements the radial shear stress is given by (2.26) found in section 2.2.3. We again substitute into this the displacements from (2.59), and so ﬁnd the radial shear stress in terms of the Boussinesq-Papkovich stress functions ξ and ψ:

1 ∂2ξ 1 ∂ξ ∂2ψ ∂ψ (1 − ν) ∂ψ σ −σ∞ = − +cos φ +(1−2ν)sin φ −2 cos φ . (2.61) rφ rφ r ∂r∂φ r2 ∂φ ∂r∂φ ∂r r ∂φ

2.3.2 Multipole solutions

A multipole solution [22] is a series solution, often arising from separation of variables, and represents a field as a function dependent upon angles, for example the polar angle, and when applied to Laplace’s equation is often combined with an expansion in powers (or inverse powers) of the radial variable. The higher the order of the term in a multipole expansion the faster the variation with angular dependence. When multipole expansions converge they often do so very quickly, so can be truncated after the first few terms for a very good approximation to the function. Multipole expansions are used CHAPTER 2. BACKGROUND 46 in a variety of topics where the wish is to study a collection of interacting objects, such as in the movement of stars and galaxies in cosmology or the interaction of molecules and atoms in chemistry. As such multipole expansions can be of use when studying composites. Often models of composite behaviour use isolated inclusion models, which provide O(c) estimate of the effective properties of the composite, where c is the volume fraction of inclusions [42]. For composites with a high volume fraction of inclusions interaction between pairs of inclusions must be considered if the accuracy of the models is to be improved. Much work on multipole theories for composites has studied spherical inclusions, starting with the now classic work of Maxwell [20] and Rayleigh [30] on thermal conductivity of composites with regularly arranged inclusions. At the basis of all of these multipole expansions is the ability to rewrite boundary conditions expressed in one local coordinate system in a global coordinate system centered about a different origin.

Addition formulae

When moving solutions to a coordinate system centered about a diﬀerent origin, from

O1 to O2 in ﬁgure 2.6 say, certain addition formulae can be very useful. During this thesis we will use those for exponential functions and Legendre polynomials as follows.

r1 r2

θ1 θ2 r O1 O2

Figure 2.6: Vector schematic of a point P with reference to two diﬀerent origins O1 and O2

iθ1 iθ2 i0 Exponential functions: If we let z1 = r1e , z2 = r2e , and O1O2 = 1e then at any point P in ﬁgure 2.6, if we assume r2 < 1, we ﬁnd:

−inθ1 ∞ e 1 1 k n + k − 1 k n = n = n = (−1) z2 , r1 z1 (z2 + 1) k k=0 since z1 = z2 + 1. Thus we may write

−inθ1 ∞ e k n + k − 1 k ikθ2 n = (−1) r2 e . (2.62) r1 k k=0 CHAPTER 2. BACKGROUND 47

−iθ1 Similarly this can be repeated for z1 = r1e to give

inθ1 ∞ e k n + k − 1 k −ikθ2 n = (−1) r2 e for r2 < 1. (2.63) r1 k k=0 Similarly if we wish to re-express multipole potentials in terms of polar coordinates based at O1, we have z2 = z1 − 1 and so

∞ e∓inθ2 n + k − 1 = (−1)n rke±ikθ1 for r < 1. (2.64) rn k 1 1 2 k =0 Legendre Polynomials: We can derive an equivalent addition theorem for multipole expansions involving Legendre polynomials as follows [?].

z P θ1 r1

θ2 r r 2

Figure 2.7: Schematic of a point P with reference to two diﬀerent origins O1 and O2

Pn(cos φ1) If we wish to express ξn = n+1 in terms of the coordinates at origin O2, then r1 from ﬁgure 2.7 and the deﬁnition of the Legendre polynomials we know that ξn is harmonic, regular at O2 and axisymmetric about the z-axis. These three conditions imply that ξn has an expansion of the form

∞ m αnm r2 Pm(cos φ2), (2.65) m =0 where we need to identify the coeﬃcients αnm. If we consider P to be on the axis of symmetry between O1 and O2 then r1 + r2 = r and φ1 = π, φ2 = 0 (so that CHAPTER 2. BACKGROUND 48

n Pn(cos φ1)= Pn(−1) = (−1) , and Pm(cos φ2)= Pm(1) = 1). Thus

n ∞ (−1) m n+1 = αnmr2 . r1 n=0

The left hand side can then be expressed in terms of r2 as

∞ (−1)n (−1)n r −n−1 (−1)n (n + m)! r m = 1 − 2 = 2 for |r | < r. (r − r )n+1 rn+1 r rn+1 n!m! r 2 2 m =0 Thus we ﬁnd (−1)n (n + m)! α = . nm rn+m+1 n!m! ˆ Pn(cos φ2) Via a similar method we can express ξn = n+1 in terms of r1 and φ1, in the r2 form ∞ m αˆnm r1 Pm(cos φ1) (2.66) m=0 where (−1)m (n + m)! αˆ = . nm rn+m+1 n!m! When r = 1 it can be useful to simplify the notation in (2.65) and (2.66) to

∞ Pn(cos φi) i m n+1 = δmr3−iPm(cos φ3−i), (2.67) ri m =0 where n n+m i (−1) n if i = 1, δm = (−1)m n+m if i = 2.  n 2.4 Single cavity problems

In order to appreciate how multiple cavities affect the macroscopic properties of a medium it is sensible to first understand the effect of one cavity. We will look at both the effect of cavities in steady state heat conduction problems and in elastic media. Due to its simplicity the steady state heat equation, Laplace’s equation, is a good starting point in understanding the methods used to solve cavity problems, which can then be applied to the more complex vector equations governing elastostatics. CHAPTER 2. BACKGROUND 49

2.4.1 Steady state heat conduction in an inﬁnite medium containing a spherical cavity

In a spherical geometry the simplest cavity problem to pose is that of steady heat emission from a spherical cavity in a conducting medium. Temperature, ψ, throughout the medium is governed by the heat equation, which under steady state conditions reduces to Laplace’s equation, ∇2ψ = 0.

x3 φ r

x1 θ

Figure 2.8: Spherical cavity in an inﬁnite medium.

In order to illustrate a solution technique that will be employed later, a situation will be considered where the boundary conditions do not have spherical symmetry. Let us therefore pose the thermal boundary condition

ψ = T cos φ on r = R (2.68) with enforced variation in φ.(Here spherical polar coordinates (r, φ, θ) are being used, as in ﬁgure 2.8.) If the cavity is the only heat source we would also expect to see the temperature decay with distance from the surface of the cavity:

ψ → 0 as r →∞. (2.69)

As in Appendix A.1, a harmonic axisymmetric solution to Laplace’s equation takes the general form: ∞ B ψ(r, φ)= A rn + n P (cos φ), (2.70) n rn+1 n n =0 where Pn(x) are Legendre Polynomials of the ﬁrst kind, as deﬁned in Appendix A.

Applying the boundary condition of decay to zero as r →∞ we see that An = 0 for all CHAPTER 2. BACKGROUND 50 n. The second boundary condition (2.68) we could then express on r = R as

∞ B T P (cos φ)= n P (cos φ). (2.71) 1 Rn+1 n n=0 The orthogonality (A.1) of the Legendre polynomials over (−1, 1) gives

2 B1 = TR , Bn = 0 for n = 1.

As such we can ﬁnally write ψ as

R2 ψ = T cos φ. r2 r Nondimensionalising on the radius of the cavity, so that r = , and the maximum R ψ temperature of the cavity surface, so that ψ = T , this solution can be written as,

cos φ ψ = . r¯2

Nondimensionalising the solution reveals its lack of dependence upon the cavity radius, R, or maximum temperature of the cavity surface, T . We expect the temperature of the material to decrease quadratically with distance from the origin, and we can see from ﬁgure 2.9 that, since the solution is dependent upon both the co-latitudinal angle φ and also the distance from the cavity origin, lines of constant temperature form in arcs around the poles of the cavity. It should also be remembered that the full spherical solution is found by rotating the contours about the z-axis. z

Figure 2.9: Temperature distribution ψ throughout the medium, on y = 0, when ψ = T cos φ is applied on the surface of the cavity (for T = 8 and cavity radius R = 1) and ψ → 0 as radial distance r tends to inﬁnity. CHAPTER 2. BACKGROUND 51

2.4.2 Spherical cavity in an elastic medium: A spherically symmetric problem

Now that we are comfortable solving the steady state heat equation in a medium we can consider the simplest spherically symmetric elastic cavity problem. Given a spherical cavity within an elastic medium, where the medium is subject to only a hydrostatic loading σrr = −p at inﬁnity and is traction free, so that σrr = σrφ = σrθ = 0, on the surface of the cavity at r = R, we want to know the stresses, strains and displacements throughout the medium.

x3 φ r

x1 θ

Figure 2.10: Spherical cavity in an inﬁnite elastic medium.

The medium is governed by the equation of equilibrium ∇ σ = 0, see (2.16), and will most easily be dealt with in spherical coordinates (r, φ, θ). Since this problem is ∂ ∂ spherically symmetric, ∂θ = ∂φ = 0, displacement is only radial, u = ur(r)er; the stress and strain tensors are diagonal; and hoop stresses and hoop strains in θ and φ must be equal. Therefore the equation of equilibrium reduces to

∂σ 2 rr + (σ − σ ) = 0. (2.72) ∂r r rr φφ

Through the use of equations (2.25)-(2.29), the stress-displacement relations in spherical coordinates, we obtain d2u 2 du 2u r + r − r = 0. dr2 r dr r2 This has the solution B u = Ar + , r r2 where A, B are to be found. Reusing the stress-displacement relations, this in turn CHAPTER 2. BACKGROUND 52 gives the radial stress throughout the medium as

2 B σ = 0 (1 + ν )A + 2(2ν − 1) , rr (1 − 2ν ) 0 0 r3 0 where 0 is the shear modulus of the elastic medium, and ν0 the Poisson ratio of the medium. We can then apply the boundary conditions at inﬁnity and at the cavity interface to ﬁnd the constants A and B. The boundary condition σrr →−p as r →∞ implies

20 B −p = lim (1 + ν0)A + 2(2ν0 − 1) , r→∞ (1 − 2ν ) r3 0 2 (1 + ν ) = 0 0 A, 1 − 2ν0 −p(1 − 2ν ) ⇒ A = 0 , 20(1 + ν0) and substituting this constant back into σrr gives

B σ = −p − 4 . rr 0 r3

Applying the second boundary condition σrr = 0 at r = R then shows that

−pR3 B = , 40 and thus throughout the medium the radial stress is given by

R3 σ = −p 1 − , rr r3 and radial displacement is

p R3 (1 + ν ) u = − (1 − 2ν )r + 0 . r 2 (1 + ν ) 0 2 r2 0 0 σ p Nondimensionalising with respect to R and respectively, so thatσ ¯ = rr , p¯ = , 0 rr u r 0 0 u¯ = r andr ¯ = , we can write the stress and displacement throughout the medium r R R as 1 σ¯ = −p¯ 1 − , rr r¯3 and p¯ (1 + ν ) u¯ = − (1 − 2ν )¯r + 0 . r 2(1 + ν ) 0 2¯r2 0 As expected in linear elasticity, the radial displacement and stress depend linearly upon CHAPTER 2. BACKGROUND 53 the far field pressure. We see that the non-dimensionalised radial stress does not depend upon the host material, and has cubic decay inr ¯ which is important near the cavity surface. The displacement depends upon both the distance and the host material. On the cavity surface we can see we have satisfied the zero traction condition on the radial stress, and that the cavity will contract. The radial displacement on the surface of the cavity is then given by 3¯p(1 − ν0) u¯r r¯=1 = − . 4(1 + ν0) Thus whilst radial stress will only decay with distance from the cavity surface, we can see that for radial displacement the relationship between radial distance and displacement is not quite so simple, but is instead a trade off between the displacement at infinity and that caused by the cavity which decays with distance away from the cavity. In a cavity free medium, under the same far field pressure, we would find the (non- ∞ ∞ dimensional) displacement and stress throughout the medium are given byσ ¯rr andu ¯r , where ∞ σ¯rr = −p,¯ and ∞ p¯(1 − 2ν0)¯r u¯r = − . 2(1 + ν0) With this we can show more clearly the effect a cavity within the medium has upon the displacement and stress, by writing the solutions as,

p¯ σ¯ − σ¯∞ = , rr rr r¯3 and p¯ u¯ − u¯∞ = − . r r 4¯r2

For a given material, say a rubber-like nearly incompressible medium of ν0 = 0.49, and where pressure is small enough that linear elasticity holds, figure 2.11 (in which p¯ = 0.1) illustrates the effect of a cavity within a material. If the cavity were not present the radial displacement would tend to zero with distance to the centre of the medium. However the presence of a cavity allows the medium to deform more near the cavity surface, as such we see that whilst the absolute displacement decays rapidly as it (1 + ν ) approaches the cavity from infinity, there is a point atr ¯ = 3 0 , in this case at 1 − 2ν0 r¯ = 4.208, at which the displacement increases again. The absolut e radial displacement can be useful as it is analogous to increased deformation. We know displacement will be negative at all points due to the nature of the pressure we have exerted upon the material, which causes any radial displacement to be towards the centre of the cavity in the negative radial direction. CHAPTER 2. BACKGROUND 54

|u¯r|

0.010 0.008 0.006 0.004 0.002

r¯ 2 4 6 8 10

Figure 2.11: Absolute value of the radial displacement in an inﬁnite elastic medium a) with a cavity of unit radius (blue) and b) with no cavity (pink).r ¯ is the radial distance from the centre of the cavity and medium.

2.4.3 Spherical cavity in an elastic medium: An axisymmetric problem

Although we have already solved the problem of a spherical cavity in an elastic medium, we here introduce a method to solve the cavity problem when far-ﬁeld displacements, which are not spherically symmetric, are applied to the medium. This method will be used later when considering multiple cavities within a host medium. If the far-ﬁeld displacements are axisymmetric then we can not apply the method used in section 2.4.2, we can however use the Boussinesq-Papkovich stress function method, as we will demonstrate.

zˆ0

yˆ0 xˆ0 φˆ rˆ

θˆ

Figure 2.12: Oﬀ center spherical cavity in an inﬁnite elastic medium.

Let us define a coordinate system, represented in Cartesian coordinates as (ˆx, y,ˆ zˆ) CHAPTER 2. BACKGROUND 55 or in spherical coordinates as (ˆr, φ,ˆ θˆ), at which a spherical cavity of radius Rˆ is centred, as in figure 2.12. We let the far field displacement be given by

ˆ ˆ ∞ pˆ L ∞ pˆL ∞ uˆrˆ = − (ˆr − cos φ), uˆφˆ = − sin φ, andu ˆθ = 0, 3kˆ0 2 6kˆ0

20(1 + ν0) Lˆ where kˆ0 = and is the vertical distance between coordinate systems in 3(1 − 2ν0) 2 2.12. (These displacements are axisymmetric and satisfy the equilibrium condition.) This offset represents a rigid body movement of the cavity in comparison to section 2.4.2, such that if Lˆ = 0 we will show that we recover the solution found in that section. These axisymmetric far field displacements imply the far field stresses are given by

∞ ∞ ∞ σˆrˆrˆ =σ ˆφˆφˆ =σ ˆθˆθˆ = −p.ˆ

As for the spherically symmetric cavity system, described in section 2.4.2, we will also impose zero traction conditions at the surface of the cavity so that

ˆ σˆrˆrˆ =σ ˆrˆφˆ = 0 onr ˆ = R.

Nondimensionalising

For simplicity we can nondimensionalise about the shear modulus of the medium and the cavity radius, for stresses and distances respectively. As such the coordinate system has nondimensional counterparts

rˆ r = , φ = φ,ˆ θ = θ.ˆ Rˆ

Angles φ and θ are nondimensional by deﬁnition, thus the ˆ notation for these variables is simply for completeness, and can be ignored if desired. Under this nondimensionali- sation displacements and stresses become

uˆrˆ uˆφ σˆrˆrˆ σˆrφˆ ur = , uφ = , σrr = , σrφ = , (2.73) Rˆ Rˆ ˆ0 ˆ0 and system parameters become

pˆ Lˆ p = , L = ,R = 1. ˆ0 Rˆ CHAPTER 2. BACKGROUND 56

This implies the displacements at inﬁnity are

∞ p(1 − 2ν0) L ∞ p(1 − 2ν0)L ∞ ur = − (r − cos φ), uφ = − sin φ and uθ = 0, 2(1 + ν0) 2 4(1 + ν0) which translate to stresses at inﬁnity of

∞ ∞ σrr = −p and σrφ = 0.

Displacements within the medium

In this situation we see the problem is independent of θ but not φ, thus the equation of equilibrium ∇ σ = 0, can not be simpliﬁed and solved as it was in section 2.4.2. However from section 2.3.1 we know we can rewrite the equation of equilibrium in terms of displacements as

∞ 2(u − u )= ∇(ξ + zψ) − 4(1 − ν0)ψez, (2.74) where ξ and ξ satisfy Laplace’s equation. Since we want stress to be constant at inﬁnity we discard solutions to Laplace’s equation that grow in r, and therefore write

∞ ∞ A C ξ = n P (cos φ), ψ = n P (cos φ), rn+1 n rn+1 n n=0 n=0 for unknown constants An and Cn.

Using the relations z = r cos φ and ez = cos φ er − sin φ eφ we substitute ξ and ψ into (2.74) to ﬁnd the components of the displacement about the origin. After using relations between the Legendre polynomials, discussed further in Appendix D, the displacements can be written as

∞ (n + 1) 2(u − u∞)= − A P r r n rn+2 n n=0 C (n + 1)P + nP + n (n + 4 − 4ν ) n+1 n−1 , rn+1 0 2n + 1 ∞ A 2(u − u∞)= n P (1) φ φ rn+2 n n=0 (1) (1) C (n + 4ν0 − 3)P + (n + 4 − 4ν0)P + n n+1 n−1 , rn+1 2n + 1

(1) ∂Pn(cos φ) where the Legendre polynomials have argument (cos φ) and Pn (cos φ) = ∂φ is the ﬁrst associated Legendre polynomial. Since P0(cos φ) = 1 this then implies that CHAPTER 2. BACKGROUND 57

(1) P0 (cos φ) = 0, meaning that the n = 0 and n = 1 terms in the expansion of uφ above have a somewhat reduced form. In order to apply the zero traction conditions on the surface of the cavity we then need to ﬁnd the radial and radial shear stresses throughout the medium.

Stress throughout the medium

In section 2.3.1 we found that the radial stress σrr is governed by equation (2.60):

∂2ξˆ ∂2ψˆ ∂ψˆ 2ν ∂ψˆ σˆ − σˆ∞ = + cos φ rˆ − 2(1 − ν ) + 0 tan φ , rr rr ∂rˆ2 ∂rˆ2 0 ∂rˆ rˆ ∂φ or nondimensionally

∂2ξ ∂2ψ ∂ψ 2ν ∂ψ σ − σ∞ = + cos φ r − 2(1 − ν ) + 0 tan φ . rr rr ∂r2 ∂r2 0 ∂r r ∂φ Now substituting for ξ and ψ yields,

∞ A σ − σ∞ = n (n + 1)(n + 2)P rr rr rn+3 n n=0 C n + 1 + n (n2 + 5n + 4 − 2ν )P + n(n + 4 − 4ν )P . rn+2 2n + 1 0 n+1 0 n−1 (2.75)

Similarly equation (2.61) yields

∞ A σ − σ∞ = − n (n + 2)P (1) rφ rφ rn+3 n n=0 2 (1) (1) C (n + 2n + 2ν0 − 1)P + (n + 1)(n + 4 − 4ν0)P + n n+1 n−1 . rn+2 2n + 1 (2.76) CHAPTER 2. BACKGROUND 58

Applying the boundary conditions

In order to ﬁnd An and Cn we apply the zero traction conditions on r = 1. Under these (2.75) and (2.76) imply

∞ ∞ −σrr = An(n + 1)(n + 2)Pn n=0 n + 1 + C (n2 + 5n + 4 − 2ν )P + n(n + 4 − 4ν )P (2.77) n 2n + 1 0 n+1 0 n−1 ∞ ∞ (1) −σrφ = − An(n + 2)Pn n=0 2 (1) (1) (n + 2n + 2ν0 − 1)P + (n + 1)(n + 4 − 4ν0)P + C n+1 n−1 . n 2n + 1 (2.78)

As in section 2.4.1 we then use the orthogonality condition of the Legendre polynomials from Appendix A to ﬁnd An and Cn for each n. Using (A.1) and (A.7),

k pδ = A (k + 1)(k +2)+ C (k2 + 3k − 2ν ) k0 k k−1 2k − 1 0 (k + 2) + C (k + 1)(k + 5 − 4ν ) (2.79) k+1 2k + 3 0 for k ≥ 0, and

(k2 + 2ν − 2) (k + 2)(k + 5 − 4ν ) 0= A (k +2)+ C 0 + C 0 (2.80) k k−1 2k − 1 k+1 2k + 3

∞ ∞ for k ≥ 1, where we impose σrr = −p,σrφ = 0. Solving simultaneously, if we take (2.79)−(k + 1)×(2.80), we ﬁnd for k ≥ 1,

(k2 − 2kν + k − ν + 1) 0 = 2C 0 0 k−1 2k − 1 implying

Ck ≡ 0 ∀k.

This incidentally implies ψ = 0 and thus

2(u − u∞)= ∇ξ. (2.81)

If we now set Ck = 0 in (2.79), this implies

δk0p = Ak(k + 1)(k + 2), CHAPTER 2. BACKGROUND 59 and thus p p A = , A = 0 for k ≥ 1 ⇒ ξ = . 0 2 k 2r Having identiﬁed the coeﬃcients of the stress functions we can therefore write the displacements and stresses about the origin. Using (2.81) we can thus derive

p L (1 + ν ) u = − (1 − 2ν )(r − cos φ)+ 0 , r 2(1 + ν ) 0 2 2r2 0 pL uφ = − (1 − 2ν0)sin φ, 4(1 + ν0) 1 σ = −p 1 − , rr r3 σrφ = 0.

This allows us to write the displacement ﬁeld u, as

p (1 + ν ) pL(1 − 2ν ) u = − (1 − 2ν )r + 0 e + 0 (cos φe − sin φe ). 2(1 + ν ) 0 2r2 r 4(1 + ν ) r φ 0 0 Results

In comparing the results here to those of a single cavity experiencing a spherically symmetric displacement field at infinity, as in section 2.4.2, it is immediately apparent that the off centre displacement field at infinity has perturbed the displacement about the cavity. If we let our solutions, obtained in section 2.4.2, for the displacement and stresses in the medium caused by a spherically symmetric displacement field be denoted sym, and the solutions obtained in this section for the axisymmetric field be denoted axi, we can see the change in displacement is caused wholly by the change in displacement conditions at infinity:

axi sym pL(1 − 2ν0) u = u + (cos φer − sin φeφ). 4(1 + ν0)

This change in displacement is the result of the rigid body motion in the z axis, since the

Cartesian unit vector ez = cos φer −sin φeφ. This rigid body displacement of the cavity is linearly dependent upon the offset L of the cavity and is linearly dependent upon the pressure applied to the material but has a more complex relationship with the material properties of host medium. As the offset increases, we see the rigid body displacement becomes the dominant term in the expression. Thus it is important when considering offset cavities to remove the rigid body displacement from displacement results prior to comparing with cavities which are not offset. For a nearly incompressible material where ν0 ≈ 0.5 we see that the rigid body displacement tends to zero and the cavity feels just hydrostatic pressure. The rigid body displacement is therefore most easily CHAPTER 2. BACKGROUND 60 seen for a very deformable material such as cork, where ν ∼ 0. As such the change in displacement between the spherically symmetric and axisymmetric case is proportional to the distance of the cavity to the centre of the medium relative to the cavity diameter and to the pressure applied to the material. This is demonstrated in figure 2.13 where a cavity which is not centred at the same point as the focus of the displacement at infinity is shifted towards the focus of the displacement.

0.4

0.2

y 0.4 0.2 0.2 0.4 0.2

0.4

Figure 2.13: Deformation of cavity in spherically symmetric (pink, dotted) and axisymmetric (blue, solid) displacement ﬁelds, where ν0 = 0.49985, L = 100, p = 2.

We can also note that since the stress conditions at inﬁnity are the same in the spherically symmetric and axisymmetric systems this results in the same radial and radial shear stresses throughout the medium

1 σaxi = σsym = −p 1 − , (2.82) rr rr r3 axi sym σrφ = σrφ = 0. (2.83) CHAPTER 2. BACKGROUND 61

2.5 Single shell problems

2.5.1 Thick spherical shell embedded in an elastic medium: Spheri- cally symmetric problem

If instead of a cavity within a medium, we have a shell, we can find solutions in much the same way as we have in sections 2.4.2 and 2.4.3, we just need to take into account the boundary between the material of the shell and that of the medium. If we have an infinite medium containing a shell of outer radius Rˆ1, inner radius Rˆ0 and shell thickness hˆ, such that Rˆ1 = Rˆ0 + hˆ, then we can construct a spherically symmetric problem by allowing the shell to be under hydrostatic pressure,p ˆ, at infinity with no radial stress upon the inner surface of the shell, Rˆ0, as in section 2.4.2. For simplicity we can consider the coordinate system (ˆr, φ,ˆ θˆ) to be located at the centre of the shell. If we consider the shell to be perfectly bonded to the medium then we must have continuity of radial stress and radial displacement across the shell-medium interface, m s m s ˆ such thatu ˆrˆ =u ˆrˆ andσ ˆrˆrˆ =σ ˆrˆrˆ onr ˆ = R1, where superscripts s or m, or subscripts 1 or 0, are used to denote values in the shell or medium respectively.

Nondimensionalising

If we nondimensionalise our problem upon the outer radius of the shell Rˆ1 and the host medium shear modulus ˆ0, so that

rˆ hˆ uˆr pˆ σˆrˆrˆ r = , h = , ur = , p = , σrr = , φ = φ,ˆ θ = θ,ˆ Rˆ1 Rˆ1 Rˆ1 ˆ0 ˆ0 then we can write the conditions designating the problem conﬁguration as

∞ σrr = −p, σrr =0 on r = R0, s m s m ur = ur on r = 1, σrr = σrr on r = 1.

Displacement and stress throughout the medium

From section 2.4.3 we know the medium is governed by the equation of equilibrium, which under spherical symmetry reduces to,

dσm 2 rr + (σm − σm ) = 0. (2.84) dr r rr φφ

The equation of equilibrium can be expressed in terms of the displacement using the stress displacement relations as the Navier-Cauchy equations, which under spherical symmetry reduce to: d2um 2 dum 2um r + r − r = 0, dr2 r dr r2 CHAPTER 2. BACKGROUND 62 with solution B um = Ar + , r r2 where A, B ∈ R. Reusing the stress-displacement relations, this then gives the radial stress throughout the medium as

2 B σm = (1 + ν )A + 2(2ν − 1) , rr (1 − 2ν ) 0 0 r3 0 where ν0 denotes the Poisson ratio of the medium.

Displacement and stress throughout the shell

The stress throughout the shell is similarly governed by the equation of equilibrium, and therefore if the shell medium has nondimensional shear modulus 1 and Poisson ratio ν1, we can express the displacement in the shell as

D us = Cr + , r r2 and the radial stress as

2 D σs = 1 (1 + ν )C + 2(2ν − 1) , rr (1 − 2ν ) 1 1 r3 1 for unknown constants C and D.

Applying the boundary and continuity conditions

In order to find the coefficients A,B,C and D we apply the stress conditions upon the shell at infinity and on the shell inner surface and the continuity conditions upon the shell-medium interface. As r →∞ we know

σrr →−p, which implies 2 (1 + ν0)A = −p, (1 − 2ν0) and thus p(1 − 2ν ) A = − 0 . 2(1 + ν0) s On the inner surface of the shell on the other hand the radial stress is zero, σrr =0 on r = R0, and thus

2 D 0= 1 (1 + ν )C + 2(2ν − 1) . (1 − 2ν ) 1 1 (R )3 1 0 CHAPTER 2. BACKGROUND 63

As such we ﬁnd 2(1 − 2ν1) D C = 3 . (1 + ν1) (R0) The continuity conditions in displacement and stress on r = 1 then imply

C + D = A + B,

1 1 ((1 + ν1)C + 2(2ν1 − 1)D)= ((1 + ν0)A + 2(2ν0 − 1)B) . (1 − 2ν1) (1 − 2ν0)

Substituting for A and C we can solve for B and D to yield

p B = − 3(1 − ν )+(1+ ν )((R )3 − 1)(1 + 2 κ ) 4ζ 1 1 0 1 0 3p(ν − 1)(1 + ν )(R )3 D = 0 1 0 , 4(1 + ν0)ζ where 3 3 ζ = (1+ ν1)(1(1 − (R0) ) + (R0) ) + 2(1 − 2ν1),

(1 − 2ν0,1) κ0,1 = . 1+ ν0,1 Substituting back into the displacement and stress, we can then write

m ur for r ≥ 1 ur = us for R ≤ r ≤ 1  r 0 where 

p 3(1 − ν )+(1+ ν )((R )3 − 1)(1 + 2 κ ) um = − κ r + 1 1 0 1 0 , r 2 0 2r2ζ 3p(ν − 1)(1 + ν )(R )3 2κ r 1 us = 0 1 0 1 + . (2.85) r 4(1 + ν )ζ (R )3 r2 0 0 Similarly we can write the radial stress as,

m σrr for r ≥ 1 σrr = σs for R ≤ r ≤ 1  rr 0 where R0 is the inner shell radius, 1 is the outer shell radius and p σm = −p + 3(1 − ν )+(1+ ν )((R )3 − 1)(1 + 2 κ ) , rr ζr3 1 1 0 1 0 3p (ν −1)(1 + ν )(R )3 1 1 σs = 1 0 1 0 − . rr (1 + ν )ζ (R )3 r3 0 0 CHAPTER 2. BACKGROUND 64

Results

Agreement with cavity solution

If 1 = 1 and ν1 = ν0 = ν we recover the solution in the cavity found in section 2.4.2, for a cavity of radius R0. Under this assumption we ﬁnd the coeﬃcients A,B,C and D simplify to,

p(1 − 2ν) A = C = − (2.86) 2(1 + ν) p(R )3 B = D = − 0 , (2.87) 4 so that p (1 − 2ν) (R )3 u = us = um = − r + 0 r r r 2 (1 + ν) 2r2 and R 3 σ = σs = σm = −p 1 − 0 , rr rr rr r as expected.

Comparison with cavity solution

Now that we can express the displacement and stress within the shell and the medium of the host we can investigate how a shell deforms in comparison to our earlier work in section 2.4.2. As in that section, we see in figure 2.14 that the inclusion affects the displacement of the entire composite, causing a decrease in deformation near the inclusion surface. However since we are now dealing with a shell, composed of a different material to the cavity, we are also able to see a change in the deformation at the shell medium boundary, as seen in 2.15. For a shell of a material with a greater shear modulus than the medium (in this case a glassy shell with ν1 = 0.25, 1 = 5 within a nearly incompressible polyurethane host with ν0 = 0.49985 and p = 1), we see the shell has a stiffening effect upon the composite. The material within the shell has a lower deformation gradient than that outside the shell, resulting in a lower deformation on the inner surface of the shell, and also causing less deformation within the medium near the cavity boundary. As in previous sections we see that the displacement is linear with regard to the pressure at infinity and thus as we increase the pressure we will only see a linear response for a fixed material system. CHAPTER 2. BACKGROUND 65

ur r 4 6 8 10

0.01

0.02

0.03

0.04

0.05

0.06

Figure 2.14: Radial displacement throughout a composite, with a host medium (ν0 = 0.49985) under far ﬁeld hydrostatic pressure (p = 1), containing: a shell (1 = 5,ν1 = 0.25) of thickness h = 0.1 with inner shell boundary R0 = 0.9 (blue, solid); a cavity of radius 0.9 (pink, dashed).

0.12

0.14

0.16

0.18

0.20

0.22 r 0.95 1.00 1.05 1.10

Figure 2.15: Radial displacement near the shell/cavity boundary in a composite (with host medium ν0 = 0.49985) under far ﬁeld hydrostatic pressure, p = 1, containing: a shell (ν1 = 0.25,1 = 5)of thickness h = 0.1 with inner shell boundary R0 = 0.9 (blue, solid) ; a cavity of radius 0.9 (pink, dashed)

Interestingly it appears that the order of magnitude of the shear modulus of the shell affects the displacement of the shell more than the Poisson ratio of the shell (in comparison to the medium). This can be seen in figures 2.16 and 2.17. As the shear modulus of the shell increases, we see less deformation at the shell-medium interface, so that for thick shells with a large shear modulus, the deformation of the shell is negligible. In comparison the Poisson ratio of the shell has a much more moderate CHAPTER 2. BACKGROUND 66 effect upon the displacement and affects thick shells far more than thin shells.

ur h 0.10 0.15 0.20 0.25 0.30

0.05

0.10

0.15

0.20

Figure 2.16: Radial displacement from undeformed shell on the shell-medium interface, R1 = 1, as shell thickness, h, changes for shells in which ν0 = ν1 = 0.49985 for shear 3 2 moduli 1 = 10 (blue, solid), 1 = 10 (pink, small dashing), 1 = 10 (yellow, dot- dashed), 1 = 1 (green, large dashing), under hydrostatic pressure p = 1.

0.10

0.15

0.20

h 0.10 0.15 0.20 0.25 0.30

Figure 2.17: Radial displacement from undeformed shell on the shell-medium interface R1 = 1 as shell thickness h changes for shells with shear modulus equal to that for the ν0 host 1 = 1 with host Poisson ratio ν0 = 0.49985 and shell Poisson ratios: ν1 = 4 ν0 3ν0 (blue, solid), ν1 = 2 (pink, small dashing), ν1 = 4 (yellow, dot-dashed), ν1 = ν0 (green, large dashing), under hydrostatic pressure p = 1. CHAPTER 2. BACKGROUND 67

Thin shell limit

Finding the displacement of an embedded spherical shell under hydrostatic pressure is relatively simple. However, when considering problems involving multiple embedded shells or other loading conditions we may wish to use approximate solutions. In fact approximations currently exist in the literature, see [10] and [17], to the solution of a single shell within an elastic medium under hydrostatic pressure. Here we compare the results of those two diﬀerent methodologies to the exact solution. If the shell is thin, h ≪ 1, then we can linearise the shell displacement about h if we assume all other parameters are of O(1). If we only want terms of up to order h we can expand terms n 2 of the form (1 − h) ∼ 1 − nh + O(h ), and so the displacement ur on the outer surface of the shell becomes

p(ν − 1)(1 + ν ) 2(1 − 2ν ) us = 0 1 1 + (1 − 3h) + O(h2). r 4(1 + ν )((1 − ν ) − h(1 − )(1 + ν )) (1 + ν ) r=1 0 1 1 1 1 (2.88) which simpliﬁes to

3p(ν − 1)((1 − ν ) − h(1 + ν )) us = 0 1 1 + O(h2). (2.89) r 4(1 + ν )((1 − ν ) − h(1 − )(1 + ν )) r=1 0 1 1 1 Fully expanding the displacement in the shell about h this could be written as

3p(ν − 1) (1 + ν ) us = 0 1 − h 1 1 + O(h2). (2.90) r 4(1 + ν ) (1 − ν ) 0 1 This will be known as the thin-shell approximation. This allows us to compare the displacement on the shell boundary predicted by the thin shell approximation (2.90) to the full thick shell solution in (2.85). For a shell in which the shear modulus is O(1) relative to that of the host medium, (as in ﬁgures 2.18 and 2.19 for example), the thin shell solution, as given in (2.90), provides a good approximation to the full solution for small h. However as the shell thickness increases the thin shell approximation underestimates the deformation of the shell slightly, which can be clearly seen in ﬁgure 2.19 in which the relative change in displacement between the exact solution and the approximate thin shell solution is depicted. CHAPTER 2. BACKGROUND 68

0.230

0.235

0.240

0.245

h 0.01 0.02 0.03 0.04 0.05

Figure 2.18: Radial displacement on the outer boundary of a shell with ν1 = 0.25 and 1 = 1.05 within a medium in which ν0 = 0.49985 and 0 = 1, plotted against the shell thickness, h. This uses the full thick shell solution (2.85) (blue, solid) and the thin shell approximation (2.90) (pink, dashed). The medium is subject to hydrostatic pressure p = 1.

ur−approx(ur) ur 0.005

0.004

0.003

0.002

0.001

h 0.01 0.02 0.03 0.04 0.05

Figure 2.19: Relative change between the radial displacement of a shell (with material parameters as in above graph) using the full thick shell solution ur and the thin shell approximation, approx(ur), as the shell thickness h changes.

Stiﬀ-thin shell limit

1 However if we try to use the thin shell approximation when 1 ∼ h we can see in ﬁgure 2.20 that whilst the full solution predicts a small contraction of the shell surface the thin shell approximation predicts a large increase in the radius of the shell, which is obviously incorrect. CHAPTER 2. BACKGROUND 69

0.15

0.10

0.05

h 0.01 0.02 0.03 0.04 0.05

0.05

Figure 2.20: Radial displacement of a shell (ν1 = 0.25,1 = 1.05h) within an inﬁ- nite medium (ν0 = 0.49985) under far-ﬁeld hydrostatic pressure p = 1, using the full thick shell solution ur (blue, solid) and the thin shell approximation, approx(ur) (pink, dashed) as shell thickness h changes.

For the thin shell approximation it was assumed that all other parameters were O(1). However some materials, such as a polyurethane medium containing glassy shells, for 3 which 1 = O(10 ), clearly invalidate this assumption. For these materials, in which

1 ≫ 1, however, it is possible to rescale the the shear modulus upon the shell thickness, −1 by deﬁning a parameter η = 1h. Thus when 1 ∼ O(h ) we ﬁnd η ∼ O(1), so that η we can simplify our results using both h ≪ 1 and = by taking a series solution in 1 h h. Under these conditions our shell displacement, (2.85), can be written

3 s 3p(ν0 − 1)(2(1 − 2ν1)+(1+ ν1)(1 − h) ) ur = η 3 , (2.91) 4(1 + ν0)(3(1 − ν1) + (1 − h )(1 + ν1)((1 − h) − 1)) −3p(ν − 1) −(ν − 1)2 + η(1 + ν )(ν − 1 + 2hν ) = 0 1 1 1 1 + O(h2). (2.92) 4(1 + ν ) (1 + η + (η − 1)ν )2 0 1 This approximation will be known as the stiff-thin shell approximation. When the stiffness of the shell is inversely proportional to the shell thickness, even when we keep all other material constants the same, we expect to see smaller displacements than we observe when 1 = O(1). Using the stiff-thin shell approximation (2.92) to the full radial displacement we can again see how well this approximation matches the full solution. CHAPTER 2. BACKGROUND 70

ur 0.0890

0.0895

0.0900

0.0905

h 0.01 0.02 0.03 0.04 0.05

Figure 2.21: Radial displacement on the boundary of a shell (ν1 = 0.25, η = 1.05) within an infinite medium (ν0 = 0.49985) under far-field hydrostatic pressure p = 1, as the shell thickness, h, changes. Using the full thick shell solution (2.85) (blue, solid) and the stiff-thin shell approximation (2.92) (pink, dashed).

ur−approx(ur) ur

0.0008

0.0006

0.0004

0.0002

h 0.01 0.02 0.03 0.04 0.05

Figure 2.22: Relative change between the radial displacement of a shell (ν1 = 0.25, η = 1.05) within an infinite medium (ν0 = 0.49985) under far-field hydrostatic pressure p = 1, using the full thick shell solution ur and the stiff-thin shell approximation, approx(ur) as shell thickness, h, changes.

We can see in ﬁgure 2.22 that the thin shell approximation is incredibly close to that of the full solution. When using the parameterisation η = h1 we keep terms, 2 such as h 1, that would otherwise have been considered negligible and discarded, as under the thin shell approximation. By retaining these terms in the stiﬀ-thin shell approximation, even when 1 is large, we provide a more accurate approximation to the full solution than the thin shell solution. CHAPTER 2. BACKGROUND 71

2.5.2 Thin spherical shell embedded in an elastic medium: Spherically symmetric problem- a second approach

We can approach ﬁnding the deformation of a thin shell in a second way. Following the work of S.L. Fok and D.J. Allwright [10] we can consider a thin spherical shell embedded within the medium under hydrostatic pressure, the geometry of which is in ﬁgure 2.23.

φ r

Figure 2.23: Spherical shell in an inﬁnite medium.

In this analysis we will assume, as in the thick shell analysis of the previous section, that the shell and the medium are isotropic; the shell is perfectly bonded to the surrounding medium and the displacements are small enough for the linear theory to still hold, but we will also assume the shell is thin in comparison to its diameter, allowing us to treat the shell as a surface situated at its mid-radius. We will again consider the medium to be under hydrostatic pressure at inﬁnity such that σˆrˆrˆ →−pˆ asr ˆ →∞ and that the inner surface of the shell is traction free. If we let the mid-radius of the shell be situated atr ˆ = Rˆ and the thickness of the shell be hˆ then this condition becomes

σˆrˆrˆ = 0 onr ˆ = Rˆ. We will also requireσ ˆrˆrˆ to be continuous across the medium-shell interface. It should be noted that by approximating the shell inner surface by the shell mid- surface this is not a consistently O(hˆ) solution, however the results of this method are widely used in industry. Therefore it is important that we consider how well this method of solution approximates the full solution, derived in the previous section. As in the previous section we will nondimensionalise stresses and pressures on the shear modulus of the medium 0, but we will nondimensionalise distance measure- ments upon the mid-radius of the shell Rˆ. The hat notation, ˆ , will be dropped for nondimensional parameters, and as such the constraints upon the shell can be written CHAPTER 2. BACKGROUND 72 as:

σrr →−p as r →∞

σrr = σrφ =0 on r = 1

σrr and σrφ continuous across the medium shell interface.

Deformation of the medium

As in the last section this problem is spherically symmetric, so displacements are purely radial u = ur(r)er and the stress and strain tensors are diagonal with σθθ = σφφ and

εθθ = εφφ. This trivially satisﬁes the boundary conditions on σrφ. Due to the spherical symmetry of the problem, the equation of equilibrium, (2.16), again reduces to (2.72), namely, dσ 2 rr + (σ − σ ) = 0, dr r rr φφ and through use of the stress-displacement relations, (2.25)-(2.29), yields the displacement B u = Ar + , r r2 and the radial stress

2 B σ = (1 + ν )A + 2(2ν − 1) , rr (1 − 2ν ) 0 0 r3 0 where ν0 is the Poisson ratio of the medium. The boundary condition at inﬁnity −p(1 − 2ν0) σrr →−p as r →∞ again shows that A = , and thus 2(1 + ν0)

B σ = −p − 4 . rr r3

We now apply the boundary condition at the interface. If we let the radial stress at the interface be an unknown constant −p˜ then at r = 1 we have

−p˜ = −p − 4B, and so (˜p − p) B = . 4 Thus throughout the medium (denoted with superscript m) we have

(p − p˜) σm = −p + , rr r3 CHAPTER 2. BACKGROUND 73

1 (1 + ν ) um = −p(1 − 2ν )r + (˜p − p) 0 . r 2(1 + ν ) 0 2r2 0 Deformation of the shell

We then need to ﬁnd the deformation of the shell. Treating the shell as a surface we have σrr ≪ σθθ and thus to leading order σrr = 0 (see Appendix C for a derivation of this approximation) and σθθ = σφφ. The hoop stresses can be related to the interface pressure by considering the forces upon the shell. Since the shell is in equilibrium the stress around the shell must be balanced by the external pressure across the cross section. Thus

σθθ2πh = −pπ,˜ and so p˜ σ = σ = − . θθ φφ 2h The strains acting upon the shell can then be found using Hooke’s law, which under spherical symmetry reduces to

(1 − ν1) εθθ = εφφ = σθθ, 21(1 + ν1) p˜(1 − ν ) = − 1 , 4h1(1 + ν1) where ν1 and 1 are the Poisson ratio and shear modulus of the shell. Under the assumption of purely radial displacement the strain-displacement relations then yield

u ε = ε = r . θθ φφ r

Thus treating the shell as a surface implies the displacement of the shell is

s p˜(1 − ν1) ur = − . 4h1(1 + ν1)

Equating the displacement of the shell and the displacement of the medium at the interface r = 1 we get

p˜(1 − ν ) 1 (1 + ν ) − 1 = −p(1 − 2ν )+(˜p − p) 0 , 4h (1 + ν ) 2(1 + ν ) 0 2 1 1 0 3(1 − ν ) p˜ = − 0 + , 4(1 + ν0) 4 3p(1 − ν ) h (1 + ν ) ⇒ p˜ = 0 1 1 . (1 + ν0) 1 − ν1 + h1(1 + ν1) CHAPTER 2. BACKGROUND 74

Therefore the displacement of the shell is

s 3p(1 − ν0)(1 − ν1) ur = − , (2.93) 4(1 + ν0) ((1 − ν1)+(1+ ν1)h1) and the displacement in the medium simpliﬁes to

p (2(1 − 2ν )(1 + ν )h + (ν − 1)(1 + ν )) um = − (1 − 2ν )r − 0 1 1 1 0 . r 2(1 + ν ) 0 2r2((1 − ν )+ h (1 + ν )) 0 1 1 1 It is noticeable at this point that the shell displacement is not the same solution we recovered in section 2.5.1 when linearising (1 − h)n in the solution for a thick shell about h ≪ 1, and assuming the shell shear modulus 1 ∼ O(1), i.e (2.89). However if we expand this solution (2.93) in powers of h we ﬁnd the solution matches equation (2.90), the thin shell approximation to the thick shell solution, to order h

s 3p(1 − ν0) (1 − ν1) ur = − , 4(1 + ν0) ((1 − ν1)+(1+ ν1)h1) 3p(1 − ν ) ν − 1+ h (1 + ν ) = − 0 1 1 1 + O(h2). 4(1 + ν0) ν1 − 1

m s As r → 1 it can be seen that ur → ur, so throughout the host material,

p (2(1 − 2ν )(1 + ν )h + (ν − 1)(1 + ν )) u = − (1 − 2ν )r − 0 1 1 1 0 . (2.94) r 2(1 + ν ) 0 2r2((1 − ν )+ h (1 + ν )) 0 1 1 1 When written as

(1 − 2ν ) 1 3h (1 + ν ) u = −p 0 r + (1 + ν ) − 1 1 , r 2(1 + ν ) 2r2 0 (1 − ν )+ h (1 + ν ) 0 1 1 1 it is obvious that as h → 0 we recover the spherical void solution in section 2.4.2.

Results

Comparison to thick shell solution We would expect the solution for a thin shell derived by the Fok-Allwright method to exhibit similar characteristics to the thick shell solution derived in section 2.5.1 when the shell thickness h is small. We wish to determine how well the Fok-Allwright solution approximates the full thick shell solution for the various parameters the solution depends upon. When parameters are fixed we will set 1 = 200, ν0 = 0.49985, ν1 = 0.25, h = 0.01 (similar to the material parameters found in Expancel microshells often used in industry) and p = 1; unless otherwise stated we will consider displacements at the shell-medium interface r = 1. If we first consider how properties vary with CHAPTER 2. BACKGROUND 75 shell thickness, we would expect the Fok-Allwright solution to be a good approximation to the full solution for thin shells but to worsen as the shell thickness increases. We can see this does indeed occur in figure 2.24. In this figure we compare the error between the full solution, (2.85), and various approximations to the solution given by the Fok-Allwright solution derived above, the thin shell solution given in (2.90) and the stiff-thin shell approximation given in (2.92). It is interesting to note that the error of the Fok-Allwright solution is also always less than h2. We can see from figure 2.24 that whilst the Fok-Allwright approximation is good, the stiff-thin shell approximation (2.92) is better, whilst the approximation provided by considering the shell as only thin, is generally worse as the the shell thickness increases. However it is worth noting that even in the worst fitting approximation the error is only around 2 percent.

|error|

0.020

0.015

0.010

0.005

h 0.02 0.04 0.06 0.08 0.10 u − approx(u ) Figure 2.24: Relative error, r r , in approximation of the full thick shell u r solution ur by the approximate solution, approx( ur), given by:(i) the Fok-Allwright solution (blue, solid),(ii) the thin shell solution (pink, dashed) and (iii) the stiﬀ-thin shell solution (yellow, dot-dashed), for a shell (1 = 200, ν1 = 0.25) in a nearly incompressible medium,(ν0 = 0.49985) under far ﬁeld hydrostatic pressure p = 1, as shell thickness, h, changes.

Similarly in figure 2.25 we see that if we fix the shell thickness at h = 0.01 and allow the shell Poisson ratio to change, the error between the solutions does increase with increasing Poisson ratio, though the error is still only O(h2) for any Poisson ratio. In comparison to the other approximations we can see that again the stiff-thin shell approximation is much more accurate than the Fok-Allwright approximation, especially as the shell material becomes less compressible. As before the thin shell approximation is a relatively poor approximation to the full solution, with a higher mean error (0.0021) than the Fok-Allwright approximation (of 0.0017), though for less CHAPTER 2. BACKGROUND 76

compressible shells, as ν1 > 0.36, the thin shell approximation is better than the Fok- Allwright approximation. Once again though it is important to note that the error for all of these approximations is always less than 0.1%.

|error| 0.0006

0.0005

0.0004

0.0003

0.0002

0.0001

ν 0.1 0.2 0.3 0.4 0.5 1 u − approx(u ) Figure 2.25: Relative error, r r , in approximation of the full thick shell u r solution ur by the approximate solution, approx( ur), given by:(i) the Fok-Allwright solution (blue, solid),(ii) the thin shell solution (pink, dashed) and (iii) the stiﬀ-thin shell solution (yellow, dot-dashed), for a shell (1 = 200, h = 0.01) in a nearly incompressible medium,(ν0 = 0.49985) under far ﬁeld hydrostatic pressure p = 1, as shell Poisson ratio, ν1, changes.

Where we see the most change between the exact solution and the Fok-Allwright approximation for thin shells is when we start varying the shell shear modulus 1. We can see in ﬁgure 2.26 that for 1 ∼ O(1) the approximation is very good, with errors 2 around O(h ), however as 1 increases and becomes inversely proportional to the shell 3 thickness, 1 ∼ 10 , the error increases to around O(h), indicating that for a large shell shear modulus the Fok-Allwright solution is a poor approximation to the exact solution. Similarly the thin shell approximation is a good approximation to the full solution when

1 ∼ O(1) however very quickly becomes useless as the shear modulus increases. The stiﬀ-thin shell approximation on the other hand is an excellent approximation for all values of the shear modulus, due to the inclusion of combinations of parameter values (such as h2) discarded in other approximations. CHAPTER 2. BACKGROUND 77

|error| 0.010

0.008

0.006

0.004

0.002

0.000 5 10 50 100 500 1000 1 u − approx(u ) Figure 2.26: Relative error, r r , in approximation of the full thick shell u r solution ur by the approximate solution, approx( ur), given by:(i) the Fok-Allwright solution (blue, solid),(ii) the thin shell solution (pink, dashed) and (iii) the stiﬀ-thin shell solution (yellow, dot-dashed), for a shell (h = 0.01, ν1 = 0.25) in a nearly incompressible medium,(ν0 = 0.49985) under far ﬁeld hydrostatic pressure p = 1, as shell shear modulus, 1, changes

2.5.3 Spherical shell embedded in an elastic medium: Axisymmetric problem

Having tackled the first individual shell problem we can demonstrate the method for solution of shell problems where the far-field displacements applied to the composite are axisymmetric. We will construct this problem to mirror that of the single void, with the L shell offset by vertical distance , representing a rigid body movement of the shell, as 2 in section 2.4.3. This will allow easier comparison between the cavity and shell cases. As such let us define the shell to have coordinates (r, φ, θ), outer radius 1 and shell thickness h, as in figure 2.27. Parameters within the problem are nondimensionalised on the shell outer radius and host medium shear modulus ˆ0. CHAPTER 2. BACKGROUND 78

z0 φ 0 r0 y0 x0 φ r

Figure 2.27: Oﬀ center spherical shell in an inﬁnite elastic medium.

If we apply far ﬁeld axisymmetric displacements, as in section 2.4.3, then

∞ p(1 − 2ν0) L ∞ p(1 − 2ν0)L ∞ ur = − (r − cos φ), uφ = − sin φ and uθ = 0, 2(1 + ν0) 2 4(1 + ν0) which imply ∞ ∞ σrr = −p and σrφ = 0.

As we are now dealing with a composite the displacements and stresses at infinity are valid only within the medium, since the shell is confined to a fixed radius. In keeping with previous problems we will also propose zero radial traction conditions, this time on the inner surface of the shell so that

σrr = σrφ =0 on r = 1 − h.

Since we are now dealing with two materials, one of the host and one of the shell, if we are to presume, as in section 2.5.2, that the shell is perfectly bonded to the host, we will also need boundary conditions across the shell-host interface, to ensure continuity of displacements and stresses. As such we set continuity conditions,

σrr,σrφ, ur, uφ continuous on r = 1.

Continuity conditions on σrθ and uθ are satisﬁed trivially.

Deformation within the medium

In section 2.4.3 we saw that for an axisymmetric problem independent of θ we could ﬁnd the displacements and stresses throughout the medium using the Boussinesq-Papkovich stress function method to solve the equation of equilibrium. We can use the same CHAPTER 2. BACKGROUND 79 method in this situation, as such displacements within the medium are given by (2.51), namely ∞ 2(u − u )= ∇(ξ + zψ) − 4(1 − ν0)ψez, where ξ and ψ satisfy Laplace’s equation. Since our medium is now composed of both the host and the shell materials let us denote, by superscripts m, or subscript 0, materials and expressions relating to the host and superscript s, or subscript 1, for materials and expressions relating to the shell. As in section 2.4.3 the stress functions in the host material are given by

∞ ∞ A C ξm = n P (cos φ), ψm = n P (cos φ), (2.95) rn+1 n rn+1 n n=0 n=0 and therefore we can write the displacements and stresses in the medium as

∞ A 2(um − u∞)= − n (n + 1)P r r rn+2 n n =0 C (n + 1)P + nP + n (n + 4 − 4ν ) n+1 n−1 , (2.96) rn+1 0 2n + 1 ∞ A 2(um − u∞)= n P (1) φ φ rn+2 n n =0 (1) (1) C (n + 4ν0 − 3)P + (n + 4 − 4ν0)P + n n+1 n−1 , (2.97) rn+1 2n + 1

∞ A σm − σ∞ = n (n + 1)(n + 2)P rr rr rn+3 n n=0 C n + 1 + n (n2 + 5n + 4 − 2ν )P + n(n + 4 − 4ν )P , (2.98) rn+2 2n + 1 0 n+1 0 n−1 ∞ A σm − σ∞ = − n (n + 2)P (1) rφ rφ rn+3 n n=0 2 (1) (1) C (n + 2n + 2ν0 − 1)P + (n + 1)(n + 4 − 4ν0)P + n n+1 n−1 . (2.99) rn+2 2n + 1

Deformation within the shell

The shear modulus and Poisson ratio of the shell material are 1 and ν1 respectively. In constructing ξs and ψs within the shell to satisfy (2.51), given in the shell by:

s s s s 21u = ∇(ξ + zψ ) − 4(1 − ν1)ψ ez, CHAPTER 2. BACKGROUND 80

n we will need to include both forms of the solution to Laplace’s equation, r Pn and −(n+1) r Pn, since the shell is of ﬁnite radius and hollow. Therefore we represent the Boussinesq-Papkovich stress functions as

∞ ∞ a c ξs = n + b rn P , ψs = n + d rn P , (2.100) rn+1 n n rn+1 n n n=0 n=0 s for unknown constants an, bn, cn and dn. If we substitute these deﬁnitions for ξ and ψs into (2.59), the displacements in terms of the stress functions in 2.3.1, we ﬁnd

∞ a 2 us = − n (n + 1)P + b nrn−1P 1 r rn+2 n n n n=0 c (n + 1)P + nP + d rn(n + 4ν − 3) − n (n + 4 − 4ν ) n+1 n−1 , n 1 rn+1 1 2n + 1 (2.101)

∞ a 2 us = n P (1) + b rn−1P (1) 1 φ rn+2 n n n n=0 (1) (1) cn n (n + 4ν1 − 3)Pn+1 + (n + 4 − 4ν1)Pn−1 + n+1 + dnr . r 2n + 1 (2.102)

Also substituting ξs and ψs into stress equations (2.60) and (2.61), in terms of the Boussinesq-Papkovich functions, yields

∞ a σs = n (n + 1)(n + 2)P + b n(n − 1)rn−2P rr rn+3 n n n n=0 c (n + 1) + n (n2 + 5n + 4 − 2ν )P + n(n + 4 − 4ν )P rn+2 2n + 1 1 n+1 1 n−1 n + d rn−1 (n + 1)(n + 4ν − 3)P + (n2 − 3n − 2ν )P , n 2n + 1 1 n+1 1 n−1 (2.103) CHAPTER 2. BACKGROUND 81

∞ a σs = − n (n + 2)P (1) − b rn−2(n − 1)P (1) rφ rn+3 n n n n=0 2 (1) (1) cn (n + 2n + 2ν1 − 1)Pn+1 + (n + 1)(n + 4 − 4ν1)Pn−1 + n+2 r 2n + 1 (1) 2 (1) n(n + 4ν1 − 3)P + (n + 2ν1 − 2)P − d rn−1 n+1 n−1 . (2.104) n 2n + 1

Applying boundary and continuity conditions

s s Starting with the zero traction conditions upon r = 1 − h in σrr and σrφ we ﬁnd

∞ 1 n+3 0= a (n + 1)(n + 2)P + b n(n − 1) (1 − h)n−2 P n 1 − h n n n n=0 1 n+2 (n + 1) + c (n2 + 5n + 4 − 2ν )P + n(n + 4 − 4ν )P n 1 − h 2n + 1 1 n+1 1 n−1 n + d (1 − h)n−1 (n + 1)(n + 4ν − 3)P + (n2 − 3n − 2ν )P , n 2n + 1 1 n+1 1 n−1 (2.105) and

∞ 1 n+3 0= − a (n + 2)P (1) − b (1 − h)n−2 (n − 1)P (1) n 1 − h n n n n =0 n+2 2 (1) (1) 1 (n + 2n + 2ν1 − 1)P + (n + 1)(n + 4 − 4ν1)P + c n+1 n−1 n 1 − h 2n + 1 (1) 2 (1) n(n + 4ν1 − 3)P + (n + 2ν1 − 2)P − d (1 − h)n−1 n+1 n−1 . (2.106) n 2n + 1 As in section 2.4.3 we then use the orthogonality relations on the Legendre and associated Legendre polynomials from Appendix A to yield,

k(k2 + 3k − 2ν ) 0 =ˆa (k + 1)(k +2)+ ˆb t2k+1k(k − 1) + c t2 1 k k k−1 2k − 1 (k + 1)(k2 − k − 2ν − 2) + d t2k+3 1 , for k ≥ 0, (2.107) k+1 2k + 3 (k2 + 2ν − 2) 0 =ˆa (k + 2) − ˆb t2k+1(k − 1) + c t2 1 k k k−1 2k − 1 k2 + 2k − 2ν − 1 − d t2k+3 1 , for k ≥ 1. (2.108) k+1 2k + 3 CHAPTER 2. BACKGROUND 82

In order to simplify the notation of these equations we have collected like terms so that

(k + 5 − 4ν ) (k + 4ν − 4) aˆ = a + c 1 , ˆb = b + d 1 and t = 1 − h. k k k+1 2k + 3 k k k−1 2k − 1

If we notice the similarities between (2.107) and (2.108) we can simplify (2.108) further.

If we multiply (2.108) by k and add to (2.107) we can eliminate the ˆbk coeﬃcient. We therefore replace (2.108) with

(k2 + 2kν + k + ν + 1) 0 =a ˆ (2k +1)+ c t2k − 2d t2k+3 1 1 . (2.109) k k−1 k+1 (k + 2)(2k + 3)

Furthermore, if we set k = 1 in (2.109) and (2.107) we ﬁnd

2(ν + 1) 0 =6â + c t2(4 − 2ν ) − 2d t5 1 , (2.110) 1 0 1 2 5 (ν + 1) 0 =3â + c t2 − 2d t5 1 , (2.111) 1 0 2 5 implying c0 ≡ 0 and so eliminating the k = 1 equation from (2.109). Therefore our inner boundary conditions are finally

k(k2 + 3k − 2ν ) 0 =ˆa (k + 1)(k +2)+ ˆb t2k+1k(k − 1) + c t2 1 k k k−1 2k − 1 (k + 1)(k2 − k − 2ν − 2) + d t2k+3 1 , for k ≥ 0, (2.112) k+1 2k + 3 (k2 + 2kν + k + ν + 1) 0 =ˆa (2k +1)+ c t2k − 2d t2k+3 1 1 , for k ≥ 2. (2.113) k k−1 k+1 (k + 2)(2k + 3)

Now looking at continuity relations; on the boundary r = 1 we wish to equate displacements and stresses inside the shell to those in the host. We therefore need

s m s m s m s m σrr = σrr, σrφ = σrφ, ur = ur , uφ = uφ on r = 1. CHAPTER 2. BACKGROUND 83

Looking ﬁrst at continuity of radial stress, we equate (2.98) with (2.103) on r = 1 to yield

∞ ∞ σrr + An(n + 1)(n + 2)Pn n=0 n + 1 + C (n2 + 5n + 4 − 2ν )P + n(n + 4 − 4ν )P n 2n + 1 0 n+1 0 n−1 ∞ = an(n + 1)(n + 2)Pn + bnn(n − 1)Pn n=0 (n + 1) + c (n2 + 5n + 4 − 2ν )P + n(n + 4 − 4ν )P n 2n + 1 1 n+1 1 n−1 n + d (n + 1)(n + 4ν − 3)P + (n2 − 3n − 2ν )P . n 2n + 1 1 n+1 1 n−1 (2.114)

We again use the orthogonality condition for Legendre polynomials to isolate interacting coeﬃcients, leaving us with

k(k2 + 3k − 2ν ) (k2 − k − 2ν − 2) aˆ (k + 1)(k +2)+ ˆb k(k − 1) + c 1 + d (k + 1) 1 k k k−1 2k − 1 k+1 2k + 3 k(k2 + 3k − 2ν ) = − δ + Aˆ (k + 1)(k +2)+ C 0 for k ≥ 0, (2.115) k0 k k−1 2k − 1

k + 5 − 4ν where Aˆ = A + C 0 . Similarly for σ continuity we equate (2.99) and k k k+1 2k + 3 rφ (1) (2.104) at r = 1 then use the associated Legendre orthogonality relation for Pn to s m ﬁnd that if σrφ = σrφ on r = 1, then

(k2 + 2ν − 2) (k2 + 2k + 2ν − 1) aˆ (k + 2) − ˆb (k − 1) + c 1 − d 1 = k k k−1 2k − 1 k+1 2k + 3 (k2 + 2ν − 2) Aˆ (k +2)+ C 0 , for k ≥ 1. (2.116) k k−1 2k − 1

We can again notice similarities between (2.115) and (2.116), which allow us to eliminate

ˆbk terms from one equation. We can therefore rewrite (2.116) as

(k2 + 2kν + k + ν + 1) Aˆ (2k +1)+ C k =a ˆ (2k +1)+ c k − 2d 1 1 for k ≥ 1. k k−1 k k−1 k+1 (k + 2)(2k + 3) (2.117) Finally comparing the k = 1 terms of both (2.115) and (2.117) whilst remembering c0 = 0 from earlier, shows that C0 is also identically zero. Therefore we eliminate the k = 1 equation from (2.117). Finally we deal with the continuity conditions upon the m s displacements. Looking at the radial displacement we want ur = ur on r = 1, therefore CHAPTER 2. BACKGROUND 84 equating (2.101) with (2.96) on the medium shell interface

∞ 1 − a (n + 1)P + b nP 2 n n n n 1 n=0 (n + 1)P + nP + (d (n + 4ν − 3) − c (n + 4 − 4ν )) n+1 n−1 n 1 n 1 2n + 1 ∞ 1 = u∞| − A (n + 1)P r r=1 2 n n n=0 (n + 1)P + nP + C (n + 4 − 4ν ) n+1 n−1 , (2.118) n 0 2n + 1

∞ (1 − 2ν0) L where ur |r=1 = −p 1 − cos φ . Applying the Legendre orthogonality 2(1 + ν0) 2 relation implies,

p(1 − 2ν0) L − (δk0 − δk1) (1 + ν0) 2 1 (k + 3 − 4ν ) (k + 1)(k + 4ν − 2) + aˆ (k + 1) − ˆb k + c k 1 − d 1 k k k−1 (2k − 1) k+1 2k + 3 1 (k + 3 − 4ν ) = Aˆ (k +1)+ C k 0 for k ≥ 0. (2.119) k k−1 2k − 1

Similarly for the continuity conditions upon uφ we ﬁnd

p(1 − 2ν ) L 1 (k + 4ν − 4) (k + 5 − 4ν ) − 0 δ + aˆ + ˆb + c 1 + d 1 (1 + ν ) 2 k1 k k k−1 2k − 1 k+1 2k + 3 0 1 (k + 4ν − 4) = Aˆ + C 0 for k ≥ 1. (2.120) k k−1 2k − 1

Once again we can eliminate bk from (2.120) and thus replace (2.120) with

1 (3k − 4kν − 2ν + 1) aˆ (2k +1)+ c k + 2d 1 1 = Aˆ (2k +1)+ C k for k ≥ 1. k k−1 k+1 2k + 3 k k−1 1 (2.121) CHAPTER 2. BACKGROUND 85

We now have 6 sets of equations to solve for 6 sets of coeﬃcients.

k(k2 + 3k − 2ν ) 0 =a ˆ (k + 1)(k +2)+ ˆb t2k+1k(k − 1) + c t2 1 k k k−1 2k − 1 (k + 1)(k2 − k − 2ν − 2) + d t2k+3 1 , for k ≥ 0 k+1 2k + 3 (k2 + 2kν + k + ν + 1) 0 =a ˆ (2k +1)+ c t2k − 2d t2k+3 1 1 , for k ≥ 2, k k−1 k+1 (k + 2)(2k + 3) k(k2 + 3k − 2ν ) aˆ (k + 1)(k +2)+ ˆb k(k − 1) + c 1 k k k−1 2k − 1 (k2 − k − 2ν − 2) + d (k + 1) 1 k+1 2k + 3 k(k2 + 3k − 2ν ) = −δ + Aˆ (k + 1)(k +2)+ C 0 for k ≥ 0, k0 k k−1 2k − 1 (k2 + 2kν + k + ν + 1) Aˆ (2k +1)+ C k =a ˆ (2k +1)+ c k − 2d 1 1 for k ≥ 2, k k−1 k k−1 k+1 (k + 2)(2k + 3) 1 (k + 3 − 4ν ) (k + 1)(k + 4ν − 2) aˆ (k + 1) − ˆb k + c k 1 − d 1 k k k−1 (2k − 1) k+1 2k + 3 1 p(1 − 2ν0) L (k + 3 − 4ν0) = (δk0 − δk1)+ Aˆk(k +1)+ Ck−1k for k ≥ 0, (1 + ν0) 2 2k − 1 1 (3k − 4kν − 2ν + 1) aˆ (2k +1)+ c k + 2d 1 1 k k−1 k+1 2k + 3 1 = Aˆk(2k +1)+ Ck−1k for k ≥ 1. (2.122)

We can only solve these equations for a ﬁnite number of coeﬃcients, therefore we need to truncate at say k = N. We can also note from the above equations that b0 never actually appears, it is always multiplied by k and therefore at k = 0 the bk term vanishes. Thus without loss of generality we can set b0 = 0. Similarly we note that d0 does not appear individually, it is not an independent variable and therefore we can set d0 = 0 as well. Within this framework we are now solving for

ak, Ak :k = 0, ..., N

bk :k = 1, ..., N

ck,Ck :k = 1, ..., N − 1

dk :k = 1, ..., N + 1, since we have b0, c0, d0 and C0 all equal to zero. It is easiest to solve these equations by transforming them into a matrix equation, and solving via Mathematica’s [49] LinearSolve function [48] . CHAPTER 2. BACKGROUND 86

Results

Agreement with axisymmetric cavity solution

When the shell and host are composed of the same material, such that 1 = 1 and

ν1 = ν0, these equations can also be solved analytically to yield

p(1 − h)3 A = a = 0 0 2 pL(1 − 2ν0) b1 = 2(1 + ν0) p(1 − 2ν0) b2 = (1 + ν0) 3p d1 = 2(1 + ν0) where all other coeﬃcients are zero. Using (2.96),(2.97), (2.101) and (2.102) we ﬁnd:

(1 − h)3 um = us = u∞ − p , (2.123) r r r 4r2 m s ∞ uφ = uφ = uφ , (2.124) which agrees with the solution found in section 2.4.3 for a cavity of radius r = 1 − h.

Thick axisymmetric shell solution

Once truncated (at N ≥ 2) we can solve the equation (2.122) symbolically using Math- ematica’s LinearSolve function to yield the coeﬃcients

3 31t (ν0 − 1)(1 + ν1) a0 = −p 3 , 2(1 + ν0)((1 − 1)(t − 1)(1 + ν1) + 3(1 − ν1)) pL (1 − 2ν0) b1 = 1 , 2 (1 + ν0) 3p1(1 − ν0)(1 − 2ν1) b2 = 3 , (1 + ν0)((1 − 1)(t − 1)(1 + ν1) + 3(1 − ν1)) 9p1(1 − ν0) d1 = 3 , 2(1 + ν0)((1 − 1)(t − 1)(1 + ν1) + 3(1 − ν1)) 3 3 21(t − 1)(1 − 2ν0)(1 + ν1)+(1+ ν0)(3(1 − ν1) + (t − 1)(1 + ν1)) A0 = p 3 , 2(1 + ν0)((1 − 1)(t − 1)(1 + ν1) + 3(1 − ν1)) where all other coeﬃcients are zero. If we notice that

2d b − 1 (1 − 2ν ) = 0, (2.125) 2 3 1 CHAPTER 2. BACKGROUND 87 we can then write the displacements, using (2.96), (2.97), (2.101) and (2.102), as

u = urer + uφeφ where A um = u ∞ − 0 for r ≥ 1 r r 2r2 ur =  a 2 (2.126) us = 1 − 0 + (2ν − 1)d r + b cos φ for 1 − h ≤ r ≤ 1,  r 21 r2 3 1 1 1  m m∞ uφ = uφ for r ≥ 1 u = (2.127) φ  s sin φ uφ = − b1 for 1 − h ≤ r ≤ 1.  21

The solution in the medium is of the same form as that found in section 2.4.3, equivalent to a radially symmetric contraction and a rigid body displacement contained within the displacement at inﬁnity. If we remember ez = cos φer − sin φeφ, then the displacement in the medium can be written as

m m m u = usymer + urigidez, where p (1 − 2ν ) A um = − 0 r + 0 , sym 2 (1 + ν ) r2 0 m pL (1 − 2ν0) urigid = . 4 (1 + ν0) We can similarly express the solution in the shell as a radial contraction and a rigid body displacement as s s s u = usymer + urigidez, where 1 2 a us = − (1 − 2ν )d r + 0 , sym 2 3 1 1 r2 1 s b1 urigid = . 21

When substituting for the constant b1 we then see that the rigid body motion of the shell equals that of the medium.

Agreement with thick shell solution at the origin

It is easiest to see the dependence upon the offset L in the solution if we substitute for the coefficients ao, b1, d1 and A0. We then see that whilst the radial symmetric parts of the solution are unaffected by the size of the offset L, the rigid body displacements CHAPTER 2. BACKGROUND 88 become: m pL (1 − 2ν0) urigid = , 4 (1 + ν0)

s pL (1 − 2ν0) urigid = . 4 (1 + ν0) As L → 0, we therefore see the rigid body displacement disappear and we recover the solution for a thick spherical shell at the global origin, as in (2.85) As L → ∞ however we must note that the rigid body displacement will become the dominant term in the displacement, and so we again see that when comparing single shells under hydrostatic pressure to those oﬀset from the focus of the medium we must remove the displacement due to rigid body motion.

2.5.4 Shell under uniaxial pressure

Having found the displacement of a shell under an axially symmetric pressure at inﬁnity in the previous section 2.5.3, we can use exactly the same method of solution to ﬁnd the displacement of a shell under uniaxial compression.

Problem construction

A shell, of mid-shell radius Rˆ and thickness hˆ, in a medium under uniaxial compression, as in ﬁgure 2.28, experiences displacements at inﬁnity, in dimensional spherical ∞ ∞ coordinates (ˆr,φ,θ), centred on the shell, of urˆ and uφ derived from the Cartesian uniaxial compression pressureσ ˆzˆzˆ = −pˆ, so that

pˆrˆ (2ν − 1) uˆ∞ = 0 P (cos φ) − 2P (cos φ) , (2.128) rˆ 6ˆ (1 + ν ) 0 2 0 0 ∞ pˆrˆ (1) uˆφ = − P2 (cos φ). (2.129) 6ˆ0 CHAPTER 2. BACKGROUND 89

p p

φˆ rˆ

θˆ

p p

Figure 2.28: Oﬀ center spherical shell in an inﬁnite elastic medium.

Upon use of the strain-displacement and stress-strain equations in section 2.2.1 these displacements yield the stresses at inﬁnity to be

pˆ σˆ = − (P (cos φ) + 2P (cos φ)), (2.130) rˆrˆ 3 0 2 pˆ σˆ = − P (1)(cos φ). (2.131) rφˆ 3 2

As in previous problems the shell must experience no radial traction on its inner surface, so that hˆ σˆ =σ ˆ = 0 onr ˆ = Rˆ = Rˆ − , rˆrˆ rφˆ 0 2 and under the presumption of perfect bonding stresses and displacements will be continuous across the shell medium interface,

hˆ σˆ , σˆ , uˆ , uˆ continuous onr ˆ = Rˆ = Rˆ + . rˆrˆ rφˆ rˆ φ 1 2

Nondimensionalising

In this case we will nondimensionalise on the mid-shell radius of the shell, Rˆ and the shear modulus of the medium, ˆ0. As such the coordinate system has nondimensional rˆ counterpart (r, φ, θ), where r = , and angles φ and θ are nondimensional by deﬁnition. Rˆ Displacements and stresses become:

uˆrˆ uˆφˆ ur = , uφ = , (2.132) Rˆ Rˆ σˆrˆrˆ σˆrˆφˆ σrr = , σrφ = (2.133) ˆ0 ˆ0 CHAPTER 2. BACKGROUND 90 and system parameters become

pˆ ˆ1 Rˆ0 Rˆ1 hˆ p = 1 = ,R0 = ,R1 = , h = . ˆ0, ˆ0 Rˆ Rˆ Rˆ

This implies the displacements at inﬁnity, valid in the medium, in local coordinates are

∞ pr (2ν0 − 1) ur = P0(cos φ) − 2P2(cos φ) , (2.134) 6 (1 + ν0) pr u∞ = − P (1)(cos φ), (2.135) φ 6 2 which translate to stresses at inﬁnity of

p σ = − (P (cos φ) + 2P (cos φ)), (2.136) rr 3 0 2 p σ = − P (1)(cos φ). (2.137) rφ 3 2

Boundary conditions upon the inner surface of the shell can then be written,

h σ = σ =0 on r = R = 1 − , rr rφ 0 2 and continuity conditions,

h σ ,σ , u and u are continuous on r = R =1+ rr rφ r φ 1 2

We can see from this problem construction that whilst the values of the displacement and stresses at inﬁnity are diﬀerent to those found in 2.5.3 the form of the problem is the same and thus the same form of solution is to be expected.

Deformation and stresses within the medium and shell

As the form of the boundary conditions is identical we can therefore use the same solution as in 2.5.3, formed using the Boussinesq-Papkovich stress function method. Hence the general form of solution is given by equations (2.95)-(2.99) within the medium, and by equations (2.100)-(2.104) within the shell. CHAPTER 2. BACKGROUND 91

Applying the boundary conditions

Using the orthogonality condition of the Legendre polynomials, zero traction on the inner surface of the shell can be simpliﬁed, as in 2.5.3, to yield two equations

k(k2 + 3k − 2ν ) 0 =ˆa (k + 1)(k +2)+ ˆb R2k+1k(k − 1) + c R2 1 k k 0 k−1 0 2k − 1 (k + 1)(k2 − k − 2ν − 2) + d R2k+3 1 , for k ≥ 0, k+1 0 2k + 3 (2.138) (k2 + 2kν + k + ν + 1) 0 =ˆa (2k +1)+ c R2k − 2d R2k+3 1 1 , for k ≥ 2, k k−1 0 k+1 0 (k + 2)(2k + 3) (2.139)

n + 5 − 4ν n + 4ν − 4 where c ≡ 0 and we usea ˆ = a + c 1 and ˆb = b + d 1 0 n n n+1 2n + 3 n n n−1 2n − 1 for notational ease. Continuity in stress again simpliﬁes to the two equations:

p k(k2 + 3k − 2ν ) − Rk+3(δ + 2δ )+ Aˆ (k + 1)(k +2)+ C R2 0 3 1 k0 k2 k k−1 1 2k − 1 k(k2 + 3k − 2ν ) =ˆa (k + 1)(k +2)+ ˆb k(k − 1)R2n+1 + c R2 1 k k 1 k−1 1 2k − 1 (k2 − k − 2ν − 2) + d R2k+3(k + 1) 1 , for k ≥ 0, k+1 1 2k + 3 (2.140)

ˆ 2 2 Ak(2k +1)+ Ck−1kR1 =ˆak(2k +1)+ ck−1kR1 (k2 + 2kν + k + ν + 1) − 2d R2n+3 1 1 , for k ≥ 2, k+1 1 (k + 2)(2k + 3) (2.141) CHAPTER 2. BACKGROUND 92

k + 5 − 4ν where C ≡ 0 and Aˆ = A + C 0 . Finally continuity in displacement 0 k k k+1 2k + 3 simpliﬁes to:

(k + 3 − 4ν ) (k + 1)(k + 4ν − 2) aˆ (k + 1) − ˆb R2n+1k + c R2k 1 − d R2n+3 1 k k 1 k−1 1 (2k − 1) k+1 1 2k + 3 pRn+3 (2ν − 1) (k + 3 − 4ν ) = − 1 0 δ − 2δ + Aˆ (k +1)+ C R2k 0 for k ≥ 0. 1 3 (1 + ν ) n0 n2 k k−1 1 2k − 1 0 (2.142) (3k − 4kν − 2ν + 1) aˆ (2k +1)+ c kR2 + 2d R2n+3 1 1 k k−1 1 k+1 1 2k + 3 ˆ 2 = 1 Ak(2k +1)+ Ck−1kR1 , for k ≥ 1. (2.143)

These 6 boundary conditions ((2.138) to (2.143)) can be solved exactly by transforming into a matrix equation, and solving via Mathematica’s LinearSolve function [48], so that

aˆ0, aˆ2, ˆb2, c1, d1, d3, A0, A2, and C1 are nonzero and all other coeﬃcients are zero. This allows the displacements in the shells to be written as

aˆ (2ν − 1) 2 us = − 0 + 2d r2 1 1 r r2 1 3 aˆ 12d c 2(5 − 4ν ) + −3 2 + 2ˆb r + 3 r3ν − 1 1 P (cos φ), (2.144) r4 2 7 1 r2 3 2 aˆ c (2ν − 1) (7 − 4ν ) 2 us = 2 + ˆb r + 2 1 1 + d r3 1 P (1)(cos φ), (2.145) 1 φ r4 2 r2 3 3 7 2 and the displacements within the medium to be

Aˆ 3Aˆ C (5 − 4ν ) 2(um − um∞)= − 0 + 2 + 2 1 0 P (cos φ) , (2.146) r r r2 r4 r2 3 2 Aˆ C 2(2ν − 1) 2(um − um∞)= 2 + 1 0 P (1)(cos φ). (2.147) φ φ r4 r2 3 2 Results

Whilst the displacement of the shell as the parameters of the problem change is useful we are more interested in the meridional hoop stress within the shell, as this will later be used, in part, to predict the buckling pressure of the shell. As such we can write the CHAPTER 2. BACKGROUND 93

meridional hoop stress σφφ, as

2 u ∂u 1 − ν ∂u ν σ = r + v r + φ + u cot φ , φφ 1 − 2ν r ∂r r ∂φ r φ where ν and take values ν1 or ν0 and 1 or 1 in the shell or the host respectively. In the shell we therefore ﬁnd

aˆ 2 aˆ σs = 0 − d r(1 + 2ν )+ 2 (−7P +1)+ ˆb (−2P + 1) φφ r3 3 1 1 r5 2 2 2 2 c d − 1 (1 − 2ν )(P +1)+ 3 r2(−14(2 + ν )P + (7 − 4ν )). (2.148) 3 r3 1 2 7 1 2 1

Using this formulation and the values of the constants previously found numerically we can investigate the eﬀect problem parameters have upon the hoop stress on the mid-surface of the shell at r = 1.

Shell parameter studies

If we first consider a stiff glassy shell in which 1 = 1750 and ν1 = 0.3 contained within a nearly incompressible medium of Poisson ratio ν0 = 0.49985, and in which p = 0.1, we can consider the effect the thickness of the shell has upon the compression of the shell. Though φ runs from 0 to π we have throughout this thesis plotted φ from 0 to 2π as an aide to visualisation of the symmetric nature of the problems. As seen in figure 2.29, for relatively thick shells the meridional hoop stress is nearly constant around the shell mid-radius, even though compression is axial at infinity. As the shell thickness decreases however the variability of the hoop stress around the mid-shell radius increases, which may indicate an decrease in buckling pressure with shell thickness, as will be detailed in later sections. CHAPTER 2. BACKGROUND 94

φ Π 3 Π 0 2 Π 2 2 Π 0

φφ 30 σ

Figure 2.29: Hoop stress σφφ on the shell-host interface for shells, in which 1 = 1750, ν1 = 0.3 and ν0 = 0.49985, under pressure p = 0.1, with meridional angle φ, for shells of various thicknesses, h, such that: (i) h = 0.001 (blue, solid), (ii) h = 0.01 (pink, dashed), (iii) h = 0.03 (yellow, dot-dashed), (iv) h = 0.05 (green, dotted).

Keeping the problem parameters as previously set, we can similarly set the shell thickness to h = 0.01, at which we saw variability in the hoop stress and we can consider the effect of the shell material. If we allow 1 to vary, as in figure 2.30, we see that not only does the hoop stress decrease in magnitude as the contrast between the shear modulus of the shell and the matrix decreases (1 = 1 implies shell and host mediums have the same shear modulus), but the amplitude of the hoop stress varies. The difference between the maximum and minimum hoop stress around the shell is smaller for both very large and small shell shear moduli than for shell shear moduli of around O(100), however for small shell shear moduli we also see a change in sign of the hoop stress, indicating that for shear moduli around O(1) to O(10), the shell can be in tension in some regions and under compression in other regions. CHAPTER 2. BACKGROUND 95

σφφ

Π φ Π 3 Π 2 Π 2 2

Figure 2.30: Hoop stress σφφ on the shell-host interface for an embedded shell in which h = 0.01, ν1 = 0.3 and ν0 = 0.49985, with meridional angle φ for various shell shear moduli: (i) 1 = 2000 (blue, solid), (ii) 1 = 1000 (pink, dashed), (iii) 1 = 100 (yellow, dot-dashed), (iv) 1 = 10 (green, dotted), (v) 1 = 1 (orange, long dashes).

Finally we can investigate the manner in which the shell Poisson ratio affects the hoop stress. The Poisson ratio of the shell has minimal effect upon the hoop stress of the shell, as seen in figure 2.31. For the most part, as the shell becomes more compressible the magnitude of the hoop stress increases, however it leaves the period and amplitude of the hoop stress curve largely unaffected.

σφφ

Π φ Π 3 Π 2 Π 2 2

Figure 2.31: Hoop stress σφφ on shell-host interface for shells in which h = 0.01, 1 = 1750 and ν0 = 0.49985, with meridional angle φ for varying values of ν1: (i) ν1 = 0.49985 (blue, solid), (ii) ν1 = 0.4 (pink, dashed), (iii) ν1 = 0.3 (yellow, dot- dashed), (iv) ν1 = 0.2 (green, dotted), (v) ν1 = 0.1 (orange, long dashes). CHAPTER 2. BACKGROUND 96

Host medium parameter studies

We could similarly investigate the way the parameter of the host medium, the host

Poisson ratio ν0, affects the shell hoop stress. In figure 2.32 we see the host Poisson ratio has minimal affect upon the shell hoop stress. As the host Poisson ratio becomes more compressible the magnitude of the hoop stress increases in a linear manner, but the amplitude and wave length upon which the shell hoop stress varies does not alter.

σφφ 3

Π φ Π 3 Π 2 Π 2 2

Figure 2.32: Hoop stress σφφ on shell-host interface with meridional angle φ for shells of thickness h = 0.01, shear modulus 1 = 1750 and Poisson ratio ν0 = 0.3, within mediums of various Poisson ratio ν0, such that: (i) ν0 = 0.49985 (blue, solid), (ii) ν0 = 0.4 (pink, dashed), (iii) ν0 = 0.3 (yellow, dot-dashed), (iv) ν0 = 0.2 (green, dotted), (v) ν0 = 0.1 (orange, long dashes).

Thin shell approximation

For a shell in which h ≪ 1 we can assume that the hoop stress throughout the shell is approximately that upon the shell mid-radius at r = 1, so that:

2 σs =ˆa − d r(1 + 2ν ) +a ˆ (−7P +1)+ ˆb (−2P + 1) φφ 0 3 1 1 2 2 2 2 2 d − c (1 − 2ν )(P +1)+ 3 (−14(2 + ν )P + (7 − 4ν )). (2.149) 3 1 1 2 7 1 2 1 CHAPTER 2. BACKGROUND 97

If all other parameters are O(1), then the constants are given by the series solution, to O(h) say, of the boundary conditions:

(ν + 1) 0 =a ˆ − d R3 1 , (2.150) 0 1 0 3 (5 − ν ) ν 0=6â + ˆb R5 + 2c R2 1 − 3d R7 1 , (2.151) 2 2 0 1 0 3 3 0 7 (7 + 5ν ) 0=5â + 2c R2 − d R7 1 , (2.152) 2 1 0 3 0 14 p (ν + 1) − R3 =a ˆ − d R3 1 − Aˆ , (2.153) 6 1 0 1 1 3 0 p (5 − ν ) ν − R5 = 6â + ˆb R5 + 2c R2 1 − 3d R7 1 3 1 2 2 1 1 1 3 3 1 7 (5 − ν ) − 6Aˆ − 2C R2 0 , (2.154) 2 1 1 3 (7 + 5ν ) 0=5â + 2c R2 − d R7 1 − 5Aˆ − 2C R2, (2.155) 2 1 1 3 1 14 2 1 1 pR3 (1 − 2ν ) 1 (1 − 2ν ) 1 0 = aˆ + 2d R3 1 − Aˆ , (2.156) 3 (1 + ν ) 0 1 1 3 0 0 1 2pR5 (1 − 2ν ) 1 (5 − 4ν ) ν 1 0 = − 3â − 2ˆb R5 + 2c R2 1 − 12d R7 1 3 (1 + ν ) 2 2 1 1 1 3 3 1 7 0 1 (5 − 4ν ) + 3Aˆ + 2C R2 0 , (2.157) 2 1 1 3 1 (7 − 10ν ) 0= 5â + 2c R2 + 2d R7 1 − 5Aˆ − 2C R2, (2.158) 2 1 1 3 1 7 2 1 1 1 h h where R0 = 1 − 2 and R1 = 1+ 2 . However if all other parameters are not O(1), such as if 1 ≫ 1, then we will need to parameterise the boundary equations for our η other parameter as well. In this case we let = , such that the boundary conditions 1 h CHAPTER 2. BACKGROUND 98 become:

(ν + 1) 0 =a ˆ − d R3 1 , (2.159) 0 1 0 3 (5 − ν ) ν 0=6â + ˆb R5 + 2c R2 1 − 3d R7 1 , (2.160) 2 2 0 1 0 3 3 0 7 (7 + 5ν ) 0=5â + 2c R2 − d R7 1 , (2.161) 2 1 0 3 0 14 p (ν + 1) − R3 =a ˆ − d R3 1 − Aˆ , (2.162) 6 1 0 1 1 3 0 p (5 − ν ) ν − R5 = 6â + ˆb R5 + 2c R2 1 − 3d R7 1 3 1 2 2 1 1 1 3 3 1 7 (5 − ν ) − 6Aˆ − 2C R2 0 , (2.163) 2 1 1 3 (7 + 5ν ) 0=5â + 2c R2 − d R7 1 − 5Aˆ − 2C R2, (2.164) 2 1 1 3 1 14 2 1 1 pR3 (1 − 2ν ) h (1 − 2ν ) 1 0 = aˆ + 2d R3 1 − Aˆ , (2.165) 3 (1 + ν ) η 0 1 1 3 0 0 2pR5 (1 − 2ν ) h (5 − 4ν ) ν 1 0 = − 3â − 2ˆb R5 + 2c R2 1 − 12d R7 1 3 (1 + ν ) η 2 2 1 1 1 3 3 1 7 0 (5 − 4ν ) + 3Aˆ + 2C R2 0 , (2.166) 2 1 1 3 h (7 − 10ν ) 0= 5â + 2c R2 + 2d R7 1 − 5Aˆ − 2C R2, (2.167) η 2 1 1 3 1 7 2 1 1 for which solutions can be kept to O(h) by taking series in h of the solutions to the constantsa ˆ0, aˆ2 etc. As such we can define two approximations to the exact meridional hoop stress from (2.148). The first approximation will be known as the thin shell approximation, in which meridional hoop stress is given by (2.149) and coefficients solve equations (2.150) to(2.158). This approximation is valid for h ≪ 1 where all other parameters are O(1). The second approximation will be known as the stiff-thin shell approximation, and in which meridional hoop stress is again given by (2.149) but in which coefficients solve equations (2.159) to (2.167). This approximation is valid for 1 1 ∼ h and h ≪ 1. The importance of recognising the order of the shell shear modulus in comparison to the shell thickness is most visible in figure 2.33, in which 1 = 200 and h = 0.01 (ν0 = 0.49985,ν1 = 0.3,p = 0.1). In this situation we find 1h ∼ O(1), and thus the stiff-thin shell approximation evaluates the exact hoop stress, (2.148), very well, whilst the thin shell approximation is a terrible approximation to the exact hoop stress. On the other hand when 1 ∼ O(1), as in 2.34, the thin shell approximation improves, though the stiff-thin shell approximation is still closer to the exact solution. CHAPTER 2. BACKGROUND 99

σφφ Π φ Π 3 Π 2 Π 0 2 2

Figure 2.33: Hoop stress σφφ on shell-host interface with meridional angle φ for shells of thickness h = 0.01, shear modulus 1 = 200 and Poisson ratio ν1 = 0.3, within mediums of Poisson ratio ν0 = 0.49985 and pressure p = 0.1 using various methods: (i) exact hoop stress (2.148) (blue, solid), (ii) stiﬀ-thin shell approximation (pink, dashed), (iii) thin shell approximation (yellow, dot-dashed).

σφφ

0.05

Π φ Π 3 Π 2 Π 2 2

0.05

0.10

0.15

Figure 2.34: Hoop stress σφφ on shell-host interface with meridional angle φ for shells of thickness h = 0.01, shear modulus 1 = 1.1 and Poisson ratio ν1 = 0.3, within mediums of Poisson ratio ν0 = 0.49985 and pressure p = 0.1 using various methods: (i) exact hoop stress (2.148) (blue, solid), (ii) stiﬀ-thin shell approximation (pink, dashed), (iii) thin shell approximation (yellow, dot-dashed). Chapter 3

Single shell buckling problems

3.1 Buckling theory

3.1.1 Previous work

The first work on buckling, calculating the maximum axial load that a long, slender, ideal column could withstand, was completed by Euler [29] in 1700s and further investigated by Lagrange [25] in the 1770’s. However it was not until the 20th century that the buckling of shells under uniform external pressure was investigated, in spherical shells by Zoelly in 1915. This analysis was restricted to linear theory of infinitesimal displacements from the fundamental spherical state and to axisymmetric deformations. A more general analysis of the problem was provided by Van der Neut, [9] in 1932. A nonlinear theory for the buckling of circular cylindrical shells was proposed by Donnell [31] in 1934, under simplifying shallow-shell assumptions, and due to its simplicity and accuracy in a wide range of thin shell problems has been widely used since. The simplifying assumptions, obtained by neglecting transverse and shear deformation limit the accuracy of Donnell’s theory to very thin shells. In-plane inertia and rotary inertia are also neglected in this theory but these are not relevant to our consideration of buckling. A complete theory on the buckling of spherical shells, however, was not produced until 1963, by Sanders [26], and, independently, by Koiter [50] in 1966. As such these equations have come to be known as the Sanders-Koiter equations [32]. In the Sanders-Koiter theory all three displacements are included in the equation of motion, though as in the Donnell theory, changes in curvature and torsion are linear. Other nonlinear shell theories in use include those by Novozhilov [43], Flugge-Lur’e- Byrne [44], and Teng and Hong [41]. The accuracy of the Sanders-Koiter, Donnell and Teng-Hong theories have recently been evaluated against finite element simulations for thin spherical shells under static uniaxial pressure by Shams and Porfiri [34]. Results indicated Sanders-Koiter theory to be more accurate than either Donnell or Teng-Hong

100 CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 101 theory for a range of shell thickness to radius ratios of O(0.01). Recently, further work has been conducted on the critical buckling pressure of embedded shells, by Fok and Allwright [10], after experimental work by Bulson [38] revealed a large difference in the buckling pressure of embedded spherical shells, of between 25% − 75%, in comparison to that predicted by the classical theory for unrestrained shells. This seems to be the first theoretical work on the buckling of embedded shells, and as such considers the simplest case of a shell embedded within an elastic medium under external hydrostatic pressure. For simplicity it was assumed that the shell was perfectly bonded to the host medium; displacements up to the point of instability obeyed linear theory; buckling was inextensional and deformations were axisymmetric. In 2007 Jones, Chapman and Allwright [17] relaxed the inextensibility constraint and extended the buckling theory to include uniaxial compression of a host medium containing a spherical shell. The extension of the buckling theory to pressures that were not spherically symmetric required a tensorial approach not previously employed by Fok and Allwright. However some simplifying assumptions within the Jones et al. theory constrained the validity of the solution to situations in which the shear modulus of the shell is similar to that of the medium, such that 1 ∼ O(1). In the proceeding work we will demonstrate that the 0 Jones et al. theory can be adjusted to increase the range of validity of the solution and we will use the same notation as Jones et al. where possible. The constrained nature of the Jones et al. analysis was also acknowledged by Shams and Porfiri [34], who noted that the simplifying assumption, of approximating the membrane resultants, was not valid for the unembedded glass micro-balloons they were considering. We will further show that it is possible to approximate the membrane resultants though care must be taken to ensure all parameters within the approximation are of an appropriate order.

3.1.2 Buckling methodology for an embedded spherical shell under external pressure

In earlier sections we have considered how shells within media deform during the linear elastic phase, where deformations are very small. However what we would really like to understand is at what pressure the deformation becomes large and the shells buckle. To do this we need to analyse the stability of the linear (pre-buckled) state, for which we will use the Trefftz criterion, and then find the point at which stability is lost using the Rayleigh-Ritz method. We will first consider the methodology for finding the buckling pressure for a thin shell, of mid-radius R and thickness h, embedded within a medium under pressure, before considering the specific cases of the thin shell under hydrostatic pressure, as in section 2.5.1 and under axial compression, as in section 2.5.4. CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 102

3.1.3 The Treﬀtz Criterion

For a system comprised of a thin shell embedded within a medium subject to pressure, we can find the pre-buckled deformation state of the system using linear elasticity, as we have for various examples in section 2.5. If we denote the pre-buckled state I with displacement u(I), we can assess the stability of this state by imposing an infinitesimal, or virtual, displacement u upon the pre-buckled displacement. The addition of this virtual displacement represents a new state for the system, state II, in which the displacement is given by u(II) = u(I) + u. The stability of this system then depends upon the change in the potential energy, W , of the system between state I and state II. Let us define the change in the potential energy of the system as

∆W = W (II) − W (I), where superscripts refer to the state of the system. This can then be expanded as a series in powers of the virtual displacement u, so that

∆W =∆W1 +∆W2 + ... where ∆W1 is linear in u, ∆W2 is quadratic, and so on for higher order terms. The pre-buckled state I is then in static equilibrium if ∆W1 = 0; this is a stable equilibrium if ∆W2 > 0 for all u, whilst the critical load for state I is the minimum load for which ∆W2 < 0 for some u, ie the minimum load at which the state becomes unstable. The Treﬀtz criterion states that for the loading parameter (often the magnitude of the pressure at inﬁnity, p, for the systems we have considered) the critical point between stability and instability is found when ∆W2 has a singular quadratic form.

Calculating the pre-buckled state

The pre-buckled state is in equilibrium when ∆W1 = 0, therefore we could calculate the pre-buckled state by ﬁnding when ∆W1 = 0 for all virtual displacements u and solving for u(I). However we have chosen to instead calculate the pre-buckled state using linear elasticity and assume that in the limit for very thin shells, h ≪ 1, u(I) will be the leading order approximation with O(h) error, which will yield changes O(h2) in potential energy ∆W1.

Applying the Treﬀtz criterion to an embedded shell

For the systems we have been considering, consisting of both a shell and a host medium, the total potential energy of the system will be composed of the potential energy within the shell and the potential energy within the host medium. CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 103

The shell potential energy

We will ﬁrst consider the potential energy in the shell. If we denote the potential energy density in the shell as Vs, then as before the change between state II and I can be written (II) (I) ∆Vs =∆Vs − ∆Vs , where Vs, depending on strain tensor γij and changes of curvature tensor ρij, is given by (2.48) in section 2.3. Since the displacement in state II is a linear combination of the displacement in state I and the virtual displacement we can rewrite ∆Vs as:

h h ∆V = Eαβλ(γ(I) + γ )(γ(I) + γ ) − Eαβλγ(I)γ(I) s 2 αβ αβ λ λ 2 αβ λ h3 h3 + Eαβλ(ρ(I) + ρ )(ρ(I) + ρ ) − Eαβλρ(I) ρ(I), 24 αβ αβ λ λ 24 αβ λ h = Eαβλ(γ(I)γ + γ γ(I) + γ γ ) 2 αβ λ αβ λ αβ λ h3 + Eαβλ(ρ(I) ρ + ρ ρ(I) + ρ ρ ). 24 αβ λ αβ λ αβ λ

If we further note that the displacement in state I was determined using linear elasticity (I) (I) we can use the linear counterpart of the middle surface strain tensor, so that γαβ ∼ θαβ, as deﬁned in (2.43) in section 2.3. We can then write the change in the potential energy as:

h ∆V = Eαβλ(θ(I) γ + γ θ(I) + γ γ ) s 2 αβ λ αβ λ αβ λ h3 + Eαβλ(ρ(I) ρ + ρ ρ(I) + ρ ρ ). 24 αβ λ αβ λ αβ λ

Within the linear theory of shells we can then relate the stress resultants nαβ and stress couples mαβ, deﬁned in equations (2.45) and (2.46) 2.3, to the strain tensor and the tensor of changes of curvature in state I in the thin shell limit via

h (I)nαβ =hEαβλθ(I) 1+ O , λ R mid h3 h (I)mαβ = Eαβλρ(I) 1+ O , 12 λ R mid and so write the change in potential energy density as

h h3 ∆V = (I)nαβγ + (I)mαβρ + Eαβλγ γ + Eαβλρ ρ . s αβ αβ 2 αβ λ 24 αβ λ CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 104

Dealing with the virtual displacement however we must remember that this has been derived using non-linear shell theory, in which

1 γ = θ + w w and ρ = w| , ij ij 2 ,i ,j ij ij where w is the normal displacement of the shell as in equation (2.3) in section 2.3. Keeping this condition in mind we can split the change in potential energy density into linear, quadratic and higher order terms represented by

∆Vs =∆V1 +∆V2 + ...

The linear terms ∆V1 are then given by

(I) αβ (I) αβ ∆V1 = n θαβ + m ραβ, whilst the quadratic terms are

(I)nαβ h h3 ∆V = w w + Eαβλθ θ + Eαβλρ ρ . 2 2 ,α ,β 2 αβ λ 24 αβ λ

It is noticeable at this point that if we had considered the virtual displacement to be linear rather than part of nonlinear shell theory, the quadratic term of the change in potential energy density would be independent of state I. Thus we can ﬁnally write the change in the potential energy of the shell ∆Ws as

(I) αβ (I) αβ ∆Ws = n θαβ + m ραβ dS Mid-shell surface (I) αβ 3 n h αβλ h αβλ + w,αw,β + E θαβθλ + E ραβρλ dS Mid-shell 2 2 24 surface + higher order terms. (3.1)

The potential energy of the medium

As discussed in section 2.2.1, we can similarly write the potential energy stored in the medium as the integral of the potential energy density within the volume V of the medium:

∆Wm = ∆VmdV, r>R1 h where R1 = R + 2 is the outer radius of the shell. As in the shell, the medium has an equilibrium solution, state I, which is perturbed by a virtual displacement in order to generate a new state II, which allows us to test the stability of the equilibrium state. In this case we denote the displacement within the medium in state I as u(I), and CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 105 the perturbation experienced by this state as u, a consequence of applying the virtual displacement v upon the shell. As such the displacement in state II can be written u(I) + u. We can then write the change in potential energy density within the medium using (2.17) in section 2.2.1 as

(II) (I) ∆Vm = Vm − Vm , 1 1 = Aijkl(e(I) + e )(e(I) + e ) − Aijkle(I)e(I), 2 ij ij kl kl 2 ij kl 1 = Aijkl(e(I)e + e e(I) + e e ). 2 ij kl ij kl ij kl

This can also be split into linear and quadratic parts, in the displacement u, so that

1 1 ∆V = Aijkl(e(I)e + e e(I))+ Aijkle e . m 2 ij kl ij kl 2 ij kl

Thus the potential energy within the medium becomes:

1 1 ∆W = Aijkl(e(I)e + e e(I)) dV + Aijkle e dV. (3.2) m 2 ij kl ij kl 2 ij kl r>R1 r>R1 Total potential energy

The total potential energy within the composite is then the sum of the energy stored in the shell and in the medium:

∆W =∆Wm +∆Ws, and can be written as linear, quadratic and higher order terms as

∆W =∆W1 +∆W2 + ..., where linear terms are given by

(I) αβ (I) αβ 1 ijkl (I) (I) ∆W1 = n θαβ + m ραβ dS + A (eij ekl + eijekl ) dV, Mid-shell r>R1 2 surface and quadratic terms given by

∆W2 = I1 + I2 + I3. CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 106

For convenience it is easiest to consider the contributions from each integral separately as,

(I)nαβ I1 = w,αw,β dS, (3.3) Mid-shell 2 surface 3 h αβλ h αβλ I2 = E θαβθλ + E ραβρλ dS, (3.4) Mid-shell 2 24 surface 1 I = Aijkle e dV. (3.5) 3 2 ij kl r>R1 As mentioned earlier, we can now ﬁnd the critical buckling stress at the stationary point of ∆W2, where we have assumed the equilibrium solution in the shell and medium satisfy ∆W1 = 0 in the thin shell limit.

3.1.4 Rayleigh-Ritz and the critical buckling stress

In order to find the stationary point of ∆W2 we will employ the Rayleigh-Ritz technique [33], in which we consider the virtual displacement to be an infinite series. In scenarios in which pressure at infinity is hydrostatic we restrict to axisymmetric deformations as it is assumed it will take less energy for a shell to buckle in one circumferential angle than both, and similarly in scenarios in which pressure is uniaxial we assume the lowest energy buckling pattern will occur in the meridonal angle over which the pressure varies, rather than the azimuthal angle over which the pressure is constant. There may be situations in which a shell buckles non-axisymmetrically, however incorporating this into our buckling methodology would be very complex, and as such will be ignored in this thesis. Hence, for the problems we are considering we are able to restrict the virtual displacement of the shell, v, to axisymmetric deformations and as such:

v = vrer + vφeφ, (3.6) where vr and vφ are independent of θ. By (2.39), we can relate the virtual displacement to the base vector components and we can use the Rayleigh-Ritz method to express the vector components, using the completeness of the Legendre polynomials and the fact that vφ = 0 at φ = 0,π, as

∞ vr = w = UnPn(cos φ), (3.7) n=0 ∞ v v = 1 = V P (1)(cos φ), (3.8) φ R n n n=0 CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 107

(1) where Pn is an associated Legendre function as deﬁned in (A.6). The constant co- eﬃcients Un and Vn are found by substituting the forms of the virtual displacement expansions, in equations (3.7)-(3.8), and then solving

∂ ∆W2 = 0, (3.9) ∂Un ∂ ∆W2 = 0. (3.10) ∂Vn

This will yield an inﬁnite set of equations, the determinant of which must be set to zero to ﬁnd a non-zero buckling deformation. The components of the eigenvectors

Un and Vn and the associated critical buckling pressure can then be found from this condition. The axisymmetric form of the virtual displacement has further repercussions when we consider the integral in (3.3) since axisymmetry in the virtual displacement ∂w implies ∂φ = 0. When we combine this with the fact that the integral I1 is integrated on the mid-shell surface, and thus does not integrate across the radius of the shell, (I) 11 (I) 11 the only stress resultant which will be included in I1 is n . Since n is the stress resultant referred to the base vectors gi in shell coordinates (see section 2.3), we need to understand how this relates to the stress σφφ, in spherical coordinates. Using equation (2.42) section 2.3, we can write the stress resultant in terms of the hoop stress as:

1 2 (I) 11 h h n = 1+ ξ σφφ dξ, R2 1 R − 2 where r = R + hξ as seen in [17], and ξ is a nondimensional parameter across the shell width.

3.2 Buckling of a spherical shell in an elastic medium

Having discussed the method used to analyse the stability of any embedded shell we can now consider an embedded shell under a variety of loading conditions. Of importance to us are the cases of hydrostatic pressure and uniaxial pressure. The equilibrium state for a shell under hydrostatic pressure was derived using various approximations in sections 2.5.1 and 2.5.2 and similarly, in section 2.5.4, we found the equilibrium state for a shell under uniaxial pressure and approximations to the equilibrium state. Here we will investigate how the form of the stress, either hydrostatic or uniaxial, and the approximation of the equilibrium state used, aﬀects the buckling pressure. CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 108

3.2.1 Hydrostatic pressure: The shell equilibrium solution and its approximations

For a shell of nondimensional thickness h and outer radius 1 in a medium under hydrostatic pressure p, we found, in section 2.5.1, that the displacement of the composite is

u = urer,

s m where ur (denoted ur in the shell and ur in the medium) is given by

p um = − (1 − 2ν )r r 2(1 + ν ) 0 0 (2 (2ν − 1) − (1 + ν ))(1 + ν )((1 − h)3 − 1) − 3(1 + ν )(1 − ν ) − 1 0 0 1 0 1 , 2r2(3(1 − ν ) + (1 − )(1 + ν )((1 − h)3 − 1)) 1 1 1 3 3 s 3p(ν0 − 1)(2(1 − 2ν1)r +(1+ ν1)(1 − h) ) ur = 2 3 . (3.11) 4r (1 + ν0)(3(1 − ν1) + (1 − 1)(1 + ν1)((1 − h) − 1))

This full solution (FS) can be approximated in various ways. We derived a thin shell approximation (TSA) to the radial displacement in (2.90), so that when h ≪ 1:

s −3p(1 − ν0)((1 − ν1) − h1(1 + ν1)) ur = . 4(1 + ν0)(1 − ν1)

In this approximation we have presumed that 1, the shear modulus of the shell nondimensionalised on the shear modulus of the medium, is of O(1). However for some materials there will be a marked contrast between the shell and medium shear moduli. As such we constructed a second approximation, the stiﬀ-thin shell approximation η 1 (STSA), in which h ≪ 1 and = ∼ O . Under these approximations the 1 h h STSA, (2.92), can be written:

−3p(ν − 1) −(ν − 1)2 + η(1 + ν )(ν − 1 + 2hν ) us = 0 1 1 1 1 . (3.12) r 4(1 + ν ) (1 + η + (η − 1)ν )2 0 1 In order to use the methodology described in section 3.1 we need to re-parameterise the displacement in the shell with respect to the shell mid-radius, Rˆ, rather than the shell outer radius, Rˆ1. Under this parameterisation we can write the displacements within CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 109 the shell as:

3 ˜3 s 3p(ν0 − 1)(2(1 − 2ν1)˜r +(1+ ν1)R0) FS: u˜r˜ = , (3.13) R˜3 4˜r2(1 + ν ) 3(1 − ν ) + (1 − )(1 + ν ) 0 − 1 0 1 1 1 ˜3 R1 1 −3p(1 − ν0) (1 − ν1)+ h˜(1 + ν1) − 1 s 2 TSA: u˜r˜ = , (3.14) 4(1 + ν0)(1 − ν1) STSA: h˜ (ν1 − 1)(1 − ν1 + (1 + ν1)) + η(1 + ν1)(ν1 − 1+ h˜(2ν1 − 1)) s −3p(ν0 − 1) 2 u˜r˜ =  2  , 4(1 + ν0) (1 + η + (η − 1)ν1)      (3.15) where s ˆ ˜ ˜ s uˆrˆ ˜ h rˆ ˜ h ˜ h u˜r˜ = , h = , r˜ = R0 = 1 − and R1 =1+ . Rˆ Rˆ Rˆ 2 2 We shall drop the tilde notation after this point, in the assumption that all lengths within equations are parameterised on the mid-shell radius, unless otherwise stated.

3.2.2 Uniaxial compression: The shell equilibrium solution and its approximations

The solutions derived in section 2.5.4 are already nondimensionalised upon the shell mid-radius and shell shear modulus. The full solution (FS) can be written:

aˆ (2ν − 1) 2 us = − 0 + 2d r2 1 1 r r2 1 3 aˆ 12d c 2(5 − 4ν ) + −3 2 + 2ˆb r + 3 r3ν − 1 1 P (cos φ), (3.16) r4 2 7 1 r2 3 2 aˆ c (2ν − 1) (7 − 4ν ) 2 us = − sin φ 2 + ˆb r + 2 1 1 + d r3 1 P ′(cos φ), (3.17) 1 φ r4 2 r2 3 3 7 2 whilst the thin shell (TSA) and stiﬀ-thin shell (STSA) approximations are approximated on the shell mid-radius, and so are of the form:

(2ν − 1) 2 us = − aˆ + 2d 1 1 r 0 1 3 12d 2(5 − 4ν ) + −3ˆa + 2ˆb + 3 ν − c 1 P (cos φ), (3.18) 2 2 7 1 1 3 2 (2ν − 1) (7 − 4ν ) 2 us = − sin φ aˆ + ˆb + 2c 1 + d 1 P ′(cos φ). (3.19) 1 φ 2 2 1 3 3 7 2 CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 110

Coefficients for the full solution, FS, solve the system of equations given in (2.138) to (2.143), whilst coefficients for the thin shell approximation, TSA, are solutions to (2.150) to (2.158) and coefficients for the stiff-thin shell approximation, STSA, are solutions to (2.159) to (2.167). The full equilibrium solution can be substituted into the linear total potential energy equation ∆W1 derived in section 3.1, to yield

∆W1 = 0, whilst approximations to the full solution yield

2 ∆W1 = O(h ), justifying our earlier assumption that the approximations to the equilibrium solution also approximately solve ∆W1 = 0.

3.2.3 Critical buckling pressure

Knowing the equilibrium solutions the critical buckling pressure can now be found by solving ∆W2 = 0, where ∆W2 is the sum of equations (3.3), (3.4) and (3.5).

The ﬁrst energy integral

We shall start by looking at the contribution to ∆W2 provided by the nondimensionalised ﬁrst energy integral (3.3):

(I)n11 ∂w 2 I1 = dS. (3.20) Mid-shell 2 ∂φ surface Since the stress resultant within the shell can be written in terms of the hoop stress, as

1 2 (I)n11 = h (1 + hξ)σs dξ, (3.21) 1 φφ − 2 (where we remember we are in a nondimensional setting in which radial distances within 1 1 the shell can be written r =1+ hξ, and ξ ∈ [− 2 , 2 ]), then in order to ﬁnd the stress s resultant we need to ﬁnd the hoop stress σφφ within the shell.

Hoop stress under hydrostatic pressure

The shell displacement can be substituted into the stress-displacement relation

2 ∂us us σs = 1 ν r + r , φφ (1 − 2ν ) 1 ∂r r 1 CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 111 to yield

3 R0 3p1(ν0 − 1)(1 + ν1) 2+ r FS: σs = . (3.22) φφ R3 2(1 + ν ) 3(1 − ν ) + (1 − )(1 + ν ) 0 − 1 0 1 1 1 R3 1 Using the same assumptions for the thin shell theory as before, that the stress throughout the shell can be approximated by that upon the shell mid-surface and that h ≪ 1 and 1 ∼ O(1), we derive the thin shell approximation for the hoop stress:

h s 3p1(ν0 − 1)(1 + ν1)(1 − 2 ) TSA: σφφ ∼ 2(1 + ν0)(1 − ν1 − h(1 − 1)(1 + ν1)) 3p (ν − 1)(1 + ν ) (1+3ν − 2 (1 + ν )) = 1 0 1 1+ 1 1 1 h + O(h2) . (3.23) 2(1 + ν )(1 − ν ) 2(1 − ν ) 0 1 1 It should be noted that the approximation used within the Jones, Chapman and All- wright methodology [17] is equivalent to taking only the leading order term, or h = 0, in our thin shell approximation. Finally we can take into account the possibility that η ≫ 1 using the parameterisation = where h ≪ 1 to ﬁnd the stiﬀ-thin shell 1 1 h approximation:

STSA: h 3pη(ν0 − 1)(1 + ν1)(1 − ) σs ∼ 2 , φφ 3h 2h(1 + ν ) (1 − ν ) − (h − η)(1 + ν ) 1 − 0 1 1 2 3pη(ν − 1)(1 + ν ) 1 (1 + 3ν + 2η(1 + ν )) = 0 1 + 1 1 2(1 + ν ) ((1 − ν )+ η(1 + ν ))h 2((1 − ν )+ η(1 + ν ))2 0 1 1 1 1 + O(h) . (3.24) Both the thin shell and stiﬀ-thin shell approximations are constant over the shell thickness and so, for these approximations, the stress resultant, (3.21), is given by

(I) 11 s n = hσφφ, CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 112 which, under each approximation, simpliﬁes to

3p (ν − 1)(1 + ν ) TSA: (I)n11 = h 1 0 1 + O(h2), (3.25) 2(1 + ν0)(1 − ν1) STSA: 3pη(ν − 1)(1 + ν ) 1 (1 + 3ν + 2η(1 + ν )) (I)n11 = 0 1 + h 1 1 2(1 + ν ) ((1 − ν )+ η(1 + ν )) 2((1 − ν )+ η(1 + ν ))2 0 1 1 1 1 + O(h2) . (3.26) Thus if we only want terms to O(h) the stress resultant, produced from our thin shell approximation, only uses the leading order term of the hoop stress and as such becomes the same approximation used by Jones, Chapman and Allwright, [17]; hereafter this will be referred to as the Jones et al. approximation or JA. To O(h), our approximations are thus given by:

3p (ν − 1)(1 + ν ) JA: (I)n11 = h 1 0 1 , (3.27) 2(1 + ν0)(1 − ν1) STSA: 3pη(ν − 1)(1 + ν ) 1 (1 + 3ν + 2η(1 + ν )) (I)n11 = 0 1 + h 1 1 . 2(1 + ν ) ((1 − ν )+ η(1 + ν )) 2((1 − ν )+ η(1 + ν ))2 0 1 1 1 1 (3.28)

The solution derived from the full shell displacement on the other hand can not be simpliﬁed in this way since the hoop stress is not constant over the shell thickness, and thus must be integrated over ξ, using (3.21). It should be noted that a solution called ‘ad hoc’ linear analysis used by Shams and Porﬁri [34] in discussion on the types of analysis that can be used to calculate the buckling pressure of unembedded spherical shells appears similar to the method used here to calculate the FS for the buckling pressure of embedded shells. Upon integration the FS yields:

FS:

1 3 2 R0 3hp1(ν0 − 1)(1 + ν1) 1 2(1 + hξ)+ 2 dξ − 2 (1 + hξ) (I)n11 = , R 3 2(1 + ν ) 3(1 − ν ) + (1 − )(1 + ν ) 0 − 1 0 1 1 1 R 1 3p (ν − 1)(1 + ν )h(12 + h2) = 1 0 1 . (3.29) R 3 4(2 + h)(1 + ν ) 3(1 − ν ) + (1 − )(1 + ν ) 0 − 1 0 1 1 1 R 1 CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 113

Hoop stress under axial compression:

We can similarly ﬁnd the stress resultant for a shell under axial compression. We know s from section 2.5.4 that the full shell hoop stress σφφ is given by (2.148),

aˆ 2 aˆ σs = 0 − d r(1 + 2ν )+ 2 (−7P +1)+ ˆb (−2P + 1) φφ r3 3 1 1 r5 2 2 2 2 c d − 1 (1 − 2ν )(P +1)+ 3 r2(−14(2 + ν )P + (7 − 4ν )), (3.30) 3 r3 1 2 7 1 2 1 where constants are solutions to equations (2.138) to (2.143), and that our two approximations are given by (2.149),

2 σs =â − d r(1 + 2ν ) +a ˆ (−7P +1)+ ˆb (−2P + 1) φφ 0 3 1 1 2 2 2 2 2 d − c (1 − 2ν )(P +1)+ 3 (−14(2 + ν )P + (7 − 4ν )), (3.31) 3 1 1 2 7 1 2 1 in which for the thin shell approximation, TSA, coefficients solve equations (2.150) to (2.158), and where for the stiff-thin shell approximation, STSA, coefficients solve equations (2.159) to (2.167). The thin shell approximation is valid for 1 = O(1) whilst 1 the stiff-thin shell approximation is valid for 1 = O h . For both approximations h ≪ 1. As in the hydrostatic case, the stress resultant can be simplified for approximations to the full hoop stress, as they are constant over the shell thickness. As such

(I) 11 s n = hσφφ, (3.32)

s where σφφ is given by (2.149), as above, for the thin shell and stiff-thin shell approximations. As expected, from the hydrostatic case, due to the form of the stress resultant, s in order to find a solution of (3.32) to O(h) we would only need σφφ to O(1). In this case the thin shell approximation is equivalent to the Jones, Chapman and Allwright approximation for an embedded shell under uniaxial compression, detailed in [17], and will be known as the JA from here on. The full solution, FS, can not be simplified in this way and must thus be fully integrated. As such integrating the full linear solution via equation (3.21) yields

−4ˆa 2 h2 16ˆa (h2 + 12) (1)n11 = h 0 − d (1 + 2ν ) + 1 − 2 (−7P + 1) (h2 − 4) 3 1 1 12 3(h2 − 4)3 2 8 c + ˆb (−2P +1)+ 1 (1 − 2ν )(P + 1) 2 2 3 (h2 − 4) 1 2 d + 3 (h2 + 4)(−14(2 + ν )P + (7 − 4ν )) . (3.33) 28 1 2 1 CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 114

A general form of the stress resultant

All approximations of the stress resultant can thus be written in the form

(I) 11 n = p(q0 + q2P2(cos φ)) where q0 and q2 depend upon the approximation chosen, so that for hydrostatic pressure q2 = 0 and in each approximation q0 is given by:

JA:

31(ν0 − 1)(1 + ν1) q0 = h , (3.34) 2(1 + ν0)(1 − ν1) STSA: 3η(ν − 1)(1 + ν ) 1 (1 + 3ν + 2η(1 + ν )) q = 0 1 + h 1 1 , (3.35) 0 2(1 + ν ) ((1 − ν )+ η(1 + ν )) 2((1 − ν )+ η(1 + ν ))2 0 1 1 1 1 FS: 2 31(ν0 − 1)(1 + ν1)h(12 + h ) q0 = 3 . (3.36) R0 4(2 + h)(1 + ν0) 3(1 − ν1) + (1 − 1)(1 + ν1) 3 − 1 R1 Whilst under axial compression, we ﬁnd

FS : h −4â 2 h2 16â (h2 + 12) q = 0 − d (1 + 2ν ) + 1 − 2 + ˆb 0 p (h2 − 4) 3 1 1 12 3(h2 − 4)3 2 8 c d + 1 (1 − 2ν )+ 3 (h2 + 4)(7 − 4ν ) , (3.37) 3 (h2 − 4) 1 28 1 h 112â (h2 + 12) q = − 2 − 2ˆb 2 p 3(h2 − 4)3 2 8 c d + 1 (1 − 2ν ) − 3 (h2 + 4)(2 + ν ) , (3.38) 3 (h2 − 4) 1 2 1 where constants are given by the solutions to equations (2.138) to (2.143). The Jones et. al approximation has q0 and q2 as,

JA : h 2 2 d q = (â − d r(1 + 2ν )+2â + bˆ − c (1 − 2ν ) − 3 (7 − 4ν )), (3.39) 0 p 0 3 1 1 2 2 3 1 1 7 1 h 2 q = − 14â − 2ˆb − c (1 − 2ν ) − 2d (2 + ν ) , (3.40) 2 p 2 2 3 1 1 3 1 CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 115 where constants are given by the first term in the series solution of equations (2.150) to (2.158) in h. Finally the stiff-thin shell approximation is given by

STSA : h 2 2 d q = (â − d r(1 + 2ν )+2â + bˆ − c (1 − 2ν ) − 3 (7 − 4ν )), (3.41) 0 p 0 3 1 1 2 2 3 1 1 7 1 h 2 q = − 14â − 2ˆb − c (1 − 2ν ) − 2d (2 + ν ) , (3.42) 2 p 2 2 3 1 1 3 1 where constants are given by the series solution in h to O(1) of equations (2.159) to (2.167). To evaluate the first energy integral (3.3), once we have the stress resultant we need to know the radial displacement within the shell. From (2.39) we know the radial shell displacement is given by the virtual radial displacement so that:

∞ w = UnPn(cos φ), n=0 from (3.7), and so ∞ ∂w = U P (1)(cos φ). ∂φ n n n=0 We can therefore write the ﬁrst energy integral as

(I)n11 ∂w 2 I1 = dS, Mid-shell 2 ∂φ surface ∞ p 2π π q = U U (q + 2 (3 cos φ2 − 1))P (1)(cos φ)P (1)(cos φ)sin φ dφdθ, 2 n m 0 2 n m n,m 0 0 =0 ∞ ∞ q n(n + 1) q 1 = 2πp(q − 2 ) U 2 + 3π 2 U U 2P (1)()P (1)() d, 0 2 (2n + 1) n 2 n m n m n=0 n,m=0 −1 ∞ q n(n + 1) = 2πp(q − 2 ) U 2 0 2 (2n + 1) n n=0 ∞ q 2n2(n + 1)(n + 2) 2(n − 1)n(n + 1)2 + 3π 2 + U 2 2 (2n + 1)2(2n + 3) (2n − 1)(2n + 1)2 n n=1 4n(n + 1)(n + 2)(n + 3) + U U , (3.43) (2n + 1)(2n + 3)(2n + 5) n n+2 where we use the recurrence relation and orthogonality conditions of the associated

Legendre polynomials during integration, and q0 and q2 have different definitions depending upon the type of pressure applied and whether the full stress resultant or Jones et al. or stiff-thin shell approximations are used. CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 116

The second energy integral

Following the process of Jones, [17] the second energy integral (3.4), can be written in nondimensional form as,

3 h αβλ h αβλ I2 = E θαβθλ + E ραβρλ dS, (3.44) Mid-shell 2 24 surface where lengths, including base vectors, have been nondimensionalised on the shell mid- radius and the elastic modulus tensor has been scaled on the shear modulus of the host medium. The second energy integral, (3.44) can then be simpliﬁed using the shallow shell approximations, provided in section 2.3, to

1 θ = (v | + v | ) − b w, ρ = w| . (3.45) αβ 2 α β β α αβ αβ αβ

If the unit normal vector to the mid-shell surface is inward facing, such that bαβ = aαβ then θαβ can be rewritten in terms of the virtual displacements as

1 θ = (v | + v | ) − a w. αβ 2 α β β α αβ

The Van der Neut substitution, originally employed by Koiter [50], can be used to write the virtual displacements vα as

λγ vα = ψ,α + εαλa χ,γ, where εαλ is the surface alternating tensor, and ψ and χ are functions to be determined. Further the assumptions of axisymmetry yield the simpliﬁcation

vα = ψ,α, where, after nondimensionalising (3.7),

∞ (1) v1 = VnPn (cos φ), n=0 and thus ∞ ψ = VnPn(cos φ), (3.46) n=0 CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 117

where V0 is set to zero without loss of generality. This allows us to ﬁnally write the linearised strain tensor as

1 θ = (ψ| + ψ| ) − wa , (3.47) αβ 2 αβ βα αβ

= ψ|αβ − waαβ, (3.48) where the order of covariant diﬀerentiation is invariant in shallow buckling, [50].

Under these simpliﬁcations the integrand of the second energy integral I2 can be written as

Using the deﬁnition of the elasticity tensor in equation (2.47),

αβλ 41(1 + ν1) E aαβaλ = , (1 − ν1) and hence the ﬁrst term in the I2 integrand becomes

2h (1 + ν )w2 1 1 . (1 − ν1)

The same deﬁnition helps simplify the second term in the I2 integrand. If we remember covariant and contravariant metric tensors in shells are so that

αβ aαβa = 2 then the second term in the I2 integrand can be written

h αβλ αβ 2h1(1 + ν1) αβ − E aαβaλψ|αβwa = − wψ|αβ a , (3.50) 2 (1 − ν1) 2h (1 + ν ) = − 1 1 w∇2ψ, (3.51) (1 − ν1) where the Laplacian on the shell surface is given by

2 αβ ∇ ψ = ψ|αβ a . CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 118

The remaining terms of the I2 integrand are of the form

αβλ E f|αβf|λ dS, (3.52) for a scalar function f. Since the elasticity tensor for shells Eαβλ is composed of contravariant metric tensors, by (2.6) we can see that the covariant derivative of the contravariant metric tensor will be zero:

αβ a |ρ = 0, and thus αβλ E |ρ = 0.

This allows us to write

αβλ αβλ E f|αβf|λdS = E f|αβf|λ Mid-shell surface αβλ αβλ − E |f|αβf|λ − E f|αβf|λ dS, αβλ αβλ = E f|αβf|λ − E f|αβf|λ dS. Mid-shell surface (3.53)

By the divergence theorem, on a closed shell, the ﬁrst part of the integral (3.53), will equate to zero, and so upon repeating the process (3.52) simpliﬁes to

αβλ E f|αβλf dS. (3.54) Mid-shell surface Remembering that in shallow shells the order of covariant differentiation is unimportant and using the definition 4 αβ λ ∇ f = a a f|αβλ, we then find that αβλ 21 4 E f|αβλ = ∇ f. (1 − ν1) As such (3.54) can be written,

2 1 f∇4f dS. (3.55) (1 − ν1) Mid-shell surface CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 119

∂f By Green’s theorem, if ∂n = f = 0 on the surface S, this can then be rewritten as:

2 1 (∇2f)2 dS. (3.56) (1 − ν1) Mid-shell surface Therefore

h αβλ h1 2 2 E ψ|αβψ|λdS = (∇ ψ) dS, (3.57) Mid-shell 2 (1 − ν1) Mid-shell surface surface 3 3 h αβλ h 1 2 2 E w|αβ w|λ = (∇ w) dS. (3.58) Mid-shell 24 12 (1 − ν1) Mid-shell surface surface

Putting all the parts of the I2 integral back together we can write (3.44) as

h1 2 2 I2 = ∇ ψ − (1 + ν1)w (1 − ν1) Mid-shell surface h2 + (1 − ν2)w2 + (∇2w)2 dS. (3.59) 1 12 Since we know the Laplacian can be given by

2 αβ ∇ f = f|αβa , which in spherical shell coordinates becomes

2 1 ∇ f = f|11 + f|22, sin2 φ and that both w and ψ are invariant with respect to θ, so that

ψ,2 = w,2 = 0, then using the definition of covariant differentiation of vectors in (2.5) we find in spherical coordinates the Laplacians for w and ψ can be written:

2 ∇ ψ = ψ,11 + cot φ ψ,1,

2 ∇ w = w,11 + cot φ w,1. CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 120

Substituting our deﬁnitions of w and ψ from (3.7) and (3.46), we ﬁnd that

∞ 2 ∇ ψ = − n(n + 1)VnPn(cos φ), (3.60) n=0 ∞ 2 ∇ w = − n(n + 1)UnPn(cos φ). (3.61) n=0

This then allows us to write I2 as

∞ 2 h 2π π I = 1 [(−n(n + 1)V − (1 + ν )U )P (cos φ)] 2 (1 − ν ) n 1 n n 1 0 0 n=0 ∞ 2 2 + (1 − ν1 ) UnPn(cos φ) n=0 ∞ 2 h2 + n(n + 1)U P (cos φ) dφ dθ, (3.62) 12 n n n=0 which using the Legendre polynomial orthogonality relations becomes:

∞ 4πh 1 I = 1 (n(n + 1)V +(1+ ν )U )2 2 (1 − ν ) 2n + 1 n 1 n 1 n=0 h2n2(n + 1)2 + (1 − ν2)U 2 + U 2 . (3.63) 1 n 12 n It is important to note that much of the preceding analysis on the buckling of the shell, in the ﬁrst and second energy integral, is based on the theory of shallow shells, and thus is only valid if the wavelength of the buckling pattern is small in comparison with the minimum principal radius of curvature of the shell, [50].

The third energy integral

The third energy integral I3, in (3.5), considers the change in energy of the host medium, rather than the shell, when the virtual displacement v, applied to the shell, causes an induced displacement u within the host. As such we need to ﬁnd the induced virtual displacement u within the host such that on the shell boundary

u|r=R1 = v,

h where we assume that the displacement on the shell boundary R1 = 1+ 2 can be approximated by that upon the shell mid-surface. We also consider that the induced stress ﬁeld vanishes as r → ∞. This problem has previously been solved by Lur’e [6] and is detailed in the thesis of Jones, [17]. By decomposing the boundary displacement CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 121 into a series vector of spherical harmonics, so that

∞ v = Y n(θ, φ), n=0 the induced displacement can be written as

∞ 1 ∇(∇ U ) u = U − (R2 − r2) −n−1 , −n−1 2 1 (3 − 4ν )(n + 1) + 2(1 − ν ) n 0 0 =0 where R n+1 U = 1 Y (θ, φ). −n−1 r n Jones then shows, [17], that we can write the vector spherical harmonics as

Y n = n(γn − αn)Pn−1 + (n + 1)γnPn+1 er (1) (1) + (αn − γn)Pn−1 + γnPn+1 eφ, (3.64) where α and γ are to be determined and the associated Legendre polynomials have argument cos φ. Thus

∞ v = (n + 1)(γn+1 − αn+1)+ nγn−1 Pner n=0 (1) + αn+1 − γn+1 + γn−1 Pn eφ. (3.65) Comparing this formulation of the virtual displacement to (3.7) and (3.8) allows us to determine γn and αn, so that:

Un = (n + 1)(γn+1 − αn+1)+ nγn−1,

Vn = αn+1 − γn+1 + γn−1, (3.66) or

U − (n − 1)V γ − α = n−1 n−1 , n n 2n − 1 U + (n + 2)V γ = n+1 n+1 . (3.67) n 2n + 3 CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 122

We can use the same decomposition of Y n within u to yield, upon rearranging indices, the components

∞ R n+1 u e = n(γ − α )P + (n + 1)γ P 1 r n n n−1 n n+1 r n=0 n + 2 r2 R n+3 (2n + 1)(n + 1)γ P − 1 − 1 n n+1 (3.68) 2 R2 r (3 − 4ν )(n + 1) + 2(1 − ν ) 1 0 0 and

∞ R n+1 u e = (α − γ )P (1) + γ P (1) 1 φ n n n−1 n n+1 r n=0 2 n+3 (1) 1 r R (2n + 1)(n + 1)γnP + 1 − 1 n+1 . (3.69) 2 R2 r (3 − 4ν )(n + 1) + 2(1 − ν ) 1 0 0

The virtual displacement u can then be rewritten in terms of Un and Vn components using (3.67) as,

∞ R n+2 R n u = A 1 + B 1 P e n r n r n r n=0 ∞ R n+2 R n + C 1 + D 1 P (1)e , (3.70) n r n r n φ n=0 where

An + Bn = Un,

Cn + Dn = Vn, (3.71) and

An = −(n + 1)Cn, 1 n(2n − 1) U + (n + 1)V C = nV −U + n n . (3.72) n 2n + 1 n n 2 (3 − 4ν )n + 2(1 − ν ) 0 0 Now that the virtual displacement within the medium is deﬁned in terms of the virtual displacement within the shell, I3 can be evaluated. Using the deﬁnition of Hooke’s Law given in (2.14), we can rewrite I3 as

1 I = σije dV, (3.73) 3 2 ij r>R1 CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 123 or, through the symmetry of the strain tensor, in (2.13),

1 I = σiju | dV. (3.74) 3 2 i j r>R1

It is then possible to rewrite I3 as

1 I = σiju dV, (3.75) 3 2 i r>R1 j since by covariant diﬀerentiation,

ij ij ij σ ui = σ |jui + σ ui|j, j and through the equation of equilibrium, (2.16), the ﬁrst term equates to zero. This formulation allows us to apply the divergence theorem to yield

1 I = σijn u dS, (3.76) 3 2 j i δV where δV is the boundary of the shell r = R1, since virtual stresses and displacements tend to zero as r →∞. Setting the normal vector to point inwards, such that n1 = −1 and all other components are zero, this expands to

1 I = − (σ11u + σ12u ) dS. (3.77) 3 2 1 2 δV From (2.20) we know we can write the stress-strain relations for these stresses which in terms of u become

∂u e = r , (3.78) 11 ∂r 1 ∂u ∂u e = r + r φ − u , (3.79) 12 2 ∂φ ∂r φ ∂u u 1 ∂u cot φ ∆= r + 2 r + φ + u . (3.80) ∂r r r ∂φ r φ

The previously found values of ur and uφ, in (3.70) are then substituted into the stress-strain relations to yield σ11 and σ12. When combined with the identities in (2.21)

u1 = ur, (3.81)

u2 = ruφ, (3.82) we can then write I3 as an integral involving squares of Legendre functions, which are CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 124 easily integrated using orthogonality relations to give

∞ U I = 4πR n nU − 2(n + 1)C 3 1 2n + 1 n n n=0 ν + 0 ((n − 2)U − 2(n + 1)C + n(n + 1)V ) 1 − 2ν n n n 0 n(n + 1)V + n ((n + 1)V + 2C −U ) , (3.83) 2(2n + 1) n n n where Cn is given in (3.72).

3.2.4 The Eigenvalue problem

Having found representations of I1, I2 and I3 in terms of the coeﬃcients Un and Vn we can now employ (3.9) and (3.10) where ∆W2 = I1 + I2 + I3. Since I1 is independent of

Vn, by (3.43), and thus

∂I 1 = 0, (3.84) ∂Vn we can simplify ∂∆W2 ∂ = (I2 + I3) = 0. (3.85) ∂Vn ∂Vn By (3.63) and (3.83):

∂I 8πh n(n + 1)(n(n + 1)V +(1+ ν )U ) 2 = 1 n 1 n , (3.86) ∂Vn (1 − ν1)(2n + 1) ∂I 4πR (n + 1) v 3 = 1 U −2E + 0 (n − 2E ) ∂V 2n + 1 n n 1 − 2v n n 0 1 + n(2(n + 1)V + 2C −U + 2E V ) , (3.87) 2 n n n n n where Cn is given by (3.72) and

∂C 1 n(n + 1)(2n − 1) E = n = n + . (3.88) n ∂V 2n + 1 2((3 − 4ν )n + 2(1 − v )) n 0 0

Substituting these into (3.85) then yields a relation between Vn and Un, so that

Vn = ςnUn, (3.89) CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 125 where

2hn(n + 1) −1 2h(1 + ν ) 2E (1 − ν ) ς = − 1 + (n + 1) + 2E 1 1 − n 0 n R (1 − ν ) n R (1 − ν ) n(1 − 2ν ) 1 1 1 1 0 1 n(2n − 1) 1 ν + − 1 − + 0 , (3.90) 2n + 1 2((3 − 4ν )n + 2(1 − ν )) 2 1 − 2ν 0 0 0 with ς0 = 0. Substituting this back into I1 to I3 gives ∆W2 in the form

∞ 2 ∆W2 = (anp + bn)Un + pcnUnUn+2, (3.91) n=0 where the coeﬃcients are given by:

n(n + 1) a = π (2q − q ) n 0 2 2n + 1 n2(n + 1)(n + 2) (n − 1)n(n + 1)2 + 3q + , (3.92) 2 (2n + 1)2(2n + 3) (2n − 1)(2n + 1)2 4πh h2n2(n + 1)2 b = 1 (n(n + 1)ς +1+ ν )2 + (1 − ν2)+ n (1 − ν )(2n + 1) n 1 1 12 1 4πR ν + 1 n − 2(n + 1)F + 0 (n − 2 − 2(n + 1)F + n(n + 1)ς ) 2n + 1 n 1 − 2ν n n 0 n(n + 1)ς + n ((n + 1)ς + 2F − 1) , (3.93) 2 n n 6πq n(n + 1)(n + 2)(n + 3) c = 2 , (3.94) n (2n + 1)(2n + 3)(2n + 5) where

C 1 n(2n − 1) 1 + (n + 1)ς F = n = nς − 1+ n . n U 2n + 1 n 2 (3 − 4ν )n + 2(1 − ν ) n 0 0

∂∆W2 We can then use this form of ∆W2 to ﬁnd , which yields ∂Un

2(anp + bn)Un + pcnUn+2 + pcn−2Un−2 = 0, (3.95) where cn = 0 for n< 0. Rewriting 1 λ = , p CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 126 allows (3.95) to be formulated as an eigenvalue system, so that

a c c − n − λ U − n−2 U − n U = 0. (3.96) b n 2b n−2 2b n+2 n n n This eigensystem can then be split into odd and even n, enabling study of symmetric and antisymmetric buckling modes.

a c c − 2n−1 − λ U odd − 2n−3 U odd − 2n−1 U odd = 0, (3.97) b n 2b n−1 2b n+1 2n−1 2n−1 2n−1 a c c − 2n−2 − λ U even − 2n−4 U even − 2n−2 U even = 0, (3.98) b n 2b n−1 2b n+1 2n−2 2n−2 2n−2 odd even for n ≥ 1 where Un = U2n−1 and Un = U2n−2. Equations (3.97) and (3.98) can thus solved numerically by truncating the inﬁnite tridiagonal matrices of the two eigenvalue problems:

(Aodd − λI)U odd = 0, (3.99) (Aeven − λI)U even = 0. (3.100)

3.2.5 Results

Agreement with Koiter’s unembedded shell

To verify the results presented in this section we wish to check that for a shell with 1 no surrounding medium (0 = 0 and ν0 = 2 ) our results for an embedded shell under hydrostatic pressure tend to those presented by Koiter [50] for an unembedded shell under hydrostatic pressure. In order to do this we need to redimensionalise our results, as they are currently nondimensionalised on the host shear modulus, which impedes our ability to ﬁnd the buckling pressure as the host shear modulus tends to 0. We need only consider the buckling pressure derived via the full shell hoop stress (FS hoop stress), as the approximations are simpliﬁcations of this solution. Redimensionalising this solution yields:

qˆ2 = 0, (3.101) h h2 3ˆ0ˆ1(ν0 − 1)(1 + ν1) 12 + Rˆ Rˆ2 qˆ0 = . (3.102) h Rˆ3 4(1 + ν ) 2+ 3(1 − ν )ˆ + (ˆ − ˆ )(1 + ν ) 0 − 1 0 ˆ 1 0 0 1 1 ˆ3 R R1 CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 127

Under q2 ≡ 0, cn = 0 for all n and the eigensystem (3.96) simpliﬁes to

b p = − n , (3.103) an or dimensionally bn pˆ = −ˆ0 , (3.104) an where an and bn are given by redimensionalising (3.92) and (3.93). In the limit as 1 0 → 0 and ν0 = 2 we ﬁnd the coeﬃcients an and bn tend to

2 hˆ −πn(n + 1) 2+ a Rˆ lim n = , (3.105) ˆ0→0 ˆ0 4(2n + 1) ˆ ˆ2 h h 2 2 4π ˆ1 n (n + 1) Rˆ 2 Rˆ2 lim bn = (1 − ν1 )+  , (3.106) ˆ0→0 (1 − ν1)(2n + 1) 12       since

(1 + ν1) lim ςn = − , (3.107) ˆ0→0 n(n + 1) and thus

hˆ n2(n + 1)2hˆ2 16ˆ1(1 + ν1) 1+ ˆ ˆ2 2 R 12R (1 − ν1 ) pˆ = 2 . (3.108) hˆ n(n + 1) 2+ Rˆ

This buckling pressure can then be compared to the Koiter buckling pressure, pK, for an unembedded shell:

2 2 2 4hˆˆ1(1 + ν1) n (n + 1) hˆ pˆK = 1+ . (3.109) ˆ ˆ2 2 Rn(n + 1) 12R (1 − ν1 )

By writing (3.108) as a series in hˆ and removing terms common to the Koiter solution, it becomes clear that the full solution for the buckling pressure of an unembedded shell matches the Koiter solution to leading order and provides higher order correction terms, so that:

ˆ ˆ ˆ2 4hˆ1(1 + ν1) h 3h 3 pˆ =p ˆK + − + + O(h ) . (3.110) Rnˆ (n + 1) Rˆ 4Rˆ2 CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 128

ˆ Therefore it is no surprise that for a glassy shell in which ν = 0.3 and h = 0.01, 1 Rˆ this buckling pressure is approximately equal to that of the Koiter buckling pressure as can be seen in figure 3.1. It should be noted that in this figure and subsequent figures buckling pressures have been calculated at integer buckling modes and then joined with smooth curves. p p0

n 10 20 30 40 50

Figure 3.1: Buckling pressure p at each buckling mode number n, nondimensionalised 2 4hˆ 1(1 + ν1) on classical buckling pressure p0 = as derived using Koiter’s buckling ˆ2 2 R 3(1 − ν1 ) pressure (blue, solid) and using the unembedded shell limit, (3.108), of the full solution buckling pressure (pink, dashed). Note: lines overlay each other.

Under hydrostatic pressure

Under hydrostatic pressure at inﬁnity we know (3.95) reduces to

2(anp + bn)Un = 0, (3.111) and thus b p = − n , (3.112) an CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 129

where bn is given by (3.93) and an simpliﬁes to

n(n + 1) a = 2πq . (3.113) n 0 2n + 1

With three possible values for q0, derived from the full solution, in (3.36), the Jones et al. and the stiﬀ-thin shell approximations to the stress resultant, in (3.34) and (3.35) respectively, we can analyse the change in the buckling pressure predicted by the diﬀer- ent approximations. We can also compare the buckling pressure found here with that predicted by Fok and Allwright [10] for an embedded shell, constrained by inextensible buckling, under hydrostatic pressure. Under the Fok-Allwright solution the buckling pressure can be given by,

4 (1 + ν )(1 + ν ) 1 1 − ν (n(n + 1) − (1 − ν )) p = 1 1 0 1+ 1 1 h3 3(1 − ν ) h 1+ ν 12(1 − ν2) 0 1 1 1 2h ((2n3 − n2 + 3n + 2) − ν (2n3 − 3n2 + 5n + 2)) + + 0 . (n − 1)(n + 2)(1 + ν ) (1 + ν )(n − 1)2(n + 2)(3n + 2 − 2ν (2n + 1)) 1 1 1 0 (3.114)

For all approximations we are most interested in how the embedding of the shell within a medium changes the buckling pressure in relation to an un-embedded shell, for which the classical buckling pressure, p0, is given, in nondimensional form, by

4h2 (1 + ν ) p = 1 1 . 0 2 3(1 − ν1 )

By considering the buckling pressure for an embedded shell in comparison to that of p the un-embedded shell, , we can evaluate whether the embedding results in the p0 shell being more or less difficult to buckle. Thus for a nearly incompressible host material, ν0 = 0.49985, containing a thin glassy shell, 1 = 200,ν1 = 0.3 and h = 0.01, we can see that our stiff-thin shell solution provides the best approximation to the buckling pressure derived from the full stress resultant, the full solution, see figure 3.2. The Fok and Allwright solution approximates the buckling pressure best for higher buckling mode numbers, but considering its simplicity actually provides a reasonable approximation for all buckling mode numbers. The Jones et al. approximation on the other hand does not provide a good approximation to the buckling pressure for any mode, drastically underestimating the pressure needed to buckle the shell within the host medium. CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 130

p p0

n 10 20 30 40 50

Figure 3.2: Buckling pressure p at each buckling mode number n, nondimensionalised on classical buckling pressure p0, for a thin glassy shell 1 = 200,ν1 = 0.3 and h = 0.01 within a nearly incompressible host ν0 = 0.49985, where the buckling pressure is derived via: (i) the full solution (blue, solid); (ii) the Jones et al. approximation (pink, dashed); (iii) the stiﬀ-thin shell approximation (yellow, dot dashed); (iv) the Fok-Allwright approximation (green, dotted). Note: blue and yellow curves overlay.

The failure of the Jones et al. approximation in our parameter range appears to be due to the fact that it does not take into account the order of magnitude of the shell shear modulus. In fact for shells in which the shear modulus of the shell and of the medium are similar, so in which 1 = 1 though ν1 is not necessarily equal to ν0, the Jones et al. approximation provides as good an approximation to the solution derived from the full stress resultant as any other solution, see ﬁgure 3.3. However it is also worth noting that when the shell and medium have the same shear modulus, 1 = 1, the buckling pressure becomes very large, with shell instability in this case corresponding to instability of the whole composite. This result is evidenced in buckling pressures around 1000 times higher than when 1 = 200, and a critical buckling mode around 10 times higher. CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 131

p p0 120 000

100 000

80 000

60 000

40 000

20 000

n 10 20 30 40 50

Figure 3.3: Buckling pressure, p, at each buckling mode number, n, nondimensionalised on classical buckling pressure, p0, for a thin shell, ν1 = 0.3 and h = 0.01, within a nearly incompressible host, ν0 = 0.49985, when shell and host medium have the same shear modulus 1 = 1, where the buckling pressure is derived via: (i) the full solution (blue, solid); (ii) the Jones et al. approximation (pink, dashed); (iii) the stiﬀ-thin shell approximation (yellow, dot dashed); (iv) the Fok-Allwright approximation (green, dotted). Note: blue, pink and yellow curves overlay.

The lowest buckling pressure over all buckling modes is known as the critical buckling pressure, this is the ﬁrst point at which the equilibrium solution becomes unstable and we will denote it pc when expressed relative to the classical buckling pressure p0, so that p p = min . c p 0 As such, for an unembedded shell, we would ﬁnd pc = 1. The mode at which the critical buckling pressure occurs is known as the critical buckling mode nc and is the mode the shell is most likely to buckle into, as it requires the least energy. For a shell in which 1 = 200, ν1 = 0.3, and h = 0.01, in a nearly incompressible medium in which

ν0 = 0.49985 the critical buckling pressures for each approximation are given in table 3.1. CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 132

Critical buckling- Buckling pressure derived via: mode nc pressure pc Full stress resultant 29 4.62984 Stiﬀ-thin shell approximation 29 4.67619 Jones et al. approximation 29 0.994888 Fok-Allwright solution 29 4.95758

Table 3.1: Critical buckling pressures and associated mode numbers derived using various methods for an embedded shell under hydrostatic pressure where 1 = 200, ν1 = 0.3, ν0 = 0.49985, h = 0.01

As such we can see that the Jones et al. solution predicts a softening in the pressure needed to buckle an embedded shell in comparison to an un-embedded shell, in direct contrast to all other approximations, however all derivations predict the same buckling mode of 29. The diﬀerence in applicability of the various solutions with 1 leads us to look at how the critical buckling pressure changes with the shell shear modulus for each of the approximations, as in ﬁgure 3.4. CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 133

50 100 150 200 1

Figure 3.4: Critical buckling pressure pc, for a thin shell, ν1 = 0.3 and h = 0.01, within a nearly incompressible host, ν0 = 0.49985, as the shell shear modulus 1 changes, where the buckling pressure is derived via: (i) the full solution (blue, solid); (ii) the Jones et al. approximation (pink, dashed); (iii) the stiﬀ-thin shell approximation (yellow, dot dashed); (iv) the Fok-Allwright approximation (green, dotted). Note: blue and yellow curves overlay, and are partially overlaid by the green curve.

Interestingly whilst for each individual shell shear modulus the Fok and Allwright solution does differ from the solution derived from the full stress resultant, the full solution, the critical buckling pressure predicted by the Fok and Allwright solution is incredibly close to that derived from the full stress resultant. On the other hand we can also see that whilst the Jones et al. solution predicts the buckling pressure curve accurately for 1 = 1, as 1 increases the critical buckling pressure predicted by Jones et al. differs drastically from the full stress resultant solution. In high contrast composites, in which 1 ∼ O(1), the Jones et al. approximation does not perform well for shells of any thickness. In figure 3.5, in which 1 = 200 it can be seen that for all h ≪ 1 the Jones et al. approximation fails to accurately predict the critical buckling pressure as derived from the full solution however the Fok and Allwright and stiff-thin shell approximations still perform well, with the stiff-thin shell approximation performing best overall. CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 134

pc 10

h 0.01 0.02 0.03 0.04 0.05

Figure 3.5: Critical buckling pressure pc, for a thin stiﬀ shell ν1 = 0.3 and 1 = 200, within a nearly incompressible host ν0 = 0.49985, as the shell thickness h changes, where the buckling pressure is derived via: (i) the full solution (blue, solid); (ii) the Jones et al. approximation (pink, dashed); (iii) the stiﬀ-thin shell approximation (yellow, dot dashed); (iv) the Fok-Allwright approximation (green, dotted).Note: blue and yellow curves overlay.

Having investigated how each approximation is affected by the shear modulus and shell thickness parameters, we can similarly look at the effect of changing the Poisson ratio of each material, but keeping the shear moduli and shell thickness fixed. In figure 3.6 we can see that all approximations yield the same qualitative behaviour of lower critical buckling pressure for a shell within a nearly incompressible host as the shell Poisson ratio increases, as the shell becomes less compressible. Similarly for fixed shell

Poisson ratio ν1 = 0.3 and shear modulus 1 = 200, altering the host Poisson ratio, see ﬁgure 3.7, causes the opposite qualitative behaviour of increasing critical buckling pressure with increasing host Poisson ratio. In both situations however the magnitude of the critical buckling pressure is much smaller than expected when using the Jones et al. approximation. The Fok and Allwright solution on the other hand provides similar results to the stiﬀ-thin shell approximation, however it generally overestimates the critical buckling pressure and becomes less accurate as the shell Poisson ratio increases. CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 135

pc 6

ν 0.2 0.3 0.4 1

Figure 3.6: Critical buckling pressure pc nondimensionalised on classical buckling pressure p0, for a thin shell h = 0.01 and 1 = 200, within a nearly incompressible host ν0 = 0.49985, as the shell Poisson ratio ν1 changes, where the buckling pressure is derived via: (i) the full solution (blue, solid); (ii) the Jones et al. approximation (pink, dashed); (iii) the stiﬀ-thin shell approximation (yellow, dot dashed); (iv) the Fok- Allwright approximation (green, dotted). Note: blue and yellow curves overlay. CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 136

ν 0.2 0.3 0.4 0

Figure 3.7: Critical buckling pressure pc nondimensionalised on classical buckling pressure p0, for an embedded thin stiﬀ shell h = 0.01,ν1 = 0.3 and 1 = 200 as the host Poisson ratio ν0 changes, where the buckling pressure is derived via: (i) the full solution (blue, solid); (ii) the Jones et al. approximation (pink, dashed); (iii) the stiﬀ-thin shell approximation (yellow, dot dashed); (iv) the Fok-Allwright approximation (green, dotted). Note: blue and yellow curves overlay.

Under uniaxial compression

Under uniaxial compression the full eigenvalue equations (3.99) and (3.100) can be truncated and solved numerically for prescribed parameters to yield the pressures at which the shell is predicted to buckle. For each combination of material parameters we truncated the full eigenvalue equations at an N for which there was negligible change (≤ O(h2)) in the buckling pressure when N was increased. The lowest positive buckling pressure will correspond to the ﬁrst point at which the 1 equilibrium solution will become unstable. Remembering that λ = , we will thus ﬁnd p the critical buckling pressure at the highest value of λ. Unlike in the hydrostatic case the critical buckling pressure does not correspond to a single buckling mode but to a buckling pattern. Each eigenvalue λ corresponds to an eigenvector, U, with elements

Un. Through equation (3.89) we can then ﬁnd the coeﬃcients Vn and thus the virtual displacements vr and vφ, using (3.7) and (3.8), yielding the buckling pattern associated with the buckling pressure p. CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 137

The eigensolution for buckling of an embedded shell under uniaxial compression depends upon constants, q0 and q2, which vary depending upon which method, either the full solution, the Jones et al. or stiﬀ-thin shell approximations (detailed in equations (3.37)-(3.42)), is used to ﬁnd the shell stress resultant.

We again deﬁne the critical buckling pressure pc to be the minimum pressure at which the shell becomes unstable, in relation to λ this is then deﬁned as:

1 1 p = min = . c λ max λ

As we split the eigensystem into odd and even modes we will be able to ﬁnd a minimum buckling pressure for each type of deformation, however the shell is most likely to deform into whichever conﬁguration has the lowest critical buckling pressure regardless of whether the deformation is symmetric or not.

We can see in figure 3.8 that for a shell, in which h = 0.01,ν1 = 0.3 and ν0 = 0.49985, whilst the stiff-thin shell approximation provides a very good approximation to the solution generated from the full stress resultant, the Jones et al. approximation again does not work well for composites with a high contrast between the shear modulus of the shell and host medium, but is a much better approximation as the material contrast lessens. However for the materials we are most interested in, those with a high contrast shear modulus 1 ∼ 200 the Jones et al. approximation vastly underestimates the pressure needed to buckle an embedded shell. We can also see in this figure, and more clearly in figure 3.9 in which the shear modulus is restricted, that for the critical buckling pressure derived from the full stress resultant, and the stiff-thin shell approximation to this, that the shells will generally buckle into the antisymmetric, odd, buckling mode as this is the lowest critical buckling pressure for each shell shear modulus. The Jones et al. approximation on the other hand finds the odd and even buckling modes to be the negligibly different, with the shell buckling into either with the same likelihood. It is worth noting that the division of buckling modes into odd and even modes is mainly due to the simplifications that this provides in the algebra, rather than an intrinsic difference in the method of buckling. CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 138

0.5

0.4

0.3

0.2

0.1

50 100 150 200 1

Figure 3.8: Critical uniaxial buckling pressure pc, for a thin shell h = 0.01, ν1 = 0.3 embedded in an elastic matrix ν0 = 0.49985 as a function of the shell shear modulus 1, where the buckling pressure is derived via: (i) the full solution (antisymmetric buckling- blue, solid; symmetric buckling- green, dotted); (ii) the Jones et al. approximation (antisymmetric buckling- yellow, solid; symmetric buckling- red, dashed); (iii) the stiﬀ- thin shell approximation (antisymmetric buckling- pink, solid; symmetric buckling- black, dotted). Note: blue and pink overlay, as do green and black similarly yellow and red.

Based on the work done for an embedded shell under hydrostatic pressure, in which the Jones et al. approximation fails for large shear moduli 1, I believe the solution derived from the full stress resultant to be a more accurate evaluation of the buckling pressure for an embedded shell under axial compression than the Jones et al. approximation. On top of the evidence built up under the hydrostatic pressure regime it is also worth noting that whilst, for all methods, the pressure needed to buckle an embedded shell decreases as the contrast between the host and shell shear modulus increases from 1, the solution derived from the full stress resultant eventually reaches a minimum point, in this case at 1 = 166, before the critical buckling pressure increases again. In the Jones et al. solution however the critical buckling pressure decreases monotonically as the contrast between the shell and medium shear modulus increases. Physically we would expect the critical buckling pressure to decrease as the shell contrast increases from 1 = 1. This is associated with the idea that for a shell and host of equal Poisson ratio as the contrast between the shells starts to increase, from 1 = 1, it becomes CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 139

0.35

0.30

0.25

0.20

0.15

0.10 30 40 50 60 70 1

Figure 3.9: Critical uniaxial buckling pressure pc, for a thin shell h = 0.01, ν1 = 0.3 embedded in an elastic matrix ν0 = 0.49985 as a function of the shell shear modulus 1, where the buckling pressure is derived via: (i) the full solution (antisymmetric buckling- blue, solid; symmetric buckling- green, dotted); (ii) the Jones et al. approximation (antisymmetric buckling- yellow, solid; symmetric buckling- red, dashed); (iii) the stiff- thin shell approximation (antisymmetric buckling- pink, solid; symmetric buckling- black, dotted). Note: blue and pink overlay, as do green and black similarly yellow and red partially overlay. easier to buckle the shell, however when the shell is far stiffer than the host medium we would expect an increase in the critical buckling pressure, therefore we would also expect a minimum critical buckling pressure to exist for some value of 1. For all methods, a similar buckling pattern is observed around the equator of the shell for high contrast composites, at 1 = 200, as can be seen in figures 3.10a, 3.10b and 3.10c. CHAPTER 3. SINGLE SHELL BUCKLING PROBLEMS 140

z 1.0

0.5

x 1.0 0.5 0.5 1.0

0.5

1.0 (a) The buckling pattern when critical buckling pressure is found via the full solution. z 1.0

0.5

x 1.0 0.5 0.5 1.0

0.5

1.0 (b) The buckling pattern when critical buckling pressure is found via the Jones et al. approximation. z 1.0

0.5

x 1.0 0.5 0.5 1.0

0.5

1.0 (c) The buckling pattern when critical buckling pressure is found via the thin-stiﬀ shell approximation.

Figure 3.10: Odd critical buckling for a thin stiﬀ shell h = 0.01,ν1 = 0.3 and 1 = 200 embedded in an elastic matrix ν0 = 0.49985, where the buckling pattern is found via various shell solutions. Chapter 4

Cavities interacting in an elastic medium

Having understood the linear elastic behaviour of isolated cavities within a medium in section 2.4 from Chapter 2 we can now look at the eﬀect of interaction between cavities within the same system under hydrostatic pressure. It is possible to look at the case of uniaxial compression, but for simplicity we restrict attention to the hydrostatic case. This is a precursor problem before attacking the more complicated (but mathematically similar) problem of the interaction of two shells, that will be considered in the next chapter. Understanding the linear elastic behaviour of two shells is necessary to attain the overall objective of this thesis of predicting the buckling pressure of interacting embedded shells.

4.1 Problem conﬁguration

With reference to ﬁgure 4.1, let us deﬁne a global coordinate system (ˆr, φ,ˆ θˆ), and situate Lˆ Lˆ two spherical cavities (1 and 2) a distance Lˆ apart, atr ˆ = , φˆ = 0 andr ˆ = , φˆ = π 2 2 with centres aligned vertically. Each cavity has a local coordinate system (ˆr1, φˆ1, θˆ) and (ˆr2, φˆ2, θˆ) and cavity radii Rˆ1 and Rˆ2, respectively, where Rˆ1 + Rˆ2 < Lˆ, such that cavities never intersect. Since the cavities are aligned vertically the azimuthal angles

θˆ1 and θˆ2 are equal at all points throughout the medium, and thus we have dropped the subscripts from both. Our medium has material properties ˆ and ν, denoting the shear modulus and Poisson ratio of the material respectively.

141 CHAPTER 4. CAVITIES INTERACTING IN AN ELASTIC MEDIUM 142

zˆ1 φˆ1 rˆ1

yˆ1

xˆ1 Lˆ φˆ rˆ2 zˆ2 2

yˆ2

xˆ 2 θˆ

Figure 4.1: Two vertically aligned spherical cavities (known as cavities 1 and 2) a distance Lˆ apart in an inﬁnite elastic medium with local coordinate systems (ˆr1, φˆ1, θˆ) and (ˆr2, φˆ2, θˆ) respectively.

4.1.1 Conditions upon the system

We subject the host medium to hydrostatic pressurep ˆ at inﬁnity, so that:

σˆrˆrˆ →−p asr ˆ →∞, and presume that cavities have no internal pressure so that on the surface of the cavities,

ˆ σˆrîrî =σ ˆrîφî = 0 onr î = Ri. (4.1) CHAPTER 4. CAVITIES INTERACTING IN AN ELASTIC MEDIUM 143

The pressure at inﬁnity in turn causes displacement of the medium, which can be expressed in the global Cartesian coordinate system as

∞ p uˆxˆ = − x,ˆ 3kˆ ∞ p uˆyˆ = − y,ˆ 3kˆ ∞ p uˆzˆ = − z,ˆ 3kˆ where uˆ = (ˆux, uˆy, uˆz) is the displacement vector for the medium and kˆ, the bulk modulus of the medium, is related to the shear modulus and Poisson ratio of the 2ˆ(1 + ν) medium by kˆ = . The global and local coordinates are related, in their 3(1 − 2ν) equivalent Cartesian systems, by

Lˆ zˆ =z ˆ + (−1)i , (4.2) i 2

xˆi =x, ˆ (4.3)

yˆi =y. ˆ (4.4)

4.1.2 Nondimensionalising

For simplicity we can nondimensionalise the system upon the inter-shell distance Lˆ and the shear modulus of the medium ˆ. Under this scaling we can write the nondimensional variables and parameters as

Rˆi rˆi uˆ pˆ Ri = , ri = , u = , p = . (4.5) Lˆ Lˆ Lˆ ˆ

Components φˆ and θˆ are nondimensional already and thus hat notation can be dropped from these variables. As such the boundary conditions are

σrr = −p as r →∞, (4.6) p p p u = − x, u = − y, u = − z as r →∞, (4.7) x 3k y 3k z 3k

σriri = σriφi =0 on ri = Ri for i = 1, 2, (4.8) where

(−1)i z = z + , y = y, x = x. (4.9) i 2 i i CHAPTER 4. CAVITIES INTERACTING IN AN ELASTIC MEDIUM 144

4.1.3 Application of the boundary conditions

In order to determine the induced deformation of the medium we will need to understand how the pressure and displacement at infinity affect the cavity boundaries. As such we need to be able to write the stress and displacement at infinity in terms of the local coordinates of each cavity. Using (4.9) we can write the displacement at infinity upon cavity i as

p p p 1 u = − x , u = − y , u = − z + (−1)i+1 as r →∞. (4.10) xi 3k i yi 3k i zi 3k i 2 i

Converting between the Cartesian displacement vector, ui = uxi exi + uyi eyi + uzi ezi , and the spherical coordinate displacement vector, ui = uri eri + uφi eφi + uθeθ, we can use the relations

xi = ri sin φi cos θ, yi = ri sin φi sin θ, zi = ri cos φ, and the transformation matrix:

uri sin φ cos θ sin φ sin θ cos φ uxi

uφi  = cos φ cos θ cos φ sin θ − sin φ uyi  , (4.11) u − sin θ cos θ 0 u  θ     zi        to ﬁnd the displacement vector in terms of spherical coordinates, for i = 1, 2:

p (−1)i+1 p u = − r + cos φ , u = (−1)(i+1) sin φ , u =0 as r →∞. ri 3k i 2 i φi 6k i θ i This can then be converted to stresses at infinity, using (2.25), (2.26), and (2.27). Since stresses are invariant with respect to a shift of coordinates its therefore no surprise that we again find the stresses at infinity can be given by

σriri = σφiφi = σθθ = −p as ri →∞ for i = 1, 2. (4.12)

4.2 Displacement of the medium

Following the work of Chen and Acrivos [2] we will use the Boussinesq-Papkovich stress function formulation, described in section 2.3.1, combined with a multipole expansion technique to ﬁnd a semi-analytical solution to the displacement and stress in the solid. CHAPTER 4. CAVITIES INTERACTING IN AN ELASTIC MEDIUM 145

4.2.1 Boussinesq-Papkovich stress function method

If we have an axisymmetric problem, symmetric about the z-axis in our case, then only two stress functions are needed and we can use the axisymmetric form of the Boussinesq-Papkovich stress function, given in (2.58). As we have a linear problem and two local coordinate systems it makes sense to modify the usual axisymmetric stress functions by splitting them up so that each stress function is singular about only one coordinate origin. Thus we can write the displacement throughout the medium as:

∞ 2u = 2u + ∇ξ + ∇(z1ψ1 + z2ψ2) − 4(1 − ν)(ψ1 + ψ2)ez (4.13)

∞ where u satisﬁes the displacement boundary conditions at inﬁnity and ξ and ψi tend to zero as r → ∞ and satisfy Laplace’s equation in each spherical coordinate system, symmetric with respect to θ, so that

∞ n+3 n+3 1 R1 2 R2 ξ = An n+1 Pn(cos φ1)+ An n+1 Pn(cos φ2) , (4.14) n=0 r1 r2 and ∞ n+2 i Ri ψi = Cn n+1 Pn(cos φi) for i = 1, 2, (4.15) n=0 ri 1 2 1 2 for unknown coeﬃcients An, An,Cn and Cn where n ≥ 0. The assumed form of ψ1 and ψ2 in (4.15) guarantees uniqueness of the solution. The form of this solution can be compared to the form of solution we would ﬁnd in elastostatics when using the method of images. If this were an elastostatics problem we could write down a solution involving images, images of images and images of images of images etc. Each image (of whatever order) would be inside either cavity 1 or 2. We would then let ψ1 be the sum of all the image terms within cavity 1 and similarly ψ2 for cavity 2. As we are doing elasticity rather than elastostatics the analogy is imperfect but is useful to explain why

ψ1 and ψ2 have their relevant forms. When similar forms of ψ1 and ψ2 are used later the same decomposition holds. The components of (4.13) can be expressed for i = 1, 2, about the origin of cavity i, as

∞ ∂ 2(uri − uri )= (ξ + ziψi + z3−iψ3−i) − 4(1 − ν)(ψi + ψ3−i) cos φi, ∂ri ∞ 1 ∂ 2(uφi − uφi )= (ξ + ziψi + z3−iψ3−i) + 4(1 − ν)(ψi + ψ3−i)sin φi, ri ∂φi

uθ ≡ 0. (4.16) CHAPTER 4. CAVITIES INTERACTING IN AN ELASTIC MEDIUM 146 where

p (−1)i+1 p u∞ = − r + cos φ , u∞ = (−1)i+1 sin φ , u = 0. ri 3k i 2 i φi 6k i θ 4.2.2 Expansion of the displacement about cavity i

Knowing the form of the solution in the medium, we then need to apply the boundary conditions on the surface of each cavity and use these conditions to ﬁnd the unknown 1 2 1 2 coeﬃcients An, An,Cn and Cn. To do this we need to write the solution, (4.13), in terms of the local coordinate system of the cavity upon which we wish to apply boundary conditions. So we can pose a solution in terms of local coordinates of each cavity in the form

∞ 2(u − u )= ∇ξ + ∇(z1ψ1 + z2ψ2) − 4(1 − ν)(ψ1 + ψ2)ezi , i = ∇(γ + ziβ) − 4(1 − ν)βezi .

Here ∂ 1 ∂ ∇ = , , 0 , ∂r r ∂φ i i i since the problem is axisymmetric,

ezi = cos φieri − sin φieφi ,

i+1 and, through z3−i = zi + (−1) ,

i i+1 γ = ξ + (−1) ψ3−i, β = ψ1 + ψ2.

i We then need to expand γ and β fully in terms of ri and φi. We can do this using the addition theorem detailed in (2.67), so for |r3−i| < 1,

∞ Pn(cos φi) i m n+1 = δmr3−iPm(cos φ3−i), ri m =0 where n n+m i (−1) n if i = 1, δm = (−1)m n+m if i = 2.  n Since we are using this addition theorem to enforce boundary conditions on the sphere surfaces, R1 + R2 < 1 we ﬁnd |r3−i| < 1 at all points where the addition theorem is CHAPTER 4. CAVITIES INTERACTING IN AN ELASTIC MEDIUM 147 required. With this we can write γi and β as

∞ n+3 ∞ i i Ri i m γ = An n+1 Pn(cos φi)+ Bmri Pm(cos φi), (4.17) n=0 ri m=0

∞ n+2 ∞ i Ri i m β = Cn n+1 Pn(cos φi)+ Dmri Pm(cos φi), (4.18) n ri m =0 =0 where in order to simplify the form of the displacement slightly we have deﬁned

∞ i 3−i i+1 3−i n+2 3−i Bm = (R3−iAn + (−1) Cn )R3−i δm , (4.19) n=0 ∞ i 3−i n+2 3−i Dm = Cn R3−i δm . (4.20) n=0 Thus in this notation the displacement is simpliﬁed to:

∞ i 2(u − u )= ∇(γ + riβ cos φi) − 4β(1 − ν)(cos φieri − sin φieφi ). (4.21)

Radial displacement about cavity i

Analysing only the radial displacement, by comparison with the solution process out- lined in section 2.3.1, our solution can be written as

∂γi ∂β 2(u − u∞)= + cos φ r + (4ν − 3)β . (4.22) ri ri ∂r i i ∂r i i Diﬀerentiating γ and β, we have

∞ ∞ ∂γi Rn+3 = −Ai (n + 1) i P + Bi mrm−1P , ∂r n n+2 n m i m i n ri m =0 =0 ∞ n+2 ∞ ∂β i Ri i m−1 = −Cn(n + 1) n+2 Pn + Dmmri Pm, ∂ri n=0 ri m=0 in which the argument of the Legendre polynomials has been dropped such that Pk = Pk(cos φi). Therefore we can write the radial displacement as

∞ n+3 ∞ ∞ i Ri i m−1 2(uri − uri )= −An(n + 1) n+2 Pn + Bmmri Pm n ri m =0 =0 ∞ n+2 ∞ i Ri i m + cos φi −Cn(n + 4 − 4ν) n+1 Pn + Dm(m + 4ν − 3)ri Pm . m=0 ri m=0 CHAPTER 4. CAVITIES INTERACTING IN AN ELASTIC MEDIUM 148

Using the recurrence relation for Legendre polynomials, from Appendix A,

(2n + 1) cos φiPn = (n + 1)Pn+1 + nPn−1, the radial displacement becomes

∞ n+3 ∞ ∞ i Ri i m−1 2(uri − uri )= −An(n + 1) n+2 Pn + Bmmri Pm n=0 ri m=0 ∞ Rn+2 (n + 1)P + nP − Ci (n + 4 − 4ν) i n+1 n−1 n rn+1 2n + 1 n=0 i ∞ (m + 1)P + mP + Di (m + 4ν − 3)rm m+1 m−1 . (4.23) m i 2m + 1 m=0 Polar displacement about cavity i

We can similarly ﬁnd displacements uφi for i = 1, 2. Using (2.59) in section 2.3.1, we can write

i ∞ 1 ∂γ ∂β 2(uφi − uφi )= + cos φi + β(3 − 4ν)sin φi. (4.24) ri ∂φi ∂φi

Differentiating Legendre polynomials with respect to φi using the chain rule we find that ∂Pn (1) = Pn , ∂φi (1) where Pn denotes the first associated Legendre polynomial with the argument cos φi i and thus differentiating γ and β with respect to φi yields

i ∞ n+3 ∞ 1 ∂γ i Ri (1) i m−1 (1) = An n+2 Pn + Bmri Pm , ri ∂φi n=0 ri m=0 ∞ n+2 ∞ ∂β i Ri (1) i m (1) = Cn n+1 Pn + Dmri Pm . ∂φi n=0 ri m=0 This allows us to write the polar displacement as

∞ n+3 ∞ ∞ i Ri (1) i m−1 (1) 2(uφi − uφi )= An n+2 Pn + Bmri Pm n=0 ri m=0 ∞ n+2 ∞ i Ri (1) i m (1) + cos φi Cn n+1 Pn + Dmri Pm n ri m =0 =0 ∞ n+2 ∞ i Ri i m − sin φi(4ν − 3) Cn n+1 Pn + Dmri Pm . (4.25) n=0 ri m=0 CHAPTER 4. CAVITIES INTERACTING IN AN ELASTIC MEDIUM 149

However we would like to express this in terms of just the associated Legendre poly- (1) (1) nomials Pk rather than a mixture of associated Legendre polynomials Pk and composite expressions cos φiPk. We can do this by diﬀerentiating the associated Legendre recurrence relation, allowing us to determine

((k + 1)P (1) + kP (1) ) cos φ P (1) = sin φP + k+1 k−1 , i k k 2k + 1 and then use the relation

(1) (1) − sin φ(2k + 1)Pk = Pk+1 − Pk−1.

Substituting these into our deﬁnition of uφi in (4.25) we ﬁnd

(1) (1) ∞ Rn+3 ∞ Rn+2 (nP + (n + 1)P ) 2(u − u∞)= Ai i P (1) + Ci i n+1 n−1 φi φi n n+2 n n n+1 2n + 1 n=0 ri n=0 ri ∞ ∞ (mP (1) + (m + 1)P (1) ) + Bi rm−1P (1) + Di rm m+1 m−1 m i m m i 2m + 1 m m =0 =0 (1) (1) (1) (1) ∞ Rn+2 (P − P ) ∞ (P − P ) + (4ν − 3) Ci i n+1 n−1 + Di rm m+1 m−1 , n n+1 2n + 1 m 1 2m + 1 n=0 ri m=0 which can be rearranged to give

∞ n+3 ∞ ∞ i Ri (1) i m−1 (1) 2(uφi − uφi )= An n+2 Pn + Bmri Pm n=0 ri m=0 (1) (1) ∞ Rn+2 ((n + 4ν − 3)P + (n + 4 − 4ν)P ) + Ci i n+1 n−1 n n+1 2n + 1 n=0 ri ∞ ((m + 4ν − 3)P (1) + (m + 4 − 4ν)P (1) ) + Di rm m+1 m−1 . (4.26) m i 2m + 1 m=0 4.3 Stresses upon cavity i

Now that we have the displacements about cavity i we can go about finding the stresses near the cavity. We are interested in finding the radial stress σriri and shear stress σriφi in order to fulfill the boundary conditions on the cavity surface.

σ 4.3.1 The radial stress riri

The radial stress σriri is given by equation (2.60) in section 2.3.1, CHAPTER 4. CAVITIES INTERACTING IN AN ELASTIC MEDIUM 150

2 i 2 ∞ ∂ γ ∂ β ∂β 2ν ∂β σriri − σ = + cos φi ri − 2(1 − ν) + tan φi . (4.27) riri ∂r2 ∂r2 ∂r r ∂φ i i i i i Now substituting for γi and β, from equations (4.17) and (4.18), yields,

∞ n+3 ∞ ∞ i Ri i m−2 σriri −σriri = An n+3 (n + 1)(n + 2)Pn + Bmm(m − 1)ri Pm n=0 ri m=0 ∞ n+2 i Ri (n + 1) 2 + C (n + 5n + 4 − 2ν)Pn+1 + n(n + 4 − 4ν)Pn−1 n rn+2 (2n + 1) n=0 i ∞ m + Di rm−1 (m + 1)(m + 4ν − 3)P + (m2 − 3m − 2ν)P . m i (2m + 1) m+1 m−1 m=0 (4.28)

σ 4.3.2 The radial shear stress riφi

The radial shear stress is governed by the equation

2 i i 2 ∞ 1 ∂ γ 1 ∂γ ∂ β ∂β (1 − ν) ∂β σriφi −σriφi = − 2 +cos φi +(1−2ν)sin φi −2 cos φi , ri ∂ri∂φi ri ∂φi ∂ri∂φi ∂ri ri ∂φi from section 2.3.1. Substituting into this our deﬁnitions of γ and β yields,

∞ n+3 ∞ ∞ i Ri (1) i (1) m−2 σriφi − σriφi = − An n+3 (n + 2)Pn + Bm(m − 1)Pm ri n=0 ri m=0 ∞ Rn+2 n2 + 2n + 2ν − 1 (n + 1)(n + 4 − 4ν) − Ci i P (1) + P (1) n rn+2 (2n + 1) n+1 (2n + 1) n−1 n=0 i ∞ m(m + 4ν − 3) (m2 + 2ν − 2) + Di rm−1 P (1) + P (1) . (4.29) m i (2m + 1) m+1 (2m + 1) m−1 m =0 4.3.3 Applying the boundary conditions

Having found the radial and radial shear stresses upon both cavities we can apply 1 1 2 2 the boundary conditions and so evaluate the unknown coeﬃcients An,Cn, An and Cn.

Looking ﬁrst at the radial stresses σriri for i = 1, 2, we have the boundary conditions (4.12) and (4.8). Applying these conditions to our solution for radial stress, (4.28), we CHAPTER 4. CAVITIES INTERACTING IN AN ELASTIC MEDIUM 151

ﬁnd:

∞ ∞ i i m−2 p = An(n + 1)(n + 2)Pn + Bmm(m − 1)Ri Pm n=0 m=0 ∞ (n + 1) + Ci (n2 + 5n + 4 − 2ν)P + n(n + 4 − 4ν)P n (2n + 1) n+1 n−1 n=0 ∞ m + Di Rm−1 (m + 1)(m + 4ν − 3)P + (m2 − 3m − 2ν)P . m i (2m + 1) m+1 m−1 m =0 (4.30)

The orthogonality of the Legendre polynomials, from Appendix A, allows this to be simpliﬁed. We can multiply (4.30) by Pk(cos φi) and integrate with respect to cos φi to yield,

i i k−2 pδk0 =Ak(k + 1)(k +2)+ Bkk(k − 1)Ri k k + 2 + Ci (k(k + 3) − 2ν)+ Ci (k + 1)(k + 5 − 4ν) k−1 (2k − 1) k+1 (2k + 3) k − 1 (k + 1) + Di Rk−2 k(k + 4ν − 4) + Di Rk ((k + 1)(k − 2) − 2ν). k−1 i (2k − 1) k+1 i (2k + 3) (4.31)

i i Substituting back for Bm and Dm, (4.20), we get a solution purely in terms of the 1 2 1 2 unknown coeﬃcients Ak, Ak,Ck and Ck , which for k ≥ 0 is

∞ i 3−i n+3 k−2 3−i pδk0 =Ak(k + 1)(k +2)+ An R3−i Ri δk k(k − 1) n=0 k k + 2 + Ci (k(k + 3) − 2ν)+ Ci (k + 1)(k + 5 − 4ν) k−1 (2k − 1) k+1 (2k + 3) ∞ (k + 1) + C3−iRn+2Rk−2 δ3−i R2 ((k + 1)(k − 2) − 2ν) n 3−i i k+1 i (2k + 3) n=0 i+1 3−i k − 1 3−i + (−1) δk k(k − 1) + k(k + 4ν − 4)δk−1 . (4.32) (2k − 1)

The ﬁnal set of boundary conditions act upon the radial shear stress, σriφi , so that

∞ σriφi =0 for i = 1, 2,

σriφi =0 on ri = Ri for i = 1, 2. CHAPTER 4. CAVITIES INTERACTING IN AN ELASTIC MEDIUM 152

Substituting into our solutions for the radial shear stress, (4.29), we have,

∞ ∞ i (1) i (1) m−2 0= An(n + 2)Pn − Bm(m − 1)Pm Ri n=0 m=0 ∞ n2 + 2n + 2ν − 1 (n + 1)(n + 4 − 4ν) + Ci P (1) + P (1) n (2n + 1) n+1 (2n + 1) n−1 n=0 ∞ m(m + 4ν − 3) (m2 + 2ν − 2) − Di Rm−1 P (1) + P (1) . (4.33) m i (2m + 1) m+1 (2m + 1) m−1 m=0 We can use the orthogonality of the associated Legendre polynomials in Appendix A, (1) by multiplying (4.33) by Pk (cos φi) and integrating with respect to cos φi, before sub- i i stituting Bm and Dm for their respective functions. This yields the final two equations, 1 1 2 2 for i = 1, 2, needed to find the unknown coefficients An,Cn, An and Cn, which for k ≥ 1 are,

∞ i 3−i n+3 k−2 3−i 0= Ak(k + 2) − An R3−i Ri δk (k − 1) n =0 k2 + 2ν − 2 (k + 2)(k + 5 − 4ν) + Ci + Ci k−1 (2k − 1) k+1 (2k + 3) ∞ ((k + 1)2 + 2ν − 2) − C3−iRn+2Rk−2 δ3−i R2 + (−1)i+1δ3−i(k − 1) n 3−i i k+1 i (2k + 3) k n =0 3−i (k − 1)(k + 4ν − 4) + δk−1 . (4.34) (2k − 1)

We can simplify (4.34) by noticing that it bears some resemblance to (4.32), and therefore if we multiply (4.34) by k and add this to (4.32) we can eliminate the sum over 3−i An . Doing this leaves us with, for k ≥ 1, the equation:

(2k + 1) 0=Ai (2k +1)+ Ci k + Ci (k + 5 − 4ν) k k−1 k+1 (2k + 3) ∞ (k2 + k + ν + 2kν + 1) − 2 C3−iRn+2Rkδ3−i . (4.35) n 3−i i k+1 (k + 2)(2k + 3) n =0 Finally we can eliminate the k = 1 equation from (4.35), by noting that for k = 1 our equations (4.32) and (4.35) become,

∞ 3 2 0 = 3Ai + Ci(2 − 1ν)+ Ci (6 − 4ν) − C3−iRn+2R δ3−i(1 + ν), 1 0 2 5 5 n 3−i i 2 n=0 CHAPTER 4. CAVITIES INTERACTING IN AN ELASTIC MEDIUM 153

∞ 3 2 0 = 3Ai + Ci + Ci (6 − 4ν) − C3−iRn+2R δ3−i(1 + ν), 1 0 2 5 5 n 3−i i 2 n=0 i implying C0 ≡ 0 for i = 1, 2. Therefore we now only need to consider (4.35) for k ≥ 2. Since each inﬁnite sum in (4.32) and (4.35) is convergent we can truncate the series so i i i i that we are solving for A0 to AN and C1 to CN−1. We can then write the equations in the form of a matrix equation

1 2 1 2 1 a1 a1 c1 c1 A F a1 a2 c1 c2 A2 F  2 2 2 2    =   , (4.36) 1 2 1 2 C1  a3 a3 c3 c3     0     2    a1 a2 c1 c2  C   0   4 4 4 4            i i i i i T i in which the column vectors A and C are given by A = (A0, ..., AN ) and C = i i T T (C1, ..., CN−1) , and the forcing term F = (p, 0,..., 0) is of length N + 1. i i For i, j = 1, 2 and m = 0, ..., N the elements of the individual matrices aj and cj are then given by: i [aj]mn = (m + 1)(m + 2)δmn for i = j and n = 0, ..., N,

i n+3 m−2 3−j [aj]mn = Ri Rj δm m(m − 1) for i = j and n = 0, ..., N,

i m [cj]mn = δm(n+1) (2m−1) (m(m + 3) − 2ν)

m+2 +δm(n−1) (2m+3) (m + 1)(m + 5 − 4ν) for i = j and n = 1, ..., N − 1,

i n+2 m−2 j+1 3−j [cj]mn = Ri Rj (−1) δm m(m − 1) 3−j 2 (m+1) +δm+1Rj (2m+3) ((m + 1)(m − 2) − 2ν)

m−1 3−j + (2m−1) m(m + 4ν − 4)δm−1 for i = j and n = 1, ..., N − 1, where δnm is the usual Kronecker delta function. For i = 1, 2, j = 3, 4 and m = 2, ..., N we have CHAPTER 4. CAVITIES INTERACTING IN AN ELASTIC MEDIUM 154

i [aj]mn = (2m + 1)δmn for i +2= j and n = 0, ..., N,

i [aj]mn =0 for i + 2 = j and n = 0, ..., N,

i [cj]mn = δm(n+1)m (2m + 1) +δ (m + 5 − 4ν) for i +2= j and n = 1, ..., N − 2, m(n−1) (2m + 3) (m2 + m + ν + 2mν + 1) [ci ] = −2 Rn+2 Rm δ3−i j mn (m + 2)(2m + 3) i 3−i m(n+1) for i + 2 = j and n = 1, ..., N − 2. In this matrix form (4.36), the equations can be solved to yield the unknown coeﬃcients 1 2 1 2 An, An,Cn and Cn, which can be substituted into (4.21) to give the displacement and stress, throughout the medium.

4.4 Results

4.4.1 Agreement with single cavity solution

As R1 → 0 we expect to recover the isolated cavity solution from section 2.4.3. From

(4.14) and (4.15), it is obvious that as R1 → 0 the matrix equations (4.32) and (4.34) decouple, for i = 2, so that for shell 2 we recover:

k pδ =A2(k + 1)(k +2)+ C2 (k(k + 3) − 2ν) k0 k k−1 (2k − 1) k + 2 + C2 (k + 1)(k + 5 − 4ν), for k ≥ 0 (4.37) k+1 (2k + 3) k2 + 2ν − 2 (k + 2)(k + 5 − 4ν) 0=A2(k +2)+ C2 + C2 , for k ≥ 1. (4.38) k k−1 (2k − 1) k+1 (2k + 3) p These have known solution A2 = where all other coeﬃcients are zero, thus yielding the 0 2 isolated cavity solution and validating the 2 cavities approach, so that the displacement throughout the medium can be expressed in the local coordinates of shell 2 as:

3 ∞ p R2 2(ur2 − ur2 )= − 2 . (4.39) 2 r2

4.4.2 Equal sized cavities

If the cavities are equal in size with R1 = R2 ≡ R, then the system of equations given by (4.32) and (4.35) simplifies significantly. For equal sized cavities we find that 1 n 2 1 n+1 2 An = (−1) An = An and Cn = (−1) Cn = Cn, thus our four sets of infinite equations CHAPTER 4. CAVITIES INTERACTING IN AN ELASTIC MEDIUM 155 collapse to only two sets, given by, for k ≥ 0,

∞ n + k pδ =A (k + 1)(k +2)+ (−1)n+kA Rn+k+1 k(k − 1) k0 k n n n=0 k k + 2 + C (k(k + 3) − 2ν)+ C (k + 1)(k + 5 − 4ν) k−1 (2k − 1) k+1 (2k + 3) ∞ n + k + 1 (k + 1) + (−1)n+kC Rn+k R2 ((k + 1)(k − 2) − 2ν) n n (2k + 3) n=0 n + k k − 1 n + k − 1 − k(k − 1) + k(k + 4ν − 4) , (4.40) n (2k − 1) n and for k ≥ 2,

(2k + 1) 0=A (2k +1)+ C k + C (k + 5 − 4ν) k k−1 k+1 (2k + 3) ∞ n + k + 1 (k2 + k + ν + 2kν + 1) − 2 (−1)n+kC Rn+k+2 . (4.41) n n (k + 2)(2k + 3) n =0 These equations can be represented by a matrix equation as

a c A F 1 1 = , (4.42) a2 c2 C 0 where again ai and ci are matrices of the multipliers of the coeﬃcients An and Cn in (4.40) and (4.41) respectively. Since (4.40) and (4.41) are both linear in p this can be scaled out of the equations so that the vector F is simply (1, 0, ..., 0)T , leaving the coeﬃcients An and Cn invariant to changes in pressure. However it must be remembered that this changes the displacement and stress equations by a factor of p, so that uri in (4.16) becomes for cavity i;

∞ ∞ ∂ n+3 Pn(cos φ1) n Pn(cos φ2) 2(uri − uri )= p AnR n+1 + (−1) n+1 ∂ri r r n=0 1 2 ∞ cos φ cos φ + C Rn+2 1 P (cos φ ) + (−1)n+1 2 P (cos φ ) n rn n 1 rn n 2 n=0 1 2 ∞ n+2 Pn(cos φ1) n+1 Pn(cos φ2) −4(1 − ν) cos φi CnR n+1 + (−1) n+1 , n r1 r2 =0 (4.43) CHAPTER 4. CAVITIES INTERACTING IN AN ELASTIC MEDIUM 156

p (−1)i+1 where u∞ = − r + cos φ and the polar displacement u , about cavity ri 3k 1 2 i φi i is then given by

∞ ∞ 1 ∂ n+3 Pn(cos φ1) n Pn(cos φ2) 2(uφi − uφi )= p AnR n+1 + (−1) n+1 ri ∂φi r r n=0 1 2 ∞ cos φ cos φ + C Rn+2 1 P (cos φ ) + (−1)n+1 2 P (cos φ ) n rn n 1 rn n 2 n=0 1 2 ∞ n+2 Pn(cos φ1) n+1 Pn(cos φ2) +4(1−ν)sin φi CnR n+1 + (−1) n+1 , (4.44) n r1 r2 =0 p where we have remembered u∞ = (−1)i+1 sin φ . φi 6k i

Interaction eﬀects in a two-cavity system

We can compare the displacement of cavities in a two-cavity system to that of a single cavity experiencing hydrostatic pressure. Since stresses within the medium are, unlike displacements, unaﬀected by shifts in coordinates we will consider how the hoop stress,

σφiφi , on a cavity boundary is changed by the introduction of a second cavity into the medium. We naturally expect that for cavities suﬃciently far apart, or suﬃciently small cavities, the hoop stress on the cavity boundary will tend to that of an isolated cavity. The hoop stress within the medium is given by (2.27) as

2 u ∂u 1 − ν ∂u ν σ = σ∞ + ri + ν ri + φi + u cot φ , (4.45) φiφi φiφi 1 − 2ν r ∂r r ∂φ r φi i i i i i i where for equal sized shells uri and uφi are given as in (4.43) and (4.44). For an isolated cavity, centered at origin i, the displacements are given as in 2.5.3 by

3 ∞ p Ri ∞ uri = uri − 2 , uφi = uφi , 4 ri yields 3 ∞ p Ri σφiφi = σφiφi − 3 , (4.46) 2 ri which is constant for any host medium at fixed radius. We should remember when considering how different parameters affect interaction between the cavities that we are modelling the interaction via linear elasticity and therefore we are considering only small deformations and infinitesimal strains. CHAPTER 4. CAVITIES INTERACTING IN AN ELASTIC MEDIUM 157

Eﬀects of changing radius

If we start by fixing the host medium to be nearly incompressible with ν = 0.49985 and the pressure to be p = 0.1, we can assess how altering the radius of the cavities affects the hoop stress on the cavity surfaces, recalling both cavities are of equal radius. For these parameters we would expect an isolated cavity to have a hoop stress of −0.15 on the cavity boundary, an example of which can be seen in figure 4.2 for small cavities of radius 0.01 and 0.1. As the cavities increase in radius the hoop stress starts to oscillate about the isolated cavity limit, with higher amplitude oscillations for larger cavities but with two zonal circles on which the stress is unchanged and whose positions are almost independent of cavity radius. The magnitude of the hoop stress is at a maximum at the nearest point between the two cavities, at φ1 = π (and through symmetry at φ2 = 0). If we look at the percentage change between the isolated cavity limit and the maximum absolute hoop stress, as in figure 4.3, we can see that the isolated cavity limit is valid for surprisingly large cavities. The maximum change between the isolated cavity hoop stress and the hoop stress created by 2 cavities is less than 1% even for cavities of radius 0.15, and rises to 5% only at r = 0.24, though we then see a sharp decline in the validity of the isolated cavity hoop stress, so that there is a change of 100% by r = 0.46. For these larger cavities it is therefore important to take into account interaction effects in order to accurately model the stress distribution throughout the medium.

σφ1φ1

0.15

0.20

0.25

φ Π Π 3 Π 2 Π 1 2 2 Figure 4.2: Hoop stress on surface of cavity 1 for a nearly incompressible medium ν = 0.49985 under hydrostatic pressure p = 0.1, for cavities of equal radius r such that: r = 0.01 (dotted, blue), r = 0.1 (small dashes, pink), r = 0.2 (large dashes, yellow), r = 0.4 (dot-dashed, green), r = 0.45 (solid, black). CHAPTER 4. CAVITIES INTERACTING IN AN ELASTIC MEDIUM 158

max percentage change 120

100

r 0.1 0.2 0.3 0.4 0.5

Figure 4.3: Percentage change between maximum absolute hoop stress (at φ1 = π) on surface of cavity 1 and isolated cavity hoop stress for a nearly incompressible medium ν = 0.49985 under hydrostatic pressure p = 0.1, for cavities of equal radius r.

Eﬀects of changing pressure

If we instead ﬁx the radius of the cavities we can alter the pressure the medium experiences. Since pressure is a linear parameter in the displacement equations (4.43) and (4.44), we know it will have a linear eﬀect upon the displacement of the cavity. Since we wish to remain within the linear elastic region of deformation we must remember to keep pressures low in comparison to the Young’s modulus of the material.

Eﬀects of changing host medium

Finally we can change the host medium in which the cavity resides by varying the Poisson ratio to be more or less compressible. As we can see in figure 4.4, the more compressible the medium the less the presence of a second cavity affects the hoop stress on the boundary of cavity 1, as the host medium effectively ‘takes up’ more of the applied force. However this is a quantitative change, rather than qualitative, and thus we still see the dramatic increase in the hoop stress on the boundary of cavity 1 as both cavities increase in radius. CHAPTER 4. CAVITIES INTERACTING IN AN ELASTIC MEDIUM 159

max percentage change

r 0.1 0.2 0.3 0.4 0.5

Figure 4.4: Percentage change between maximum absolute hoop stress (at φ1 = π) on the surface of cavity 1 and the isolated cavity hoop stress for a medium under hydrostatic pressure p = 0.1, for cavities of equal radius r, where the medium has Poisson ratio: ν = 0.49985 (blue, solid); ν = 0.4 (pink, dashed); ν = 0.25 (yellow, dotted); ν = 0.1 (green, dot-dashed).

4.4.3 Cavities with diﬀering radii

If we allow the two cavities, 1 and 2, to have different radii R1 and R2, then we need 1 2 1 2 to solve the full matrix equation in (4.36) to find the coefficients An, An,Cn and Cn for n = 0, ...N. We still need to choose an N at which to truncate the series of coefficients, though even for large cavities this can occur at a relatively low N as the series converges quite quickly. Solving for all four sets of coefficients allows us to see how cavities of very different sizes affect each other. For example:

• Is a large cavity aﬀected at all by a small cavity in close proximity?

• If it is, is there a certain radius for the large cavity at which interaction could be said to occur?

• Does the Poisson ratio of the host medium affect cavities of different sizes in the same way that it affects cavities of the same size?

Cavity interacting with a neighbour of varying radius

We will first consider how, in a nearly incompressible host medium (ν = 0.49985) a cavity at position 1 is affected by a neighbouring cavity, at position 2, of a different CHAPTER 4. CAVITIES INTERACTING IN AN ELASTIC MEDIUM 160 radius. We can compare the cavity at position 1 to an isolated cavity in a medium under hydrostatic pressure, as we did when analysing cavities of equal radius.

Changing radius

If we let the medium be constant, and set p = 0.1, but allow the radii of the cavities to change independently of each other we can find for every cavity, with radius R1, a corresponding radius, R2, of a near neighbour cavity which will significantly affect the hoop stress of the first cavity. We can do this by fixing the radius R1 and adjusting

R2 whilst mapping the maximum change in the hoop stress over the cavity surface, at r1 = R1, in comparison to the hoop stress of an isolated cavity of radius R1. Doing this we see in ﬁgure 4.5 that as the radius of cavity 1 increases, the hoop stress on the surface of this cavity deviates from the isolated cavity limit for smaller near neighbour cavities. max percentage change

R 0.1 0.2 0.3 0.4 0.5 2

Figure 4.5: Absolute percentage change of the hoop stress on the surface of cavity 1 at radius R1, from when cavity 1 has a near neighbour cavity of radius R2 (varying with R2 axis) to when cavity 1 is isolated. The medium is nearly incompressible (ν = 0.49985) and experiences pressure p = 0.1, various radii of R1 are given by: R1 = 0.01 (blue, dotted), R1 = 0.1 (pink, small dashes), R1 = 0.2 (yellow, medium dashes), R1 = 0.3 (green, dot-dashed), R1 = 0.4 (black, large dashes), R1 = 0.45 (brown, solid).

Changing the host medium

From our work with cavities of equal size we also expect the Poisson ratio of the host to affect the deformation of the cavities and so alter the radii at which interaction occurs. As for equal sized shells we see, in figure 4.6, that as the host Poisson ratio decreases the radii at which cavities interact increases, as is demonstrated by the decrease in the difference between the hoop stress on the boundary of cavity 1 and that of an isolated CHAPTER 4. CAVITIES INTERACTING IN AN ELASTIC MEDIUM 161 cavity.

max percentage change

R 0.1 0.2 0.3 0.4 0.5 2

Figure 4.6: Absolute percentage change of the hoop stress on the surface of cavity 1 at radius R1 = 0.4, from when cavity 1 has a near neighbour cavity of radius R2 to when cavity 1 is isolated. The medium experiences pressure p = 0.1 and is of various Poisson ratios given by: ν = 0.49985 (blue, dotted), ν = 0.4 (pink, dashes), ν = 0.25 (yellow, dot-dashed), ν = 0.1 (green, solid). Chapter 5

Shells interacting in an elastic medium

Extending the two-cavity problem we get to the final linearly elastic problem we will look at: that of two shells in an elastic medium. The two shells will be positioned as in section 4 and will experience the equivalent boundary conditions in order to make comparisons between the two situations simpler. Thus, as in figure 5.1 we will have have two spherical hollow shells (1 and 2) under hydrostatic pressurep ˆ a distance Lˆ Lˆ apart, with centers aligned vertically with the globalz ˆ axis and positioned atz ˆ = 2 Lˆ andz ˆ = − respectively. Shells 1 and 2, respectively, have local coordinate systems 2 (ˆr1, φˆ1, θˆ) and (ˆr2, φˆ2, θˆ), shell thicknesses hˆ1 and hˆ2 and outer shell radii Rˆ1 and Rˆ2 so that Rˆ1 + Rˆ2 < Lˆ. Shell i has material parameters î and νi, the shear modulus and

Poisson ratio respectively, for i = 1, 2, whilst the medium has shear modulus ˆ0 and

Poisson ratio ν0. As before we will nondimensionalise distances with respect to Lˆ and stresses with respect the shear modulus of the host medium, ˆ0. We will drop the hat notation for nondimensional parameters.

162 CHAPTER 5. SHELLS INTERACTING IN AN ELASTIC MEDIUM 163

z1 φ1 r1

x1 z y 1 x φ2 z2 r2

x2 θ

Figure 5.1: Two spherical shells in an inﬁnite elastic host medium.

As the boundary conditions upon the shells are the equivalent of those upon the cavities in section 4 we will again have zero traction upon the inner surface of the shells so that for i = 1, 2

σriri = σriφi =0 on ri = Ri − hi, and the pressure applied at inﬁnity will be,

∞ ∞ ∞ σriri = σφiφi = σθθ = −p, for i = 1, 2.

As in section 2.5.3, the displacements at inﬁnity are valid only within the host medium, so that

(1 − 2ν ) cos φ u∞ = −p 0 r + (−1)i+1 i , ri 2(1 + ν ) i 2 0 ∞ i+1 (1 − 2ν0) uφi = (−1) p sin φi, for i = 1, 2, 4(1 + ν0) ∞ uθ = 0.

Throughout this chapter will use m superscripts or 0 subscripts to denote material constants and parameters belonging to the host medium and s superscripts or i subscripts to denote parameters of the shells for shells i = 1 and i = 2. Since we now have shells CHAPTER 5. SHELLS INTERACTING IN AN ELASTIC MEDIUM 164 which may be constructed from a diﬀerent material to the surrounding host medium we also need continuity conditions upon the displacement and stresses on the shell-matrix interface. Since the problem is again invariant with respect to θ we can assume uθ ≡ 0, so the only continuity conditions we will need will be

uri , uφi ,σriri ,σriφi continuous on ri = Ri.

5.1 Displacement and stresses throughout the composite

We can solve the equation of equilibrium via the Boussinesq-Papkovich stress functions from section 2.3.1. As in section 4 we can superimpose the solutions for the displacement in each coordinate system so that it is given by

∞ 2(u − u )= ∇(ξ + z1ψ1 + z2ψ2) − 4(1 − ν)(ψ1 + ψ2)ez, where ψ1 and ψ2 satisfy Laplace’s equation in the local coordinates of shells 1 and 2 respectively and ξ is a linear combination of functions which satisfy Laplace’s equation in each of the two local coordinate systems. However, as we are dealing with different materials, valid in different regions of space, it will be convenient to define the material constants and stress functions for each section individually.

In the host

The material constants of the host are given by = 1 (due to nondimensionalising) and ν = ν0. Since the displacement in the host must decay with r1 and r2 we can use (A.10) in Appendix A.1 to represent the stress functions as

∞ n+3 n+3 m 1 R1 2 R2 ξ = An n+1 Pn(cos φ1)+ An n+1 Pn(cos φ2) , n=0 r1 r2 ∞ n+2 m i Ri ψi = Cn n+1 Pn(cos φi), for i = 1, 2. (5.1) n ri =0 It should be noticed at this point that the stress functions within the host are the same as those used in section 4 to describe the host medium and thus we expect solutions of the same form.

In shell i

∞ The material constants of shell i are given by = i and ν = νi, and u = 0. As the shell is a bounded region which does not include the origin of either local coordinate system we need all parts of the axisymmetric solution to Laplace’s equation, as in CHAPTER 5. SHELLS INTERACTING IN AN ELASTIC MEDIUM 165

(A.10) in Appendix A.1. Therefore the form of the solution will mirror that found in (2.100) from section 2.5.3. We can therefore write for i = 1, 2:

∞ n+3 n s i Ri i ri ξ = an n+1 + bn n−2 Pn(cos φi), n=0 ri Ri ∞ n+2 n s i Ri i ri ψi = cn n+1 + dn n−1 Pn(cos φi), n ri Ri =0 s ψ3−i = 0 (5.2)

5.1.1 Stresses and displacements within the host

The stresses and displacements within the host medium are analogous to those in section 4 and therefore with slight notational amendment we can import them from section 4. In the notation of the host medium we can write the Boussinesq-Papkovich solution as

m ∞ m m m 2(u − u )= ∇(γi + ziβ ) − 4(1 − ν0)β ezi , (5.3) where

∞ n+3 ∞ m m i+1 m i Ri i m γi = ξ + (−1) ψ3−i = An n+1 Pn(cos φi)+ Bmri Pm(cos φi), (5.4) n ri m =0 =0 ∞ n+2 ∞ m m m i Ri i m β = ψ1 + ψ2 = Cn n+1 Pn(cos φi)+ Dmri Pm(cos φi), (5.5) n=0 ri m=0 given

∞ i 3−i i+1 3−i n+2 3−i Bm = (R3−iAn + (−1) Cn )R3−i δm , (5.6) n=0 ∞ i 3−i n+2 3−i Dm = Cn R3−i δm . (5.7) n =0 and n n+m i (−1) n if i = 1, δm = (−1)m n+m if i = 2.  n  CHAPTER 5. SHELLS INTERACTING IN AN ELASTIC MEDIUM 166

As such we will then have the displacements

∞ n+3 m ∞ i Ri i n−1 2(uri − uri )= −An(n + 1) n+2 + Bnnri Pn n=0 ri n+2 i Ri i n (n + 1)Pn+1 + nPn−1 + −C (n + 4 − 4ν0) + D (n + 4ν0 − 3)r , n rn+1 n i 2n + 1 i (5.8)

∞ n+3 m ∞ i Ri i n−1 (1) 2(uφi − uφi )= An n+2 + Bnri Pn n=0 ri n+2 (1) (1) R ((n + 4ν0 − 3)P + (n + 4 − 4ν0)P ) + Ci i + Di rn n+1 n−1 , (5.9) n rn+1 n i 2n + 1 i and stresses

∞ n+3 m ∞ i Ri i n−2 σriri − σriri = An n+3 (n + 1)(n + 2)Pn + Bnn(n − 1)ri Pn n=0 ri n+2 i Ri (n + 1) 2 + C (n + 5n + 4 − 2ν0)Pn+1 + n(n + 4 − 4ν0)Pn−1 n rn+2 (2n + 1) i n + Di rn−1 (n + 1)(n + 4ν − 3)P + (n2 − 3n − 2ν )P , n i (2n + 1) 0 n+1 0 n−1 (5.10)

∞ n+3 m ∞ i Ri (1) i (1) n−2 σriφi − σriφi = − An n+3 (n + 2)Pn − Bn(n − 1)Pn ri n ri =0 Rn+2 n2 + 2n + 2ν − 1 (n + 1)(n + 4 − 4ν ) + Ci i 0 P (1) + 0 P (1) n rn+2 (2n + 1) n+1 (2n + 1) n−1 i n(n + 4ν − 3) (n2 + 2ν − 2) − Di rn−1 0 P (1) + 0 P (1) , (5.11) n i (2n + 1) n+1 (2n + 1) n−1

(1) where Pn = Pn(cos φi) and Pm is an associated Legendre polynomial of order 1.

5.1.2 Stresses and displacements within shell i

The form of the Boussinesq-Papkovich stress functions within the shells mirrors that found within an isolated shell under axisymmetric displacement conditions at inﬁnity, as seen in section 2.5.3. As such we know we can write the displacements and stresses CHAPTER 5. SHELLS INTERACTING IN AN ELASTIC MEDIUM 167 in each shell i as

∞ n+3 i s i Ri bn n−1 2iuri = −an(n + 1) n+2 + n−2 nri Pn n=0 ri Ri n+2 n i Ri i ri (n + 1)Pn+1 + nPn−1 + −c (n + 4 − 4νi) + d (n + 4νi − 3) , n rn+1 n Rn−1 2n + 1 i i (5.12)

∞ n+3 n−1 s i Ri i ri (1) 2iuφi = an n+2 + bn n−2 Pn n=0 ri Ri n+2 n (1) (1) R r ((n + 4νi − 3)P + (n + 4 − 4νi)P ) + ci i + di i n+1 n−1 , (5.13) n rn+1 n Rn−1 2n + 1 i i

∞ n+3 n−2 s i Ri i ri σriri = an n+3 (n + 1)(n + 2)Pn + bn n−2 n(n − 1)Pn n=0 ri Ri n+2 i Ri (n + 1) 2 + c (n + 5n + 4 − 2νi)Pn+1 + n(n + 4 − 4νi)Pn−1 n rn+2 (2n + 1) i n−1 i ri n 2 + d (n + 1)(n + 4νi − 3)Pn+1 + (n − 3n − 2νi)Pn−1 , n Rn−1 (2n + 1) i (5.14)

∞ n+3 n−2 s i Ri (1) i ri (1) σriφi = − an n+3 (n + 2)Pn − bn n−2 (n − 1)Pn n ri Ri =0 Rn+2 n2 + 2n + 2ν − 1 (n + 1)(n + 4 − 4ν ) + ci i i P (1) + i P (1) n rn+2 (2n + 1) n+1 (2n + 1) n−1 i rn−1 n(n + 4ν − 3) (n2 + 2ν − 2) − di i i P (1) + i P (1) . (5.15) n Rn−1 (2n + 1) n+1 (2n + 1) n−1 i 5.2 Boundary conditions upon shell i

On each shell we have 6 boundary conditions; two deal with the stresses on the inner surface of the shell and 4 are continuity conditions on stresses and displacements on the shell-matrix interface. The stresses and displacements upon each inclusion are piecewise CHAPTER 5. SHELLS INTERACTING IN AN ELASTIC MEDIUM 168 functions given by

m f (ri, φi, ) ri ≥ Ri, f(ri, φi)= f s(r , φ ),R − h ≤ r ≤ R ,  i i i i i i  where f(ri, φi) can be any of σriri ,σriφi , uri or uφi . Within the shell the zero traction conditions upon the radial and radial shear stresses are therefore given by

s s σriri = σriφi =0 on ri = Ri − hi, and the conditions ensuring continuity in each piecewise function are given by

m s σriri = σriri m s σriφi = σriφi  on ri = Ri. (5.16) um = us  ri ri  m s  uφi = uφi   Applying these boundary conditions to our expansions (5.8)-(5.11) and (5.12)-(5.15) (1) we have 6 equations dependent on Pn(cos φi) or Pn (cos φi). The φi dependence can be eliminated from these equations using the orthogonality relations of Legendre polynomials, and associated Legendre polynomials, as described in Appendix A. If we look ﬁrst at the boundary conditions upon the inner surface of the shell, we ﬁnd the radial stress condition becomes, for k ≥ 0,

k+3 k−2 i Ri i (Ri − hi) 0=ak k+3 (k + 1)(k +2)+ bk k−2 k(k − 1) (Ri − hi) Ri k+1 i Ri k 2 + ck−1 k+1 (k + 3k − 2νi) (Ri − hi) (2k − 1) k+3 i Ri (k + 5 − 4νi) + ck+1 k+3 (k + 1)(k + 2) (Ri − hi) (2k + 3) k−2 i (Ri − hi) (k + 4νi − 4) + dk−1 k−2 k(k − 1) Ri (2k − 1) k i (Ri − hi) k + 1 + dk+1 k ((k + 1)(k − 2) − 2νi). (5.17) Ri (2k + 3)

Ri−hi We can simplify this notationally by letting ti = Ri and noticing that certain coef- i i ﬁcients, such as ak and ck+1, appear in tandem within this equation. If we therefore CHAPTER 5. SHELLS INTERACTING IN AN ELASTIC MEDIUM 169 let

(k + 5 − 4ν ) aˆi = ai + ci i , (5.18) k k k+1 2k + 3 (k + 4ν − 4) ˆbi = bi + di i , (5.19) k k k−1 2k − 1 (5.20) then (5.17) can be rewritten as

i ˆi 2k+1 0 =ˆak(k + 1)(k +2)+ bkk(k − 1)ti k k + 1 + ci t2 (k2 + 3k − 2ν )+ di t2k+3 (k2 − k − 2ν − 2), (5.21) k−1 i (2k − 1) i k+1 i (2k + 3) i for k ≥ 0. Using the same notation the radial shear stress boundary condition, under the action of the orthogonality condition of the associated Legendre polynomials, can be written as

i ˆi 2k+1 0 =a ˆk(k + 2) − bk(k − 1)ti (k2 + 2ν − 2) (k2 + 2k + 2ν − 1) + ci t2 i − di t2k+3 i , (5.22) k−1 i 2k + 3 k+1 i 2k + 3 for k ≥ 1. It is useful to note here that the k = 1 equations of (5.21) and (5.22) imply i that c0 ≡ 0. This is clearly demonstrated by setting k = 1 in both equations to ﬁnd

2 0=6ˆai + ci t2(4 − 2ν ) − di t5(2ν + 2), 1 0 i i 5 2 i i 1 0=3ˆai + ci t2(2ν − 1) − di t5(2ν + 2). 1 0 i i 5 2 i i

i Therefore if we set c0 = 0 for i = 1, 2 in all other boundary conditions we can eliminate ˆi the equation (5.22) for k = 1. Similarly we can eliminate bk from (5.22) by multiplying (5.22) by k and adding to (5.21). This allows us to replace (5.22) by

(k2 + 2kν + k + ν + 1) 0 =a ˆi (2k +1)+ ci t2k − 2di t2k+3 i i for k ≥ 2. (5.23) k k−1 i k+1 i (k + 2)(2k + 3)

˜i ˜i If we similarly deﬁne Ak and Bk as

(k + 5 − 4ν ) A˜i = Ai + Ci 0 , (5.24) k k k+1 2k + 3 (k + 4ν − 4) B˜i = Bi + Di 0 , (5.25) k k k−1 2k − 1 CHAPTER 5. SHELLS INTERACTING IN AN ELASTIC MEDIUM 170 then applying the continuity conditions on the shell-host interface and the appropriate Legendre orthogonality condition yields the ﬁnal four equations. For continuity of radial displacement we ﬁnd for k ≥ 0,

i+1 (1 − 2ν0) (−1) −p (δk0 + δk1) (1 + ν0) 2Ri 1 (k + 3 − 4ν ) (k + 1)(k + 4ν − 2) + aˆ (k + 1) − ˆb k + c k i − d i k k k−1 (2k − 1) k+1 2k + 3 i k(k + 3 − 4ν ) (k + 4ν − 2) =A˜i (k + 1) − B˜i kRk−2 + Ci 0 − Di (k + 1) 0 Rk, k k i k−1 2k − 1 k+1 2k + 3 i (5.26) whilst continuity of polar displacement, for k ≥ 1, implies

(1 − 2ν ) 1 1 (k + 4ν − 4) (k + 5 − 4ν ) (−1)ip 0 ( δ )+ aˆ + ˆb + c i + d i (1 + ν ) 2R k1 k k k−1 2k − 1 k+1 2k + 3 0 i 1 (k + 4ν − 4) (k + 5 − 4ν ) =A˜i + B˜i Rk−2 + Ci 0 + Di Rk 0 . (5.27) k k i k−1 2k − 1 k+1 i 2k + 3

ˆi ˜i We can again eliminate bk and Bk components from (5.27) by multiplying it by k and adding to (5.26). We can then replace (5.27) for k ≥ 1 with

(3k − 4kν − 2ν + 1) 0 aˆi (2k +1)+ ci k + 2di i i k k−1 k+1 2k + 3 i (3k − 4kν − 2ν + 1) =A˜i (2k +1)+ Ci k + 2Di Rk 0 0 . (5.28) k k−1 k+1 i 2k + 3

Continuity of radial stress across the interface yields for k ≥ 0,

k(k2 + 3k − 2ν ) (k2 − k − 2ν − 2) aˆi (k + 1)(k +2)+ ˆbi k(k − 1) + ci i + di (k + 1) i k k k−1 2k − 1 k+1 2k + 3 k(k2 + 3k − 2ν ) = − pδ + A˜i (k + 1)(k +2)+ B˜i k(k − 1)Rk−2 + Ci 0 k0 k k i k−1 2k − 1 (k2 − k − 2ν − 2) + Di Rk(k + 1) 0 , (5.29) k+1 i 2k + 3 and continuity of radial shear stress for k ≥ 1 implies

(k2 + 2ν − 2) (k2 + 2k + 2ν − 1) aˆi (k + 2) − ˆbi (k − 1) + ci i − di i k k k−1 2k − 1 k+1 2k + 3 (k2 + 2ν − 2) (k2 + 2k + 2ν − 1) = A˜i (k + 2) − B˜i (k − 1)Rk−2 + Ci 0 − Di Rk 0 . k k i k−1 2k − 1 k+1 i 2k + 3 (5.30) CHAPTER 5. SHELLS INTERACTING IN AN ELASTIC MEDIUM 171

i Remembering that c0 ≡ 0 allows elimination of the k = 1 equation from (5.30), as if we set k = 1 in both (5.29) and (5.30) we ﬁnd that

(2ν + 2) (2ν + 2) 6ˆai − 2di i = 6A˜i + Ci(4 − 2ν ) − 2Di R 0 , 1 2 5 1 0 0 2 i 5 (2ν + 2) (2ν + 2) 3ˆai − di i = 3A˜i + Ci(2ν − 1) − Di R 0 . (5.31) 1 2 5 1 0 0 2 i 5

i Therefore C0 ≡ 0 for i = 1, 2 and the k = 1 equation in (5.30) can be removed. We can further simplify (5.30) by multiplying it by k and adding the resulting equation to (5.29), to produce for k ≥ 2,

(k2 + 2kν + ν + 1) aˆi (2k +1)+ ci k − 2di i i k k−1 k+1 (k + 2)(2k + 3) (k2 + 2kν + ν + 1) = A˜i (2k +1)+ Ci k − 2Di Rk 0 0 . (5.32) k k−1 k+1 i (k + 2)(2k + 3)

In conclusion this leaves us with 6 boundary conditions for each shell, i = 1, 2, given by:

i ˆi 2k+1 0 =a ˆk(k + 1)(k +2)+ bkk(k − 1)ti k k + 1 + ci t2 (k2 + 3k − 2ν )+ di t2k+3 (k2 − k − 2ν − 2), for k ≥ 0, k−1 i (2k − 1) i k+1 i (2k + 3) i (5.33) (k2 + 2kν + k + ν + 1) 0 =a ˆi (2k +1)+ ci t2k − 2di t2k+3 i i for k ≥ 2, k k−1 i k+1 i (k + 2)(2k + 3) (5.34)

1 (k + 3 − 4ν ) (k + 1)(k + 4ν − 2) aˆ (k + 1) − ˆb k + c k i − d i k k k−1 (2k − 1) k+1 2k + 3 i i+1 (1 − 2ν0) (−1) ˜i ˜i k−2 − p (δk0 + δk1)= Ak(k + 1) − BkkRi (1 + ν0) 2Ri k(k + 3 − 4ν ) (k + 4ν − 2) + Ci 0 − Di (k + 1) 0 Rk, fork ≥ 0, k−1 2k − 1 k+1 2k + 3 i (5.35) 1 (3k − 4kν − 2ν + 1) aˆi (2k +1)+ ci k + 2di i i k k−1 k+1 2k + 3 i (3k − 4kν − 2ν + 1) = A˜i (2k +1)+ Ci k + 2Di Rk 0 0 for k ≥ 1, k k−1 k+1 i 2k + 3 (5.36) CHAPTER 5. SHELLS INTERACTING IN AN ELASTIC MEDIUM 172

k(k2 + 3k − 2ν ) (k2 − k − 2ν − 2) aî (k + 1)(k +2)+ ci i + di (k + 1) i k k−1 2k − 1 k+1 2k + 3 î ˜i ˜i k−2 + bkk(k − 1) = −pδk0 + Ak(k + 1)(k +2)+ Bkk(k − 1)Ri k(k2 + 3k − 2ν ) (k2 − k − 2ν − 2) + Ci 0 + Di Rk(k + 1) 0 , for k ≥ 0, k−1 2k − 1 k+1 i 2k + 3 (5.37) and (k2 + 2kν + ν + 1) aî (2k +1)+ ci k − 2di i i k k−1 k+1 (k + 2)(2k + 3) (k2 + 2kν + ν + 1) = A˜i (2k +1)+ Ci k − 2Di Rk 0 0 , for k ≥ 2. k k−1 k+1 i (k + 2)(2k + 3) (5.38)

5.2.1 Solving the boundary conditions

Since the Boussinesq-Papkovich solution expresses the displacement within the system in terms of infinite series, so each of the boundary conditions, for i = 1 and 2, are valid for infinitely many k. Therefore, to solve the system we must truncate the original series at a suitable point for each of the unknown families of coefficients. We already i i know that c0 and C0 = 0 for i = 1, 2, but we can also notice from (5.33)-(5.38) that i i i without loss of generality b0 and d0 can be set to zero for i = 1, 2. This is because b0 i does not appear in any boundary condition in (5.33)-(5.38) whilst d0, a component of î i b1, only appears in conjunction with b1 and so is also unnecessary. If we truncate each equation of (5.33)- (5.38) at k = N, we find we have 2(6N + 1) equations to solve, and therefore must find 12N + 2 coefficients for the system to be consistent. For i = 1, 2 i i i i i we can then determine: an and An for n = 0, .., N; bn for n = 1, ..., N; cn and Cn for i n = 1, ..., N − 1 and dn for n = 1, ..., N + 1. This allows us to represent (5.33)- (5.38) as a matrix equation, as was done in section 4.3.3, and so numerically solve, using the Mathematica LinearSolve function [48], for the unknown coefficients.

5.3 Results

5.3.1 Interaction of shells

Once again we would like to know how interaction between shells in an elastic matrix changes the distribution of stress felt by each shell, in comparison to the constant stress experienced by an isolated shell, as in section 2.5.3. We will again consider the hoop stress upon the shell, due to the invariance of the hoop stress to shell position when shells are isolated and the role of the hoop stress in shell buckling, as seen in section

3.2. For an isolated shell of outer radius Ri, shell thickness hi, situated at origin i CHAPTER 5. SHELLS INTERACTING IN AN ELASTIC MEDIUM 173

and under hydrostatic pressure p, the hoop stress, σφiφi , given by (2.27), is piecewise continuous so that

3 Ri 2 ˜ −a˜0 3 − (1 + ν1)d1 Ri − hi ≤ ri ≤ Ri, ri 3 σφiφi =  3 (5.39)  ∞ ˜ Ri σφiφi − A0 3 ri ≥ Ri, ri   where the coeﬃcientsa ˜0, d˜1 anda ˜0 are given as in (2.127), by

3 31t (ν0 − 1)(1 + ν1) a˜0 = −p 3 , 2(1 + ν0)((1 − 1)(t − 1)(1 + ν1) + 3(1 − ν1)) ˜ 9p1(1 − ν0) d1 = 3 , 2(1 + ν0)((1 − 1)(t − 1)(1 + ν1)+ 0(1 − ν1)) 3 3 ˜ 21(t − 1)(1 − 2ν0)(1 + ν1)+(1+ ν0)(3(1 − ν1) + (t − 1)(1 + ν1)) A0 = p 3 , 2(1 + ν0)((1 − 1)(t − 1)(1 + ν1) + 3(1 − ν1))

hi ∞ where t = 1 − and σφiφi = −p. Since displacements are also piecewise continuous Ri within the two-shell system, as in section 5.1, we also ﬁnd the hoop stress is piecewise continuous, such that from the origin of shell i,

2 u ∂u (1 − ν) ∂u ν σ = ri + ν ri + φi + u cot φ , (5.40) φiφi 1 − 2ν r ∂r r ∂φ r φi i i i i i i where i,νi Ri − hi ≤ ri ≤ Ri,  ,ν = 0,ν0 ri ≥ Ri, r3−i ≥ R3−i, (5.41)   3−i,ν3−i R3−i − h3−i ≤ r3−i ≤ R3−i,  and  1 ∂ (ξs + z ψs + z ψs ) 2 ∂r i i 3−i 3−i  i i  s s  −4(1 − νi)(ψ1 + ψ2) cos φi ,Ri − hi ≤ ri ≤ Ri,    ∞ 1 ∂ m m m uri + (ξ + ziψi + z3−iψ3−i)  2 ∂ri uri =  (5.42)  m m  −4(1 − ν0)(ψ1 + ψ2 ) cos φi , ri ≥ Ri, r3−i ≥ R3−i, 1 ∂  (ξs + z ψs + z ψs ) 2 ∂r i i 3−i 3−i  3−i i   −4(1 − ν )(ψs + ψs) cos φ ,R − h ≤ r ≤ R ,  3−i 1 2 i 3−i 3−i 3−i 3−i    and similarly for uφi , as in section 5.1. CHAPTER 5. SHELLS INTERACTING IN AN ELASTIC MEDIUM 174

Changing radius

If we consider shells of a given material, such that νi = 0.25,i = 200, within a hi polyurethane medium (ν0 = 0.49985) with a fixed shell thickness to radius ratio, , Ri then for an isolated shell changing the radius of the shell has no effect upon the stress felt on the shell-matrix interface, as can be seen in (5.39). Thus, in the two-shell system, when both shells are small we expect to recover the isolated shell hoop stress for each shell. We also expect to recover the isolated shell hoop stress when one shell is very small and the other shell is large, as we expect the small shell to have a negligible interaction effect upon the dominant large shell. What we are most interested in therefore, is a) for shells of equal radius at what radius does the hoop stress on each shell alter significantly from the isolated shell limit, and b) for shells of differing radii when does the smaller shell start to move the hoop stress of the larger shell away from the isolated shell limit.

We will first consider shells of equal radius first; in this situation, let hi = h and Ri = R h for both shells (i = 1, 2), we will fix the shell thickness to radius ratio to be = 0.01 R and pressure at infinity to be p = 0.1. As the shells are equal we only need look at the hoop stress upon one shell, as the second shell will mirror this behaviour. Under these conditions an isolated shell would have hoop stress σφiφi = −3.85382 on the shell-matrix interface. In figure 5.2, which shows the hoop stress upon the shell-matrix interface of shell 1, we can see that for small shells we recover this isolated shell limit. However as the shells increase in radius we see the hoop stress upon shell 1 decreases near the neighbouring shell, near φ1 = π, suggesting the second shell acts as a buffer taking some of the stress that we would usually be propagated through the host medium to shell

1. The hoop stress has maximum magnitude at φ1 = 0, and the decrease in this hoop stress as the shell radius increases also suggests that the second shell is decreasing the impact of the hydrostatic pressure at infinity. This is very different to the behaviour we saw when considering two equal sized cavities within a host medium, where we saw the hoop stress increase at φ1 = π, and suggests that the material of the shell may play a large role in determining the shell hoop stress, as will be investigated later. Shells of radius R ≤ 0.4 all show similar behaviours, with decreasing hoop stress as radius increases, however by R = 0.45 we find the hoop stress oscillates with higher frequency in φ1; it is unclear why this is the case, but must be presumed to coincide with the narrowing gap between the two shells. CHAPTER 5. SHELLS INTERACTING IN AN ELASTIC MEDIUM 175

σφ1φ1 3.50 3.55 3.60 3.65 3.70 3.75 3.80

φ Π Π 3 Π 2 Π 1 2 2

Figure 5.2: Hoop stress σφ1φ1 upon shell 1 for equal sized shells R1 = R2 with ﬁxed shell h1 h2 thickness to radius ratio, = = 0.01. The shells are stiﬀ and glassy (ν1,ν2 = R1 R2 0.25,1,2 = 200) whilst the host medium is nearly incompressible ν0 = 0.49985. Various radii are exhibited: R1 = 0.01 (blue, solid), R1 = 0.1 (pink, dashed), R1 = 0.2 (yellow, dot-dashed), R1 = 0.3 (green, dotted), R1 = 0.4 (black, dashed), R1 = 0.45 (red, solid).

Whilst we can look at shells of various radii it may also be useful to look at the maximum diﬀerence between the hoop stress on shell 1 and that for an isolated shell, which we can see in ﬁgure 5.2 is when the hoop stress on shell 1 is at its minimum magnitude. Figure 5.3 thus demonstrates that the minimum hoop stress moves away from the isolated shell limit quite quickly, by R = 0.1, as both shells increase in radius and that this growth is nonlinear with respect to the increase in radius. CHAPTER 5. SHELLS INTERACTING IN AN ELASTIC MEDIUM 176

σφ1φ1 3.50 3.55 3.60 3.65 3.70 3.75 3.80 R ,R 0.1 0.2 0.3 0.4 1 2

Figure 5.3: Minimum hoop stress, in magnitude, σφ1φ1 upon shell 1 for equal sized shells h1 h2 R1 = R2 as radius changes, with ﬁxed shell thickness to radius ratio, = = 0.01. R1 R2 The shells are stiﬀ and glassy (ν1,ν2 = 0.25,1,2 = 200) whilst the host medium is nearly incompressible ν0 = 0.49985.

For the second case in which the two shells can have different radii, though we still fix all other parameters, we see a less dramatic reaction in the hoop stress to increasing the size of shell 2 whilst fixing the size of shell 1 than we saw when increasing both shell radii. We can see in figure 5.4 that the hoop stress on the shell-matrix interface of shell 1 changes quantitatively rather than qualitatively as we increase the size of the near neighbour shell 2 for each fixed radius R1. For a fixed R1, as usual, we have constant hoop stress when the near neighbour cavity is very small, demonstrated by the blue, solid line in figure 5.4. However as we increase the radius of the near neighbour cavity we decrease the hoop stress upon shell 1. We find both a reduction in the average hoop stress over the shell-matrix interface and that the magnitude of oscillation of the hoop stress increases. We also find the highest magnitude of hoop stress to be experienced at the furthest point from the near neighbour shell φ1 = 0. Where the hoop stress is at its smallest magnitude it does change with the radius of shell 1, but is not affected by the radius of shell 2. The smaller the radius of R1 the closer the minimum hoop π 3π stress becomes to φ = and φ = , however as R increases in size we see the 1 2 1 2 1 minimum points gather closer to φ1 = π. For a fixed R1 we then see that the range of the magnitude of the hoop stress increases as R2 increases. From this behaviour we can see that even very small shells, see figure 5.4a, interact with their near neighbour in a significant way, placing less pressure upon the shell-matrix interface than the isolated π 3π shell solution would predict, especially around the shell meridian on φ = , . In 1 2 2 these small shells the decrease in pressure on the shell at the meridian is not however matched at the poles, both φ1 = 0 and φ1 = π experience the highest hoop stress on CHAPTER 5. SHELLS INTERACTING IN AN ELASTIC MEDIUM 177 the shell, no matter the size of the near neighbour shell. For larger shells however, see figures 5.4c and 5.4d, the behaviour the point closest to the near neighbour shell at

φ1 = π, is also altered as the neighbouring shell absorbs more of the pressure exerted upon shell 1.

σφ1φ1 σφ1φ1 3.70 3.70

3.75 3.75

3.80 3.80

φ φ Π Π 3 Π 2 Π 1 Π Π 3 Π 2 Π 1 2 2 2 2

(a) R1 =0.01 (b) R1 =0.1

σ σφ1φ1 φ1φ1 3.55 3.70 3.60 3.65 3.75 3.70 3.75 3.80 3.80 φ φ Π Π 3 Π 2 Π 1 0 Π Π 3 Π 2 Π 1 2 2 2 2

Figure 5.4: Each figure shows the hoop stress σφ1φ1 upon shell 1 at the shell-matrix h interface for variously sized shell 2, with fixed shell thickness to radius ratios, 1 = R1 h2 = 0.01. The shells are stiff and glassy (ν1,ν2 = 0.25,1,2 = 200) whilst the host R2 medium is nearly incompressible ν0 = 0.49985. Various radii are exhibited in each figure: R2 = 0.01 (blue, solid), R2 = 0.1 (pink, dashed), R2 = 0.2 (yellow, dot-dashed), R2 = 0.3 (green, dotted), R2 = 0.4 (black, dashed), R2 = 0.45 (red, solid).

Changing shell thickness

If we fix shell radius, at say R1 = R2 = 0.4 at which we saw interaction occurring in figure 5.3, we can look instead at the effect shell thickness has upon the hoop stress of the shells. We can see in figure 5.5 that as the thickness of the shell increases the magnitude of the hoop stress upon the shell decreases, as does the magnitude by which the hoop stress differs from that of an isolated shell. This is better seen in figure 5.6 CHAPTER 5. SHELLS INTERACTING IN AN ELASTIC MEDIUM 178 which shows the difference between the hoop stress on shell 1 when it is part of the two- shell system in which R2 = 0.4 and when shell 1 is considered to be an isolated shell. This shows that there is a more complex relationship between shell thickness and hoop stress in the two-shell system. Whilst in general the change in hoop stress between the two-shell and isolated-shell systems does decrease as shell thickness increases we see that for very thin shells the change between the two systems varies significantly with angle φ1.

σφ1φ1 1

φ Π Π 3 Π 2 Π 1 2 2

Figure 5.5: Hoop stress σφ1φ1 upon shell 1 for equal sized shells R1 = R2 = 0.4. The shells are stiﬀ and glassy (ν1,ν2 = 0.25,1,2 = 200) whilst the host medium is nearly incompressible ν0 = 0.49985. Various shell thicknesses are exhibited: h1, h2 = 0.005R1 (blue, solid), h1, h2 = 0.01R1 (pink, dashed), h1, h2 = 0.02R1 (yellow, dot-dashed), h1, h2 = 0.03R1 (green, dotted), h1, h2 = 0.04R1 (black, dashed), h1, h2 = 0.05R1 (red, solid). CHAPTER 5. SHELLS INTERACTING IN AN ELASTIC MEDIUM 179

∆|σφ1φ1 | φ Π Π 3 Π 2 Π 1 2 2 0.1

0.2

0.3

0.4

Figure 5.6: Diﬀerence between hoop stress σφ1φ1 upon shell 1 for equal sized shells R1 = R2 = 0.4 and the hoop stress of an isolated shell when R1 = 0.4. The shells are stiﬀ and glassy (ν1,ν2 = 0.25,1,2 = 200) whilst the host medium is nearly incompressible ν0 = 0.49985. Various shell thicknesses are exhibited: h1, h2 = 0.005R1 (blue, solid), h1, h2 = 0.01R1 (pink, dashed), h1, h2 = 0.02R1 (yellow, dot-dashed), h1, h2 = 0.03R1 (green, dotted), h1, h2 = 0.04R1 (black, dashed), h1, h2 = 0.05R1 (red, solid).

Changing materials

Similarly we can look at what happens to the radii at which shells interact when we change the host or shell materials.

Shell materials

For shells of equal radius and fixed shell thickness to radius ratio, if we keep the host medium as nearly incompressible but allow the Poisson ratio of the shells to change we see in figure 5.7 that apart from the magnitude of the hoop stress decreasing as the shells become more compressible, there is little change in the stress pattern over the shell interface from that seen in figure 5.2 for these shell radii. A small change occurs as the shells become more compressible in the point at which the shell experiences π 3π the smallest hoop stress; this moves away from φ = π towards φ = and φ = , 1 1 2 1 2 indicating the shell may be experiencing less support from the neighbouring cavity when the shells are more compressible. For small shells which are basically noninteracting we naturally see only change in magnitude of the hoop stress as the shells become more or less compressible, with no change in the nearly constant nature of the hoop stress over the shell interface. CHAPTER 5. SHELLS INTERACTING IN AN ELASTIC MEDIUM 180

σφiφi

3.4

3.6

3.8

4.0

4.2 φ Π Π 3 Π 2 Π 1 2 2

Figure 5.7: Hoop stress σφ1φ1 on the shell-matrix interface of shell 1 for two identical shells of radius R1 = R2 = 0.4, thickness h1 = h2 = 0.01R1 and shear modulus 1 = 2 = 200 under hydrostatic pressure p = 0.1. The shells have various Poisson ratios: ν1 = ν2 = 0.1 (blue, solid); ν1 = ν2 = 0.2 (pink, dotted); ν1 = ν2 = 0.3 (yellow, dot-dashed); ν1 = ν2 = 0.4 (green, dotted); ν1 = ν2 = 0.49985 (black, dashed).

However if we change the shear modulus of the shells, as in figure 5.8, we see not only a change in the magnitude of the hoop stress on the shell-matrix interface but also a qualitative change in the hoop stress near the neighbouring shell. In comparison to the isolated shell, see figure 5.9, we see that when the shear modulus of the shell equals that of the host medium, the pressure upon the shell oscillates around that of the isolated 3π 5π shell, larger near φ = π but smaller near φ = and φ = and only slightly 1 1 4 1 4 above that of the isolated shell at φ1 = 0. As the shear modulus of the shell increases slightly the amplitude of these oscillations increases too, however this changes when the shell is much stiffer than the host medium, at which point we see an overall drop in the hoop stress upon the host medium and a decline in the amplitude of the oscillations.

Instead we see a softening of the hoop stress occur around φ1 = π, eventually leading to the maximal change in the hoop stress seen at φ = π when 1 = 500. CHAPTER 5. SHELLS INTERACTING IN AN ELASTIC MEDIUM 181

σφ1φ1 φ Π Π 3 Π 2 Π 1 2 2 1

Figure 5.8: Hoop stress σφ1φ1 on the shell-matrix interface of shell 1 for shells of radius R1 = R2 = 0.4, ν1 = ν2 = 0.25 in a nearly incompressible host medium ν0 = 0.49985 for shells of various shear moduli: 1 = 2 = 1 (blue solid), 1 = 2 = 10 (pink, dashed), 1 = 2 = 50 (yellow, dot-dashed), 1 = 2 = 150 (green, dotted), 1 = 2 = 500 (black, dashed).

∆|σφ1φ1 | 0.2

0.1

φ Π Π 3 Π 2 Π 1 2 2 0.1

0.2

Figure 5.9: Change in the hoop stress on the shell-matrix interface of shell 1 in a two- two iso shell system from an isolated shell, ∆|σφ1φ1 | = |σφ1φ1 |−|σφ1φ1 |, for shells of radius R1 = R2 = 0.4 and ν1 = ν2 = 0.25 in a nearly incompressible host medium ν0 = 0.49985 for shells of various shear moduli: 1 = 2 = 1 (blue solid), 1 = 2 = 10 (pink, dashed), 1 = 2 = 50 (yellow, dot-dashed), 1 = 2 = 150 (green, dotted), 1 = 2 = 500 (black, dashed).

Host medium materials

If we keep the shells within the host as glassy and stiﬀ such that ν1,2 = 0.25 and

1,2 = 200 then we can see whether changing the Poisson ratio of the host medium affects the hoop stress on the shell-matrix interface. From figure 5.10 we see that, as the CHAPTER 5. SHELLS INTERACTING IN AN ELASTIC MEDIUM 182 host medium becomes more compressible, as the Poisson ratio decreases, the magnitude of the hoop stress experienced by shell 1 increases, and that, as when changing the material parameters in the shell, the main change to the hoop stress occurs in the region π 3π of the neighbouring shell, between φ = and φ = . From figure 5.11 we can see 1 2 1 2 that in comparison to an isolated shell, shell 1 in the two-shell system experiences either higher or lower hoop stress across the entirety of the shell circumference. For our current parameters shell 1 experiences higher average hoop stresses when the Poisson ratio of the host medium is less than ν0 = 0.4 and lower average hoop stresses above this. This indicates that for more compressible hosts the shell comes under more pressure when a neighbouring cavity is present whilst for nearly incompressible hosts the opposite reaction is seen.

∆|σφ1φ1 | 4

φ Π Π 3 Π 2 Π 1 2 2 Figure 5.10: Change in the hoop stress on the shell-matrix interface of shell 1 in a two iso two-shell system from an isolated shell, ∆|σφ1φ1 | = |σφ1φ1 | − |σφ1φ1 |, for shells of radius R1 = R2 = 0.4, ν1 = ν2 = 0.25 and 1 = 2 = 200 in hosts of various Poisson ratio: ν1 = ν2 = 0.49985 (blue solid), ν1 = ν2 = 0.4 (pink, dashed), ν1 = ν2 = 0.3 (yellow, dot-dashed), ν1 = ν2 = 0.25 (green, dotted), ν1 = ν2 = 0.2 (black, dashed), ν1 = ν2 = 0.1 (red, solid). CHAPTER 5. SHELLS INTERACTING IN AN ELASTIC MEDIUM 183

σφ1φ1

0.6

0.4

0.2

φ Π Π 3 Π 2 Π 1 2 2 0.2

Figure 5.11: Hoop stress σφ1φ1 on the shell-matrix interface of shell 1 for shells of radius R1 = R2 = 0.4, ν1 = ν2 = 0.25 and 1 = 2 = 200 in hosts of various Poisson ratio: ν1 = ν2 = 0.49985 (blue solid), ν1 = ν2 = 0.4 (pink, dashed), ν1 = ν2 = 0.3 (yellow, dot-dashed), ν1 = ν2 = 0.25 (green, dotted), ν1 = ν2 = 0.2 (black, dashed), ν1 = ν2 = 0.1 (red, solid). Chapter 6

Buckling of interacting shells in an elastic medium

6.1 Buckling methodology

We have previously investigated how individual shells buckle, in section 3.1, and understood the linear elastic deformation of embedded interacting shells, in section 5. Using this work we can now progress to understanding the buckling patterns of interacting shells embedded with an elastic medium. We will use a similar method to that used in section 3.1, extended to take into account the interaction inherent in the problem. This involves looking at the stability of the linear state using the Trefftz criterion and then finding the point at which stability is lost using the Rayleigh-Ritz method. We will concentrate on finding the buckling pressure of two shells embedded within an infinite medium. The host medium material has Poisson ratio ν0 and shear modulus ˆ0. Each Lˆ shell will be offset from the centre of the coordinate system by distance 2 , such that the centre of shell 1 is related to the centre of shell 2 byr ˆ1 cos φ1 + Lˆ =r ˆ2 cos φ2, where

(ˆr1, φ1,θ), and (ˆr2, φ2,θ2) are the local coordinates of shell 1 and shell 2 respectively.

Each shell has material parameters νi and î and has mid-radius Rî and thickness hî. From here onwards distances will be nondimensionalised with respect to the inter-shell distance Lˆ and forces with respect to the host medium shear modulus ˆ0. Hats notation will be dropped for nondimensionalised variables.

6.1.1 The Treﬀtz Criterion

As when studying the buckling of single shells, the Treﬀtz criterion states that for the loading parameter, the pressure p at inﬁnity in our case, the critical point between stability and instability is found at a stationary point of ∆W2.

We can ﬁnd ∆W2 using the pre-buckled deformation state of the system: the linear

184 CHAPTER 6. BUCKLING OF INTERACTING SHELLS 185 elastic deformation found in section 5. If we denote the prebuckled state I with displacement u(I), and impose a virtual displacement u upon the prebuckled displacement we will uncover a new state for the system, state II, in which the displacement is given by u(II) = u(I) + u. The stability of this system then depends upon the change in the potential energy, W , of the system between state I and state II, represented by

∆W = W (II) − W (I), where superscripts refer to the state of the system. If the change in potential energy is then expanded as a series about the virtual displacement u, we can deﬁne ∆W2 to be the quadratic term in the expansion, so that

∆W =∆W1 +∆W2 + ... and ∆W1 is linear in u. As in single shell buckling the prebuckled state I is then stable if ∆W2 > 0 for all u, whilst the critical load for state I is the minimum load for which

∆W2 < 0 for some u, ie the minimum load at which the state becomes unstable.

Applying the Treﬀtz criterion to interacting embedded shells

For an embedded two-shell system, the total potential energy of the system will be composed of the potential energy stored within each shell and the potential energy within the host medium.

The potential energy within each shell

If we denote the potential energy density in shell i, V si , then we can replicate the process undertaken in section 3.1, replacing Vs, the potential energy density in a single shell system, with V si . This yields the quadratic change in the potential energy density within each shell to be

(I)nαβ h h3 ∆V si = i wi wi + i Eαβλθi θi + i Eαβλρi ρi , 2 2 ,α ,β 2 i αβ λ 24 αβ λ

(I) αβ i where ni , the stress resultants of shell i in state I, w , the normal displacement of i the shell i and the changes of curvature tensor ραβ and linear middle surface strain i tensor, θαβ, for shell i are deﬁned as in section 2.3. Thus we can ﬁnally write the change in the potential energy of shell i, which is si quadratic in the virtual displacement, as ∆W2 :

(I) αβ 3 si ni i i hi αβλ i i hi αβλ i i ∆W2 = w,αw,β + E θαβθλ + E ραβρλ dS, (6.1) Mid-shell 2 2 24 surface CHAPTER 6. BUCKLING OF INTERACTING SHELLS 186

2 where dS = Ri sin φidφidθ and the abbreviation ‘mid-shell’ is taken to mean integration on the mid-shell surface.

The potential energy of the host medium

As in section 3.1, we can similarly write the potential energy stored in the host medium as the integral of the potential energy density within the volume V of the host medium:

∆Wm = ∆Vm dV, ∂V where ∂V is the boundary of the host created by the outer radius of the two shells. As in the shell, the host medium has an equilibrium solution, state I, which is perturbed by a virtual displacement in order to generate a new state II, which allows us to test the stability of the equilibrium state. In this case we denote the displacement within the host in state I as u(I), and the perturbation experienced by this state as u, a consequence of applying a virtual displacement vi upon each shell. As such the displacement in state II can be written u(I) + u. We can then write the change in potential energy density within the host medium using (2.17) in section 2.2.1 as

(II) (I) ∆Vm = Vm − Vm , 1 1 = Aijkl(e(I) + e )(e(I) + e ) − Aijkle(I)e(I), 2 ij ij kl kl 2 ij kl 1 = Aijkl(e(I)e + e e(I) + e e ), 2 ij kl ij kl ij kl where ejk is the strain experienced by the host medium. This can also be split into linear and quadratic parts, in the displacement u, so that the quadratic change in the potential energy within the host becomes:

1 ∆W m = Aijkle e dV. (6.2) 2 2 ij kl ∂V Total potential energy

The quadratic term of the total potential energy within the composite is then the sum of the energy stored in each shell and in the host medium:

m s1 s2 ∆W2 =∆W2 +∆W2 +∆W2 .

The quadratic terms can be more succinctly given by

2 i i ∆W2 = (I1 + I2)+ I3, i =1 CHAPTER 6. BUCKLING OF INTERACTING SHELLS 187 in which, for convenience, it is easiest to consider the contributions from each integral separately as,

(I) αβ i ni i i I1 = w,αw,β dS, (6.3) Mid-shell 2 surface 3 i hi αβλ i i hi αβλ i i I2 = E θαβθλ + E ραβρλ dS, (6.4) Mid-shell 2 24 surface 1 I = Aijkle e dV. (6.5) 3 2 ij kl ∂V 6.1.2 Rayleigh-Ritz and the critical buckling stress

In order to ﬁnd the stationary point of ∆W2 we will again employ the Rayleigh-Ritz technique, of considering the virtual displacement to be an inﬁnite series. For the problems we are considering we are able to consider the virtual displacement in each shell vi to be axisymmetric and as such:

i v = vri eri + vφi eφi , (6.6)

where vri and vφi are independent of θ. By (2.39), this is related to the base vector i components wi and v1 and can be expressed using the Rayleigh-Ritz method as:

∞ i i vri = w = UnPn(cos φi), , (6.7) n=0 i ∞ v1 i (1) vφi = = VnPn (cos φi), (6.8) Ri n=0 (1) where Ri is the mid-surface radius of shell i and Pn is an associated Legendre function. i i The constant coeﬃcients Un and Vn are then found by solving

∂ i ∆W2 = 0, (6.9) ∂Un ∂ i ∆W2 = 0, (6.10) ∂Vn for i = 1 and 2. As in single shell buckling the determinant of this infinite set of equations must be set to zero to find the non-zero buckling deformation of each shell. The critical buckling pressure of each shell can then be found from this condition. Similarly, as in single shell buckling, the axisymmetry of the virtual displacements CHAPTER 6. BUCKLING OF INTERACTING SHELLS 188 allows simplification of integral (6.3):

(I) αβ i ni i i I1 = w,αw,β dS Mid-shell 2 surface

∂wi The axisymmetry in the virtual displacement implies wi = = 0. When we combine ,2 ∂θ i this with the fact that the integral I1 is integrated on the mid-shell surface, and thus does not integrate across the radius of the shell, the only stress resultant which will be i (I) 11 (I) 11 included in I1 is ni . Since ni is the stress resultant referred to the base vectors i gj in shell coordinates (see section 2.3), we need to understand how this relates to the stress in spherical coordinates. Using section 2.3, we can write the stress resultant in terms of the hoop stress as:

1 2 (I) 11 hi hi ni = 2 1+ ξ σφiφi dξ, (6.11) R 1 Ri i − 2 where ri = Ri + hiξ, and ξ is a nondimensional parameter across the shell width.

6.2 Energy Integrals

6.2.1 The First Energy Integral

i Using the axisymmetry of the system the ﬁrst energy integral I1 can be written as

(I)n11 Ii = i wi wi dS 1 2 ,1 ,1 shelli where shell i is understood to be the mid-shell surface of shell i. In this energy integral the derivative of the virtual radial displacement is given by

∞ i i i (1) w,1 = w,φi = UnPn (cos φi), n =0 and the stress resultant by (6.11). In order to calculate the ﬁrst energy integral it is therefore necessary to calculate the linear elastic hoop stress σφiφi within each shell. Through use of the linear elastic shell displacement, ui, as found in section 5, and the shell stress-displacement relation (2.27) we can write the hoop stress in shell i as:

2 u ∂u 1 − ν ∂u ν σ = r + ν r + φ + u cot φ . (6.12) φiφi 1 − 2ν r ∂r r ∂φ r φ In section 5 we used the Bousinessq-Papkovitch stress function method to ﬁnd the linear elastic displacement of the shells and the host within this two shells system CHAPTER 6. BUCKLING OF INTERACTING SHELLS 189

s s under hydrostatic pressure p. The shell displacements uri and uφi , which we need for (6.12), are thus given by

∞ + n+2 n−1 s + i Ri i ri 2iuri = Ri −an(n + 1) + bnn + Pn ri R n=0 i n+1 R+ r n (n + 1)P + nP − ci (n + 4 − 4ν ) i − di i (n + 4ν − 3) n+1 n−1 , n i r n R i 2n + 1 i i (6.13)

∞ + n+2 n−1 s + i Ri i ri (1) 2iuφi = Ri an + bn + Pn ri n=0 Ri + n+1 n (1) (1) i Ri i ri ((n + 4νi − 3)Pn+1 + (n + 4 − 4νi)Pn−1) + cn + dn + , ri R 2n + 1 i (6.14)

+ ± hi where Ri is the outer radius of shell i, such that Ri = Ri ± 2 , and the coeﬃcients i i i i an, bn, cn and dn are known, having been found through solution of the linear elastic i i system (5.38) in section 5. The coeﬃcients an to dn are functions of the material parameters of both the shell and the host and are linear in the applied hydrostatic pressure p. Substituting (6.13) and (6.14) into (6.12) this yields the hoop stress σφiφi :

∞ n+3 n+1 p R+ r n−2 R+ (n + 4ν − 4) σ = aî i + ˆbi i +c î i i φiφi 2 n r n + n−1 r 2n − 1 n i Ri i =0 r n (n + 5 − 4ν ) + dî i i P (2) n+1 R+ 2n + 3 n i + n+3 n−2 i Ri î ri − aˆn (n + 1)(n +2)+ bnn(n − 1) + ri R i n+1 R+ (n2 − 4nν − n − 2) +c î i n i n−1 r 2n − 1 i n 2 i ri (n + 4nνi + 3n + 4νi + 4) + dˆ (n + 1) Pn , (6.15) n+1 R+ 2n + 3 i where we have used the linearity of the coefficients with respect to the pressure p to redefine

1 (n + 5 − 4ν ) ci aî = ai + ci i , cî = n , n p n n+1 2n + 3 n p 1 (n + 4ν − 4) di ˆbi = bi + di i , dî = n . (6.16) n p n n−1 2n − 1 n p CHAPTER 6. BUCKLING OF INTERACTING SHELLS 190

(I) 11 The hoop stress (6.15) can then be used to calculate the stress resultant, ni , via (6.11). By deﬁning a function F (k) such that

+ k + 2−k − 2−k (Ri ) (Ri ) (Ri ) − , hi = 0, k = 2, hiRi 2 − k 2 − k 1 k 2 +  + k h Ri (Ri ) + − Fk = 1+ ξ dξ =  ln(Ri ) − ln(Ri ) , k = 2, hi = 0, − 1 Ri ri  hiRi 2  + k Ri , hi = 0,  R  i  (6.17)  where ri = Ri + hiξ, we can write the stress resultant as:

∞ (I) 11 phi 0 2 (2) ni = 2 qℓ Pℓ + qℓ Pℓ , (6.18) 2Ri ℓ=0 in which

2 0 i ˆi i ℓ(ℓ − 4ℓνi − ℓ − 2) qℓ = − aˆℓ(ℓ + 1)(ℓ + 2)Fℓ+3 + bℓℓ(ℓ − 1)F2−ℓ +c ˆℓ−1 Fℓ+1 2ℓ − 1

î (ℓ + 1) 2 + dℓ+1 (ℓ + 4ℓνi + 3ℓ + 4νi + 4)F−ℓ , 2ℓ + 3 (ℓ + 4ν − 4) (ℓ + 5 − 4ν ) q2 =a î F + ˆbi F +c î i F + dî i F . (6.19) ℓ ℓ ℓ+3 ℓ 2−ℓ ℓ−1 2n − 1 ℓ+1 ℓ+1 2ℓ + 3 −ℓ

We can thus write the ﬁrst energy integral as

∞ 2π π i phi i i 0 2 (2) (1) (1) 2 I1 = 2 UnUm (qℓ Pℓ + qℓ Pℓ )Pn Pm Ri sin φi dφidθ. (6.20) 4Ri 0 0 n=0,m=0,ℓ=0

Under the substitution zi = cos φi this can be notationally simpliﬁed to

∞ pπh 1 Ii = i U i U i (q0P + q2P (2))P (1)P (1) dz . (6.21) 1 2 n m ℓ ℓ ℓ ℓ n m i n=0, −1 m=0, ℓ=0

0 2 where qℓ and qℓ are given by (6.19) and the Legendre polynomials have argument zi. Finally we can use Gaunt’s formula to integrate over the triple associated Legendre polynomial product. CHAPTER 6. BUCKLING OF INTERACTING SHELLS 191

Gaunt’s formula

Gaunt’s formula [21] for the triple product integral of associated Legendre polynomials is given by:

1 (u) (v) (w) Pℓ Pm Pn dz = G[ℓ, m, n, u, v, w] −1 (m + v)!(n + w)!(2s − 2n)!s! = 2(−1)s−m−w (m − v)!(s − ℓ)!(s − m)!(s − n)!(2s + 1)! q (ℓ + u + t)!(m + n − u − t)! × (−1)t t!(ℓ − u − t)!(m − n + u + t)!(n − w − t)! t=p (6.22) when s ∈ N, m + n ≥ ℓ ≥ m − n and when the conditions below hold, otherwise the integral equals 0. Other quantities within (6.22) are given by

2s = ℓ + m + n, p = max(0, n − m − u), q = min(m + n − u, ℓ − u, n − w).

This formula only applies when the following are true:

1. the degrees are non-negative integers ℓ,m,n ≥ 0,

2. all three orders are non-negative integers u, v, w ≥ 0,

3. the orders sum up u = v + w

4. the degrees obey m ≥ n

In order to use Gaunt’s formula on the ﬁrst energy integral (6.21) we need to rearrange the integrals into a suitable form. Looking ﬁrst at

∞ 1 i i 2 (2) (1) (1) UnUmqℓ Pℓ Pm Pn dz, (6.23) n=0, −1 m=0, ℓ=0 we can see that the orders and degrees are non-negative integers and the orders sum up. However m n at all points within the sum. As such we need to split the sum (6.23) into

∞ ∞ ∞ 1 i i 2 (2) (1) (1) UnUmqℓ Pℓ Pm Pn dz n m n −1 ℓ=0 =0 = +1 ∞ ∞ n 1 i i 2 (2) (1) (1) + UnUmqℓ Pℓ Pm Pn dz, (6.24) n=0 m=0 −1 ℓ=0 CHAPTER 6. BUCKLING OF INTERACTING SHELLS 192 where the ﬁrst part of the sum obeys m ≥ n. Relabelling and rearranging the second integral we can also force the second half of the sum to obey m ≥ n. We can then apply Gaunt’s formula to the following:

∞ ∞ ∞ 1 i i 2 (2) (1) (1) UnUmqℓ Pℓ Pm Pn dz ℓ n=0 m=n+1 −1 =0 ∞ ∞ m 1 i i 2 (2) (1) (1) + UmUnqℓ Pℓ Pm Pn dz. (6.25) m n −1 ℓ=0 =0 =0 According to Gaunt’s formula the triple integral equals zero unless m + n ≥ ℓ ≥ m − n, as such we can also reduce the sum over ℓ to:

∞ ∞ m+n i i 2 UnUmqℓ G[ℓ,m,n, 2, 1, 1] n=0 m=n+1 ℓ m n =− ∞ m m+n i i 2 + UmUnqℓ G[ℓ,m,n, 2, 1, 1], (6.26) m=0 n=0 ℓ=m−n where we remember

G[ℓ,m,n, 2, 1, 1] = 0 unless ℓ + m + n even.

We can simplify this form of solution further by noticing

∞ m ∞ ∞ = , m=0 n=0 n=0 m=n which still preserves the important factor that m ≥ n at all points within the sum. As such we can write

∞ 1 ∞ ∞ m+n i i 2 (2) (1) (1) i i 2 Un Um qℓ Pℓ Pm Pn dz =2 Un Um qℓ G[ℓ,m,n, 2, 1, 1] n=0, −1 n=0 m=n ℓ=m−n m=0, ℓ=0 ∞ 2n i 2 2 − (Un) qℓ G[ℓ,n,n, 2, 1, 1]. (6.27) n=0 ℓ =0 The remaining part of (6.21), i.e.

∞ 1 i i 0 (1) (1) UnUmqℓ Pn Pm Pℓ dz, (6.28) n=0, −1 m=0, ℓ=0 CHAPTER 6. BUCKLING OF INTERACTING SHELLS 193 needs to be similarly rearranged into a form on which Gaunt’s formula can be used. Once again we ﬁnd the orders and degrees of the triple product integral are non-zero integers, and the integral can be written so that the orders sum up. We again need to split the integral such that m ≥ ℓ at each point of the summation. Note in this case m ≥ ℓ, rather than n, due to the rearrangement of the triple product integral. Upon splitting the sum over ℓ, and rearranging, we can write (6.28) as:

∞ ∞ m 1 i i 0 (1) (1) UnUmqℓ Pn Pm Pℓ dz n=0 m=0 ℓ −1 =0 ∞ ∞ ∞ 1 i i 0 (1) (1) + UnUℓqm Pn PmPℓ dz, (6.29) n=0 ℓ m ℓ −1 =0 = +1 where m ≥ ℓ for each part of the sum. In this formulation we can now apply Gaunt’s formula to (6.29) to yield

∞ ∞ m ∞ ∞ ∞ i i 0 i i 0 Un Um qℓ G[n,m,ℓ, 1, 1, 0] + Un Uℓ qm G[n,m,ℓ, 1, 0, 1]. n m n =0 =0 ℓ=0 =0 ℓ=0 m=ℓ+1 (6.30)

Since Gaunt’s formula implies G[ℓ, m, n, u, v, w] = 0 unless m + n ≥ ℓ ≥ m − n, we similarly ﬁnd G[n, m, ℓ, u, v, w] = 0 unless m + ℓ ≥ n ≥ m − ℓ, and can so restrict the sum over n so that

∞ 1 ∞ m m+ℓ i i 0 (1) (1) i i 0 UnUmqℓ Pn Pm Pℓ dz = Un Um qℓ G[n,m,ℓ, 1, 1, 0] n=0, −1 m=0 ℓ=0 n=m−ℓ m=0, ℓ=0 ∞ ∞ m+ℓ i i 0 + Un Uℓ qm G[n,m,ℓ, 1, 0, 1], ℓ m ℓ n m ℓ =0 = +1 =− (6.31) CHAPTER 6. BUCKLING OF INTERACTING SHELLS 194 where G = 0 unless n+m+ℓ ∈ N. Therefore we can write (6.21) using Gaunt’s formula as

∞ pπh 1 Ii = i U i U i (q0P + q2P (2))P (1)P (1) dz 1 2 n m ℓ ℓ ℓ ℓ n m n=0, −1 m=0, ℓ=0 ∞ ∞ m+n pπh = i 2 U i U i q2 G[ℓ,m,n, 2, 1, 1] 2 n m ℓ n=0 m=n ℓ=m−n ∞ 2n i 2 2 − (Un) qℓ G[ℓ,n,n, 2, 1, 1] n=0 ℓ=0 ∞ m m+ℓ i i 0 + Un Um qℓ G[n,m,ℓ, 1, 1, 0] m =0 ℓ=0 n=m−ℓ ∞ ∞ m+ℓ i i 0 + Un Uℓ qm G[n,m,ℓ, 1, 0, 1] , (6.32) ℓ m ℓ n m ℓ =0 = +1 =− where G[ℓ, m, n, u, v, w] = 0 unless m + ℓ + n is even.

6.2.2 The Second Energy Integral

The second energy integral, I2, with a slight change in notation, is identical to that calculated in section 3.2 in (3.44). As such we know that (6.4):

i=2 3 hi αβλ i i hi αβλ i i I2 = E θαβθλ + E ραβρλ dS, Mid-shell 2 24 i =1 surface evaluates to

i=2 ∞ 4πhii 1 i i 2 I2 = n(n + 1)Vn +(1+ νi)Un (1 − νi) 2n + 1 i=1 n=0 2 2 2 2 h n (n + 1) 2 + (1 − ν2)U i + i U i , (6.33) i n 12R2 n i as the same reasoning can be applied as in 3.2.3.

6.2.3 The Third Energy Integral

This integral considers the change in energy within the host medium, rather than within each shell, and thus must take into account the double shell nature of the system. In order to calculate (6.5): CHAPTER 6. BUCKLING OF INTERACTING SHELLS 195

1 I = Aijkle e dV, 3 2 ij kl ∂V we therefore need to understand how the virtual displacement vi, which we imposed upon each shell, will aﬀect the virtual displacement of the host medium. If we let the induced virtual displacement be denoted u, then we wish to ﬁnd u so that on the shell boundaries i u|ri=Ri = v , (6.34)

+ hi for i = 1, 2, where we assume that the displacement on the shell boundary Ri = Ri + 2 can be approximated by that upon the shell mid-surface, Ri. We also consider that, due to the inﬁnitesimal nature of the virtual displacement, the induced stress ﬁeld vanishes as ri → ∞. This virtual displacement problem can then be solved using the Bousinessq-Papkovich stress function method. Using the axisymmetric form of the Bousinessq-Papkovich stress function method from (2.58) we can write the virtual displacement u as

2u = ∇(ξ + ri cos φiψi + r3−i cos φ3−iψ3−i) − 4(1 − ν0)(ψi + ψ3−i)ezi , (6.35)

∂ 1 ∂ where ∇ = ∂ri , ri ∂φi , 0 , ∞ n+3 n+3 1 R1 2 R2 ξ = An n+1 Pn(cos φ1)+ An n+1 Pn(cos φ2) , n=0 r1 r2 ∞ n+2 i Ri ψi = Cn n+1 Pn(cos φi) for i = 1, 2, (6.36) n ri =0 and r1 and r2 are related by r1 cos φ1 +1 = r2 cos φ2 and r1 sin φ1 = r2 sin φ1. This form of the displacement takes into account the condition that the stress field must vanish as ri → ∞. In order to represent the virtual displacement within the host medium in terms of the virtual displacement within the shell we wish to find the coefficients i i An and Cn (for i = 1, 2) in terms of the coefficients for the virtual shell displacement i i Un and Vn (i = 1, 2), for which we will need to apply the boundary condition (6.34) upon each shell. The Bousinessq-Papkovich stress formulation allows us to write each displacement component, in shell i coordinates, as

∞ n+3 i Ri i n−1 2uri = −An(n + 1) n+2 + Bnnri Pn n ri =0 n+2 i Ri i n (n + 1)Pn+1 + nPn−1 + −C (n + 4 − 4ν0) + D (n + 4ν0 − 3)r , n rn+1 n i 2n + 1 i (6.37) CHAPTER 6. BUCKLING OF INTERACTING SHELLS 196

∞ n+3 i Ri i n−1 (1) 2uφi = An n+2 + Bnri Pn n=0 ri n+2 (1) (1) R ((n + 4ν0 − 3)P + (n + 4 − 4ν0)P ) + Ci i + Di rn n+1 n−1 , (6.38) n rn+1 n i 2n + 1 i where

∞ i 3−i i+1 3−i n+2 i Bm = (R3−iAn + (−1) Cn )R3−i δm, (6.39) n=0 ∞ i 3−i n+2 i Dm = Cn R3−i δm. (6.40) n=0 and m n+m i (−1) n if i = 1, δm = (−1)n n+m if i = 2.  n Applying the boundary conditions on ri = Ri, then implies

∞ ∞ i i i n−2 2 UnPn = Ri −An(n +1)+ BnnRi Pn n=0 n=0 (n + 1)P + nP + −Ci (n + 4 − 4ν )+ Di (n + 4ν − 3)Rn−1 n+1 n−1 , n 0 n 0 i 2n + 1 (6.41)

∞ ∞ i (1) i i n−2 (1) 2 VnPn = Ri An + BnRi Pn n=0 n=0 (1) (1) i i n−1 ((n + 4ν0 − 3)Pn+1 + (n + 4 − 4ν0)Pn−1) + Cn + DnRi , (6.42) 2n + 1 where the virtual shell displacements vri and vφi are given by (6.7) and (6.8). The orthogonality of Legendre, and associated Legendre, polynomials can then be used to i i i relate the coeﬃcients of the matrix An and Cn for i = 1, 2 to those of the shell Un and CHAPTER 6. BUCKLING OF INTERACTING SHELLS 197

i Vn for i = 1, 2. The orthogonality of the Legendre polynomials thus implies,

2 i ˆi ˆi n−2 Un = −An(n +1)+ BnnRi Ri n(n + 3 − 4ν ) (n + 4ν − 2)(n + 1) − Ci 0 + Di Rn 0 , for n ≥ 0, n−1 2n − 1 n+1 i 2n + 3 (6.43)

2 i î î n−2 i (n + 4ν0 − 4) i n (n + 5 − 4ν0) Vn = An + BnRi + Cn−1 + Dn+1Ri for n ≥ 1, Ri 2n − 1 2n + 3 (6.44) where we define

(n + 5 − 4ν ) (n + 4ν − 4) Aˆi = Ai + Ci 0 , and Bˆi = Bi + Di 0 . (6.45) n n n+1 2n + 3 n n n−1 2n − 1

As these equations are true for all n ≥ 0 or n ≥ 1 respectively, we must truncate the system at a suitable integer N in order to obtain a solution. We can thus build the matrix system Ma = u, where 1 1 2 2 1 1 2 2 u = (U0 ,...,UN ,U0 ,...,UN ,V1 ,...,VN ,V1 ,...,VN ) is the vector of virtual displacement coeﬃcients for the two shells,

1 1 2 2 1 1 2 2 a = (A0,...,AN , A0,...,AN ,C0 ,...,CN−1,C0 ,...,CN−1) is the vector of virtual displacement coefficients for the host medium, and M is a block matrix comprising the coefficients of the virtual displacement coefficients in the host medium. As such M can be expressed as

a1 b1 c1 d1 a2 b2 c2 d2 M = (6.46) a3 b3 c3 d3   a3 b4 c4 d4     CHAPTER 6. BUCKLING OF INTERACTING SHELLS 198 where

[a1]ij = [b2]ij = −(j + 1)δij , i + j [b ] = iRi−2Rj+3(−1)j , 1 ij 1 2 i i + j [a ] = iRi−2Rj+3(−1)i , 2 ij 2 1 i (i + 1)(i + 5 − 4ν ) i(i + 3 − 4ν ) [c ] = [d ] = − 0 δ − 0 δ , 1 ij 2 ij (2i + 3) (i+1)j (2i − 1) (i−1)j i + j (i + 4ν − 4) i − 1+ j [d ] = Rj+2(−1)j iRi−2 + 0 1 ij 2 1 i (2i − 1) i − 1 (k + 4ν − 2) i + j + 1 + Ri (k + 1) 0 , 1 (2k + 3) i + 1 i + j (i + 4ν − 4) i − 1+ j [c ] = Rj+2(−1)i+1 iRi−2 + 0 2 ij 1 2 i (2i − 1) i − 1 (k + 4ν − 2) i + j + 1 + Ri (k + 1) 0 , 2 (2k + 3) i + 1

[a3]ij = [b4]ij = δij, i + j [a ] = Ri−2Rj+3(−1)j , 4 ij 1 2 i i + j [b ] = Ri−2Rj+3(−1)i , 3 ij 2 1 i (i + 5 − 4ν ) (i + 4ν − 4) [c ] = [d ] = 0 δ + 0 δ , 3 ij 4 ij (2i + 3) (i+1)j (2i − 1) (i−1)j i + j (i + 4ν − 4) i − 1+ j [d ] = Rj+2(−1)j Ri−2 + 0 3 ij 2 1 i (2i − 1) i − 1 (i + 5 − 4ν ) i + j + 1 + Ri 0 , 1 (2i + 3) i + 1 i + j (i + 4ν − 4) i − 1+ j [c ] = Rj+2(−1)i+1 Ri−2 + 0 4 ij 1 1 i (2i − 1) i − 1 (i + 5 − 4ν ) i + j + 1 + Ri 0 . 2 (2i + 3) i + 1 Now that the virtual displacement within the host can be expressed in terms of the virtual displacement within the shell, and the virtual displacements are, approximately, continuous on the shell boundary, we can ﬁnd the contribution of the matrix to the energy of the system. As when considering single shell buckling we can use the deﬁnition CHAPTER 6. BUCKLING OF INTERACTING SHELLS 199

of Hooke’s Law given in (2.14), to rewrite I3 as

1 I = σije dV, (6.47) 3 2 ij ∂V which via the symmetry of the stress tensor, in (2.13), becomes

1 I = σiju | dV. (6.48) 3 2 i j ∂V

It is then possible to rewrite I3 as

1 I = σiju dV, (6.49) 3 2 i ∂V j since by covariant diﬀerentiation and the equation of equilibrium, (2.16),

ij ij ij ij σ ui = σ |jui + σ ui|j = σ ui|j. j This formulation allows us to apply the divergence theorem to the host medium, to yield

1 I = σijn u dS, (6.50) 3 2 j i δV where δV is the boundary of the shells r1 = R1 and r2 = R2. As in the single shell case we do not have a contribution from the inﬁnite boundary of the host medium since virtual stresses and displacements tend to zero as r1, r2 → ∞. We can therefore split the integral up so that

1 1 I = σijn u dS + σijn u dS, (6.51) 3 2 j i 2 j i δV1 δV2 where δV1 is the boundary of r1 = R1 and δV2 is the boundary of r2 = R2. Setting i the normal vector to point inwards on each shell boundary, such that n1 = −1 and all other components are zero, this expands to

1 1 I = − (σ11u1 + σ12u1) dS − (σ11u2 + σ12u2) dS . (6.52) 3 2 1 1 1 2 1 2 2 1 2 2 2 δV1 δV2 From (2.20), in section 2.2.1 we know we can write the stress-strain relations in the CHAPTER 6. BUCKLING OF INTERACTING SHELLS 200 host medium, in coordinates about shell i, as:

11 2ν0 i σi = ∆i + 2e11, (6.53) 1 − 2ν0 12 2 i σi = 2 e12, (6.54) ri where i 1 i 1 i ∆i = e11 + 2 e22 + 2 2 e33. ri ri sin φi In terms of u these strains then become

i ∂uri e11 = , (6.55) ∂ri 1 ∂u ∂u ei = ri + r φi − u , (6.56) 12 2 ∂φ i ∂r φi i i ∂uri uri 1 ∂uφi cot φi ∆i = + 2 + + uφi , (6.57) ∂ri ri ri ∂φi ri

11 12 and hence we can write the stresses σi and σi as

∂u 2ν u 1 ∂u cot φ σ11 = 2 ri + 0 2 ri + φi + i u , (6.58) i ∂r 1 − 2ν r r ∂φ r φi i 0 i i i i 12 1 ∂uri ∂uφi σ = + ri − u i . (6.59) i r2 ∂φ ∂r φ i i i

The Boussinesq-Papkovich stress function form of uri and uφi , given by (6.35) and 11 12 (6.36), can then be substituted into the stress-strain relations to yield σi and σi . When combined with the identities in (2.21) in section 2.2.1:

i u1 = uri , (6.60) i u2 = riuφi , (6.61) we can then write I3 as an integral in which all parts are known in terms of the shell i i virtual displacements Un and Vn as:

2 π 2 11 12 I3 = − π Ri sin φi σi uri + Riσi uφi |ri=Ri dφi, i=1 0 (6.62) CHAPTER 6. BUCKLING OF INTERACTING SHELLS 201

or under the substitution zi = cos φi

2 1 2 11 12 I3 = − π Ri σi uri + Riσi uφi |ri=Ri dzi. (6.63) i=1 −1 In terms of the virtual displacement in the host medium, as deﬁned in (6.37) and (6.38), we ﬁnd

∞ n(n2 + 3n − 2ν ) σ11| = Aî (n + 1)(n +2)+ Bî n(n − 1)Rn−2 + Ci 0 i ri=Ri n n i n−1 2n − 1 n=0 (n2 − n − 2 − 2ν ) + Di (n + 1)Rn 0 P , n+1 i 2n + 3 n ∞ (n2 + 2ν − 2) σ12| = − Aî (n +2)+ Bî Rn−2(n − 1) − Ci 0 i ri=Ri n n i n−1 2n − 1 n =0 2 (1) (n + 2n + 2ν − 1) Pn + Di Rn 0 , (6.64) n+1 i 2n + 3 R i and by (6.34)

∞ i uri |ri=Ri = UnPn, (6.65) n=0 ∞ i (1) uφi |ri=Ri = VnPn . (6.66) n=0 Thus using the orthogonality of the Legendre polynomials the third energy integral becomes

2 ∞ U i I = −2π R2 n Aî (n + 1)(n +2)+ Bî n(n − 1)Rn−2 3 i 2n + 1 n n i i n=0 =1 n(n2 + 3n − 2ν ) (n2 − n − 2 − 2ν ) + Ci 0 + Di (n + 1)Rn 0 n−1 2n − 1 n+1 i 2n + 3 n(n + 1) + Vi − Aî (n +2)+ Bî Rn−2(n − 1) 2n + 1 n n n i (n2 + 2ν − 2) (n2 + 2n + 2ν − 1) − Ci 0 + Di Rn 0 , n−1 2n − 1 n+1 i 2n + 3 (6.67) CHAPTER 6. BUCKLING OF INTERACTING SHELLS 202 or equivalently

2 ∞ (n + 1)(n + 2) I = −2π R2 Aˆi (U i − nVi ) 3 i n 2n + 1 n n i n=0 =1 n(n − 1) + Bˆi Rn−2 (U i + (n + 1)Vi ) n i 2n + 1 n n Ci n + n−1 (U i (n2 + 3n − 2ν ) −Vi (n + 1)(n2 + 2ν − 2)) (2n − 1) (2n + 1) n 0 n 0 i n Dn+1Ri (n + 1) i 2 i 2 + (Un(n − n − 2 − 2ν0)+ Vnn(n + 2n + 2ν0 − 1)) . (2n + 1)(2n + 3) (6.68)

6.3 The Eigenvalue Problem

Having found the three energy integrals which constitute ∆W2, we can express (6.10) i i in terms of Un and Vn. Looking ﬁrst at

∂ i ∆W2 = 0, (6.69) ∂Vs

1 2 1 2 it can be noted that I1 = I1 + I1 , (6.21), is independent of both Vs and Vs , and further

∂ ∂ i i (I2)= i (I2). ∂Vs ∂Vs

This allows equation (6.69) to be reduced to

∂ i i (I2 + I3) = 0, (6.70) ∂Vs

∂ i for i = 1, 2, where from (6.33), we can write i (I2) as ∂Vs

i ∂I2 8πhii s(s + 1) i i i = s(s + 1)Vs +(1+ νi)Us , (6.71) ∂Vs (1 − νi) 2s + 1 CHAPTER 6. BUCKLING OF INTERACTING SHELLS 203 and from (6.68), we ﬁnd:

∞ ∂I ∂Aî (n + 1)(n + 2) 3 = −2πR2 n (U i − nVi ) ∂Vi i ∂Vi 2n + 1 n n s n=0 s î ∂Bn n−2 n(n − 1) i i + i Ri (Un + (n + 1)Vn) ∂Vs 2n + 1 i ∂Cn−1 1 n i 2 i 2 + i (Un(n + 3n − 2ν0) −Vn(n + 1)(n + 2ν0 − 2)) ∂Vs (2n − 1) (2n + 1) i n ∂Dn+1 Ri (n + 1) i 2 i 2 + i (Un(n − n − 2 − 2ν0)+ Vnn(n + 2n + 2ν0 − 1)) ∂Vs (2n + 1)(2n + 3) 2 2πRi î î s−2 − − Ass(s + 1)(s +2)+ BsRi s(s − 1)(s + 1) 2s + 1 i i s Cs−1 2 Ds+1Ri (s + 1) 2 − s(s + 1)(s + 2ν0 − 2) + s(s + 2s + 2ν0 − 1) (2s − 1) (2s + 3) ∞ (n + 1)(n + 2) − 2πR2 DAˆ3−iVi (U 3−i − nV3−i) 3−i n s 2n + 1 n n n =0 ˆ3−i ∂Bn n−2 n(n − 1) 3−i 3−i + i R3−i (Un + (n + 1)Vn ) ∂Vs 2n + 1 3−i ∂Cn−1 1 n 3−i 2 3−i 2 + i (Un (n + 3n − 2ν0) −Vn (n + 1)(n + 2ν0 − 2)) ∂Vs (2n − 1) (2n + 1) 3−i n ∂Dn+1 R3−i(n + 1) 3−i 2 3−i 2 + i (Un (n − n − 2 − 2ν0)+ Vn n(n + 2n + 2ν0 − 1)) . ∂Vs (2n + 1)(2n + 3) (6.72)

Since I1 is the only integral featuring the pressure at inﬁnity, p, the reduced equation 1 2 1 2 (6.70) can be solved numerically to ﬁnd Vn and Vn in terms of Un and Un. These values can then be substituted into ∂ i ∆W2 = 0, (6.73) ∂Us i i i 3−i which we can again simplify. Since both I1 and I2 are functions of Un but not Un , we can reduce (6.73) to ∂ i i i (Ii + I2 + I3) = 0, (6.74) ∂Us CHAPTER 6. BUCKLING OF INTERACTING SHELLS 204 where

∞ m+s s s+n ∂Ii pπh 1 = i 2 U i q2 G[ℓ,m,s, 2, 1, 1] + 2 U i q2 G[ℓ,s,n, 2, 1, 1] ∂U i 2 m ℓ n ℓ s m=s ℓ m s n=0 ℓ s n =− =− 2s s s+ℓ i 2 i 0 − 2Us qℓ G[ℓ,s,s, 2, 1, 1] + Un qℓ G[n,s,ℓ, 1, 1, 0] ℓ=0 ℓ=0 n=s−ℓ ∞ m ∞ m+s i 0 i 0 + Um qℓ G[s,m,ℓ, 1, 1, 0] + Un qm G[n,m,s, 1, 0, 1] m m s n=m−s =0 ℓ=|s−m| = +1 ∞ ∞ i 0 + Uℓ qm G[s,m,ℓ, 1, 0, 1] , (6.75) ℓ=0 m=ℓ+1 i where G[ℓ, m, n, u, v, w] = 0 unless m + ℓ + n is even. To derive the derivative of I1 we have used a number of summation equivalencies such as

∞ ∞ ∞ δns = , n=0 m=n m=s ∞ ∞ s δms = , n m=n n =0 =0 ∞ m m+ℓ ∞ m δns = , m m =0 ℓ=0 n=m−ℓ =0 ℓ=|s−m| ∞ ∞ m+ℓ ∞ s+l δns = . (6.76) ℓ=0 m=ℓ+1 n=m−ℓ ℓ=0 m=max(s−l,l+1)

i i The diﬀerential of I2 with respect to Us is given by,

∂Ii 8πh 1 2 = i i (1 + ν ) s(s + 1)V i +(1+ ν )U i ∂U i (1 − ν ) 2n + 1 i s i s s i h2s2(s + 1)2 + (1 − ν2)U i + i U i , (6.77) i s 12R2 s i CHAPTER 6. BUCKLING OF INTERACTING SHELLS 205

i and the diﬀerential of I3 with respect to Us is given by

∞ ∂I ∂Aî (n + 1)(n + 2) 3 = −2πR2 n (U i − nVi ) ∂U i i ∂U i 2n + 1 n n s n s =0 î ∂Bn n−2 n(n − 1) i i + i Ri (Un + (n + 1)Vn) ∂Us 2n + 1 i ∂Cn−1 1 n i 2 i 2 + i (Un(n + 3n − 2ν0) −Vn(n + 1)(n + 2ν0 − 2)) ∂Us (2n − 1) (2n + 1) i n ∂Dn+1 Ri (n + 1) i 2 i 2 + i (Un(n − n − 2 − 2ν0)+ Vnn(n + 2n + 2ν0 − 1)) ∂Us (2n + 1)(2n + 3) 2 2πRi î î s−2 − As(s + 1)(s +2)+ BsRi s(s − 1) 2s + 1 i i s Cs−1 2 Ds+1Ri (s + 1) 2 + s(s + 3s − 2ν0)+ (s − s − 2 − 2ν0) (2s − 1) (2s + 3) ∞ ∂Aˆ3−i (n + 1)(n + 2) − 2πR2 n (U 3−i − nV3−i) 3−i ∂U i 2n + 1 n n n s =0 ˆ3−i ∂Bn n−2 n(n − 1) 3−i 3−i + i R3−i (Un + (n + 1)Vn ) ∂Us 2n + 1 3−i ∂Cn−1 1 n 3−i 2 3−i 2 + i (Un (n + 3n − 2ν0) −Vn (n + 1)(n + 2ν0 − 2)) ∂Us (2n − 1) (2n + 1) 3−i n ∂Dn+1 R3−i(n + 1) 3−i 2 3−i 2 + i (Un (n − n − 2 − 2ν0)+ Vn n(n + 2n + 2ν0 − 1)) . ∂Us (2n + 1)(2n + 3) (6.78)

i i We can then use the solution to (6.70) for Vs to construct (6.74) purely in terms of Us, and construct the eigenvalue problem

(Ap + B)U = 0,

U U 1 U 2 1 1 2 2 where = + = (U0 ,..., UN , U0 ,... UN ). Matrix A is constructed from the coef- i ∂ i ﬁcients of U for i = 1, 2 found in i I , given in (6.75), and similarly B represents the ∂Us 1 ∂ i contribution from i (I +I ), for i = 1, 2, given by (6.77) and (6.78) after substitution ∂Us 2 3 by the solution to (6.70).

6.4 Results

Given two identical shells, with outer radius R+ and thickness h, with material parameters ν1 = ν2 = 0.3 and 1 = 2 = 200, embedded within a polyurethane medium, CHAPTER 6. BUCKLING OF INTERACTING SHELLS 206

in which ν0 = 0.49985, we can investigate how the buckling pressure of the system is aﬀected by the shell radius or shell thickness. For identical shells the critical buckling pressure of the system will indicate the minimum pressure at which at least one shell i i will buckle, whilst the coeﬃcients Un and Vn for i = 1, 2 derived from the eigenvectors of the system will demonstrate the buckling pattern of each shell. We expect all shells of small radius to buckle independently, and thus to recover a critical buckling pressure for the system equal to that of a single embedded shell, which was derived in section 3.2. We also expect the eigenvectors of the system to have a dominant mode equivalent to the critical buckling mode of the independent embedded shell. However, as the radius of the shells increases we expect that the shells will start to interact, altering the buckling pressure of the system. This interaction may increase or decrease the buckling pressure of the system, may cause both shells to buckle and may occur at various shell radii depending upon the thickness of the shell.

6.4.1 Agreement with single shell buckling - dilute limit

If we have two small equal sized shells, with fixed shell thickness to radius ratio, we would expect these to be non-interacting and moreover to yield the same solution as found in section 3.2 for the buckling of a single shell. This is indeed what we recover. Within an elastic host medium such that ν0 = 0.49985, for two glassy shells, + (νi = 0.3, i = 200), a fixed distance apart with Ri = 0.01005 and hi = 0.0001, hi such that = 0.01, we recover the single shell critical buckling pressure pc = 4.62984 Ri (as found in table 3.1) where it is remembered that pc is nondimensionalised on the classical buckling pressure for an unembedded isolated shell p0. This is similarly true for shells of various thicknesses, as seen in table 6.1, where we can see that for small + shells, where Ri = 0.01, the buckling pressure for the dilute two-shell system equals the buckling pressure predicted by the full stress resultant solution for single shells under hydrostatic pressure 3.112. The buckling patterns of small shells further reveals that at the critical buckling pressure only one shell will buckle, with an equal likelihood for either shell. In each case the buckling pattern we recover is negligibly different to that found for a single shell, with a pattern dominated by the Legendre polynomial of the 29th mode for h h h = 0.01 and by the 12th mode for = 0.03. For very thin shells = 0.01 R+ R+ R+ this leads to a buckling pattern which is symmetric about the poles of the shell, but not symmetric about the shell equator, as can be seen in figure 6.1, for small shells in + + + + which R1 = R2 = R = 0.01 and h1 = h2 = 0.01R , however for thicker shells, see figure 6.2, the buckling pattern is completely spherically symmetric. As this buckling theory is based upon that of shallow shells, it is only valid if the wavelength of the buckling pattern is small in comparison with the minimum principal CHAPTER 6. BUCKLING OF INTERACTING SHELLS 207

h Critical buckling pressure in: Shell thickness ratio R+ single shells two-shell system 0.01 4.60457 4.604575 0.02 2.3408 2.3408 0.03 1.73829 1.73829 0.04 1.47411 1.47411 0.05 1.32506 1.32506

h Table 6.1: Critical buckling pressures for single shells of thickness ratio and equal R+ h small shells of radius R+ = 0.01 and thickness ratio derived using the single R+ shell buckling pressures for shells under hydrostatic pressure (3.112), and the two-shell system. Shell parameters are 1 = 200, ν1 = 0.3, host parameter ν0 = 0.49985.

0.010 0.010

0.005 0.005

0.010 0.005 0.005 0.010 0.010 0.005 0.005 0.010

0.005 0.005

0.010 0.010

(a) Buckling pattern of shell 1 (b) Buckling pattern of shell 2 h Figure 6.1: Buckling patterns of two identical shells of = 0.01 and R+ = 0.01 at R+ critical buckling pressure pc = 4.604575. Amplitude of buckling is arbitrary. Shell 1 (pink) is situated a unit distance above shell 2 (blue). radius of curvature of the shell, [50]. As all buckling modes are excited, to varying amplitudes, in this two-shell system, we do not always have an obvious length scale for the wavelength of the buckling pattern. In the dilute case, however, we do have a h dominant mode. For = 0.01 the 29th buckling mode is dominant and as such the R+ wavelength of the buckling pattern is very small, as can be seen in ﬁgure 6.1, however as the shell thickness increases the wavelength of the buckling pattern increases, as h the dominant buckling mode decreases, so that by = 0.05 the 8th buckling mode R+ is dominant and thus the wavelength of the buckling pattern increases. However, as 1 the principal radius of curvature k is , and we are dealing with very small shells R+ such that R+ = 0.01, we ﬁnd that the wavelength of the buckling pattern is still much h smaller than k even at = 0.05, and as such shallow shell buckling theory still holds. R+ CHAPTER 6. BUCKLING OF INTERACTING SHELLS 208

0.0010 0.0010

0.0005 0.0005

0.0010 0.0005 0.0005 0.0010 0.0010 0.0005 0.0005 0.0010

0.0005 0.0005

0.0010 0.0010 (a) Buckling pattern of shell 1 (b) Buckling pattern of shell 2 h Figure 6.2: Buckling patterns of two identical shells of = 0.03 and R+ = 0.01 at R+ critical buckling pressure pc = 0.993704. Amplitude of buckling is arbitrary. Shell 1 (pink) is situated a unit distance above shell 2 (blue).

As is noted in the buckling pattern ﬁgures the amplitude of buckling is arbitrary, the amplitude of the waves in the buckling pattern could be multiplied by any constant factor, given that after the moment of buckling deformation of the shell will be non- linearly elastic.

6.4.2 Equal sized shells

Moving away from the dilute limit we wish to understand what happens to the buckling pressure as the shells start to interact. The simplest case is thus to look at the effect radius and shell thickness have upon identical shells. We wish to understand for what radius the critical buckling pressure of the dilute system is valid, and how this changes with shell thickness, as well as what happens to the buckling pressure when that predicted by the dilute limit is not accurate. In figure 6.3 we have plotted the critical buckling pressure of the two-shell system pc for identical shells of varying radii for various shell thicknesses. The dilute critical buckling pressure is equal to that given by the critical buckling pressure of the two-shell system at small R+, providing a useful measure against which to judge the affect of interaction. As can be seen in figure 6.3 the dilute critical buckling pressure is valid for a large range of radii for all shell thicknesses. As shell thickness increases it appears, however, that two effects are taking place: that the critical buckling pressure moves away from the dilute limit earlier, and that the magnitude of this move decreases with h shell thickness, such that by = 0.05 these effects cancel out and the presence of R+ a second shell within the system appears negligible. Looking at each of the critical buckling profiles in detail adds to this perspective somewhat. Figure 6.4 supports the CHAPTER 6. BUCKLING OF INTERACTING SHELLS 209 perspective that the critical buckling pressure moves away from the dilute limit at smaller radii as shell thickness increases and that the magnitude of this move decreases with shell thickness. We also see, however, that the critical buckling pressure is not monotonically decreasing from the dilute limit, but rather that interaction stiffening, where shell interaction causes a stiffening effect within the composite, is displayed. This is most pronounced for thicker shells but exists for all shell thicknesses. Thus, whilst on the macroscale interaction may appear negligible for thicker shells, we actually find that over a small range the critical buckling pressure varies greatly as shell radius increases. In manufacturing these microvoided composites it should therefore be considered whether the buckling pressure for the composite is actually that predicted by the dilute limit, or whether this may have been changed by the volume fraction of shells within the composite. For thin shells an increase in buckling pressure can be seen for specific h intervals of shell radii, however it is worth noting that for = 0.05 interaction R+ always causes stiffening, as the dilute critical buckling pressure is the minimum buckling pressure over all shell radii.

pc 4.5

4.0

3.5

3.0

2.5

2.0

1.5

R+ 0.1 0.2 0.3 0.4 0.5

Figure 6.3: Critical buckling pressure pc of interacting shells, each of outer radius + R , shell parameters ν1 = ν2 = 0.3, 1 = 2 = 200 and host material ν0 = 0.49985 h h for various shell thickness: = 0.01 (blue, solid), = 0.03 (pink, dashed), and R+ R+ h = 0.05 (yellow, dot-dashed). R+ CHAPTER 6. BUCKLING OF INTERACTING SHELLS 210

4.6 4.6 1.745 1.331 4.55 4.55 1.74 1.33 4.5 4.5 c c c c 1.329 1.735 p p p 4.45 p 4.45 1.328 4.4 1.73 4.4 1.327 4.35 4.35 1.725 1.326 4.3 4.3 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 R+ R+

1.745 1.331

1.74 1.33

c 1.329 c 1.735 p p 1.328 1.73 1.327

1.725 1.326

0.0 0.1 0.2 0.3 0.4 0.5 R+

+ Figure 6.4: Critical buckling pressure pc of interacting shells, each of outer radius R , shell parameters ν1 = ν2 = 0.3, 1 = 2 = 200 and host material ν0 = 0.49985 for h h h various shell thickness: = 0.01 (blue), = 0.03 (pink), and = 0.05 (yellow). R+ R+ R+

Whilst for dilute buckling we expect shells to buckle independently, as we increase the radii of the shells we find that both shells collapse at the system critical buckling pressure, see figures 6.5 and 6.6. For very thin shells, this appears to constrain the buckling amplitude of each shell on the surfaces furthest from the neighbouring shell, resulting for large shells, such as in figure 6.5d, in each shell buckling only at the pole nearest to the neighbouring shell. For thicker shells, dominated by lower buckling modes, we find slightly different behaviour. Buckling patterns are still constrained away from a near neighbour, in comparison to dilute buckling patterns, however shells tend to buckle symmetrically about both the poles and equator of each shell. This leads to each shell buckling in a different manner, as can be seen in figure 6.6, so that one shell buckles inwards at the poles, whilst the other buckles outwards. For large, thick shells it is again worth remembering that this is a shallow shell theory and is only valid when the wavelength of the buckling pattern is much smaller than the principal radius of curvature of the shell. Thus for shells of radius R+ = 0.49, h we need wavelengths to be much less than 2, however at = 0.05 the wavelength is R+ still dominated by a Legendre polynomial of the 8th mode, and thus has a wavelength of π/4 which, whilst smaller than the radius of curvature, may be close to the boundary of the region in which the solution is valid. CHAPTER 6. BUCKLING OF INTERACTING SHELLS 211

0.2 0.3 0.2 0.1 0.1

0.2 0.1 0.1 0.2 0.3 0.2 0.1 0.1 0.2 0.3 0.1 0.1 0.2

0.2 0.3 (a) Buckling pattern at R+ =0.2 (b) Buckling pattern at R+ =0.3

0.4 0.4 0.2 0.2 0.4 0.2 0.2 0.4 0.4 0.2 0.2 0.4 0.2 0.2 0.4

0.4 0.6 (c) Buckling pattern at R+ =0.4 (d) Buckling pattern at R+ =0.49 h Figure 6.5: Buckling patterns of two identical shells of = 0.01 for various shell R+ radii at critical system buckling pressure pc . Amplitude of buckling is arbitrary. Shell 1 (pink) is situated a unit distance above shell 2 (blue). CHAPTER 6. BUCKLING OF INTERACTING SHELLS 212

0.3 0.10 0.2 0.05 0.1

0.10 0.05 0.05 0.10 0.3 0.2 0.1 0.1 0.2 0.3 0.1 0.05 0.2

0.10 0.3 (a) Buckling pattern at R+ =0.1 (b) Buckling pattern at R+ =0.3 0.4 0.4

0.2 0.2

0.4 0.2 0.2 0.4 0.4 0.2 0.2 0.4 0.2 0.2

0.4 0.4 (c) Buckling pattern at R+ =0.4 (d) Buckling pattern at R+ =0.49 h Figure 6.6: Buckling patterns of two identical shells of = 0.05 for various shell R+ radii at critical system buckling pressure pc . Amplitude of buckling is arbitrary. Shell 1 (pink) is situated a unit distance above shell 2 (blue). CHAPTER 6. BUCKLING OF INTERACTING SHELLS 213

6.4.3 Unequal shells

Whilst equal sized shells buckle simultaneously this may not be the case for shells of differing sizes. We can again calculate the buckling pressure for the two-shell system, but whilst this reveals the pressure at which at least one shell will buckle we also need to combine this with the shell buckling patterns to find out which shells buckle at this critical pressure. From the results for equal sized shells we know thin shells are more sensitive to changes in buckling pressure than thicker shells, and as such we will only consider shells of thickness to radius ratio 0.01 in this section. We expect that if the shells are sufficiently small, and as such basically noninteracting, the buckling pressure of the system will equal that for a single shell, such that pc = 4.60457 when the shell thickness to radius ratio of each shell is 0.01. This is indeed what we see in figure

6.7, when both R1 and R2 are small, for example when R1 = 0.01 or 0.1, the critical buckling pressure is equal to or slightly greater than that of the single shell until around

R2 = 0.23. In this regime the larger of the two shells in the system will buckle, as can be seen in figure 6.8. However we can see in figure 6.7 that even very small shells can influence the buckling pressure of the system when coupled with a second larger shell.

This is seen in the case when R1 = 0.3 or 0.4, where near neighbour shells can be very small, such that R2 = 0.01, but the buckling pressure of the system is lower than that of a single shell. When the buckling pressure of the system is slightly smaller than that of the of a single shell we see a second regime, in which the smaller of the two interacting shells buckles. This is seen both when R1 is large but R2 small, and when

R2 is large but R1 small. Thus, there is a radius at which for small R1 the buckling patterns switch from the larger to the smaller of the two shells buckling, which appears to be around R2 = 0.24 for R1 ≤ 0.1. This buckling regime is shown in ﬁgure 6.9 at

R1 = 0.1, and R1 = 0.4 for R2 = 0.3 and R2 = 0.1 respectively. There then appears to be a final third regime, which occurs when the two interacting shells are very close to each other. At this point the buckling pressure of the system decreases again and both shells buckle simultaneously. This can be seen in figure 6.7 as the gradient of the buckling pressure of the system increases dramatically, and the buckling patterns can be seen in figure 6.10. These changes in buckling pressure and buckling regime indicate that the isolated shell buckling solution, in chapter 3.2, is not valid in the two-shell system unless the shells are far enough apart to be considered non-interacting. Whilst we know the isolated shell buckling pressure is not valid once one shell has radius 0.3, for these system parameters, pinpointing exactly where the single shell buckling pressure breaks down is not in the remit of this thesis. It would be recommended that whenever shells may possibly be interacting within a composite the buckling pressure between near neighbour shells is calculated to find whether the single shell buckling pressure is valid or not. CHAPTER 6. BUCKLING OF INTERACTING SHELLS 214

pc ææ à 4.6 à ò ò

4.4

4.2

4.0

R 0.2 0.4 0.6 0.8 1.0 2

Figure 6.7: Critical buckling pressure pc of a system of interacting shells of outer radii R1 and R2, for diﬀering R1 as R2 varies, with shell parameters h1 = 0.01R1, h2 = 0.01R2, ν1 = ν2 = 0.3, and 1 = 2 = 200 embedded in host material ν0 = 0.49985 for various radii R1: R1 = 0.01 (blue), R1 = 0.1 (pink, dashed), R1 = 0.3 (yellow, dot- dashed), and R1 = 0.4 (green, dotted). Markers denote points where buckling patterns are shown below.

0.2 0.10

0.1 0.05

0.10 0.05 0.05 0.10 0.2 0.1 0.1 0.2

0.05 0.1

0.10 0.2

(a) Buckling pattern at R1 = 0.1, (b) Buckling pattern at R1 = 0.1, R2 =0.01 R2 =0.2

Figure 6.8: Buckling patterns of a system of interacting shells of outer radii R1 and R2, with shell parameters h1 = 0.01R1, h2 = 0.01R2, ν1 = ν2 = 0.3, and 1 = 2 = 200 embedded in host material ν0 = 0.49985 at the critical buckling pressure pc denoted by the black circle in ﬁgure 6.7. Shell 1 (pink) is situated a unit distance directly above shell 2 (blue) CHAPTER 6. BUCKLING OF INTERACTING SHELLS 215

0.4 0.3

0.2 0.2 0.1

0.4 0.2 0.2 0.4 0.3 0.2 0.1 0.1 0.2 0.3 0.1 0.2 0.2

0.4 0.3

(a) Buckling pattern at R1 = 0.4, (b) Buckling pattern at R1 = 0.1, R2 =0.1 R2 =0.3

Figure 6.9: buckling patterns of system of interacting shells of outer radii R1 and R2, with shell parameters h1 = 0.01R1, h2 = 0.01R2, ν1 = ν2 = 0.3, and 1 = 2 = 200 embedded in host material ν0 = 0.49985 at the critical buckling pressure pc denoted by the black square in ﬁgure 6.7. Shell 1 (pink) is situated a unit distance directly above shell 2 (blue).

0.4 0.4

0.2 0.2

0.4 0.2 0.2 0.4 0.4 0.2 0.2 0.4

0.2 0.2

0.4 0.4

(a) Buckling pattern at R1 = 0.3, (b) Buckling pattern at R1 = 0.4, R2 =0.5 R2 =0.5

Figure 6.10: buckling patterns of system of interacting shells of outer radii R1 and R2, with shell parameters h1 = 0.01R1, h2 = 0.01R2, ν1 = ν2 = 0.3, and 1 = 2 = 200 embedded in host material ν0 = 0.49985 at the critical buckling pressure pc denoted by the black triangles in ﬁgure 6.7. Shell 1 (pink) is situated a unit distance directly above shell 2 (blue). Chapter 7

Summary and Further Work

7.1 Summary

Buckling, in both isolated shells and interacting shells, has been the theme of this thesis. Chapter 3 extended the theories around buckling in isolated shells, adjusting the work of Jones et al. [17] so that their theories could be expanded to composites containing highly contrasting materials. This allowed comparisons to be drawn between the work of Fok and Allwright [10] and the new adjusted Jones et al. work, yielding more accurate predictions for the buckling pressure of isolated shells under hydrostatic pressure than previously published. This also allowed predictions for the buckling pressure of isolated shells under uniaxial pressure to be evaluated for high contrast composites, yielding very different buckling pressures than those previously predicted by Jones et al. This correction of the buckling pressures of embedded shells under non-spherically symmetric loading then allowed consideration of the buckling of interacting shells within an elastic host medium. The first effects of interaction are seen in the linear elastic stress profiles of near neighbour shells as shell stresses loose their spherical symmetry, weakening the shells at the point nearest their neighbour. For very thin shells this leads to a decrease in the critical pressure needed to cause at least one of the shells to buckle, in comparison to that needed to buckle either shell in isolation. The effects of interaction are weaker for thicker shells but can lead to small stiffening effects, though in general thicker shells can be considered noninteracting as the critical buckling pressure needed to buckle either shell differs negligibly to that for isolated shells. For thin stiff shells however the decrease in pressure between interacting shells may need to be taken into account during the manufacturing process, especially when the volume of micro-shells within a host medium may be high. High volume fractions of micro-shells indicate near neighbour shells are in close proximity and thus that the critical buckling pressure at which one of these neighbours will buckle may be significantly reduced, possibly compromising the stiffness of the composite.

216 CHAPTER 7. SUMMARY AND FURTHER WORK 217

Often of concern within manufacturing processes is the robustness of micro-shell composites to perturbations in thickness or radius of the micro-shells. This work demonstrates that shell buckling pressures are robust to perturbations in shell thickness or radius, with small changes in shell thickness or radius only yielding small changes in critical buckling pressures. In summary, where near neighbour interaction between microspheres is suspected within a composite, due to high volume fractions or poor mixing, the buckling pressure of the microspheres will be lower than that of isolated shells provided the shells are thin and stiﬀ in comparison to the host medium, and this may change the eﬀective properties of the composite .

7.2 Further Work

If further work on the effect of interaction between shells were to be undertaken it would be useful to classify for what radii, shell thickness, or shell materials, the shells should be considered as interacting or non-interacting, whether this is a function of the shell buckling pressures in relation to the isolated shell buckling pressure, or whether the change in the buckling patterns of the shells should herald the start of interaction. Further work could also look into the effect of cavities upon shells. The work of this thesis shows that, unless both shells are of equal size, at the critical buckling pressure only one shell will buckle. Once a shell has buckled this effectively removes any stiffening effect of the shell and thus effectively leaves a cavity within the host medium. For further analysis of the overall characteristics of the composite it would be of use to know the effect of this cavity upon the remaining shells within the host medium, for example whether the cavity increases or decreases the buckling pressure of the remaining shells. Whilst some limitations of the current work are easily overcome, for example we model shells with no internal pressure but to change this would require minimal adjust- ments to the algebra, the assumption of axisymmetric buckling would be very complex to extend but may provide further useful insight into the phenomenon of interacting buckling shells. Whilst we have here concentrated on two-shell systems it would also be of use to model multi-shell systems. If the composite is poorly mixed during the manufacturing process it is possible for agglomerations of micro-shells to occur which may dramatically affect the local effective properties of the composite. The lack of symmetry in this type of multi-shell system would increase the mathematical complexity of the problem and may require a different approach to that taken during this thesis. This thesis is a wholly theoretical piece of work and therefore one of the most useful extensions to this work would be to experimentally validate the results. Currently CHAPTER 7. SUMMARY AND FURTHER WORK 218 buckling of embedded micro-shells has not been observed, as many methods that might be used to investigate the structure of a composite post buckling destroy the structure they are trying to observe. The lack of experimental evidence on embedded micro-shell buckling may also suggest that other effects, such as delamination, play a larger role in the micro-mechanics of these types of composites than buckling. As such it may be of interest to compare pressures at which buckling and delamination of embedded shells occur, before taking forward further research on the buckling of embedded micro-shells. Appendices

219 Appendix A

Solution methods in linear elasticity

When considering some of the classic linear elasticity problems it is useful to be familiar with both the type of solutions they provide and some methods of arriving at these solutions.

Legendre polynomials [15]:

The Legendre functions of the ﬁrst and second kind, Pℓ(x) and Qℓ(x), are linearly independent solutions to the Legendre diﬀerential equation

d dy (1 − x2) + ℓ(ℓ + 1)y = 0, dx dx which can also be written as

d2y dy (1 − x2) − 2x + ℓ(ℓ + 1)y = 0. dx2 dx

If ℓ is an integer the Legendre functions of the ﬁrst kind reduce to a polynomial known as the Legendre polynomial. These polynomials satisfy the recurrence relation

(2ℓ + 1)xPℓ(x) = (ℓ + 1)Pℓ+1(x)+ ℓPℓ−1(x), and orthogonality relations

1 2 P (x)P (x)dx = δ , (A.1) ℓ m 2ℓ + 1 mℓ −1 1 2ℓ(ℓ + 1) (1 − x2)P ′(x)P ′ (x)dx = δ , (A.2) ℓ m 2ℓ + 1 mn −1

220 APPENDIX A. SOLUTION METHODS IN LINEAR ELASTICITY 221 where prime denotes diﬀerentiation with respect to the argument.

The associated Legendre polynomials [14]:

The associated Legendre diﬀerential equation is a generalisation of the Legendre diﬀer- ential equation and can be written

d dy m2 (1 − x2) + ℓ(ℓ + 1) − y = 0, (A.3) dx dx 1 − x2 or setting x = cos φ as

d2y cos φ dy m2 + + ℓ(ℓ + 1) − y = 0. (A.4) dφ2 sin φ dφ sin2 φ (m) The solutions, Pℓ (x), are known as the associated Legendre polynomials (when ℓ is an integer), or the associated Legendre functions of the ﬁrst kind (if ℓ is not an integer), though when m is odd they are not in fact polynomials. The second solution (m) to the associated Legendre diﬀerential equation is given by Qℓ (x) and is known as an associated Legendre function of the second kind. The associated Legendre polynomials satisfy the recurrence relations

(m) (m) (m) (ℓ − m)Pℓ (x)= x(2ℓ − 1)Pℓ−1 (x) − (ℓ + m − 1)Pℓ−2 (x). (A.5)

The associated Legendre polynomials can be related to the Legendre polynomials by

m (m) m m d P (x) = (−1) (1 − x2) 2 P (x), (A.6) ℓ dxm ℓ and equate with the Legendre polynomials when m = 0. The associated Legendre polynomials obey the orthogonality relation

1 (m) (m) 2 (ℓ + m)! P (x)P ′ (x) dx = δ ′ . (A.7) ℓ ℓ 2ℓ + 1 (ℓ − m)! ℓℓ −1 See Appendix D for further identities involving Legendre functions.

A.1 Laplace’s equation and solutions

Laplace’s equation, ∇2ψ = 0, is integral to many areas of mathematics, such as electro- magnetism, ﬂuid dynamics and the study of heat conduction, where it is used describe the behaviour of potentials. It also appears as part of the equation of equilibrium for APPENDIX A. SOLUTION METHODS IN LINEAR ELASTICITY 222 elastostatics, when it is posed in terms of displacements. Laplace’s equation is the homogeneous form of the Poisson equation and in the study of heat conduction is known as the steady-state heat equation.

Laplace operator in spherical coordinates

The Laplace operation is given in tensor form by ∇2. In spherical polar coordinates, see ﬁgure 2.2, the Laplace operator can be written:

1 ∂ ∂ 1 ∂ ∂ 1 ∂2 ∇2 = r2 + sin φ + . (A.8) r2 ∂r ∂r r2 sin φ ∂φ ∂φ r2 sin2 φ ∂θ2 As this is separable, if we let ψ = R(r)Θ(θ)Φ(φ) then Laplace’s equation can be rearranged to sin2 φ ∂ ∂R sin φ ∂ ∂Φ 1 ∂2Θ r2 + sin φ = − . (A.9) R ∂r ∂r Φ ∂φ ∂φ Θ ∂θ2 As such each side of the equation must equal a constant, say k2, thus

∂2Θ + k2Θ = 0 ∂θ2 implies Θ = e±ikθ for k = 0, 1, 2,.... The other half of (A.9) can then be rearranged to

1 ∂ ∂R 1 ∂ ∂Φ k2 r2 = − sin φ + . R ∂r ∂r sin φΦ ∂φ ∂φ sin2 φ Again equating each side of the equation to a constant, m(m + 1) by convention, the radial equation in R is then

∂2R ∂R r2 + 2r = Rm(m + 1), ∂r2 ∂r and the equation in Φ becomes

∂2Φ ∂Φ k2 + cot φ + m(m + 1) − Φ = 0, ∂φ2 ∂φ sin2 φ the associated Legendre diﬀerential equation for x = cos φ and k = 0, ..., m [12]. By ∞ n+c posing a series solution R(r) = n=0 anr for the radial equation, it can be shown m −(m+1) that R(r)= Amkr + Bmkr , and the associated Legendre equation has solutions (k) Pm (cos φ), the associated Legendre polynomials of the ﬁrst kind. Therefore the general solution for ψ is

∞ m m −(m+1) (|k|) −ikθ ψ(r, φ, θ)= [Amkr + Bmkr ]Pm (cos φ)e . m =0 k=−m APPENDIX A. SOLUTION METHODS IN LINEAR ELASTICITY 223

If the problem is axisymmetric, i.e. if ψ is invariant with respect to θ, this is equivalent to setting k = 0 in the general solution. Thus for ψ(r, φ) we have the solution,

∞ m −(m+1) ψ(r, φ)= [Amr + Bmr ]Pm(cos φ), (A.10) m=0 where, as introduced above, Pm are Legendre polynomials of the ﬁrst kind. Appendix B

Expressing displacements at inﬁnity about a diﬀerent origin

To express a displacement, in spherical coordinates centered about one origin, in a new set of spherical coordinates centered about a diﬀerent origin, it is simplest to ﬁrst convert spherical coordinates to Cartesian coordinates; rewrite in terms of Cartesian coordinates at the second origin and convert back to spherical coordinates at that origin. We will run through an example for clarity. Given a displacement u expressed in spherical coordinates (ˆr, φ,ˆ θˆ) about origin Oˆ , so that,

uˆ =r ˆerˆ.

We wish to express the displacement of any point in the medium, with respect to the origin O, in terms of the spherical coordinates (r, φ, θ), as seen in ﬁgure B.1.

224 APPENDIX B. EXPRESSING DISPLACEMENTS AT INFINITY 225

xˆ3

φˆ Oˆ xˆ2 xˆ1 er θ rˆ eθ

erˆ L x3 eφ eφˆ φ r O x2 θ x1

Figure B.1: Two vertically aligned spherical coordinate systems with origins a distance L apart.

In this case origin Oˆ is aligned vertically with origin O, at a distance L away, such that θˆ = θ for any point in the medium.

We can ﬁrst express uˆ about origin Oˆ in terms of Cartesian coordinates (ˆx1, xˆ2, xˆ3) as

uˆ =x ˆ1e1ˆ +x ˆ2e2ˆ +x ˆ3e3ˆ. In Cartesian coordinates the unit vectors are the same with respect to any coordinate origin, and thus

e1ˆ = e1, e2ˆ = e2, e3ˆ = e3. This allows us to express the displacement in Cartesian coordinates about origin O as:

u =x ˆ1e1 +x ˆ2e2 +x ˆ3e3.

Since the Oˆ origin is only shifted vertically from the O origin, we can relate the two

Cartesian coordinate systems by,x ˆ1 = x1, xˆ2 = x2, xˆ3 = x3 − L, and so write the displacement in terms of only coordinates centered at origin O as

u = x1e1 + x2e2 + (x3 − L)e3.

We wish to express this displacement as

u = urer + uφeφ + uθeθ. APPENDIX B. EXPRESSING DISPLACEMENTS AT INFINITY 226

The components in the Cartesian and spherical coordinate systems are related by

ur sin φ cos θ sin φ sin θ cos φ x1 uφ = cos φ cos θ cos φ sin θ − sin φ x2 , u − sin φ cos φ 0 x  θ     3       where x1 = r sin φ cos θ,x2 = r sin φ sin θ,x3 = r cos φ. This leads to the solution in spherical coordinates

u = (r − L cos φ)er + L sin φeφ. Appendix C

Thin shell approximation for radial stress

For a spherically symmetric problem thin shell theory sets the radial stress σrr to zero when there is no loading within the shell. The radial stress is approximated to zero as it is signiﬁcantly smaller than the hoop stresses. If we actually calculate the radial stress throughout the shell the equation of equilibrium ∇ σ = 0 implies that

A 2B σ = E − rr s 1 − 2ν (1 + ν )r3 s s where Es and νs are the Young’s modulus and Poisson ratio of the shell. If we consider h the shell to have no loading on its inner surface R − 2 and an applied pressure −p on h h its outer surface R + 2 , then at R − 2

A 2B 0= Es − , 1 − 2ν h 3 s (1 + νs)(R − 2 ) 2B(1 − 2ν ) ⇒ A = s . h 3 (1 + νs)(R − 2 )

h At R + 2 we then have

2BEs 1 1 −p = 3 − 3 , 1+ νs h h R − 2 R + 2

− p(1 + νs) ⇒ B = 1 1 2Es − . h 3 h 3 R − 2 R + 2

227 APPENDIX C. THIN SHELL APPROXIMATION FOR RADIAL STRESS 228

Therefore throughout the shell,

1 1 − h 3 r3 R − 2 σrr = −p , 1 1 − h 3 h 3 R − 2 R + 2 ∼ O(p)

p(R + O(h)) In comparison to σ = − we can see that for small h, σ ≫ σ and θθ 2h θθ rr thus σrr can be ignored. Appendix D

Useful identities for Legendre polynomials

Throughout this thesis we manipulate Legendre polynomials to simplify expressions. The identities we use most and how these identities are derived are listed below. To derive these identities we will make use of the Legendre diﬀerential equation, which for

Pn(cos φ) can be written as

2 ′′ ′ sin φPn = 2 cos φPn − n(n + 1)Pn,

′ where Pn indicates diﬀerentiation with respect to cos φ, and the iterative property of Legendre polynomials, given by

(2n + 1) cos φPn = (n + 1)Pn+1 + nPn−1.

A basic identity linking Legendre polynomials and their derivatives is given by

1 P = (P ′ − P ′ ). (D.1) n 2n + 1 n+1 n−1

2 ′ We also make use of identities involving sin φPn. These identities can be derived using (D.1) (and its differential), the Legendre differential equation and the iterative properties of the Legendre polynomials. For example, we can find

n(n + 1) sin2 φP ′ = − (P − P ), n 2n + 1 n+1 n−1

229 APPENDIX D. USEFUL IDENTITIES FOR LEGENDRE POLYNOMIALS 230 through,

sin2 φ sin2 φP ′ = (P ′′ − P ′′ ), n 2n + 1 n+1 n−1 1 = 2 cos φP ′ − (n + 1)(n + 2)P − 2 cos φP ′ + n(n − 1)P , 2n + 1 n+1 n+1 n−1 n−1 2 cos φ (n + 1)(n + 2)P − n(n − 1)P = (P ′ − P ′ ) − n+1 n−1 , 2n + 1 n+1 n−1 2n + 1 (n + 1)(n + 2)P − n(n − 1)P = 2 cos φP − n+1 n−1 , n 2n + 1 2 (n + 1)(n + 2)P − n(n − 1)P = ((n + 1)P + nP ) − n+1 n−1 , 2n + 1 n+1 n−1 2n + 1 n(n + 1) = − (P − P ). 2n + 1 n+1 n−1

This allows us to derive

(n + 1) n(n + 1) (n + 1) cos φP − sin2 φP ′ = ((n + 1)P + nP )+ (P − P ), n n 2n + 1 n+1 n−1 2n + 1 n+1 n−1

= (n + 1)Pn+1, and

n n(n + 1) n cos φP + sin2 φP ′ = ((n + 1)P + nP ) − (P − P ), n n 2n + 1 n+1 n−1 2n + 1 n+1 n−1

= nPn−1. References

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