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 Copyright 19 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 ap- proximations ............................. 108 3.2.2 Uniaxial compression: The shell equilibrium solution and its ap- proximations ............................. 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 µ1 = 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 hydro- static pressure (3.112), and the two-shell system. Shell parameters are µ1 = 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 (µ1 = 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,µ1 = 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 µ1 = 10 (blue, solid), µ1 = 10 (pink, small dashing), µ1 = 10 (yellow, dot-dashed), µ1 = 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 µ1 = 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 µ1 = 1.05 within a medium in which ν0 = 0.49985 and µ0 = 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