Universit`A Di Trento Dottorato Di Ricerca in Ingegneria Dei Materiali

Universit`adi Trento Dottorato di Ricerca in Ingegneria dei Materiali e delle Strutture XV ciclo

BOUNDARY ELEMENT TECHNIQUES IN FINITE ELASTICITY

MICHELE BRUN

Relatore: Prof. Davide Bigoni Correlatore: Dr. Domenico Capuani

Trento, Febbraio 2003 Universit`adegli Studi di Trento Facolt`adi Ingegneria Dottorato di Ricerca in Ingegneria dei Materiali e delle Strutture XV ciclo

Esame ﬁnale: 14 febbraio 2003

Commissione esaminatrice: Prof. Giorgio Novati, Universit`adegli Studi di Trento Prof. Francesco Genna, Universit`adegli Studi di Brescia Prof. Alberto Corigliano, Politecnico di Milano

Membri esperti aggiunti: Prof. John R. Willis, University of Cambridge, England Prof. Patrick J. Prendergast, Trinity College, Dublin Acknowledgements

The research reported in this thesis has been carried out in the Department of Mechanical and Structural Engineering at the University of Trento during the last three years. I would like to gratefully acknowledge my supervisor, Prof. Davide Bigoni, for all the stimulating and brilliant discussions, for his non-stop encourage- ments, remarks, corrections and suggestions. I would like to thank Dr. Domenico Capuani for his accurate, precise and fruitful help. I wish to express my sincere gratitude to Prof. Giorgio Novati, and to all members of the group of mechanics at the University of Trento: Prof. Marco Rovati, Prof. Antonio Cazzani (and Lavinia), Dr. Roberta Springhetti (and child) and little scientists Arturo di Gioia, Giulia Frances- chini, Massimilano Margonari and Andrea Piccolroaz. I am indebted to Massimiliano Gei for his suggestions about the topic of my research and to Katia Bertoldi for the help in the development of numerical applications. Last but not the least, I thank my family and the unique Elisa for their love, patience and fully support.

Trento, January 2003

Michele Brun

1 2 Contents

0.1 Introduction...... 6 0.2 Outlineofthethesis ...... 7 0.2.1 A discussion of some of the obtained results ...... 7 0.2.2 Openproblems ...... 8

1 Linear-isotropic incompressible elasticity 9 1.1 Basicequations ...... 9 1.2 Thefundamentalsolution ...... 10 1.2.1 Thestreamfunction ...... 10 1.2.2 TheGreen’sfunction...... 13 1.2.3 TheGreen’sin-planepressure...... 14

2 Integral representations and symmetric Galerkin formulation for incompressible elasticity and Stokes ﬂow 17 2.1 Integral equation formulation ...... 18 2.2 Integral representation at the boundary ...... 24 2.2.1 Examples ...... 32 2.3 Symmetric formulation of the boundary element method . . . . 39

3 Elements of Continuum Mechanics 45 3.1 Kinematics ...... 45 3.1.1 Motions ...... 50 3.1.2 Material and Spatial Derivatives ...... 50 3.1.3 Rate of Deformation ...... 51 3.2 Stress ...... 52 3.3 Invariance of material response ...... 53 3.3.1 Objectiverates ...... 56

3 4 Contents

3.4 Constitutiveequations ...... 57 3.4.1 Isotropic materials ...... 58 3.4.2 Hyperelastic materials ...... 60 3.4.3 Some elastic potentials ...... 63

4 Incremental Deformations 67 4.1 Incremental constitutive equations ...... 67 4.1.1 Incrementalmoduli...... 69 4.1.2 Mooney-Rivlin material ...... 72 4.1.3 Ogdenmaterial...... 72 4.1.4 Hypoelasticity and the loading branch of elastoplastic constitutivelaws ...... 73 4.1.5 J2 material: hyperelastic and hypoelastic approaches . . 74 4.1.6 The loading branch of non-associative, elastoplastic law 74 4.1.7 The general form of constitutive equations for plane, incompressible, incremental deformations ...... 76

5 Green’s function for incremental non-linear elasticity 79 5.1 Theequilibriumequations ...... 79 5.2 Theregimeclassification ...... 80 5.3 Thestreamfuction ...... 82 5.4 Thevelocityfield ...... 84 5.5 Thevelocitygradient...... 85 5.6 Theincrementalstressfield ...... 85 5.7 The gradients of the Green’s tensor set ...... 87 5.7.1 The velocity gradient ...... 87 5.7.2 The second-gradient of velocity ...... 88 5.7.3 Thegradientofpressurerate ...... 90

6 Boundary elements formulation 91 6.1 Integral representation ...... 92 6.2 Boundary discretization ...... 93 6.3 Two numerical examples: non-linear elasticity without domain integrals...... 97 6.3.1 In-plane tension and compression ...... 97 6.3.2 Simpleshear ...... 98 6.4 Conclusion ...... 99 Contents 5

7 Numerical examples 101 7.1 Bifurcation of elastic structures ...... 103 7.1.1 Elasticblock ...... 103 7.1.2 Layered elastic material ...... 109 7.1.3 Cracked elastic blocks ...... 112 7.1.4 An application to rubber bearings ...... 113 7.2 Shear bands within the elliptic range ...... 116

A Remarks on the notation and theorems 125

B Biot’s expression of incremental moduli 129

C Out-of-plane stress component for the Ogden hyperelastic material 131

Bibliography 131 6 Contents

0.1 Introduction

Large strain effects influence stiffness of structures, induce mechanical anisotropy in materials, affect decay rates of self-equilibrated loads —connected to the Saint Venant’s principle— and influence wave propagation and dynamical response. Moreover, the large strain formalism is the key for the analysis of bifurcation phenomena and material instabilities. The latter has attracted an intense research effort in recent years and has been often approached under small strain hypotheses. However, a finite strain formulation is the only complete and fully consistent way to analyze material instabilities, strain localization, and bifurcations. In a number of engineering problems large strain effects are of chief im- portance. Microelectromechanical Systems are subject to severe state of pre- stress that, taken alone, may lead to failure of the device (Elwenspoek and Wiegerink, 2001). Pre-stress affects the behaviour of geological formations (Triantafyllidis and Lehner, 1993; Triantafyllidis and Leroy, 1994), biological systems (Demiray, 1996; Holzapfel et al., 2000), and is a concern in a number of structural elements, including seismic insulators and rubber bearings (D’Ambrosio et al., 1995; Kelly, 1997). The need for applications attracted a research effort so broad and intense that many problems can nowadays be considered concluded. For instance, commercial finite element codes embody user-friendly routines for large strain analysis of various elastic materials and produce quite reliable results, at least when local instabilities such as strain localization are not involved. However, compared to finite element methods, the boundary element techniques —well developed for the infinitesimal theory— have been much less explored for large strain problems. The usual approach to boundary elements when large strains are involved consists in the use of the fundamental solution referred to infinitesimal isotropic elasticity. Since incremental strains involve anisotropic stiffness and effects related to the pre-stress, it is to be expected that such an approach is essentially unsatisfactory. General procedures for obtaining Green’s function and boundary integral equations for incremental problems of non-linear elasticity are known (Willis, 1991). However, only recently an explicit formulation for incremental, plane strain, incompressible elasticity has been given (Bigoni and Capuani, 2002), thus providing a new perspective for boundary elements applications. In addition, Bigoni and Capuani also provided a strategy to analyze strain localization in terms of perturbation, a way also never explored. Many problems were left unanswered in that work. For instance, can the boundary integral equa- 0.2 Outline of the thesis 7 tions be implemented and result competitive with finite element codes? May the incremental solution be used as a basis to analyze large strains? Can the resolution of internal fields be sufficient to analyze localized phenomena such as shear band formation? May the shear band formation be analyzed employing a perturbative technique within the elliptic range, even for boundary value problems involving finite bodies? The present thesis provides a contribution to the clarification of these and related issues.

0.2 Outline of the thesis

The thesis is organized as follows. Small strain incompressible elasticity is considered in Chapters 1 and 2. The merits in the formulation lie in the formal analogy to the theory of slow, viscous flow of a fluid. Results may therefore find application in fluid mechanics and serve to introduce the more complex problems of incremental non-linear incompressible elasticity, addressed in Chapters 3–7. However, a number of new results are found and presented in Chapter 2 . Particularly, a symmetric Galerkin boundary element formulation is given for two-dimensional, steady and incompressible flow. The formulation requires the derivation of certain per se relevant integral representations at the boundary for velocity gradient and pressure, which turn out to be coupled at angular points of the contour profile. A syntetic review of large strain elasticity, incremental formulation and the Green’s function is presented in Chapters 3, 4 and 5, respectively. The discretization and numerical implementation is detailed in Chapter 6 together with two numerical applications for non-linear homogeneous deformations, whereas numerical examples relative to small deformations superimposed upon a given homogeneous strain are presented in Chapter 7. The examples provide a systematic investigation of shear band formation, analyzed using a perturbative technique within the elliptic range. In the numerical approach, the incremental solution is employed as the basic step to build a large strain analysis.

0.2.1 A discussion of some of the obtained results The numerical examples show that the boundary element technique is particularly suitable to the analysis of bifurcation phenomena, including strain localization. Moreover, the fact that strain localization can be traced when ellipticity holds implies that regularization methods are not needed. Obviously, the shear band thickness becomes function of a characteristic dimension of the perturbation employed to trigger the localization. The current literature 8 Contents is abundant of proposals of regularization techniques for strain localization problems. A considerable number of these are not clearly motivated from mechanical point of view and can be considered mere remedia. We have presented an eﬀort to overcome the problem in a non-traditional way. The boundary element technique is in essence diﬀerent from the available formulations. A consequence of this is that domain integrals in the usual sense are avoided.

0.2.2 Open problems A number of problems have not been completed in this work. First, the boundary integral equations for pressure, deformation and stress presented in Chapter 2 for incompressible infinitesimal elasticity can be extended to the incremental context. Second, the shear band analysis should be extended to a postcritical range, so that a full evidence of localization development can be given. Third, the numerical approach to large strain was intended to represent a first proposal. It can therefore be refined in different directions. Finally, all the formulation can be generalized, for instance, finding the Green’s function for dynamic loading or for three-dimensional situations. Chapter 1

Linear-isotropic incompressible elasticity

In this Section a brief review of the constitutive laws for incompressible elasticity (and slow viscous ﬂow) is presented, together with the inﬁnite body Green’s function for two-dimensional deformations. The formulation is classical, see Ladyzhenskaya (1963), and poses the necessary basis for integral formulations and large deformations.

1.1 Basic equations

An incompressible, linear-elastic, isotropic solid occupying a two-dimensional domain Ω of closed boundary ∂Ω is considered. Due to incompressibility, the displacement u at a generic point x satisﬁes the condition div u = 0. (1.1) The constitutive equation for the stress σ is given by

σ = 2µE + pI, (1.2) where µ is the shear modulus, E is the inﬁnitesimal strain tensor 1 E = u + uT , (1.3) 2 ∇ ∇ and p is the pressure In the presence of a body force b(x), the equilibrium equation is

div σ + b = 0, (1.4)

9 10 Linear-isotropic incompressible elasticity which can be written in terms of displacements as

µ u + p + b = 0. (1.5) 4 ∇ In a mixed boundary-value problem, eqn. (1.5) is accompanied by displacements u and tractions τ = σn assigned on non-overlapping portions ∂Ωu and ∂Ωτ of the boundary:

u = u¯, on ∂Ωu, τ = τ¯, on ∂Ωτ . (1.6)

1.2 The fundamental solution

As will be shown in the next chapter, to the general solution of the boundary value problem (1.5)-(1.6) can be given an integral representation by using the Green’s function for the inﬁnite body. This is the so-called ”fundamental solution” which is associated to a concentrated force acting at point y, and which satisﬁes the equation:

µ ug + pg + egδ(x y)= 0, (1.7) 4 ∇ − g where δ(x) is the Dirac delta function and ei = δig. Here and in the following, the apex g is used to indicate quantities relevant to the Green’s state. In order to determine the Green’s function ug, a stream function ψg is introduced which satisﬁes the relations

ug = ψg , ug = ψg , (1.8) 1 ,2 2 − ,1 together with condition (1.1). Expressions of ψg, ug and pg are carried out in the sub-sections below.

1.2.1 The stream function

It is useful to re-write eqn. (1.7) for both the directions x1 and x2 of a cartesian reference system:

g g g µ u1,11 + µ u1,22 + p,1 + δ1g δ(r) = 0, (1.9) g g g µ u2,22 + µ u2,11 + p,2 + δ2g δ(r) = 0, where (.) represents differentation along the direction x , and r = x y ,i i − with the point y playing the role of a parameter. Note that -unless otherwise stated- differentiation is assumed to be with respect to the field variable xi. 1.2 The fundamental solution 11

By substituting eqn. (1.8) into eqns. (1.9)1 and (1.9)2, diﬀerentiating them with respect to x2 and x1 respectively, and subtracting the results, it is possible to eliminate the pressure obtaining

g g g µ ψ,1111 + 2 ψ,1122 + ψ,2222 + δ1g δ(r),2 + δ2g δ(r),1 = 0. (1.10) Introducing the linear diﬀerential operator

∂4( ) ∂4( ) ∂4( ) ( )= µ · + 2 · + · , (1.11) L · ∂x4 ∂x2x2 ∂x4 1 1 2 2 eqn. (1.10) can be written as

∂δ(r) ∂δ(r) ψg + δ δ = 0. (1.12) L 1g ∂x − 2g ∂x 2 1 Since the plane wave expansion of the δ function is (Courant and Hilbert, 1962; Gel’fand and Shilov, 1964)

1 dω δ(r)= , (1.13) −4π2 (ω · r)2 ωI=1 | | where ω is an unit vector (see ﬁg. [1.1]), the analogous Radon transformation for the stream function ψg is

1 ψg(r)= ψ˜g(ω · r) dω. (1.14) −4π2 ωI=1 | | Therefore, in the transformed domain, eqn. (1.12) becomes

∂ 1 ∂ 1 ψ˜g(ω · r)+ δ δ = 0, (1.15) L 1g ∂x (ω · r)2 − 2g ∂x (ω · r)2 2 1 and so δ ω δ ω ψ˜g = 2 1g 2 − 2g 1 . (1.16) L (ω · r)3 Employing the chain rule of diﬀerentation

∂ g g 0 ψ˜ (ω · r)= ωk(ψ˜ ) , (1.17) ∂xk 12 Linear-isotropic incompressible elasticity

1 r x a q y

O x1

Figure 1.1: Vectors ω, r and angles α and θ.

where a prime denote diﬀerentation with respect to the argument ω · r, eqn. (1.16) becomes 0000 δ ω δ ω L(ω)(ψ˜g) = 2 1g 2 − 2g 1 , (1.18) (ω · r)3 where 2 2 2 L(ω)= µ ω1 + ω2 = µ. (1.19) The integration of (1.18) yields

g 3 µ ψ˜ = (δ1gω2 δ2gω1)(ω · r)(log ω · ˆr 1) + C1(ω · r) − | |− (1.20) 2 +C2(ω · r) + C3(ω · r)+ C4, where ˆr represents a dimensionless measure of r (i.e. ˆr = r/r0, with r0 being an arbitrary distance). The coeﬃcients of the integration constants C1, C3, turn out to be zero in the transform (1.14), whereas C2 and C4 can be set equal to zero for the properties of the Green’s displacement and stress ﬁelds. Hence, the stream function ψ˜g can be cast in the form δ ω δ ω ψ˜g(ω · r)= 1g 2 − 2g 1 (ω · r)(log ω · ˆr 1). (1.21) µ | |− According to eqn. (1.14), in a reference system with polar coordinates r(= r ) and θ singling out the position of the generic point x with respect | | 1.2 The fundamental solution 13 to the application point y of the concentrated force, the stream function is determined by the antitransform

2π g r π ψ = 2 sin α+θ + (1 g) cos α [log( rˆcos α ) 1] dα, (1.22) −4π µ 0 − 2 | | − Z h i which may be expanded to yield π g r π ψ = 2 (log(ˆr) 1) sin α + θ + (1 g) cos α dα −2π µ − 0 − 2 Z h i π/2 π + log(cos α) sin α+θ + (1 g) cos α dα (1.23) 0 − 2 Z h i π π + log( cos α) sin α+θ + (1 g) cos α dα , π/2 − − 2 ) Z h i where the ﬁrst integral can be easily evaluated in a closed-form while the second and the third integral, with singular integrands, need some algebraic manipulations. By using a change of variable α = π α in the third integral 0 − of eqn. (1.23) and the relation sin(a + b) sin(a b) = 2 cos a sin b we obtain − − r π π ψg = (log(ˆr) 1) sin θ + (1 g) −2π2µ − 2 − 2 n h i (1.24) π π/2 + 2sin θ + (1 g) log(cos α) cos2α dα , − 2 0 ) h i Z which corresponds to

r(3 g) rˆ 1 ψg(r) = (2g 3) − log , (1.25) − 4πµ 2 − 2 where r1 = r cos θ and r2 = r sin θ. Eqns. (1.21) and (1.25) give the stream function of the fundamental solution in the transformed and antitransformed domain respectively.

1.2.2 The Green’s function According to eqns. (1.8) and (1.21), the displacement of the Green’s state in the transformed domain is given by

(eg ω ωgI) ω u˜g(ω · r)= ⊗ − log ω · ˆr , (1.26) µ | | 14 Linear-isotropic incompressible elasticity

g where ei = δig (see Appendix A). Consistently with eqn. (1.13), the antitransform is given by

1 ug(r)= u˜g(ω · r) dω, (1.27) −4π2 ωI=1 | | which leads to π g 1 π π ui = −2 logr ˆ sin α + (1 g) sin α + (1 i) dα 2π µ 0 − 2 − 2 Z h i h i π π π + log( cos α )sin α + θ + (1 g) sin α + θ + (1 i) dα , −0 | | − 2 − 2 Z h i h i (1.28) where denotes the Cauchy principal value. From eqn. (1.28), using the − relations R π/2 π cos2α log(cos α) dα = (1 log 4), 0 8 − Z (1.29) π/2 π sin2α log(cos α) dα = (1+ log 4), − 8 Z0 and neglecting constant terms, the Green’s function is obtained 1 r ug(r)= eg log r g r . (1.30) −4πµ − r2 It is worth noting that the same result can be obtained by diﬀerentiating eqn. (1.25) and using the chain rule:

∂(.) ∂(.) sin θ ∂(.) = cos θ , ∂x1 ∂r − r ∂θ (1.31) ∂(.) ∂(.) cos θ ∂(.) = sin θ + . ∂x2 ∂r r ∂θ

1.2.3 The Green’s in-plane pressure

g In order to determine p , eqns. (1.9)1 and (1.9)2 are diﬀerentiated with respect to x1 and x2, respectively, and, taking the incompressibility constraint into account, are summed to give

pg = eg · δ(r). (1.32) 4 − ∇ 1.2 The fundamental solution 15

According to (1.13), the Green’s in-plane pressure can be written as 1 pg(r)= p˜g(ω · r) dω, (1.33) −4π2 ωI=1 | | and from (1.32)-(1.33), it follows that

g ωg (˜p )00 = 2 . (1.34) (ω · r)3 Integraton of eqn. (1.34) gives ω p˜g = g + C (ω · r)+ C , (1.35) ω · r 5 6 where the term with C5 turns out to be zero in the transform (1.33) and the constant C6 can be set equal to zero, leading to ω p˜g(ω · r)= g . (1.36) ω · r From eqns. (1.33) and (1.36), the following equation is obtained

1 π sin θ + (1 g) π π pg = cos θ + (1 g) − 2 tan α dα , (1.37) −2πr ( − 2 − π −0 ) h i Z which leads to the final expression 1 r pg(r)= g . (1.38) −2π r2 eqns. (1.30) and (1.38) provide the expressions of displacement and pressure of the Green’s function set ug,pg . { } Finally, by differentiating expressions (1.30) and (1.38), and by using the constitutive relation (1.2), gradient fields and stress σg of the Green’s state can be obtained: 1 r r ug (r)= r eg δ r r ek + 2 g k r , (1.39) ,k −4πµr2 k − gk − g r2 1 r pg(r)= eg 2 g r , (1.40) ∇ 2πr2 − r2 1 r σg(r)= g r r, (1.41) −π r4 ⊗ 16 Linear-isotropic incompressible elasticity

1 r r σg (r)= r ek r + r r ek + δ r r 4 g k r r . (1.42) ,k −πr4 g ⊗ g ⊗ gk ⊗ − r2 ⊗ Chapter 2

Integral representations and symmetric Galerkin formulation for incompressible elasticity and Stokes ﬂow

In this Chapter, a boundary integral formulation is given for small strain incompressible elasticity. The merits of the formulation lie in the formal analogy to the theory of viscous flow of a fluid (Stokes flow). Results may therefore find application in fluid mechanics and serve to introduce the more complex problems of incremental non-linear incompressible elasticity, which will be addressed in the next Chapters. Integral equations and related numerical techniques are classical in linear, solid and fluid mechanics (Lamb, 1932). Among these numerical techniques, the boundary element method permits a successfull treatment of flow incompressibility constraint and has been therefore thoroughly employed (Youngren and Acrivos, 1975; Wu and Wahbah, 1976, and, more recently, Pozrikidis, 1992, 1997a, 2001). However, direct boundary element methods involve an unsymmetrical coefficient matrix of the final solving algebraic system. This is certainly a drawback of the method, particularly when boundary elements are coupled to finite elements. Following an approach known in solid mechanics, symmetry is recovered in the so-called symmetryc Galerkin boundary element method (see Bonnet et al. 1998 and reference quoted therein), a technique apparently not developed for incompressible solids and in hydrodynamics. This development is the focus of the present chapter. To this purpose, integral representations of pressure and stress tensor at the boundary are derived, representing generalizations of formulations valid in interior points given by Ladyzhenskaya (1963). In particular, integral representations at corner points

17 Integral representations and symmetric Galerkin formulation 18 for incompressible elasticity and Stokes flow of the boundary are obtained, providing explicit expression for the so-called ‘free-term matrices’. Interestingly, the integral representations for pressure and displacement gradient turn out to be coupled at corner points and — in the hypersingular form— also involve terms depending on the boundary curvatures at the corner. Analyses of corner points is believed to be relevant in relation to a number of problems in incompressible elasticity and Stokes flow: for instance, inclusion or dislocation (Willis, 1968; Mura, 1987), flow past non-smooth rigid particles (Pakdel and Kim, 1996) or cusp formation at fluid interfaces (Pozrikidis, 1997b; 1998). In the special case of smooth boundary (the so-called ‘Lyapunov surfaces’), the equations de-couple and the integral formulations obtained by Pozrikidis (2001) and by Liron and Barta (1992) are recovered when the boundary conditions pertain to flow past a rigid particle of arbitrary shape. The obtained integral equations are finally employed to estabilish a symmetryc Galerkin formulation.

2.1 Integral equation formulation

In Section (1.1), the differential equations governing the static behaviour of an incompressible, isotropic, linear elastic solid have been given. In particular, the interior problem has been formulated, with reference to a finite domain Ω of closed boundary ∂Ω. Obviously, the same equations hold for the exterior problem, defined on an unbounded domain Ω containing a closed boundary ∂Ω. This may be the case of an infinite medium with a cavity on which general boundary conditions (1.6) are prescribed. Of course, the solution of the exterior problem is required to satisfy decay conditions far from the cavity or inclusion. Both the interior and the exterior problem can be given an integral formulation which, for the latter, keeps the advantage of taking the decay condition at infinity intrinsically into account. Here, the integral formulation is given for the exterior problem, considering the unperturbed displacement and pressure fields u∞ and p∞ of the infinite medium without any disturbance (cavity or inclusion), and the corresponding perturbation fields uD and pD originating in the medium from the presence of a cavity or inclusion. In this case, the total displacement u and pressure p in any point of the medium surrounding the cavity (or inclusion) ∂Ω, can be written as

D D u = u∞ + u , p = p∞ + p . (2.1) 2.1 Integral equation formulation 19

The integral equations given by Ladyzenskaya (1963) for u and p, are

g · ug(y)= αug∞(y)+ u (x, y) σ(x) n(x) dlx Z ∂Ω (2.2) u(x) · σg(x, y) n(x) dl , − x ∂ZΩ

g p(y)= αp∞(y) [σ(x) n(x)] p (x, y) dl − g x Z ∂Ω (2.3) + 2µ u (x) pg(x, y) · n(x) dl . g ∇ x ∂ZΩ In eqns. (2.2), (2.3), α is a parameter which allows to switch from the exterior to the interior problem. In the exterior problem, α = 1 and y is a point of the inﬁnite medium surrounding ∂Ω; in the interior problem, α = 0 and y is a point of the medium enclosed by ∂Ω. In both cases of exterior and interior problems, n is the unit outward normal to ∂Ω (pointing out of the solid). The procedure for obtaining eqns. (2.2), (2.3), is outlined in the following for the exterior problem. As is shown in ﬁg. [2.1b], the region Φ is considered, delimited by contours ∂Ω, Γ and Θ , where Γ and Θ are circles centered in y, of radius ε (ε 0) ε ρ ε ρ → and ρ (ρ ), respectively. →∞ Due to the equilibrium equations

divσD(x)= 0 and divσg(x, y)= 0, (2.4) the following identity holds

divσg(x, y) · uD(x) divσD(x) · ug(x, y) dx = 0. (2.5) − Z Φ Under the integral sign, the following relations can be carried out

g · D g D g D divσ u = σij ui σij ui,j, (2.6) ,j − h i divσD · ug = σD ug σD ug . (2.7) ij i ,j − ij i,j Recalling the constitutive equation Integral representations and symmetric Galerkin formulation 20 for incompressible elasticity and Stokes ﬂow

Figure 2.1: Geometry, boundary conditions and countours considered.

σ = E( u)+ pI, (2.8) ∇ where

Eijkl = µ(δikδjl + δilδjk), (2.9) taking the major simmetry Eijkl = Eklij into account, together with relation (1.1), eqn. (2.5) can be reduced to

g D D g σij (x, y) ui (x) σij (x) ui (x, y) dx = 0. (2.10) − ,j ZΦ h i Applying the divergence theorem to the left-hand side of eqn. (2.10), this becomes

σg (x, y) n uD σDn ug (x, y) dl = 0. (2.11) ij j i − ij j i x ∂ZΦh i 2.1 Integral equation formulation 21

By separating the diﬀerent contributions on ∂Ω, Γε and Θρ, eqn. (2.11) is re-written as

τ g(x, y) uD τ D ug (x, y) dl = τ g(x, y) uD τ D ug(x, y) dl , − i i − i i x j j − j j x Z Z Γε+Θρ ∂Ωh i (2.12) D D g g with τi = σij nj and τi = σij nj. As for the integral on Γε at the right-hand side of eqn. (2.12), recalling expressions (1.30), (1.41) of ug and σg, and taking the zeroth-order series expansions

τ (x)= τ (y)+ ( r ), (2.13) ◦ | |

uD(x)= uD(y)+ ( r ), (2.14) ◦ | | in the limit ε 0, it gives →

D g lim τi u (x, y) dlx = 0, (2.15) ε 0 i → ΓZε

g D D lim τ ui (x, y) dlx = Cigui , (2.16) ε 0 i → ΓZε where

g Cig = lim σ (x, y) nj (x) dlx, (2.17) ε 0 ij → ΓZε is the so-called C-matrix. It involves a regular integral, since σg is proportional to 1/ε, and dlx = ε dθ on Γε. For interior points, it turns out to be Cig = δig. As for the integral on Θρ at the right-hand side of eqn. (2.12), it has to satisfy the decay condition at inﬁnity

D g g D lim τ (x) u (x, y) τ (x, y) u (x) dlx = 0. (2.18) ρ i i − i i →∞ Z Θρ Hence, eqn. (2.12) reduces to

uD = τ D·ug(x, y) τ g(x, y) ·uD dl . (2.19) g − x Z ∂Ω Integral representations and symmetric Galerkin formulation 22 for incompressible elasticity and Stokes ﬂow

The corresponding integral equation for the unperturbed ﬁeld u∞, for the region enclosed by ∂Ω, external to Φ, is

g g [τ ∞·u (x, y) τ (x, y) ·u∞] dl = 0, (2.20) − x ∂ZΩ which, subtracted to eqn. (2.19), gives eqn. (2.2). Finally, the procedure to obtain the integral equation (2.3) for pressure p, is described. To this purpose, eqn. (2.11) is diﬀerentiated twice with respect to the variable y and constitutive tensor E is applied, yielding

E σD n ug (x, y) σg (x, y) n uD dl = 0. (2.21) knsg ij j i,sn − ij,sn j i x ∂ZΦ h i Under the integral sign, the ﬁrst term can be expressed taking the equilibrium eqn. (2.4)2 into account

pi, = E ui = E ug , (2.22) − k knsg g,sn knsg i,sn so that E σD n ug (x, y)= pi (x, y) (σDn) . (2.23) knsg ij j i,sn − ,k i To evaluate the second term in eqn. (2.21), taking the constitutive equation g (2.8) for σij,sn, and substituting the derivative of eqn. (2.22), yields

E σg n uD = E pp + E pg δ n uD, (2.24) knsg ij,sn j i − ijqp ,kq knsg ,sn ij j i where E p j i i ijqp p,kq = µ p,ik + p,jk = 2 µp,jk, (2.25) whereas the quantity

E pg = µ pk + pn = 2 µ pk, (2.26) knsg ,sn ,nn ,kn 4 is zero for points x = y, in view of eqn. (1.32). 6 Hence, eqn. (2.21) transforms into

2 µn uD pi (x, y)+ σD n pi(x, y) dl = 0, (2.27) − j i ,j ij j ,k x Z ∂Φ 2.1 Integral equation formulation 23 which can be integrated to yield

2 µ uD pg(x, y) · n + (σD n) pg(x, y) dl = 0. (2.28) − g ∇ g x Z ∂Ω+Γε+Θ ρ Accounting for the property

pg(x, y) · n(x) dl = pg dx = 0, (2.29) ∇ x 4 ∂IΦ ΦZ which holds for any region Φ which does not contain the singular point y due to eqn. (1.32), in the limiting process we have

D g · D g · lim ug (x) p (x, y) n dlx = ug (y) lim p (x, y) n dlx = 0, (2.30) ε 0 ∇ ε 0 ∇ → → ΓZε ΓZε

2π D g D n(θ) n(θ) lim (σ n) (x) p (x, y) dlx = σ (y) · ⊗ dθ. (2.31) ε 0 g 2π → ΓZε Z0 The last integral in eqn. (2.31) can be explicitly computed, yielding

2π n(θ) n(θ) σD(y) · ⊗ dθ = σD(y) · I = pD(y). (2.32) 2π Z0 Due to a decay condition at inﬁnity analogous to (2.18), the integral along Θρ has to vanish by taking the limit for ρ tending to inﬁnity. Hence eqn. (2.28) reduces to

pD(y)= (σD n) pg(x, y)dl + 2 µ uD pg(x, y) · n dl . (2.33) − g x g ∇ x ∂ZΩ ∂ZΩ

The corresponding integral equation for the unperturbed ﬁeld p∞, for the region enclosed by ∂Ω, is

g g (σ∞ n) p (x, y)dl 2 µ u∞ p (x, y) · n dl = 0, (2.34) g x − g ∇ x ∂ZΩ ∂ZΩ which, summed to eqn. (2.33), gives eqn. (2.3), as pD = p p . − ∞ Integral representations and symmetric Galerkin formulation 24 for incompressible elasticity and Stokes ﬂow

Integral equations for u and σ in interior points can be obtained from ∇ diﬀerentiation of eqn.(2.2) with respect to the variable y, and from constitutive eqn. (1.2), yielding

g g u (y)= αu∞ (y) u (x, y) · σn dl + u · σ (x, y)n dl , (2.35) g,k g,k − ,k x ,k x ∂ZΩ ∂ZΩ and

g g k σ (y)= ασ∞(y) [σ (x, y) σ n] dl + u· σ (x, y)+ σ (x, y) n dl . gk gk − k x ,k ,g x ∂ZΩ ∂ZΩ h i (2.36)

2.2 Integral representation at the boundary

The displacement ﬁeld at the boundary can be obtained from eqns. (2.11) and (2.20), moving the source point y on the boundary ∂Ω, considering the integration contours sketched in ﬁg. [2.2], and taking the limit for ε 0 and → for ρ . →∞ Correspondingly, eqns. (2.11) and (2.20) transform to

σg (x, y) n · uD σDn · ug (x, y) dl = 0, (2.37) − x ∂Ω ∂ΩεZ+Γε+Θρ − and g g [σ (x, y) n · u∞ σ∞n · u (x, y)] dl = 0, (2.38) − x Z ∂Ω ∂Ωε+Γε − where Γε is the intersection of the circle of radius ε centered at y with the region Ω occupied by the continuum, Γε the remaining part of the circle, and ∂Ωε the boundary length intercepted by Γε, as indicated in ﬁg. [2.2d]. The limiting process, for ε 0 and ρ , of eqns. (2.37) and (2.38), → → ∞ yields

CuD(y) = ug(x, y) · σDn dl uD · σg(x, y)n dl , (2.39) g x −− x Z Z ∂Ω ∂Ω

g g [(C I)u∞(y)] = u (x, y) · σ∞n dl u∞ · σ (x, y)n dl , (2.40) − g x −− x ∂ZΩ ∂ZΩ 2.2 Integral representation at the boundary 25

Figure 2.2: Geometry of the problem and countours considered.

where denotes the Cauchy principal value and the C-matrix is deﬁned by − eqn. (2.17). By summing eqns. (2.40) and (2.39) we get the ﬁnal equation R

g g g C u (y)= αu∞(y)+ u (x, y) · σn dl u · σ (x, y)n dl . (2.41) i i g x −− x ∂ZΩ ∂ZΩ Integral representations and symmetric Galerkin formulation 26 for incompressible elasticity and Stokes ﬂow

In particular, using eqn. (1.41) we get an explicit form for the C-matrix

1 C = [(θ θ )I + (n(θ ) t(θ )) (n(θ ) t(θ )) ] , (2.42) 2π 1 − 0 1 ⊗ 1 sym − 0 ⊗ 0 sym where (.)sym denotes the symmetric part of a tensor, (n, t) are the unit vectors of a right-handed Cartesian reference system (ﬁg. [2.2b-d]) and θ0 and θ1 are the angular coordinates of the half-tangents to the boundary at point y. Note that at a smooth point of the boundary θ1 = θ0 + π, and n(θ1) = n(θ0), 1 t(θ1)= t(θ0), C = 2 I. It may be interesting to obtain a form of eqn. (2.41) which does not contain the Cauchy principal integral. To this purpose, we note that, for a closed contour of a region Φ not enclosing the singular point y, as in ﬁg. [2.2c], we have g g divσ dx = σ n dlx = 0, (2.43)

ZΦ ∂Ω ∂ΩεZ+Γε+Θρ − where g lim σ nj dlx = δgi, (2.44) ρ ij →∞ ΘZρ yielding g g g C = lim (σ n) dlx = αδig (σ n) dlx. (2.45) i ε 0 i −− i → ΓZε ∂ZΩ Employing eqn. (2.45) in eqn. (2.41), we get the regularized version

g g α u (y) u∞(y) = u (x, y) · σn dl [u(x) u(y)] · σ (x, y)n dl . g − g x − − x Z Z ∂Ω ∂Ω (2.46) holding true for points y lying internally, externally or at the (possibly non- smooth) boundary.

In order to obtain a boundary integral representation for the pressure p, we make use of eqn. (2.29) to regularize eqn. (2.28) which, for the contour shown in ﬁg. [2.2c], may be written as

σDn (x) pg(x, y) 2µ pg(x, y) · n(x) uD(x) uD(y) dl = 0. g − ∇ g − g x ∂Ω ∂ΩεZ+Γhε+Θρ i − (2.47) 2.2 Integral representation at the boundary 27

Taking the limit for ρ tending to inﬁnity, and assuming that σD decays and D n that u does not grow faster than ρ , with n < 1, the integral along Θρ turns out to vanish. In order to evaluate the integral along Γε in the limit ε 0, we introduce the ﬁrst-order and zeroth-order series expansions for the → displacement uD(x) and the stress σD(x)

uD(x) uD(y)= uD(y)(x y)+ o( x y 2), − ∇ − | − | (2.48) σD(x)= σD(y)+ o( x y ). | − | Using eqns. (2.48) and taking the limit ε 0, eqn. (2.47) can be re-written → as θ θ 1 − 0 pD(y) + 2µC · uD(y) 2π ∇ (2.49) = (σDn) pg dl + 2µ pg · n uD(x) uD(y) dl , −− g x − ∇ g − g x Z Z ∂Ω ∂Ω where C is the C-matrix defined by eqn. (2.42), and n n pg n = pg n r = g i on Γ . (2.50) i ,j j i 2 π ε ε The unperturbed field p is now analyzed, considering the contour ∂Ω ∞ − ∂Ωε + Γε. Writing the analogous of the eqn. (2.47) for the unperturbed fields and for the considered contour, gives in the limit ε 0 → θ1 θ0 − 1 p∞(y) + 2µC · u∞(y) 2π − ∇ (2.51) g g = (σ∞n) p dl + 2µ p · n u∞(x) u∞(y) dl , −− g x − ∇ g − g x Z Z ∂Ω ∂Ω so that, summing to (2.49), yields

θ θ 1 − 0 p(y) + 2µC · u(y) 2π ∇ (2.52) g g = αp∞(y) (σn) p dl + 2µ p · n [u (x) u (y)] dl . −− g x − ∇ g − g x ∂ZΩ ∂ZΩ

Eqn. (2.52) is the integral equation representing the pressure p at points y of the boundary. In this equation, the boundary values of p are coupled with Integral representations and symmetric Galerkin formulation 28 for incompressible elasticity and Stokes ﬂow those of the velocity gradient but, at smooth points of the contour, where θ θ = π and C = 1 I, the integral equation simpliﬁes to 1 − 0 2

1 g g p(y)= αp∞(y) (σn) p dl + 2µ p · n [u (x) u (y)] dl . 2 −− g x − ∇ g − g x ∂ZΩ ∂ZΩ (2.53) An alternative form of eqn. (2.52), but involving hypersingular integrals, will be useful later, and is simply obtained as follows. Considering the contour reported in ﬁg. [2.2c-d], we obtain when ρ →∞

pg(x, y) · n(x) dl ∇ x ∂Ω ∂ΩεZ+Γε+Θρ − θ1(ε) (2.54) 1 = pg(x, y) · n(x) dl + n (θ) dθ = 0, ∇ x 2πε g ∂Ω Z∂Ωε θ0Z(ε) −

where θ0(ε) and θ1(ε) are the angles singling out the initial and ﬁnal edges of the arc Γε (see the detail in ﬁg. [2.2]). A Taylor series expansion of the integral yields

θ1(ε) 2 n(θ) dθ = n(θ1)+ n(θ0)+ ε θ10 (0)t(θ1)+ θ0(0)t(θ0) + o(ε ), (2.55) Z θ0(ε) where θ0 = θ0(0) and θ1 = θ1(0) are the angular coordinates of the half- tangents to the boundary at point y (ﬁg. [2.2b-d]), and a prime denotes differentiation with respect to the argument. In the limit ε 0, we obtain →

g 1 = p · n dl = θ0 (0)t (θ )+ θ0 (0)t (θ ) , (2.56) ∇ x −2π 1 g 1 0 g 0 Z ∂Ω where =denotes the Hadamard ﬁnite part of the integral (Courant and Hilbert, 1962). Employing eqn. (2.56) into eqn. (2.52) we arrive at the expression R

θ1 θ0 µ − p(y) + 2µC · u(y) θ0 (0)t(θ )+ θ0 (0)t(θ ) · u(y) 2π ∇ − π 1 1 0 0 (2.57) g g = αp∞(y) (σn) p dl + 2µ = p · n u dl . −− g x ∇ g x ∂ZΩ ∂ZΩ 2.2 Integral representation at the boundary 29

Note that in the case of piecewise linear boundary θ10 (0) = θ0(0) = 0; whereas for smooth boundary, where θ θ = π, θ (0) = θ (0) and t(θ ) = t(θ ), 1 − 0 10 − 0 1 0 we get

1 g g p(y)= αp∞(y) (σn) p dl + 2µ = p · n u dl , (2.58) 2 − − g x ∇ g x ∂ZΩ ∂ZΩ which is alternative to (2.53), involving hypersingular kernels. An integral equation for the displacement gradient at boundary points can be obtained starting from eqn. (2.35). In particular, we begin noting that for a closed contour not enclosing the singularity point y, the following condition holds g σ,k(x, y) n(x) dlx = 0. (2.59) I Restricting the derivation to the case of inﬁnite domain, applying eqn. (2.35) and keeping into account eqn. (2.59), we get for the perturbed and unperturbed ﬁelds

σDn · ug σg n · uD(x) uD(y) dl = 0, (2.60) ,k − ,k − x ∂Ω ∂ΩεZ+Γεh+Θρ i −

g g σ∞n · u σ n · (u∞(x) u∞(y)) dl = 0, (2.61) ,k − ,k − x Z ∂Ω ∂Ωεh+Γε i − g g where, according to expression (1.39) and (1.42), u,k and σ,k are singular as 1/r and 1/r2 respectively, when r tends to zero. Following a procedure analogous to that illustrated for the derivation of pressure representation and taking the limits for ρ and ε 0 , the combination of eqns. (2.60)-(2.61) →∞ → gives

[H u(y)+ F E(y)+ Fp(y)] ∇ gk g g (2.62) = α u∞ (y) σ n · u dl + σ n · [u(x) u(y)] dl , g,k −− ,k x − ,k − x ∂ZΩ ∂ZΩ where g Hgkim = lim σ nj (xm ym) dlx, (2.63) − ε 0 ij,k − → ΓZε Integral representations and symmetric Galerkin formulation 30 for incompressible elasticity and Stokes ﬂow

g Fgkim = 2 µ lim nmu dlx, (2.64) ε 0 i,k → ΓZε

g Fgk = lim n · u dlx. (2.65) ε 0 ,k → ΓZε Substituting the expressions for the velocity and stress gradients of the Green state given by eqns. (1.39) and (1.42) and taking the relation r = rn on Γ − ε into account, eqns. (2.63)-(2.65) yield

H u(y)+ F E(y) ∇ 1 = uC + uT C C u 5 C uT 6(C · u)I + 16 C u , 4 ∇ ∇ − ∇ − ∇ − ∇ ∇ (2.66) θ θ 1 F = 1 − 0 I + C, (2.67) − 4πµ 2µ where θ1 1 C = n n n n dθ. (2.68) π ⊗ ⊗ ⊗ θZ0 In the special case of a smooth (Lyapunov) boundary

1 1 C = I and C = (I I + S) , (2.69) 2 8 ⊗ where S the symmetrizing fourth-order tensor (see Appendix A), so that

1 1 F E = E, H u = 3 u uT , (2.70) 4 ∇ 8 ∇ −∇ and 1 H u + F E = u, F = 0. (2.71) ∇ 2∇ In this case, eqn. (2.62) reduces to

1 g g u (y)= α u∞ (y) σn · u dl + σ n · [u(x) u(y)] dl . (2.72) 2 g,k g,k − − ,k x − ,k − x ∂ZΩ ∂ZΩ 2.2 Integral representation at the boundary 31

By means of eqns. (2.53) and (2.72), the constitutive eqn. (1.2) provides the boundary integral representation for the stress

1 g σ (y)= ασ∞(y) (σ σ n) dl 2 gk gk − − k x ∂ZΩ + µ [u(x) u(y)]· σg + σk n dl + 2 µ δ [u(x) u(y)] pi · n dl . − − ,k ,g x gk − − i∇ x ∂ZΩ ∂ZΩ (2.73) An alternative form of (2.62) involving hypersingular integrals can be obtained considering the contour of ﬁg. [2.2c-d] and taking the limit for ρ , →∞ so that

θ1(ε) 1 σg (x, y) n(x) dl + n ek + δ n 3 n n n dθ = 0. (2.74) ,k x πε g gk − g k ∂Ω Z∂Ωε θ0Z(ε) − Expanding the second integral in Taylor series, taking the limit for ε 0, → and substituting into eqn. (2.62), lead to

[ u(y)+ H u(y)+ F E(y)+ F p(y)] B ∇ gk g g (2.75) = α u∞ (y) σ n · u dl + = σ n · u dl , g,k − − ,k x ,k x ∂ZΩ ∂ZΩ where 1 = θ0 (0) ( t(θ1) I + I t(θ1) 3 t(θ1) t(θ1) t(θ1) ) B −π 1 ⊗ ⊗ − ⊗ ⊗ (2.76) + θ0 (0) ( t(θ ) I + I t(θ ) 3 t(θ ) t(θ ) t(θ ) ) . 0 0 ⊗ ⊗ 0 − 0 ⊗ 0 ⊗ 0 The tensors , D, F, and F collect the free-terms of the integral equation B representing the velocity gradient at points on the boundary. Note that B depends on the curvatures of the boundary and vanishes both for smooth and piecewise rectilinear boundary. This dependence on the curvature has a correspondence in elasticity (Guiggiani, 1995). At smooth points of the contour, where F turns out to be zero, eqn. (2.75) simpliﬁes as follows:

1 g g u (y)= α u∞ (y) σn · u dl + = σ n · u dl . (2.77) 2 g,k g,k − − ,k x ,k x ∂ZΩ ∂ZΩ Integral representations and symmetric Galerkin formulation 32 for incompressible elasticity and Stokes ﬂow

Once the velocity gradient and the pressure are given, the integral representation of the stress tensor on smooth points of the boundary follows from eqn. (1.2):

1 g σ (y)= ασ∞(y) (σ σ n) dl 2 gk gk − − k x Z ∂Ω (2.78) + µ = u · σg + σk n dl + 2 µ δ = u pi · n dl . ,k ,g x gk i ∇ x ∂ZΩ ∂ZΩ In closure of this Section, we note that eqns. (2.77) and (2.78) permit the complete determination of velocity gradient and stress at the boundary when tractions and velocities are here known.

2.2.1 Examples Compressible gas bubble in Stokes-flow We consider the special case of a bubble suspended in an ambient viscous liquid analyzed by Pozrikidis (2001). In this case we adopt the sign conventions relative to fluid with positive pressure in compression and the unit normal n to the boundary is assumed to be inward, pointing into the fluid. When ∂Ω represents the boundary (clearly smooth) of a bubble, the traction is given by σn = [ p + γ κ(x)] n, (2.79) − B where pB is the pressure, γ the surface tension and κ is the curvature of the bubble, so that condition (2.53) becomes equivalent to the integral equation obtained by Pozrikidis (2001). Note that for a closed contour ∂Φ of a region Φ not enclosing the singularity point y, the divergence theorem implies that the following conditions hold

div ug dx = ug · n dl = ug(x, y) · n(x) dl = 0, − x x ZΦ ∂IΦ ∂Ω ∂ZΩε+Γε −

pg dx = pg(x, y) n (x) dl = 0, − ,g g x ΦZ ∂Ω ∂ZΩε+Γε − (div ug) dx = ug (x, y) · n(x) dl = 0, − ,k ,k x ZΦ ∂Ω ∂ZΩε+Γε − 2.2 Integral representation at the boundary 33

divσ dx = σg(x, y) n(x) dl = 0, − x ΦZ ∂Ω ∂ZΩε+Γε − g g g as div u = 0, and div σ = p,g = 0 in Φ, so that 1 1 ug n dl = e , pg n dl = , − i i x −2 g − g x 2 Z Z ∂Ω ∂Ω (2.80) 1 ug n dl = σg n dl = e e , − i,k i x − ki i x −2 g k ∂ZΩ ∂ZΩ where eg and ek are versors relative to the reference system (x1,x2). Therefore substituting eqn. (2.79) into (2.41), (2.53), (2.72) and (2.73), and using (2.80) yield

1 1 u (y)= u∞(y) [p + γ κ(y)] e 2 g g − 2 B g (2.81) [κ(x) κ(y)] ug(x, y) · n dl + u · σg(x, y)n dl , − − x − x ∂ZΩ ∂ZΩ

1 1 p(y)= p∞(y) [p γ κ(y)] 2 − 2 B − (2.82) + γ [κ(x) κ(y)] n pg dl 2µ pg · n [u (x) u (y)] dl , − g x − − ∇ g − g x ∂ZΩ ∂ZΩ

1 1 u (y)= u∞ (y)+ [p + γ κ(y)] e e 2 g,k g,k 2 B g k (2.83) + [κ(x) κ(y)] n · ug dl σg n · [u(x) u(y)] dl , − ,k x −− ,k − x ∂ZΩ ∂ZΩ and

1 1 g σ (y)= σ∞(y)+ [p + γ κ(y)] e e + [κ(x) κ(y)] (σ n) dl 2 gk gk 2 B g k − k x ∂ZΩ µ [u(x) u(y)]· σg + σk n + 2 µ δ [u(x) u(y)] pi · n dl . − − − ,k ,g gk − − i∇ x ∂ZΩ ∂ZΩ (2.84) Integral representations and symmetric Galerkin formulation 34 for incompressible elasticity and Stokes ﬂow

Keeping into account condition [3.11] of Pozrikidis (2001), eqn. (2.82) coincides with eqn. [3.18] of Pozrikidis (2001).

Rigid inclusion We consider now the case of a rigid particle inserted in a deformable solid. The displacement at the contact between solid and particle satisﬁes Poisson’s theorem of rigid body motion, so that it can be expressed in the form:

u(x)= u(y)+ ω (x y), (2.85) × − where u and ω are the translational and angular velocities, respectively, x and y are generic points on the rigid boundary ∂Ω, so that x y = r in our − case. From eqn. (2.45) we obtain

[(I C) u(y)] = u(y) · σg(x, y) n dl , (2.86) − g − x ∂ZΩ furthermore, ω r is orthogonal to r, and × 1 r σg n = g (r · n) r, (2.87) −π r4 is parallel to r, so that, if we consider a contour along the boundary of the particle, but ruling out the point y with a small circle Γε of radius ε, taking the limit ε 0 we get → ω r · σgn dl = 0, (2.88) × x ∂ZΩ and, so [(I C) u(y)] = [u(y)+ ω r] · σg(x, y) n dl . (2.89) − g − × x ∂ZΩ Taking eqn. (2.89) into account, eqn. (2.41) simpliﬁes to

g · ug(y)= ug∞(y)+ u (x, y) σn dlx, (2.90) ∂ZΩ which holds true also for corner points. 2.2 Integral representation at the boundary 35

If we consider the pressure, we can see that along the arc Γ , r = rn, so ε − that r pg · n = g , (2.91) ∇ −2 π r3 is parallel to r, and, hence,

ω r · ( pg · n) dl = 0. (2.92) − × ∇ x ∂ZΩ From eqns. (2.56) and (2.92) we get

g 1 = [u(y)+ ω r] p (x, y) ·n dl = θ0 (0) t(θ )+ θ0 (0) t(θ ) · u(y), × g ∇ x 2π 1 1 0 0 Z ∂Ω (2.93) for points x and y lying on ∂Ω. Taking eqn. (2.93) into account, eqn. (2.57) (and eqn.(2.52)) reduce to

θ1 θ0 g − p(y) + 2µC · u(y)= p∞(y) (σn) p dl , (2.94) 2π ∇ −− g x ∂ZΩ which for smooth boundary becomes:

1 g p(y)= p∞(y) (σn) p dl , (2.95) 2 −− g x ∂ZΩ a formula implicitly derived by Liron and Barta (1992). In addition to the above, we note now that for smooth boundaries

[u(x) u(y)] · σg + σk n dl = 0, (2.96) − − ,k ,g x ∂ZΩ so that eqns. (2.73) and (2.78) become

1 g σ (y)= σ∞(y) (σ σn) dl , (2.97) 2 gk gk −− k x ∂ZΩ from which multiplication by the normal n(y) yields the expression proposed by Liron and Barta (1992)

1 g τ (y)= τ ∞(y) n(y) · (σ σn) dl , (2.98) 2 g g − − x ∂ZΩ referred to the Lyapunov boundary. Integral representations and symmetric Galerkin formulation 36 for incompressible elasticity and Stokes ﬂow

Elastic inclusion We consider now an elastic inclusion occupying the domain ΩI , as indicated in fig. [2.3]. The constitutive laws for the two elastic materials differ in the stiffness coefficients µO and µI , related to the external and internal body, respectively.

Figure 2.3: Elastic body with elastic inclusion.

The velocity u(x) is identiﬁed with the velocity uO(x) u (x)+uD,O(x) ≡ ∞ of the exterior body if x ΩO, while u(x) uI (x) for x ΩI ; on the ∈ ≡ ∈ boundary ∂Ω we impose

u(x) uO(x) uI (x). (2.99) ≡ ≡ The integral equation (2.41) applied at y ∂Ω, for the inﬁnite body and ∈ for the inclusion are

O g,O O g C u(y) = u∞(y)+ u (x, y) · σ n dl u · σ (x, y)n dl , (2.100) g g x −− x Z Z ∂Ω ∂Ω and

I g,I I g C u(y) = u (x, y) · σ n0 dl u · σ (x, y)n0 dl , (2.101) g x −− x Z Z ∂Ω ∂Ω respectively, where CI = I CO. − 2.2 Integral representation at the boundary 37

By summing eqns. (2.100) and (2.101), and taking into account the inter- face condition O I σ n = σ n0, (2.102) − where n and n0 are indicated in ﬁg. [2.3], we obtain

g,O g,I O u (y)= u∞(y)+ u (x, y) u (x, y) · σ n dl , (2.103) g g − x Z ∂Ω which holds true for points y lying at smooth and non-smooth boundary, and that, if µO = µI , reduces to

ug(y)= ug∞(y). (2.104)

. In the case of Lyapunov boundary we obtain

O I σgk(y)+ σgk(y) = σ∞(y) 2 gk (2.105) + µO µI = u · σg + σk n + 2δ u pi · n dl , − ,k ,g gk i ∇ x Z ∂Ω h i in which the left-hand side term can be considered an average value of the stress at y. Finally, by projecting along n and n0 the integral eqn. (2.73) for the inﬁnite medium and the inclusion, respectively, and subtracting the results, we get

O τg (y)= τg∞(y) (2.106) + µO µI = u · σg + σk n n + 2 n u pi · n dl . − ,k ,g k g i ∇ x Z ∂Ω h i

Fractures Here, we consider a fracture, which is traction free as in ﬁg. [2.4a], so that

τ = 0 on ∂Ω. (2.107) Integral representations and symmetric Galerkin formulation 38 for incompressible elasticity and Stokes ﬂow

Figure 2.4: Fracture: (a) traction free, (b) zero thickness.

In this case the integral equations simplify as

g [C u(y)] = u∞(y) u · σ n dl , (2.108) g g −− x ∂ZΩ

θ θ 1 − 0 p(y) + 2µC · u(y) 2π ∇ (2.109) g = p∞(y) + 2µ p · n [u (x) u (y)] dl , − ∇ g − g x ∂ZΩ

g [H u(y)+F E(y)+Fp(y)] = u∞ (y)+ σ n· [u(x) u(y)] dl , (2.110) ∇ gk g,k − ,k − x ∂ZΩ and, for Lyapunov boundary

1 g k σ (y)= σ∞(y)+ µ [u(x) u(y)]· σ + σ n 2 gk gk − − ,k ,g Z ∂Ω (2.111) +2 µ δ [u(x) u(y)] pi · n dl , gk − − i∇ x ∂ZΩ 2.3 Symmetric formulation of the boundary element method 39 where only the regularized forms on the boundary are reported. If we consider a fracture of zero thickness as in fig. [2.4b], the boundary + + can be subdivided into two parts: ∂Ω , with normal n , and ∂Ω−, with normal n = n+. Moreover, at any point x on the fracture faces, we have − − σ+(x) n+(x)= σ (x) n (x) and − − − + D [[u(x)]] = u (x) u−(x) = [[u (x)]], (2.112) − where [[.]] defines the jump of a field. All the integrals along ∂Ω can be splitted in the sum ∂Ω = ∂Ω+ + ∂Ω− , yielding

R R R D g + [C u(y)] = u∞(y) [[u (x)]] · σ n dl , (2.113) g g −− x ∂ΩZ+ θ θ 1 − 0 p(y) + 2µC · u(y) 2π ∇ (2.114) g + D = p∞(y) + 2µ = p · n [[u (x)]] dl , ∇ g x ∂ΩZ+

g + D [H u(y)+ F E(y)+ F p(y)] = u∞ (y)+ = σ n · [[u (x)]] dl , (2.115) ∇ gk g,k ,k x ∂ΩZ+ and, for Lyapunov boundary

1 D g k + σ (y)= σ∞(y)+ µ = [[u (x)]]· σ + σ n dl 2 gk gk ,k ,g x Z+ ∂Ω (2.116) +2 µ δ = [[uD(x)]] pi · n+ dl . gk i ∇ x ∂ΩZ+

2.3 Symmetric formulation of the boundary element method

Owing to eqns. (2.41) and (2.78), the representations for the displacement (velocity for the ﬂuid) u and the traction τ = σ n, (exerted on the solid by the boundary, and by the ﬂuid on the boundary, respectively) on a Lyapunov boundary are given by:

1 g g u (y)= α u∞(y)+ u (x, y) · σn dl u · σ (x, y)n dl , (2.117) 2 g g x − − x ∂ZΩ ∂ZΩ Integral representations and symmetric Galerkin formulation 40 for incompressible elasticity and Stokes ﬂow

1 g τ (y)= ατ ∞(y) n(y) · (σ σ n) dl 2 g g − − x ∂ZΩ (2.118) + µn (y) = u · σg + σk n dl + 2 δ = u pi · n dl . k  ,k ,g x gk i ∇ x ∂ZΩ ∂ZΩ   It is worth noting that either eqn. (2.117) or eqn. (2.118) can be employed to solve any boundary value problem through a collocation technique, thus obtaining in the former case the so-called direct method (that will be employ in presence of finite deformations) and, in the latter, the hypersingular boundary element method. In both cases the coefficient matrix of the solving system of algebraic equations is non-symmetric. A way to symmetrize the solving system is obtained below employing a Galerkin approach, in analogy to methodologies used in compressible solid mechanics (Bonnet et al. 1998). Let δu and δτ be virtual velocity and traction fields satisfying the boundary conditions

δu = 0, on ∂Ωu, δτ = 0, on ∂Ωτ , (2.119) where ∂Ωu and ∂Ωτ are the portions of the boundary where velocities and tractions are assigned, respectively. Taking the scalar product of eqns. (2.117) and (2.118) by the virtual ﬁelds δτ and δu, respectively, and integrating over the contour, the following equations are obtained: 1 u(y) · δτ (y) dl α u∞(y) · δτ (y) dl 2 y − y ∂ΩZu ∂ZΩu (2.120) = δτ (y) ug · τ dl u · τ g dl dl , g  x −− x y ∂ΩZu ∂ZΩ ∂ZΩ   1 τ (y) · δu(y) dl α τ ∞(y) · δu(y) dl 2 y − y ∂ZΩτ ∂ΩZτ

= δu (y) n(y) · u · σgτ dl − g  − x (2.121) ∂ZΩτ  ∂ZΩ

µn (y)= u· σg + σk n + 2δ u pi · n dl dl , − k ,k ,g gk i∇ x y ∂ZΩ h i   2.3 Symmetric formulation of the boundary element method 41

which, separating unknowns and data, can be re-written as

δτ (y) ug · τ dl u · τ g dl dl g  − x − x y ∂ΩZu ∂ΩZu ∂ZΩτ   1 = u(y) · δτ (y) dl α u∞(y) · δτ (y) dl (2.122) 2 y − y ∂ΩZu ∂ΩZu

+ δτ (y) u · τ g dl ug · τ dl dl , g  x −− x y ∂ΩZu ∂ΩZu ∂ZΩτ  

µ δu (y) n (y)= u · σg + σk n + 2 δ u pi · n dl dl g  k ,k ,g gk i ∇ x y ∂ZΩτ  ∂ZΩτ h i   1  δu (y) n(y) · σgτ dl dl = τ (y) · δu(y) dl − g  − x y 2 y ∂ZΩτ ∂ΩZu ∂ZΩτ   g α τ ∞(y) · δu(y) dl + δu (y) n(y) · σ τ dl dl − y g  − x y ∂ΩZu ∂ZΩτ ∂ZΩτ   µ δu (y) n (y)= u · σg + σk n + 2 δ u pi · n dl dl . − g  k ,k ,g gk i∇ x y ∂ZΩτ  ∂ΩZu h i  (2.123) Eqns. (2.122) and (2.123) are the starting point to derive a symmetric formulation of the boundary element method. To this purpose, the boundary e u τ ∂Ω is divided into ne elements ∂Ω (e = 1, ,ne), with subsets ne and ne · · · u τ belonging respectively to ∂Ωu and ∂Ωτ (clearly ne = ne + ne ). Within each boundary element ∂Ωe, the following representations for velocities and tractions are chosen:

ui(x)= ϕα(x)u ¯iα (α = 0, , Θα), · · · (2.124) τ (x)= ϕ (x)τ ¯ (β = 0, , Θ ), i β iβ · · · β whereu ¯iα andτ ¯iβ are the nodal values of velocities and tractions, respectively and ϕα and ϕβ are the relevant shape functions. Assuming for δug and δτg the same shape functions, namely Θα = Θβ = Θ, and taking into account that Integral representations and symmetric Galerkin formulation 42 for incompressible elasticity and Stokes ﬂow eqns. (2.122), (2.123) hold true for every δug and δτg, the following linear algebraic system can be obtained:

A B u p = , (2.125) DC τ q representing the governing equations of the discrete model. In eqn. (2.125), vectors u, τ collect the unknown nodal values of velocity and tractions, the system matrix is obtained by assembling the element sub-matrices

αβ g k i Agi = µ ϕα(y) nk(y)= σij,k + σij,g nj + 2 δgkp,j nj ϕβ(x)dlx dly, e  e  ∂ZΩτ  ∂ZΩτ h i 

  Bαβ = ϕ (y) n (y) σg ϕ (x) dl dl , gi − α  j − ij β x y ∂ZΩe ∂ΩZe τ  u    αβ g Cgi = ϕα(y)  ui ϕβ(x) dlx dly, ∂ΩZe ∂ΩZe u  u    Dαβ = ϕ (y) σg n ϕ (x) dl dl , gi − α  − ij j β x y ∂ΩZe ∂ZΩe u  τ    (2.126) where, in every submatrix 2(α 1)+g and 2(β 1)+i are the row and column − − position, respectively. Vectors on the right-hand side are given, for node α of each element, by

α 1 p = ϕ (y)τ (y) dl α ϕ (y)τ ∞(y) dl g 2 α g y − α g y e e ∂ZΩτ ∂ΩZτ

+ ϕ (y) n (y) σg τ dl dl α  i − ij j x y e ∂ZΩτ ∂ZΩτ   µ ϕ (y) n (y)= u σg + σk n + 2 δ u pi n dl dl , − α  k i ij,k ij,g j gk i ,j j x y e ∂ZΩτ  ∂ΩZu h i  (2.127)   2.3 Symmetric formulation of the boundary element method 43

α 1 q = ϕ (y)u (y) dl α ϕ (y)u∞(y) dl g 2 α g y − α g y e e ∂ΩZu ∂ΩZu

+ ϕ (y) u τ g dl ugτ dl dl , α  − i i x − i i x y e ∂ΩZu ∂ΩZu ∂ZΩτ   (2.128) where 2(α 1) + g is the generic component of each vector. Simple algebra, − omitted for brevity, shows that A = AT , C = CT and D = BT so that system matrix appearing in (2.125) turns out to be symmetric, a feature characteristic of the Galerkin boundary element formulation. The formulation is well known in the context of compressible elasticity (Bonnet el al. 1998), but was never extended to include the incompressible limit. Integral representations and symmetric Galerkin formulation 44 for incompressible elasticity and Stokes ﬂow Chapter 3

Elements of Continuum Mechanics

A brief review of the fundamental equations governing large deformations of solid bodies is presented below. The interested reader may refer to Truesdell and Noll (1965), Malvern (1969), Gurtin (1981), Ogden (1984) and Holzapfel (2000) for further references.

3.1 Kinematics

A body occupies a regular region B of the Euclidean point space . We B E will refer to such a region as a configuration at a certain instant time t. Let B be a fixed configuration assumed as reference. ◦

xo x

Bo B

Figure 3.1: Conﬁgurations of a continuum body.

The so-called material (or Lagrangean) description is performed with re-

45 46 Elements of Continuum Mechanics spect to the material points x defined in B , whereas the spatial (or Eulerian) ◦ ◦ description refers to spatial points x defined in B. The deformation is a one-to-one mapping carrying material points x into ◦ spatial points x: ϕ : B B, x = ϕ(x ). (3.1) ◦ → ◦ 1 The (unique) inverse function ϕ− maps each point of the current configuration x into the correspondent point x in the reference configuration: ◦ 1 1 ϕ− : B B , x = ϕ− (x). (3.2) → ◦ ◦ The deformation gradient is defined as

F(x )= ϕ (x ), (3.3) ◦ ∇ ◦ where the symbol denotes the gradient of material fields, whereas Div des- ∇ ignates the material divergence; similarly, we define the spatial gradient and the spatial divergence, grad and div, to be the gradient and the divergence of spatial fields. Tensor F is a ‘two-point tensor’, since it associates geometrical elements in the reference configuration to the corresponding elements in the current configuration. To avoid non-physical effects, the determinant of the deformation gradient must be strictly positive

J = det(F) > 0. (3.4)

The vector field u (x )= ϕ(x ) x , (3.5) ◦ ◦ ◦ − ◦ represents the displacement field at a given material point. The material gradient of the displacement field is given by

H(x )= u = F I. (3.6) ◦ ∇ ◦ − In general, F is a function of x , when F is independent of x the defor- ◦ ◦ mation is homogeneous. A particular class of homogenous deformations is the rigid deformation, where the distance between two particles in the reference conﬁguration is preserved in any conﬁguration. In a rigid translation any particle shifts of the same magnitude and direction, hence u is not a function of the material point, but it is constant in the ◦ whole body. A rigid body translation is described by the deformation

x = ϕ(x )= x + α, (3.7) ◦ ◦ 3.1 Kinematics 47 so that F = I. In a rigid rotation the whole body rotates with respect to a ﬁxed point y , ◦ in this case the deformation is described by

x = ϕ(x )= y + Q(x y ), Q Orth+, (3.8) ◦ ◦ ◦ − ◦ ∈ so that F = Q. Since F Lin+, it can always be decomposed according to the (unique) ∈ polar decomposition F = RU = VR, (3.9) where R Orth+ is a rotation, U and V are the right and left stretch tensors, ∈ respectively. They are deﬁned as

U = √FT F, U Sym+, (3.10) ∈ V = √FFT , V Sym+. (3.11) ∈ The tensors U and V are positive deﬁnite. Let λi (i = 1, 2, 3) denote the positive eigenvalues of U and ui (i = 1, 2, 3) the corresponding unit eigenvectors; therefore, the spectral representation of U is

3 U = λ u u . (3.12) i i ⊗ i Xi=1 From eqn. (3.9) the following relation is obtained

3 V = RURT = λ (Ru Ru ), (3.13) i i ⊗ i Xi=1 so that V and U have the same eigenvalues, while their eigenvectors are rotated by R. The symmetric tensors

3 C = FT F = U2 = λ2 u u , (3.14) i i ⊗ i X1=1 and 3 B = FFT = V2 = λ2 (Ru Ru ), (3.15) i i ⊗ i 1=1 X 48 Elements of Continuum Mechanics are the right and left Cauchy-Green tensors, respectively, and are both positive definite. The unit extension n of the linear element dx is defined as ◦ dx dx = | | − | ◦| = √n Cn 1, (3.16) n dx · − | ◦| where n = dx / dx , and quantifies the length variation of a linear element ◦ | ◦| dx . ◦Let n and m be the orthogonal directions of the material vectors dx and ◦ dy (namely n = dx / dx , and m = dy / dy ), and θ the angle between ◦ ◦ | ◦| ◦ | ◦| the corresponding spatial vectors dx and dy. The angle variation γ = π/2 θ is defined through the relation mn − dx dy n Cm sin (γmn)= · = · . (3.17) dx dy √n Cn√m Cm | | | | · · The volume V generated by the three vectors u, v, and w is measured by the scalar triple product

(u v) · w = u · (v w) = (w u) · v = V, (3.18) × × × as indicated in ﬁg. [3.2].

u x v w

Figure 3.2: Volume determined by u, v and w.

The inﬁnitesimal material volume dV , generated by dx , dy and dz , ◦ ◦ ◦ maps into the inﬁnitesimal volume ◦

dV = dx (dy dz)= Fdx (Fdy Fdz ). (3.19) · × ◦ · ◦ × ◦ In the two or three-dimensional Euclidean space, the following relation holds

T + Aa Ab = (det A)A− (a b), A Lin , a, b , (3.20) × × ∀ ∈ ∀ ∈V 3.1 Kinematics 49 so that applying eqn. (3.20), eqn. (3.19) becomes

T dV = Fdx [det(F)F− (dy dz )] = J dV . (3.21) ◦ · ◦ × ◦ ◦ Therefore, if a deformation is isochoric (i.e. dV = dV ) ◦ J = det(F) = 1. (3.22)

The volumetric expansion coefficient is defined as dV dV Θ= − ◦ = det(F) 1. (3.23) dV − ◦ An oriented surface element in the reference configuration is given by

dA = n dA = dx dy , (3.24) ◦ ◦ ◦ ◦ × ◦ where n is the unit normal to dA , as indicated in ﬁg. [3.3]. Due to the ◦ ◦

no dAo

dyo

dAo

dxo

Figure 3.3: Parallelogram determined by dx◦ and dy◦. deformation ϕ(x ), dA transforms according to the Nanson’s formula ◦ ◦ T ndA = J F− n dA . (3.25) ◦ ◦ Note that the relations between n and n are given by ◦ T T F− n F n ◦ n = T , n = T . (3.26) F− n ◦ F n | ◦| | | The Green-Lagrange deformation tensor is deﬁned as 1 G = (C I), G Sym+, (3.27) 2 − ∈ so that in the undeformed conﬁguration G 0. ≡ 50 Elements of Continuum Mechanics

3.1.1 Motions The motion is a smooth one-parameter family of deformations, where the parameter is the time t. The position of the particle x at time t is ◦ x = ϕ(x ,t), (3.28) ◦ while the trajectory described by a material point is = (x ,t) x B(t),t R . (3.29) T { | ∈ ∈ } 1 Since ϕ is a one-to-one function, a unique inverse function ϕ− exists, such that 1 x = ϕ− (x,t). (3.30) ◦ The description of motions given by relation (3.30) is called Eulerian (or spatial); in this case attention is paid to a point in space. In continuum mechanics a Lagrangean description of motions is usually employed, while in fluid mechanics an Eulerian description of motions is preferred. The velocity and the acceleration fields are obtained taking the first and second derivative of the motion ϕ with respect to time t, at x fixed. The ◦ material descriptions of the velocity and the acceleration fields are therefore ∂ϕ(x ,t) x˙ (x ,t)= ◦ , ◦ ∂t (3.31) ∂2ϕ(x ,t) x¨(x ,t)= ◦ . ◦ ∂t2 The spatial description of the velocity and acceleration fields is obtained as 1 x˙ (x ,t)= x˙ [ϕ− (x,t),t]= v(x,t), ◦ (3.32) 1 x¨(x ,t)= x¨[ϕ− (x,t),t]= a(x,t). ◦ It is important to note that ∂v a(x,t)= + (gradv) v. (3.33) ∂t 3.1.2 Material and Spatial Derivatives Let f(x ,t) be a smooth material field and g(x,t) a smooth spatial field. The ◦ material time derivative of f, denoted by f˙ is the derivative of f with respect to time t, holding x fixed ◦ ∂f(x ,t) f˙(x ,t)= ◦ . (3.34) ◦ ∂t 3.1 Kinematics 51

The spatial time derivative of g, denoted by ∂g/∂t is the derivative of g with respect to time t, holding x fixed. In order to perform the material time derivative of the spatial field g, first we express g as a function of x , ◦ g(x,t)= g[ϕ(x ,t),t], (3.35) ◦ then we compute the material time derivative

∂g(ϕ(x ,t),t) g˙ = ◦ , (3.36) ∂t and ﬁnally we ‘push forward’ the obtained result to the spatial description, so that ∂g(ϕ(x ,t),t) g˙ = ◦ . (3.37) ∂t −1 x◦=ϕ (x,t) Equivalently, if the chain rule is applied, we obtain

∂g(x,t) g˙(x,t) = + grad g v(x,t). (3.38) ∂t ·

3.1.3 Rate of Deformation Since F˙ = x˙ (x ,t), (3.39) ∇ ◦ the chain rule gives ∂v ∂x F˙ = = LF, (3.40) ∂x · ∂x ◦ where L = grad v is the spatial velocity gradient. The tensor L can be decomposed according to L = D + W, (3.41) where D is the rate of strain tensor:

1 D = (L + LT ), D Sym, (3.42) 2 ∈ and W is the spin tensor:

1 W = (L LT ), W Skw. (3.43) 2 − ∈ 52 Elements of Continuum Mechanics

Computing the material time derivative of J, we determine how the in- ﬁnitesimal volume dV = JdV changes with time during the motion. If we ◦ consider the relation

J = det(F) = det(U)= λ1λ2λ3, (3.44) where λi (i = 1, 2, 3) are the principal stretches, it follows that

λ˙ 1 λ˙ 2 λ˙ 3 J˙ = (det F)· = J + + . (3.45) λ1 λ2 λ3 !

1 By introducing the scalar product U˙ · U− we obtain the useful relations

T 1 J˙ = J tr(U˙ U− )= J (trL)= J div v, (3.46) and, in particular, a motion is isochoric if, and only if, one of the following relations is satisﬁed at every instant

J = 1, dV = const, J˙ = 0, (3.47) 1 div v = 0, trL = trD = 0, U˙ · U− = 0.

3.2 Stress

Contact forces between separate parts of a body are deﬁned through a surface force density s depending on the surface (unit) normal n and on every pair (x,t) in the trajectory: s(x, n,t). (3.48)

Equilibrium of generic parts of a continuum body is governed by the P Euler axioms b dV + s dA = 0,

Z ∂Z P P (3.49) r b dV + r s dA = 0, × × Z ∂Z P P where b are the body forces per unit volume, and r is the position vector of the generic spatial point. 3.3 Invariance of material response 53

A necessary and sufficient condition that the Euler axioms be satisfied is the existence of a spatial tensor field σ, the Cauchy stress, such that s(n)= σ n,

σ = σT , (3.50)

div σ + b = 0. A spatial stress field satisfying div σ = 0 is called self-equilibrated. The Cauchy tensor σ is defined in the spatial configuration B, which is usually unknown. Thus, it is useful to work with a stress tensor that is referred to the reference configuration. The contact force df acting on the surface element dA may be written as

df = s dA = s dA , (3.51) ◦ ◦ where s is the nominal traction vector, describing the surface force acting on ◦ the current configuration, but referred to B . ◦ Defining s = tT n , where t is the nominal stress tensor and tT is the first ◦ ◦ Piola-Kirchhoff stress tensor, and using eqns. (3.48) and (3.51), it follows that σ n dA = tT n dA . (3.52) ◦ ◦ If the Nanson’s formula (3.25) is employed, we get

1 t = JF− σ. (3.53) Hence, tT is a two-point tensor. In general t is not a symmetric tensor, however it satisﬁes the relation Ft = tT FT . (3.54)

3.3 Invariance of material response

Considering a moving body , two different observers, O and O+, sharing B the same reference configuration B , will record, in general, different motions, ◦ indicated by ϕ(x ,t) and ϕ+(x+,t+), respectively. Such motions are called ◦ equivalent if the relative distances◦ between points and time intervals between events under observation are preserved, so that x x = x+ x+ , | − ◦| | − ◦ | (3.55) t t = t+ t+. − ◦ − ◦ 54 Elements of Continuum Mechanics

The requirement (3.55)1 is satisﬁed by the spatial mapping

x+ x+ = Q(t)(x x ), (3.56) − ◦ − ◦ where Q Orth+ is the rotation between the reference systems associated ∈ with the two observers. Eqns. (3.55) may be rewritten as

x+ = c(t)+ Q(t) x, and t+ = t + α, (3.57) where c(t)= x+ Q(t) x, and α = t+ t , (3.58) ◦ − ◦ − ◦ are the space and time shifts between the two observers, respectively. Thus, at each time t the deformation x+(x ,t) may be viewed as a certain rigid-body ◦ motion superimposed on x(x ,t). ◦ A spatial tensor ﬁeld of order n (namely q q q ) is called 1 ⊗ 2 ⊗···⊗ n objective Eulerian if during any change of observer transforms according to

(q q q )+ = Q q Q q Q q . (3.59) 1 ⊗ 2 ⊗···⊗ n 1 ⊗ 2 ⊗···⊗ n In particular, if n = 1, for an objective spatial vector ﬁeld q eqn. (3.59) reduces to q+(x+,t+)= Q(t) q(x,t). (3.60) For an objective spatial second-order tensor ﬁeld A(x,t)= q (x,t) q (x,t), 1 ⊗ 2 eqn. (3.59) becomes

A(x+,t+)= Q(t) A(x,t) QT (t), (3.61)

Clearly, an objective spatial scalar field is not affected by a change of observer ψ+(x+,t+)= ψ(x,t). (3.62) A material tensor field of order n is said to be objective Lagrangean if it is unaffected by an observer transformation. Referring to the representation (3.59), the Lagrangean objectivity is expressed by the relation

(q q q )+ = q q q . (3.63) 1 ⊗ 2 ⊗···⊗ n 1 ⊗ 2 ⊗···⊗ n Application of eqn. (3.63) for scalar, vector and second-order tensor fields gives ψ+(x+,t+)= ψ (x ,t), q+(x+,t+)= q (x ,t), ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ (3.64) and A+(x+,t+)= A (x ,t), ◦ ◦ ◦ ◦ 3.3 Invariance of material response 55 respectively, where ψ , q and A are generic material fields. ◦ ◦ ◦ For two different observers the deformation gradient F is ∂x ∂x+ F = , and F+ = . (3.65) ∂x ∂x+ ◦ ◦ Employing eqn. (3.57) we obtain the relation between F and F+

∂x+ ∂x F+ = = Q = QF. (3.66) ∂x ∂x ◦ ◦ which gives the transformation rule for an objective two-point tensor. The polar decomposition for the observer O+ at point x+ B+ is given ∈ by F+ = R+U+ = V+R+, (3.67) and applying eqn. (3.66), we get

QRU = R+U+ and QVR = V+R+. (3.68)

Since the tensor QR is orthogonal, it follows that

R+ = QR, (3.69) U+ = U, and V+ = QVQT , (3.70) hence, U is objective Lagrangean and V objective Eulerian. The Cauchy-Green tensors C and B are also objective Lagrangean and Eulerian, respectively,

C+ = C and B+ = QBQT . (3.71)

An example of a non-objective tensor is given by the velocity gradient 1 L = FF˙ − , a spatial tensor which transforms according to

L+ = QLQT + QQ˙ T , (3.72) where QQ˙ T represents the spin of the reference frame of O relative to the reference frame of O+. Since L = D + W we obtain

D+ = QDQT , and W+ = QQ˙ T + QWQT , (3.73) 56 Elements of Continuum Mechanics so that D is objective Eulerian. Let s = σ n be the Cauchy traction vector that the ﬁrst observer measures at the istant time t. Considering that s+(n+)= Q s(n), we get

σ+ n+ = Q σ n. (3.74)

Using the relation n+ = Q n, we obtain

σ+ = Q σ QT ; (3.75) therefore the stress tensor σ is objective Eulerian. Finally, the ﬁrst Piola-Kirchhoﬀ stress tensor transforms according to

tT + = QtT . (3.76)

3.3.1 Objective rates We consider a spatial vector q and a spatial tensor A that transform according to A+ = QAQT , q+ = Q q, (3.77) so that they are objective Eulerian. Their material derivatives do no longer follow these rules since

q˙ + = Qq˙ + Qq˙ , A˙ + = QAQ˙ T + QAQ˙ T + QAQ˙ T. (3.78)

Therefore q˙ and A˙ are not objective ﬁelds. Objective time derivatives are unaﬀected by rigid rotations. Eqn. (3.73)2 yields Q˙ = W+Q QW and a substitution in (3.78) gives −

+ + ∇q = Q ∇q, A∇ = Q AQ∇ T , (3.79) where ∇q= q˙ Wq, A∇ = A˙ WA + AW, (3.80) − − are the definitions of the co-rotational or Jaumann rates of a vector and a tensor, respectively. Notice that objective Lagrangean fields are unaffected by a change of observer, and so their material derivatives behave in the same way, namely

q˙ + = q˙ , A˙ + = A˙ . (3.81) ◦ ◦ ◦ ◦ 3.4 Constitutive equations 57

3.4 Constitutive equations

A material is Cauchy elastic if the state of stress in the current conﬁguration depends solely on the state of deformation. The mathematical expression of the constitutive equation for an elastic material is therefore

σ(x,t)= σˆ(F(x ,t), x ), (3.82) ◦ ◦ or, simply σ = σˆ(F), (3.83) where σˆ is the response function. Since the material response is independent of the observer, it must be expressed as σ+ = σˆ(F+), (3.84) and by introducing the transformation laws for F and σ, we get

σˆ(QF)= Qσˆ(F)QT , Q Orth+. (3.85) ∈ Our aim is now to determine alternative objective forms for the constitutive equations: since R is a particular rotation and F = RU, relation (3.83) may be rewritten as σ = σˆ(RU)= Rσˆ(U)RT , (3.86)

1 and so, by inserting R = FU− ,

1 T T T σ = FU− σˆ(U)U− F = Fσ˘(U)F , (3.87)

1 T where σ˘(U)= U− σˆ(U)U− represents the response function in terms of U. In the same way we may derive a response function in terms of C, through the relations

σˆ(F)= Rσ˜(C)RT or σˆ(F)= Fσ˜(C)FT . (3.88)

On the other hand, if we introduce the nominal stress tensor we obtain

1 t = ˆt(F)= J F− σˆ(F). (3.89) 58 Elements of Continuum Mechanics

3.4.1 Isotropic materials

To introduce the concept of isotropy we consider a body in the reference configuration B and deform it, by applying a deformation gradient F. At the ◦ same time we consider the same body , and, starting from the same reference B configuration B , we apply first a rotation Q and then the same deformation ◦ as before, so that the total gradient of the deformation is FQ. If the response of the material is the same for any rotation Q, the material is said to be isotropic. Therefore,

σˆ(F)= σˆ(FQ), Q Orth+, (3.90) ∀ ∈ for at least one reference conﬁguration. T 1 By setting Q = R = F− V we get

σ = σˆ(F)= σˆ(FRT )= σˇ(V), (3.91) or, since B = V2 σ = σ¯(B). (3.92)

With the left Cauchy-Green B as independent variable in the response function σ¯, the invariance of the response function for an isotropic material takes the form σ¯(QBQT )= Qσ¯(B)QT . (3.93)

Due to the symmetry of B, the representation theorem for symmetric isotropic tensor functions (see Appendix A) holds. In this case, the isotropic response function σ¯ given in eqn. (3.93) admits the explicit form

1 σ = β0I + β1B + β2B− , (3.94) where β0, β1 and β2 are functions of the three invariants of B. Moreover, if the material is incompressible, we must impose the internal constraint h(F) J 1 = 0, (3.95) ≡ − applied at every point of the body B. Consideration of material frame-indiﬀerence implies that eqn. (3.82) takes the form ∂h(F) σ = σˆ(F) q FT , (3.96) − ∂F 3.4 Constitutive equations 59 where q is a Lagrange multiplier, representing an additional unknown of the problem, associated with the constraint equation (3.95). In our case

∂h(F) ∂(det F) T T = = (det F) F− = F− . (3.97) ∂F ∂F Hence, if the Cauchy-elastic material is incompressible, relation (3.94) becomes (Truesdell and Noll, 1965)

1 σ = qI + β B + β B− , (3.98) − 0 1 where β0 and β1 are isotropic scalar functions of B, so that they can be expressed as functions of the ﬁrst two invariants of B (see Appendix A)

β0 = β0(I1,I2), β1 = β1(I1,I2), (3.99) where I and I are given by eqn. (A.5) in Appendix A, and I 1 for an 1 2 3 ≡ incompressible material. The parameter q in (3.98) may be determined by applying the Cayley- Hamilton theorem. In this way, we ﬁnd out that q is connected to the hydrostatic pressurep ˆ = trσ/3, by the relation 1 q = pˆ + β I + β I2 I . (3.100) − 0 1 2 1 1 − 2 Two particular cases of (3.98) are the Mooney-Rivlin material,

1 σ = qI + αB + βB− , (3.101) − where α and β are taken constant, and the neo-Hookean material,

σ = qI + αB, (3.102) − where, in addition, β is equal to zero. The constitutive equation (3.98) implies coaxiality of tensors B and σ, so that these share (at least) one principal reference system —the Eulerian principal axes— and, recalling the spectral decomposition of B, we may write

3 3 B = λ2v v , σ = σ v v , (3.103) i i ⊗ i i i ⊗ i Xi=1 Xi=1 in which λi > 0 (i = 1, 2, 3) are the principal stretches, satisfying the incompressibility constraint λ1λ2λ3 = 1, (3.104) 60 Elements of Continuum Mechanics and vi (i = 1, 2, 3) are the eigenvectors of B. Therefore, the functions β0 and β1 can be solved by expressing eqn. (3.98) in the Eulerian principal reference system

1 (σ σ )λ2 (σ σ )λ2 β = 1 − 3 1 2 − 3 2 , 0 λ2 λ2 λ2 λ2 − λ2 λ2 1 − 2 1 − 3 2 − 3 (3.105) 1 σ σ σ σ β = 1 − 3 2 − 3 , 1 λ2 λ2 λ2 λ2 − λ2 λ2 1 − 2 1 − 3 2 − 3 representing two equations that can be alternatively expressed employing, on the right hand side of eqn. (3.105), every permutation of 1, 2 and 3 as indices.

Newtonian fluids Results presented in this thesis may find applications in slow viscous flow of a fluid. We therefore introduce the constitutive equation describing a Newto- nian, incompressible fluid σ =p Î + 2µD, (3.106) where the Cauchy stress is related to the Eulerian strain rate through the viscous coefficient µ and the pressure in the fluidp ˆ = trσ/3. Therefore, for plane flow orthogonal to the axis 3

Di3 = D3i = 0 (i = 1, 2, 3), (3.107) and so σ11 + σ22 σi3 = 0 (i = 1, 2), σ33 = , 2 (3.108) σ σ = 2µ (D D ) , σ = µD . 11 − 22 11 − 22 12 12 3.4.2 Hyperelastic materials When an elastic potential, the strain-energy function W = W (F), exists such that (Ogden, 1984) ∂W tT = [F], (3.109) ∂F the material is called hyperelastic. The necessary and sufficient condition for W to be objective, choosing Q = RT , is W (F)= W (RT F)= W (U), (3.110) 3.4 Constitutive equations 61 where we have used the same letter W , denoting different functions (the same abbreviation of notation will be employed below). The strain energy function W may be also expressed in terms of the left stretch tensor V, the Cauchy- Green tensors C and B, or the Green-Lagrange strain tensor G. Alternative forms for the constitutive equation (3.109) may be found from the relation ∂W (F) T ∂W (C) W˙ = tr F˙ = 2 tr FT F˙ , (3.111) ∂F ∂C " # which implies ∂W (F) T ∂W (C) = 2 FT . (3.112) ∂F ∂C For example, we have: T T ∂W (C) 1 ∂W (F) 1 ∂W (C) T t = 2F , or σ = J − F = 2J − F F . ∂C ∂F ∂C (3.113) If W is an isotropic function, the following relation is satisfied W (B)= W (QBQT ). (3.114) In this case we can express the constitutive eqn. (3.109) in the form (Holzapfel 2000) 1 ∂W (B) σ = 2J − B. (3.115) ∂B Since, due to the isotropy, W does not depend on directional properties, recalling the spectral decomposition of B and the representation theorem for scalar isotropic functions, we obtain

3 3 ∂W (B) ∂W 1 ∂W = 2 vi vi = vi vi, (3.116) ∂B ∂λi ⊗ 2λi ∂λi ⊗ Xi=1 Xi=1 so that, considering eqn. (3.115), we conclude that

3 3 1 ∂W σ = σivi vi, = J − λi vi vi. (3.117) ⊗ ∂λi ⊗ Xi=1 Xi=1 If the material is incompressible, namely J 1, we may postulate a form ≡ of the strain-energy function as W = W (B) pˆ(J 1), (3.118) − − 62 Elements of Continuum Mechanics where the pressurep ˆ = trσ/3 serves as a Lagrange multiplier. The elastic potential W is still an objective isotropic function, but deﬁned for J 1; in this case only two principal stretches λ are independent, since it ≡ i must be λ3 = 1/(λ1λ2). Therefore the strain-energy function can be expressed as 1 Wˆ (λ1, λ2)= W (λ1, λ2, ), (3.119) λ1λ2 and by applying the chain rule, we get

∂Wˆ ∂W ∂W λi = λi λ3 , i = 1, 2. (3.120) ∂λi ∂λi − ∂λ3 Further, if the deformation is plane 1 λ = λ, λ = and λ = 1, (3.121) 1 2 λ 3 the elastic potential takes the form

1 W˘ (λ)= Wˆ λ, . (3.122) λ If the chain rule is now employed, we obtain

∂W˘ ∂Wˆ ∂Wˆ λ = λ1 λ2 . (3.123) ∂λ ∂λ1 − ∂λ2 Finally, we get

∂W ∂W ∂Wˆ ∂Wˆ ∂W˘ σ1 σ2 = λ1 λ2 = λ1 λ2 = λ , − ∂λ1 − ∂λ2 ∂λ1 − ∂λ2 ∂λ ∂W ∂W ∂Wˆ σ1 σ3 = λ1 λ3 = λ1 , (3.124) − ∂λ1 − ∂λ3 ∂λ1 ∂W ∂W ∂Wˆ σ2 σ3 = λ2 λ3 = λ2 . − ∂λ2 − ∂λ3 ∂λ2 It is clear that the elastic potential may be expressed also as a function of the invariants of B (see Appendix A), where I 1 for an incompressible 3 ≡ material. Hence the elastic potential takes the form

W¯ (I1,I2)= Wˆ (λ1, λ2), (3.125) 3.4 Constitutive equations 63 and if the chain rule

∂Wˆ ∂W¯ ∂I ∂W¯ ∂I = 1 + 2 , (3.126) ∂λi ∂I1 ∂λi ∂I2 ∂λi is employed, the following relations are obtained

∂W¯ ∂W¯ σ σ = 2 λ2 λ2 + 2 λ2 + λ2 , 1 − 2 1 − 2 ∂I 1 2 ∂I 1 2 ¯ ¯ 2 1 ∂W 2 1 ∂W σ1 σ3 = 2 λ1 + 2 λ1 + , (3.127) − − λ2λ2 ∂I λ2λ2 ∂I 1 2 1 1 2 2 1 ∂W¯ 1 ∂W¯ σ σ = 2 λ2 + 2 λ2 + . 2 − 3 2 − λ2λ2 ∂I 2 λ2λ2 ∂I 1 2 1 1 2 2

3.4.3 Some elastic potentials

Mooney-Rivling material

A simple explicit form of strain-energy function for isotropic rubber-like elastic media was proposed by Mooney (1940) in the form

µ µ W¯ (I ,I )= 1 (I 3) 2 I2 I 6 , (3.128) 1 2 2 1 − − 4 1 − 2 − or

µ 1 µ 1 1 Wˆ (λ , λ )= 1 λ2 + λ2 + 3 2 + + λ2λ2 3 , 1 2 2 1 2 λ2λ2 − − 2 λ2 λ2 1 2 − 1 2 1 2 (3.129) where µ and µ are material parameters and µ = µ µ represents the 1 2 0 1 − 2 shear modulus in the unstressed state. With reference to the representation (3.98), we obtain:

β0 = µ1, β1 = µ2, (3.130) and, in the case of plane strain, eqn. (3.129) simpliﬁes to

µ µ 1 W˘ (λ)= 1 − 2 λ2 + 2 . (3.131) 2 λ2 − 64 Elements of Continuum Mechanics

Ogden material The following class of strain-energy functions was proposed by Ogden (1972) to ﬁt experimental results on rubber:

N µ 1 ˆ i αi αi W (λ1, λ2)= λ1 + λ2 + αi 3 , (3.132) αi (λ1λ2) − Xi=1 or N µ 1 ˘ i αi W (λ)= λ + α 2 , (3.133) αi λ i − Xi=1 for plane strain, where µi and αi are material parameters, subjected to the constraints: N 2µ = µ α , with µ α > 0 (i = 1, , N), (3.134) 0 i i i i · · · Xi=1 in which N is a positive integer determining the number of terms in the strain-energy function, µi are constant shear moduli, and αi are dimensionless parameters, with i = 1, , N. In particular, Mooney-Rivlin material can be · · · recovered as a particular case taking N=2, α =2 and α = 2. 1 2 − An excellent correlation with experimental data relative to simple-tension, equibiaxial tension and pure shear of vulcanised rubber is obtained employing the values (Treloar, 1944; 1975; Ogden, 1972): α = 1.3 µ = 6.3 105 N/m2, 1 1 · 5 2  α2 = 5.0 µ2 = 0.012 10 N/m , (3.135)  ·  α = 2.0 µ = 0.1 105 N/m2, 3 − 3 − ·  yielding µ = 4.225 105 N/m2. 0 · Coeﬃcients in the representations (3.98) can be obtained from eqns. (3.105) and (3.132) in the form:

N αi αi αi αi 1 λ (λ λ )− λ (λ λ )− β = µ 1 − 1 2 λ2 2 − 1 2 λ2 , 0 2 2 i 2 2 1 2 2 2 λ1 λ2 " λ1 (λ1λ2)− − λ2 (λ1λ2)− # − Xi=1 − − N αi αi αi αi 1 λ (λ λ )− λ (λ λ )− β = µ 1 − 1 2 2 − 1 2 , 1 2 2 i 2 2 2 2 λ1 λ2 " λ (λ1λ2)− − λ (λ1λ2)− # − i=1 1 − 2 − X (3.136) 3.4 Constitutive equations 65 or

2 N αi λ λ 1 2 1 β0 = µi − λ , λ4 1 λ2 1 − λαi − Xi=1 − N (3.137) λ2 λαi 1 λ2 β1 = µi − 1 , λ4 1 λ2 1 − λαi − Xi=1 − for plane strain.

J deformation theory of plasticity 2− AJ deformation theory of plasticity was proposed by Hutchinson and Neale 2− (1978) (see also Hutchinson and Tvergaard, 1981; Neale, 1981) in the framework of hyperelasticity, to model metals subject to proportional loading. With reference to the representation (3.98), the constitutive-law in the J deformation theory of plasticity can be expressed as: 2− 2 σi = Esi +p ˆ (i = 1, 2, 3), 3 (3.138) 1 + 2 + 3 = 0, where i = log λi are the logarithmic strains. In the same equation Es is the secant modulus to the curve representing the eﬀective stress σE versus eﬀective strain E

2 2 2 2 3 2 2 2 E = 1 + 2 + 3 , σE = S1 + S2 + S3 , (3.139) r3 r2 where Si (i = 1, 2, 3) are the principal component of the deviatoric stress S = σ (trσ/3)I. The curve is assumed to be determined by − N 1 Es = KE − , (3.140) where N ]0, 1] is an hardening exponent, K is a positive constitutive param- ∈ eter. The strain-energy function results therefore to take the form

K W = N+1. (3.141) N + 1 E 66 Elements of Continuum Mechanics

The above framework corresponds to the following choice of parameters q, β0 and β1 (see eqns. (3.98) and (3.138)):

4 2 2 2 4 2 2 2 2 λ2 λ2− λ1 + λ3 1 λ1 λ1− λ2 + λ3 2 q = pˆ + Es − − − , − 3 λ2 λ2 λ2 λ2 λ2 λ2 1 − 2 1 − 3 2 − 3 2 2 2 1 (1 3) λ (2 3) λ β = E − 1 − 2 , 0 3 s λ2 λ2 λ2 λ2 − λ2 λ2 1 − 2 1 − 3 2 − 3 2 1 β = E 1 − 3 2 − 3 . 1 3 s λ2 λ2 λ2 λ2 − λ2 λ2 1 2 1 3 2 3 − − − (3.142) For plane strain 3 = log λ3 = 0, (3.143) so that the incompressibility requires

= = = log λ, (3.144) 1 − 2 and the eﬀective strain E becomes 2 E = . (3.145) √3 If we introduce N 1 2 − K˜ = K , (3.146) √3 the following relations hold ˜ N 1 4 K N+1 E = K˜ − and W = . (3.147) s E 3 N + 1 In this case, with reference to the representation (3.98), we obtain:

q = p,ˆ − (3.148) 2 E β = β = s . 0 − 1 3 λ2 λ 2 − − Chapter 4

Incremental Deformations

A general form of constitutive equations for plane, incompressible, incremental deformations in terms of the material derivative of the nominal stress tensor t, that will be adopted in the boundary element formulation, is presented. In particular, Biot (1965) has shown that ﬁrst-order incremental plane strain deformations superimposed upon a given homogeneous strain are governed by two incremental shear moduli, functions of the current stretch. In this Section, referring to incompressible materials, it is shown that the Biot assumption embraces a broad class of material behaviour, including hyper-elastic, hypo-elastic, Newtonian ﬂuid and the loading branch of plastic constitutive laws.

4.1 Incremental constitutive equations

An updated Lagrangean formulation is considered: the current conﬁguration is taken as the reference for the incremental analysis. Therefore F = I, and F˙ = L. We recall that the constitutive equation for a Cauchy-elastic, incompressible and isotropic material can be written as

1 σ = qI + β B + β B− , (4.1) − 0 1 where β0 and β1 are generic functions of the two invariants of B. Taking the material derivative of (4.1) yields

1 1 σ˙ = q˙I + β B˙ + β B− · + β˙ B + β˙ B− , (4.2) − 0 1 0 1 67 68 Incremental Deformations where B˙ = DB + BD + WB BW, (4.3) − and 1 1 1 1 1 B− · = B− D DB− B− W + WB− , (4.4) − − − in which D is the Eulerian strain rate and W the spin tensor. Keeping into account now eqn. (4.1) and the deﬁnition of the Jaumann derivative σ∇= σ˙ Wσ + σW, (4.5) − the constitituve equation (4.2) becomes

1 1 1 σ∇ +q ˙I = β (DB + BD) β (B− D + DB− )+ β˙ B + β˙ B− . (4.6) 0 − 1 0 1 Noting that

∂βi ∂βi β˙i = trB˙ + 2 B · B˙ with βi = βi(I1,I2) (i = 0, 1), (4.7) ∂I1 ∂I2 and trB˙ = 2B · D, B · B˙ = 2B2 · D, or ˆ ˆ ˙ ∂βi ∂βi βî = λ˙ 1 + λ˙ 2 with βî = βî(λ1, λ2) (i = 0, 1), (4.8) ∂λ1 ∂λ2 where βî = βî(λ1, λ2) (i = 0, 1) are coefficients β0 and β1 expressed as function of the principal stretches, we arrive at

σ∇ +q ˙I = β (DB + BD) β (B 1D + DB 1) 0 − 1 − −

∂β0 2 ∂β0 ∂β1 2 ∂β1 1 + 2 B·D + 2B ·D B + 2 B·D + 2B ·D B− , ∂I1 ∂I2 ∂I1 ∂I2 (4.9) which is equivalent to

σ∇ +q ˙I = β (DB + BD) β (B 1D + DB 1) 0 − 1 − − ˆ ˆ ˆ ˆ (4.10) ∂β0 ∂β0 ∂β1 ∂β1 1 + λ˙ 1 + λ˙ 2 B + λ˙ 1 + λ˙ 2 B− . ∂λ1 ∂λ2 ! ∂λ1 ∂λ2 ! The incremental constitutive equations (4.9) and (4.10) are valid for three- dimensional, incompressible Cauchy elasticity. 4.1 Incremental constitutive equations 69

4.1.1 Incremental moduli

We are interested here in the particularization of (4.9) to incremental plane strain deformations superimposed upon a generic state of homogeneous deformation. In the Eulerian principal reference system

2 2 1 B = λ1 e1 e1 + λ2 e2 e2 + 2 2 e3 e3, ⊗ ⊗ λ1λ2 ⊗ (4.11)

Di3 = D3i = 0 (i = 1, 2, 3), so that the out-of-plane stress rate components can be determined as

∇σ3i = ∇σi3= 0 (i = 1, 2), (4.12) and

1 ∂β0 ∂β0 ∇σ = q˙ + λ2 λ2 + 2 λ2 + λ2 33 − 1 − 2 λ2λ2 ∂I 1 2 ∂I 1 2 1 2 (4.13) ∂β ∂β + λ2λ2 1 + 2 λ2 + λ2 1 (D D ) , 1 2 ∂I 1 2 ∂I 11 − 22 1 2 or

1 ∂βˆ0 ∂βˆ0 ∇ σ33 = q˙ + 2 2 λ1 λ2 − "λ1λ2 ∂λ1 − ∂λ2 ! (4.14) ˆ ˆ 2 2 ∂β1 ∂β1 (D11 D22) + λ1λ2 λ1 λ2 − . ∂λ1 − ∂λ2 !# 2

Finally, the in-plane rate components can be expressed in the Biot (1965) form as

∇σ12 = 2µD12,   σ∇ ∇σ = 2µ (D D ) , (4.15)  11 22 11 22  − ∗ − D11 + D22 = 0,   where µ and µ are two incremental moduli corresponding respectively to ∗ shearing parallel to, and at 45◦ to, the Eulerian principal axes. These can be 70 Incremental Deformations expressed as functions of the invariants of B λ2 + λ2 β µ = 1 2 β 1 , 2 0 − λ2λ2 1 2 2 2 2 2 2 λ1 + λ2 λ1 λ2 ∂β0 2 2 ∂β0 µ = β0 + − + 2 λ1 + λ2 ∗ 2 2 ∂I ∂I 1 2 2 1 λ2 + λ2 λ2 λ2 ∂β ∂β 1 2 β + 1 − 2 1 + 2 λ2 + λ2 1 , − λ2λ2 2 1 2 ∂I 1 2 ∂I 1 2 ( 1 2 ) (4.16) or as functions of the principal stretches 2 2 ˆ λ1 + λ2 ˆ β1 µ = β0 2 2 , 2 − λ1λ2 ! 2 2 2 2 ˆ ˆ λ1 + λ2 λ1 λ2 ∂β0 ∂β0 µ = βˆ0 + − λ1 λ2 (4.17) ∗ 2 4 ∂λ1 − ∂λ2 ! 2 2 2 2 ˆ ˆ 1 λ1 + λ2 ˆ λ1 λ2 ∂β1 ∂β1 2 2 β1 + − λ1 λ2 , − λ1λ2 " 2 4 ∂λ1 − ∂λ2 !# since λ˙ i = λiDii, i not summed. (4.18) An alternative expression for the two incremental moduli µ and µ , related ∗ to the existence of a strain-energy function, was proposed by Biot (1965) (see Appendix B) in the form 2 2 1 λ1 + λ2 ∂W ∂W µ = 2 2 λ1 λ2 , 2 λ λ ∂λ1 − ∂λ2 1 − 2 (4.19) 2 2 2 1 ∂W ∂W 2 ∂ W 2 ∂ W ∂ W µ = λ1 + λ2 + λ1 + λ2 λ1λ2 . ∗ 4 ∂λ ∂λ ∂λ2 ∂λ2 − ∂λ ∂λ 1 2 1 2 1 2 As it will become clear later, constitutive relations of the form (4.15) with generic coeﬃcients µ and µ embrace a much broader class of material behaviours than Cauchy, isotropic∗ elasticity.

Istantaneous moduli for plane strain We are interested here in the particularization of (4.16) and (4.17) to incremental plane-strain deformation superimposed upon a plane homogeneous deformation. 4.1 Incremental constitutive equations 71

In plane strain, the left Cauchy-Green strain tensor and its inverse in the Eulerian principal reference system can be expressed as

1 diag B = λ2, , 1 , λ2 (4.20) 1 1 2 diag B− = , λ , 1 . λ2 Substitution of (4.20) in (4.9) yields

∇σ3i = ∇σi3= 0 (i = 1, 2), (4.21) and 1 ∂ ∇ 2 σ33 = q˙ + λ 2 (β0 + β1) − − λ ∂I1 (4.22) 1 ∂ + 2 λ2 + (β + β ) (D D ) , λ2 ∂I 0 1 11 − 22 2 while the incremental moduli µ and µ become functions of the invariants of ∗ B 1 1 µ = λ2 + (β β ) , 2 λ2 0 − 1 2 1 2 1 1 2 1 µ = λ + (β0 β1) + λ (4.23) ∗ 2 λ2 − 2 − λ2 ∂ 1 ∂ (β β ) + 2 λ2 + (β β ) . ∂I 0 − 1 λ2 ∂I 0 − 1 1 2 If we introduce the coeﬃcients

1 β˜ (λ)= βˆ λ, (i = 0, 1), (4.24) i i λ the incremental moduli, functions of the principal stretch, take the form

1 1 µ = λ2 + β˜ β˜ , 2 λ2 0 − 1 (4.25) 1 2 1 λ 2 1 ∂ µ = λ + β˜0 β˜1 + λ β˜0 β˜1 . ∗ 2 λ2 − 4 − λ2 ∂λ − 72 Incremental Deformations

The expressions of µ and µ related to the strain-energy function W˘ (λ) are ∗ λ λ4 + 1 ∂W˘ µ = , 2 λ4 1 ∂λ − (4.26) λ ∂ ∂W˘ µ = λ . ∗ 4 ∂λ ∂λ !

4.1.2 Mooney-Rivlin material For a Mooney-Rivlin material we simply obtain

1 2 2 µ2 µ = µ = λ1 + λ2 µ1 , (4.27) ∗ 2 − λ2λ2 1 2 or 1 λ4 + 1 µ = µ = (µ1 µ2) , (4.28) ∗ 2 λ2 − for plane strain. Further, the out-of-plane stress increment is

∇σ = q.˙ (4.29) 33 − 4.1.3 Ogden material For an Ogden material, coeﬃcients in the representations (4.15) can be obtained from eqn. (3.132) in the form

2 2 N 1 λ1 + λ2 αi αi µ = 2 2 µi (λ1 λ2 ), 2 λ1 λ2 − − Xi=1 (4.30) N 1 αi αi µ = αiµi (λ1 + λ2 ), ∗ 4 Xi=1 or N 1 λ4 + 1 λ2 αi 1 µ = µi − , 2 λ4 1 λαi − Xi=1 (4.31) N 1 λ2 αi + 1 µ = αiµi , ∗ 4 λαi Xi=1 for plane strain.

The component σ∇33 is given in Appendix C. 4.1 Incremental constitutive equations 73

4.1.4 Hypoelasticity and the loading branch of elastoplastic constitutive laws

We consider a general incremental costitutive equation relating the Jaumann derivative of the Cauchy stress tensor to the Eulerian strain rate D through a generic tensor T Sym in the form ∈

σ∇ = q˙I + γ D + (γ T · D + γ T2 · D)I + (γ T · D + γ T2 · D)T − 1 2 3 4 5 2 2 2 2 + (γ6T · D + γ7T · D)T + γ8(DT + TD)+ γ9(DT + T D), (4.32) where coefficients γ (i = 1, , 9) are polynomial functions of the invariants i · · · of T. In the particular case in which T is identified with the Cauchy stress σ, the constitutive equation (4.32) describes an incompressible, hypoelastic material (Truesdell and Noll, 1965). However, even if T does not represent the Cauchy stress and the coefficients γ (i = 1, , 9) remain completely i · · · unspecified (but independent of D), in a principal reference system of T and for plane, incremental deformation, we get

∇σi3 = ∇σ3i= 0, i = 1, 2,

∇σ = q˙ + (T T ) γ + γ T + γ T 2 (4.33) 33 − 1 − 2 2 4 3 6 3 2 + (T1 + T2) γ3 + γ5T3 + γ7T3 D11, where Ti (i = 1, 2, 3) denote the principal values of T and ∇σ12, ∇σ21 and ∇σ ∇σ can be expressed in the form (4.15), with 11 − 22

2 2 γ1 + γ8 (T1 + T2)+ γ9 T + T µ = 1 2 , 2 1 2 2 µ = γ1 + (T1 T2) γ4 + (T1 + T2) (γ5 + γ6) + (T1 + T2) γ7 ∗ 2 − n 2 h 2 i + (T1 + T2) γ8 + T1 + T2 γ9 . (4.34) 74 Incremental Deformations

4.1.5 J2 material: hyperelastic and hypoelastic approaches

For a J2 material, the incremental shear moduli in the representations (4.15) can be obtained from eqn. (3.138) in the form: 1 µ = E ( ) coth ( ), 3 s 1 − 2 1 − 2 (4.35) 1 Es 2 2 µ = 2 3(1 + 2) + N(1 2) , ∗ 9 E − h i or 2 µ = E coth (2 ), 3 s (4.36) 1 µ = Es N, ∗ 3 for plane strain. The out-of-plane stress increment is given by

σ∇11 + σ∇22 ∇σ = pˆ˙ = . (4.37) 33 2 Diﬀerently from Hutchinson and Neale (1978), St¨oren and Rice (1975) present the following hypoelastic law (their eqn. [26]) to model elastoplastic materials subject to proportional loading:

1 N S · σ∇ S∇ = 2h D − S, (4.38) 1 − N S · S where S = σ (trσ/3)I is the deviatoric stress, N is a work-hardening ex- − ponent and h1 is the secant modulus on the shear stress-strain curve. Eqn. (4.38) can be written as

S · D σ∇ = pˆ˙ I + 2h D (1 N) S , (4.39) 1 − − S · S where pˆ˙ = trσ˙ /3, which is, evidently, a particular case of (4.32) and can therefore be cast in the form (4.15).

4.1.6 The loading branch of non-associative, elastoplastic law A generic constitutive equation for an incompressible isotropic-elastic, plastic material depending on a generic collection of state variables can be written K 4.1 Incremental constitutive equations 75 in the form 4µ2 2µD < Q · D > P if f(σ, ) = 0 , σ∇ pˆ˙ I = − H K (4.40)  − 2µD if f(σ, ) < 0 ,  K where f is the yield function, Q Sym is the yield function gradient and  ∈ P Sym the plastic potential gradient. The Macaulay brackets < > apply ∈ · to every scalar α in such a way that <α>= (α + α )/2 and introduce the | | incremental non-linearity, typical of plasticity. The scalar H > 0 is the plastic modulus, related to the hardening modulus h through

H = h + he with he = 2µQ · P. (4.41) Finally, tensors D, P and Q are subject to the incompressibility constraints trD = 0, trP = 0, trQ = 0, (4.42) so that pˆ˙ = tr σ∇ /3. Restricting the constitutive equation (4.40) to its loading branch is synony- mous of assuming f(σ, ) = 0 and eliminating the incremental non-linearity, K thus obtaining 4µ2 σ∇ = pˆ˙ I + 2µD (Q · D)P. (4.43) − H Let us assume now that P and Q are coaxial (and not necessarily coaxial with the Cauchy stress). In the principal reference system of P and Q and for plane incremental deformations, eqn. (4.43) can be cast in the form (4.15) with µ2 µ = µ (Q1 Q2) (P1 P2) . (4.44) ∗ − H − − The constitutive fourth-order tensor in (4.43) contains the non-symmetric term P Q, where Q = P corresponds to the so-called non-associative elasto- ⊗ 6 plasticity. However, due to the incompressibility and plane strain constraints and without altering the material response, we can add a term P Q + A (I 3 e e ) , (4.45) ⊗ ⊗ − 3 ⊗ 3 where A is a symmetric tensor coaxial to P and Q and e3 is the unit vector singling the out-of-plane direction. Now, the choice diagA = P Q P Q + α, α, P Q + P Q 2α , (4.46) { 2 1 − 1 2 − 2 1 1 2 − } taken in the principal reference system of P and Q, symmetrizes the constitutive operator for every α R, so that ∈ 76 Incremental Deformations

non-associativity does not yield an unsymmetric tangent constitutive operator for incompressible, isotropic-elastic, plane-strain plasticity with coaxial yield function and plastic potential gradients.

4.1.7 The general form of constitutive equations for plane, incompressible, incremental deformations We have shown in the previous sections that a broad class of incremental material behaviours —including Cauchy elasticity, hyperelasticity, hypoelasticity, the loading branch of coaxial elastoplasticity, and Newtonian ﬂuids— can be described by constitutive equations (4.15). It is expedient now to transform the constitutive equation (4.15) in terms of material derivative of the nominal stress tensor t. 1 Recalling the deﬁnition of the nominal stress tensor t˙ = JF− σ, its material time derivative is equal to

1 1 1 t˙ = J˙F− σ + J(F− )·σ + JF− σ˙ , (4.47) where 1 1 (F− )· = F− L. (4.48) − Since incompressible incremental deformations are considered and the current conﬁguration is assumed as the reference conﬁguration, relation (4.47) becomes t˙ = σ˙ Lσ, (4.49) − By introducing the Jaumann derivative we obtain

t˙ = σ∇ σW Dσ, (4.50) − − A substitution of eqn. (4.15) in eqn. (4.50) yields, in the Eulerian principal reference system, t˙ = (2µ σ )D σ W , 12 − 2 12 − 1 12 t˙ = (2µ σ )D σ W , 21 − 1 21 − 2 21 t˙11 t˙22 = 2µ (D11 D22) D11σ1 + D22σ2 − ∗ − − (4.51) = (4µ σ1 σ2)D11, ∗ − − t˙i3 = t˙3i = 0,

t˙33 = σ∇3=σ ˙ 3. 4.1 Incremental constitutive equations 77

In addition, from equation (4.49) we obtain

t˙ + t˙ =σ ˙ +σ ˙ σ D σ D =σ ˙ +σ ˙ + (σ σ )D , (4.52) 11 22 1 2 − 1 11 − 2 22 1 2 2 − 1 11 and therefore

σ˙ 1 +σ ˙ 2 t˙11 = (2µ σ1)D11 + , (4.53) ∗ − 2 σ˙ 1 +σ ˙ 2 t˙22 = (2µ σ2)D22 + . (4.54) ∗ − 2

Since L = D+W = gradv, constitutive equations (4.15) may be expressed as

t˙ij = Kijklvl,k +pδ ˙ ij = K˜ ijklvl,k +πδ ˙ ij, vi,i = 0, (4.55) where δij is the Kronecker delta,

σ˙ +σ ˙ t˙ + t˙ σ σ p˙ = 1 2 andπ ˙ = 11 22 =p ˙ 1 − 2 v (4.56) 2 2 − 2 1,1 are the in plane hydrostatic stress rate (positive in tension), related respectively to the Cauchy and to the nominal stress rates, vi is the velocity, and Kijkl represents the instantaneous moduli, possessing the major symmetry Kijkl = Kklij and having the form (Hill and Hutchinson, 1975) σ K1111 = µ p, K1122 = µ , K1112 = K1121 = 0, ∗ − 2 − − ∗ σ K2211 = µ , K2222 = µ + p, K2212 = K2221 = 0, − ∗ ∗ 2 − σ σ K = µ + , K = K = µ p, K = µ , 1212 2 1221 2112 − 2121 − 2 with σ + σ σ = σ σ , p = 1 2 . (4.57) 1 − 2 2

Note that the only components of K˜ ijkl diﬀerent from Kijkl are:

K˜ 1111 = K˜ 2222 = µ p, (4.58) ∗ − so that K˜ ijkl also possesses the major symmetry. Only the forms (4.57) and (4.58) of the constitutive equation (4.15) will be needed in the following. 78 Incremental Deformations

The components of the material derivative of the nominal stress are given in the following σ t˙11 = 2µ p v1,1 +p, ˙ ∗ − 2 − σ t˙ = (µ p)v + µ + v , 12 − 1,2 2 2,1 σ t˙21 = (µ p)v2,1 + µ v1,2, − − 2 (4.59) σ t˙22 = 2µ + p v2,2 +p, ˙ ∗ 2 − t˙i3 = t˙3i = 0 (i = 1, 2),

t˙33 =σ ˙ 3. Chapter 5

Green’s function for incremental non-linear elasticity

The Bigoni and Capuani (2002) Green’s function, that will be crucial for boundary element formulation, is brieﬂy presented. In particular, an elastic, incompressible, inﬁnite body is considered subject to plane and homogeneous deformation. At a certain value of the loading, when the material is still in the elliptic range, an incremental concentrated line load is considered acting at an arbitrary location in the body and extending orthogonally to the plane of deformation. This plane strain problem has been solved by Bigoni and Capuani who have therefore obtained a Green’s function for incremental, non- linear elastic deformation.

5.1 The equilibrium equations

Let us consider an hyperelastic, incompressible and initially isotropic solid under plane strain conditions, for which the constitutive equation is given by eqn. (4.55). When an incremental force with components (f˙1, f˙2) acts at the point x = y along the principal stress axes, the incremental equilibrium equation becomes

t˙ij,i + f˙j δ(r) = 0, (5.1) where δ is the two-dimensional Dirac delta function, and r = x y. −

79 80 Green’s function for incremental non-linear elasticity

Substitution of (4.59) in (5.1) yields σ (2µ p)v1,11 + (µ p)v2,12 + µ v1,22 + f˙1 δ(r)= π˙ ,1, ∗ − − − 2 − σ (5.2) (2µ p)v2,22 + (µ p)v1,21 + µ + v2,11 + f˙2 δ(r)= π˙ ,2, ∗ − − 2 − whereπ ˙ is the in-plane hydrostatic stress rate given in eqn. (4.56). Since the incompressibility requires div v = 0, the velocity ﬁeld can be described by a stream function ψ(x1,x2), such that

v = ψ , v = ψ . (5.3) 1 ,2 2 − ,1

By diﬀerentiating equations (5.2)1 and (5.2)2 with respect to x1 and x2, respectively, and subtracting the result, we obtain σ σ µ + ψ,1111 + 2(2µ µ)ψ,1122 + µ ψ,2222 + f˙1 δ(x),2 f˙2 δ(x),1 = 0. 2 ∗ − − 2 − (5.4)

5.2 The regime classiﬁcation

If we introduce a measure of the current state of stretch

2 2 σ λ1 λ2 k = = 2 − 2 , (5.5) 2µ λ1 + λ2 the characteristic equation associated to (5.4) becomes

ω4 µ µω4 (1 + k) 1 + 2 2 1 + (1 k) = 0, (5.6) 2 ω4 µ − − 2 ∗ which admits

four real solutions ω /ω in the hyperbolic regime (H); • 1 2 two real solutions ω /ω in the parabolic regime (P); • 1 2 no real solutions ω /ω in the elliptic regime (E); • 1 2 where the different regimes are reported in fig. [5.1] 5.2 The regime classification 81

m* /m

P 1 P

EC 0.5

H H k -1 1

Figure 5.1: Elliptic (imaginary and complex), parabolic and hyperbolic regimes.

The formulation in the present work is restricted to the elliptic range (E), 2 2 where the roots for ω1/ω2 µ 1 2 ∗ √∆ 2 − µ ± 2 µ µ γ = , ∆= k 4 ∗ + 4 ∗ , (5.7) 1,2 1+ k − µ µ are both negative or complex. The elliptic range may be further sub-divided into elliptic-imaginary (EI) and elliptic-complex (EC) regimes. These are deﬁned as follows: ∆ > 0, so that γ and γ are both negative in (EI), • 1 2 ∆ < 0, so that γ and γ are a conjugate pair in (EC). • 1 2 It may be important to remark that shear bands, corresponding to the ap- pearance of discontinuous strain rates, is formally excluded within the elliptic range, i.e. in the context analyzed here. However, as shown by Bigoni and Capuani (2002) and Radi et al. (2002), formation of zones of concentrated strains in response to a perturbation become possible when the boundary of the elliptic regime is approached. This will be also demonstrated with numerical examples referring to van Hove conditions in Chapter 7. The boundary of the elliptic range is characterized by the following conditions: elliptic imaginary/parabolic (EI/P) boundary is attained when k = 1 • ± and µ /µ 1/2, corresponding to γ1 = 0; ∗ ≥ 82 Green’s function for incremental non-linear elasticity

elliptic complex/hyperbolic (EC/H) boundary is attained when ∆ = 0 • and µ /µ < 1/2, corresponding to γ1 = γ2. ∗ 5.3 The stream fuction

Due to the linearity of the incremental problem, the solution regarding a generic point load f can be obtained by the superposition of the two solutions for the two components f1 and f2 (Willis 1971, 1972, 1973). If we assume f˙i = δig and we introduce the linear diﬀerential operator

σ ∂4( ) ∂4( ) σ ∂4( ) ( )= µ + 4· + 2(2µ µ) 2 · 2 + µ 4· , (5.8) L · 2 ∂x1 ∗ − ∂x1x2 − 2 ∂x2 the equilibrium equation (5.4) becomes

∂δ(r) ∂δ(r) ψg + δ δ = 0. (5.9) L 1g ∂x − 2g ∂x 2 1 Since the plane wave expansion of the δ function is (Courant and Hilbert, 1962; Gel’fand and Shilov, 1964) 1 dω δ(r)= , (5.10) −4π2 (ω · r)2 ωI=1 | | where ω is an unit vector, the analogous transformation for the stream function ψg is 1 ψg(r)= ψ˜g(ω · r)dω. (5.11) −4π2 ωI=1 | | Therefore, in the domain ω · r the equation (5.9) becomes

∂ 1 ∂ 1 ψ˜g(ω · r)+ δ δ = 0, (5.12) L 1g ∂x (ω · r)2 − 2g ∂x (ω · r)2 2 1 and so δ ω δ ω ψ˜g = 2 1g 2 − 2g 1 . (5.13) L (ω · r)3 Employing the chain rule of diﬀerentation

∂ g g 0 ψ˜ (ω · r)= ωk(ψ˜ ) , (5.14) ∂xk 5.3 The stream fuction 83

1 r x a q y

O x1

Figure 5.2: Vectors ω, r and angles α and θ. where a prime denote differentiation with respect to ω · r, the equation (5.13) becomes 0000 δ ω δ ω L(ω)(ψ˜g) = 2 1g 2 − 2g 1 , (5.15) (ω · r)3 where σ 4 2 2 σ 4 L(ω) = µ + ω1 + 2(2µ µ)ω1ω2 + µ ω2 2 ∗ − − 2 (5.16) = µ(1 + k) ω2 γ ω2 ω2 γ ω2 , 1 − 1 2 1 − 2 2 with γ1 and γ2 given in (5.7). Equation (5.16) is always positive in the elliptic regime (E). The integration of (5.16) yields g 3 L(ω)ψ˜ = (δ1gω2 δ2gω1)(ω · r)(log ω · ˆr 1) + C1(ω · r) − | |− (5.17) 2 + C2(ω · r) + C3(ω · r)+ C4, where ˆr represents a dimensionless measure of length. As demonstrated in Section (1.2) coefficients C1, C2, C3 and C4 have to be set equal to zero. In the domain (r, θ), where r and θ are the polar coordinates singling out the generic point with respect to the position y of the concentrated force, the stream function is determined by the antitransform 1 δ ω δ ω ψg = 1g 2 − 2g 1 (ω · r)(log ω · ˆr 1)dω, (5.18) −4π2 L(ω) | |− ωI=1 | | 84 Green’s function for incremental non-linear elasticity and since ω · r = r cos θ, we obtain 2π g r cos α sin [α+θ + (1 g)π/2] ψ = 2 − [log(ˆr cos α) 1] , −4π µ(1 + k) 0 Λ(α+θ) − Z (5.19) where Λ(α) = (cos2 α γ sin2 α)(cos2 α γ sin2 α) > 0. (5.20) − 1 − 2 5.4 The velocity field

According to (5.10) the velocity in the domain (ω · r) transforms to 1 vg(r)= v˜g(ω · r)dω (5.21) i −4π2 i ωI=1 | | and, if eqns. (5.3) and (5.3) are employed, we obtain log ω · ˆr v˜g = ω 2δ ω ω | |. (5.22) i | | ig − i g L(ω) g The antitransform ofv ˜i into the domain (r, θ) yields 1 π δ log r vg = ig i 2π2µ(1 + k) [(2 i) γ + 1 i] √ γ + [(2 i) γ + 1 i] √ γ − 2 − − 1 − 1 − − 2 π 2 [Kg(α+θ)+(2δ 1) Kg(α θ)] log (cos α) dα , − i ig − i − Z0 ) (5.23) where sin [α + (1 i) π ] sin [α + (1 g) π ] Kg(α)= − 2 − 2 . (5.24) i Λ(α)

It is important to note that, due to the major symmetry of Kijkl, 2 1 v1 = v2, (5.25) g and that the Green’s velocity vi can be expressed as g ig ig vi (r, θ)= ξ1 log r + ξ2 (θ), (5.26) where the dependence on the current state is condensed in the coefficients ξig and ξig. The former coefficient satisfies ξig = 0 if i = g, while the latter 1 2 1 6 contains the directional dependence on θ. 5.5 The velocity gradient 85

5.5 The velocity gradient

The gradient of the Green’s set may be determined through a derivation in the transformed domain ω · r or in the domain (r, θ). The derivation is simpler in the transformed domain, where however a singularity of higher order is found, which needs to be integrated. Hence, to avoid the singularity, the gradients are obtained through a derivation in the domain (r, θ). The gradient of the Green’s velocity may be obtained directly from eqn. (5.23) as ∂vg ∂vg sin θ ∂vg i = cos θ i i , ∂x ∂θ − r ∂θ 1 (5.27) ∂vg ∂vg cos θ ∂vg i = sin θ i + i , ∂x2 ∂θ r ∂θ and can be written as

1 (3 2i) π cos θ + (1 g) π vi = − − 2 i,g 2π2µ(1 + k)r [(2 i) γ + 1 i] √ γ + [(2 i) γ + 1 i] √ γ ( − 2 − − 1 − 1 − − 2 π π 2 π + sin θ+(1 g) Σ α+θ+(i 1) , α+θ log(cos α) dα − 2 0 − 2 h iZ π π 2 π sin θ+(1 g) Σ α θ (i 1) , α+θ log(cos α) dα , − − 2 − − − 2 Z0 ) h i (5.28) where index i is not summed and

sinα [2 cosα Λ(β) sinα Λ0(β)] ∂Λ(β) Σ (α, β)= − , Λ0(β)= . (5.29) Λ2(β) ∂β

The four components of the velocity gradient not included in eqn. (5.28) can be obtained using incompressibility and symmetry of Green’s tensor:

v1 = v2 = v1 , v1 = v2 = v2 . (5.30) 2,2 1,2 − 1,1 2,1 1,1 − 2,2 5.6 The incremental stress ﬁeld

In order to determineπ ˙ , we diﬀerentiate the equations (5.2)1 and (5.2)2 with respect to x1 and x2, respectively, and we sum them to get σ π˙ 11 +π ˙ 22 = 2(µ µ)(v1,111 +v2,222)+ (v1,111 v2,222) f˙1δ,1 f˙2δ,2. (5.31) − ∗ − 2 − − − 86 Green’s function for incremental non-linear elasticity

According to (5.10), the transform of the Green in-plane hydrostatic nominal stress is 1 π˙ g(r)= π˜˙ g(ω · r)dω, (5.32) −4π2 ωI=1 | | so that the substitution of the Green velocity ﬁeld in the domain ω · r into eqn. (5.31), with f˙i = δig yields

˙ g 00 3 g 000 3 g 000 σ 3 g 000 3 g 000 (π˜ ) = 2(µ µ) ω1(˜v1) + ω2(˜v2 ) + ω1(˜v1) ω2(˜v2 ) − ∗ − 2 − h i h i ω δ + ω δ + 2 1 1g 2 2g . (ω · r)3 (5.33) The integration of equation (5.33) leads to

˙ g 3 g 0 3 g 0 σ 3 g 0 3 g 0 π˜ = 2(µ µ) ω1(˜v1) + ω2(˜v2) + ω1(˜v1 ) ω2(˜v2 ) − ∗ − 2 − h i h i (5.34) ω δ + ω δ + 1 1g 2 2g , (ω · r) where, according to (5.22)

2 0 ω δ ω ω 1 (˜vg) = | | ig − i g , (5.35) i L(ω) ω · r and inessential contributions have been cancelled. Then,

ω (1 ω2) ˙ g ωg g g 2 2 σ π˜ = + (2g 3) − 2(µ µ)(ω1 ω2) , (5.36) ω · r − (ω · r)L(ω) ∗ − − − 2 h i which in the domain (r, θ) becomes

1 π 1 π K˜ (α+θ) π˙ g = cos θ (g 1) + g dα , (5.37) −2πr ( − − 2 π(1 + k) 0 cos α ) h i Z where g π K˜ (α) = Kg (α) cos α + (g 1) Γ(α), g − 2 (5.38) µ 2 Γ(α) = 2 ∗ 1 2 cos α 1 k. µ − − − 5.7 The gradients of the Green’s tensor set 87

From Green’s velocity gradient and pressure rateπ ˙ g we obtain the in-plane hydrostatic stress rate σ p˙ g =π ˙ g vg , (5.39) − 2 1,1 and the associated incremental nominal stress (by using constitutive equations (4.55)) ˙g g g ˙g g g t11 = (2µ p) v1,1 +π ˙ , t12 = (µ p) v1,2 + (µ + µk) v2,1, ∗ − − ˙g g g ˙g g g t21 = (µ p) v2,1 + (µ µk) v1,2, t22 = (2µ p) v1,1 +π ˙ . − − − ∗ − (5.40) We remark that the gradient of the Green’s velocity, the in-plane hydrostatic stress rate and thus the incremental stress rate can be expressed as a function of θ divided by r.

5.7 The gradients of the Green’s tensor set

The gradients of the Green’s tensor set vg, π˙ g , that will be used in the { } boundary elements formulation, are listed hereafter. They are obtained by deriving vg andπ ˙ g in the domain (r, θ), and using the chain rule (5.27).

5.7.1 The velocity gradient The components of the Green’s velocity gradient are explicitly reported below 1 π cos θ v1 = 1,1 2π2µ(1 + k)r γ √ γ + γ √ γ 2 − 1 1 − 2 π + sin θ Σ(α+θ, α+θ) log(cos α) dα ), Z0 1 π sin θ v2 = 2,2 2π2µ(1 + k)r √ γ + √ γ − 1 − 2 π π + cos θ Σ α+θ+ , α+θ log(cos α) dα , 0 2 Z 1 π sin θ v1 = 1,2 2π2µ(1 + k)r γ √ γ + γ √ γ 2 − 1 1 − 2 π cos θ Σ(α+θ, α+θ) log(cos α) dα , − Z0 88 Green’s function for incremental non-linear elasticity and 1 π cos θ v2 = 2,1 −2π2µ(1 + k)r √ γ + √ γ − 1 − 2 π π sin θ Σ α+θ+ , α+θ log(cos α) dα , − 0 2 Z where sinα [2 cosα Λ(β) sinα Λ (β)] Σ (α, β)= − 0 , Λ2(β) Λ(β) = sin4 β(cot2 β γ )(cot2 β γ ), − 1 − 2 ∂Λ(β) Λ (β)= = sin(2β) cos2 β (2+ γ + γ )+sin2 β (2γ γ + γ + γ ) . 0 ∂β − 1 2 1 2 1 2 The four components of the velocity gradient not reported can be obtained using incompressibility and symmetry of vg, v1 = v2 = v1 , 1,1 − 1,2 − 2,2 v2 = v2 = v1 . 2,2 − 1,1 − 2,1 5.7.2 The second-gradient of velocity The components of the second-gradient of the Green’s velocity are provided below sin 2θ 1 v1 = 1,12 − 2πµ(1 + k)r2 γ √ γ + γ √ γ 2 − 1 1 − 2 cos 2θ π + Σ(α+θ, α+θ) log(cos α) dα 2π2µ(1 + k)r2 Z0 sin θ cos θ π + Σ 0(α+θ, α+θ) log(cos α) dα , 2π2µ(1 + k)r2 Z0 cos 2θ 1 v1 = 1,22 2πµ(1 + k)r2 γ √ γ + γ √ γ 2 − 1 1 − 2 sin 2θ π + Σ(α+θ, α+θ) log(cos α) dα 2π2µ(1 + k)r2 Z0 cos2 θ π Σ 0(α+θ, α+θ) log(cos α) dα , − 2π2µ(1 + k)r2 Z0 5.7 The gradients of the Green’s tensor set 89

sin 2θ 1 v2 = 1,11 −2πµ(1 + k)r2 √ γ + √ γ − 1 − 2 cos 2θ π π 2 2 Σ α+θ+ , α+θ log(cos α) dα − 2π µ(1 + k)r 0 2 Z sin θ cos θ π π 2 2 Σ 0 α+θ+ , α+θ log(cos α) dα , − 2π µ(1 + k)r 0 2 Z sin2 θ cos2 θ 1 v1 = − 1,11 2πµ(1 + k)r2 γ √ γ + γ √ γ 2 − 1 1 − 2 sin 2θ π Σ(α+θ, α+θ) log(cos α) dα − 2π2µ(1 + k)r2 Z0 sin2 θ π Σ 0(α+θ, α+θ) log(cos α) dα , − 2π2µ(1 + k)r2 Z0 and cos2 θ sin2 θ 1 v2 = − 2,11 − 2πµ(1 + k)r2 √ γ + √ γ − 1 − 2 sin 2θ π π 2 2 Σ α+θ+ , α+θ log(cos α) dα − 2π µ(1 + k)r 0 2 Z sin2 θ π π 2 2 Σ 0 α+θ+ , α+θ log(cos α) dα , − 2π µ(1 + k)r 0 2 Z where ∂Σ (α, β) 2 cos(2α) Λ (β)sin2 α + 2Λ (β) sin(2α) Σ (α, β) = = 00 0 0 ∂β Λ(β) − Λ2(β) 2Λ 2(β)sin2 α + 0 , Λ3(β) Λ (β) = 2 cos(2β) (γ γ 1) 2 cos(4β)(1 + γ )(1 + γ ). 00 1 2 − − 1 2 The eleven components not included in previous relations can be obtained using the incompressibility and the symmetry of vg v2 = v1 = v2 = v2 , 1,11 2,11 − 2,21 − 2,12 v1 = v1 = v1 = v2 = v2 = v2 , (5.41) 1,11 − 2,12 − 2,21 − 1,12 2,22 1,21 v1 = v1 = v1 = v2 . 1,12 1,21 − 2,22 − 1,22 90 Green’s function for incremental non-linear elasticity

5.7.3 The gradient of pressure rate The components of the gradient of the pressure rate are given by

cos(2θ) 1 π F (α, θ) cos θ + F (α, θ)sin θ π˙ 1 = 0 dα, ,1 2πr2 − 2π2r2(1 + k) cos α Z0 sin(2θ) 1 π F (α, θ)sin θ F (α, θ) cos θ π˙ 1 = − 0 dα, ,2 2πr2 − 2π2r2(1 + k) cos α Z0 sin(2θ) 1 π G(α, θ) cos θ + G (α, θ)sin θ π˙ 2 = + 0 dα, ,1 2πr2 2π2r2(1 + k) cos α Z0 and cos(2θ) 1 π G(α, θ)sin θ G (α, θ) cos θ π˙ 2 = + − 0 dα, ,2 − 2πr2 2π2r2(1 + k) cos α Z0 where sin2(α+θ) cos(α+θ)Γ(α+θ) F (α, θ)= , Λ(α+θ) sin(α+θ) cos2(α+θ)Γ(α+θ) G(α, θ)= , Λ(α+θ)

µ 2 Γ(α) = 2 ∗ 1 (2 cos α 1) k, µ − − − and ∂F Γ(α+θ) F (α, θ) = = 2 cos2(α+θ) sin(α+θ) sin3(α+θ) 0 ∂θ Λ(α+θ) − 2 cos(α+θ)sin (α+θ) Γ(α+θ)Λ 0(α+θ) + Γ 0(α+θ) , Λ(α+θ) − Λ(α+θ) ∂G Γ(α+θ) G (α, θ) = = cos3(α+θ) 2sin2(α+θ) cos(α+θ) 0 ∂θ Λ(α+θ) − 2 cos (α+θ) sin(α+θ) Γ(α+θ)Λ 0(α+θ) + Γ 0(α+θ) , Λ(α+θ) − Λ(α+θ) ∂Γ µ Γ (α) = = 4 ∗ 1 sin(2α). 0 ∂α µ − Chapter 6

Boundary elements formulation

A general numerical scheme is formulated in this Section to handle generic boundary value problems with prescribed nominal tractions and/or displacements. When restricted to perturbations of homogeneously deformed, incompressible solids, our boundary elements technique retains all well-known advantages of the small strain formulation. These are:

discretization only of the boundary of the body; • automatic satisfaction of the incompressibility constraint; • possibility of describing singularities arising near corner points of the • boundary;

possibility of employing meshes thoroughly varying in size throughout • the body.

Several attempts can be found in the literature to analyze non-linear problems using boundary elements techniques. In some cases the non-linearities were related to the material (Bui, 1978; Maier, 1983; Telles, 1983; Venturini, 1983; Bonnet and Mukherjee, 1996), in other cases to large elastic (Novati and Brebbia, 1982; Tran-Cong et al., 1990; Polizzotto, 2000) or elastoplastic (Chandra and Mukherjee, 1986a; 1986b; Okada et al. 1988; 1989; 1990; Jin et al. 1989; Chen and Ji, 1990; Foerster and Kuhn, 1994) strains. In all cases, in addition to the usual boundary integrals, a domain integral is introduced, leading to the so-called ‘ﬁeld-boundary element method’. The introduction of this term nulliﬁes a main advantage of BEM and originates from the discrep- ancy between the non-linear character of the equations governing the problem

91 92 Boundary elements formulation and the employed fundamental solution (usually referring to linear, isotropic elasticity). In the present Section, the focus is on incrementally-linear problems, so that domain integrals do not appear in the formulation that will be presented. However, we believe that the solution of incrementally linear problems should be regarded as the ﬁrst step toward the analysis of fully non-linear situations and, in particular, we anticipate with a few simple examples that when employed as a tool to analyze large (thus non-linear) deformations, our incremental method naturally leads to a volume discretization diﬀerent —in essence— from all already known.

6.1 Integral representation

We restrict the presentation to mixed boundary value problems in which velocities and (incremental) nominal tractionsτ ˙ are prescribed functions deﬁned on separate portions ∂B and ∂B of the boundary ∂B = ∂B ∂B v τ v ∪ τ

vi =v ¯i, on ∂Bv, t˙ijni = τ¯˙j on ∂Bτ , (6.1) of a solid B, currently in a state of homogeneous, ﬁnite deformation. In this context, two integral representations exist relating the velocity and the pressure rate in interior points of the body to the boundary values of nominal traction rates and velocities (Bigoni and Capuani, 2002). These are

v (y)= t˙ (x) n (x) vg(x, y) t˙g (x, y) n (x) v (x) dl , (6.2) g ij i j − ij i j x ∂BZ h i and

p˙(y) = t˙ (x) n (x)p ˙g(x, y) dl + n (x) v (x) K p˙g (x, y) dl − ig i x i j ijkg ,k x ∂BZ ∂BZ h i 2 2 σ 1 4µµ 4µ + µσ 2µ σ ni(x) vi(x) v1,11(x, y)dlx − ∗ − ∗ − ∗ − 2 ∂BZ σ σ µ + n (x) v (x) v2 (x, y)dl , − 2 i i 2,11 x ∂BZ (6.3) If the point y is on the boundary, eqn. (6.2) becomes (Bigoni and Capuani, 2002; Bonnet 1995) 6.2 Boundary discretization 93

Cgv (y) = τ˙ (x) vg(x, y) dl τ˙ g(x, y) v (x) dl , j j j j x −− j j x (6.4) ∂BZ ∂BZ where denotes the Cauchy principal value and −

R g g C = lim τ˙ (x, y) dlx (6.5) i ε 0 i → ∂CZε is the so-called C matrix, depending on the material parameters, state of pre- stress and the geometry of the boundary (in the case of a smooth boundary, g 1 Ci = 2 δgi). Note that symbol ∂Cε introduced in (6.5) denotes the intersection between a circle of radius ε centred at y and the domain B.

6.2 Boundary discretization

The boundary equation (6.4) is the starting point to derive the collocation boundary element method. To this purpose, the boundary ∂B is divided into m elements Γe (e = 1, ,m), with subsets m and m belonging respectively · · · v τ to ∂Bv and ∂Bτ (clearly m = mv + mτ ). Following the usual methodology of boundary elements, the same discretization is assumed for velocity and traction rates at the boundary. In particular, inside each boundary element Γe we use

vi(x)= ϕα(x)v ¯iα, (6.6) τ˙ (x)= ϕ (x) τ¯˙ , α = 0, , Θ, i α iα · · · wherev ¯iα, τ¯˙iα are the nodal values of velocities and nominal traction rates, respectively, and ϕα are the relevant shape functions, selected as polynomials of degree Θ. The discretized form of eqn. (6.4), collocating the point y at y(¯e,α¯), corresponding to the nodeα ¯ of the elemente ¯, is:

m m g e¯ e g (¯e,α¯) e g (¯e,α¯) Ci v¯iα¯ + v¯iα ϕα(x)τ ˙i (x, y )dlx = τ¯˙iα ϕα(x) vi (x, y )dlx, Xe=1 ΓZe Xe=1 ΓZe (6.7) where indices α and i are summed and range between 0 Θ and 1 2, re- − − spectively. 94 Boundary elements formulation

An analysis of eqn. (6.7) reveals that the number of unknowns is nunk = e e 2(mv +mτ )Θ = 2mΘ, beingv ¯iα prescribed on ∂Bv and τ¯˙iα on ∂Bτ . On collocating now the eqn. (6.7) at mΘ nodes along the two directions x1 and x2, yields the following algebraic system: Hvˆ = Gτ,ˆ (6.8) e ˙ e where vˆ andτ ˆ are the vector forms ofv ¯iα and τ¯iβ, defined as: e e vˆ2Θ(e 1)+2α+i =v ¯iα, τˆ2Θ(e 1)+2α+i = τ¯˙iα, (6.9) − − where Θ is not a free index, but the fixed number which specifies the degree of the interpolating functions. Solution of system (6.8), after separation of data and unknowns, gives the e e nodal velocitiesv ¯iα on ∂Bτ and the nominal traction rates τ¯˙iα on ∂Bv. After system (6.8) has been solved, the fields v(y) andp ˙(y) can be evaluated at internal points y by applying the discretized forms of eqns. (6.2) and g (6.3). In particular, eqn. (6.2) reduces to eqn. (6.7) with Ci = δgi and the integration is straightforward, as r is always different from zero. We limit the presentation to discretization of the boundary into linear elements and linear shape functions, so that Θ = 1 and nunk = 2m. Integrals are computed numerically, using Gaussian quadrature formulae with, when not otherwise specified, 12 points for integrals in the Green’s functions (5.23)–(5.40) and 18 points in the discretized eqn. (6.7). The singular integrals in (6.7), computed over the two elements adjacent to the node e in which the equation is collocated, are evaluated analytically. (ig,e) In particular, the strongly singular integrals Istrong in the left hand side of eqn. (6.7) are equal to

(ig,e) g (e 1,1) g (e,0) Istrong = ϕ1(x)τ ˙i (x, y − ) dlx + ϕ0(x)τ ˙i (x, y ) dlx, (6.10)

ΓeZ−1 ΓZe which, with reference to the geometry sketched in ﬁg. [6.1], can be transformed into 1 1 (ig,e) g (e) g (e) Istrong = le 1 s0 τ˙i (r ,θ1) ds0 + le (1 s00)τ ˙i (r ,θ2) ds00, (6.11) − − Z0 Z0 so that the change of variable

η = le 1(1 s0)= les00 (6.12) − − 6.2 Boundary discretization 95 yields

le−1 le (ig,e) η g η g Istrong = 1 τ˙i (η,θ1) dη + 1 τ˙i (η,θ2) dη. (6.13) − le 1 − le Z0 − Z0 As far as the two elements e 1 and e are concerned, the incremental Green’s −

e+1 q1

n(q2) q2 e-1 s' h=s" h n(q 1) e le

le-1

Figure 6.1: Geometry at node e. tractions can be computed as:

µ∗ χig(k, ) τ˙ g(r(e)) = ( 1)e µ , (6.14) i − r which are independent of θ, so that, in the collocation node e, χig can be calculated as ( 1)eτ g(1). In all our examples we have always found numerically − i that χ11 = χ22 = 0, a result that can be analytically checked in the special case of Newtonian ﬂuid, i.e. when k = 0 and µ = µ, but still requires a proof ∗ in the general case. Introducing eqn. (6.14) into eqn. (6.13), we obtain an explicit formula for the singular integrals (6.10) in the form

(ig,e) e le Istrong = ( 1) χig log . (6.15) − le 1 − 96 Boundary elements formulation

(ig,e) The weakly singular integrals Iweak in the right hand side of eqn. (6.7) is equal to

(ig,e) g (e 1,1) g (e,0) Iweak = ϕ1(x) vi (x, y − ) dlx + ϕ0(x) vi (x, y ) dlx, (6.16)

ΓeZ−1 ΓZe which becomes (ﬁg. [6.1])

le−1 le (ig,e) η g η g Iweak = 1 vi (η,θ1) dη + 1 vi (η,θ2) dη. (6.17) − le 1 − le Z0 − Z0

2 1 Taking in account eqn. (5.26), which in the speciﬁc case i = g gives v1 = v2 = ig 6 ξ2 (θ), the integrals (6.16) can be analytically evaluated and the resulting four components are listed in Table (6.1).

Table 6.1: Analytic expression for the weakly singular integrals (6.16) at the right hand side of eqn. (6.7) (ig,e) i g Integral Iweak le−1 2 log(le−1) 3 1 le 2 log(le) 3 1 − − 1 1 2 A11 + v1(1,θ1) + 2 A11 + v1(1,θ2)

h le−1 2 i le h2 i 1 2 2 v1 (θ1)+ 2 v1 (θ2)

le−1 1 le 1 2 1 2 v2 (θ1)+ 2 v2 (θ2)

le−1 2 log(le−1) 3 2 le 2 log(le) 3 2 2 2 − + v (1,θ ) + − + v (1,θ ) 2 − A22 2 1 2 − A22 2 2 h i h i A11 = 4µπ(1 + k)(γ2√ γ1 + γ1√ γ2), A22 = 4µπ(1 + k)(√ γ1 + √ γ2) − − − −

We note, in passing, that the use of polynomials of higher-order for the shape functions would be straightforward, since the singularities in the above integrals arise from the constant part of the interpolating functions. Finally, the ﬁrst term in eqn. (6.7) is non-singular and can be computed through the expression

θ2 g g Ci = fi (θ) dθ. (6.18)

θZ1 6.3 Two numerical examples: non-linear elasticity without domain integrals 97

6.3 Two numerical examples: non-linear elasticity without domain integrals

An elastic block subject to homogeneous plane strain, increasing deformation is considered, with the purpose of illustrating with simple examples that our formulation does not involve domain integrals.

6.3.1 In-plane tension and compression An elastic block is constrained to plane deformations, and starting from an unstressed square conﬁguration is subjected to increasing tension or compression in one direction. We refer to Mooney-Rivlin (3.128) and Ogden (3.132) non-linear elastic laws. The trivial, analytical solution to this problem is given by Ogden (1982) ∂Wˆ ∂Wˆ σ1 σ2 = λ1 λ2 , (6.19) − ∂λ1 − ∂λ2 so that for uniaxial stress (σ2 = 0) and plane strain deformations (λ1 = 1/λ2 = λ, λ3 = 1), from (3.131) and (3.132) we get

N 2 2 αi αi σ = (µ µ ) λ λ− , and σ = µ λ λ− , (6.20) 1 1 − 2 − 1 i − i=1 X for Mooney-Rivlin and Ogden materials, respectively. Within our framework, the solution may be obtained by integrating the incremental equations, using a forward Euler scheme. Since the basic step of the method is a linear increment superimposed upon a homogeneous configuration domain integrals or volume discretizations are completely avoided. It is worth mentioning that in the special case of homogeneous deformations, the domain integrals can be brought to the boundary even within the usual framework (see for instance Maier et al. (1992)) with reference to elastoplasticity. In that case, however, specific manipulations of the domain integrals are requested to transform these to boundary integrals, while in our method domain integrals are simply absent. Results are presented in fig. [6.2], where for the Mooney-Rivlin material we have taken µ0 = 0.35412 MPa, whereas for the Ogden material we have referred to the values listed in (3.135). The analytical solution is compared to the results given by the numerical procedure with a uniform mesh of 16 boundary elements. Gaussian quadrature formulae have been employed with 12 integration points for Green’s functions (5.23)–(5.40) and 18 points for integrals in the discretized eqn. (6.7). 98 Boundary elements formulation

Mooney-Rivlin Ogden s/m s/m 6 0 6 0

4 4

Analytical Analytical 2 BEM 2 BEM

0 l 0 l 0 1 2 3 0 1 2 3

-2 -2

-4 -4

Figure 6.2: Uniaxial deformation of a Mooney-Rivlin and Ogden elastic blocks. Non- dimensionalized Cauchy stress σ versus stretch λ.

6.3.2 Simple shear

An elastic block is constrained to plane deformations, and starting from an unstressed square conﬁguration is subjected to a simple shear as indicated in ﬁg. [6.3]. The problem admits an analytical solution in terms of Cauchy stress and shearing strain γ given, for instance, by Gurtin (1981). In particular we get 2 σ12 = Γ(γ ) γ, (6.21) σ σ = γσ , 11 − 22 12 where Γ(γ2) = β (γ2) β (γ2) is the generalized shear modulus and β and 0 − 1 0 β1 refer to the constitutive law given in eqn. (3.98). We consider Mooney-Rivlin (3.128) and Ogden (3.132) non-linear elastic law. In this case (see eqns. (3.130) and (3.137)) the generalized shear modulus takes the form Γ= µ µ , (6.22) 1 − 2 and N λ2 λ2αi 1 Γ= µi − , (6.23) λ4 1 λαi − Xi=1 respectively. 6.4 Conclusion 99 Du

x0,2 x2 x 1 g x0,1 b

Figure 6.3: Simple shear deformation of an elastic block.

The numerical solution is obtained by integrating the incremental equations, using a forward Euler scheme. At every incremental step the solution is calculated with respect to the current principal system (x1,x2) (see fig. [6.3]) which is rotating during deformation. Results, reported in the reference system (x0,1,x0,2) (see fig. [6.3]), are presented in figs. [6.4] and [6.5], where for the Mooney-Rivlin and Ogden material we have referred to the values of the previous example. The analytical solution is compared to the results given by the numerical procedure with a uniform mesh of 80 boundary elements. Gaussian quadrature formulae have been employed with 12 integration points for Green’s functions (5.23)–(5.40) and 18 points for integrals in the discretized eqn. (6.7).

6.4 Conclusion

In conclusion of this Section we may speculate on a more general procedure allowing for increments superimposed upon a generic, non-homogeneous deformation. This would necessarily lead to some volume discretization to de- scribe the material inhomogeneity, but this should be regarded as diﬀerent —in essence— from the usual domain discretization (e.g. Polizzotto, 2000). The demonstration is provided by the above example, showing that our method 100 Boundary elements formulation

s -s s/ m 11 22 12 0 3 m0 1.6

1.2 2

0.8

Analytical 1 BEM 0.4

0 g 0 g 0 0.4 0.8 1.2 1.6 2 0 0.4 0.8 1.2 1.6 2

Figure 6.4: Mooney-Rivlin elastic blocks. Non-dimensionalized Cauchy stress σ versus shear strain γ.

s11 -s22 s1.6/ m 12 0 3 m0

1.2

0.8

Analytical 1

0.4 BEM

0 g 0 g 0 0.4 0.8 1.2 1.6 2 0 0.4 0.8 1.2 1.6 2

Figure 6.5: Ogden elastic blocks. Non-dimensionalized Cauchy stress σ versus shear strain γ.

does not need any domain discretization when the material properties are homogeneous. Chapter 7

Numerical examples

A Fortran 95 code with dynamic allocation of memory has been implemented to develop applications of the boundary element technique object of the present thesis. Routines for the evaluation of the internal fields: velocity, velocity gradient, pressure rate and stress rate have also been implemented. Several examples are provided below demonstrating the capabilities of the method. Since the formulation fully embodies large strain effects, it allows the determination of bifurcation loads and modes. A systematic investigation of these is the focus of this Section, with emphasis on special situations including multilayered and cracked bodies. The results may find broad applications and, in particular, numerical simulations are presented which may be useful in the design of rubber bearings employed in earthquake-resistant design of buildings. Two peculiar kinds of bifurcations are given special attention: corresponding to the so-called surface instability and shear banding, the latter associated to the condition of loss of ellipticity. Both bifurcations represent ill-posedness of the incremental problem and are characterized by the fact that the bifurcation mode embodies an arbitrarily-short wave length. This gives rise to well-known difficulties in finite element simulations. However, while there is an immense literature on numerical investigation of shear bands, numerical treatments of surface instability are scarce (Hutchinson and Tvergaard, 1980; Tvergaard, 1982; Bardet, 1990). This is rather surprising, since both instabilities are local and involve similar problems. Many routes have been proposed to numerically regularize a problem be- yond the elliptic range. In particular, a class of approaches consists in modify- ing the constitutive models to include an intrinsic characteristic length. Non-

101 102 Numerical examples local constitutive models (Bazant et al. 1984; Pijaudier-Cabot and Bazant, 1987; Leblond et al. 1994), Cosserat continua (Mühlhaus and Vardoulakis, 1987; de Borst and Sluys, 1991), visco-plastic models (Needleman, 1988; Loret and Prevost, 1990), and higher-order gradient models (Coleman and Hogdon, 1985; Triantafyllidis and Aifantis, 1986; Vardoulakis and Aifantis, 1991; Fleck and Hutchinson, 1993; 1997; Menzel and Steinmann, 2000; Sluys and Estrin, 2000) fall within this class. In another class of approaches, numerical regularization is pursued by introducing ad hoc special interpolating functions at element level (Pietruszczak and Mróz, 1981; Belytschko et al., 1988; Nacar et al., 1989; Larsson et al. 1993), or by embedding strong discontinuities (Simo et al. 1993; Armero and Garikipati, 1996; Oliver, 2000; Armero, 2001; Borja, 2002). All the above-mentioned approaches are based on finite element techiques, while the only analysis with the boundary element method is quite recent (Benallal et al. 2002). This is again surprising, since BEM are known to be an efficient tool in problems involving concentration of deformations, such as for instance in fracture mechanics (Cruse, 1988; Aliabadi, 1997). It is clear from the above discussion that analysis of strain localization in solids has been a topic for years in the numerical community. Nevertheless, the problem is far from being solved, so that while on one hand commercial codes do not still incorporate features for automatic shear band analysis, on the other hand many fundamental aspects —such as for instance propagation conditions at the shear band tip— still remain almost completely unexplored. A common denominator of the former class of the above-mentioned strategies is that they represent a restoration of ellipticity of governing equations (Benallal et al. 1988; 1993). In contrast to this general situation, two different approaches were recently initiated by Petryk and Thermann (1996; 2002) and Bigoni and Capuani (2002), characterized by the fact that, for completely different reasons, the restoration of ellipticity is avoided. In the former approach, devoted to incrementally non-linear materials, path-stability criterion (Petryk, 1985) is sufficient to determine the volume fraction and thus the overall behaviour of material elements in a post-critical range. In the latter approach, strain localization is analyzed in the proximity of the ellipticity boundary, as induced by a perturbation, still inside the region of ellipticity. As a result of perturbation, introducing in a sense a length scale, localized deformation patterns emerge. For instance, Bigoni and Capuani (2002) 7.1 Bifurcation of elastic structures 103 have shown that localized deformations may be observed in a Mooney-Rivlin material, which is known to remain within the elliptic range. A goal of the present study is to continue the investigation by extending the approach to boundary value problems and employing the boundary element technique developed in Chapter 6. Obviously, special numerical strategies are completely avoided since the analysis is performed within the elliptic range. Results demonstrate that the boundary element technique is particularly suitable to the analysis of surface instability and shear band formation.

7.1 Bifurcation of elastic structures

A Fortran 95 code with dynamic allocation of memory has been implemented to develop applications of the boundary element technique presented in Chap- ter 6. Integrals are computed numerically, using Gaussian quadrature formulae with 12 integration points for Green’s functions (see Chapter 5) and 18 points for integrals in the discretized boundary integral equation (see Section 6.2), unless otherwise speciﬁed. Several examples —considering small strains superimposed upon homogeneous, arbitrarly large deformations— are provided below demonstrating the capabilities of the method.

7.1.1 Elastic block An elastic block is considered, in a square (the edge length of the block has been taken equal to 2b), stressed current conﬁguration, for a Mooney-Rivlin material. A uniaxial state of stress is prescribed in terms of the non-dimensional parameter k deﬁned as

2 2 σ σ1 σ2 λ1 λ2 k = = − = 2 − 2 , (7.1) 2µ 2µ λ1 + λ2 where σ1, assumed zero in the following, and σ2 are the principal Cauchy stresses —so that σ is the current deviatoric, in-plane stress— µ is the incremental modulus for shear parallel to the principal Eulerian axes and λ1, λ2 are the in-plane stretches. Bifurcations from this state were analyzed by Biot (1965), Hill and Hutchin- son (1975) and Young (1976), considering a smooth bilateral constraint at the 104 Numerical examples two edges normal to the direction of the uniaxial stress. For Mooney-Rivlin material, bifurcations are only possible in compression, where it turns out that the first bifurcation occurs at k 0.522, corresponding to an anti-symmetric ≈ mode with a ratio λ/(2b) = 2 between wave length λ and edge length. Above this bifurcation value, an infinite set of critical values of k follows, corresponding to antisymmetric bifurcations. The accumulation point of these values defines the surface instability, occurring at k 0.839, solution, with reversed ≈ sign, of the equation (see Radi et al. 2002, their eqn. [3.16] with ξ = 1)

k 1 k 1 − 1 = 0, (7.2) 2 − r1+ k ! − and corresponding to the limit λ/(2b) = 0. For values of k greater than the surface instability threshold, an inﬁnite set of symmetric bifurcation becomes possible, bounded by k 0.926 for ≈ λ/2b = 2/3, as a linear combination of wave modes

iκ(√γj x1+x2) iκ(√γj x1+x2) vl = al e , p˙ = c e , (7.3) where v is the velocity,p ˙ is the in-plane pressure rate, i = √ 1, a and c are l − l complex amplitudes, γj (j = 1, 2) are the roots of the characteristic equation (see Section 5.3) and κ is the wave number of the bifurcation mode. Therefore, the surface instability ‘separates’ the two infinite sets of antisymmetric and symmetric bifurcation modes. Some critical values of k and the corresponding modal wavelength λ/2b are reported in the second and third column of Tab. 7.1. For the above-described geometry, bifurcation points can be numerically traced by analyzing the eigenvalue problem associated to the boundary element discretization. In our case, two uniform meshes of 72 and 144 elements (denoted as ‘coarse’ and ‘fine’ in the following) have been chosen. The upper and lower edges of the block have been constrained with smooth rigid boundaries (the lower central node has been fixed to eliminate possibility of rigid body translations) and the determinant of the solving system has been analyzed at different values of pre-stress k. The results are reported in the fourth and fifth column of Tab. 7.1. We note that while the initial and final critical values k are computed with an excellent accuracy, values getting close to the accumulation point can be hardly detected numerically. However, bifurcations can also be analyzed using a perturbation technique. In particular, we may start from the above geometry, with the velocities prescribed to be zero along the lower edge. In this situation we may assign 7.1 Bifurcation of elastic structures 105

Table 7.1: Bifurcation pre-stress k and relative mode for a square elastic, Mooney-Rivlin block Analytical Numerical mode λ/2b k k for ﬁne mesh k for coarse mesh

anti 2 0.522 0.522 0.523 anti 1 0.732 0.733 0.735 anti 2/3 0.796 0.797 0.802 anti 1/2 0.821 0.824 0.832 anti 2/5 0.832 0.836 — anti 1/3 0.836 — — anti 2/7 0.838 — — anti ...... — — — 0 0.839 — — sym ...... — — sym 2/7 0.841 — — sym 1/3 0.843 0.849 — sym 2/5 0.849 0.854 0.869 sym 1/2 0.866 0.867 0.880 sym 2/3 0.926 0.930 0.939

symmetric or antisymmetric perturbations in terms of dead loading along the free edges. It may be anticipated that the symmetry of the perturbation will trigger a correspondingly symmetric bifurcation mode. Let us analyze the two kinds of perturbations in detail.

Antisymmetric perturbation and Euler-type instability As is shown in fig. [7.1], the lateral faces of the block, initially free, are subsequently subject to perturbations, in terms of anti-symmetric, normal incremental dead-load. Referring to a Mooney–Rivlin material and to the coarse and fine meshes 106 Numerical examples already employed, the portion of the edges subject to incremental load has been taken equal to 1/9 of the total edge length. The results of numerical investigation are reported in figs. [7.1] and [7.2].

The velocity vc at the upper point of the right edge [non-dimensionalized as µvc/(bτ˙), where b is the half-length of the edge andτ ˙ is the applied nominal traction rate] is plotted versus the pre-stress k in fig. [7.1], where the computed values are marked for k = 0.8, 0.6, 0.4, 0.2, 0, 0.2, 0.4, 0.45, 0.5, 0.5125 . {− − − − } The profiles of velocity components [multiplied by µ/(bτ˙)] along the vertical edge are shown in fig. [7.2] for different values of k. Comparisons are included with results obtained using ABAQUS-Standard (Ver. 6-2-Hibbitt, Karlsson & Sorensen Inc.), with plane-strain, 4-nodes bilinear, hybrid elements (CPE4H). An analysis of fig. [7.1] reveals that stiffness in the incremental response varies significantly as a function of the pre-stress. In particular, traction (corresponding to negative values of k) increases stiffness, whereas compression (corresponding to positive values of k) induces stiffness degradation. The latter becomes dramatic when a critical value of k is reached. This is found to range between 0.5336 and 0.5344 (at which value the stiffness becomes negative) for the coarse mesh, between 0.5289 and 0.5297 for the fine mesh and between 0.5281 and 0.5289 for Finite Element analysis. Though relative to a slightly different boundary condition at the lower edge, this result agrees with the bifurcation analysis (Tab. 7.1). Profiles of velocities components along the edge are shown in fig. [7.2] for k = 0.4, 0, 0.4, {− 0.5125 . Results of ABAQUS are in excellent agreement with those ob- } tained with our boundary element technique up to k = 0.4; in the case when k = 0.5125 the qualitative behaviour is still well captured. Incremental displacement fields are reported in fig. [7.3], where we note that the qualitative deformations are similar, but the quantitative incremental displacements tend to blow up, when the bifurcation point is approached. We can also note that the bifurcation mode is similar to a Euler-type deformation, corresponding to a beam restrained to rotate at the edges. Internal fields can be obtained using boundary integral equations for the velocity (6.2) and for the in-plane pressure rate (6.3). In particular, results obtained employing a discretized version of the boundary integral equation (6.3) for the in-plane pressure rate are reported in fig. [7.4], showing the in-plane pressure rate field (normalized through division by µ) at inner points, with coordinates x /b = 0.75, 0.25, 0.25, 0.75 and x /b = 1 {− − } 2 0.75, 0.25, 0.25, 0.75 . {− − } 7.1 Bifurcation of elastic structures 107

m pre-stress v bt& c 13.1 c . . 12 12.2 t t

x2 10.2

x1 b 8

BEM Coarse Mesh 4 BEM Fine Mesh ABAQUS 0.53

0 k

-0.8 -0.4 0 0.4

Figure 7.1: Non-dimensionalized velocity of the corner point c versus pre-stress k.

The points having the same co-ordinate x2 are connected by lines. Four levels of pre-stress k = 0.4, 0, 0.4, 0.5125 have been considered and the {− } results are compared to those calculated using ABAQUS with constant pressure elements. The fair agreement of the results represents the ﬁrst numerical validation of the boundary equation for in-plane pressure rate obtained by Bigoni and Capuani (2002).

Symmetric perturbation and surface instability

Let us consider now the symmetric perturbation sketched in fig. [7.5], considering the same setting of the previous, antisymmetric situation. It is clear from fig. [7.5] that the finer is the mesh, the closest is the zero-stiffness asymptote to the surface instability, k = 0.839. The wavy deformation mode associated to a localized surface bifurcation is made visible 108 Numerical examples

BEM Coarse Mesh x2 x2 BEM Fine Mesh 0.8 b 0.8 b ABAQUS

0.4 0.4

0 0

k=-0.4 -0.4 -0.4 m -0.8 -0.8 m v v & 1 & 2 bt-0.4 -0.3 -0.2 -0.1 0 -0.01 0 0.01 0.02 0.03bt

x2 0.8 b

0.4

-0.4 k=0 m -0.8 m v v & 1 & 2 bt-0.6 -0.4 -0.2 0 0 0.02 0.04 0.06 0.08bt

x2 0.8 b

0.4

k=0.4 -0.4 m -0.8 m v v & 1 & 2 bt-2 -1.6 -1.2 -0.8 -0.4 0 0 0.1 0.2 0.3 0.4 bt

x2 0.8 b

0.4

k=0.512 -0.4 m -0.8 m v v & 1 & 2 bt-16 -12 -8 -4 0 0 1 2 3bt 4

Figure 7.2: Velocity profiles along the vertical edge, for the geometry specified in the detail of fig. [7.1].

in fig. [7.6], where incremental displacement fields are reported. Differently from the antisymmetric situation (fig. [7.1]) we may note here that the deformation mode changes qualitatively, in the sense that waviness increases when the bifurcation is approached. 7.1 Bifurcation of elastic structures 109

k=-.400 k=0.0 k=.400 k=.5125

Figure 7.3: Qualitative deformations at different values of pre-stress k, for antisymmetric perturbation. The geometry is specified in the detail of fig. [7.1]

For a ﬁner non-uniform mesh of 36 144 elements the surface instability × is approached at k = 0.842 and the qualitative deformation shown in ﬁg. [7.7] displays nine wavelengths.

7.1.2 Layered elastic material The layered elastic structure sketched in the detail of ﬁg. [7.8] is now considered. The structure is made up of three layers, with a ‘material 1’ common to the outer layers and a diﬀerent ‘material 2’ forming the core. Three cases are considered for the ratios of incremental shear moduli (µ /µ)1, (µ /µ)2 and µ1/µ2 of the two materials and relevant values are re- ∗ ∗ ported in Tab. 7.2.

Table 7.2: Bifurcation stress k for the layered elastic structure Case Shear modulus ratios Bifurcation stress k ( ) (µ /µ)1 (µ /µ)2 µ1/µ2 Analytical ∗ Numerical ∗ ∗ 1 1.0 0.5 0.5 0.4722 0.4852 : 0.4859 − 2 2/3 1.0 1.5 0.4386 0.4469 : 0.4477 − 3 0.5 1.0 2.0 0.3714 0.3789 : 0.3797 − (*) The analytical results refer to slightly diﬀerent boundary conditions than those employed for the numerical analysis.

All layers are speciﬁed to undergo the same homogeneous, plane strain deformation with the principal directions of deformation aligned normal and parallel to the layers, as considered by Bigoni et al. (1997) and Bigoni and 110 Numerical examples

k=-.400 k=0.0

p/m p/m x2 /b=3/4 x2 /b=3/4 0.4 0.2

0.2 x2 /b=1/4 x2 /b=1/4

x2 /b=-1/4 0 0 x1 /b x1 /b -0.8 -0.4 0 0.4 0.8 -0.8 -0.4 0 0.4 0.8 x2 /b=-1/4

x2 /b=-3/4 -0.2

x /b=-3/4 -0.2 2 BEMFine Mesh -0.4 ABAQUS

k=.400 k=.5125

1.5p/m p/m x /b=3/4 8 x2 /b=3/4 2

4 0.5 x2 /b=1/4 x2 /b=1/4

0 0 x1 /b x1 /b -0.8 -0.4 0 0.4 0.8 -0.8 -0.4 0 0.4 0.8

-0.5 x2 /b=-1/4 -4 x2 /b=-1/4

-1 -8 x2 /b=-3/4 -1.5 x2 /b=-3/4

Figure 7.4: Values of internal in-plane pressure rate at different values of pre-stress k, for the geometry specified in the detail of fig. [7.1]. Comparison with results obtained with ABAQUS is also reported.

Gei (1999). Therefore, a uniaxial state of traction or compression prevails in the laminate. Starting from this pre-stressed state, an incremental, anti-symmetric loadingτ ˙ is prescribed (orthogonal to the external vertical edges of the structure) applied on a loading zone equal to 2b/15, with b denoting the half-length of the laminate edge (see the detail in ﬁg. [7.8]). Calculations have been performed with a uniform mesh of 80 elements for each layer and results are reported in ﬁg. [7.8], where the velocity [normalized 7.1 Bifurcation of elastic structures 111

pre-stress m v & c bt 4

3.74 3.47

. x2 t 3 x c 1 . b t

2 2.03 b

BEM Coarse Mesh 1 BEM Fine Mesh ABAQUS 0.85 0 k -0.8 -0.4 0 0.4 0.8

Figure 7.5: Non-dimensionalized velocity of the middle point c versus pre-stress k.

k=-.800 k=0.0 k=.800 k=.8406

Figure 7.6: Qualitative deformations at different values of pre-stress k, for symmetric perturbation. The geometry is specified in the detail of fig. [7.5]

through multiplication by µ1/(bτ˙)] is plotted versus the pre-stress k. It should be noted that the values of k are independent of the material, since they depend only on the in-plane stretch (7.1), identical in all layers. The bifurcation values of k, obtained by the boundary element method, are also reported in Tab. 7.2, where they are compared with the analytical values given by Bigoni and Gei (1999), under the slightly diﬀerent boundary condition of smooth rigid contact in the lower edge. 112 Numerical examples

k=.842

Figure 7.7: Qualitative deformations near to surface instability.

7.1.3 Cracked elastic blocks

A pre-stressed, rectangular (3b 2b) elastic block is considered, containing × cracks parallel to the free edges. Three cracked configurations are chosen as sketched in the particular of fig. [7.9] and the response to a symmetric perturbation is analyzed at different levels of pre-stress k. The perturbation consists of a uniform nominal loading rate along the entire lateral edges which are free until the instant of the perturbation, inducing Mode I near-tip fields. For this situation, no analytical solutions are available, with the exception of the asymptotic near-tip representation obtained by Radi et al. (2002). The incremental displacements at a characteristic point of each cracked geometry are plotted in fig. [7.9], where the asymptotes correspond to the first bifurcation value k = 0.112, 0.150, 0.397 . { } The deformed configurations of the cracked bodies, for values of k close to bifurcation are illustrated in fig. [7.10]. In the analysis the unilateral contact of the crack faces has been taken into account. A peculiar effect is visible in the geometry II of fig. [7.10], where the upper crack remains closed, due to the high value of k. Analyses not reported here have shown that the same crack opens for values of k 0.0625. ≤ 7.1 Bifurcation of elastic structures 113

m 1 v pre-stress bt& c

c . . 16 t t x2

x1 12 b 1 2 1

2b 2b 2b 3 3 3 8

Case 1 Case 2 4 Case 3

0.485

0 k -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.379 0.447

Figure 7.8: Non-dimensionalized velocity of the corner point c versus pre-stress k for an elastic laminate.

7.1.4 An application to rubber bearings

Steel-laminated elastomeric bearings are structural elements widely used in bridges and in buildings as seismic isolators (Kelly, 1997). These are made up of alternate steel and rubber laminates, bonded together. The presence of steel layers increases the vertical stiffness, while the rubber layers permit large shear deformations. The design of bearings is strongly conditioned by stability considerations, so that bridge specifications usually limit their height-to-width ratio. For the evaluation of stability of rubber bearings a number of models at different levels of accuracy has been proposed (Stanton et al. 1990; Kelly, 1997; D’Ambrosio et al. 1995; Imbimbo and Kelly, 1998; Iizuka, 2000; Tsai and Hsueh, 2001). As a practical application of the present boundary element formulation, we 114 Numerical examples

m v compression b & i t 1000 20

150 x2 . . 800 16 3b t x1 t

600 12 2b 100

3 400 8 3/5b 3/2b 3/5b 1 50 6/5b 2 200 4 I II III

0.397

0 0 0 k

-0.4 -0.2 0 0.2 0.4 0.112 0.150

Figure 7.9: Non-dimensionalized velocity (versus pre-stress k) at a characteristic point of each cracked conﬁguration (points 1, 2, 3 for conﬁgurations I, II, III, respectively).

develop here a simple model to evaluate the incremental, horizontal stiffness of elastomeric bearings as a function of the vertical applied load. In particular, we assume a plane strain situation in which the vertical load produces a homogeneous deformation of the rubber layers corresponding to uniaxial compressive stress. Although this assumption is generally violated due to the bonding between steel and rubber, we find reasonable predictions. As shown in fig. [7.11], we consider a rubber bearing —similar to that employed by Imbimbo and De Luca (1998, their fig. [2]), but in plane strain— made up of three steel layers (thickness 3 mm) and two rubber layers (thickness 8 mm), with a total height h of 25 mm and we investigate two aspect ratios 7.1 Bifurcation of elastic structures 115

K=.397 K=.150 K=.112 3

Figure 7.10: Deformed conﬁgurations in proximity of bifurcation of the cracked elastic blocks.

h/b = 1/16 and h/b = 1/8. The rubber layers are Mooney-Rivlin materials,

pre-stress

t. t. 1 2 h=8mm h = 25 mm 1 h=3mm2

b=200,400mm

Figure 7.11: Geometry of the analyzed rubber bearings. deﬁned by the strain energy function

µ µ W (I ,I )= 1 (I 3) 2 I2 I 6 , (7.4) 1 2 2 1 − − 4 1 − 2 − with µ = 0.599 MPa and µ = 0.108 MPa. Horizontal stiffness has been 1 2 − obtained calculating the horizontal displacement induced by a horizontal antisymmetric loading applied over the thickness of the upper (steel) layer (fig. [7.11]). Results are reported in fig. [7.12], where the horizontal stiffness (normalized with respect to the maximum stiffness of the wider bearing Smax = 5.403 MPa) versus applied vertical load (normalized with respect to the maximum vertical load of the wider bearing P = 2.025 106 N/m) are reported. max × 116 Numerical examples

S/Smax

0 .8

0 .6

0 .4

0 .2 h/b=1/16 h/b=1/8

0 P/Pmax 0 0.2 0.4 0.6 0.8 1

Figure 7.12: Inﬂuence of the vertical load on the horizontal stiﬀness.

7.2 Shear bands within the elliptic range

Let us consider an incompressible body characterized by the constitutive equation introduced in eqn. (4.55). It is well-known that shear bands represent an extreme form of material instability, corresponding to failure of ellipticity

niKijhknhgk = 0, subject to nkgk = 0, (7.5) for at least one unit vector nk and non-null orthogonal vector gk. Condition (7.5) can be shown to yield the classical regime classiﬁcation, as a matter of fact in a two-dimensional setting can be rewritten as

gjniKijhknhgk = 0, (7.6) with n = cos α, sin α , g = sin α, cos α . (7.7) { } { } { } {− } Introducing now the explicit expression for Kijkl given in eqn. (4.57) condition (7.7) yields the two equivalent equations

1 k + (2k 4 + 4µ /µ) cos2 α + 4(1 µ /µ) cos4 α = 0, − − ∗ − ∗ (7.8) µ sin4 α (1 + k) cot4 α + 2(2µ /µ 1) cot2 α + 1 k = 0, ∗ − − the latter of which is equal to eqn. (5.6), that is the usual basis for the regime classiﬁcation (Hill and Hutchinson, 1975; Bigoni and Capuani, 2002). 7.2 Shear bands within the elliptic range 117

In a continuous loading program, loss of ellipticity can occur after various bifurcation thresholds are attained. For instance, in the examples presented in the previous Section shear bands may only occur well after the detected bifurcation points. As a consequence, shear banding must be analyzed when the structure is in a post-critical range (a circumstance usually overlooked in the literature). This is not an easy task, since in that case the current state is inhomogeneous. However, there is a special case where shear bands may occur as the ﬁrst possible bifurcation. This is the so-called ‘van Hove condition’, in which the solid is subject to prescribed displacements over the entire boundary and the current state (deformation and stress) is homogeneous (van Hove, 1947). More in detail, the incremental solution is unique —unless an arbitrary uniform pressure— until the strong ellipticity condition holds

gjniKijhknhgk > 0, subject to nkgk = 0, (7.9) for all pair of orthogonal vectors nk and gk. However, it is clear from the definitions (7.5) and (7.9) that strong ellipticity = ellipticity, ⇒ and that for K possessing the major symmetry1, the first failure of strong ellipticity in a continuous loading program corresponds to failure of ellipticity and shear band formation. Therefore, we consider van Hove conditions, assuming the geometric setting shown in fig. [7.13], where a square elastic block is considered, homogeneously deformed in a state of uniaxial tension or compression. Displacements are prescribed on the entire boundary, so that the solution is known unless an arbitrary value of homogeneous pressure. We assume µ /µ = 0.25, corresponding to the elliptic complex regime, and we ∗ perturb this configuration prescribing the triangular distributions of velocity sketched in fig. [7.13]. The perturbation is characterized by the ratio of the maximum assigned velocity to the length of the application zone. This ratio has been assumed equal to 9/20 and 9/40 on the left and right edge, respectively, of the block in fig. [7.13]. Results of computations —in terms of level sets of the velocity modulus— are reported in fig. [7.14] and [7.16], the former relative to c/b = 1/2, the latter to c/b = 4/9. Level sets of the modulus of the in-plane deviatoric stress increment (normalized through division by µ) σ˙ σ˙ σ˙ | | = | 1 − 2|, (7.10) √2 µ √2 µ 1The more general case in which the constitutive operator is not symmetric is treated in detail by Bigoni (2000). 118 Numerical examples

c x2 prescribed x1 velocity pre-stress

velocity

prescribed b

Figure 7.13: Loading geometry in van Hove conditions.

are also reported in figs. [7.15], [7.17] and referred to as ‘von Mises stress increment’ in the following. It may be interesting to note that the in-plane deviatoric stress increment coincides for the J deformation theory of plastic- 2− ity (see Sections 3.4.3 and 4.1.5) with the three-dimensional deviatoric stress increment trσ˙ devσ˙ = σ˙ I, (7.11) − 3 since the out-of-plane stress increment is equal to (σ ˙ 1 +σ ˙ 2)/2 for the J2- deformation theory. A 144-element uniform mesh has been employed to discretize the boundary and a 324-point uniform grid has been used to evaluate the interior velocity field. The Gauss points for the numerical integrations have been increased to 48. Three different situations are reported in the figures, corresponding to three different values of pre-stress k = 0.859, 0, 0.859 . The values 0.859 {− } ± are close to the boundary of loss of ellipticity, occurring at k = 0.866025. ± When the elliptic boundary is attained, shear bands become possible, inclined at an angle η solution of the equation (Hill and Hutchinson, 1975)

2 1 + 2 µ /µ (1 µ /µ) tan η = ∗ − ∗ , (7.12) 1 2µ /µ p − ∗ which, in the special case of µ /µ = 0.25, gives a band inclination η = 27.367◦, ∗ with respect to the direction of the maximum in-plane stress component. It can be seen from ﬁgs. [7.14] and [7.16] that when the elliptic boundary is approached, the velocity tends to localize along well-deﬁned shear band 7.2 Shear bands within the elliptic range 119

u/u 1 0.8 0.8 0.8 0.9 0.6 0.6 0.6 0.8 0.4 0.4 0.4 ~63° 0.7

0.8

0.2 0.2 0.2 0.6

0.4 ~27° 0.2 0.6 0

-0.2 0 0 0 -0.4 x2 /b x2 /b -0.6 -0.8 0.5 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -0.2 -0.2 -0.2 ~63° 0.4 -0.4 -0.4 -0.4 0.3

-0.6 -0.6 -0.6 0.2

-0.8 K=-0.859 -0.8 K=0.0 -0.8 K=0.859 0.1

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 0

x1 /b x1 /b x1 /b

Figure 7.14: Van Hove conditions: level sets of velocity modulus at diﬀerent values of pre-stress k (c/b = 1/2 in ﬁg. [7.13].

Ssvm /s vm 101.00 0.8 0.8 0.8 9 80.80 7 0.6 0.6 0.6 60.60 5 0.4 0.4 0.4 ~27° 40.40 ~63° 3.5 0.2 0.2 0.2 0.8 0.6 30.30 0.4

0.2

-0.2

-0.4 2.5

-0.6

-0.8 0 x2 /b 0 x2 /b 0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 20.20 1.8 -0.2 ~63° -0.2 -0.2 1.60.16 1.4 -0.4 -0.4 -0.4 1.20.12 1 -0.6 -0.6 -0.6 0.90.10 0.7 -0.8 K=-0.859 -0.8 K=0.0 -0.8 K=0.859 0.50.05 0.3 0.10.00 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

x1 /b x1 /b x1 /b

Figure 7.15: Van Hove conditions: level sets of von Mises stress increment at diﬀerent values of pre-stress k (c/b = 1/2 in ﬁg. [7.13].

patterns. They highlight the inclinations of the discontinuity bands formally possible only at the elliptic boundary. The fact that strain localization can be observed within the elliptic range employing a perturbation approach agrees with ﬁndings by Bigoni and Ca- puani (2002). On the other hand, it may provide an explanation of the fact that shear banding is a preferred instability when compared to other diﬀuse bifurcations, possible at loss of ellipticity under van Hove conditions (Ryzhak, 1999). The van Hove conditions are very peculiar and provide the maximum pos- 120 Numerical examples

u/u 1 0.8 0.8 0.8 0.9 0.6 0.6 0.6 0.8 0.4 0.4 0.4 ~27° 0.7

~63° ~63° 0.8 0.2 0.2 0.2 0.6

0.4 ~27° 0.2 0.6 0

-0.2 0 0 0 -0.4 x2 /b x2 /b -0.6 -0.8 0.5 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -0.2 -0.2 -0.2 ~63° 0.4 -0.4 -0.4 -0.4 0.3

-0.6 -0.6 -0.6 0.2 -0.8 K=-0.859 -0.8 K=0.0 -0.8 K=0.859 ~27° 0.1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 0

x1 /b x1 /b x1 /b

Figure 7.16: Van Hove conditions: level sets of velocity modulus at diﬀerent values of pre-stress k (c/b = 4/9 in ﬁg. [7.13]).

Ssvm /s vm 101.00 0.8 0.8 0.8 9 80.80 7 0.6 0.6 0.6 60.60 5 0.4 0.4 0.4 ~27° 40.40 ~63° 3.5 0.2 0.2 0.2 0.8 ~63° 0.6 30.30 0.4 0.2 ~27° 0

-0.2

-0.4 2.5

-0.6

-0.8 0 x2 /b 0 x2 /b 0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 20.20 1.8 -0.2 ~63° -0.2 -0.2 1.60.16 1.4 -0.4 -0.4 -0.4 1.20.12 1 -0.6 -0.6 -0.6 0.90.10 ~27° 0.7 -0.8 K=-0.859 -0.8 K=0.0 -0.8 K=0.859 0.50.05 0.3 0.10.00 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

x1 /b x1 /b x1 /b

Figure 7.17: Van Hove conditions: level sets of von Mises stress increment at diﬀerent values of pre-stress k (c/b = 4/9 in ﬁg. [7.13]).

sible ‘confinement’ to a material sample. Referred to the case of compressible materials, Ryzhak (1993, 1994) has shown that the van Hove theorem can be extended to a less restrictive condition, that will be called ‘weak van Hove’ in the following. In particular, the material must be homogeneous and orthotropic, with orthotropy axes parallel and orthogonal to the given loading direction. In- stead of the usual prescription on displacement, now two parallel edges can be in smooth (bilateral) contact with a rigid constraint (the lubricated ends employed by Biot to explain the so-called ‘internal instabilities’). This is the situation sketched in fig. [7.18] which we employ as a current 7.2 Shear bands within the elliptic range 121 configuration to be perturbed with two assigned, triangular velocity distributions. As in van Hove conditions, the current situation is again defined unless

prescribed x2 velocity x2

x1 x1 pre-stress pre-stress prescribed 2b velocity

b b (a) (b)

Figure 7.18: Loading geometries in weak van Hove conditions. an arbitrary value of homogeneous pressure. Level sets of the velocity are plotted in figs. [7.19] and [7.21], for different values of pre-stress k = 0, 0.7, 0.857 corresponding to compression parallel { } to x2. Level sets of the modulus of the in-plane deviatoric stress increment (7.10) are reported in figs. [7.20] and [7.22]. A 216-elements, uniform mesh has been employed for the boundary and 648 points for the evaluation at the internal points. Again, the Gauss points have been increased to 48 for the evaluation of integrals. We may note that until k = 0.7 there is no much evidence of shear banding, but this becomes evident when the boundary of loss of ellipticity is approached with k = 0.857. The examples of figs. [7.19-7.22] show that peculiar deformation patterns emerge, due to ‘reflection’ of shear bands at the boundary. This feature of localized deformation has been observed in different con- texts (behaviour of porous plastic materials, Tvergaard, 1982; dynamics of visco-plastic solids, Deb et al., 1996a,1996b) and could be exploited to explain pattern formation in living tissues or in geological structures. In par- 122 Numerical examples

1.0

0.9

~27° 0.8

~27° 0.7

0.6 ~27° 0.5 ~27° 0.4

0.3 ~27° ~27° 0.2

0.1 K=0.0 K=0.7 K=0.857 0

Figure 7.19: Weak van Hove conditions: level sets of velocity modulus at diﬀerent values of pre-stress k (ﬁg. [7.18a]).

1.0

0.9

~27° 0.8

~27° 0.7

0.6 ~27° 0.5 ~27° 0.4

0.3 ~27° ~27° 0.2

0.1 K=0.0 K=0.7 K=0.857 0

Figure 7.20: Weak van Hove conditions: level sets of von Mises stress increment at diﬀerent values of pre-stress k (ﬁg. [7.18a]).

ticular, the adaptative substructuring of trabecular bone shown by Huiskes et al. (2000, their ﬁg. [4]) displays similarities to the pattern of ﬁg. [7.21], whereas the deformation patterns in granular media evidenced by Desrues and 7.2 Shear bands within the elliptic range 123

Chambon (2002, their ﬁg. [4]) exhibit a similarity to the pattern of ﬁg. [7.19].

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2 ~27° ~27° 0.1 K=0.0 K=0.7 K=0.857 0

Figure 7.21: Weak van Hove conditions: level sets of velocity modulus at diﬀerent values of pre-stress k (ﬁg. [7.18b]).

1.0

0.9

~27° 0.8

~27° 0.7

0.6

0.5

0.4

0.3

0.2 ~27° ~27° 0.1 K=0.0 K=0.7 K=0.857 0

Figure 7.22: Weak van Hove conditions: level sets of von Mises stress increment at diﬀerent values of pre-stress k (ﬁg. [7.18b]). 124 Numerical examples

Figure 7.23: Growth of a trabecular bone, results of computer simulations after Huiskes et al. (2000).

Figure 7.24: Strain localization in a thick specimen of sand tested in a plain strain biaxial apparatus. Four diﬀerent loading increments are presented (Desrues and Chambon, 2002). Appendix A

Remarks on the notation and theorems

Notation A brief account of the notation is given in this section. We refer generally to Gurtin’s (1981) notation. In particular, indicating by the Euclidean three-dimensional space and by the associated vector E V space, boldface lower-case letters (a, b, . . . ) and upper-case letters (A, B, . . . ) denote points of and elements of Lin, the set of second-order tensors, E respectively, whereas scalars are represented with lowercase (preferably Greek) letters (α, β, . . .). Lin+ is the set of all the second-order tensors with positive determinant, Sym is the symmetric restriction of Lin, Orth is the set of all the orthogonal second-order tensors, and Orth+ is the set of all rotations. A fourth-order tensor, that, except N and R (respectively sets of natural and real numbers), will be designated by a blackboard letters (A, B,. . . ), is a linear mapping from Lin into Lin; S is the symmetrizing fourth-order tensor defined as: S[A] = (A + AT )/2, S = (δ δ + δ δ )/2. If u, v, w ijhk ih jk ik jh ∈ V and A, B, C Lin, then: u · v and A · B define the inner products be- ∈ tween elements of and Lin, respectively; AB indicates the product between V second-order tensors, such that (AB)u = A(Bu); (u v)w = u(v · w) and ⊗ (A B)C = A(B · C); tr denotes the trace and grad the gradient, div u = ⊗ tr(grad u) the divergence of a vector field and (div A) · u = div(AT u) defines the divergence of a second-order tensor. We make use of index notation and we adopt the summation convention, · j in which repeated indexes are summed, for example a b = i=1 ai bi = ai bi, with j = 2, 3 depending on the dimension of the domain. In some cases we P

125 126 Remarks on the notation and theorems adopt also a mixed tensorial-indicial notation, in which only free indexes are speciﬁed, i.e., αi = bi · Ac. The Kronecker delta δij is deﬁned as

1, if i = j, δ (A.1) ij ≡ 0, if i = j, 6 i and in the mixed tensorial-indicial notation is indicated as (e )j = δij.

Divergence theorem

Let be a bounded, regular region of , and let v : , and A: Lin R E R→V R→ be smooth ﬁelds. Then

v · n dA = div v dV, An dA = div A dV, (A.2)

∂Z Z ∂Z Z R R R R where n is the outward unit normal to ∂ . R

Localization theorem

Let Φ be a continuous scalar or vector ﬁeld on an open set in . Then, R E given any x 0 ∈ R 1 Φ(x0) = lim Φ dV, (A.3) δ 0 V (Ω ) → δ ΩZδ where Ω is a closed ball of radius δ centered at x , and ( ) is the volume δ 0 V R of . R

Principal invariants of a second-order tensor

Let A Lin. We deﬁne the principal invariants of A the scalar functions ∈ IA = tr A,

IIA = 1 (trA)2 tr(A2) = trA 1 det A, (A.4) 2 − − h i IIIA = det A. 127

An alternative deﬁnition of invariants is A A I1 = I = tr A,

IA = 2 IIA (IA)2 = tr(A2), (A.5) 2 − A A I3 = III = det A.

Cayley-Hamilton theorem Every tensor A Lin satisﬁes its own characteristic equation: ∈ A A A A3 I A2 + II A III = 0. (A.6) − − Representation theorem for isotropic scalar functions A function φ: Sym R, (A.7) A⊂ → is isotropic if and only if there exists a function φI : ( ) R, where ( )= A A A I A → I A (I ,I ,I ) A , such that { 1 2 3 | ∈ A} I A A A φ(A)= φ (I1 ,I2 ,I3 ), (A.8) for every A . ∈A Representation theorem for symmetric isotropic tensor functions A function S: Sym Sym (A.9) A⊂ → is isotropic if and only if three isotropic scalar functions βi (i = 0, 1, 2) exist such that 1 S(A)= β0(A) I + β1(A) B + β2(A) B− , (A.10) for every A . ∈A 128 Remarks on the notation and theorems Appendix B

Biot’s expression of incremental moduli

To obtain the expression (4.19)1 of the incremental mudulus µ, let us begin by considering the spectral representations of the left Cauchy-Green strain tensor B, and of the Cauchy stress σ

2 2 2 B = λ1 e1 e1 + λ2 e2 e2 + λ3 e3 e3, ⊗ ⊗ ⊗ (B.1) σ = σ e e + σ e e + σ e e . 1 1 ⊗ 1 2 2 ⊗ 2 3 3 ⊗ 3 where (e1, e2, e3) denote the Eulerian principal axes. The time derivative of (B.1) is

B˙ = 2λ λ˙ e e + 2λ λ˙ e e + 2λ λ˙ e e + λ2 (e˙ e + e e˙ ) 1 1 1 ⊗ 1 2 2 2 ⊗ 2 3 3 3 ⊗ 3 1 1 ⊗ 1 1 ⊗ 1 + λ2 (e˙ e + e e˙ )+ λ2 (e˙ e + e e˙ ) , 2 2 ⊗ 2 2 ⊗ 2 3 3 ⊗ 3 3 ⊗ 3 σ˙ =σ ˙ e e +σ ˙ e e +σ ˙ e e + σ (e˙ e + e e˙ ) 1 1 ⊗ 1 2 2 ⊗ 2 3 3 ⊗ 3 1 1 ⊗ 1 1 ⊗ 1

+ σ2 (e˙ 2 e2 + e2 e˙ 2)+ σ3 (e˙ 3 e3 + e3 e˙ 3) . ⊗ ⊗ ⊗ ⊗ (B.2) Taking into account that for incremental plane strain deformations is e˙ 3 = 0 and that e · e˙ = e˙ · e , the in-plane out-of-diagonal components result to 1 2 − 1 2 be ˙ 2 2 · B12 = λ1 λ2 e˙1 e2, − (B.3) σ˙ = (σ σ )e˙ · e . 12 1 − 2 1 2 129 130 Biot’s expression of incremental moduli

On the other hand, eqns. (4.3) and (4.5) give B˙ = λ2 + λ2 D λ2 λ2 W , 12 1 2 12 − 1 − 2 12 (B.4) σ˙ =σ∇ (σ σ ) W , 12 12 − 1 − 2 12 and considering eqns. (B.4) and (B.3) yields · 2 2 (e˙1 e2 + W12) (λ1 λ2) D12 = 2 2 − , λ1 + λ2 (B.5) σ∇ = (e˙ · e + W ) (σ σ ), 12 1 2 12 1 − 2 from which, eliminating the term e˙1 · e2 + W12, we get λ2 + λ2 σ∇ = 1 2 (σ σ ) D . (B.6) 12 λ2 λ2 1 − 2 12 1 − 2 A comparison between eqn. (4.15)1 and eqn. (B.6) gives the Biot’s expression of µ, eqn. (4.19)1. The incremental moduli µ can be obtained by taking the time derivative ∗ of (3.124) written for the in-plane components and taking into account that λ˙ 3 = 0 for plane strain incremental deformations 2 2 ˙ ∂W ˙ ∂ W ˙ ∂ W σ˙ 1 σ˙ 2 = λ1 + λ1λ1 2 + λ1λ2 − ∂λ1 ∂λ1 ∂λ1∂λ2 2 2 (B.7) ˙ ∂W ˙ ∂ W ˙ ∂ W λ2 λ2λ1 λ2λ2 2 . − ∂λ2 − ∂λ1∂λ2 − ∂λ2

If instead of the potential W (λ1, λ2, λ3) the potential Wˆ (λ1, λ2) (see eqn.(3.124)) is used, the same relation (B.7) is obtained, except that Wˆ replaces W . Considering eqns. (4.3), (B.2) and (4.5) for the diagonal components, we get λ˙ i = Diiλi, i = 1, 2, (B.8) σ∇ σ∇ =σ ˙ σ˙ , 22 − 11 11 − 22 which, used in eqn. (B.7), yield 2 2 2 ∂W ∂W ∂ W ∂ W ∂ W D11 D22 σ∇ σ∇ = λ +λ +λ2 + λ2 2λ λ − . 22 − 11 1 ∂λ 2 ∂λ 1 ∂λ2 2 ∂λ2 − 1 2 ∂λ ∂λ 2 1 2 1 2 1 2 (B.9) A comparison between eqn. (4.15)2 and eqn. (B.9) gives the Biot’s expression of µ , eqn. (4.19)2. ∗ Appendix C

Out-of-plane stress component for the Ogden hyperelastic material

The only non-zero component of the Jaumann derivative of the Cauchy stress in the out-of-plane direction is:

N i i i g1(λ1, λ2)+ g2(λ1, λ2)+ g3(λ1, λ2) αi+1 ∇σ33= µi λ (C.1) 2 2 2 2 2 2 2 3 " (λ1 λ2) (λ1 + λ2) λ λ λ λ # Xi=1 − 1 − 3 2 − 3 where gi (λ , λ ) = λαi λ2(αi+1) λ2 λ2 2 λ (λ λ ) λ2 +λ2 1 1 2 1 2 1 − 3 1 1 − 2 1 3 2 2 2 2 4 2 2 6 α λ λ 2 λ + λ 2λ λ +λ λ 1+λ− , i 2 − 3 − 2 3 − 3 1 2 − 3 3 gi (λ , λ ) = (λ + λ ) (λ λ )3λ6 α λ2 λ2 λ2 λ2 1+λ6 2 1 2 1 2 1 − 2 3 i 1 − 3 2 − 3 3 1 3 9 3 5 2 2 + 2λ− λ 1 + λ− λ + 1 + λ− λ +λ , 3 3 − 3 3 3 1 2

i (1+2αi) αi 2 2 4 5 2 5 7 g (λ , λ ) = λ λ λ λ 2λ 1 λ λ + 2λ− 1 λ λ 3 1 2 1 2 2 − 3 2 − 1 2 2 − 1 2 4 4 2 2 2 2 2 + 2λ− λ− (λ λ ) +2λ (λ λ ) λ +λ + λ +λ 1 2 1 − 2 2 2 − 1 1 3 2 3 2 2 2 2 2 + αiλ2 (λ2 λ1) λ3 λ2 + λ1− λ3− λ3− , − − − (C.2) in which, due to incompressibility: 1 λ3 = . (C.3) λ1λ2

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