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

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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 aniso- tropy 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 un- der small strain hypotheses. However, a finite strain formulation is the only complete and fully consistent way to analyze material instabilities, strain lo- calization, 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 num- ber 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 ad- dition, Bigoni and Capuani also provided a strategy to analyze strain localiza- tion 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 employ- ing 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 con- sidered 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 there- fore find application in fluid mechanics and serve to introduce the more com- plex 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 elastic- ity, 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 de- formations 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
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