A course on finite elasticity

Description

Cover Preface Contents 1 Concept of an elastic material 2 Observers and invariance 3 Mechanical power and hyperelasticity 3.1 Elasticity and energy 3.2 Work inequality 4 Material symmetry 4.1 Stress response 4.2 Strain energy 4.3 Isotropy 5 Fiber symmetry 6 Stress response in the presence of local constraints on the deformation 6.1 Local constraints 6.2 Constraint manifolds and the Lagrange multiplier rule 6.3 Material symmetry in the presence of constraints 7 Some boundary-value problems for uniform isotropic incompressible materials 7.1 Problems exhibiting radial symmetry with respect to a fixed axis 7.1.1 Pressurized cylinder 7.1.2 Azimuthal shear 7.1.3 Torsion of a solid circular cylinder 7.1.4 Combined extension and torsion 7.2 Problems exhibiting radial symmetry with respect to a fixed point 7.2.1 Integration of the equation 7.2.2 Pressurized shells, cavitation 8 Some examples involving uniform, compressible isotropic materials 8.1 Spherical symmetry, revisited 8.2 Plane strain 8.3 Radial expansion/compaction 9 Material stability, strong ellipticity and smoothness of equilibria 9.1 Small motions superposed on finitely deformed equilibrium states 9.2 Smoothness of equilibria 9.3 Incompressibility 10 Membrane theory 10.1 General theory 10.2 Pressurized membranes 10.3 Uniqueness of the director 10.4 Isotropic materials 10.5 Axially symmetric deformations of a cylindrical membrane 10.6 Bulging of a cylinder 11 Stability and the energy criterion 11.1 The energy norm 11.2 Instability 11.3 Quasiconvexity 11.4 Ordinary convexity 11.4.1 Objections to ordinary convexity 11.5 Polyconvexity 11.6 Rank-one convexity 11.7 Equilibria with discontinuous deformation gradients 11.8 The Maxwell–Eshelby relation 11.8.1 Example: alternating simple shear 12 Linearized theory, the second variation and bifurcation of equilibria 13 Elements of plasticity theory 13.1 Elastic and plastic deformations 13.2 Constitutive response 13.3 Energy and dissipation 13.4 Invariance 13.5 Yielding, the work inequality and plastic flow 13.6 Isotropy 13.7 Rigid-plastic materials 13.8 Plane strain of rigid-perfectly plastic materials: slip-line theory 13.8.1 State of stress, equilibrium 13.8.2 Velocity field Supplemental notes 1 The cofactor 2 Gradients of scalar-valued functions of tensors 3 Chain rule 4 Gradients of the principal invariants of a symmetric tensor 5 Relations among gradients 6 Extensions 7 Korn's inequality 8 Poincaré's inequality Index