# Non-Linear Theory of Elasticity

**By**

- A.I. Lurie

This book examines in detail the Theory of Elasticity which is a branch of the mechanics of a deformable solid. Special emphasis is placed on the investigation of the process of deformation within the framework of the generally accepted model of a medium which, in this case, is an elastic body. A comprehensive list of Appendices is included providing a wealth of references for more in depth coverage. The work will provide both a stimulus for future research in this field as well as useful reference material for many years to come.

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Published: November 1990

Imprint: North-holland

ISBN: 978-0-444-87439-9

## Contents

- A selection of contents:
**1. Deformation of a Continuous Medium.**Material coordinates. Spatial coordinates. Vector bases. Deformation gradients. Cauchy-Green and almansi strain measures. Orthogonal tensors accompanying deformation. Dilatation. The oriented elementary area. Variation of the state of strain. The second derivative of a scalar function of a tensor argument. Kinematic relations. Rigid motions. Frame-indifferent tensors. The objective derivative of a tensor. The Rivlin-Ericksen tensors. Tensors of affine deformation.**2. Stress in A Continuous Medium.**Body and surface forces. The Cauchy stress tensor. The equations of motion of a continuous medium. The tensor of stress functions. On polar media. Alternative definitions of the stress tensor. The incremental work.**3. The State Equations.**The simple body. The principle of material frame-indifference. Elastic materials. The symmetry group of a material. Orthogonal transformation. Isotropic material. The solid body. An isotropic solid material. An elastic fluid.**4.****The Equations of Nonlinear Theory of Elasticity and the Statement of Problems.**The specific stored energy of deformation. The state equation of an elastic isotropic material. Variation of the stress state. The equilibrium equations for a varied stress state. The relaxation tensor of an isotropic medium. Equations of motion and equilibrium for an isotropic elastic body. Methods of analysis of equilibrium boundary-value problems for a nonlinearly elastic body. The theorem of Ericksen. The principle of stationary complementary energy. The Hamilton-Ostrogradskii principle.**5. The State Equations for a Nonlinearly Elastic Material.**On the choice of a state equation for an isotropic elastic body. Seth's body. Signorini's body. Murnaghan's material. A quasi-linear John material. The energy of dilatation and distortion. Empirical criterion. Convexity of the specific strain energy.**6. Problems of the Nonlinear Theory of a Compressible Elastic Medium.**The affine transformation of a reference configuration. The uniaxial stretching of a rod. Simple shear. A quasi-linear material. Lamé's problem for a cylinder and sphere. The circular membrane. Bending of a strip into a right cylindrical form. Second order effects. The construction of a solution. The dilatation of body subjected to distortion. Second order effects in: The problem of torsion and stretching of a rod; in the plane problem for a quasi-linear material. On the "Physically Nonlinear" theory of elasticity.**7. Incompressible****Elastic Material.**An elastic material with superposed constraints. Second order effects in an incompressible elastic body. The plane deformation of an incompressible material. The universal deformations of an incompressible material. The exact general solutions. Torsion, stretching and change of diameter of a circular cylinder. Lamé's problem for a hollow cylinder. The everted cylinder. Lamé's problem for a hollow sphere. Bending of a plane sheet into a right cylindrical form. The universal solutions of the equations of motion. The Lagrangian differential equations for the parameters in universal solutions. Radial vibrations of cylindrical tube. The antiplane deformation of an incompressible material.**8. Small Deformation of an Initially Loaded Body.**The strain energy and the defining equation in a Vx-configuration. The stationary value principle in a Vx-configuration. Torsional deformation superposed on a uniaxial state of stress. Neutral equilibrium of a quasi-linear material. Application to the problem of the stability of a compressed rod. The secure loading, Holden's estimates. Korn's inequality. The compressed rod (Euler's Column). Quasi-Linear material. An incompressible elastic body. Convexity with respect to a gradient. Hadamard's condition. Hadamard's conditon and stability. A deformable disk in a rigid band. Hadamard's criterion for a uniformly stressed incompressible elastic medium.**9. Thermodynamic Relations. The Equations of Thermoelasticity.**The energy balance equation. The first law of thermodynamics. The second law of thermodynamics. Free energy. The dissipative inequality. Representations in terms of the specific internal energy. The heat conduction equation. Isothermal and adiabatic processes. The equations of thermoelasticity. Appendices. Bibliography. Subject index.