Linear and Non-Linear Continuum Solid Mechanics

Deformable solids, that is to say, those which undergo changes in geometry when subjected to external loads or other types of solicitations, as well as other related topics are the focus of this book. Within the main field, this text deals with advanced linear elasticity and plasticity approaches and the behavioural study of more complex types of materials. This includes composites of more recent manufacture and others whose material characterisation has only recently been possible. It also describes how linear elastic behaviour extends to anisotropic materials in general and how deformations can result in small or large strain components. The information on plastic behaviour expands to include strain hardening of the materials. Amongst other new topics incorporated into this volume are studies of hyperelastic materials, which can represent elastomeric and some types of biological materials. A section of the book deals with viscoelastic materials, i.e. those who deform when subjected to long-term loads. The behaviour of viscoplasticity, as well as elasto-viscoplasticity, describes well other types of materials, including those present in many geotechnical sites. The objective of this volume is to present material that can be used for teaching continuum mechanics to students of mechanical, civil or aeronautical engineering. In order to understand the contents the reader only needs to know linear algebra and differential calculus. Examples have been included throughout the text and at the end of each chapter, exercises are presented which can be used to check on comprehension of the theoretical information presented.

Table of Contents:
Introduction Continuum Mechanics; The Continuum Deformable Solid Relationships Between Displacements and Strains in Deformable Bodies. Kinematic Equations Displacements Field in Deformable Bodies; Strain Tensors; Tensors in large strain theory; Tensors in small strain theory: Cauchy strain tensor; Another definition of strain: Cauchy-Green strain tensor; Principal Directions of Strain and Maximum Angular Strain; Ellipsoid of Strains; Indicatrix Quadric of Strains; Directrix Quadric of Strains; Mohr's Circles of Strain; Compatibility Equations in Theory of Small Strains Equilibrium Equations in Deformable Bodies Stress Tensor at a Point; Internal Equilibrium Equations; Equations of Equilibrium on the Boundary; Principal Stresses and Principal Stress Directions; Maximum Shear Stresses; Quadrics Associated with the Stress Tensor; Ellipsoid of stress; Indicatrix quadric of stresses; Directrix quadric of stresses Relationship Between Stresses and Strains in Deformable Bodies. Constitutive Equations Behaviour of Continuous Bodies. Constitutive Equations; Linear Elastic Materials. Generalized Hooke's Law; Strain Energy Plane Linear Elasticity. Plane Strain and Plane Stress Plane Strain Field; Plane Stress Field; Principal Directions in Plane Elasticity; Mohr's Circle in Plane Elasticity; Stress Function in Plane Elasticity. Airy Function; Plane strain field; Plane stress field; Representative Points and Lines in Plane Elasticity; Plane Elasticity in Polar Coordinates;Equilibrium equations and kinematic relationships; Plane stress and plane strain fields; Airy function in polar coordinates; Examples of Elastic Problems in Polar Coordinates; Circular tube solicited by radial pressure;Wedge and semi-infinite continuum with a concentrated vertical load; Semi-infinite domain with a distributed load; Thin plate with hole; Hyperelasticity Hyperelastic Materials; Models of Incompressible Hyperelastic Materials; Mooney-Rivlin model; Neo-Hooke model; Yeoh model; Ogden model; Strain Energy in Hyperelastic Materials; Obtaining the Stress Tensor in Hyperelastic Materials; Stress Tensors in Incompressible Hyperelastic Materials; Uniaxial Forces; Stress in the Mooney-Rivlin model; Stress in the Neo-Hooke model; Compressible Hyperelastic Materials; Mooney-Rivlin model; Neo-Hooke model; Ogden model; Yeoh model Plasticity Introduction; Plastification Criteria in Metallic Materials; Beltrami-Haigh criterion; Von Mises-Hencky criterion; Tresca criterion; Plastification Criteria in Soil Mechanics; Mohr-Coulomb criterion; Drucker-Prager criterion; Cambridge model. Cam Clay and modified Cam Clay; Elastoplastic Behaviour of Materials; Elastoplastic behaviour without strain hardening; Elastoplastic behaviour with strain hardening Linear Viscoelasticity Introduction; Models of Viscoelastic Behaviour; Two-parameter models. Maxwell and Voigt models; Three-parameter models. Maxwell's and Voigt's standard linear models; Generalized standard linear models. Maxwell and Voigt models Linear Elasto-viscoplasticity Introduction; Model with Two Parameters; Models with Three Parameters. Bingham Models; Bingham's Four-Parameter Model; Non-linear Viscosity Models. Norton's Model