Synopses & Reviews
This volume provides insight into modelling and ultimate limit computation of complex structures, with their components represented by solid deformable bodies. The book examines practically all the important questions of current interests for nonlinear solid mechanics: plasticity, damage, large deformations, contact, dynamics, instability, localisation and failure, discrete models, multi-scale, multi-physics and parallel computing, with special attention given to finite element solution methods. The presentation of topics is structured around different aspects of typical boundary value problems in nonlinear solid mechanics, which provides the best pedagogical approach while keeping the book size reasonable despite its very broad contents. Other strong points are the exhaustive treatment of subjects, with each question studied from different angles of mechanics, mathematics and computation, as well as a successful merger of scientific cultures and heritage from Europe and the USA. The book content and style is also the product of rich international experience in teaching Master and Doctoral level courses, as well as the courses organized for participants from industry (IPSI courses) in France, and similar courses in Germany and Italy. Every effort was made to make the contents accessible to non-specialists and users of computer programs. The original French edition published by Hermes Science - Lavoisier Paris in 2006 was nominated for the Roberval Award for University Textbooks in French.
Review
From the reviews: "Examining modelling and ultimate limit computation of complex structures, the text ... focuses on finite element solution methods. Who is it for? Undergraduates and postgraduates in the field of physics, materials science, chemistry and electrical engineering disciplines. ... it is perfectly complemented by numerous illustrations and worked examples. Would you recommend it? Definitely for anyone studying solid state theory." (Times Higher Education, December, 2009)
Synopsis
The main purpose of this book is to present all the ingredients for constructing the numerical models for representing the complex nonlinear behavior of structures and their components, which are represented as deformable solid bodies. Nonetheless, the book will also prove useful for those mostly interested in linear problems of mechanics, since the sure way to obtain a sound theoretical formulation of a linear problem goes through the consistent linearization of the more general nonlinear problem. The original French edition published by Hermes Science - Lavoisier Paris in 2006 was nominated for the Roberval Award for University Textbooks in French.
Synopsis
This book offers a recipe for constructing the numerical models for representing the complex nonlinear behavior of structures and their components, represented as deformable solid bodies. Its appeal extends to those interested in linear problems of mechanics.
Table of Contents
1 Introduction; 1.1 Motivation and objectives; 1.2 Outline of the main topics; 1.3 Further studies recommendations; 1.4 Summary of main notations; 2 Boundary value problem in linear and nonlinear elasticity; 2.1 Boundary value problem in elasticity with small displacement gradients; 2.1.1 Domain and boundary conditions; 2.1.2 Strong form of boundary value problem in 1D elasticity; 2.1.3 Weak form of boundary value problem in 1D elasticity and the principle of virtual work; 2.1.4 Variational formulation of boundary value problem in 1D elasticity and principle of minimum potential energy; 2.2 Finite element solution of boundary value problems in 1D linear and nonlinear elasticity; 2.2.1 Qualitative methods of functional analysis for solution existence and uniqueness; 2.2.2 Approximate solution construction by Galerkin, Ritz and finite element methods; 2.2.3 Approximation error and convergence of finite element method; 2.2.4 Solving a system of linear algebraic equations by Gauss elimination method; 2.2.5 Solving a system of nonlinear algebraic equations by incremental analysis; 2.2.6 Solving a system of nonlinear algebraic equations by Newton's iterative method; 2.3 Implementation of finite element method in ID boundary value problems; 2.3.1 Local or elementary description; 2.3.2 Consistence of finite element approximation; 2.3.3 Equivalent nodal external load vector; 2.3.4 Higher order finite elements; 2.3.5 Role of numerical integration; 2.3.6 Finite element assembly procedure; 2.4 Boundary value problems in 2D and 3D elasticity; 2.4.1 Tensor, index and matrix notations; 2.4.2 Strong form of a boundary value problem in 2D and 3D elasticity; 2.4.3 Weak form of boundary value problem in 2D and 3D elasticity; 2.5 Detailed aspects of the finite element method; 2.5.1 Isoparametric finite elements; 2.5.2 Order of numerical integration; 2.5.3 The patch test; 2.5.4 Hu-Washizu (mixed) variational principle and method of incompatible modes; 2.5.5 Hu-Washizu (mixed) variational principle and assumed strain method for quasi-incompressible behavior; 3 Inelastic behavior at small strains; 3.1 Boundary value problem in thermomechanics; 3.1.1 Rigid conductor and heat equation; 3.1.2 Numerical solution by time-integration scheme for heat transfer problem; 3.1.3 Thermo-mechanical coupling in elasticity; 3.1.4 Thermodynamics potentials in elasticity; 3.1.5 Thermodynamics of inelastic behavior: constitutive models with internal variables; 3.1.6 Internal variables in viscoelasticity; 3.1.7 Internal variables in viscoplasticity; 3.2 1D models of perfect plasticity and plasticity with hardening; 3.2.1 1D perfect plasticity; 3.2.2 1D plasticity with isotropic hardening; 3.2.3 Boundary value problem for 1D plasticity; 3.3 3D plasticity; 3.3.1 Standard format of 3D plasticity model: Prandtl-Reuss equations; 3.3.2 J2 plasticity model with von Mises plasticity criterion; 3.3.3 Implicit backward Euler scheme and operator split for von Mises plasticity; 3.3.4 Finite element numerical implementation in 3D plasticity; 3.4 Refined models of 3D plasticity; 3.4.1 Nonlinear isotropic hardening; 3.4.2 Kinematic hardening; 3.4.3 Plasticity model dependent on rate of deformation or viscoplasticity; 3.4.4 Multi-surface plasticity criterion; 3.4.5 Plasticity model with nonlinear elastic response; 3.5 Damage models; 3.5.1 1D damage model; 3.5.2 3D damage model; 3.5.3 Refinements of 3D damage model; 3.5.4 Isotropic damage model of Kachanov; 3.5.5 Numerical examples: damage model combining isotropic and multisurface criteria; 3.6 Coupled plasticity-damage model; 3.6.1 Theoretical formulation of 3D coupled model; 3.6.2 Time integration of stress for coupled plasticitydamagemodel; 3.6.3 Direct stress interpolation for coupled plasticitydamagemodel; 4 Large displacements and deformations; 4.1 Kinematics of large displacements; 4.1.1 Motion in large displacements; 4.1.2 Deformation gradient; 4.1.3 Large deformation measures; 4.2 Equilibrium equations in large displacements; 4.2.1 Strong form of equilibrium equations; 4.2.2 Weak form of equilibrium equations; 4.3 Linear elastic behavior in large displacements: Saint-Venant- Kirchhoff material model; 4.3.1 Weak form of Saint-Venant-Kirchhoff 3D elasticity model and its consistent linearization; 4.4 Numerical implementation of finite element method in large displacements elasticity; 4.4.1 1D boundary value problem: elastic bar in large displacements; 4.4.2 2D plane elastic membrane in large displacements; 4.5 Spatial description of elasticity in large displacements; 4.5.1 Finite element approximation of spatial description of elasticity in large displacements; 4.6 Mixed variational formulation in large displacements and discrete approximations; 4.6.1 Mixed Hu-Washizu variational principle in large displacements and method of incompatible modes; 4.6.2 Mixed Hu-Washizu variational principle in large displacements and assurned strain methods for quasi-incompressible behavior; 4.7 Constitutive models for large strains; 4.7.1 Invariance restrictioris on elastic response; 4.7.2 Constitutive laws for large deformations in terms of principal stretches; 4.8 Plasticity and viscoplasticity for large deformations; 4.8.1 Multiplicative decomposition of deformation gradient; 4.8.2 Perfect plasticity for large deformations; 4.8.3 Isotropic and kinematic hardening in large deformation plasticity; 4.8.4 Spatial description of large deformation plasticity; 4.8.5 Numerical implementation of large deformation plasticity; 5 Changing boundary conditions: contact problems; 5.1 Unilateral 1D contact problem; 5.1.1 Strong form of ID elasticity in presence of unilateral contact constraint; 5.1.2 Wcak form of unilatcral 1D contact problcm and its finite element solution; 5.2 Contact problems in 2D and 3D; 5.2.1 Contact between two deformable bodies in 2D case; 5.2.2 Mortar element method for contact; 5.2.3 Numerical examples of contact problems; 5.2.4 Refinement of contact model; 6 Dynamics and time-integration schemes; 6.1 Initial boundary value problem; 6.1.1 Strong form of elastodynamics; 6.1.2 Weak form of equations of motion; 6.1.3 Finite element approximation for mass matrix; 6.2 Time-integration schemes; 6.2.1 Central difference (explicit) scheme; 6.2.2 Trapezoidal rule or average acceleration (implicit) scheme; 6.2.3 Mid-point (implicit) scheme and its modifications for energy conservation and energy dissipation; 6.3 Mid-point (implicit) scheme for finite deformation plasticity; 6.4 Contact problem and time-int.egration schemes; 6.4.1 Mid-point (implicit) scheme for contact problem in dynamics; 6.4.2 Central difference (explicit) scheme arid impact problem; 7 Thermodynamics and solution methods for coupled problems; 7.1 Thermodynamics of reversible processes; 7.1.1 Thermodynamical coupling in ID elasticity; 7.1.2 Thermodynamics coupling in 3D elasticity and constitutive relations; 7.2 Initial-boundary value problem in thermoelasticity and operator split solution method; 7.2.1 Weak form of initial-boundary value problem in 3D elasticity and its discrete approximation; 7.2.2 Operator split solution method for 3D thermoelasticity; 7.2.3 Numerical examples in thermoelasticity; 7.3 Thermodynamics of irreversible processus; 7.3.1 Thermodynamics coupling for 1D plasticity; 7.3.2 Thermodynamics coupling in 3D plasticity; 7.3.3 Operator split solution method for 3D thermoplasticity; 7.3.4 Numerical example: thermodynamics coupling in 3D plasticity; 7.4 Thermomechanical coupling in contact; 8 Geometric and material instabilities; 8.1 Geometric instabilities; 8.1.1 Buckling, nonlinear instability and detection criteria; 8.1.2 Solution methods for boundary value problem in presence of instabilities; 8.2 Material instabilities; 8.2.1 Detection criteria for material instabilities; 8.2.2 Illustration of finite element mesh lack of objectivity for localization problems; 8.3 Localization limiters; 8.3.1 List of localization limiters; 8.3.2 Localization limiter based on mesh-dependent softening rriodulus - 1D case; 8.3.3 Localization limiter based on viscoplastic regularization - 1D case; 8.3.4 Localization limiter based on displacement or deformation discontinuity - 1D case; 8.4 Localization limiter in plasticity for massive structure; 8.4.1 Theoretical formulation of limiter with displacement discontinuity - 2D/3D case; 8.4.2 Numerical implementation within framework of incompatible mode method; 8.4.3 Numerical examples for localization problems; 8.5 Localization problem in large strain plasticity; 9 Multi-scale modelling of inelastic behavior; 9.1 Scale coupling for inelastic behavior in quasi-static problems; 9.1.1 Weak coupling: nonlinear homogenization; 9.1.2 Strong coupling micro-macro; 9.2 Microstructure representation; 9.2.1 Microstructure representation by structured mesh with isoparametric finite elements; 9.2.2 Microstructure representation by structured mesh with incompatible mode elements; 9.2.3 Microstructure representation with uncertain geometry and probabilistic interpretation of size effect for dominant failure mechanism; 9.3 Conclusions and remarks on current research works; References; Index.