Numerical analysis and simulation of plasticity.

*(English)*Zbl 0930.74001
Ciarlet, P. G. (ed.) et al., Numerical methods for solids (Part 3). Numerical methods for fluids (Part 1). Amsterdam: Elsevier (ISBN 0-444-82569-X). Handbook of Numerical Analysis 6, 183-499 (1998).

This large work, which can be considered as a self-contained book, describes classical models of plasticity in the framework of nonlinear continuum mechanics, and various numerical algorithms for their integration. The material is divided into five chapters. Chapter 1, entitled “The classical models”, presents both rate-independent theories (local evolution equations, elastoplastic tangent moduli, multisurface plasticity) and rate-dependent models (viscoplastic regularization, \(J_2\)-theory). Additionally, the author gives here a weak formulation of dynamic plasticity and discusses contractivity, uniqueness and dissipativity of elastoplastic flows.

Chapter 2 “Integration algorithms” provides a broad overview of numerical methods used currently in the simulation of plastic models. Among other things, the author describes here backward difference and implicit Runge-Kutta return mapping algorithms, finite element discretization, algorithms for computation of closed-form projection, consistent algorithmic elastoplastic moduli, iso-error maps etc. A discussion of numerical stability and accuracy, and illustrative numerical simulations complete the chapter.

Chapter 3 “Nonlinear continuum mechanics” is a short course in general methods used for the derivation of constitutive equations and for the analysis of finite deformations. Basic kinematic notions and stress tensors are introduced together with objective transformations and frame invariance. The general theory is applied to finite elasticity and to multiplicative plasticity at finite strains, completed by a variational formulation of momentum equations. The chapter concludes with a detailed presentation of total and incremental initial-boundary problems.

Chapter 4 “The discrete initial-boundary value problems” deals with the generalization of algorithms described in chapter 2 to the full finite deformation problems. The emphasis lays here on an important particular case of \(J_2\)-flow theory and on the applications of frame invariance and certain conservation laws. Finally, chapter 5 “The coupled thermomechanical problems” extends the classical plasticity models to the full thermomechanical regime, and discusses the corresponding extensions of numerical methods. A vast list of references (circa 300) completes the presentation.

In summary, this work can be recommended both to experienced practitioners and novice users as an excellent reference text on modern plasticity models and related numerical methods.

For the entire collection see [Zbl 0905.00032].

Chapter 2 “Integration algorithms” provides a broad overview of numerical methods used currently in the simulation of plastic models. Among other things, the author describes here backward difference and implicit Runge-Kutta return mapping algorithms, finite element discretization, algorithms for computation of closed-form projection, consistent algorithmic elastoplastic moduli, iso-error maps etc. A discussion of numerical stability and accuracy, and illustrative numerical simulations complete the chapter.

Chapter 3 “Nonlinear continuum mechanics” is a short course in general methods used for the derivation of constitutive equations and for the analysis of finite deformations. Basic kinematic notions and stress tensors are introduced together with objective transformations and frame invariance. The general theory is applied to finite elasticity and to multiplicative plasticity at finite strains, completed by a variational formulation of momentum equations. The chapter concludes with a detailed presentation of total and incremental initial-boundary problems.

Chapter 4 “The discrete initial-boundary value problems” deals with the generalization of algorithms described in chapter 2 to the full finite deformation problems. The emphasis lays here on an important particular case of \(J_2\)-flow theory and on the applications of frame invariance and certain conservation laws. Finally, chapter 5 “The coupled thermomechanical problems” extends the classical plasticity models to the full thermomechanical regime, and discusses the corresponding extensions of numerical methods. A vast list of references (circa 300) completes the presentation.

In summary, this work can be recommended both to experienced practitioners and novice users as an excellent reference text on modern plasticity models and related numerical methods.

For the entire collection see [Zbl 0905.00032].

Reviewer: O.Titow (Berlin)

##### MSC:

74-02 | Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids |

74Cxx | Plastic materials, materials of stress-rate and internal-variable type |

74Sxx | Numerical and other methods in solid mechanics |