##
**Plasticity. Mathematical theory and numerical analysis.**
*(English)*
Zbl 0926.74001

Interdisciplinary Applied Mathematics. 9. New York, NY: Springer. xiii, 371 p. (1999).

The monograph reflects the recent confluence of developments in mechanics and mathematics regarding plasticity, differential equations and variational inequalities. Written for both engineering scientists and mathematicians, the book presents the material in a convex analysis setting with applied numerical analysis adapted to boundary value problems. To this end, the authors use earlier contributions of G. Duvaut, J.-L. Lions, R. Temam, J. J. Moreau, Y. C. Simo and others.

The book consists of three parts. The first part includes, besides historical references, the basic elements of continuum mechanics, linear elasticity and plasticity (yield criteria, hardening, flow). The theorems of convex analysis are reformulated as fundamental properties of plastic flow laws. The second part presents variational problems of elasto-plasticity preceded by an overview of results from functional analysis (normed and Banach spaces, operators and functionals, convergence, compactness, Hilbert functional and dual spaces, etc.). The equations refer to elliptic boundary value problems and to parabolic initial-boundary value problems because the latter ones present formal similarity with the former ones. The basic variational problem is based on a flow law with a prescribed dependence on input data and estimates. Linear kinematic and isotropic hardening with von Mises yield condition illustrate the theory. The qualitative analysis leads to the study of the well-posedness of an associated abstract problem. Stability is also studied. The last chapter of this part examines the dual variational problem of elasto-plasticity related to the flow law, yield surface and normality law. The study refers to the stress and the full stress-strain problem, and the presentation differs from presentations available in the literature.

The third part entitled “Numerical analysis of variational problems” studies approximations of the above problems and overviews the mathematics of finite elements. For this purpose, the authors reformulate the boundary value problem as an equivalent variational one, the domain of independent variables is partitioned in finite parts, the finite element is built up as a collection of piecewise smooth functions, the variational problem is projected to the finite element space, and the resulting finite element system is solved by an iterative method. Various conclusions are drawn from the solution. The results obtained lead to a priori and posteriori error estimates and to the superconvergence. The main purpose is to introduce basic aspects as an interpolation theory. Subsequent chapters furnish approximations of elliptic variational problems of different kinds, numerical analysis with the use of predictor-correctors, etc.

This monograph appears as a forward looking and an impressive synthesis of contributions in a field of great interest in engineering sciences and mathematics. The high-level presentation enjoys the clarity and completeness and makes the book attractive for anyone interested in plasticity.

The book consists of three parts. The first part includes, besides historical references, the basic elements of continuum mechanics, linear elasticity and plasticity (yield criteria, hardening, flow). The theorems of convex analysis are reformulated as fundamental properties of plastic flow laws. The second part presents variational problems of elasto-plasticity preceded by an overview of results from functional analysis (normed and Banach spaces, operators and functionals, convergence, compactness, Hilbert functional and dual spaces, etc.). The equations refer to elliptic boundary value problems and to parabolic initial-boundary value problems because the latter ones present formal similarity with the former ones. The basic variational problem is based on a flow law with a prescribed dependence on input data and estimates. Linear kinematic and isotropic hardening with von Mises yield condition illustrate the theory. The qualitative analysis leads to the study of the well-posedness of an associated abstract problem. Stability is also studied. The last chapter of this part examines the dual variational problem of elasto-plasticity related to the flow law, yield surface and normality law. The study refers to the stress and the full stress-strain problem, and the presentation differs from presentations available in the literature.

The third part entitled “Numerical analysis of variational problems” studies approximations of the above problems and overviews the mathematics of finite elements. For this purpose, the authors reformulate the boundary value problem as an equivalent variational one, the domain of independent variables is partitioned in finite parts, the finite element is built up as a collection of piecewise smooth functions, the variational problem is projected to the finite element space, and the resulting finite element system is solved by an iterative method. Various conclusions are drawn from the solution. The results obtained lead to a priori and posteriori error estimates and to the superconvergence. The main purpose is to introduce basic aspects as an interpolation theory. Subsequent chapters furnish approximations of elliptic variational problems of different kinds, numerical analysis with the use of predictor-correctors, etc.

This monograph appears as a forward looking and an impressive synthesis of contributions in a field of great interest in engineering sciences and mathematics. The high-level presentation enjoys the clarity and completeness and makes the book attractive for anyone interested in plasticity.

Reviewer: M.Mişicu (Bucureşti)

### 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 |

74S05 | Finite element methods applied to problems in solid mechanics |

49J40 | Variational inequalities |