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**Quasistatic contact problems in viscoelasticity and viscoplasticity.**
*(English)*
Zbl 1013.74001

AMS/IP Studies in Advanced Mathematics. 30. Providence, RI: AMS, American Mathematical Society/ International Press. xvii, 442 p. (2002).

The book is devoted to analytical and numerical solving of quasistationary variational inequalities describing various types of contacts between deformable bodies or between deformable and rigid bodies. The authors consider the material behaviour with elastic, viscoelastic or viscoplastic constitutive laws. The contact is modeled with various conditions, including Signorini nonpenetration condition, normal compliance and normal damped response conditions. The friction is modeled using Coulomb’s and Tresca’s friction laws or with laws involving a dissipative frictional potential.

The book is divided into four parts and eighteen chapters. The first part contains the basic notions and results from functional analysis, as well as finite difference and finite element approximations. These results are applied to the study of stationary and nonstationary variational inequalities. The second part is devoted to mechanical background of contact problems. It summarizes the basic notions and general principles of mechanics of continua together with preliminary results on variational theory and numerical analysis for contact elastic problems. The third and the fourth parts represent the main material given in the book. They deal with quasistatic problems for Kelvin-Voigt viscoelastic materials and rate-type viscoplastic materials, respectively. These parts contain the original results of the authors, concerning various contact and frictional or frictionless boundary conditions, providing variational analysis and numerical approximations of the corresponding evolutionary variational inequalities. The convergence of semi-discrete and fully discrete approximations is verified, and several results of numerical simulations are shown.

The book deserves recommendation to all who are interested in the theory and applications of evolutionary variational inequalities, especially modelling the contact problems for viscoelastic and viscoplastic bodies.

The book is divided into four parts and eighteen chapters. The first part contains the basic notions and results from functional analysis, as well as finite difference and finite element approximations. These results are applied to the study of stationary and nonstationary variational inequalities. The second part is devoted to mechanical background of contact problems. It summarizes the basic notions and general principles of mechanics of continua together with preliminary results on variational theory and numerical analysis for contact elastic problems. The third and the fourth parts represent the main material given in the book. They deal with quasistatic problems for Kelvin-Voigt viscoelastic materials and rate-type viscoplastic materials, respectively. These parts contain the original results of the authors, concerning various contact and frictional or frictionless boundary conditions, providing variational analysis and numerical approximations of the corresponding evolutionary variational inequalities. The convergence of semi-discrete and fully discrete approximations is verified, and several results of numerical simulations are shown.

The book deserves recommendation to all who are interested in the theory and applications of evolutionary variational inequalities, especially modelling the contact problems for viscoelastic and viscoplastic bodies.

Reviewer: Igor Bock (Bratislava)

### MSC:

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

74M15 | Contact in solid mechanics |

49J40 | Variational inequalities |