Variational methods for problems from plasticity theory and for generalized Newtonian fluids.

*(English)*Zbl 0964.76003
Lecture Notes in Mathematics. 1749. Berlin: Springer. vi, 269 p. (2000).

The monograph is devoted to a rigorous mathematical analysis of variational problems describing the equilibrium configuration of certain types of perfect elastoplastic solids, and also the stationary flows of some incompressible generalized Newtonian fluids. The authors concentrate on variational problems from the deformation theory of plasticity and on fluid models whose stress-strain relation can be formulated in terms of dissipative potential. The mathematical form of both problems is very close, and is reduced to the study of variational integrals with convex integrands depending only on the symmetric part of gradients of unknown vector-valued functions. The integrands have the linear growth, and the problem of finding suitable classes of admissible deformations is not trivial. Another difficulty of the considered problems consists in their formulations in non-reflexive spaces.

The authors attack these problems by formulating a dual variational problem for the stress tensor, and by the relaxation method. Chapter 1 contains a general scheme for the relaxation of convex variational problems being coercive on some non-reflexive spaces like \(L^1\) or \(W^1_1\). The authors give a detailed description of the approach to the perfect plasticity. Besides plasticity with power hardening and perfect plasticity, the authors investigate here the model with logarithmic hardening. Hence the integrand is not of power growth, and it is necessary to introduce new function spaces for localization of solutions. Chapter 2 presents an analysis of regularity of weak solutions. The problems of regularity for integrands with linear growth and for vector-valued functions have been till now discussed only in a few papers. The approach of the authors is based on the study of dual variational problem for stresses. They obtain additional regularity for stress tensor, and with the help of duality relations establish the regularity of displacement field.

The local regularity of solutions to variational problems describing stationary flow of generalized Newtonian fluids is studied in chapter 3. The presentation includes also the viscoplastic fluids of Bingham type. The authors derive partial regularity in spite of the fact that the dissipative potential is not smooth. All investigations are limited to incompressible flows, i.e. only fields with vanishing divergence are considered. Chapter 4 presents the regularity theory for the logarithmic case. Here the authors introduce special functional spaces, and for plasticity with logarithmic hardening they investigate directly the minimizing deformation field. A lot of various results on function spaces is proved in the appendices A and B, and a special chapter is devoted to inevitable tools from functional analysis. The book presents a very complex view-point on the plasticity and fluid mechanics problems. The mechanical and physical phenomena are explained by rigorous mathematical methods, and the potential reader is acquainted with a unified approach to plasticity and fluid mechanics problems in their full complexity.

The authors attack these problems by formulating a dual variational problem for the stress tensor, and by the relaxation method. Chapter 1 contains a general scheme for the relaxation of convex variational problems being coercive on some non-reflexive spaces like \(L^1\) or \(W^1_1\). The authors give a detailed description of the approach to the perfect plasticity. Besides plasticity with power hardening and perfect plasticity, the authors investigate here the model with logarithmic hardening. Hence the integrand is not of power growth, and it is necessary to introduce new function spaces for localization of solutions. Chapter 2 presents an analysis of regularity of weak solutions. The problems of regularity for integrands with linear growth and for vector-valued functions have been till now discussed only in a few papers. The approach of the authors is based on the study of dual variational problem for stresses. They obtain additional regularity for stress tensor, and with the help of duality relations establish the regularity of displacement field.

The local regularity of solutions to variational problems describing stationary flow of generalized Newtonian fluids is studied in chapter 3. The presentation includes also the viscoplastic fluids of Bingham type. The authors derive partial regularity in spite of the fact that the dissipative potential is not smooth. All investigations are limited to incompressible flows, i.e. only fields with vanishing divergence are considered. Chapter 4 presents the regularity theory for the logarithmic case. Here the authors introduce special functional spaces, and for plasticity with logarithmic hardening they investigate directly the minimizing deformation field. A lot of various results on function spaces is proved in the appendices A and B, and a special chapter is devoted to inevitable tools from functional analysis. The book presents a very complex view-point on the plasticity and fluid mechanics problems. The mechanical and physical phenomena are explained by rigorous mathematical methods, and the potential reader is acquainted with a unified approach to plasticity and fluid mechanics problems in their full complexity.

Reviewer: Igor Bock (Bratislava)

##### MSC:

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

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

74G65 | Energy minimization in equilibrium problems in solid mechanics |

76A05 | Non-Newtonian fluids |

76M30 | Variational methods applied to problems in fluid mechanics |

74C05 | Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials) |