Coercive limits for constitutive equations of monotone-gradient type
The authors consider the state equations of an inelastic body, which include the equilibrium equation including the dynamic d'Alembert term, and the constitutive equation including terms which describe the dependence of stress tensor on the history of the deformation gradient. They introduce a vector of variables which depends on the plastic strain and various properties of the material. The system of the type considered here was studied by Ionescu and Sofinonea in their 1993 monograph [\textit{I. R. Ionescu} and \textit{M. Sofonea}, Functional and Numerical Methods in Viscoplasticity. Oxford University Press, New York (1993; Zbl 0787.73005)]. They assumed that the function defining the mapping between these internal variables and their time derivatives is Lipschitz continuous. More recent study of materials with memory can be found in Alber's monograph [\textit{A. D. Alber}, Materials with memory. Initial-Boundary Value Problems for Constitutive Equations with Internal Variables. Lecture Notes in Mathematics, vol. 1682, Springer Verlag, Berlin (1998; Zbl 0977.35001)], where models (with internal variables) were discussed which do not safisfy the Lipschitz condition. Existence of free energy came as a consequence of thermodynamic considerations. Another assumption that has been frequently made regards coerciveness. Such assumption implies existence of global solutions. Unfortunately, as the authors observe, this assumptions is false in many importunt models presendy used in engineering practice, such as perfect elastoplasticity. The state equations are rewritten as an evolution equation system. The main feature of this system is monotonicity with respect to a specific bilinear form. The rest of this work is devoted to models in which the function describing the relation between the vector of internal variables and its derivative can be written as a gradient of a convex function. The main theorem asserts existence of a strong solution global in time, provided the initial data has coercive limit property.
scientific article; zbMATH DE number 1740687