×

On Kelvin-Voigt model and its generalizations. (English) Zbl 1371.74067

Summary: We consider a generalization of the Kelvin-Voigt model where the elastic part of the Cauchy stress depends non-linearly on the linearized strain and the dissipative part of the Cauchy stress is a nonlinear function of the symmetric part of the velocity gradient. The assumption that the Cauchy stress depends non-linearly on the linearized strain can be justified if one starts with the assumption that the kinematical quantity, the left Cauchy-Green stretch tensor, is a nonlinear function of the Cauchy stress, and linearizes under the assumption that the displacement gradient is small. Long-time and large data existence, uniqueness and regularity properties of weak solution to such a generalized Kelvin-Voigt model are established for the non-homogeneous mixed boundary value problem. The main novelty with regard to the mathematical analysis consists in including nonlinear (non-quadratic) dissipation in the problem.

MSC:

74D10 Nonlinear constitutive equations for materials with memory
74H30 Regularity of solutions of dynamical problems in solid mechanics
35B65 Smoothness and regularity of solutions to PDEs
35D30 Weak solutions to PDEs
35Q74 PDEs in connection with mechanics of deformable solids
PDFBibTeX XMLCite
Full Text: DOI