Hiromichi Itou

A generalization of the Kelvin–Voigt model that can represent viscoelastic materials whose moduli depend on the mechanical pressure is derived from an implicit constitutive relation in which both the Cauchy stress and the linearized strain appear linearly. For consistency with the assumption of small deformation, a thresholding approach is applied. The proposed mixed variational problem is investigated for its well‐posedness within the context of maximal monotone and coercive graphs. For isotropic extension or compression, a semi‐analytic solution of the generalization of the Kelvin–Voigt problem under stress control is presented. The corresponding numerical simulation for monotone and cyclic pressure loading is carried out, and the results then compared against the linearized model.

We consider dynamic motion of a linearized elastic body with a crack subject to a modified contact law, which we call the Signorini contact condition of dynamic type , and to the Tresca friction condition. Whereas the modified contact law involves both displacement and velocity, it formally includes the usual non-penetration condition as a special case. We prove that there exists a unique strong solution to this model. It is remarkable that not only existence but also uniqueness is obtained and that no viscosity term that serves as a parabolic regularization is added in our model. This article is part of the theme issue ‘Non-smooth variational problems and applications’.