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International audience; We study the existence of solutions arising from the modelling of elastic materials using generalized theories of continua. In view of some evidence from physics of metamaterials, we focus our effort on two recent nonstandard relaxed micromorphic models including novel micro-inertia terms. These novel micro-inertia terms are needed to better capture the band-gap response. The existence proof is based on the Banach fixed-point theorem.

We consider the quasi-static evolution of the thermo-plasticity model in which the evolution equation law for the inelastic strain is given by the Prandtl–Reuss flow rule. The thermal part of the Cauchy stress tensor is not linearized in the neighbourhood of a reference temperature. This nonlinear thermal part is imposed to add a damping term to the balance of the momentum, which can be interpreted as external forces acting on the material. In general, the dissipation term occurring in the heat equation is an integrable function only and the standard methods can not be applied. Combining truncation techniques and Boccardo-Gallouët approach with monotone methods, we prove an existence of renormalized solutions.

The main aim of this paper is to prove the existence and uniqueness of solutions to an initial-boundary value problem corresponding to the Biot model. The existence theorem is proved by Galerkin method and the passage to the limit in the approximation process is shown in a standard way. Assuming that the given data satisfy some natural regularity requirements a better regularity of solutions is obtained than it could be found in the literature.