ALEX 2018 Workshop: Abstracts

Dynamical problems in continuum mechanics of solids at large strains

Tomáš Roubíček

Mathematical Institute, Charles University in Prague (Czech Republic) and

Institute of Thermomechanics, Czech Academy of Sciences (Czech Republic)

Deformable solids at large strains uses material reference (Lagrange) description which leads or may lead to mathematically amenable problems. In dynamical situations, one must rely on differential equations rather than on variational principles for some functionals as in the (quasi)static situations, which brings many difficulties.

Various gradient theories are thus to be employed. In particular, the concept of nonsimple (possibly nonlocal) materials seems essential to obtain global existence results for frame-indifferent models, possibly accounting for at least local non-interpenetration. Besides the desired regularization analytical effects, in linearized situations these higher-order terms lead to anomalous or possibly also normal dispersion of elastic waves.

In some situations, the reference configuration does not have any real meaning, and only the actual deformed configuration is relevant. Then all transport tensors (like mobility of diffusants in poroelastic materials or heat conductivity) must be pulled back. Yet, the analysis of such coupled systems is known only in particular situations [1], cf. e.g. the only quasistatic model in [3] of thermal coupling in Kelvin-Voigt materials while full dynamical variant seems difficult.

When there are some internal variables considering with gradient theories (as plasticity or damage or capillarity in the Cahn-Hilliard model), these gradients are also to be considered rather as pulled back. This gives rise to a Korteweg-like stress and to additional difficulties, not always successfully solved so far.