Relaxed Potentials and Evolution Equations for Inelastic Microstructures

Abstract

We consider microstructures which are not inherent to the material but occur as a result of deformation or other physical processes. Examples are martensitic twin-structures or dislocation walls in single crystals and microcrack-fields in solids. An interesting feature of all those microstructures is, that they tend to form similar spatial patterns, which hints at a universal underlying mechanism. For purely elastic materials this mechanism has been identified as minimisation of global energy. For non-quasiconvex potentials the minimisers are not anymore continuous deformation fields, but small-scale fluctuations related to probability distributions of deformation gradients, so-called Young measures. These small scale fluctuations correspond exactly to the observed microstructures of the material. The particular features of those, like orientation or volume fractions, can now be calculated via so-called relaxed potentials. We develop a variational framework which allows to extend these concepts to inelastic materials. Central to this framework will be a Lagrange functional consisting of the sum of elastic power and dissipation due to change of the internal state of the material. We will obtain time-evolution equations for the probability-distributions mentioned above. In order to demonstrate the capabilities of the formalism we will show an application to crystal plasticity.