Scales and Shapes in Continuum Thermomechanics

The behavior of solids undergoing mechanical deformations gives rise to challenging mathematical problems. These are inherently nonlinear and often formulated in variational terms as balances between energy-storage and dissipation mechanisms. Correspondingly, the mathematical setting is that of calculus of variations and partial differential equations. This project aims at tackling a number of issues in nonlinear continuum thermomechanics by focusing on model reduction. In particular, we will address linearized models in thermoviscoelasticity and viscoplasticity by means of variational convergence techniques. We will also investigate lower-dimensional models as limit models of three-dimensional bulk structures. Another research focus will be topology optimization problems in elastoplasticity, damage, and brittle or microstructured materials. Here, the goal is to identify the optimal shape of a body with respect to its expected mechanical response. Based on these results, new optimal control problems will be formulated and analyzed. Finally, we will develop efficient numerical approaches and computational simulations for the above mentioned problems, which will complete our effort in establishing rigorous but also practically applicable models.

The behavior of solids under mechanical deformations gave rise to challenging mathematical problems. These problems are inherently nonlinear and were often formulated in a variational manner as equilibrium conditions between energy storage and dissipation mechanisms. Accordingly, the mathematical framework encompassed the calculus of variations and the theory of partial differential equations. The aim of this project was to address a range of questions in nonlinear continuum thermomechanics, with a particular focus on model reduction. In particular, linearized models of viscoelasticity and plasticity were investigated using techniques from variational convergence. A further research focus concerned topology optimization for elastoplasticity, damage, and microstructured materials. In this context, the optimal shape of a component was identified with respect to its expected mechanical response. Based on these results, new optimal control problems were formulated and analyzed.