Variational Structures in Thermomechanics of Solids

The accurate description of the complex thermomechanical behavior of solids requires the efficient treatment of strongly nonlinearly coupled partial differential equations systems. These stem from the combination of balance and constitutive equations, which in turn can be often rephrased in a variational setting from the specification of suitable equilibrium and dissipation potentials. The aim of the project is to set up a unified approach to such variationally-based evolutionary models for the thermomechanical evolution of deformable solids, in particular in phase transition of shape-memory and elastoplastic materials. The focus will be on innovative modeling and analytic techniques, as well as on real-scale computations. We will develop a new class of qualitatively more efficient and rigorously supported models in Thermomechanics and investigate specific applications in Engineering, Materials Science, and Natural Sciences. The ambitious research plan will be realized within an international collaborative initiative between the Faculty of Mathematics of the University of Vienna and the Institute of Information Theory and Automation of the Czech Academy of Sciences. The cooperation of two teams will merge together the expertises of researchers in mathematical modeling, mathematical analysis, and computational mechanics. State-of-the-art tools from nonlinear partial differential equations and inequalities and the calculus of variations will be exploited and combined with three-dimensional Finite Elements approximations of inelastic solids behavior.

The project focused on the mathematical modeling of different material systems, the corresponding analysis, and numerical simulations. More specifically, we addressed the mechanical behavior of different classes of complex materials showing a combination of inelastic and thermal effects. The overarching aim was that of providing macroscopic theories of a variational character. Among the many research directions explored, let us mention finite plasticity, thermoplasticity, Peridynamics, and Eulerian-Lagrangian formulations. These achievements have been obtained in the frame of a strict collaboration between the Czech and the Austrian teams.