Variational Formulations in the Non-Isothermal Thermo-Chemo-Mechanics of Nonlinear Materials


Prof. Dipl.-Ing. Björn Kiefer, Ph.D., Prof. Dr. rer. nat. Oliver Rheinbach


Dr.-Ing. Stefan Prüger


In this project, which is part of DFG Priority Programme 2256, complex material behavior is modeled, in which interacting thermo-chemo-mechanical processes play a dominant role. Our approach is a close co-design of modeling, based on variational formulations, and numerical solution methods for the arising discretized nonlinear and linearized problems, that employ state-of-the art parallel iterative solvers from domain decomposition. The focus of this work will be on capturing micro-scale mechanisms such as stress- and temperature-biased solid-solid phase transformations and chemical component diffusion, often coupled to chemical reactions, that directly influence effective macroscopic material behavior. The simulation framework we seek to establish therefore has great relevance to a wide spectrum of challenges in mechanics, particularly with respect to multi-scale response predictions of modern engineering materials, smart and multi-functional materials, biomechanics and other fields.We will consider geometrically nonlinear, dissipative, thermo-chemo-mechanical processes occurring in mixtures of multiple chemical components diffusing among multiple transforming solid phases, based on non-equilibrium continuum thermodynamics, mixture theory, and constitutive variational principles that build on a generalized notion of standard dissipative materials. The effect of microstructure evolution will be addressed in two different scenarios: (i) in a homogenized sense and (ii) with spatially-resolved microstructure. For both scenarios, appropriate nonlinear solvers for unstructured grids will be applied.To achieve our goals, we plan to take the following steps: On the modeling side, we will investigate suitable variational settings in a co-design with numerical methods, formulate variationally-consistent homogenization schemes, and extend the theory to non-isothermal processes. The focus on the numerical side is place on developing new nonlinear solvers, based on nonlinear domain decomposition methods, nonlinear saddle point solvers, and energy-aware algorithms. Solver methods that require no grid hierarchy are of particular interest, i.e., where coarse levels can be constructed without or with little geometric information. To test the accuracy, efficiency and robustness of the developed algorithms, a number of benchmark problems from the literature will be considered, e.g., related to spinodal decomposition or austenite-martensite phase transformation. Once the model is sufficiently verified, we will be in a position to address challenging novel problems that are currently of high interest in designing future engineering applications, such as reactive oxid layers growth in multifunctional steel filters or hydrogen diffusion related embrittlement in steels and shape memory alloys. The co-design strategy between engineering and mathematics that defines this project is supported through cross-sectional activity, where also the software environments are adapted in an ongoing process.




  • Kiefer, B., Rheinbach, O., Roth, S., Röver, F.: Variational Methods and Parallel Solvers in Chemo-Mechanics, PAMM - Proceedings in Applied Mathematics (DOI: 10.1002/pamm.202000272)