Julian Kochmann

Thermodynamic Model Formulations for Inhomogeneous Solids with Application to Non-isothermal Phase Field Modelling

Abstract The purpose of the current work is the comparison of thermodynamic model formulations for chemically and structurally inhomogeneous solids at finite deformation based on “standard” non-equilibrium thermodynamics [SNET: e. g. S. de Groot and P. Mazur, Non-equilibrium Thermodynamics, North Holland, 1962] and the general equation for non-equilibrium reversible–irreversible coupling (GENERIC) [H. C. Öttinger, Beyond Equilibrium Thermodynamics, Wiley Interscience, 2005]. In the process, non-isothermal generalizations of standard isothermal conservative [e. g. J. W. Cahn and J. E. Hilliard, Free energy of a non-uniform system. I. Interfacial energy. J. Chem. Phys. 28 (1958), 258–267] and non-conservative [e. g. S. M. Allen and J. W. Cahn, A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27 (1979), 1085–1095; A. G. Khachaturyan, Theory of Structural Transformations in Solids, Wiley, New York, 1983] diffuse interface or “phase-field” models [e. g. P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. Modern Phys. 49 (1977), 435–479; N. Provatas and K. Elder, Phase Field Methods in Material Science and Engineering, Wiley-VCH, 2010.] for solids are obtained. The current treatment is consistent with, and includes, previous works [e. g. O. Penrose and P. C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Phys. D 43 (1990), 44–62; O. Penrose and P. C. Fife, On the relation between the standard phase-field model and a “thermodynamically consistent” phase-field model. Phys. D 69 (1993), 107–113] on non-isothermal systems as a special case. In the context of no-flux boundary conditions, the SNET- and GENERIC-based approaches are shown to be completely consistent with each other and result in equivalent temperature evolution relations.

Efficient Multiscale FE-FFT-Based Modeling and Simulation of Macroscopic Deformation Processes with Non-linear Heterogeneous Microstructures

The purpose of this work is the prediction of micromechanical fields and the overall material behavior of heterogeneous materials using an efficient and robust two-scale FE-FFT-based computational approach. The macroscopic boundary value problem is solved using the finite element (FE) method. The constitutively dependent quantities such as the stress tensor are determined by the solution of the local boundary value problem. The latter is represented by a periodic unit cell attached to each macroscopic integration point. The local algorithmic formulation is based on fast Fourier transforms (FFT), fixed-point and Newton-Krylov subspace methods (e.g. conjugate gradients). The handshake between both scales is defined through the Hill-Mandel condition. In order to ensure accurate results for the local fields as well as feasible overall computation times, an efficient solution strategy for two-scale full-field simulations is employed. As an example, the local and effective mechanical behavior of ferrit-perlit annealed elasto-viscoplastic 42CrMo4 steel is studied for three-point-bending tests. For simplicity, attention is restricted to the geometrically linear case and quasi-static processes.