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Modeling and simulation of non-isothermal rate-dependent damage processes in inhomogeneous materials using the phase-field approach
We present a continuum model that incorporates rate-dependent damage and fracture, a material order parameter field and temperature. Different material characteristics throughout the medium yield a strong inhomogeneity and affect the way fracture propagates. The phasefield approach is employed to describe degradation. For the material order parameter we assume a Cahn Larché-type dynamics, which makes the model in particular applicable to binary alloys. We give thermodynamically consistent evolution equations resulting from a unified variational approach. Diverse coupling mechanisms can be covered within the model, such as heat dissipation during fracture, thermal-expansion-induced failure and elastic-inhomogeneity effects. We furthermore present an adaptive Finite Element code in two space dimensions that is capable of solving such a highly nonlinear and non-convex system of partial differential equations. With the help of this tool we conduct numerical experiments of different complexity in order to investigate the possibilities and limitations of the presented model. A main feature of our model is that we can describe the process of micro-crack nucleation in regions of partial damage to form macro-cracks in a unifying approach.
Analysis and optimal control theory for a phase field model of Caginalp type with thermal memory
A nonlinear extension of the Caginalp phase field system is considered that takes thermal memory into account. The resulting model, which is a first-order approximation of a thermodynamically consistent system, is inspired by the theories developed by Green and Naghdi. Two equations, resulting from phase dynamics and the universal balance law for internal energy, are written in terms of the phase variable (representing a non-conserved order parameter) and the so-called thermal displacement, i.e., a primitive with respect to time of temperature. Existence and continuous dependence results are shown for weak and strong solutions to the corresponding initial-boundary value problem. Then, an optimal control problem is investigated for a suitable cost functional, in which two data act as controls, namely, the distributed heat source and the initial temperature. Fréchet differentiability between suitable Banach spaces is shown for the control-to-state operator, and meaningful first-order necessary optimality conditions are derived in terms of variational inequalities involving the adjoint variables. Eventually, characterizations of the optimal controls are given.