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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.
Entropic solutions to a thermodynamically consistent PDE system for phase transitions and damage
In this paper we analyze a PDE system modelling (non-isothermal) phase transitions and damage phenomena in thermoviscoelastic materials. The model is thermodynamically consistent: in particular, no small perturbation assumption is adopted, which results in the presence of quadratic terms on the right-hand side of the temperature equation, only estimated in L1. The whole system has a highly nonlinear character. We address the existence of a weak notion of solution, referred to as entropic, where the temperature equation is formulated with the aid of an entropy inequality, and of a total energy inequality. This solvability concept reflects the basic principles of thermomechanics as well as the thermodynamical consistency of the model. It allows us to obtain global-in-time existence theorems without imposing any restriction on the size of the initial data. We prove our results by passing to the limit in a time discretization scheme, carefully tailored to the nonlinear features of the PDE system (with its entropic formulation), and of the a priori estimates performed on it. Our time-discrete analysis could be useful towards the numerical study of this model.
A temperature-dependent phase-field model for phase separation and damage
In this paper we study a model for phase separation and damage in thermoviscoelastic materials. The main novelty of the paper consists in the fact that, in contrast with previous works in the literature (cf., e.g., [21, 22]), we encompass in the model thermal processes, nonlinearly coupled with the damage, concentration and displacement evolutions. More in particular, we prove the existence of entropic weak solutions, resorting to a solvability concept first introduced in [10] in the framework of Fourier-Navier-Stokes systems and then recently employed in [9, 38] for the study of PDE systems for phase transition and damage. Our global-intime existence result is obtained by passing to the limit in a carefully devised time-discretization scheme.