Phase-Field Modeling of Defect Dynamics: Interplay Between Inertia and Viscous Stress

Free boundary problems categorize a class of problems where the region in which the problem is to be solved is unknown in advance and must be found as part of the solution. Such problems arise in a diverse range of scenarios eg. fracture, phase transformations etc. In each of these scenarios, typically, regions with uniform phases are separated by evolving boundaries comprising of sharp interfaces. The existence of sharp interfaces make numerical computations challenging, as the interfaces need to be explicitly tracked. Smoothing out sharp interfaces is an effective way of circumventing the need to explicitly track interfaces, and reduce computational complexity. One of the most widely used models for such problems is the phase-field model. When combined with Griffith’s fracture theory, phase-field model is also a leading approach for modelling crack propagation. 

This work includes inertial evolution of microstructures, phase interfaces, and cracks propagating at intersonic to supersonic speeds. However, conventional phase-field models coupled with elastodynamics fall short in providing accurate models, even qualitatively, for supersonic propagation of interfaces. Motivated by the limitations of phase-field models the first study conducted is of a simple 1D interface propagation problem where the shortcomings pertaining to the physics of standard phase-field models are identified to be: (1) the absence of higher-order stresses to balance unphysical stress singularities, and (2) the ability of the model to access unphysical regions of the energy landscape. Based on these observations, this work proposes an augmented phase-field model to introduce the missing physics. The augmented model adds: (1) a viscous stress to the momentum balance, in addition to the dissipative phase-field evolution, to regularize singularities; and (2) an augmented driving force which restricts accessing unphysical phases in the energy landscape. When coupled with elastodynamics, the augmented model correctly describes both subsonic and supersonic interface motion. 

Given the success of the augmented 1-d dynamic phase-field model, the rest of the work focuses on applying those augmented terms to 2-d problems. Specifically, 2-d dynamic phasefield fracture were studied and it was found that the addition of viscous stress the system had a profound effect on the crack propagation behavior. In the regime of subsonic crack velocities, addition of viscous stress would affect the way the crack branches and in the intersonic to supersonic regimes, the presence of viscous stress is necessary for the crack to reach supersonic velocities. 

Higher dimension dynamic interface propagation problems were also studied in which there is a propagating twin interface within a 2-d domain. This part of the work seeks to capitalize on the augmented ’driving force term’ and utilize it to control the nucleation of phases as a function of predetermined conditions. The results in this section clearly highlights the benefits of working with a dynamic phase-field model in which nucleation and kinetics may be transparently prescribed.