■ Abstract The phase-field method has recently emerged as a powerful computational approach to modeling and predicting mesoscale morphological and microstructure evolution in materials. It describes a microstructure using a set of conserved and nonconserved field variables that are continuous across the interfacial regions. The temporal and spatial evolution of the field variables is governed by the Cahn-Hilliard nonlinear diffusion equation and the Allen-Cahn relaxation equation. With the fundamental thermodynamic and kinetic information as the input, the phase-field method is able to predict the evolution of arbitrary morphologies and complex microstructures without explicitly tracking the positions of interfaces. This paper briefly reviews the recent advances in developing phase-field models for various materials processes including solidification, solid-state structural phase transformations, grain growth and coarsening, domain evolution in thin films, pattern formation on surfaces, dislocation microstructures, crack propagation, and electromigration.

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Phase-Field Models for Microstructure Evolution

https://www.crm.sns.it/media/course/3060/miehe+hofacker+welschinger10.pdf

A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits

A phase-field description of dynamic brittle fracture

The computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies. This can be overcome by a diffusive crack modeling based on the introduction of a crack phase field. In this paper, we outline a thermodynamically consistent framework for phase field models of crack propagation in elastic solids, develop incremental variational principles and consider their numerical implementations by multi- field finite element methods. We start our investigation with an intuitive and descriptive derivation of a regularized crack surface functional that -converges for vanishing length-scale parameter to a sharp crack topology functional. This functional provides the basis for the definition of suitable convex dissipation functions which govern the evolution of the crack phase field. Here, we propose alternative rate-independent and viscous over-force models which ensure the local growth of the phase field. Next, we define an energy storage function whose positive tensile part degrades with increasing phase field. With these constitutive functionals at hand, we derive the coupled balances of quasi-static stress equilibrium and gradient-type phase field evolution in the solid from the argument of virtual power. Here, we consider a canonical two-field setting for rate-independent response and a time-regularized three-field formulation with viscous over-force response. It is then shown that these balances follow as the Euler equations of incremental variational principles that govern the multi-field problems. These principles make the proposed formulation extremely compact and provide a perfect base for the finite element implementation, including features such as the symmetry of the monolithic tangent matrices. We demonstrate the performance of the proposed phase field formulations of fracture by means of representative numerical examples.

Thermodynamically-Consistent Phase Field Models of Fracture: Variational Principles and Multi-Field

In this contribution we address the issue of efficient finite element treatment for phase-field modeling of brittle fracture. We start by providing an overview of the existing quasi-static and dynamic phase-field fracture formulations from the physics and the mechanics communities. Within the formulations stemming from Griffith's theory, we focus on quasi-static models featuring a tension-compression split, which prevent cracking in compression and interpenetration of the crack faces upon closure, and on the staggered algorithmic implementation due to its proved robustness. In this paper, we establish an appropriate stopping criterion for the staggered scheme. Moreover, we propose and test the so-called hybrid formulation, which leads within a staggered implementation to an incrementally linear problem. This enables a significant reduction of computational cost--about one order of magnitude--with respect to the available (non-linear) models. The conceptual and structural similarities of the hybrid formulation to gradient-enhanced continuum damage mechanics are outlined as well. Several benchmark problems are solved, including one with own experimental verification.

A review on phase-field models of brittle fracture and a new fast hybrid formulation

We introduce a phenomenological continuum model for the mode III dynamic fracture that is based on the phase-field methodology used extensively to model interfacial pattern formation. We couple a scalar field, which distinguishes between "broken" and "unbroken" states of the system, to the displacement field in a way that consistently includes both macroscopic elasticity and a simple rotationally invariant short-scale description of breaking. We report two-dimensional simulations that yield steady-state crack motion in a strip geometry above the Griffith threshold.

Tip splitting instability in a phase field model of mode III dynamic fracture

Extending Phase Field Models of Solidification to Polycrystalline Materials

Crack propagation is studied numerically using a continuum phase-field approach to mode III brittle fracture. The results shed light on the physics that controls the speed of accelerating cracks and the characteristic branching instability at a fraction of the wave speed.

Unsteady crack motion and branching in a phase-field model of brittle fracture.

Phase field model of premelting of grain boundaries

http://www.princeton.edu/~mhaataja/PAPERS/misha_actamat2004.pdf

Alexander E. Lobkovsky

Papers

Tip splitting instability in a phase field model of mode III dynamic fracture

Extending Phase Field Models of Solidification to Polycrystalline Materials

Unsteady crack motion and branching in a phase-field model of brittle fracture.

Phase field model of premelting of grain boundaries

Grain shape, grain boundary mobility and the Herring relation