Phase Field Modeling of Fast Crack Propagation

Abstract

We present a continuum theory which predicts the steady state propagation of cracks. The theory overcomes the usual problem of a finite time cusp singularity of the Grinfeld instability by the inclusion of elastodynamic effects which restore selection of the steady state tip radius and velocity. We developed a phase field model for elastically induced phase transitions; in the limit of small or vanishing elastic coefficients in the new phase, fracture can be studied. The simulations confirm analytical predictions for fast crack propagation.PACS numbers: 62.20. Mk, 46.50.+a, 47.54.+r Understanding the day-to-day phenomenon of fracture is a major challenge for solid state physics and materials science. Starting with the early idea of Griffith [1], who realized that crack growth is a competition between a release of elastic energy and an increase of surface energy, various approaches have been developed to describe the striking features of cracks [2]. Usually, the motion of cracks is understood on the level of breaking bonds at sharp tips, and obviously theoretical predictions depend sensitively on the underlying empirical models of the atomic properties (see for example [3]). Plastic effects, however, lead to extended crack tips (finite tip radius r 0 ), and it is conceivable that for example fracture in gels can be described macroscopically. Then a full modeling of fracture should not only determine the crack speed but also the crack shape self-consistently.Recent phase field models go beyond the microscopic limit of discrete models with broken rotational symmetry, and encompass much of the expected behavior of cracks [4,5]; these models are close in spirit though different in details with respect to earlier approaches [6]. However, the scale of the appearing patterns is always dictated by the phase field interface width, and thus these models have problems in the sharp interface limit. Other descriptions are based on macroscopic equations of motion but suffer from inherent finite time singularities which do not allow steady state crack growth unless the tip radius is limited by the phase field interface width [7]. Numerical approaches which are not based on a phase field provide a selection mechanism by the introduction of complicated nonlinear terms in the elastic energy for high strains in the tip region [8], requiring additional parameters.It is therefore highly desirable to look for minimal models of fracture which are free from microscopic details and which are based on well established thermodynamical concepts. This is also motivated by experimental results showing that many features of crack growth are rather generic [9]; among them is the saturation of the steady state velocity appreciably below the Rayleigh speed and a tip splitting for high applied tension.Already in our previous publication [10] we emphasized a connection between fracture mechanics and elastically induced surface diffusion processes: the AsaroTiller-Grinfeld (ATG) instability [11] appears to be a good starting point for the quest f...

Multi-phase-field analysis of short-range forces between diffuse interfaces

“…Here we limit our study to the effect of a uniaxial stress applied along an axis perpendicular to the grain boundary. We couple phase change and elasticity by modeling the liquid as a solid with vanishing shear modulus, as in previous studies of solidification under stress [28], the Asaro-Tiller-Grinfeld instability [31], and fracture associated with a phase change [30]. In this approach, the local displacement vector (with two degrees of freedom in two dimensions) is also represented in the liquid, whereas the local state in this phase is fully characterized by the hydrostatic pressure.…”

Phase-field modeling of fracture in liquid

“…(2) The evolution of the system is described by the phase-field equation for g that leads to formation of finite-width (diffuse) interfaces between regions with different (meta) stable states, in contrast to discontinuity surfaces in a sharp-interface approach. While for evolution of cracks in solids phase-field theory was recently intensively developed and applied to various problems, [14][15][16][17][18] and in more advanced form (but for quasistatic deformation) in Refs. 19 and 20, we are not aware of any similar work for fracture of liquid.…”

“…FEM was used for problem solution (in contrast to Refs. [14][15][16][17][18][19][20], which allows one to easily include arbitrary boundary conditions, large displacement and strains, dynamic formulation, and more sophisticated constitutive equations (when required). Problems on cavitation in spherical and ellipsoidal particles with different aspect ratios after compressive pressure at its surface sharply dropped are solved.…”

Phase field modeling of crack propagation

“…Steady state growth of a crack with finite tip radius, determined through the interface kinetic coefficient and the sound speed. Taken from [103].…”