Dependence of equilibrium Griffith surface energy on crack speed in phase-field models for fracture coupled to elastodynamics

Abstract

Phase-field models for crack propagation enable the simulation of complex crack patterns without complex and expensive tracking and remeshing as cracks grow. In the setting without inertia, the crack evolution is obtained from a variational energetic starting point, and leads to an equation for the order parameter coupled to elastostatics. Careful mathematical analysis has shown that this is consistent with the Griffith model for fracture. Recent efforts to include inertia in this formulation have replaced elastostatics by elastodynamics. In this brief note, we examine the elastodynamic augmentation, and find that it effectively causes the Griffith surface energy to depend on the velocity of the crack. That is, considering two identical specimens that are each fractured by a single crack that grows at different velocities in the two specimens, it is expected that the final equilibrium configurations are nominally identical; however, the phase-field fracture models augmented with elastodynamics achieve final configurations—in particular, the Griffiths surface energy contributions—that depend on the crack velocity. The physical reason is that the finite relaxation time for the stresses in the elastodynamic setting enables the cracked region to widen, beyond the value observed in the quasistatic setting. Once the crack widens, the “no-healing” condition prevents it from relaxing even after the specimen reaches equilibrium. In phase-field models, crack width in the reference configuration is unrelated to the physical opening of the crack but is instead a measure of Griffiths surface energy. This observation suggests that elastodynamic phase-field fracture models should not be used in settings where the crack velocity is large.

Appendix: Numerical method and values of parameters

In the expression for the energy:

$$\begin{aligned} E[{{\varvec{u}}},\phi ]= & {} \int \limits _{\varOmega }(\phi ^2 + \eta _\epsilon )W(\nabla {{\varvec{u}}}) \mathrm {d}\varOmega \nonumber \\&+\, {\mathcal {G}}_c\int \limits _{\varOmega }\left[ \frac{(1-\phi )^2}{4\epsilon } + \epsilon |\nabla \phi |^2\right] \mathrm {d}\varOmega \end{aligned}$$

(A.1)

we use $\eta _\epsilon = 10^{-8}$, ${\mathcal {G}}_c = 0.8$, $\epsilon = 10^{-4}$, and linear isotropic elasticity is assumed with Lame constants $\lambda = 10^4$, $\mu =8 \times 10^3$ in non-dimensionalized units. The mass density is $10^4$.

The computational domain is a square of size 1. We define the origin at the bottom left corner. As described in the main text, the initial condition includes a notch—at $x_1=0.5$ running from $x_2=0$ to $x_2=0.15$—to initiate the growth of the crack. A notch is defined here as a region over which $\phi \eqsim 0$.

Tractions ${{\varvec{t}}}$ are imposed on the left and right boundaries at $t=0$ to cause tensile elastic waves to initiate and travel towards the center of the domain. The boundary conditions are as follows:

$$\begin{aligned} x_1= & {} L \ (\text {Right face}): t_1 = P, t_2 = 0 \end{aligned}$$

(A.2)

$$\begin{aligned} x_1= & {} 0 \ (\text {Left face}): t_1 = -P, t_2 = 0 \end{aligned}$$

(A.3)

$$\begin{aligned} x_2= & {} L \ (\text {Top face}): t_1 = 0, u_2 = 0 \end{aligned}$$

(A.4)

$$\begin{aligned} x_2= & {} 0 \ (\text {Bottom face}): t_1 = 0, t_2 = 0 \end{aligned}$$

(A.5)

When these waves hit the notch, they cause crack growth to commence. We have used $P=750$ and $P=1000$ to get the different crack speeds described in the main document. The displacement at $(x_1,x_2) = (L/2, L)$ is set to zero to prevent rigid-body translations in the 1 direction.

For the time evolution of the momentum equation, we use finite differences in time with an explicit central difference scheme; this is a standard approach for elastodynamics. In the particular context of phase-field fracture approaches, a related Newmark method is used by Bleyer et al. (2017). The time step used is $\varDelta t = 5 \times 10^{-5}$ for the material parameters above.

The spatial discretization uses standard finite elements with uniform rectangular elements and piecewise-linear shape functions for ${{\varvec{u}}}$ and $\phi$. There are 200 elements in both directions.

At each time step, we minimize with respect to $\phi$. The minimization is performed using a gradient descent algorithm. To satisfy the irreversibility constraint (“no healing”), following Bourdin et al. (2011), we perform the minimization while restricting $\phi =0$ on the set where $\phi < \phi _0$ with $\phi _0=0.01$.