Deep learning-based phase-field modelling of brittle fracture in anisotropic media

Abstract

This work presents a variational physics-informed deep learning framework for phase-field modelling of brittle crack propagation in anisotropic media. Previous Deep Ritz Method (DRM) approaches have focused on second-order, isotropic phase-field fracture formulations. In contrast, the present work introduces, for the first time within a variational deep learning setting, a family of higher-order anisotropic phase-field models through a generalised crack density functional. The resulting fracture problem is solved by minimising the total energy using the DRM. The trial space is enriched with higher-order B-spline basis functions to represent higher-order gradients accurately and stably, thereby eliminating the need for conventional automatic differentiation. The methodology is assessed for isotropic, cubic, and orthotropic fracture surface energy densities. Numerical examples demonstrate direction-dependent crack growth in anisotropic cases, highlighting the capability of the method to accurately capture this behaviour.

Figures (12)

• Figure 1: (a) Solid body $\Omega$ with crack path $\Gamma$ and (b) phase field approximation of the crack path $\Gamma$.
• Figure 2: \ref{['fig:gc_theta:a']} Polar representation of the fracture surface energy density $\mathcal{G}_{c}(\theta)$ and \ref{['fig:gc_theta:b']} its reciprocal $1/\mathcal{G}_{c}(\theta)$ for cubic material symmetry as a function of crack orientation $\theta$. The red portions of the polar plots denote the forbidden orientations associated with $\mathcal{S}^{r}(\theta) < 0$, while the blue portions indicate energetically stable paths with $\mathcal{S}^{r}(\theta) > 0$. $\mathcal{S}^{r}(\theta)$ is defined in Section \ref{['sec:convex_nonconvex']}.
• Figure 3: Schematic illustration of the DRM employed in this work. Spatial coordinates are input to the NN to obtain approximations of the displacement- and phase-fields. These are used within a quadrature scheme to evaluate the total potential energy, which is subsequently minimised by an optimiser until a convergence criterion is reached. Parameters of the network are explained in Section \ref{['sec:deep_neural_network']}.
• Figure 4: \ref{['fig:activation_function_scaling']} Effect of the scaling coefficient $r$ on the $\tanh$ activation function. Smaller values extend the quasi-linear regime. \ref{['fig:nonsmooth_sigmoid']} Non-smooth piecewise constraint used to enforce admissible phase-field values.
• Figure 5: Displacement field enforcement via distance functions. The shaded regions represent non-zero values, while the white regions are zero. The yellow field represents the loading distance function, indigo represents the fixed boundary conditions, and magenta represents the NN predictions, all scale $\sim1$. The light orange field represents the output scaled by the prescribed displacement, satisfying the prescribed loading and boundary conditions. The regions bounded by both the loaded and the fixed distance functions are enforced to be zero by inverting and composing with the network prediction. The exact prescribed displacement is then obtained by reintroducing the loaded area and scaling by the prescribed displacement $\mathbf{U}_p$.