Why planar cracks fragment into echelon cracks

TL;DR

The paper addresses why planar cracks fragment into echelon patterns under out-of-plane shear by showing that traditional Griffith-based energy minimization, as implemented in classical variational phase-field models, cannot predict echelon crack nucleation. It introduces a strength-constrained phase-field formulation that imposes a Drucker-Prager-like strength surface, producing crack growth only when local stress states exceed material strength. This approach reproduces echelon cracking in both hard graphite and soft PDMS without introducing disorder or ad hoc assumptions, and identifies two key non-dimensional controls, $\sigma_{ss}/\sigma_{ts}$ and $H/l_{ch}^{ss}$, that govern orientation and fragmentation. The framework reconciles energy- and stress-based criteria within a single theory and provides a general mechanism for crack initiation and propagation in brittle materials, with potential extension to anisotropic media.

Abstract

Predicting the growth of large cracks in brittle materials is a fundamental unresolved problem in fracture mechanics. Under out-of-plane shear loading, an initially planar crack may fragment into multiple cracks, forming an echelon crack pattern. Explaining this phenomenon is essential for developing a general theory of crack growth. Although numerous empirical criteria have been proposed in the literature, none provide a unified explanation of all observed features and are largely restricted to two-dimensional growth in linear elastic isotropic materials. In this Letter, we confront a classical set of echelon crack growth experiments using two phase-field approaches: the classical variational model and a strength-constrained model. We show that, contrary to prevailing views, the variational model based solely on Griffith's energetic competition between elastic and fracture energies is fundamentally incomplete even for predicting the growth of large cracks. By incorporating a material strength surface that constrains the regions in which a crack can grow, the resulting model accurately predicts echelon crack growth without invoking any ad hoc assumptions about material or geometrical disorder. Results are presented for both soft and hard materials, confirming the model's general applicability to any brittle material. We further identify two governing non-dimensional parameters that control crack orientation and morphology and demonstrate that one of them, the ratio of shear to tensile strength, determines whether crack paths are more influenced by energy-based or stress-based empirical criteria, thereby reconciling these criteria within a single framework.

Figures (5)

• Figure 1: (a) Schematic of the tearing test over a thick notched plate from Knauss knauss1970observation. (b) Experimental observation of echelon crack growth. (c) Crack path predicted by classical variational phase-field model (\ref{['BFM00']}). (d) Path predicted by the strength-constrained phase-field model (\ref{['phase-field-equations']}).
• Figure 2: (a) Schematic of a smaller geometry used for comprehensive analysis (all dimensions are in mm; applied displacement in $z$-direction). (b) First invariant of the stress tensor (left axis) and maximum principal stress (right axis) plotted as a function of $z$ coordinate in front of the crack. (c) Crack path contour obtained from the variational model for a hard brittle material, Graphite, and (d) contour for soft brittle material, PDMS. (e) Plot of the determinant of the deformation gradient tensor over the propagating crack. (f) Plot of the energy release rate for a planar and echelon crack growth.
• Figure 3: Simulations with the strength-constrained phase field model. Crack path contour for three values of applied displacement, $u$, for (a) Graphite, and (b) PDMS.
• Figure 4: (a) Crack contours for five increasing values of the shear-to-tensile strength ratio, $\sigma_{\texttt{ss}}/\sigma_{\texttt{ts}}$, along with the corresponding compressive-to-tensile strength ratios, $\sigma_{\texttt{cs}}/\sigma_{\texttt{ts}}$. (b) Orientation angle for the largest daughter crack as a function of $\sigma_{\texttt{cs}}/\sigma_{\texttt{ts}}$ for two values of crack extension $\Delta a$. (c) Contour plot around the crack front of the regions of the specimen where the stress field exceeds the strength surface ($\mathcal{F}({\bf S})=0$) at three different locations (coordinate in mm) along the thickness and applied displacement $u=0.075$ mm. (d) A 2D cut of the strength surface plotted in terms of the principal stresses $\sigma_1$ and $\sigma_2$ corresponding to the $\sigma_3 = -0.75 \, \sigma_2$ plane.
• Figure 5: Number of daughter cracks as a function of $H/ l_{\rm ch}^{ss}$. Inset panels (top) show results obtained by varying $l_{\rm ch}^{ss}$ via changes in $G_c$, $E$, or $\sigma_{\texttt{ss}}$; inset panels (bottom) show results obtained by varying $H$. Error bars indicate the range of observed crack counts. No echelon cracks observed in the gray shaded region.