Forbidden Directions for the Fracture of Thin Anisotropic Sheets: An Analogy with the Wulff Plot

Abstract

It is often postulated that quasistatic cracks propagate along the direction allowing fracture for the lowest load. Nevertheless, this statement is debated, in particular for anisotropic materials. We performed tearing experiments in anisotropic brittle thin sheets that validate this principle in the case of weak anisotropy. We also predict the existence of forbidden directions and facets in strongly anisotropic materials, through an analogy with the description of equilibrium shapes in crystals. However, we observe cracks that do not necessarily follow the easiest direction but can select a harder direction, which is only locally more advantageous than neighboring paths. These results challenge the traditional description of fracture propagation, and we suggest a modified, less restrictive criterion compatible with our experimental observations.

Phase‐field modeling and simulation of fracture in brittle materials with strongly anisotropic surface energy

“…If these functions are smooth and recalling Equation (1), the optimality condition results in dG=d D dG c =dÂ, which can be identified as a configurational torque balance [16,17]. Recent experiments on thin anisotropic films have interrogated the MERR criterion in materials with anisotropic fracture toughness, finding results inconsistent with the global maximization of G.Â/=G c .Â/ [7]. By examining strongly anisotropic materials, these experiments established a close analogy with crystal growth and the Wulff construction for the equilibrium shape of a crystal in materials with strongly anisotropic surface energy, which exhibit forbidden orientations and faceted surfaces.…”

“…However, many of the interesting features of fracture in anisotropic materials, such as sawtooth crack patterns or forbidden crack directions, are directly related to the nonconvexity of strongly anisotropic surface energies [7]. Here, we start from a regularized variational theory of brittle fracture [27] and modify it by formulating a strongly anisotropic surface energy inspired by phase-field models of crystal growth.…”

“…The variational nature of this model suggests that the underlying crack-path selection principle is related to the MERR criterion, and in fact, it has been shown through asymptotic analysis that cracks propagate obeying the configurational equilibrium dG=d D dG c =d in a weakly anisotropic phase-field model [17]. We implement numerically the proposed model with local maximum entropy (LME) meshfree approximants [28], which allow us to directly deal with the high-order nature of the PDE [29,30], and explore the fracture behavior of the model in the light of the experiments by [7].…”

“…While these criteria provide similar predictions for homogeneous isotropic materials (in fact, (1) and (2) coincide under certain conditions [15,26]), they greatly differ when generalized to materials with anisotropic surface energy, in which the fracture toughness G c .Â/ is orientation dependent. When generalizing MERR, Griffith's propagation Equation (1) is first met along a direction  such that G.Â/=G c .Â/ is maximized [7,15], which provides a criterion to select the crack direction. If these functions are smooth and recalling Equation (1), the optimality condition results in dG=d D dG c =dÂ, which can be identified as a configurational torque balance [16,17].…”

“…While most theoretical and computational studies have focused on elastic anisotropy [1][2][3], the anisotropy of the fracture toughness influences more strongly the crack propagation of a wide variety of materials including single crystals [4][5][6], extruded polymers [7], geological materials [8,9], including sedimentary [10] and granitic rocks [11], or apple flesh [12]. The issue of brittle crack propagation in materials with anisotropic surface energy deeply interrogates our understanding of fracture and is receiving increasing attention from a variety of points of view, such as molecular dynamics [13], continuum mechanics [14,15], phase-field modeling [16,17], and experiments [7,18]. Here, by exploiting the analogy with crystal growth, we develop and implement numerically a phase-field model for brittle fracture of materials with strongly anisotropic surface energy and interpret our numerical results in the light of recent theories and experiments.…”