Simulating curvilinear crack propagation in two dimensions with universal meshes

Abstract

SUMMARYWe formulate a class of delicately controlled problems to model the kink-free evolution of quasistatic cracks in brittle, isotropic, linearly elastic materials in two dimensions. The evolving crack satisfies familiar principles-Griffith's criterion, local symmetry, and irreversibility. A novel feature of the formulation is that in addition to the crack path, the loading is also treated as an unknown. Specifically, a scaling factor for prescribed Dirichlet and Neumann boundary conditions is computed as part of the solution to yield an always-propagating and ostensibly kink-free crack and a continuous loading history beyond the initial step. A dimensionless statement of the problem depends only on the Poisson's ratio of a homogeneous material, and is in particular, independent of its Young's modulus and fracture toughness.Numerical resolution of the formulated problem relies on two new ideas. The first is an algorithm to compute triangulations conforming to cracked domains by locally deforming a given background mesh in the vicinity of evolving cracks. The algorithm is robust under mild assumptions on the sizes and angles of triangles near the crack and its smoothness. Hence, a subset, if not the entire family, of cracked domains realized during the course of a simulation can be discretized with the same background mesh; we term the latter a universal mesh for such a family of domains. Universal meshes facilitate adopting a discrete representation for the crack (as splines in our examples), preclude the need for local splitting/retriangulation operations, liberate the crack from following directions prescribed by the mesh, and enable the adoption of standard FEMs to compute the elastic fields in the cracked solid. Second, we employ a method specifically designed to approximate the stress intensity factors for curvilinear cracks. We examine the performance of the resulting numerical method with detailed examples, including comparisons with an exact solution of a crack propagating along a circular arc (which we construct) and comparisons with experimental fracture paths. In all cases, we observe convergence of computed paths, their derivatives, and loading histories with refinement of the universal mesh.

Computing stress intensity factors for curvilinear cracks

“…When the crack faces are loaded, a boundary integral over the faces has to be carried out irrespective of the auxiliary fields. For this particular problem it may be appealing to choose a pairing with β DFC such as (35) and (36). Doing so reduces the numerical complexity of the interaction integral as λ greatly simplifies (and vanishes identically in the absence of body forces).…”

“…The analytical solution was derived in [34], and a recapitulation of the solution can be found in [35]. The resulting stress intensity factors for uniform far field tension loading are (see, e.g., [36])…”

“…This proposition is not directly applicable to the discontinuous Galerkin method in §5.1, because in this case it is also necessary to account for the use of an approximation space that does not conform to H 1 . Finally, the two functionals G(u) = I[∇u, β DFC , δγ] in (35) and (36) are not continuous in H 1 (B C ; R 2 ), because of the evaluations of ∇u on the crack faces, so we cannot directly apply the above result.…”

An embedded fracture modeling framework for simulation of hydraulic fracturing and shear stimulation

“…Previous work has been done in the area of developing models that apply adaptive grid refinement as fractures propagate [12,33]. This requires a significant level of computational overhead, and the numerical results have been observed to be grid-dependent.…”

Modeling the connectivity and intersection of hydraulically loaded cracks with in situ fractures in rock

“…However, given that crack surfaces do not run through the element, no crack‐enrichment functions are needed in our approach. The idea of aligning element boundaries with crack surfaces is also used in formulating universal meshes . However, the latter method is primary used for quasi‐static fracture modeling, and the formulation of mesh adaptivity is drastically different from that of the aSDG method.…”