The cohesive element approach to dynamic fragmentation: the question of energy convergence

Abstract

SUMMARYThe cohesive element approach is getting increasingly popular for simulations in which a large amount of cracking occurs. Naturally, a robust representation of fragmentation mechanics is contingent to an accurate description of dissipative mechanisms in form of cracking and branching. A number of cohesive law models have been proposed over the years and these can be divided into two categories: cohesive laws that are initially rigid and cohesive laws that have an initial elastic slope. This paper focuses on the initially rigid cohesive law, which is shown to successfully capture crack branching mechanisms in simulations. The paper addresses the issue of energy convergence of the finite-element solution for high-loading rate fragmentation problems, within the context of small strain linear elasticity. These results are obtained in an idealized one-dimensional setting, and they provide new insight for determining proper cohesive zone spacing as function of loading rate. The findings provide a useful roadmap for choosing mesh sizes and mesh size distributions in two and three-dimensional fragmentation problems. Remarkably, introducing a slight degree of mesh randomness is shown to improve by up to two orders of magnitude the convergence of the fragmentation problem.

Elastic damage to crack transition in a coupled non-local implicit discontinuous Galerkin/extrinsic cohesive law framework

“…Concerning the properties related to the crack propagation, as the method corresponds to an extrinsic CZM, for which the crack path and the dissipated energy converge with the mesh size for unstructured meshes [40,41],the hybrid scheme also inherits these properties as it has recently been orally discussed in [60]. We will demonstrate in the numerical application that the results are insensitive with the mesh size.…”

“…(36)(37)(38)(39)(40)(41). Peerlings [5] has derived a semi-analytical solution in the particular case of a damage law equivalent to perfect plasticity, i.e.…”

“…The value of D hom in the system of Eqs. (39)(40)(41)(42) is thus a known value during the strain softening stage. Note that the damage in the damage zone keeps increasing during the strain softening stage.…”

“…We assume a uniform monotonic increasing strain ε, which indicates a monotonic increasingẽ = e = ε before strain localization in the 1D bar occurs. The maximum diffuse damage D hom in equation (41) corresponds to the damage value reached at the peak stress -strain softening onset-in Fig. 4(a), and can be computed by solving dσ dε = 0, leading to max(ε hom ) = 0.0238 and D hom = 0.5093 .…”

“…However the DG approach suffers from a higher number of degrees of freedom, which is strengthened by the fine mesh required by the use of CZM. Indeed as the crack has to follow the element boundary, a fine mesh is required to ensure convergence of the crack path and dissipation energy, at least for unstructured meshes [40,41]. However a fine mesh is always required to capture complex crack propagation, whatever the discretization technique used, and in the particular case of the DG method this drawback is mitigated by the high scalability of the method.…”