Coupling damage and plasticity for a phase-field regularisation of brittle, cohesive and ductile fracture: One-dimensional examples

Abstract

Plasticity and damage are two fundamental phenomena in nonlinear solid mechanics associated to the development of inelastic deformations and the reduction of the material stiffness. Alessi et al. [4] have recently shown, through a variational framework, that coupling a gradient-damage model with plasticity can lead to macroscopic behaviours assimilable to ductile and cohesive fracture. Here, we further expand this approach considering specific constitutive functions frequently used in phase-field models of brittle fracture. A numerical solution technique of the coupled elastodamage-plasticity problem, based on an alternate minimisation algorithm, is proposed and tested against semi-analytical results. Considering a one-dimensional traction test, we illustrate the properties of four different regimes obtained by a suitable tuning of few key constitutive parameters. Namely, depending on the relative yield stresses and softening behaviours of the plasticity and the damage criteria, we obtain macroscopic responses assimilable to (i) brittle fracturè a la Griffith, (ii) cohesive fractures of the Barenblatt or Dugdale type, and (iii) a sort of cohesive fracture including a depinning energy contribution. The comparisons between numerical and analytical results prove the accuracy of the proposed numerical approaches in the considered quasi-static time-discrete setting, but they also emphasise some subtle issues occurring during time-discontinuous evolutions.

Phase-field modeling of fracture for quasi-brittle materials

“…The weak form of each of these equations is naturally obtained and can be discretized using the finite element method. It is noteworthy that the definition of a total energy quantity is not always straightforward: it strongly depends on the form of the dissipated work D(α) [34].…”

“…This section presents a straightforward implementation of the numerical solution of Eqs. (33) and (34).…”

Phase field modeling of chemomechanical fracture of intercalation electrodes: Role of charging rate and dimensionality

“…These models have been validated by theoretical analyses [24] and comparisons of predicted and observed crack paths in non-trivial geometries [25]. They have been used to reproduce complex experimental observations in brittle fracture including thin-film fracture [26], thermal fracture [20], mixed mode fracture [27], dynamic fracture [27,28,29], fracture in colloidal systems [30], ductile fracture [31,32,33,34], and fatigue fracture [35,36,37]. Given their potential over the past few years, researchers have extended the use of these models to chemo-mechanical fracture in battery particles [38,39,40,41].…”

Damage model for plastic materials at finite strains

“…Under small strains, the first local energy‐based model for coupled damage‐plasticity was introduced by Ju . Later on, Alessi et al . developed its non‐local extension by including gradients of a damage variable into the stored energy, in the spirit of variational models for regularized brittle fracture and gradient damage developed in the mathematical, e.g., [] and engineering, e.g., [] literature.…”

“…Under small strains, the first local energy-based model for coupled damage-plasticity was introduced by JU. [29] Later on, ALESSI et al [2][3][4] developed its non-local extension by including gradients of a damage variable into the stored energy, in the spirit of variational models for regularized brittle fracture and gradient damage developed in the mathematical, e.g., [9,10,24,40] and engineering, e.g., [22,33,44,45] literature. Such enrichment introduces an additional length scale into the energy functional to characterize the regions to which damage localizes; see also [1] for an overview and comparison of available formulations.…”