A phase-field description of dynamic brittle fracture

In this contribution we address the issue of efficient finite element treatment for phase-field modeling of brittle fracture. We start by providing an overview of the existing quasi-static and dynamic phase-field fracture formulations from the physics and the mechanics communities. Within the formulations stemming from Griffith's theory, we focus on quasi-static models featuring a tension-compression split, which prevent cracking in compression and interpenetration of the crack faces upon closure, and on the staggered algorithmic implementation due to its proved robustness. In this paper, we establish an appropriate stopping criterion for the staggered scheme. Moreover, we propose and test the so-called hybrid formulation, which leads within a staggered implementation to an incrementally linear problem. This enables a significant reduction of computational cost--about one order of magnitude--with respect to the available (non-linear) models. The conceptual and structural similarities of the hybrid formulation to gradient-enhanced continuum damage mechanics are outlined as well. Several benchmark problems are solved, including one with own experimental verification.

A review on phase-field models of brittle fracture and a new fast hybrid formulation

Phase-field modeling of brittle fracture in elastic solids is a well-established framework that overcomes the limitations of the classical Griffith theory in the prediction of crack nucleation and in the identification of complicated crack paths including branching and merging. We propose a novel phase-field model for ductile fracture of elasto-plastic solids in the quasi-static kinematically linear regime. The formulation is shown to capture the entire range of behavior of a ductile material exhibiting $$J_2$$J2-plasticity, encompassing plasticization, crack initiation, propagation and failure. Several examples demonstrate the ability of the model to reproduce some important phenomenological features of ductile fracture as reported in the experimental literature.

http://congress.cimne.com/coupled2015/admin/files/fileabstract/a482.pdf

Phase-field modeling of ductile fracture

In its numerical implementation, the variational approach to brittle fracture approximates the crack evolution in an elastic solid through the use of gradient damage models. In this article, we first formulate the quasi-static evolution problem for a general class of such damage models. Then, we introduce a stability criterion in terms of the positivity of the second derivative of the total energy under the unilateral constraint induced by the irreversibility of damage. These concepts are applied in the particular setting of a one-dimensional traction test. We construct homogeneous as well as localized damage solutions in a closed form and illustrate the concepts of loss of stability, of scale effects, of damage localization, and of structural failure. Considering several specific constitutive models, stress

https://hal-polytechnique.archives-ouvertes.fr/hal-00549530/document

Gradient Damage Models and Their Use to Approximate Brittle Fracture

Being one of the most promising candidates for the modeling of localized failure in solids, so far the phase-field method has been applied only to brittle fracture with very few exceptions. In this work, a unified phase-field theory for the mechanics of damage and quasi-brittle failure is proposed within the framework of thermodynamics. Specifically, the crack phase-field and its gradient are introduced to regularize the sharp crack topology in a purely geometric context. The energy dissipation functional due to crack evolution and the stored energy functional of the bulk are characterized by a crack geometric function of polynomial type and an energetic degradation function of rational type, respectively. Standard arguments of thermodynamics then yield the macroscopic balance equation coupled with an extra evolution law of gradient type for the crack phase-field, governed by the aforesaid constitutive functions. The classical phase-field models for brittle fracture are recovered as particular examples. More importantly, the constitutive functions optimal for quasi-brittle failure are determined such that the proposed phase-field theory converges to a cohesive zone model for a vanishing length scale. Those general softening laws frequently adopted for quasi-brittle failure, e.g., linear, exponential, hyperbolic and Cornelissen et al. (1986) ones, etc., can be reproduced or fit with high precision. Except for the internal length scale, all the other model parameters can be determined from standard material properties (i.e., Young’s modulus, failure strength, fracture energy and the target softening law). Some representative numerical examples are presented for the validation. It is found that both the internal length scale and the mesh size have little influences on the overall global responses, so long as the former can be well resolved by sufficiently fine mesh. In particular, for the benchmark tests of concrete the numerical results of load versus displacement curve and crack paths both agree well with the experimental data, showing validity of the proposed phase-field theory for the modeling of damage and quasi-brittle failure in solids.

A unified phase-field theory for the mechanics of damage and quasi-brittle failure

This paper presents a modified regularized formulation of the Ambrosio–Tortorelli type to introduce the crack non-interpenetration condition in the variational approach to fracture mechanics proposed by Francfort and Marigo [1998. Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46 (8), 1319–1342]. We focus on the linear elastic case where the contact condition appears as a local unilateral constraint on the displacement jump at the crack surfaces. The regularized model is obtained by splitting the strain energy in a spherical and a deviatoric parts and accounting for the sign of the local volume change. The numerical implementation is based on a standard finite element discretization and on the adaptation of an alternate minimization algorithm used in previous works. The new regularization avoids crack interpenetration and predicts asymmetric results in traction and in compression. Even though we do not exhibit any gamma-convergence proof toward the desired limit behavior, we illustrate through several numerical case studies the pertinence of the new model in comparison to other approaches.

Hanen Amor

Papers

Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments

Gradient Damage Models and Their Use to Approximate Brittle Fracture