Gradient Damage Models and Their Use to Approximate Brittle Fracture

Abstract

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 the present paper, 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. That allows us to construct homogeneous as well as localized damage solutions in a closed form and to illustrate the concepts of loss of stability, of scale effects, of damage localization, and of structural failure. Considering several specific constitutive models, stress-displacement curves, stability diagrams, and energy dissipation provide identification criteria for the relevant material parameters, such as limit stress and internal length. Finally, the one-dimensional analytical results are compared with the numerical solution of the evolution problem in a two dimensional setting.

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

“…The following analysis has strong analogies with the work in [55]. The reader is therefore invited to refer therein for additional details.…”

“…if fp ≤ 0 then no plastic increment p k = p k−1 ; else projection: erties have been widely discussed in previous works, when considering damage-gradient model as phase-field approximation of brittle fracture [55,49]. Here we recall and discuss some important facts, useful for later discussions.…”

“…The functions (21) are different from those considered in [4], where they were chosen to allow for explicit analytical results. The present model reduces to the ones commonly used in a phase-field regularisation of brittle fracture [55,47,24] when neglecting plasticity (σ p → ∞). The underlying damage model is that of a strongly brittle material [58], for which the energy necessary to break the material is finite, while the underlying plastic model is that of perfect plasticity.…”

“…The function H(σ, α, C(σ)) vanishes for α = 0 and α = α * (σ) with 0 < α * (σ) < 1, [55]. Moreover, the damage profile has a maximum inx such that α(x) = α * (σ) and is symmetric, hence α (x) = 0.…”

“…The variational approach of brittle fracture and the associated regularised formulations [22,21,23,38] are nowadays widely adopted in computational mechanics. The mechanical interpretation of the regularised models as damage models added a further insight on the prediction of crack nucleation in the quasi-brittle setting, giving a precise connection between the regularising length-scale and the maximum allowable stress before failure [55,24].…”