Gradient Damage Models and Their Use to Approximate Brittle Fracture

By K. Pham, Hanen Amor, Jean-Jacques Marigo, C. Maurini. 2011

767 citations recorded by the source corpus.

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—displacement curves, stability diagrams, and energy dissipation provide identification criteria for the relevant material parameters, such as limit stress and internal length. Finally, the 1D analytical results are compared with the numerical solution of the evolution problem in a 2D setting.

Subfields: Materials Chemistry, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/00ddebbdf08badfca96e03ca786cafbb3158ae9d

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Materials Chemistry

In Mechanics of Materials

Citations in the corpus

Phase‐field modeling of ductile fracture at finite strains: A robust variational‐based numerical implementation of a gradient‐extended theory by micromorphic regularization (2017, methodology): “[30] apply gradient damage models to the regularized description of Griffith-type brittle fracture.”
A phase-field formulation for fracture in ductile materials: Finite deformation balance law derivation, plastic degradation, and stress triaxiality effects (2016, background): “[38] has noted the fact that the quadratic degradation function leads to a model with no linear elastic regime in the stress-strain response.”
A numerical assessment of phase-field models for brittle and cohesive fracture: Γ-Convergence and stress oscillations (2015, background): “Pham et al.(2011); Borden et al.(2012) andPham and Marigo (2013) also state that the length scale parameterℓ in the model byBourdin et al. (2000) can be interpreted as a material parameter since it influences the critical stress.”
Crack nucleation in variational phase-field models of brittle fracture (2018, unknown): “…of the energy E , i.e. there exists where (cid:107) · (cid:107) C × D We brieﬂy summarize the solution of the uniaxial tension of a homogeneous bar [77, 80], referring the reader to the recent review [67] for further details: As one increases the applied strain, the damage ﬁeld remains 0 and the…”
Toughness or strength? Regularization in phase-field fracture explained by the coupled criterion (2020, methodology): “Pham and Marigo (2010a,b) theorized that this parameter should be identified separately as it represents the material’s internal length (Freddi and Royer-Carfagni, 2010; Pham et al., 2011; Piero, 2013).”
A framework to model the fatigue behavior of brittle materials based on a variational phase-field approach (2018, methodology): “(5) The fracture energy density is written as [18, 31, 32]”
Phase Field Modeling of Fracture in Multi-Physics Problems. Part II. Coupled Brittle-to-Ductile Failure Criteria and Crack Propagation in Thermo-Elastic-Plastic Solids (2015, unknown): “[31] and Miehe [28].”
Phase-field modeling of anisotropic brittle fracture including several damage mechanisms (2018, unknown): “Standard phase-ﬁeld models for isotropic media are known to exhibit a solution in which damage localizes on a narrow band orthogonal to the loading direction when the bar is suﬃciently long [4].”
A phase-field formulation for dynamic cohesive fracture (2018, background): “ phase-eld approximiation from Bourdin et al. (2008), the damage eld enters the energy through a linear term. Similar formulations have been discussed, amongst others, in Fremond and Nedjar (1996), Pham et al. (2011), Miehe (2011) and Lorentz and Godard (2011); Lorentz et al. (2012), in the context ”