A variational model for fracture and debonding of thin films under in-plane loadings

Abstract

We study fracture and delamination of a thin stiff film bonded on a rigid substrate through a thin compliant bonding layer. Starting from the three-dimensional system, upon a scaling hypothesis, we provide an asymptotic analysis of the three-dimensional variational fracture problem as the thickness goes to zero, using Γ-convergence. We deduce a two-dimensional limit model consisting of a brittle membrane on a brittle elastic foundation. The fracture sets are naturally discriminated between transverse cracks in the film (curves in 2D) and debonded surfaces (two-dimensional planar regions). We introduce the vectorial plane-elasticity case, applying the rigorous results established for scalar displacement fields, in order to numerically investigate the typical cracking scenarios encountered in applications. To this end, we formulate a reduced-dimension, rate-independent, irreversible evolution law for transverse fracture and debonding of thin film systems. Finally, we propose a numerical implementation based on a regularized formulation of the fracture problem via a gradient damage functional. We provide an illustration of the capabilities of the formulation exploring complex crack patterns in one and two dimensions, showing a qualitative comparison with geometrically involved real life examples.

A non-intrusive global/local approach applied to phase-field modeling of brittle fracture

“…and we also substitute u G for u C in the third and fourth integrals in (25). This yields the desired global/local representation (approximation) of the reference energy functional E in (1), namely,…”

“…We mention the papers by Amor et al [7], Miehe et al [8,9], Kuhn and Müller [10], Pham et al [11], Borden et al [12], Mesgarnejad et al [13], Kuhn et al [14], Ambati et al [15], Wu et al [16], where various formulations are developed and validated. Recently, the framework has been also extended to ductile (elasto-plastic) fracture [17][18][19][20][21][22], pressurized fracture in elastic and porous media [23,24], fracture in films [25] and shells [26][27][28], and multi-field fracture [29][30][31][32][33][34][35][36]. Non-intrusive global/local approaches have also been applied to a quite large number of situations: the computation of the propagation of cracks in a sound model using the extended finite element method (XFEM) [37], the computation of assembly of plates introducing realistic non-linear 3D modeling of connectors [38], the extension to non-linear domain decomposition methods [39] and to explicit dynamics [40,41] with an application to the prediction of delamination under impact using Abaqus [42].…”

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

“…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.…”

Numerical investigation of dynamic brittle fracture via gradient damage models

“…We observe that in both cases the jump takes place at x ≈ 0.9 somewhat prior to the fracture toughness discontinuity x = 1. We suspect that this is due to the damage regularization of cracks with a half-band D = 2 = 0.1 using the constitutive laws of (4). If this effect is ignored, the crack length after the jump is recorded in Table 3 for each case.…”

“…The gradient damage model can thus be regarded as an elliptic regularization of the previous sharp-interface variational fracture model and the internal length serves as a purely numerical parameter which should be as small as possible. This is the interpretation undertaken among others by [3][4][5] where the gradient damage model is applied to drying, thin films debonding, and other fracture mechanics problems.…”