Perturbation analysis for an imperfect interface crack problem using weight function techniques

Abstract

Vellender, A., Mishuris, G., Piccolroaz, A. (2013). Perturbation analysis for an imperfect interface crack problem using weight function techniques. International Journal of Solids and Structures, 50 (24), 4098-4107We analyse a problem of anti-plane shear in a bi-material plane containing a semi-infinite crack situated on a soft imperfect interface. The plane also contains a small thin inclusion (for instance an ellipse with high eccentricity) whose influence on the propagation of the main crack we investigate. An important element of our approach is the derivation of a new weight function (a special solution to a homogeneous boundary value problem) in the imperfect interface setting. The weight function is derived using Fourier transform and Wiener-Hopf techniques and allows us to obtain an expression for an important constant of (which may be used in a fracture criterion) that describes the leading order of tractions near the crack sigma((0))(0) for the unperturbed problem. We present computations that demonstrate how sigma((0))(0) varies depending on the extent of interface imperfection and contrast in material stiffness. We then perform perturbation analysis to derive an expression for the change in the leading order of tractions near the tip of the main crack induced by the presence of the small defect, whose sign can be interpreted as the inclusion's presence having an amplifying or shielding effect on the propagation of the main crack. (C) 2013 Elsevier Ltd. All rights reserved.Peer reviewe

Boundary integral formulation for interfacial cracks in thermodiffusive bimaterials

“…The reciprocity identity (3.18) then becomes 22) where the symbol denotes the convolution with respect to both x 1 and x 3 , which is defined as follows [36]: ] and U are the symmetrical and skew-symmetrical weight functions matrices defined and derived in closed form in [32], whereas the term Σ 2 stands for the traction along the x 1 -axis corresponding to the singular auxiliary displacements U. The integral equation (3.22) is the generalization to thermoelastic diffusive media of the Betti identity derived in [20,21], and it relates the physical solution u, σ 2 , θ, χ to the auxiliary singular solution U, Σ 2 .…”

“…The general approach recently proposed in Piccolroaz & Mishuris [20], Morini et al [21], Vellender et al [22] and Mishuris et al [23] for interfacial crack problems in isotropic and anisotropic elastic bimaterials, based on Betti's reciprocal theorem and weight functions theory, is extended in order to study fracture processes in presence of thermodiffusion. The volume integral terms present in the reciprocity identity, associated with the temperature and mass concentration effects [24], are converted into surface integrals through an exact transformation based on the notion of Lamé elastic potentials [25] while assuming that the temperature and mass concentration fields are harmonic in the domain.…”

Boundary integral formulation for cracks at imperfect interfaces

“…In this paper we adopt the approach based on weight functions in the imperfect interface setting found by Vellender et al [33]. In particular, the weight functions U and Σ must satisfy the following transmission conditions…”

“…They can be computed from the general relationships, which hold for the symmetric and skew-symmetric weight functions (Vellender et al, 2013)…”

“…Recently, weight functions have been derived for strip and plane geometries containing cracks and imperfect interfaces [32,33]; these weight functions allow for the evaluation of constants that may be used in fracture criteria and have been used to obtain other information about a structure's behaviour [34].…”