Boundary integral formulation for cracks at imperfect interfaces

Abstract

We consider an infinite bi-material plane containing a semi-infinite crack situated on a soft imperfect interface. The crack is loaded by a general asymmetrical system of forces distributed along the crack faces. On the basis of the weight function approach and the fundamental reciprocal identity, we derive the corresponding boundary integral formulation, relating physical quantities. The boundary integral equations derived in this paper in the imperfect interface setting show a weak singularity, in contrast to the perfect interface case, where the kernel is of the Cauchy type. We further present three alternative variants of the boundary integral equations which offer computationally favourable alternatives for certain sets of parameters.

Boundary integral formulation for interfacial cracks in thermodiffusive bimaterials

“…Consequently, the crack process is favoured and the value of K I becomes larger as does the crack opening (figure 3b). Observing both figure 6a,b, it can be noted that the stress intensity factor increases as the ratio γ 23) and the traction ahead of the tip (5.20) becomes (b) Symmetrically distributed temperature profile at the interface…”

“…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.…”

Integral identities for fracture along imperfectly joined anisotropic ceramic bimaterials

“…They have been used in the analysis of crack problems in complex domains containing an arbitrary number of wedges and layers separated by imperfect interfaces (Mishuris, 1997a,b); the resulting singular integral equations with fixed point singularities have been analysed by Duduchava (1979), based on the theory of linear singular operators (Gohberg and Krein, 1960). More recently, singular integral equations have been applied to problems involving interfacial cracks in both isotropic (Piccolroaz and Mishuris, 2013;Mishuris et al, 2014) and anisotropic bimaterials (Yu and Suo, 2000;Morini et al, 2013a) and also in thermodiffusive bimaterials Morini and Piccolroaz (2015). This paper extends the singular integral equation approach to fracture in an anisotropic bimaterial containing an imperfect interface.…”

“…The problem considered here is the anisotropic equivalent of that seen in Mishuris et al (2014), which considered solely isotropic bimaterials. Besides this, perhaps the key novel feature in the present manuscript from a methodology viewpoint, is that known weight functions derived for the perfect interface problem are used in the derivation of singular integral equations for the soft imperfect interface case.…”

“…Besides this, perhaps the key novel feature in the present manuscript from a methodology viewpoint, is that known weight functions derived for the perfect interface problem are used in the derivation of singular integral equations for the soft imperfect interface case. This differs from previous approaches; for instance Mishuris et al (2014) used specially-derived weight functions that took into account the local crack-tip behaviour brought about by the presence of imperfect interface transmission conditions, whereas the approach employed here uses existing perfect interface weight functions, which have fundamentally different behaviour near the crack tip to the physical solution in the imperfect interface problem. The derived identities have potential applications in the modelling of piezoceramics, solid oxide fuel cells and other ceramic devices.…”