Symmetric and skew-symmetric weight functions in 2D perturbation models for semi-infinite interfacial cracks

Abstract

Piccolroaz, A; Mishuris, G; Movchan AB. Symmetric and skew-symmetric weight functions in 2D perturbation models for semi-infinite interfacial cracks. Journal of the Mechanics and Physics of Solids. 2009, 57(9), 1657-1682In this paper we address the vector problem of a 2D half-plane interfacial crack loaded by a general asymmetric distribution of forces acting on its faces. It is shown that the general integral formula for the evaluation of stress intensity factors, as well as high-order terms, requires both symmetric and skew-symmetric weight function matrices. The symmetric weight function matrix is obtained via the solution of a Wiener-Hopf functional equation, whereas the derivation of the skew-symmetric weight function matrix requires the construction of the corresponding full field singular solution. The weight function matrices are then used in the perturbation analysis of a crack advancing quasi-statically along the interface between two dissimilar media. A general and rigorous asymptotic procedure is developed to compute the perturbations of stress intensity factors as well as high-order terms.Peer reviewe

Stroh formalism in analysis of skew-symmetric and symmetric weight functions for interfacial cracks

“…For cracks in homogeneous elastic materials, in the two-dimensional setting the skew-symmetric loading does not contribute to stress intensity factors, whereas it becomes relevant and it must to be accounted for in three-dimensional solids (Bueckner, 1985;Meade & Keer, 1984). The situation is different when the crack is placed at the interface between two dissimilar elastic materials: even for two-dimensional problems the skew-symmetric loads generate a non-zero contribution to stress intensity factors (Bercial-Velez et al, 2005;Lazarus & Leblond, 1998;Piccolroaz et al, 2009). In particular, for Mode III interfacial cracks the stress components do not oscillate, but a non-vanishing skew-symmetric component of the weight function still has to be accounted (Piccolroaz et al, 2009(Piccolroaz et al, , 2010.…”

“…The situation is different when the crack is placed at the interface between two dissimilar elastic materials: even for two-dimensional problems the skew-symmetric loads generate a non-zero contribution to stress intensity factors (Bercial-Velez et al, 2005;Lazarus & Leblond, 1998;Piccolroaz et al, 2009). In particular, for Mode III interfacial cracks the stress components do not oscillate, but a non-vanishing skew-symmetric component of the weight function still has to be accounted (Piccolroaz et al, 2009(Piccolroaz et al, , 2010. In the case of isotropic media, weight functions for semi-infinite cracks can be defined as singular non-trivial solutions of the homogeneous boundary problem with zero tractions on the crack faces but unbounded elastic energy (Willis & Movchan, 1995).…”

“…In the case of isotropic media, weight functions for semi-infinite cracks can be defined as singular non-trivial solutions of the homogeneous boundary problem with zero tractions on the crack faces but unbounded elastic energy (Willis & Movchan, 1995). For interfacial cracks between dissimilar isotropic elastic media, the weight functions are well discussed in literature (Lazarus & Leblond, 1998;Piccolroaz et al, 2009Piccolroaz et al, , 2010. The problem is generally reduced to a functional equation of Wiener-Hopf type, and its solution gives the symmetric weight function matrix (Antipov, 1999), while the skew-symmetric component is obtained by the construction of the corresponding full-field singular solution of the elasticity boundary value problem discussed in Piccolroaz et al (2009).…”

“…For interfacial cracks between dissimilar isotropic elastic media, the weight functions are well discussed in literature (Lazarus & Leblond, 1998;Piccolroaz et al, 2009Piccolroaz et al, , 2010. The problem is generally reduced to a functional equation of Wiener-Hopf type, and its solution gives the symmetric weight function matrix (Antipov, 1999), while the skew-symmetric component is obtained by the construction of the corresponding full-field singular solution of the elasticity boundary value problem discussed in Piccolroaz et al (2009). For interfacial cracks between anisotropic dissimilar elastic media, although weight functions have been derived by Gao (1991Gao ( , 1992 and Ma & Chen (2004), the results on skew-symmetric weight functions are not readily available.…”

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 .…”