Analysis of Plane-Strain Crack Problems (Mode-I & Mode-II) in the Presence of Surface Elasticity

Abstract

We consider plane deformations of a linearly elastic solid in the case where either a mode-I or mode-II crack is present but, perhaps more significantly, when surface effects are included in the mechanics of the crack faces. The surface effects lead to a more accurate description of deformation and are incorporated using a version of the continuum based surface/interface model of Gurtin and Murdoch. We obtain a semi-analytic solution valid throughout the entire domain of interest (including at the crack tips) via two series of coupled Cauchy singular integro-differential equations which are solved numerically using an adapted collocation technique. It is shown that, among various other interesting phenomena, when the solid incorporates a traction-free crack face and is subjected to uniform far-field stresses (tension and in-plane shear), the surface effects result in the elastic response and corresponding stress fields being size-dependent. In particular, we note that, in contrast to classical linear elastic fracture mechanics, our model allows for finite stresses at the (sharp) crack tip.

Interface cracks with surface elasticity in anisotropic bimaterials

“…By using the Green's function method, the original boundary value problem is finally reduced to a system of Cauchy singular integro-differential equations. When the bulk materials are isotropic, our obtained singular integro-differential equations reduce to those by Kim et al (2010Kim et al ( , 2011a. Thus the correctness of the proposed Green's function method is verified.…”

“…If a homogeneous plane is elastically isotropic with shear modulus µ and Poisson's ratio ν and the crack surface is also isotropic with surface Lame constants s λ and s µ , we have (Ting, 1996;Kim et al, 2010Kim et al, , 2011a:…”

“…The surface/interface theory of Gurtin and Murdoch has been incorporated into the analysis of the anti-plane shear and plane strain problems associated with a crack in isotropic homogeneous media (Kim et al, 2010(Kim et al, , 2011aAntipov and Schiavone, 2011) and an interfacial crack in isotropic bimaterials (Kim et al, 2011b,c). The complex variable method and the Fourier transform were used in their analysis.…”

Existence of solutions for the anti-plane stress for a new class of “strain-limiting” elastic bodies

“…Recently, there have been several papers (see Kim, Schiavone and Ru [21,23,24], Kim, Ru and Schiavone [22], Antipov and Schiavone [1]) that study fracture in brittle materials appealing to an approach by Murdoch and Gurtin [19] that introduces surface elasticity at the surface of cracks. These studies modify the classical linearized elasticity theory by allowing for elastic surface energy.…”

Untitled

“…Ru's solution indicates that, when the asymptotic behavior of the harmonic materials satisfies the constitutive restriction proposed by Knowles and Sternberg [1975], the oscillatory singularity again disappears. Most recently, various authors (see, for example, [Kim et al 2010b;2011a;2011b;2011c;Antipov and Schiavone 2011;Wang 2015]) have incorporated the continuum-based surface/interface theory of Gurtin and Murdoch [1975;1978;Gurtin et al 1998] into the fracture analysis of linearly elastic solids. It was shown that the incorporation of the Gurtin-Murdoch surface model can suppress the classical strong square-root stress/strain singularity at the crack tip predicted in linear elastic fracture mechanics (LEFM) to the weaker logarithmic singularity [Walton 2012;Kim et al 2013].…”