Analysis of a mode-III crack in the presence of surface elasticity and a prescribed non-uniform surface traction

Abstract

We consider an elastic solid incorporating a mode-III crack in which the crack faces incorporate the effects of surface elasticity and are further subjected to prescribed non-uniform surface tractions. The surface elasticity is modelled using the continuum-based model of Gurtin and Murdoch. Using complex variable techniques, the corresponding problem is reduced to the solution of a first order Cauchy singular integro-differential equation which, in turn, leads to the complete solution of the aforementioned crack problem valid everywhere in the domain of interest (including at the crack tip). Finally, we note that, as a particular case of our analysis, the classical decomposition of a mode-III crack problem in linear elasticity continues to hold even in the presence of surface elasticity. (2000). 45E05 · 74B99. Mathematics Subject Classification

Displacement field in an elastic solid with mode-III crack and first-order surface effects

“…This result is in agreement with the recent papers [Kim et al 2013;Walton 2012] where it was noted that by imposing the further condition θ ≡ 0 at the crack tips (to achieve finite stress [Kim et al 2010a; 2010b]) we effectively overdetermine the problem so that, contrary to the results reported in these two references, there can be no solutions of (2.2)-(2.4) with finite stress at the crack tips. In fact, this is clear from the DISPLACEMENT FIELD IN AN ELASTIC SOLID WITH CRACK AND SURFACE EFFECTS 789 uniqueness theorem proved above: once the unique displacement field is determined in the body one cannot then arbitrarily assign values to the derivatives of the displacement field at the crack tips.…”

“…In the case of antiplane deformations of a linearly elastic and homogeneous isotropic solid containing a sharp crack, Kim et al [2010a; have incorporated surface effects on the faces of the crack using the Gurtin-Murdoch surface elasticity model [Gurtin and Murdoch 1975;1978;Ru 2010]. The resulting mathematical model gave rise to a nonstandard boundary-value problem for the (harmonic) antiplane displacement.…”

“…In this paper we undertake a rigorous formulation and analysis of the displacement boundary-value problem first mentioned in [Kim et al 2010a;. We establish a uniqueness result for the displacement field and use complex variable methods to reduce the problem to a series of integral equations which are solved numerically using an efficient and stable, yet straightforward numerical method.…”

A crack with surface elasticity in finite plane elastostatics

“…Most recently, various authors (see, for example, [Kim et al 2010b;2010a;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].…”

Plane-Strain Fracture with Curvature-Dependent Surface Tension: Mixed-Mode Loading

“…A number of recent papers have adopted the Gurtin-Murdoch surface elasticity paradigm for modeling the constitutive behavior of the surface stress tensor [12,8,10,9,11]. However, it has been shown (first in [15] and subsequently corroborated in [7]) that while this choice of crack-surface stress tensor constitutive relation does remove the strong (square-root) cracktip stress/strain singularity, it is replaced by a weak, logarithmic one.…”