Bridged cracks of mode III with surface elasticity

• Incorporated the effects of surface elasticity into a mode III bridged crack with variable bridging stiffness.
• Stresses exhibit a weak logarithmic singularity at the crack tips.
• Both surface elasticity and crack bridging can toughen the solid.
• When crack is much shorter than intrinsic material lengths, toughening is due to surface elasticity.
• When the crack is much longer than two material lengths, toughening is due to crack bridging.

We examine the contribution of surface elasticity to a fully or partially bridged finite mode III crack. The surface effect is incorporated using the continuum-based surface/interface model of Gurtin and Murdoch. We assume that the bridging force is proportional to the crack opening displacement while the bridging stiffness is allowed to vary arbitrarily along the crack. By considering a distribution of screw dislocations on the crack, the problem is reduced to a first-order Cauchy singular integro-differential equation. After the expansion of both the unknown dislocation density and the known variable bridging stiffness into Chebyshev polynomials, the integro-differential equation is solved numerically using a collocation technique. A complete solution valid throughout the solid (including at the crack tips) is obtained. We identify two intrinsic material lengths: one associated with the surface elasticity and the other with the crack bridging. We find that the surface elasticity plays a predominant role when the crack is significantly shorter than the two intrinsic lengths. At the other extreme, the crack bridging plays a predominant role when the crack is considerably longer than the two intrinsic lengths. Numerical results are presented to demonstrate the significance of the solution. Finally, a consequence of our analysis is that the solution of the problem involving the Dugdale model of a mode III crack with surface elasticity is also readily available. Our results in this respect show that the determination of the size of the plastic zone is independent of the parameter characterizing the surface elasticity.