Saint-Venant torsion of a circular bar with a bridged radial crack incorporating surface elasticity

Abstract

We examine the contribution of crack bridging and surface elasticity to the Saint-Venant torsion problem of a circular cylinder containing a radial crack. The surface effect on the crack faces is incorporated by using the continuum-based surface/interface model of Gurtin and Murdoch. The bridging force is assumed proportional to the crack opening displacement, while the bridging stiffness is allowed to vary arbitrarily along the crack. Both an internal crack and an edge crack are studied. By using the Green’s function method, the boundary value problem is reduced to a first-order Cauchy singular integro-differential equation. After the utilization of the Gauss–Chebyshev integration formula and the expansion of both the unknown dislocation density and the known variable bridging stiffness into Chebyshev polynomials, the singular integro-differential equation is solved numerically by using the collocation method. The stresses exhibit a weak logarithmic singularity rather than a strong square root singularity at the crack tips due to the surface effect. The strengths of the logarithmic singularity at the crack tips, the reduction in torsional rigidity and the jump in warping function across the crack faces are calculated. Our results also show that both surface elasticity and crack bridging will stiffen the cracked cylinder and that the stiffening effect of crack bridging is remarkable in the case when the crack is very long or is extremely close to the lateral surface of the cylinder.