Crack growth for the double slip plane crack model-I. The R-curve

Abstract

The double slip plane (DSP) crack model of Weertman, Lin and Thomson has been used to obtain crack growth equations for the mode II or III crack under a monotonically increasing stress (the R-curve) in this paper. [In a companion paper the growth under cyclic stress (the Paris fatigue crack growth equation) is determined.] The success of the analysis depends upon the fact that the DSP crack closely approximates a Bilby-Cottrell-Swinden (BCS) crack when the slip zone is large compared with the slip plane to crack plane spacing. Consequently the dislocation distribution on the slip planes approximates the BCS crack one ahead of the crack tip. Behind the crack tip a result fortuitous for the analysis is found that the gradient of the dislocation density on a slip plane is proportional to the shear stress on the slip plane. The results obtained are: if the friction stress on a slip plane is constant the crack never propagates catastrophically. Instead crack extension occurs which is proportional to K 2 where rK is the stress intensity factor. Were the surface energy of a solid to be suddenly reduced, as it might be by the sudden introduction of an active environment, the distance the tip of a stressed stationary crack jumps is proportional to K. The distance jumped, for a large drop in surface energy, is smaller than the crack advance that occurs if the active environment were always present and the crack is monotonically loaded to the same value of K. If work hardening of the friction stress takes place stable crack growth takes place up to a critical K c value. The R-curve equation is given by an integral which is simple in form but requires a numerical integration or series expansion. The critical K c value is proportional to the critical K cb stress intensity factor of a Griffith crack raised to the power ( m + 1 )/2 m where m is the power exponent of the simulated plastic stress-strain curve. We believe that this paper demonstrates the double slip plane crack model is the first crack model since the Griffith crack model and its variants that can give an explicit fracture equation starting from first principles.