Dislocation-based approximate solution of the plastic zone of the mode II crack in an elastic perfectly plastic solid

Abstract
A dislocation-based (approximate) solution is found for the stress-strain field of the plastic zone (in small-scale yielding) of the stationary mode II crack in an elastic perfectly plastic and incompressible solid. General dislocation equations applicable for plane strain elastic-plastic conditions are presented and used to solve the problem. These equations have broader application than to the particular problem of the paper. The solution is obtained with use of dislocation crack tip shielding and the dislocation crack extension force. Derived dislocation boundary conditions which play an important role in the analysis areBt= 0 at an elastic–plastic boundary and, at elastic–plastic, plastic–plastic and crack plane boundaries, the jump condition [(1 –v)/2G) {∂σtt/∂xn}jump= {t· B}jump+ ∂Bn/∂xt, whereGis the shear modulus, σttis the non-traction stress,vis Poisson’s ratio.Bis the (area) dislocation density vector.Bis the surface dislocation density vector andtandnare the tangential and normal directions to a boundary. The strain compatibility equation is [G/(1 –v)] (∇ xB)z= ∇2½(σnn+ σtt). The near tip strain and stress contours of fan sectors are given by the equationr=rch(θ), wherercis a constant and the azimuthal functionh(θ)is given by the equationh'"+ 9h'= (p0– 2θ) (h" + h), wherep0is a constant and a prime denotes ∂/∂θ. The (approximate) elastic region stress field solution is presented in the companion paper to this one. A mode I crack solution, similar in its structure to the mode II crack solution, also is presented in the paper. This latter solution is shown in the companion paper to be flawed.

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