Radial Dependence of Near-Tip Continuum Fields for Plane Strain Tensile Crack Growth in Elastic-Ideally Plastic Solids

This paper is an extension of work by Drugan et al. (1982) who derive the stress and deformation fields at the tip of a plane strain tensile crack that grows quasi-statically, under general nonsteady conditions, in an elastic-ideally plastic solid. Here I perform a higher-order analysis of the near-tip fields for this growing crack problem. My principal objectives are to determine the radial variation of the near-tip stress field and elucidate the structure of the deformation fields in the 90-deg sector ahead of the growing crack; this information was not provided by the lowest-order solution of Drugan et al. (1982). I also derive a crucial asymptotic expression for the normal radial component of the deformation rate tensor in a moving “centered fan” plastic sector, which was given without complete proof by Rice (1982). The analysis presented herein differs from typical perturbation analyses in that I am able to derive the higher-order structure of the continuum fields rather than having to assume expansions for them. Among the results, normal polar components of deviatoric stress are shown to vary as (ln r)⁻¹, while the in-plane polar shear component varies as (ln r)⁻², for small r > 0 in moving “centered fan” plastic sectors, r denoting distance from the (moving) crack tip. Further, in-plane strains proportional to ln|ln r| as r → 0 appear not to be precluded in the 90-deg sector ahead of the growing crack.