Small-Scale Yielding at the Tip of a Through-Crack in a Shell

Deformation theory is used to model plastic deformation at the tip of a through-crack in a thin shell. In the vicinity of the crack the shell is subjected to both stretching and bending, but stretching is assumed to dominate. Thus the stresses are tensile, but with a nonuniform distribution through the thickness, which depends on the material properties as well as on the geometry. The nonlinear near-tip fields (which are singular) have been analyzed asymptotically. Cracks in shallow shells and spherical shells have been investigated in some detail. It is shown that the angular variations are the same as for generalized plane-stress plate problems. Assuming small-scale yielding a path-independent integral, which is valid in a region close to the crack edge, is used to connect the nonlinear near-tip fields with the corresponding singular parts of the linear fields. It is shown that the nonlinear behavior significantly affects the through-the-thickness variations of the near-tip fields. The singular parts of the membrane stresses tend to become more uniform through the thickness of the shell with a flatter strain-hardening curve.