In Part I we found that although the near tip fields of cracks on bimaterial interfaces do not have a separable form of the HRR type, they appear to be nearly separable in an annular zone within the plastic zone. Furthermore, the fields bear strong similarities to mixed mode HRR fields for homogeneous medium. Based on our numerical results, we have been able to identify a clear mathematical structure. We found that the small-scale yielding crack tip fields are members of a family parameterized by a near tip phase angle ξ, and that the fields nearly scale with the value of the J-integral. In Part II, the original derivation of the mathematical structure of the small-scale yielding fields is elaborated upon. The issue of crack face contact is addressed and the phenomenology is described in terms of the phase parameter ξ. Crack tip plastic deformation results in an open crack for a range of ξ which is nearly symmetric about the state corresponding to pure remote tension. Plane-strain plastic zones and crack tip fields for the complete range of ξ are presented. Over distances comparable to the size of the dominant plastic zone, the stress levels that can be achieved are limited by the yield stress of the weaker (lower yield strength) material. On the other hand, the stresses well within the plastic zone are governed by the strain-hardening behavior of the more plastically compliant (lower strain-hardening) material. We observe that the extent of the annular zone where the fields are nearly separable (i.e., of the HRR form) is dependent on the remote load combinations and the material combination. When the tractions on the interface are predominantly tensile, there are no indications of crack face contact over any length scale of physical relevance. Instead, the crack tip opens smoothly and crack tip fields as well as the crack opening displacement are scaled by the J-integral. The paper concludes with a discussion on the range of load combinations which could be applied to two fracture test specimen geometries to obtain valid fracture toughness data.
Elastic-plastic analysis of combined mode I and III crack-tip fields under small-scale yielding conditions