On the crack-tip stress singularity in the Mooney-Rivlin material

Large deformation fracture of hyperelastic materials is a straightforward extension of the linear elastic fracture under small deformation, which is, however, more complicated and abundant for the crack-tip field. In this paper, the apparent singularity of crack-tip stress of the Mooney-Rivlin material is studied through the large deformation analysis. Firstly, the theoretical results of Cauchy and first Piola-Kirchhoff (PK1) asymptotic stress fields at the mode-I crack-tip are summarized for typical hyperelastic materials such as the neo-Hookean solid, the generalized neo-Hookean material and the exponentially hardening material. Next, a procedure is suggested for numerically computing the apparent singularity. Variations of the apparent singularities in the Cauchy stress ${\sigma }_{22}$ and the PK1 stress $P_{22}$ are finally obtained with the far-field applied strain for the Mooney-Rivlin material. The results show that the apparent singularity in ${\sigma }_{22}$ is monotonically intensified from 1/2 (the asymptotic one at the infinitesimal strain) to 1 (the asymptotic one at the infinite strain), and it has a narrow interval of weakening oscillation in $P_{22}$, both of which appear in the range of small applied strains. These outcomes are explained by the mechanism that the zone of large deformation near the tip is rapidly enlarged during the change of far-field loading. The present work suggests that different hyperelastic models possess their apparent singularities of stress per se at the crack tip. This is helpful in establishing a criterion related to crack-tip singularity for precisely studying the constitutive law of hyperelastic materials.