Anti-plane crack solutions in higher-order elasticity

The asymptotic solutions for cracks in a linear elastic medium under plane or anti-plane state were obtained by Williams (1957) nearly sixty years ago. However, solutions for cracks in second-order elasticity are unavailable, in contrast to those based on the neo-Hookean models, e.g., Knowles (1997). This paper addresses the formulation and solution of crack problems under finite anti-plane deformation using higher-order elasticity. It is found that: (1) a combined second- and third-order elasticity is necessary to ensure that full equilibrium is satisfied, (2) the equilibrium equations are non-homogeneous partial differential equations (pde's) with variable coefficients, and (3) exact particular solutions can be obtained while the homogeneous pde's reduce to nonlinear eigenvalue problems that can be solved numerically. The results show that: (1) the displacement or stress dependence on the radial coordinate is generally a function of the elastic constants, (2) the Piola-Kirchhoff stress matrix is fully populated, with induced normal stresses, in-plane as well as out-of-plane shear stresses, (3) singular stresses at the crack tip may exist, depending on the eigenvalues, and (4) normal stresses are predicted on the crack faces, which may lead to anomalous mechanical behavior in soft biological materials.