Stress singularity of a notch in a higher-order elastic solid under anti-plane deformation

The problem of the stress singularity of a notch in a nonlinear solid under finite anti-plane deformation is investigated using higher-order elasticity. The equilibrium equations are written in terms of the first Piola-Kirchhoff stresses, which are replaced by displacements up to the third-order. The resulting variable-coefficient partial differential equations are solved numerically, subject to vanishing out-of-plane shear tractions on the notch faces. The key results are: (i) the stress exponent characterizing the variation of stress with distance from the notch tip may be positive or negative, implying that the stresses can be non-singular or singular, (ii) the stress exponent becomes more positive with a decrease in the notch angle, (iii) a single dimensionless reduced elastic parameter determines the stress exponent, and (iv) the stress exponent varies with the elastic constants, unlike the case in linear elasticity. Specifically, for a given notch angle the stress exponent becomes more negative with decrease in the first Lamé constant λ and the third-order elastic constant n, and with the increase in the magnitude of the negative third-order constant m, while it varies non-monotonically with the second Lamé constant (shear modulus) µ.These results have significant implications for notch-like defects in soft solids, e.g., replacement tissues, industrial robots and devices in biomedical applications.