The calculation of singular orders for composite material anti-plane propagating V-notches

Abstract

Because of geometric or material discontinuities, stress singularities can occur around the vertex of a V-notch. The singular order is an important parameter for characterizing the degree of the stress singularity. The present paper focuses on the calculation of the singular order for the anti-plane propagating V-notch in a composite material structure. Starting from the governing equation of elastodynamics and the displacement asymptotic expansion, an ordinary differential eigen equation with respect to the singular order is proposed. The interpolating matrix method is then employed to solve the established eigen equation to conduct the singular orders. The effects of the material principal axis direction, shear modulus, and propagation velocity and acceleration on the singular orders of the V-notch are respectively investigated, and some conclusions are drawn. The singular orders of the V-notches decrease with an increase in the material principal axis direction angle, except for the crack, whose singular orders do not change with the principal axis direction. The singular orders increase with an increase in the shear modulus $G_{13}$, while they decrease with an increase in the shear modulus $G_{23}$. The singular orders increase as the magnitude of the propagation velocity increases, while they decrease as the direction of the propagation velocity increases. The singular orders increase with an increase in the value of the propagation acceleration, while they decrease with an increase in the direction of the propagation acceleration. The singular orders become larger when the ratio of $G_{13}^{(2)} /G_{13}^{(1)}$ increases, while they become smaller when the ratio of $G_{23}^{(2)} /G_{23}^{(1)}$ increases, and they increase with an increase in the mass density ratio for the bi-material V-notch.