Investigation of singularity orders and eigen-angular functions for V-notches in radially inhomogeneous materials

ABSTRACT

The singularities of V-notches in a material whose modulus of elasticity varies as a power function of the r-coordinate are investigated. The eigen-equations are derived by substituting the asymptotic expansions of displacement fields around a notch vertex into the elasticity equilibrium equations. The radial boundary conditions are also presented by a combination of stress singularity orders and eigen-angular functions. The singularity orders and eigen-angular functions can be calculated simultaneously by solving a set of eigen ordinary differential equations with variable coefficients, which are solved by the interpolating matrix method developed by some of the authors before without the iterative process. The accuracy of the proposed method is verified by comparing the present results with the reference ones when the inhomogeneous material is degraded into a homogeneous one. Then, the stress singularities of V-notches in the radially inhomogeneous material under the plane and anti-plane loadings are investigated, respectively. The results show that the stress singularity of a V-notch in the radially inhomogeneous material under the plane loading is more serious than the one under the anti-plane loading. The plane V-notch under the clamped–clamped boundary condition presents the stress singularity at a smaller notch angle α than the one under the free–free boundary condition. The radially inhomogeneous bulgy V-notch even presents singularity. In addition, the stress singularity becomes stronger with the increase of the exponent c in the variation function of the elasticity modulus.

KEYWORDS:

• Asymptotic expansion
• radially inhomogeneous material
• singularity analysis
• V-notch

Funding

This work was supported by the National Natural Science Foundation of China, China (No.11372094).