A complex potential method for the asymptotic solution of wedge problems using first-order shear deformation plate theory

Abstract

Abstract In the present work a complex potential approach is derived in order to investigate stress singularities in wedges or sharp notches using first-order shear deformation plate theory. The focus is on the calculation of the singularity exponent as a fundamental quantity in fracture mechanics. Isotropic homogeneous and bi-material wedges are considered. The effect of different boundary conditions along the notch faces and the influence of the elastic contrast on the singularity exponent are discussed in detail. Within an asymptotic analysis, the governing PDE-system is solved introducing three holomorphic potentials. The requirement that the boundary and continuity conditions must be fulfilled leads to an eigenvalue problem determining the singularity exponent and the angular distribution of the field variables. The underlying eigenvalue equation is a highly non-linear transcendental equation which in general is solved numerically. For specific bi-material configurations closed-form analytical solutions are obtained. The findings are compared with results from literature and with finite element calculations. It is shown, that the proposed complex potential method is a very efficient approach to study the asymptotic solution behaviour. In contrast to methods based on real-valued eigenfunction expansions the physical requirement that all field variables can take only real values is fulfilled automatically due to the nature of the complex potential formalism. The obtained near fields allow for further applications such as an embedding in numerical methods.