Analysis of a Power-Law Material Containing a Single Hole Subjected to a Uniaxial Tensile Stress Using the Complex Pseudo-Stress Function

The two-dimensional compatibility equation for time-dependent materials described by the power law is expressed in terms of the second derivative of the stress function with respect to complex conjugate variables. The equation is solved by introducing the pseudo-stress function which satisfies the biharmonic equation resulting from the compatibility equation. The relationship between the second derivative of the stress function and the pseudo-stress function is established. The mixed derivative of the stress function associated with the dilatational stress is expressed by an integral of the complicated pseudo-stress function. The velocity and strain-rate components are expressed in terms of the pseudo-stress function. Therefore, responses of power-law creep materials subjected to various boundary conditions can be obtained. Using the pseudo-stress function, the stress distribution in power-law materials, containing a single hole under plane strain and subjected to a uniaxial tensile stress, is found. The Stress Concentration Factors (SCF) on the hole surface, obtained by using the pseudo-stress function, are compared with those under plane stress obtained by another investigator. The results show that the SCF under plane stress is approximately 8 percent higher than that obtained by using the analysis techniques described herein. The maximum tangential stress for m < 0.5 is obtained away from the hole whereas for 0.5 ≤ m ≤ 1 the maximum stress is found at the hole for θ = π/2.