Asymptotic Solutions for Multi-Hole Problems: Plane Strain versus Plane Stress Boundary Conditions in Borehole Applications

Abstract

The elastic response of circular cylindrical holes in elastic plates is analyzed using the linear superposition method (LSM) to assess the impact of plate thickness on the stress state for the thin- and thick-plate solutions. Analytical solutions for stress accumulations near holes in elastic plates are relevant for a wide range of practical applications. For example, detailed analyses of the stress concentrations near boreholes piercing rock formations are needed during drilling operations to avoid premature fracturing due to tensile and shear failure. Stress concentrations near tiny holes in very thick plates approach the solution of a plane strain boundary condition; for large holes in very thin plates, the solution of a plane stress boundary condition will apply. For most practical cases, the response will be intermediate between the plane stress and plane strain end members, depending on the relative dimensions of the thickness of the elastic volume penetrated and the hole diameter. A nondimensional scaling parameter is introduced to quantify for which hole radius to plate thickness ratio occurs the transition between the two types of solutions (plane strain versus plane stress). Moreover, in this study, we consider the case of the presence of the internal pressure load in the analysis of the stress concentrations near boreholes. This consideration is important to carefully assess the magnitude of the elastic stress concentrations and their orientation near the hole in the rock formation when the pressure load of the mud is added to the borehole during drilling operations. For holes subjected to an internal pressure only, there is no difference between the plane stress (thin-plate solution) and plane strain solutions (thick-plate solutions). For cases with far-field stress, the plane strain solution is more sensitive to the Poisson’s ratio than the plane stress solution. Multi-hole problems are also evaluated with LSM and the results are benchmarked against known solutions of different methods.