Linear superposition method (LSM) for solving stress tensor fields and displacement vector fields: Application to multiple pressure-loaded circular holes in an elastic plate with far-field stress

This study presents a novel, linear superposition method (LSM) to compute the stress tensor field and displacement vector field in a homogeneous elastic medium with an unlimited (but finite) number of circular cylindrical holes. The displacement field and the associated stress concentrations are due to a far-field stress. The method allows for the hole-centers to occur in arbitrary locations, and the hole-radii may vary over a wide range (but holes may not overlap). The holes may also induce additional elastic displacement due to internal pressure loading that will affect the local stress field, which is fully accounted for in the method. Each hole may be loaded by either equal or individual pressure loads. The underlying algorithms and solution methodology are explained and examples are given for a variety of cases. Selected case study examples show excellent matches with results obtained via independent methods (photo-elastics, complex analysis, and discrete volume solution methods). The LSM provides several advantages over alternative methods: (1) Being closed-form solutions, infinite resolution is preserved throughout, (2) Being grid-less, no time is lost on gridding, and (3) fast computation times. The specific examples of LSM applications to the multi-hole problem developed here, allow for an unlimited number of holes, with either equal or varying radii, in arbitrary constellations. The solutions further account for variable combinations of far-field stress and pressure loads on individual holes. The method can be applied for either plane strain or plane stress boundary conditions. A constitutive equation for linear elasticity controls the stress field solutions, which can be scaled for the full range of Poisson's ratios and Young moduli possible in linear elastic materials.