On Singularities in Linear Elasticity

This thesis investigates stress singularities within the framework of linear elasticity, focusing on their occurrence, interpretation, and implications in two fundamental cases: an infinite elastic wedge under a moment at its vertex and an elastic disk subjected to sectorial thermal expansion. Stress singularities, where stresses theoretically approach infinity, represent critical regions that influence material failure. Although singularities are mathematical idealizations, they highlight the limitations of classical elasticity and provide insights into real-world structural behavior. The first study addresses the wedge problem, where an infinite elastic wedge is subjected to a concentrated couple at its vertex. Classical solutions, such as Carothers’ formulation, predict a quadratic singularity in the stress field. However, at a critical wedge angle, this solution exhibits spurious behavior known as the wedge paradox. This work proposes a novel interpretation of the stress states by reframing the problem in terms of auxiliary wedges and dipole forces, offering a generalized solution that remains valid for all wedge angles. This approach resolves the paradox by demonstrating that the singular stress state corresponds to a system of dipoles with no resultant moment at the critical angle. The second study explores thermal stress singularities in an infinite elastic disk with a sector experiencing uniform temperature rise. The thermal mismatch between the heated sector and the surrounding material generates stress discontinuities, leading to a logarithmic singularity at the sector boundary. While classical analysis of this infinite geometry yields divergent solutions, introducing a finite length scale, such as the disk radius, constrains the stress field. The study employs dislocation arrays to model thermal mismatches and validates the theoretical predictions using finite element simulations. This approach bridges the gap between idealized infinite models and finite geometries encountered in engineering applications, such as glass panels subjected to differential heating. This research emphasizes that stress singularities, while mathematically infinite, can be interpreted to provide physically meaningful insights into structural reliability and failure mechanisms. By combining analytical, numerical, and dimensional analysis techniques, the study offers robust frameworks for addressing singularities in practical engineering contexts, including fracture mechanics, thermal stress analysis, and material design.