Revisiting the Stress Field Inside an Elastic Sphere Subjected to a Concentrated Load

TL;DR

This paper provides a comprehensive analytical solution for the stress distribution inside a homogeneous elastic sphere under surface concentrated loads, including generalizations for multiple load positions.

Contribution

It offers explicit formulas for stress fields inside an elastic sphere under point loads, derived from elastodynamic equations and extended via superposition.

Findings

Explicit closed-form stress expressions inside the sphere.

Solution derived from elastodynamic equations and static limit.

Generalization to multiple load positions through superposition.

Abstract

We present a complete analytical solution for the stress field inside a homogeneous, inside a homogeneous, linearly elastic solid sphere subjected to a concentrated normal load applied on its surface. Starting from the three-dimensional linearized elastodynamic equations, the displacement and stress fields are derived using scalar and vector potential representations combined with spherical harmonic expansions. All expansion coefficients are determined explicitly by enforcing the traction boundary conditions. The static elastic solution is obtained rigorously as the long-time limit of the dynamical formulation. Closed-form expressions for all components of the stress tensor are provided, enabling direct evaluation of the principal stresses and their differences throughout the interior of the sphere. The analytical solution is further generalized to arbitrary loading positions by means of rotational transformations, allowing systematic treatment of multiple concentrated loads through superposition.