Elastic theory for the deformation of a solid or layered spheroid under axisymmetric loading

The theory for the deformations of a spheroidal particle is of great scientific interest in numerous physical and biological problems ranging from fracture analysis of plain solids to the compression of biological cells in an atomic force microscope or during micropipette aspiration. Using a formulation in terms of Papkovich–Neuber potentials, we derive the deformations of a prolate, elastic spheroid under known axisymmetric loading. The internal stresses to which the object is subjected are deduced from Hooke’s law of elasticity in prolate spheroidal coordinates. The generalisation to layered spheroids with viscoelastic properties is also discussed. Since for isotropic objects the surface displacements and stresses are directly related by the elastic modulus and Poisson’s ratio alone, the presented, closed-form, analytical solutions may be applied to deduce these important elastic constants from standard stress-deformation experiments. We illustrate the versatility of the findings by analysing the surface displacements and stress states of spheroids with small and large aspect ratios in the presence of both normal and shear surface tractions. Of particular interest in this study is the influence of Poisson’s ratio on the deformation of a near-spherical particle, for instance a soft cancer cell, which is subjected to surface stresses of the kind that can be found in optical traps, like the optical stretcher.