Analytical expressions for the two basic modes of surface displacement and overall deformation of a free-standing or elastically embedded sphere

TL;DR

This paper derives explicit analytical formulas for the primary surface deformation modes of elastic spheres, considering both free-standing and embedded conditions, aiding in material optimization for soft actuators.

Contribution

It provides the first explicit analytical expressions for the main deformation modes of elastic spheres under internal forces, covering both embedded and free-standing cases.

Findings

Explicit formulas for surface deformation modes

Applicable to embedded and free-standing spheres

Facilitates computational optimization of soft actuators

Abstract

Calculating by analytical theory the deformation of finite-sized elastic bodies in response to internally applied forces is a challenge. Here, we derive explicit analytical expressions for the amplitudes of modes of surface deformation of a homogeneous, isotropic, linearly elastic sphere. The modes under consideration correspond to overall changes in volume and to relative uniaxial elongation or contraction. Both situations of an elastic sphere embedded under no-slip conditions in a linearly elastic, homogeneous, isotropic, infinitely extended background medium and of a free-standing elastic sphere are considered. For example, our analytical expressions are important as a basis for computational material optimization in the context of spherical soft actuators.