Shape transitions in a soft incompressible sphere with residual stresses

32/2017

Shape transitions in a soft incompressible sphere with residual stresses

Thursday 29th June 2017

Riccobelli, D.; Ciarletta, P.

Residual stresses may appear in elastic bodies due to the formation of misfits in the micro-structure, driven by plastic deformations, thermal or growth processes. They are especially widespread in living matter, resulting from the dynamic remodelling processes aiming at optimizing the overall structural response to environmental physical forces. From a mechanical viewpoint, residual stresses are classically modelled through the introduction of a virtual incompatible configuration that maps the natural state of the body. In this work, we instead employ an alternative approach based on a strain energy function that constitutively depends on both the deformation gradient and the residual stress tensor. In particular, our objective is to study the morphological stability of an incompressible sphere, made of a neo-Hookean material and subjected to given distributions of residual stresses. The boundary value elastic problem is studied with analytic and numerical tools. Firstly, we perform a linear stability analysis on the pre-stressed sphere using the method of incremental deformations. The marginal stability conditions are given as a function of a control parameter, being the dimensionless variable that represents the characteristic intensity of the residual stresses. Secondly, we perform finite element simulations using a mixed formulation in order to investigate the post-buckling morphology in the fully nonlinear regime. Considering different initial distributions of the residual stresses, we find that different morphological transitions are all localized around the material domain where the hoop residual stress reaches its maximum compressive value. The loss of spherical symmetry is found to be controlled by the mechanical and geometrical properties of the sphere, as well as on the spatial distribution of the residual stress. The results provide useful guidelines in order to design morphable soft spheres, for example by controlling the residual stresses through active deformations. They finally open a pathway for the non-disruptive characterization of residual stresses in soft tissues, such as solid tumours.

This report, or a modified version of it, has been also submitted to, or published on
Mathematics and Mechanics of Solids