Diffusion-induced stress in a functionally graded incompressible elastic sphere

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Diffusion-induced stress is investigated for an isotropic and elastic incompressible solid sphere which is charged by radial diffusion. The interaction between a diffused solute and the arising stress field is accounted for by a chemo-mechanical potential widely employed for describing the process of charge (or discharge) in lithium-ion batteries and which depends on the hydrostatic part of the stress field. Both linear and nonlinear theories of elasticity are considered in the model at their limit of incompressibility. Such a constraint, by relating the displacement and the stress fields analytically to the unknown solute concentration, allows us to derive analytical asymptotic behaviours near the centre of the particle as well as to simplify the equations, solved here by a simple finite difference numerical scheme. Linear and nonlinear predictions are compared for a spherical electrode subject to large deformations. Furthermore, in light of the great interest in engineering science applications, the equations are derived for the most general case of functionally graded materials. It is shown how the space varying elastic coefficient can be used as a tuning mechanism for controlling stress evolution, which in most cases might represent the main reason for degradation phenomena in solids.