Exact solutions for an elastic damageable hollow sphere subjected to isotropic mechanical loadings

• Original exact solutions are derived for mechanical fields in a hollow sphere with an elastic damage matrix.
• These solutions allow us to discuss the relevance of the considered models.
• Unloading regime is also fully investigated.

In this paper, we first recall some available Eshelby-based homogenization schemes applied to microcracked materials. An emphasis is put on models accounting for interacting opened or closed microcracks and their spatial distribution. On the basis of these schemes, we briefly present a class of isotropic damage models. The main part of the study is devoted to the derivation of exact solutions for mechanical fields (damage distribution, displacement, stress fields) in a hollow sphere subjected to a radial loading and made up of an elastic damageable material. The solutions, discussed in link with the different homogenization schemes, may serve as a reference for assessment of numerical predictions of brittle damage models. Interestingly, it is shown that the existence of physically meaningful solutions strongly depends on the model under consideration. Finally, we establish explicit solution to the hollow sphere problem in unloading regime.