Gal Debotton (Ben-Gurion University, Israel)

Thursday 21st November, 2019 14:00-15:00 Maths 311B


We examine the mechanical response of incompressible composites with spherical inclusions within the framework of finite deformation elasticity. The closed-from analytical expressions resulting from our four-steps process demonstrate excellent agreement with corresponding finite element simulations. Firstly, we solve the problem of a neo-Hookean thin-wall composite sphere microstructure subjected to homogeneous boundary conditions, and determine the stored strain energy. On the basis of this result, we derive a closed-form expression for the strain energy-density function (SEDF) of a composite sphere assemblage with a dilute concentration of the matrix phase. To the best of our knowledge, this is the first time where the precise dependence of an effective SEDF on the two invariants I1 and I2 is explicitly determined for an isotropic composite. We further demonstrate that the dependence on I2 is small and, accordingly, bound it from above and below by two pseudo neo-Hookean strain energy-density functions. Next, making use of the dilute iterated homogenization method we determine the response of a composite sphere assemblage with finite volume fraction of the two phases. At the end of this third step we end up with closed-form expressions for an upper bound and a lower estimate for the effective SEDF of a neo-Hookean particulate iterated composite sphere assemblage. In the limit of infinitesimal deformations our solution converges to the well-known Hashin-Shtrikman bound, which is slightly higher than our lower estimate, and at infinitely large deformations it agrees with an existing lower bound. Lastly, exploiting the result for the neo-Hookean composites, we follow the non-linear comparison method and obtain straightforward estimates of composites with more general I1 phases (e.g., Ogden, Gent, Arruda-Boyce). We compare our findings with the few available closed-form results for finitely deforming composites with rigid inclusions and with corresponding finite element simulations.

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