Iterative nonlinear homogenization methods and applications in finite elasticity

M. I. IDIART; O. LOPEZ-PAMIES

Puerto España

Congreso; XII Pan American Congress of Applied Mechanics; 2012

American Academy of Mechanics

We present a homogenization theory to model the mechanical response of hyperelastic solids with random microstructures of the particulate type such as particle-reinforced composites and porous materials. The central idea is to devise a special class of microstructures --- by means of an iterated homogenization procedure in finite elasticity together with an exact dilute result for sequential laminates --- that allows to compute exactly the macroscopic response of the resulting heterogeneous materials. The proposed framework incorporates direct microstructural information up to the two-point correlation functions, and requires the solution to a Hamilton-Jacobi equation with the fiber concentration and the macroscopic deformation gradient playing the role of ``time'' and ``spatial'' variables, respectively. In addition to providing constitutive models for the macroscopic response of heterogeneous materials, the proposed theory also gives information about the local fields in the matrix and inclusion phases, which can be used to study the evolution of microstructure and the development of instabilities. By way of example we present closed-form results for the case of Neo-Hookean solids reinforced by a transversely isotropic distribution of anisotropic fibers, and for the case of Neo-Hookean solids weakened by voids.