Constitutive modeling of hyperelastic solids reinforced by spheroidal particles under large deformations

This paper presents a homogenization-based constitutive model for the mechanical behavior of particle-reinforced elastomers with random microstructures subjected to finite deformations. The model is based on a recently developed homogenization method (Avazmohammadi and Ponte Castaneda 2013; J. Elasticity 112, 1828–1850) for two-phase, hyperelastic composites, and is able to directly account for the shape, orientation, and concentration of the particles. We describe the application of this method to composites consisting of an incompressible rubber reinforced by aligned, spheroidal, rigid particles, undergoing generally non-aligned, three-dimensional loadings. While the results are valid for finite particle concentrations, in the dilute limit they can be viewed as providing a generalization of Eshelby’s results in linear elasticity. In particular, we provide analytical estimates for the overall response and microstructure evolution of the particle-reinforced composites with generalized neo-Hookean matrix phases under non-aligned loadings. For the special case of aligned pure shear and axisymmetric shear loadings, we give closed-form expressions for the effective stored-energy function of the composites with neo-Hookean matrix behavior. Moreover, we investigate the possible development of “macroscopic” (shear band-type) instabilities in the homogenized behavior of the composite at sufficiently large deformations. These instabilities whose wavelengths are much larger than the typical size of the microstructure are detected by making use of the loss of strong ellipticity condition for the effective stored-energy function of the composites. The analytical results presented in this paper will be complemented in Part II of this work by specific applications for several representative microstructures and loading configurations.

This research has been published in two parts in Philosophical Magazine available at the following links:

http://www.tandfonline.com/doi/abs/10.1080/14786435.2013.878048#.U_U3m_ldWYA

http://www.tandfonline.com/doi/abs/10.1080/14786435.2013.878049#.U_U4fvldWYA