| The determination of the effective or macroscopic properties of composite materials from the corresponding local properties of the constituent phases and the underlying sub-structure constitutes the fundamental problem in Mechanics of Composites. This problem is motivated from the remarkable physical observation that the constitutive properties of these materials appear to be homogeneous or uniform at the length scale of practical applications, despite the sharp variation of their local properties at the length scale of the heterogeneity and the fairly complex spatial distribution of their phases. This dissertation is concerned with the macroscopic mechanical properties of "multi-scale" viscoplastic composites, i.e., composite systems with viscoplastic constituents exhibiting heterogeneity at more than one, well-separated length-scales. Semi-crystalline polymers, such as polyethylene, polypropylene, Nylon-6, etc., are prominent examples of two-scale systems, and constitute the largest class of polymers used in technological applications. At the larger length-scale, a semi-crystalline polymer is an aggregate of a large number of randomly distributed grains that, at the smaller length scale, are made up of alternating layers of an amorphous and a crystalline phase, both of which exhibit nonlinear material behavior. During processing, these materials are often subjected to large deformations that may lead to highly anisotropic mechanical properties as a result of the evolution of the underlying sub-structure.;Primarily motivated by applications to semi-crystalline polymers, this work develops general variational methods for the estimation of the effective behavior of multi-scale viscoplastic composites. This general theory is applied to several two-scale material systems with increasing degree of complexity and sophistication. The predictions and some important features of these methods are first investigated in the context of two-dimensional model problems. Then, the general variational methods are used to develop homogenization-based constitutive models for the macroscopic response and texture evolution of semi-crystalline polymers under arbitrary finite-strain loading histories. Finally, the theory is specialized to two-scale polycrystals with granular structures at the meso-scale level and lamellar structures at the micro-scale level. In this context, we model the macroscopic instantaneous plastic anisotropy of gamma-TiAl-based polysynthetically twinned crystals and the rolling textures of (alpha + beta) Ti alloys. |
