Macroscopic response and stability in lamellar nanostructured elastomers with "oriented" and "unoriented" polydomain microstructures

We propose a homogenization-based framework to construct constitutive models for the macroscopic response of polycrystalline hyperelastic solids. The theory is presented in a rather general context, but attention is primarily focused on its specialization to lamellar thermoplastic elastomers (TPEs). The proposed framework incorporates direct information on the constitutive properties of the soft and hard blocks, the lamellar nanostructure, as well as the complete orientation distribution (i.e., the lamination directions) and average shape of the grains. In addition to providing constitutive models for the macroscopic response of lamellar TPEs, the proposed theory also provides information about the evolution of the underlying nano- and micro-structure - including the crystallographic texture - and the associated development of macroscopic instabilities. It is found that for sufficiently large stiffness contrast (between the hard and soft blocks) and sufficiently high Poisson's ratio of the soft blocks there is a sudden change in the deformation mode of "unfavorably" oriented (perpendicular to the tensile direction) layers - from a high-energy triaxial deformation mode to a lower-energy rotation and shear-along-the-layers mode, which is responsible for a reduction in the overall stiffness of the granular aggregate. This reduction in stiffness can lead to the development of shear-band-type instabilities, which are perpendicular to the "unfavorably" oriented layers, and may be precursors to the chevron-type instabilities that have been observed in these material systems. The effect of the constitutive properties of the blocks and of the initial microstructure on the overall behavior, microstructure evolution and the possible development of these elastic instabilities is investigated in some detail.