This project addresses multi-scale homogenization (averaging) techniques for fiber-reinforced polymer systems, including long-fiber reinforced polymers, and thermoplastic elastomers with cylindrical structures. The goal is to develop general and computationally efficient constitutive models that are capable of handling coupled elasto-visco-plastic behavior and the effect of microstructure--and its evolution--at two different length scales. The possible development of macroscopic and microscopic instabilities will also be investigated. The techniques to be developed in this work will be of broad application to large classes of polymeric, metallic, biological and geological material systems, and will lend themselves to numerical implementation in constitutive subroutines for use with standard finite element method packages. Once fully developed, they will replace current models based on linear-elastic analyses, requiring the use of large safety factors in industrial applications.
This project includes two international collaborations, one with Pierre Suquet (CNRS, Marseilles) and the other with Javier Llorca (IMDEA, Madrid), on complementary aspects of the proposed work. They will allow the students involved in this project to acquire valuable international experience through participation in multi-country, multi-laboratory collaborative projects. The principal investigator will teach a course on Homogenization Methods, and is also writing a book that is based on this course, which aims to make the recent progress in nonlinear homogenization available to a larger audience. Improvements in the modeling and performance of long-fiber-reinforced polymers should lead to significant energy savings in the aerospace and other industries, while better understanding of thermoplastic elastomers is highly desirable because of their recyclable properties, and consequent positive implications for the environment.
In this project, we have developed multi-scale homogenization techniques for fiber-reinforced material systems with nonlinear viscoelastic constitutive behaviors. Although the methods and tools can be used more generally, including for long-fiber reinforced polymers, complex fluids and thermoplastic elastomers, we have focused on short-fiber reinforced elastomers and suspensions of elastic particles in a Newtonian fluids. The models account for (i) the coupled, strongly nonlinear elasto-viscoplastic response of the constituent phases, (ii) finite strains and the concomitant evolution of the microstructure with the deformation , and (iii) the possible development of macroscopic instabilities through loss of ellipticity of the associated homogenized constitutive relations. The first major finding from this work was a generalization of Eshelby's celebrated solution for ellipsoidal particles embedded in a linear-elastic medium into the (three-dimensional) finite-strain regime. Closed-form analytical results have been obtained for certain special cases, including for spheroidal particle subjected to axisymmetric loading conditions. An important novel feature of the results is the finite rotation of the fibers, when the loading axes are not aligned with the geometric axis of the fibers. In such cases, the particles rotate tending to align their long axis with the tensile loading axis. Interestingly, the macroscopic response of the medium can be significantly affected by such fiber rotations, leading to softening—and even to loss of ellipticity. In addition, we have been able to show that the elastomers reinforced by aligned fibers undergo a nematic- to smectic-type transition at the point of the loss of ellipticity instability, when they are loaded in compression along the long fiber direction. This instability is similar to instabilities that are known to take place in liquid crystal elastomers, but had not yet been documented in purely elastic systems. The second result consisted in the development of new constitutive theories accounting for large-deformation viscoelastic coupling in the inclusion phase of a fiber-reinforced viscoplastic material. In the limit of dilute suspensions of viscoelastic particles in a Newtonian viscous fluid, an exact solution has been obtained, which shows the existence of "normal stresses" as a consequence of the evolving anisotropy in the medium due to the change in shape and orientation of the particles. In addition, it has led to the discovery of a new type of ``trembling'' instability in these systems (under certain specific initial and loading conditons). For more general conditions, the method provides variational approximation for the macroscopic response of such material systems undergoing coupled elasto-viscoplastic response at finite strains. The third major result is the development of a general incremental strategy for generating improved bounds and estimates for nonlinear composites using iterated homogenization. This new methodology resolves a long-standing issue concerning the high-triaxiality limit of the earlier "linear comparison" variational estimates for porous metals, but is expected to have implications more broadly, in particular for ductile failure through void nucleation, growth and coalescence. This project included an international collaboration with Professor Pierre Suquet (CNRS, Marseilles, France) on the use of the linear comparison composite method developed by the Principal Investigator to handle non-uniform thermal strain (or eigen-strains) in the phases of the composite.
