A vectorial variational model for phase separation within a periodically heterogeneous composite material is studied. We consider the case when the scale of the phase separation interface is much smaller than the scale of the periodic heterogeneity. We are able to characterize a first order gamma convergence expansion of the energy where W is a double well potential. Due to many technical details, please see the preprint for the limiting functionals.
Our research is novel in two ways. Firstly, the interaction of phase separation and periodic homogenization in this regime has only been studied before in the 1D-scalar case with a particular choice of potential W, by Braides and Zeppieri. Our paper extends these results. Secondly, we allow our potential to be nonisothermal (the wells can depend on position). As a corollary, we strengthen recent results by Cristoferi and Gravina in the theory of nonisothermal Modica-Mortola functionals without homogenization.
We develop a homogenization analysis for the effective behavior of a special class of nonclassical materials, metamaterials, that exhibit a periodic strong-difference (high contrast) between the material properties of the different components. A cross-section of such material is idealized in the figure where the (blue) periodic inclusions can be considered to be soft inside a (red) stiff matrix. Furthermore, we consider our material to be thin and would like to derive an effective membrane energy for the material.
Our research is novel in that the interaction between homogenization, dimension reduction, and high contrast has not been considered before. Furthermore, we are able to do the analysis even when the potentials satisfy the physical non-interpenetrability condition, i.e. the energy is infinite for deformations with nonpositive determinant. This condition prevents the self-intersection of matter in continuum mechanics but it is very hard to deal with mathematically. Many classical relaxation theorems that are known to hold for energies with polynomial growth have yet to be proven under the non-interpenetrability condition. This research is currently in the process of being typed up.
Citation: Ganedi, L., Oza, A. U., Shelley, M., & Ristroph, L. (2018). Equilibrium shapes and their stability for liquid films in fast flows. Physical review letters, 121(9), 094501.
Consider the blowing of a soap bubble. The airflow shapes the soap film but the changes in the soap film shape also sculpt the flow. In this paper, we derive a model for the equilibrium for the fluid-structure interaction between liquid films and laminar flow in the high Reynolds number regime. We give a condition for when no equilibrium solutions exist (e.g. the film detaches into a bubble). We solve the model numerically and show experimental agreement in both shapes and drag laws. I was able to work on all facets of this project and it has given me a deep appreciation of the experimental challenges that must be overcome in order to make precise measurements and verify numerical and theoretical results.