Coupled energetic models for incompressible nematic elastomers

Andrea C Rubiano, Purdue University

Abstract

We investigate, through methods in the Calculus of Variations, mathematical energetic models for incompressible nematic elastomers. These models are based on the coupling between the neo-classical energy density, developed by Bladon, Warner and Terentjev as an extension of the rubber elasticity theory, with the classical energy density from the Landau-de Gennes theory for uniaxial nematic liquid crystals. A unit-length molecular director of the nematic elastomer and an incompressible deformation are the unknown functions, minimizers of the coupled energy. In contrast to previous mathematical work in this field, the molecular director is not assumed to be constant throughout the domain. After establishing a suitable generalized energetic model for working in Sobolev spaces, we prove lower semi-continuity of the energy. Considering generalized shear deformations motivated by physical experiments on thin film domains, we show the existence of minimizers, and keeping the restriction of incompressibility on the deformation and unit length of the director, we derive weak Euler Lagrange equations satisfied by the minimizers. Additionally, we consider the reduction of the model to a 2-dimensional one and deduce existence results for non-convex energy densities involving terms related to the constraint of volume’s preservation. In this case we also find weak Euler-Lagrange equations and prove a partial regularity result.

Degree

Ph.D.

Advisors

Bauman, Purdue University.

Subject Area

Applied Mathematics|Mathematics|Energy

COinS