MAGIC113: Introduction to Mathematical Modelling of Liquid Crystal Elastomers

Liquid crystal elastomers (LCEs) are advanced multifunctional materials that combine elasticity with orientational order. Specifically, mechanical strains give rise to changes in liquid crystalline order and, conversely, changes in the orientational order generate mechanical stresses and strains.

The quest for responsive materials with the ability to mimic living systems or to enable green energy production and conversion processes is one of the major challenges for modern materials design. Because of their large reversible deformations and complex material responses in the presence of natural stimuli like heat or light, and electric or magnetic fields, LCEs are suitable for a wide range of applications in science, manufacturing, and medical research. Moreover, biodegradable, recyclable and reprocessable LCEs have also been achieved.

This course offers an introduction to core concepts in the mathematical modelling of LCEs by linking nonlinear elasticity with liquid crystal theory in a unified and coherent manner. There are many similarities between LCEs and conventional rubber on the one hand, and between solid mechanics and the analysis of liquid crystals on the other hand, and all similarities or differences need to be examined carefully.

The study of LCEs is an interdisciplinary subject involving chemistry, physics, materials science, mathematics and computational simulations. For this course, a general background in applied mathematics is required, including algebra, calculus, geometry, differential equations and continuum mechanics.

1. 1. Introduction
2. 1.1. Over view of tensor theory
3. 2. Isotropic nonlinear elasticity 
4. 2.1. Hyperelastic models 
5. 2.2. Stresses and strains
6. 3. Liquid crystal theory
7. 3.1. Order parameters
8. 3.2. Isotropic to nematic phase transition
9. 4. Nematic liquid crystal elastomers
10. 4.1. Ideal nematic elastomers
11. 4.2. Stresses and stress-free configurations
12. 4.4. Shear striping instability
13. 4.5. Variational formulation
14. 5.  Photo-mechanical effects