Pierluigi Cesana (Kyushu University, Japan): Modeling the microstructure of active membranes
am Mittwoch, 31.05.2017 16:15 im Raum M6
ABSTRACT: Microstructure and pattern formation occurs in a large family of materials undergoing solid-to-solid phase-transformations whose technological applications span from biophysics to metallurgy.
Mathematically, microstructure is described by using nonlinear elasticity models and is associated to deep questions of the multi-dimensional calculus of variations (some of which still unsolved) such as lower semicontinuity and quasiconvexity. In this talk I will present a case study arising from Nematic Liquid Crystal Elastomers (NLCEs). This is a class of optically active biopolymers and gels with applications in aeronautics in which a complex interplay of material and structural non-linearities is observed. I will focus on the geometrically constrained problem for thin membranes of NLCEs. In this regime, membranes can display fine-scale features both due to wrinkling that one expects in thin elastic membranes and oscillations of the local optical axis that are typical of NLCEs.
Existence of solutions to Boundary Value Problems turns out to be an extremely delicate and challenging problem in the calculus of variations due to the intricate coupling of the optical microstructure with the high curvature and high stress regions in the deformed membrane.
Gamma-convergence and Relaxation theory shed some light on the analysis of the dimension reduction problem: I show existence of a regime where one has shear strain but no shear stress and all the fine-scale features are in-plane with no wrinkling. This may act as a mechanism preventing formation of wrinkles in active membranes under complex boundary conditions.