Gamma convergence and Cosserat curvy shell models
The Encyclopedia Britannica describes elasticity as the “ability of a deformed material body to return to its original shape and size when the forces causing the deformation are removed”. The theory of (nonlinear) elasticity provides a mathematical framework to model the response of different elastic materials under given loads. The elastic body is generally modeled as a continuum, thus the theory is concerned with the macroscopic effects rather than the underlying microscopic mechanism of atomic bonding that causes the elastic behavior. To this day, there exists no mathematical model which correctly describes the entirety of elastic behavior. In fact, precise description of physical materials as ideally elastic is not even possible because of various interfering effects like thermodynamics, plasticity, and microscopic structures. Thus in a real-world scenario, there is no material response which is a purely elastic continuum behavior. Therefore, for the concept of elasticity, it is crucial to find a reasonable simplification that is capable of expressing an approximate description of the actual physical behavior. For an ideal theoretical elastic material, all other mentioned effects are left out and the act of deforming a body is reduced to its resulting configuration to omit time-dependency. Shell theory is the basic of the first Part in this dissertation. Shells are formed from two curvy layers with a common inner surface in between with a very small thickness. We note that the thickness in one direction is much more smaller than the two other dimensions which are orthogonal to the direction of the thickness. The theory of shells is similar to the theory of plates with more specification like curvature. The thickness of the shell can play a role to effect the boundary and external loads. In the first part of this dissertation we consider a three dimensional Cosserat model and derive a curvy shell model with small thickness. By applying the nonlinear scaling for both deformation field ϕ and the microrotation R, we obtain the homogenized membrane energy Wmp. The homogenized curvature energy Wcurv is obtained separately in the second chapter of the first Part. A combination of these homogenized energies and applying the concept of Γ-convergence will lead us to find a minimizer for the sequence of energies as the thickness leads to zero. In the second part, we discuss some properties of the scenes of minimal surfaces. Minimal surfaces are defined as surfaces with zero mean curvature, which a parametrized minimal surface satisfies in Lagrane’s equation. For many years the only known complete, embedded minimal surfaces of finite topology were the catenoid and helicoid which we disscused about them in section 7.5. In this chapter we also assume that an in-plane drill rotation is the deformation mapping from a smooth regular shell surface to another which is parameterized on the same domain and we show that this is not possible unless all rotations at a portion of the boundary are fixed.