Patterns in an elastic bar

Student thesis: Doctoral Thesis

Abstract

We consider Yip's formulation of the Ericksen model for an elastic bar on an elastic foundation [63] which leads to the Euler-Lagrange equation for the functional ε(u) = ∫ between 0 and 1 (γu²xx +W(ux) + ɑu²)dx, where x is an element of the set (0, 1). with double Dirichlet boundary conditions. Here the potential W(p) = ((|p| - 1)²), is not differentiable at p = 0.We define and prove existence and uniqueness of periodic solutions with any number n ≥ 0 of internal zeroes for all ɑ, γ > 0 and discuss the existence of non-periodic solutions.The Euler-Lagrange equation contains conditions that make it diffcult to track, and then dropping one of them we obtain a weak formulation for this reduced problem,which we then prove it has a unique solution. Next, we use a combination of two numerical methods, namely the Finite Elements Method (FEM) to approximate the model and the Derivative Free Optimization (DFO) to find the location of the jump.
Date of Award18 May 2020
Original languageEnglish
Awarding Institution
  • University Of Strathclyde
SupervisorMichael Grinfeld (Supervisor) & Gabriel Barrenechea (Supervisor)

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