Local and Nonlocal Models in Thin-Plate and Bridge Dynamics

Authors

Jeremy Trageser, University of Nebraska-LincolnFollow

Date of this Version

7-2015

Comments

A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfilment of Requirements For the Degree of Doctor of Philosophy, Major: Mathematics, Under the Supervision of Professors Petronela Radu and Daniel Toundykov. Lincoln, Nebraska: July, 2015

Copyright (c) 2015 Jeremy E. Trageser

Abstract

This thesis explores several models in continuum mechanics from both local and nonlocal perspectives. The first portion settles a conjecture proposed by Filippo Gazzola and his collaborators on the finite-time blow-up for a class of fourth-order differential equations modeling suspension bridges. Under suitable assumptions on the nonlinearity and the initial data, a finite-time blowup is demonstrated as a result of rapid oscillations with geometrically growing amplitudes. The second section introduces a nonlocal peridynamic (integral) generalization of the biharmonic operator. Its action converges to that of the classical biharmonic as the radius of nonlocal interactions---the ``horizon"---tends to zero. For the corresponding steady state problem, which represents a peridynamic analog of a hinged or clamped plate under load, the existence and uniqueness are shown. By utilizing a compactness result devised by Jean Bourgain, Haim Brezis, and Petru Mironescu and employing a method developed by Qiang Du and Tadele Mengesha, it is demonstrated that as the horizon tends to zero, the solutions of the nonlocal boundary value problems converge strongly in L² to the solutions of the corresponding classical elliptic problems.

Advisers: Petronela Radu and Daniel Toundykov