Local and nonlocal models in thin-plate and bridge dynamics

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.