From a thin membrane to an unbounded solid : dynamics and instabilities in radial motion of nonlinearly viscoelastic spheres

Abstract

In this thesis, a theoretical investigation of the dynamic motion of spherically symmetric bodies is presented, considering nonlinear viscoelastic material responses. To explore the stability thresholds of the dynamic motion and to compare them with available formula for the quasi-static limit, the present formulation employs a generalized constitutive relation and accounts for different loading scenarios. Specifically for instantaneously applied load, by studying the entire spectrum of radii ratios of the spherical body, ranging from a thin membrane to an unbounded medium, we show that geometric effects can significantly reduce the dynamic stability limit while viscoelasticity has a stabilizing effect. Additionally, we show that in finite spheres, rate-dependence can induce a bifurcation of the long-time response. The stability thresholds derived in this thesis, together with their geometric and constitutive sensitivities, can inform the design of more resilient material systems that employ soft materials in dynamic settings, with examples including seismic bearings that are designed to absorb shocks but often fail due to rupture of internal cavities, and thin inflatable membrane structures like rubber balloons, which may exhibit snap-through instabilities and consequent ruptures. By accounting for rate-dependence, the results of this thesis also shed light on the response of biological materials to dynamic load and the possible instabilities that can lead to injury in vulnerable organs, such as the brain and the lungs. Moreover, while modern therapeutic ultrasound techniques intentionally generate cavities within the tissue, the present investigation of the material response to harmonic excitations across various frequencies can lead to safer practice.