Longitudinal shock waves in a class of semi-infinite stretch-limited elastic strings

Abstract

In this paper, we initiate the study of wave propagation in a recently proposed mathematical model for stretch-limited elastic strings. We consider the longitudinal motion of a simple class of uniform, semi-infinite, stretch-limited strings under no external force with finite end held fixed and prescribed tension at the infinite end. We study a class of motions such that the string has one inextensible segment, where the local stretch is maximized, and one extensible segment. The equations of motion reduce to a simple and novel shock front problem in one spatial variable for which we prove existence and uniqueness of local-in-time solutions for appropriate initial data. We then prove the orbital asymptotic stability of an explicit two-parameter family of piece-wise constant stretched motions. If the prescribed tension at the infinite end is increasing in time, our results provide an open set of initial data launching motions resulting in the string becoming fully inextensible and tension blowing up in finite time.

On an elastic strain-limiting special Cosserat rod model

“…• elastic strings [48,49]. However, this paper is the first work studying a set of strain-limiting constitutive relations for special Cosserat rods.…”

“…Thus, such bodies must be described either by relations such as (1.3) or even the more general class (1.2), with these relations not being invertible. Recently, Rodriguez studied such stretch-limiting constitutive relations for elastic strings in both static [48] and dynamic [49] settings, and Rajagopal and Rodriguez studied certain inelastic behavior for the quasistatic motion of an inextensible, unshearable, viscoelastic rod [45].…”