Dynamic stability of one-dimensional nonlinearly viscoelastic bodies

Abstract

is introduced both as a better approximation to reality and to obtain tractable (parabolic rather than hyperbolic) differential equations .Under this assumption , we show that the des irable theorems for the nonlinear equations : existence , uniqueness, and stability , follow from the analogous theorems for the linearized equations .

Nonlinear dynamical theory of the elastica

“…On the other hand, (1.1)-(1.3) is not the only physically realistic dynamical system associated with potential (2.12), and the equivalence of the Liapounov definition and the energy criterion can properly be discussed within other dynamical systems. Indeed, progress has recently been made in a general justification of this equivalence within the context of viscoelasticity, see [6], [1], and [13]. The first two of these papers discuss stability in very general rod models; in particular, their stress-strain laws are not restricted to being linear as is assumed here.…”

Stability Estimates for a Twisted Rod Under Terminal Loads: A Three-dimensional Study

“…We study the stability of a naturally straight Kirchhoff rod, aligned along the x-axis, subject to clamped boundary conditions, controlled end-rotation and terminal loads. We start with a general rod-energy in (6) and show that the diagonal rod-energy in (15) is the most general rod-energy compatible with any planar equilibrium state. We approach the stability analysis from a variational perspective and study the second variation of the diagonal rod-energy, including three-dimensional effects into the problem formulation.…”

“…Comment: We note that for the rod-energy (6) to have any straight equilibrium state characterized by θ(s) = C 1 and φ(s) = C 2 , for constants C 1 and C 2 , we must have M 13 = M 23 = 0.…”

“…Within the framework of this second method, one studies the time evolution of small perturbations around both the trivial and bifurcated solutions for the linearized problem [28,1,17]. The spectrum of the linearized operator provides information about the exponential instability of some of these solutions which, for some prescribed dynamics, can yield rigorous results [6]. The third approach is a variational approach whereby we describe stable equilibria as local minimizers of a suitably defined elastic energy [7].…”

Asymptotic stability for equilibria of nonlinear semiflows with applications to rotating viscoelastic rods, Part I

“…Early work in this direction was done by Browne [1978], who considered the problem of existence, uniqueness and stability for the quasilinear partial differential equations governing the motion of nonlinearly viscoelastic onedimensional bodies.…”