William J. Hrusa

The Cauchy problem in one-dimensional nonlinear viscoelasticity

Abstract The initial value problem for a nonlinear hyperbolic Volterra equation which models the motion of an unbounded viscoelastic bar is studied. Under physically motivated assumptions, the existence of a unique, globally defined, classical solution is established provided the initial data are sufficiently smooth and small. Boundedness and asymptotic behavior are also discussed. This analysis is based on energy estimates in conjunction with properties of strongly positive definite kernels.

On formation of singularities in one-dimensional nonlinear thermoelasticity

It is well known that smooth motions of nonlinear elastic bodies generally will break down in finite time due to the formation of shock waves. On the other hand, for thermoelastic materials, the conduction of heat provides dissipation that competes with the destabilizing effects of nonlinearity in the elastic response. The work of Coleman & Gurtin [2] on the growth and decay of acceleration waves provides a great deal of insight concerning the interplay between dissipation and nonlinearity in one-dimensional nonlinear thermoelastic bodies. Assuming that the elastic modulus, specific heat, and thermal conductivity are strictly positive, the stress-temperature modulus is nonzero, and that the elastic response is genuinely nonlinear they show that acceleration waves of small initial amplitude decay but waves of large initial amplitude can explode in finite time. In other words, thermal diffusion manages to restrain waves of small amplitudes but nonlinearity in the elastic response is dominant for waves of large amplitudes.

A model equation for viscoelasticity with a strongly singular kernel

In much of the mathematical work on nonlinear viscoelasticity it is assumed that the kernel (or memory function) is smooth on

The Lavrentiev Gap Phenomenon in Nonlinear Elasticity

[0,\infty )

On a class of quasilinear partial integrodifferential equations with singular kernels

. There are, however, theoretical and experimental indications that certain viscoelastic materials may be described by equations involving kernels that are singular at zero. In this paper we establish local (in time) existence of smooth solutions to a nonlinear integrodifferential equation with a singular kernel. This equation provides a model for the motion of a certain class of viscoelastic materials. Our analysis is based on energy estimates and properties of positive definite kernels.

On perturbations of differentiable semigroups

The main objective of this paper is to present examples of the Lavrentiev phenomenon within the framework of two-dimensional nonlinear elasticity. Loosely speaking, this phenomenon is associated with the sensitivity of the infimum in a variational problem to the regularity required of the competing mappings. We provide a physically natural stored energy density and reasonable, though nontraditional, boundary conditions such that the energy functional exhibits the Lavrentiev phenomenon with admissible classes that are subsets of the continuous deformations. The stored-energy density W that we produce is smooth, materially homogeneous, frame-indifferent, isotropic and polyconvex. Furthermore, the corresponding minimization problem is such that existence of a continuous minimizer follows from known results. The basis for our examples is a convex integrand W0 for which the Euler-Lagrange equations have a very special form. We show that the functional associated with this W0 exhibits the Lavrentiev phenomenon for certain problems; by making a perturbation to W0, we create the stored-energy density W described in the previous paragraph. With other perturbations to the integrand W0 and modifications of the boundary conditions, we are able to produce additional examples of the Lavrentiev phenomenon. Finally, we note that the integrand we use is just one of a family of integrands that can be used to produce examples of the phenomenon.

The Lavrentiev Phenomenon in Nonlinear Elasticity

Abstract We prove local and global existence theorems for a model equation in nonlinear viscoelasticity. In contrast to previous studies, we allow the memory function to have a singularity. We approximate the equation by equations with regular kernels and use energy estimates to prove convergence of the approximate solutions.

Initial Value Problems in Viscoelasticity

AbstractLetX be a Banach space and letA be the infinitesimal generator of a differentiable semigroup {T(t) |t ≥ 0}, i.e. aC0-semigroup such thatt ↦T(t)x is differentiable on (0, ∞) for everyx εX. LetB be a bounded linear operator onX and let {S(t) |t ≥ 0} be the semigroup generated byA +B. Renardy recently gave an example which shows that {S(t) |t ≥ 0} need not be differentiable. In this paper we give a condition on the growth of ‖T′(t)‖ ast ↓ 0 which is sufficient to ensure that {S(t) |t ≥ 0} is differentiable. Moreover, we use Renardy’s example to study the optimality of our growth condition. Our results can be summarized roughly as follows:(i)If lim supt→0+t log‖T′(t)‖/log(1/2) = 0 then {S(t) |t ≥ 0} is differentiable.(ii)If 0<L=lim supt→0+t log‖T′(t)‖/log(1/2)<∞ thent ↦S(t ) is differentiable on (L, ∞) in the uniform operator topology, but need not be differentiable near zero(iii)For each function α: (0, 1) → (0, ∞) with α(t)/log(1/t) → ∞ ast ↓ 0, Renardy’s example can be adjusted so that limsupt→0+t log‖T′(t)‖/α(t) = 0 andt →S(t) is nowhere differentiable on (0, ∞). We also show that if lim supt→0+tp ‖T′(t)‖<∞ for a givenp ε [1, ∞), then lim supt→0+tp‖S′(t)‖<∞; it was known previously that if limsupt→0+tp‖T′(t)‖<∞, then {S(t) |t ≥ 0} is differentiable and limsupt→0+t2p–1‖S′(t)‖<∞.

Lavrentiev's phenomenon for totally unconstrained variational problems in one dimension

The main objective of this paper is to present examples of the Lavrentiev phenomenon within the framework of two-dimensional nonlinear elasticity. Loosely speaking, this phenomenon is associated with the sensitivity of the infimum in a variational problem to the regularity required of the competing mappings. We provide a physically natural stored energy density and reasonable, though nontraditional, boundary conditions such that the energy functional exhibits the Lavrentiev phenomenon with admissible classes that are subsets of the continuous deformations. The stored-energy density W that we produce is smooth, materially homogeneous, frame-indifferent, isotropic and polyconvex. Furthermore, the corresponding minimization problem is such that existence of a continuous minimizer follows from known results. The basis for our examples is a convex integrand W0 for which the Euler-Lagrange equations have a very special form. We show that the functional associated with this W0 exhibits the Lavrentiev phenomenon for certain problems; by making a perturbation to W0, we create the stored-energy density W described in the previous paragraph. With other perturbations to the integrand W0 and modifications of the boundary conditions, we are able to produce additional examples of the Lavrentiev phenomenon. Finally, we note that the integrand we use is just one of a family of integrands that can be used to produce examples of the phenomenon.

Propagation of Discontinuities in Linear Viscoelasticity

Abstract : This document reviews some recent mathematical results concerning integrodifferential equations that model the motion of one-dimensional nonlinear viscoelastic materials. In particular, the authors discuss global (in time) existence and long-time behavior of classical solutions, as well as the formation of singularities in finite time from smooth initial data. Although the mathematical theory is comparatively incomplete, some remarks are more concerning the existence of weak solutions (i.e., solutions with shocks). Some relevant results from linear wave propagation will also be discussed. Keywords: integrodifferential equations; mathematical models; Nonlinear viscoelasticity, materials with fading memory, viscoelastic fluids, linear wave propagation, acceleration waves, smooth kernels, singular kernels, Laplace transforms, hyperbolic equations, global existence, smooth solution.