On VOIGT and KELVIN Matrix Notations of Second‐Order Tensors

Stability of an abstract-wave equation with delay and a Kelvin-Voigt damping

Tensors for matrix differentiation

Let’s setup one useful form of tensor notation, which incorporates the matrix and inner product, the outer product, the Hadamard (MATLAB .* or ◦) product diag and diag−1. These will be denoted using different combinations of pairs of up-stairs and down-stairs indices. If we have only 2 order tensors (and lower) we want to be able to easily convert the result into matrix representation. We have ...

The Lack of Exponential Stability in Certain Transmission Problems with Localized Kelvin-Voigt Dissipation

Abstract. In this paper we consider the transmission problem of a material composed of three components; one of them is a Kelvin–Voigt viscoelastic material, the second is an elastic material (no dissipation), and the third is an elastic material inserted with a frictional damping mechanism. The main result of this paper is that the rate of decay will depend on the position of each component. W...

An Exact Solution for Kelvin-Voigt Model Classic Coupled Thermo Viscoelasticity in Spherical Coordinates

In this paper, the classic Kelvin-Voigt model coupled thermo-viscoelasticity model of hollow and solid spheres under radial symmetric loading condition is considered. A full analytical method is used and an exact unique solution of the classic coupled equations is presented. The thermal and mechanical boundary conditions, the body force, and the heat source are considered in the most general fo...

Stability of a Nonlinear Axially Moving String with the Kelvin-Voigt Damping

In this paper, a nonlinear axially moving string with the Kelvin-Voigt damping is considered. It is proved that the string is stable, i.e., its transversal displacement converges to zero when the axial speed of the string is less than a certain critical value. The proof is established by showing that a Lyapunov function corresponding to the string decays to zero exponentially. It is also shown ...