Precise energy decay rates for some viscoelastic and thermo-viscoelastic rods

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Virginia Tech


Energy dissipation in systems with linear viscoelastic damping is examined. It is shown that in such viscoelastically damped systems the use of additional dissipation mechanisms (such as boundary velocity feedback or thermal coupling) may not improve the rate of energy decay. The situation where the viscoelastic stress relaxation modulus decreases to its (positive) equilibrium modulus at a subexponential rate, e.g., like (1 + t)-x + E, where α > 0, E > 0 is examined. In this case, the nonoscillatory modes (the so-called creep modes) dominate the energy decay rate. The results are in two parts.

In the first part, a linear viscoelastic wave equation with infinite memory is examined. It is shown that under appropriate conditions on the kernel and initial history, the total energy is integrable against a particular weight if the kinetic energy component of the total energy is integrable against the same weight. The proof uses energy methods in an induction argument. Precise energy decay rates have recently been obtained using boundary velocity feedback. It is shown that the same decay rates hold for history value problems with conservative boundary conditions provided that an a priori knowledge of the decay rate of the kinetic energy term is assumed.

In the second part, a simple linear thermo-viscoelastic system, namely, a viscoelastic wave equation coupled to a heat equation, is examined. Using Laplace transform methods, an integral representation formula for W(x,s), the transform of the displacement w(x, t), is obtained. After analyzing the location of the zeros of the appropriate characteristic equation, an asymptotic expansion for the displacement w(O,t) is obtained which is valid for large t and the specific kernel g(t) = g(–) + δtη-1 [over]Î (η), 0 < η < 1. With this expansion it is shown that the coupled system tends to its equilibrium at a slower rate than that of the uncoupled system.