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Well-Posedness of the Fractional Zener Wave Equation for Heterogeneous Viscoelastic Materials by Ljubica Oparnica; Endre Süli is a Mathematics article available to read on EtoBox.

What is Well-Posedness of the Fractional Zener Wave Equation for Heterogeneous Viscoelastic Materials about?

Abstract Zener’s model for viscoelastic solids replaces Hooke’s law σ = 2με(u) + λ tr(ε(u)) I, relating the stress tensor σ to the strain tensor ε(u), where u is the displacement vector, μ > 0 is the shear modulus, and λ ≥ 0 is the first Lamé coefficient, with the constitutive law (1 + τDt) σ = (1 + ρDt)[2με(u) + λ tr(ε(u)) I], where τ > 0 is the characteristic relaxation time and ρ ≥ τ is the characteristic retardation time. It is the simplest model that predicts creep/recovery and stress relaxation phenomena. We explore the well-posedness of the fractional version of the model, where the first-order time-derivative Dt in the constitutive law is replaced by the Caputo time-derivative $\begin{array}{} D\_t^\alpha \end{array} $ with α ∈ (0, 1), μ, λ belong to L∞(Ω), μ is bounded below by a positive constant and λ is nonnegative. We show that, when coupled with the equation of motion ρü = Div σ + f, considered in a bounded open Lipschitz domain Ω in R3 and over a time interval (0, T], where ρ ∈ L∞(Ω) is the density of the material, assumed to be bounded below by a positive constant, and f is a specified load vector, the resulting model is well-posed in the sense that the associated i

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Author

Ljubica Oparnica; Endre Süli

Publisher

Springer Science and Business Media LLC

Published

2020

Language

EN

Field

Mathematics (Physical Sciences)