Waves in fractional Zener type viscoelastic media

Abstract

Classical wave equation is generalized for the case of viscoelastic materials obeying fractional Zener model instead of Hooke's law. Cauchy problem for such an equation is studied: existence and uniqueness of the fundamental solution is proven and solution is calculated.

Fractional Burgers wave equation

“…In particular, it is found that both solid-like and fluid-like materials can have either infinite or finite wave speed. Singularity propagation properties of the memory and non-local type fractional wave equations are investigated in [17,18] using the tools of microlocal analysis, supporting the results obtained in [20]. Wave propagation phenomena in viscoelastic bodies, modeled by integer and fractional order models, including the question of wave speed and energy dissipation properties are analyzed in [8,9].…”

“…The thermodynamical restriction (20) Therefore, again by (20), one has that b − a3 a1 |sin((µ−β−α)ϕ)| sin((µ−β+α)ϕ) > 0, which, along with the positivity of all other terms in (51), implies that f ρ (ϕ) > 0 if ϕ ∈ 0, π 2 .…”

“…obtained and analyzed in [2] for thermodynamical consistency and used in [22] as constitutive equations in wave propagation modeling. Namely, the results of [20,21], where the wave propagation speed is found via the conic solution support, i.e., |x| < ct, in the case of the fractional Zener model and its generalization, respectively given by…”

Distributed-order fractional constitutive stress–strain relation in wave propagation modeling

“…, with X * = T * E/ρ and T * determined differently for each particular constitutive model, for example according to one of the generalized time constants in φ ε . We refer to [43] for more details. Thus, we shall look for solutions to system of equations (11) -(13) which satisfy initial and boundary conditions:…”

“…Generalizations of the classical wave equation are considered on infinite domain in [43,44], where the constitutive equation, representing a single class of thermodynamically consistent linear fractional constitutive equation (4), is chosen to be either the fractional Zener model, or its generalization having arbitrary number of fractional derivatives of same orders acting on both stress and strain. Wave propagation speed is obtained from the support property of fundamental solution in [43,44], while in [42] tools of microlocal analysis were employed in order to examine the singularity propagation properties in the case of fractional Zener wave equation. Fractional wave equation, with power type distributed order model (5) as the constitutive equation, is considered on finite domain in [7,8].…”