Well-Posedness of the Fractional Zener Wave Equation for Heterogeneous Viscoelastic Materials

Zener’s model for viscoelastic solids replaces Hooke’s law σ = 2 μ ε ( 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 + τ D t ) σ = (1 + ρ D t )[2 μ ε ( 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 D t in the constitutive law is replaced by the Caputo time-derivative D t 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 ϱ ü = Div σ + f , considered in a bounded open Lipschitz domain Ω in ℝ 3 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 initial-boundary-value problem, with initial conditions u (0, x ) = g ( x ), u̇ (0, x ) = h ( x ), σ (0, x ) = S ( x ), for x ∈ Ω, and a homogeneous Dirichlet boundary condition, possesses a unique weak solution for any choice of g ∈ [H 1 0 (Ω)] 3 , h ∈ [L 2 (Ω)] 3 , and S = S T ∈ [L 2 (Ω)] 3×3 , and any load vector f ∈ L 2 (0, T ; [L 2 (Ω)] 3 ), and that this unique weak solution depends continuously on the initial data and the load vector.