Wellposedness of the fractional zener wave equation for heterogeneous viscoelastic materials
 Author
 Ljubica Oparnica (UGent) and Endre Süli
 Organization
 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 wellposedness of the fractional version of the model, where the firstorder timederivative Dt in the constitutive law is replaced by the Caputo timederivative Dαt 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 wellposed in the sense that the associated initialboundaryvalue 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 ∈ [H10(Ω)]3, h ∈ [L2(Ω)]3, and S = ST ∈ [L2(Ω)]3×3, and any load vector f ∈ L2(0, T; [L2(Ω)]3), and that this unique weak solution depends continuously on the initial data and the load vector.
 Keywords
 Applied Mathematics, Analysis, MODEL, fractional Zener model, existence, uniqueness, viscoelastic materials
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Please use this url to cite or link to this publication: http://hdl.handle.net/1854/LU8654031
 MLA
 Oparnica, Ljubica, and Endre Süli. “WellPosedness of the Fractional Zener Wave Equation for Heterogeneous Viscoelastic Materials.” FRACTIONAL CALCULUS AND APPLIED ANALYSIS, vol. 23, no. 1, 2020, pp. 126–66, doi:10.1515/fca20200005.
 APA
 Oparnica, L., & Süli, E. (2020). Wellposedness of the fractional zener wave equation for heterogeneous viscoelastic materials. FRACTIONAL CALCULUS AND APPLIED ANALYSIS, 23(1), 126–166. https://doi.org/10.1515/fca20200005
 Chicago authordate
 Oparnica, Ljubica, and Endre Süli. 2020. “WellPosedness of the Fractional Zener Wave Equation for Heterogeneous Viscoelastic Materials.” FRACTIONAL CALCULUS AND APPLIED ANALYSIS 23 (1): 126–66. https://doi.org/10.1515/fca20200005.
 Chicago authordate (all authors)
 Oparnica, Ljubica, and Endre Süli. 2020. “WellPosedness of the Fractional Zener Wave Equation for Heterogeneous Viscoelastic Materials.” FRACTIONAL CALCULUS AND APPLIED ANALYSIS 23 (1): 126–166. doi:10.1515/fca20200005.
 Vancouver
 1.Oparnica L, Süli E. Wellposedness of the fractional zener wave equation for heterogeneous viscoelastic materials. FRACTIONAL CALCULUS AND APPLIED ANALYSIS. 2020;23(1):126–66.
 IEEE
 [1]L. Oparnica and E. Süli, “Wellposedness of the fractional zener wave equation for heterogeneous viscoelastic materials,” FRACTIONAL CALCULUS AND APPLIED ANALYSIS, vol. 23, no. 1, pp. 126–166, 2020.
@article{8654031, 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 wellposedness of the fractional version of the model, where the firstorder timederivative Dt in the constitutive law is replaced by the Caputo timederivative Dαt 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 wellposed in the sense that the associated initialboundaryvalue 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 ∈ [H10(Ω)]3, h ∈ [L2(Ω)]3, and S = ST ∈ [L2(Ω)]3×3, and any load vector f ∈ L2(0, T; [L2(Ω)]3), and that this unique weak solution depends continuously on the initial data and the load vector.}}, author = {{Oparnica, Ljubica and Süli, Endre}}, issn = {{13110454}}, journal = {{FRACTIONAL CALCULUS AND APPLIED ANALYSIS}}, keywords = {{Applied Mathematics,Analysis,MODEL,fractional Zener model,existence,uniqueness,viscoelastic materials}}, language = {{eng}}, number = {{1}}, pages = {{126166}}, title = {{Wellposedness of the fractional zener wave equation for heterogeneous viscoelastic materials}}, url = {{http://dx.doi.org/10.1515/fca20200005}}, volume = {{23}}, year = {{2020}}, }
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