Advanced search
1 file | 631.05 KB Add to list

Well-posedness of the fractional zener wave equation for heterogeneous viscoelastic materials

Author
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 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 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 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 ∈ [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

Downloads

  • AuthorsCopy Oparnica-E Sull.pdf
    • full text (Published version)
    • |
    • open access
    • |
    • PDF
    • |
    • 631.05 KB

Citation

Please use this url to cite or link to this publication:

MLA
Oparnica, Ljubica, and Endre Süli. “Well-Posedness 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/fca-2020-0005.
APA
Oparnica, L., & Süli, E. (2020). Well-posedness of the fractional zener wave equation for heterogeneous viscoelastic materials. FRACTIONAL CALCULUS AND APPLIED ANALYSIS, 23(1), 126–166. https://doi.org/10.1515/fca-2020-0005
Chicago author-date
Oparnica, Ljubica, and Endre Süli. 2020. “Well-Posedness of the Fractional Zener Wave Equation for Heterogeneous Viscoelastic Materials.” FRACTIONAL CALCULUS AND APPLIED ANALYSIS 23 (1): 126–66. https://doi.org/10.1515/fca-2020-0005.
Chicago author-date (all authors)
Oparnica, Ljubica, and Endre Süli. 2020. “Well-Posedness of the Fractional Zener Wave Equation for Heterogeneous Viscoelastic Materials.” FRACTIONAL CALCULUS AND APPLIED ANALYSIS 23 (1): 126–166. doi:10.1515/fca-2020-0005.
Vancouver
1.
Oparnica L, Süli E. Well-posedness 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, “Well-posedness 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 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 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 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 ∈ [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         = {{1311-0454}},
  journal      = {{FRACTIONAL CALCULUS AND APPLIED ANALYSIS}},
  keywords     = {{Applied Mathematics,Analysis,MODEL,fractional Zener model,existence,uniqueness,viscoelastic materials}},
  language     = {{eng}},
  number       = {{1}},
  pages        = {{126--166}},
  title        = {{Well-posedness of the fractional zener wave equation for heterogeneous viscoelastic materials}},
  url          = {{http://dx.doi.org/10.1515/fca-2020-0005}},
  volume       = {{23}},
  year         = {{2020}},
}

Altmetric
View in Altmetric
Web of Science
Times cited: