Thermoviscoelasticity in Kelvin–Voigt Rheology at Large Strains

Mielke, Alexander; Roubíček, Tomáš

doi:10.1007/s00205-020-01537-z

Cited by 44 publications

(100 citation statements)

References 48 publications

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“…He claims that his approach applies to all boundary conditions involving pinned or stress-free endpoints that could be held at constant temperature or insulated. [63]. In this work, the authors formulate a thermodynamically consistent frame-indifferent viscoelastic model by using the second grade non-simple materials.…”

Section: Governing Equationsmentioning

confidence: 99%

“…Similarly, Dafermos & Hsiao [57], Dafermos [58], Chen & Hoffmann [59], Blanchard & Guibé [60], and Racke & Zheng [61] (see also [62]) investigated thermoviscoelasticity. The most recent work on thermoviscoelasticity is by Mielke & Roubíček [63]. In this work, the authors formulate a thermodynamically consistent frame-indifferent viscoelastic model by using the second grade non-simple materials.…”

Section: Governing Equationsmentioning

confidence: 99%

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Nonlinear viscoelasticity of strain rate type: an overview

Şengül

2021

Proc. R. Soc. A.

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There are some materials in nature that experience deformations that are not elastic. Viscoelastic materials are some of them. We come across many such materials in our daily lives through a number of interesting applications in engineering, material science and medicine. This article concerns itself with modelling of the nonlinear response of a class of viscoelastic solids. In particular, nonlinear viscoelasticity of strain rate type, which can be described by a constitutive relation for the stress function depending not only on the strain but also on the strain rate, is considered. This particular case is not only favourable from a mathematical analysis point of view but also due to experimental observations, knowledge of the strain rate sensitivity of viscoelastic properties is crucial for accurate predictions of the mechanical behaviour of solids in different areas of applications. First, a brief introduction of some basic terminology and preliminaries, including kinematics, material frame-indifference and thermodynamics, is given. Then, considering the governing equations with constitutive relationships between the stress and the strain for the modelling of nonlinear viscoelasticity of strain rate type, the most general model of interest is obtained. Then, the long-term behaviour of solutions is discussed. Finally, some applications of the model are presented.

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Section: Governing Equationsmentioning

confidence: 99%

Section: Governing Equationsmentioning

confidence: 99%

Nonlinear viscoelasticity of strain rate type: an overview

Şengül

2021

Proc. R. Soc. A.

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“…We refer, e.g., to [28] for local intime existence or to [15] for the existence of measure-valued solutions. To date, weak solutions in a general finite strain setting [21,31] can only be guaranteed by using the concept of second-grade nonsimple materials where the elastic properties additionally depend on the second gradient of the deformation [39,40]. At the same time, let us mention that a main justification of the model investigated in [21] lies in the observation that, in the small strain limit, the problem leads to the standard system of linear viscoelasticity without second gradient.…”

Section: Introductionmentioning

confidence: 99%

Derivation of a one-dimensional von Kármán theory for viscoelastic ribbons

Friedrich

Machill

2022

Nonlinear Differ. Equ. Appl.

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We consider a two-dimensional model of viscoelastic von Kármán plates in the Kelvin’s-Voigt’s rheology derived from a three-dimensional model at a finite-strain setting in Friedrich and Kružík (Arch Ration Mech Anal 238: 489–540, 2020). As the width of the plate goes to zero, we perform a dimension-reduction from 2D to 1D and identify an effective one-dimensional model for a viscoelastic ribbon comprising stretching, bending, and twisting both in the elastic and the viscous stress. Our arguments rely on the abstract theory of gradient flows in metric spaces by Sandier and Serfaty (Commun Pure Appl Math 57:1627–1672, 2004) and complement the $$\Gamma $$ Γ -convergence analysis of elastic von Kármán ribbons in Freddi et al. (Meccanica 53:659–670, 2018). Besides convergence of the gradient flows, we also show convergence of associated time-discrete approximations, and we provide a corresponding commutativity result.

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“…This in turn forces σ vi to be strongly nonlinear. In the case of non-simple materials, such nonlinear viscosity is mathematically tractable, although its analysis is very delicate [40]. In combination with the Maxwellian rheology (that is, creep), this combination is sometimes referred as the Jeffreys rheology.…”

Section: Remark 21 (Kelvin-voigt Viscosity)mentioning

confidence: 99%

Finite thermoelastoplasticity and creep under small elastic strains

Roubíček

Stefanelli

2018

Mathematics and Mechanics of Solids

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A mathematical model for an elastoplastic continuum subject to large strains is presented. The inelastic response is modeled within the frame of rate-dependent gradient plasticity for nonsimple materials. Heat diffuses through the continuum by the Fourier law in the actual deformed configuration. Inertia makes the nonlinear problem hyperbolic. The modelling assumption of small elastic Green-Lagrange strains is combined in a thermodynamically consistent way with the possibly large displacements and large plastic strain. The model is amenable to a rigorous mathematical analysis. The existence of suitably defined weak solutions and a convergence result for Galerkin approximations is proved.

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Thermoviscoelasticity in Kelvin–Voigt Rheology at Large Strains

Cited by 44 publications

References 48 publications

Nonlinear viscoelasticity of strain rate type: an overview

Nonlinear viscoelasticity of strain rate type: an overview

Derivation of a one-dimensional von Kármán theory for viscoelastic ribbons

Finite thermoelastoplasticity and creep under small elastic strains

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