A thermodynamically consistent stress-rate type model of one-dimensional strain-limiting viscoelasticity

Erbay, H. A.; Şengül, Yasemin

doi:10.1007/s00033-020-01315-7

Cited by 16 publications

(26 citation statements)

References 24 publications

(46 reference statements)

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“…In [105], the authors proved local-in-time existence of strong solutions to the same model introduced in [104]. Very recently, Erbay & Şengül [106] also investigated constitutive equations with stress rate dependence and proposed a new model for limiting strain behaviour. They showed not only that the one-dimensional model they proposed is thermodynamically consistent but also that the travelling wave solutions coincide with that of the strain rate model they studied before.…”

Section: Applicationsmentioning

confidence: 99%

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: Applicationsmentioning

confidence: 99%

Nonlinear viscoelasticity of strain rate type: an overview

Şengül

2021

Proc. R. Soc. A.

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“…In the case when F is bounded, it would be interesting to investigate the existence and uniqueness of solutions to (2.1), where the constitutive relation is now replaced by (5.1). Rudimentary comparisons of solutions to the stress-rate problem and the strain-rate problem are performed in [10], but results concerning the existence of global weak solutions to multidimensional stress-rate type models are still lacking.…”

Section: Conclusion and Open Problemsmentioning

confidence: 99%

Existence of large-data global weak solutions to a model of a strain-limiting viscoelastic body

Bulíček¹,

Patel²,

Şengül³

et al. 2020

Preprint

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We prove the existence of a unique large-data global-in-time weak solution to a class of models of the form u tt = div T + f for viscoelastic bodies exhibiting strain-limiting behaviour, where the constitutive equation, relating the linearised strain tensor ε(u) to the Cauchy stress tensor T, is assumed to be of the form ε(u t ) + αε(u) = F (T), where we definea T, for constant parameters α ∈ (0, ∞) and a ∈ (0, ∞), in any number d of space dimensions, with periodic boundary conditions. The Cauchy stress T is shown to belong to L 1 (Q) d×d over the space-time domain Q. In particular, in three space dimensions, if a ∈ (0, 2 7 ), then in fact T ∈ L 1+δ (Q) d×d for a δ > 0, the value of which depends only on a.

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“…As done by Rajagopal and Srinivasa [63] one can also incorporate thermal effects into the constitutive relation to describe the response of thermoviscoelastic solids. Also, in the light of (28), other thermodynamically consistent models for viscoelastic solids in the context of strain-limiting theory are possible to explore (see e.g., [14]).…”

Section: Elasticitymentioning

confidence: 99%