Global existence of solutions for the one-dimensional response of viscoelastic solids within the context of strain-limiting theory
type: Item
instance of: scholarly article
title: Global existence of solutions for the one-dimensional response of viscoelastic solids within the context of strain-limiting theory
zbMATH Open document ID: 1506.74078
DOI: 10.1007/978-3-031-04496-0_14
author: Yasemin Şengül
publication date: 2022-10-09T00:00:00Z
review text: This paper deals with the global existence of solutions for the one-dimensional response of viscoelastic solids within the context of the strain-limiting theory. Following the work of Rajagopal, for the strain-limiting model in the one-dimensional case and its extension to rate-type viscoelastic materials by proposing an implicit constitutive equation and including an additional term corresponding to linearized strain rate, the author discusses a constitutive relation which is a generalization of the Kelvin-Voigt viscoelastic model. This outlines the strength of this constitutive relation that allows for creep and the shortcoming that it does not allow for stress relaxation. The research and study of the limiting strain models has attracted the attention of many investigators. In the present paper, the author discusses the existence of the displacement after a careful review of local-in-time existence results. It is shown that the energy diminishes with time. The proves, in addition to other results, the existence of the global solutions utilizing the investigation about the energy and a lemma where the blow-up conditions are stated and proven. The paper is quite mathematical and derives the required results by employing a number of theorems and lemmas. For the entire collection see [Zbl 1495.74001].
Mathematics Subject Classification ID: 74D10
Mathematics Subject Classification ID: 74A20
Mathematics Subject Classification ID: 74H20
Mathematics Subject Classification ID: 35Q74
zbMATH DE Number: 7598425
zbMATH Keywords: generalized Kelvin-Voigt viscoelastic model
zbMATH Keywords: blow-up condition
zbMATH Keywords: Helmholtz energy
zbMATH Keywords: energy decay
reviewed by: Vinod K. Arya
MaRDI profile type: Publication
full work available at URL: 978-3-031-04496-0 14
OpenAlex ID: W4312690536
cites work: On elastic solids with limiting small strain: modelling and analysis
cites work: Analysis and approximation of a strain-limiting nonlinear elastic model
cites work: Existence of large-data global weak solutions to a model of a strain-limiting viscoelastic body
cites work: Some topics on a new class of elastic bodies
cites work: A thermodynamically consistent stress-rate type model of one-dimensional strain-limiting viscoelasticity
cites work: Local existence of solutions to the initial-value problem for one-dimensional strain-limiting viscoelasticity
cites work: Contacting crack faces within the context of bodies exhibiting limiting strains
cites work: On the states of stress and strain adjacent to a crack in a strain-limiting viscoelastic body
cites work: Crack problem within the context of implicitly constituted quasi-linear viscoelasticity
cites work: Unsteady motions of a new class of elastic solids
cites work: On strong ellipticity for implicit and strain-limiting theories of elasticity
cites work: A note on the wave equation for a class of constitutive relations for nonlinear elastic bodies that are not Green elastic
cites work: On Constitutive Equations For Anisotropic Nonlinearly Viscoelastic Solids
cites work: A finite element analysis of some boundary value problems for a new type of constitutive relation for elastic bodies
cites work: A new class of quasi-linear models for describing the nonlinear viscoelastic response of materials
cites work: A numerical study of a plate with a hole for a new class of elastic bodies
cites work: On implicit constitutive theories.
cites work: The elasticity of elasticity
cites work: A note on a reappraisal and generalization of the Kelvin-Voigt model
cites work: On a new class of models in elasticity
cites work: Nonlinear elastic bodies exhibiting limiting small strain
cites work: Conspectus of concepts of elasticity
cites work: On the nonlinear elastic response of bodies in the small strain range
cites work: An implicit constitutive relation for describing the small strain response of porous elastic solids whose material moduli are dependent on the density
cites work: Shear Waves in a Class of Nonlinear Viscoelastic Solids
cites work: A thermodynamic framework for rate-type fluid models
cites work: Modeling fracture in the context of a strain-limiting theory of elasticity: a single anti-plane shear crack
cites work: Viscoelasticity with limiting strain
description: scientific article; zbMATH DE number 7598425