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Global Existence of Solutions for the One-Dimensional Response of Viscoelastic Solids Within the Context of Strain-Limiting Theory

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Research in Mathematics of Materials Science

Part of the book series: Association for Women in Mathematics Series ((AWMS,volume 31))

Abstract

In this work global-in-time existence of solutions for the initial-value problem based on a model explaining nonlinear response of one-dimensional viscoelastic solids exhibiting limiting strain behaviour is proved. In this model, a nonlinear constitutive relation depending on the rate of change of the linearized strain is considered. This constitutive equation is obtained from implicit constitutive theory under the smallness assumption of the strain and the strain rate. Local-in-time existence of solutions for this problem was obtained in Erbay et al. (J Differ Equ 269:9720–9739, 2020) under certain assumptions on the nonlinearity. In this work, firstly, local existence for the displacement is given. Then, the blow-up condition is stated and an equivalent condition is formulated. Finally, an energy inequality is proven which is used to show that the local solutions, in fact, exist globally.

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Author information

Authors and Affiliations

  1. School of Mathematics, Cardiff University, Cardiff, UK

    Yasemin Şengül

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  1. Yasemin Şengül
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Correspondence to Yasemin Şengül .

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Editors and Affiliations

  1. School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, USA

    Malena I. Español

  2. Mathematics Department, University of Pittsburgh, Pittsburgh, PA, USA

    Marta Lewicka

  3. Department of Mathematics, Heriot-Watt University, Edinburgh, UK

    Lucia Scardia

  4. Institute of Mathematics, Universität Würzburg, Würzburg, Germany

    Anja Schlömerkemper

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Şengül, Y. (2022). Global Existence of Solutions for the One-Dimensional Response of Viscoelastic Solids Within the Context of Strain-Limiting Theory. In: Español, M.I., Lewicka, M., Scardia, L., Schlömerkemper, A. (eds) Research in Mathematics of Materials Science. Association for Women in Mathematics Series, vol 31. Springer, Cham. https://doi.org/10.1007/978-3-031-04496-0_14

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