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Quasistatic Nonlinear Viscoelasticity and Gradient Flows

Abstract

We consider the equation of motion for one-dimensional nonlinear viscoelasticity of strain-rate type under the assumption that the stored-energy function is \(\lambda \)-convex, which allows for solid phase transformations. We formulate this problem as a gradient flow, leading to existence and uniqueness of solutions. By approximating general initial data by those in which the deformation gradient takes only finitely many values, we show that under suitable hypotheses on the stored-energy function the deformation gradient is instantaneously bounded and bounded away from zero. Finally, we discuss the open problem of showing that every solution converges to an equilibrium state as time \(t \rightarrow \infty \) and prove convergence to equilibrium under a nondegeneracy condition. We show that this condition is satisfied in particular for any real analytic cubic-like stress-strain function.

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Acknowledgments

We are grateful to Gero Friesecke, Bob Pego and Endre Süli for useful discussions. We also thank the referee for valuable comments. The research of both authors was partly supported by EPSRC Grant EP/D048400/1 and by the EPSRC Science and Innovation award to the Oxford Centre for Nonlinear PDE (EP/E035027/1). The research of JMB was also supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant Agreement No 291053 and by a Royal Society Wolfson Research Merit Award. The research of YŞ was also supported by TÜBİTAK fellowship 2213.

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Correspondence to J. M. Ball.

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Dedicated to the memory of Klaus Kirchgässner.

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Ball, J.M., Şengül, Y. Quasistatic Nonlinear Viscoelasticity and Gradient Flows. J Dyn Diff Equat 27, 405–442 (2015). https://doi.org/10.1007/s10884-014-9410-1

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Keywords

  • Viscoelasticity
  • Gradient flows
  • Nonlinear partial differential equations
  • Infinite-dimensional dynamical systems

Mathematics Subject Classification

  • 35A01
  • 35A02
  • 74D10
  • 82B26