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Emergence of Coherent Localized Structures in Shear Deformations of Temperature Dependent Fluids

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Abstract

Shear localization occurs in various instances of material instability in solid mechanics and is typically associated with Hadamard-instability for an underlying model. While Hadamard instability indicates the catastrophic growth of oscillations around a mean state, it does not by itself explain the formation of coherent structures typically observed in localization. The latter is a nonlinear effect and its analysis is the main objective of this article. We consider a model that captures the main mechanisms observed in high strain-rate deformation of metals, and describes shear motions of temperature dependent non-Newtonian fluids. For a special dependence of the viscosity on the temperature, we carry out a linearized stability analysis around a base state of uniform shearing solutions, and quantitatively assess the effects of the various mechanisms affecting the problem: thermal softening, momentum diffusion and thermal diffusion. Then, we turn to the nonlinear model, and construct localized states—in the form of similarity solutions—that emerge as coherent structures in the localization process. This justifies a scenario for localization that is proposed on the basis of asymptotic analysis in Katsaounis and Tzavaras (SIAM J Appl Math 69:1618–1643, 2009).

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Authors and Affiliations

  1. Computer, Electrical, Mathematical Sciences and Engineering Division (CEMSE), King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia

    Theodoros Katsaounis & Athanasios E. Tzavaras

  2. Department of Mathematics and Applied Mathematics, University of Crete, Heraklion, Greece

    Theodoros Katsaounis

  3. Centre de Mathématiques et Informatique, Aix-Marseille Université, Marseille, France

    Julien Olivier

Authors
  1. Theodoros Katsaounis
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  2. Julien Olivier
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  3. Athanasios E. Tzavaras
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Corresponding author

Correspondence to Athanasios E. Tzavaras.

Additional information

Communicated by A. Bressan

Research partially supported by the EU FP7-REGPOT project “Archimedes Center for Modeling, Analysis and Computation” and the “DIKICOMA” Project of the Hellenic Secretariat of Research and Technology. Part of this work was completed at the Department of Applied Mathematics, University of Crete, Greece.

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Katsaounis, T., Olivier, J. & Tzavaras, A.E. Emergence of Coherent Localized Structures in Shear Deformations of Temperature Dependent Fluids. Arch Rational Mech Anal 224, 173–208 (2017). https://doi.org/10.1007/s00205-016-1071-2

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  • DOI: https://doi.org/10.1007/s00205-016-1071-2

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