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Finite Scale Microstructures in Nonlocal Elasticity

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Advances in Continuum Mechanics and Thermodynamics of Material Behavior

Abstract

In this paper we develop a simple one-dimensional model accounting for the formation and growth of globally stable finite scale microstructures. We extend Ericksen’s model [9] of an elastic “bar” with nonconvex energy by including both oscillation-inhibiting and oscillation-forcing terms in the energy functional. The surface energy is modeled by a conventional strain gradient term. The main new ingredient in the model is a nonlocal term which is quadratic in strains and has a negative definite kernel. This term can be interpreted as an energy associated with the long-range elastic interaction of the system with the constraining loading device. We propose a scaling of the problem allowing one to represent the global minimizer as a collection of localized interfaces with explicitly known long-range interaction. In this limit the augmented Ericksen’s problem can be analyzed completely and the equilibrium spacing of the periodic microstructure can be expressed as a function of the prescribed average displacement. We then study the inertial dynamics of the system and demonstrate how the nucleation and growth of the microstructures result in the predicted stable pattern. Our results are particularly relevant for the modeling of twined martensite inside the austenitic matrix.

Supported in part by NSF grant DMS-9703727.

Supported in part by NSF grants DMS-9501433, DMS-9803572.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Utah State University, Logan, UT, 84322, USA

    Xiaofeng Ren

  2. Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN, 55455, USA

    Lev Truskinovsky

Authors
  1. Xiaofeng Ren
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  2. Lev Truskinovsky
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Editor information

Editors and Affiliations

  1. University of Illinois, Urbana-Champaign, USA

    Donald E. Carlson

  2. University of Houston, USA

    Yi-Chao Chen

Additional information

Dedicated to Roger Fosdick.

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© 2000 Springer Science+Business Media Dordrecht

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Ren, X., Truskinovsky, L. (2000). Finite Scale Microstructures in Nonlocal Elasticity. In: Carlson, D.E., Chen, YC. (eds) Advances in Continuum Mechanics and Thermodynamics of Material Behavior. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0728-3_18

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  • DOI: https://doi.org/10.1007/978-94-010-0728-3_18

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-3837-9

  • Online ISBN: 978-94-010-0728-3

  • eBook Packages: Springer Book Archive

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