Journal of elasticity and the physical science of solids

, Volume 59, Issue 1, pp 319–355

Finite Scale Microstructures in Nonlocal Elasticity


  • Xiaofeng Ren
    • Department of Mathematics and StatisticsUtah State University
  • Lev Truskinovsky
    • Department of Aerospace Engineering and MechanicsUniversity of Minnesota

DOI: 10.1023/A:1011003321453

Cite this article as:
Ren, X. & Truskinovsky, L. Journal of Elasticity (2000) 59: 319. doi:10.1023/A:1011003321453


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.

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© Kluwer Academic Publishers 2000