On a Model of Nonlocal Continuum Mechanics Part II: Structure, Asymptotics, and Computations |
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Authors: | Roger L Fosdick Darren E Mason |
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Institution: | (1) Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN, 55455, U.S.A.;(2) Center for Nonlinear Analysis, Department of Mathematics, Carnegie Mellon University, Pittsburgh, PA, 15213, U.S.A |
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Abstract: | We consider the asymptotic behavior and local structure of solutions to the nonlocal variational problem developed in the
companion article to this work, On a Model of Nonlocal Continuum Mechanics Part I: Existence and Regularity. After a brief
review of the basic setup and results of Part I, we conduct a thorough analysis of the phase plane related to an integro-differential
Euler--Lagrange equation and classify all admissible structures that arise as energy minimizing strain states. We find that
for highly elastic materials with relatively weak particle-particle interactions, the maximum number of internal phase boundaries
is two. Moreover, we also develop explicit bounds for the number of internal phase boundaries supported by any material and
show that this bound is essentially inversely related to the particle size. To understand the question of asymptotics, we
utilize the Young measure and show that in the sense of energetics and averages, minimizers of the full nonlocal problem converge
to minimizers of two limiting problems corresponding to both the large and small particle limits. In fact, in the small particle
limit, we find that the minimizing fields converge, up to a subsequence in strong-Lp, for 1 ≤ p < ∞, to fields that support either a single internal phase boundary, or two internal phase boundaries that are
distributed symmetrically about the body midpoint. We close this work with some computations that illustrate these asymptotic
limits and provide insight into the notion of nonlocal metastability.
This revised version was published online in August 2006 with corrections to the Cover Date. |
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Keywords: | phase transformations nonlocality calculus of variations |
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