ON A MODEL OF NONLOCAL CONTINUUM-MECHANICS PART II - STRUCTURE, ASYMPTOTICS, AND COMPUTATIONS

We consider the asymptotic behavior and local structure of solutions t o the nonlocal variational problem developed in the companion article to this work, On a Model of Nonlocal Continuum Mechanics Part I: Exist ence and Regularity. After a brief review of the basic setup and resul ts of Part I, we conduct a thorough analysis of the phase plane relate d to an integro-differential Euler-Lagrange equation and classify all admissible structures that arise as energy minimizing strain states. W e find that for highly elastic materials with relatively weak particle -particle interactions, the maximum number of internal phase boundarie s is two. Moreover, we also develop an explicit upper bound on the num ber of internal phase boundaries supported by any material and show th at this bound is essentially proportional to the particle size. To und erstand the question of asymptotics, we utilize the Young measure and show that in the sense of energetics and measure, minimizers of the fu ll nonlocal problem converge to minimizers of two limiting problems co rresponding to both the large and small particle limits. In fact, in t he small particle limit, we find that the minimizing fields converge, up to a subsequence in strong-L-p, for 1 less than or equal to p < inf inity, to fields that support either a single internal phase boundary, or two internal phase boundaries that are distributed symmetrically a bout the body midpoint. We close this work with some computations that illustrate these asymptotic limits and provide insight into the notio n of nonlocal metastability.