  In this paper we study the asymptotic behavior of solutions to systems of
strongly coupled integral equations with oscillatory coefficients. The system
of equations is motivated by a peridynamic model of the deformation of
heterogeneous media that additionally accounts for short-range forces. We
consider the vanishing nonlocality limit on the same length scale as the
heterogeneity and show that the system's effective behavior is characterized by
a coupled system of local equations that are elliptic in the sense of
Legendre-Hadamard. This effective system is characterized by a fourth-order
tensor that shares properties with Cauchy elasticity tensors that appear in the
classical equilibrium equations for linearized elasticity.
