  We present a nonlocal formulation of contact mechanics that accounts for the
interplay of deformations due to multiple contact forces acting on a single
particle. The analytical formulation considers the effects of nonlocal
mesoscopic deformations characteristic of confined granular systems and,
therefore, removes the classical restriction of independent contacts. This is
in sharp contrast to traditional contact mechanics theories, which are strictly
local and assume that contacts are independent regardless the confinement of
the particles. For definiteness, we restrict attention to elastic spheres in
the absence of gravitational forces, adhesion or friction. Hence, a notable
feature of the nonlocal formulation is that, when nonlocal effects are
neglected, it reduces to Hertz theory. Furthermore, we show that, under the
preceding assumptions and up to moderate macroscopic deformations, the
predictions of the nonlocal contact formulation are in remarkable agreement
with detailed finite-element simulations and experimental observations, and in
large disagreement with Hertz theory predictions---supporting that the
assumption of independent contacts only holds for small deformations. The
discrepancy between the extended theory presented in this work and Hertz theory
is borne out by studying periodic homogeneous systems and disordered
heterogeneous systems.
