Stochastic analysis of one-dimensional heterogeneous solids with long-range interactions

Random mass distribution in one-dimensional (1D) elastic solids in the presence of long-range interactions is studied in
this paper. Besides the local Cauchy contact forces among adjacent elements, long-range forces depending on the product
of interacting masses, as well as on their relative displacements, are considered. In this context, the random fluctuations
of the mass distribution involve a stochastic model of the nonlocal interactions, and the random displacement field of the
body is provided as the solution of a stochastic integro-differential equation. The presence of the random field of mass
distribution is reflected in the random kernel of the solving integro-differential equation with deterministic static and
kinematic boundary conditions, since the long-range interactions have no effects at the borders. Numerical applications
are reported to highlight the effects of fluctuations of the mass field along the body on the long-range forces and the
mechanical response of the 1D elastic body considered.