  As a nonlocal extension of continuum mechanics, peridynamics has been widely
and effectively applied in different fields where discontinuities in the field
variables arise from an initially continuous body. An important component of
the constitutive model in peridynamics is the influence function which weights
the contribution of all the interactions over a nonlocal region surrounding a
point of interest. Recent work has shown that in solid mechanics the influence
function has a strong relationship with the heterogeneity of a material's
micro-structure. However, determining an accurate influence function
analytically from a given micro-structure typically requires lengthy
derivations and complex mathematical models. To avoid these complexities, the
goal of this paper is to develop a data-driven regression algorithm to find the
optimal bond-based peridynamic model to describe the macro-scale deformation of
linear elastic medium with periodic heterogeneity. We generate macro-scale
deformation training data by averaging over periodic micro-structure unit cells
and add a physical energy constraint representing the homogenized elastic
modulus of the micro-structure to the regression algorithm. We demonstrate this
scheme for examples of one- and two-dimensional linear elastodynamics and show
that the energy constraint improves the accuracy of the resulting peridynamic
model.
