  Nonlocal gradient operators are basic elements of nonlocal vector calculus
that play important roles in nonlocal modeling and analysis. In this work, we
extend earlier analysis on nonlocal gradient operators. In particular, we study
a nonlocal Dirichlet integral that is given by a quadratic energy functional
based on nonlocal gradients. Our main finding, which differs from claims made
in previous studies, is that the coercivity and stability of this nonlocal
continuum energy functional may hold for some properly chosen nonlocal
interaction kernels but may fail for some other ones. This can be significant
for possible applications of nonlocal gradient operators in various nonlocal
models. In particular, we discuss some important implications for the
peridynamic correspondence material models.
