Nonlocal gradient operators are basic elements of nonlocal vector calculusthat play important roles in nonlocal modeling and analysis. In this work, weextend earlier analysis on nonlocal gradient operators. In particular, we studya nonlocal Dirichlet integral that is given by a quadratic energy functionalbased on nonlocal gradients. Our main finding, which differs from claims madein previous studies, is that the coercivity and stability of this nonlocalcontinuum energy functional may hold for some properly chosen nonlocalinteraction kernels but may fail for some other ones. This can be significantfor possible applications of nonlocal gradient operators in various nonlocalmodels. In particular, we discuss some important implications for theperidynamic correspondence material models.
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