Stability of nonlocal Dirichlet integrals and implications for peridynamic correspondence material modeling
For researchers in nonlocal modeling and peridynamics, this clarifies conditions for stability of nonlocal gradient-based energy functionals, correcting earlier assumptions.
This work analyzes the coercivity and stability of a nonlocal Dirichlet integral based on nonlocal gradients, finding that stability depends on the choice of interaction kernel, contrary to previous claims. This has implications for peridynamic correspondence material models.
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.