We present an existence theory based on minimization of the nonlocal energies appearing in peridynamics, which is a nonlocal continuum model in solid mechanics that avoids the use of deformation gradients. We employ the direct method of the calculus of variations in order to find minimizers of the energy of a deformation. Lower semicontinuity is proved under a weaker condition than convexity, w...

Abstract In this paper, we present numerical studies of a recently proposed scalar nonlocal nonlinear conservation law in one space dimension. The nonlocal model accounts for nonlocal interactions over a finite horizon and enjoys maximum principle, monotonicity-preserving and entropy condition on the continuum level. Moreover, it has a well-defined local limit given by a conventional local cons...

Modeling of the evolution of distributed damage such as microcracking, void formation, and softening frictional slip necessitates strain-softening constitutive models. The nonlocal continuum concept has emerged as an effective means for regularizing the boundary value problems with strain softening, capturing the size effects and avoiding spurious localization that gives rise to pathological me...

We propose a new hybrid quasicontinuum model which captures both long and short-wave instabilities of crystal lattices and combines the advantages of weakly nonlocal (gradient) and strongly nonlocal continuum models. To illustrate the idea we consider a simple one-dimensional lattice exhibiting commensurate and incommensurate short-wave instabilities and compute stability limits for the homogen...

We study the condition number of the stiffness matrix arising from finite element discretizations of nonlocal integral operators with singular and integrable kernels. Such operators are used in nonlocal diffusion, peridynamics formulation of continuum mechanics, image processing, and phase transition. We quantify of the extremal eigenvalues with respect to all underlying parameters; the horizon...

Most physical processes are the result of collective interactions across disparate length and time scales. The dynamic fracture of brittle solids is a particularly interesting collective interaction connecting large and small length scales. With the application of enough stress or strain to a sample of brittle material, atomistic-scale bonds will eventually snap, leading to fracture of the macr...