The peridynamics model was proposed, in 2000, by Stuart Silling of the Sandia National Laboratories as a nonlocal reformulation of solid mechanics. By nonlocal, we mean that continuum points separated by a finite distance may exert force upon each other. The peridynamics theory is based on integral equations and does not assume even the weak differentiability of the displacement field, in contrast with classical continuum mechanics. The dependence upon differentiability of the displacement field limits the direct applicability of classical elasticity continuum mechanics, whereas discontinuous displacements represent no mathematical or computational difficulty for peridynamics. Consequently, peridynamics has frequently been applied in the study of material failure. Peridynamics is a member of
a larger class of nonlocal formulations of solid mechanics.
Our preliminary work on peridynamics (PD) has focused on three aspects. First, we have begun the development, analysis, and implementation of finite element methods for the discretization of a variational formulation of the PD model. So far, we have studied continuous and discontinuous Galerkin (CG and DG) methods for 1D problems. The use of DG methods is especially interesting because the offer the possibility of accurately resolving discontinuous phenomena, e.g., cracks. Two significant advantages of PD over classical elasticity (CE) is that, because it is free of spatial derivatives, PD models admit discontinuous solutions and that, for PD, DG methods can be implemented without the need for introducing additional edge terms to account for fluxes across element boundaries.
We have also shown that PD can be cast as an upscaling of molecular dynamics (MD). Specifically, we address the extent to which the solutions of MD simulations can be recovered by PD. Analytical comparisons of the equations of motion and dispersion relations for MD and PD have been derived along with supporting computational results comparing PD and CE with MD that at MD length scales and nearby length scales, PD is much better suited, compared to CE, for capturing the nonlocal behavior of MD.
We have also undertaken, beginning with the development of a nonlocal calculus for nonlocal boundary value problems, a systematic study of variational formulations of PD, including well posedeness, boundary conditions, and the characterization of function spaces and admissible solutions. In particular, we have developed a nonlocal Gauss's theorem, nonlocal Green's identities, and nonlocal Green's functions and applied them to Dirichlet and Neumann problems for nonlocal scalar convection-diffusion-reaction problems. Our preliminary results provide a fairly complete picture for the 1D linear PD setting.
Current work is directed at extending the nonlocal calculus to vector-valued mappings and then using it as a tool for the further analysis of PD models in 2D and 3D. We are also investigating PD models with random data and also the use of PD as a replacement for CE in atmomistic to continuum coupled materials model, with the quadrature rule-quasicontinuum method replacing MD. In the latter effort, our goal is to help eliminate the gap that seems to exist between the validity of CE and the tractability of MD in AtC coupled models based on those two constituents.
Comparison of the dispersion between MD, PD, and CE
Comparison between MD, PD, and CE of the evolution of a discontinuous initial condition
Discontinuous finite element approximations corresponding to a discontinuous exact solution
M. Gunzburger and R. Lehoucq; A nonlocal vector calculus with application to nonlocal boundary value problems, submitted.
X. Chen and M. Gunzburger; Finite element methods for a peridynamics model of mechanics, in preparation.
K. Zhou, Q. Du, M. Gunzburger, and R. Lehoucq; A vector nonlocal calculus with application to the peridynamic model of materials, in preparation.
Last updated: 12/29/09 by Max Gunzburger