Mathematical Analysis for the Peridynamic Nonlocal Continuum

We develop a functional analytical framework for a linear peridynamic model of a spring network system in any space dimension. Various properties of the peridynamic operators are examined for general micromodulus functions. These properties are utilized to establish the well-posedness of both the stationary peridynamic model and the Cauchy problem of the time dependent peridynamic model. The connections to the classical elastic models are also provided. Mathematics Subject Classification. 45A05, 46N20, 74B99. Received September 3rd, 2009. Revised March 21, 2010. Published online August 2, 2010. Introduction The peridynamic (PD) model proposed by Silling [19] is an integral-type nonlocal continuum theory. It depends crucially upon the nonlocality of force interactions and does not explicitly involve the notion of deformation gradients. On one hand, it provides a more general framework than the classical theory for problems involving discontinuities or other singularities in the deformation; on the other hand, it can also be viewed as a continuum version of molecular dynamics. Although a relatively recent development, the effectiveness of PD model has already been demonstrated in several sophisticated applications, including the fracture and failure of composites, crack instability, fracture of polycrystals, and nanofiber networks. Yet, from a rigorous mathematical point of view, many important and fundamental issues remain to be studied. In this work, we intend to formulate a rigorous functional analytical framework of the PD models so as to provide a better understanding of the PD model and to guide us in the development and analysis of the numerical algorithms. This in turn will help us utilize the PD theory for multiscale materials modeling. Indeed, PD can be effectively used in the multiscale modeling of materials in different ways: it can serve as a bridge between molecular dynamics (MD) and continuum elasticity (CE) to help mitigate the difficulties encountered when one attempts to couple MD and CE directly [4,5,11,15,18] and, in some situations, PD can be used as a stand-alone model to capture the behavior of materials over a wide range of spatial and temporal scales. For example, to study problems involving defects, one can use the same equations of motion over the entire body and no special treatment is needed near or at defects [6,22].