Mathematical Analysis on Peridynamic Models

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This award supports the ongoing research program of the Principal Investigator on the close analytical examination of "nonlocal models," a type of mathematical model of relatively recent vintage that has proved to be very effective for modeling certain challenging phenomena, such as fracture in solid mechanics. For example, understanding how materials behave, their failure as well as their strength when deformed, is crucial for their proper usage and also for the design of new materials with potential impact on manufacturing, materials engineering, and related technologies. For this purpose, models with varied levels of success have been proposed in the past. This research project aims to develop basic mathematical techniques that will deliver proper analytical footing for the recently proposed "peridynamic model" of continuum mechanics. The findings will also be applicable to other nonlocal models of similar structure, with applications in social and biological sciences. The activities not only will contribute to the success and effectiveness of attempts on modeling development and experimental validation but also will ensure that future modeling and simulation efforts based on these nonlocal theories will be more quantitative and reliable. The research project will provide opportunities and support for the training of graduate students. The Principal Investigator will integrate the findings of the project into classroom teaching and other educational endeavors.This project concerns the development of theory and techniques for nonlocal models in general and for the peridynamic model in particular. These models are characterized by their effective description of continuous as well as discontinuous fields within a single mathematical framework by using integral equations in lieu of differential equations. The models have been successfully applied to better describe jump stochastic processes, anomalous diffusion, and spontaneous formation and propagation of cracks in solids, to name a few applications. However, the models have also presented the scientific community with new mathematical challenges. This research is devoted to exploring some analytical issues while at the same time laying the necessary mathematical foundation for future analyses on nonlocal and peridynamic models. Issues to be addressed include research activities that advance knowledge on the analysis of linearized peridynamic models of practical interest; regularity properties of solutions of nonlocal equations as a function of applied force, initial data, and coefficients; and understanding of peridynamic-based nonlinear behavior. The approaches involve various tools that lead to extensions of classical mathematical concepts and techniques to the nonlocal setting, including perturbation methods, calculus of variations, and nonlinear functional analysis. Furthermore, the basic mathematical infrastructures that will be worked out are likely to impact the development of effective and reliable finite element methods and other numerical schemes to solve complex engineering problems via peridynamics. The research will make nonlocal and peridynamics-based modeling and simulation more mathematically consistent, and it will contribute to making such modeling and simulation more quantitative and predictive in practical applications.