A Qualitative Study of Nonlocal Models in Mechanics

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This project supports the principal investigator's ongoing research program on the close analytical examination of nonlocal models in mechanics. These mathematical models are of relatively recent vintage and have 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. The PI aims to develop basic mathematical techniques that will deliver a sound analytical footing for the recently proposed "peridynamic model" as well as other nonlocal models of continuum mechanics. The findings will also be applicable to additional models having similar structure, with applications in social and biological sciences. The activities will not only contribute to the success and effectiveness of attempts on modeling development and experimental validation but will also ensure that future modeling and simulation efforts based on these nonlocal theories will be more quantitative and reliable. This 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 mechanics 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 instead 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 studies of nonlocal and peridynamic models. Issues to be addressed include demonstratiing well posedness of linearized nonlocal models of practical interest, establishing a rigorous connection with well-studied strain-gradient models, proving regularity properties of solutions of nonlocal equations as a function of the data, and implementing variational techniques to study some aspects of nonlocal nonlinear problems. 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 infrastructure that will be worked out is likely to impact the development of effective and reliable finite element methods and other numerical schemes to solve complex engineering problems that involve nonlocality. The research will make nonlocal and peridynamics-based modeling and simulation more mathematically consistent, quantitative, and predictive in practical applications.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.