Analytical Aspects of Nonlocal Models in Applications

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Nonlocal models in mechanics and other application areas are of relatively recent vintage but have proved to be very effective for modeling certain challenging phenomena, such as fracture in solid mechanics. Understanding how materials behave, their failure as well as their strength when deformed, is crucial for their proper usage and for the design of new materials, with potential impact in manufacturing, materials engineering, and related technologies. This project aims to develop fundamental mathematical techniques that will deliver a sound analytical footing for the so-called peridynamic model, as well as other nonlocal models of continuum mechanics. The findings are expected to be applicable to other models having similar structure, with applications in engineering sciences. The activities will contribute to improved effectiveness of these models and ensure that future modeling and simulation efforts based on these nonlocal theories will be more quantitative and reliable. The project will provide research training opportunities for doctoral students. The project centers on a rigorous study of nonlocal models in applications, primarily in mechanics. The models under consideration 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. These models have been successfully applied to better describe jump stochastic processes, super- and sub- diffusion, and spontaneous formation and propagation of cracks in solids, to name a few applications. However, they have also presented the scientific community with new mathematical challenges. This project is devoted to exploring some analytical issues while at the same time laying the necessary mathematical foundation for future studies of nonlocal models. Issues to be addressed include well-posedness of linearized nonlocal models that involve anisotropy and heterogeneity, asymptotic analysis of linear and nonlinear nonlocal models with oscillatory coefficients, development of a regularity theory of solutions as a function of applied force and material coefficients, and optimal design and optimal control problems in systems governed by nonlocal models. The mathematical approaches considered include perturbation methods, calculus of variations, and nonlinear functional analysis. The research aims to 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.