Variational Analysis of Nonlocal Models in Applications

University Of Tennessee Knoxville, Knoxville TN

Investigators

Abstract

This project aims to develop mathematical techniques for analyzing nonlocal models, which have proven effective in modeling complex natural, scientific, and engineering phenomena such as material fractures and anomalous diffusion processes. In mechanics, understanding material behavior, failure, and strength under deformation is crucial for their proper application and the design of new materials, with significant implications for manufacturing, materials engineering, and related technologies. This project provides a robust analytical foundation for the peridynamic model, widely used in the engineering community, as well as other nonlocal models employed in scientific research. The research activities will enhance the effectiveness of these models and ensure that future modeling and simulation efforts based on nonlocal theories are more quantitative and reliable. Additionally, the project offers valuable training opportunities for students and young researchers. The research topics provide a rich training ground and research experience that draw on ideas from various interdisciplinary areas. The project research activities aim to establish the necessary mathematical foundation for the recently proposed continuum-kinematics-inspired peridynamics model, which addresses many limitations of the well-known bond-based peridynamics. The principal investigator (PI) demonstrates the well-posedness of both nonlinear and linear models, provide rigorous derivations of simpler linearized models, and establish connections with local models. This research presents significant technical challenges not encountered in traditional local approaches. To address these challenges, the PI employs various methodologies and utilizes analytical tools from the theory of integral and differential equations, linear and nonlinear functional analysis, and calculus of variations. Implementing these approaches requires novel insights to adapt standard classical tools to the nonstandard nonlocal setting, thereby advancing knowledge in this area. The rigorous variational analysis and mathematical frameworks developed by the PI will significantly impact the creation of effective and reliable finite element methods and other numerical schemes for solving complex engineering problems using enhanced peridynamics models. 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.