A Qualitative Study of Nonlocal Models in Mechanics

该项目支持主要研究者正在进行的关于力学非局部模型的密切分析检查的研究计划。这些数学模型是相对较新的年份，并已被证明是非常有效的建模某些具有挑战性的现象，如固体力学断裂。例如，了解材料的行为，它们的失效以及变形时的强度，对于它们的正确使用以及对制造，材料工程和相关技术具有潜在影响的新材料的设计至关重要。为此目的，在过去已经提出了具有不同程度成功的模型。PI旨在开发基本的数学技术，为最近提出的“周向模型”以及连续介质力学的其他非局部模型提供良好的分析基础。这些发现也将适用于具有类似结构的其他模型，并应用于社会和生物科学。这些活动不仅将有助于成功和有效的尝试建模开发和实验验证，但也将确保未来的建模和模拟工作的基础上，这些非本地理论将更加定量和可靠。该项目将为培养研究生提供机会和支持。主要研究者将把本项目的研究成果融入课堂教学和其他教育工作中。本项目涉及一般力学中非局部模型的理论和技术的发展，特别是周向模型。这些模型的特点是在一个单一的数学框架内，通过使用积分方程，而不是微分方程的连续以及不连续的字段的有效描述。该模型已成功地应用于更好地描述跳跃随机过程，异常扩散，和自发形成和固体中的裂纹的传播，仅举几个应用。然而，这些模型也给科学界带来了新的数学挑战。本研究致力于探索一些分析问题，同时为今后的非局部和周期性模型的研究奠定必要的数学基础。要解决的问题包括论证适定性的线性化非局部模型的实际利益，建立一个严格的连接，以及研究应变梯度模型，证明作为数据的函数的非局部方程的解的正则性，并实施变分技术来研究非局部非线性问题的某些方面。这些方法涉及各种工具，导致经典的数学概念和技术的扩展到非局部设置，包括扰动方法，变分法，非线性泛函分析。此外，将制定的基本数学基础设施可能会影响有效和可靠的有限元方法和其他数值方案的发展，以解决涉及非局部性的复杂工程问题。该研究将使基于非局部和周期性的建模和仿真在实际应用中更加数学一致、定量和预测。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.