Mathematical Analysis on Peridynamic Models

该奖项支持首席研究员正在进行的研究计划，对“非局部模型”进行密切的分析检查，这是一种相对较新的数学模型，已被证明对某些具有挑战性的现象建模非常有效，例如固体力学中的断裂。 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。 本研究计划旨在发展基本的数学技术，为最近提出的连续介质力学的“周向模型”提供适当的分析基础。 研究结果也将适用于其他类似结构的非局部模型，在社会和生物科学中的应用。 这些活动不仅将有助于建模开发和实验验证尝试的成功和有效性，而且还将确保基于这些非局部理论的未来建模和模拟工作更加定量和可靠。 该研究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，通过使用积分方程代替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，同时为未来的非局部和周期性模式的分析奠定必要的数学基础。 要解决的问题包括研究活动，推进知识的分析线性化peridendicic模型的实际利益;规律性的非局部方程的解决方案作为一个功能的作用力，初始数据和系数;和peridendic-based非线性行为的理解。 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，以解决复杂的工程问题，通过周波。 该研究将使基于非局部和周期性的建模和仿真在数学上更加一致，并将有助于使这种建模和仿真在实际应用中更加定量和预测。

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.