A Qualitative Study of Nonlocal Models in Mechanics

力学非局部模型的定性研究

• 批准号: 1910180
• 负责人: Tadele Mengesha
• 金额: $ 18.44万
• 依托单位: University of Tennessee Knoxville
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-08-15 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1910180&HistoricalAwards=false
• 关键词: Qualitative; Study; Nonlocal; Models; Mechanics

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.

该项目支持首席研究员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，并且已被证明对于模拟某些具有挑战性的现象非常有效，例如固体力学中的断裂。例如，了解材料在变形时的行为、失效和强度，对材料的正确使用以及对制造、材料工程和相关技术有潜在影响的新材料的设计至关重要。为此目的，过去提出了各种成功程度不同的模型。PI旨在发展基本的数学技术，为最近提出的“周期动力学模型”以及其他非局部连续介质力学模型提供可靠的分析基础。这些发现也将适用于具有类似结构的其他模型，并在社会和生物科学中得到应用。这些活动不仅有助于建模开发和实验验证的成功和有效性，而且还将确保未来基于这些非局部理论的建模和仿真工作将更加定量和可靠。该项目将为研究生的培养提供机会和支持。首席研究员将把研究结果整合到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，特别是对于周动力模型。这些模型的特点是用积分方程代替微分方程，在单一的数学框架内有效地描述连续和不连续场。这些模型已经成功地应用于更好地描述跳跃随机过程、异常扩散以及固体中裂纹的自发形成和扩展，仅举几例。然而，这些模型也给科学界带来了新的数学挑战。本研究在探索一些解析性问题的同时，也为今后非局部和周动力模型的研究奠定了必要的数学基础。要解决的问题包括证明实际兴趣的线性化非局部模型的良好定态性，建立与充分研究的应变梯度模型的严格联系，证明非局部方程解作为数据函数的正则性，以及实施变分技术来研究非局部非线性问题的某些方面。这些方法涉及到将经典数学概念和技术扩展到非局部环境的各种工具，包括微扰方法、变分法和非线性泛函分析。此外，将制定的基本数学基础设施可能会影响有效可靠的有限元方法和其他数值方案的发展，以解决涉及非定域性的复杂工程问题。该研究将使非局部和基于周围动力学的建模和仿真在实际应用中更具数学一致性、定量和预测性。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Calderón-Zygmund theory for non-convolution type nonlocal equations with continuous coefficient

连续系数非卷积型非局部方程的Calderón-Zygmund理论

• DOI: 10.1007/s42985-022-00161-8
• 发表时间: 2022
• 期刊: Partial Differential Equations and Applications
• 作者: Fall, Mouhamed Moustapha;Mengesha, Tadele;Schikorra, Armin;Yeepo, Sasikarn
• 通讯作者: Yeepo, Sasikarn

Connections between nonlocal operators: from vector calculus identities to a fractional Helmholtz decomposition. F

非局部算子之间的连接：从向量微积分恒等式到分数亥姆霍兹分解。

• 发表时间: 2022
• 期刊: Fractional Calculus Applied Analysis
• 作者: D'Elia, M.;Gulian, M.;Mengesha, T.;Scott, J. M.
• 通讯作者: Scott, J. M.

Asymptotic Analysis of a Coupled System of Nonlocal Equations with Oscillatory Coefficients

• DOI: 10.1137/19m1288085
• 发表时间: 2019-09
• 期刊: Multiscale Model. Simul.
• 作者: J. Scott;T. Mengesha

A fractional Korn-type inequality for smooth domains and a regularity estimate for nonlinear nonlocal systems of equations

光滑域的分数科恩型不等式和非线性非局部方程组的正则估计

• DOI: 10.4310/cms.2022.v20.n2.a5
• 发表时间: 2022
• 期刊: Communications in Mathematical Sciences
• 影响因子: 1
• 作者: Mengesha, Tadele;Scott, James M.
• 通讯作者: Scott, James M.

The solvability of a strongly-coupled nonlocal system of equations

强耦合非局部方程组的可解性

• DOI: 10.1016/j.jmaa.2020.123919
• 发表时间: 2020
• 期刊: Journal of mathematical analysis and applications
• 影响因子: 1.3
• 作者: Mengesha, Tadele;Scott, James M.
• 通讯作者: Scott, James M.