Science at the Triple Point between Mathematics, Mechanics and Materials Science

数学、力学和材料科学之间的三重点科学

• 批准号: EP/J014494/1
• 负责人: John Ball
• 金额: $ 3.87万
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2012
• 资助国家: 英国
• 起止时间: 2012 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FJ014494%2F1
• 关键词: Science; Triple; Point; between; Mathematics

In 2010 the US National Science Foundation (NSF) awarded a 'Partnerships for International Research and Education' (PIRE) grant for "Science at the Triple Point between Mathematics, Mechanics and Materials Science" from a highly competitive field of hundreds of applications. The grant was awarded to form a network of international working groups around major research areas in Mathematics, Mechanics and Materials Science. The University of Oxford is one of four named European Partners working with US Participating Institutions to develop research collaborations that will benefit both PIRE-funded participants and European researchers alike. This proposal is directed to issues in applied mathematics and mechanics which arise from materials science. Many contemporary problems in new and advanced materials are related to the variety of length and time scales and heterogeneities inherent in their fabrication and function. Predictive theories for these complex systems require new advanced mathematics whose discovery will be enhanced by international collaboration. New mathematics can abstract methods developed in one area and apply them to other areas, facilitating far-reaching cross-fertilization and the discovery of unanticipated linkages. The complementary strengths and combined expertise of the team will push applied analysis to these new frontiers.The proposal concerns four main subjects:1. Pattern Formation from Energy Minimization. Several members of the PIRE team have organized their careers around problems at the interface between materials science and the calculus of variations. We have identified four topics that seem ripe for near-term development. Each is an area where several team members have expertise; therefore the enhanced communication associated with this PIRE will greatly accelerate our progress.(a) Elastic sheets, leaves, and flowers.(b) Dimension reduction(c) Stressed epitaxial films(d) Dislocation microstructures in crystal plasticity.A recurring theme is the search for ansatz-free lower bounds. Guessing the minimum-energy state is usually easy (nature gives us a hint). Understanding why the guess is right - why no other state can do better - is typically much more difficult.2. Challenges in Atomistic to Continuum Modeling and Computing. Localized defects such as dislocations, crack tips, or grain boundaries interact across large length scales though elastic fields. Accurate simulation of localized defects requires an atomistic model - which however is too computationally demanding to be used for the entire system. Hence the attraction of atomistic-to-continuum coupling, which permits one to use the computationally intensive atomistic model only near the defects. Far away, where the deformation is nearly uniform, a continuum elastic model provides adequate resolution.3 Prediction of Hysteresis.For a solid-to-solid phase transformation, thermal hysteresis refers to a transformation temperature on cooling that differs from that on heating. Hysteresis also occurs during stress-induced transformation, with the stress needed to induce the forward transformation being different from that causing the reverse transformation. Similar effects occur in ferromagnetism and ferroelectricity. Recently this topic has acquired fresh significance in connection with materials for energy conversion, since the efficiency of a conversion process often depends on the size of an associated hysteresis loop.4 Pattern Dynamics and Evolution of Material Microstructure. Cellular and granular networks are ubiquitous in nature. They exhibit behaviour on many different length and time scales and are often found to be metastable. The energetics and connectivity of the ensemble of the grain and the boundary network during evolution play a crucial role in determining the properties of a material across a wide range of scales.

2010年，美国国家科学基金会(NSF)授予“数学、力学和材料科学三叉点科学”国际研究与教育伙伴关系(PERR)奖，该奖项来自一个竞争激烈的领域，共有数百项申请。这笔拨款是为了围绕数学、力学和材料科学的主要研究领域建立一个国际工作组网络。牛津大学是四个指定的欧洲合作伙伴之一，他们与美国的参与机构合作，发展研究合作，这将使帝国资助的参与者和欧洲研究人员都受益。这项建议是针对材料科学中出现的应用数学和力学问题。新材料和先进材料中的许多当代问题与长度和时间尺度的多样性以及它们的制造和功能所固有的异质性有关。这些复杂系统的预测理论需要新的高等数学，这些数学的发现将通过国际合作得到加强。新的数学可以抽象出在一个领域开发的方法，并将其应用于其他领域，促进深远的相互影响和发现意想不到的联系。该团队的互补优势和综合专业知识将把应用分析推向这些新的领域。该提案涉及四个主要主题：1.从能量最小化形成模式。帝国团队的几名成员围绕材料科学和变分法之间的接口问题组织了他们的职业生涯。我们已经确定了四个近期发展似乎成熟的主题。每个领域都有几个团队成员拥有专业知识；因此，与该PERE相关的加强交流将极大地加速我们的进展。(A)弹性片材、树叶和花朵。(B)降维(C)应力外延薄膜(D)晶体塑性中的位错微结构。一个反复出现的主题是寻找无反饱和的下限。猜测最低能量状态通常很容易(大自然给了我们一个提示)。理解为什么这个猜测是正确的--为什么没有其他州能做得更好--通常要困难得多。原子论对连续体建模和计算的挑战。位错、裂纹尖端或晶界等局域缺陷通过弹性场在大尺度上相互作用。对局部缺陷的准确模拟需要一个原子模型--然而，该模型对计算要求太高，不能用于整个系统。因此，原子到连续介质的耦合很有吸引力，它允许人们只在缺陷附近使用计算密集型原子模型。在变形接近均匀的远处，连续弹性模型提供了足够的分辨率。3滞后的预测。对于固-固相变，热滞是指冷却时的相变温度不同于加热时的相变温度。在应力诱导相变过程中也会出现滞后现象，诱导正向相变所需的应力与引起反向相变的应力不同。类似的效应也发生在铁磁性和铁电性上。最近，这一主题在能源转换材料方面具有了新的意义，因为转换过程的效率往往取决于相关滞后环的大小。4材料微结构的模式动力学和演化。蜂窝和颗粒状网络在自然界中无处不在。它们在许多不同的长度和时间尺度上表现出行为，并且经常被发现是亚稳定的。在演化过程中，颗粒和边界网络集合的能量学和连通性在决定材料在广泛范围内的性质方面发挥着至关重要的作用。

项目成果

An investigation of non-planar austenite-martensite interfaces

非平面奥氏体-马氏体界面的研究

• DOI: 10.1142/s0218202514500122
• 发表时间: 2014
• 期刊: Mathematical Models and Methods in Applied Sciences
• 影响因子: 3.5
• 作者: Ball J

Incompatible Sets of Gradients and Metastability

• DOI: 10.1007/s00205-015-0883-9
• 发表时间: 2014-07
• 期刊: Archive for Rational Mechanics and Analysis
• 影响因子: 2.5
• 作者: J. Ball;Richard D. James

Geometry of polycrystals and microstructure

多晶的几何形状和微观结构

• DOI: 10.1051/matecconf/20153302007
• 发表时间: 2015
• 期刊: MATEC Web of Conferences
• 作者: Ball J

Kink pair production and dislocation motion.

• DOI: 10.1038/srep39708
• 发表时间: 2016-12-22
• 期刊: Scientific reports
• 影响因子: 4.6
• 作者: Fitzgerald SP

Partial regularity and smooth topology-preserving approximations of rough domains

粗糙域的部分正则性和光滑拓扑保持近似

• DOI: 10.1007/s00526-016-1092-6
• 发表时间: 2017
• 期刊: Calculus of Variations and Partial Differential Equations
• 影响因子: 2.1
• 作者: Ball J