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

2010年，美国国家科学基金会（NSF）授予“国际研究和教育合作伙伴关系”（PIRE）资助“数学，力学和材料科学之间的三重点科学”，该领域竞争激烈，有数百项申请。该资助旨在围绕数学、力学和材料科学的主要研究领域建立国际工作组网络。牛津大学是与美国参与机构合作的四个指定欧洲合作伙伴之一，以发展研究合作，这将使Pire资助的参与者和欧洲研究人员都受益。这一建议是针对应用数学和力学的问题，其中所产生的材料科学。许多当代先进材料的问题都与其制造和功能中固有的长度和时间尺度以及异质性有关。这些复杂系统的预测理论需要新的高等数学，其发现将通过国际合作得到加强。新的数学可以抽象出一个领域的方法，并将其应用于其他领域，促进深远的交叉施肥和发现意想不到的联系。团队的优势互补和专业知识的结合将把应用分析推向这些新的前沿。能量最小化的图案形成。PIRE团队的几位成员围绕材料科学和变分法之间的界面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已经成熟的主题。每一个领域都有几个团队成员拥有专业知识;因此，与此PIRE相关的增强沟通将大大加快我们的进展。(a)弹性床单，树叶，还有花。(b)降维（c）应力外延膜（d）晶体塑性中的位错微观结构。一个反复出现的主题是寻找无ansatz下限。猜测最小能量状态通常很容易（自然界给了我们一个提示）。理解为什么猜测是正确的-为什么没有其他国家可以做得更好-通常要困难得多。从原子到连续体建模和计算的挑战。局部缺陷，如位错，裂纹尖端，或晶界相互作用跨越大的长度尺度，通过弹性场。局部缺陷的精确模拟需要一个原子模型-然而，这是太计算要求用于整个系统。因此，吸引力的原子到连续耦合，这使得人们可以使用计算密集的原子模型，只有附近的缺陷。在远处，变形几乎是均匀的，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，热滞后是指冷却时的相变温度与加热时的相变温度不同。在应力诱导相变过程中也会出现滞后现象，其中诱导正向相变所需的应力不同于引起反向相变的应力。类似的效应也发生在铁磁性和铁电性中。最近，这个主题已经获得了新的意义与材料的能量转换，因为转换过程的效率往往取决于相关的磁滞回线的大小。细胞和颗粒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态的。在演化过程中，晶粒和边界网络的能量学和连通性在确定材料在各种尺度上的性质方面起着至关重要的作用。

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.