Non-Convex Homogenization and Applications to (Ferromagnetic) Shape-Memory Polycrystals

非凸均质化及其在（铁磁）形状记忆多晶中的应用

• 批准号: 1108847
• 负责人: Pedro Ponte Castaneda
• 金额: $ 23.25万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-10-01 至 2015-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1108847&HistoricalAwards=false
• 关键词: Non; Convex; Homogenization; Applications; Ferromagnetic

Ponte CastanedaDMS-1108847 The main objective of this proposal is the development of mathematical tools for modeling the shape-memory effect and pseudo-elasticity in shape-memory polycrystals. For this purpose, the investigator makes use of suitable generalizations of the linear comparison homogenization methods for these material systems, incorporating the effects both of crystallographic texture and of morphological texture (as determined by the two-point correlations of the microstructures). The main mathematical challenge arises as a consequence of the lack of convexity (or, more precisely, quasi-convexity) of the relevant energy functions. It is known that this breakdown of convexity leads to the development of additional microstructures at the single crystal level, which lies at the heart of the shape-memory effect. Because of this, linear comparison estimates are also developed for the "relaxation" (or quasi-convexification) of single-crystal shape-memory alloys (SMAs), exploiting a recently uncovered connection between homogenization at the polycrystal level and relaxation at the single-crystal level. The project results in novel and highly efficient multi-scale modeling techniques capable of handling complex microstructures, coupled strongly nonlinear response, microstructure evolution, and the possible development of instabilities. The mathematical techniques developed in this work are of broad application to large classes of polymeric, metallic, biological, and geological material systems, including multi-functional materials, and lend themselves to numerical implementation in constitutive subroutines for use with standard numerical packages. The shape-memory effect is the ability of certain materials to recover, upon heating, apparently permanent deformation sustained below a certain critical temperature. This effect is usually accompanied by pseudo-elasticity whereby single-crystal samples of these materials are observed to undergo fairly large strains (in the order of 10%) at nearly constant stress, which is fully recovered upon unloading. These two properties make shape-memory alloys (SMAs) very attractive as low-frequency, robust actuators and sensing devices for a variety of novel technological applications. However, SMAs are normally used in polycrystalline form, typically as wires, or thin strips and films, where the shape-memory effect is much reduced. For this reason, it is essential to understand and model the relations between the single-crystal behavior, the microstructure, and the macroscopic behavior of the polycrystals. Improvements in the modeling of SMA polycrystals should result in improved performance for these materials, which are being used extensively in industry, as actuators, sensors, couplings, and electrical connectors, as well as in an ever increasing number of medical and robotics applications.

Ponte Castanedadms-1110847该建议的主要目的是开发用于对形状内存效应和伪弹性建模的数学工具。 为此，研究者利用了这些物质系统的线性比较均质化方法的合适概括，并结合了晶体学纹理和形态纹理的影响（由微结构的两点相关性确定）。 主要的数学挑战是由于缺乏相关能量函数的凸性（或更确切地说是准跨性别）的结果。 众所周知，这种凸度的分解导致在单晶水平上开发其他微观结构，这是形状记忆效应的核心。 因此，也为单晶形状 - 内存合金（SMA）的“弛豫”（或准转换）开发了线性比较估计，从晶体水平。 该项目导致了能够处理复杂的微观结构，强烈的非线性响应，微结构演变以及不稳定性的可能发展的新型和高效的多尺度建模技术。 这项工作中开发的数学技术广泛应用于大量的聚合物，金属，生物学和地质材料系统，包括多功能材料，并借用了本构型子例程中的数值实施，可与标准数值包一起使用。 形状记忆效应是某些材料在加热后恢复的能力，显然是在一定临界温度以下的永久变形。 这种效果通常伴随着伪弹性，从而观察到这些材料的单晶样品在几乎恒定的应力下会经历相当大的菌株（以10％的速度），这在卸载后已完全恢复。 这两种特性使形状内存合金（SMA）非常有吸引力，例如低频，健壮的执行器和传感设备，用于各种新型技术应用。 但是，SMA通常以多晶形式使用，通常用作电线，或细条和薄膜，其中形状 - 记忆效果大大降低。 因此，必须了解和建模单晶行为，微结构和多晶的宏观行为之间的关系。 SMA多晶建模的改进应改善这些材料的性能，这些材料在行业中广泛使用，作为执行器，传感器，耦合和电气连接器，以及越来越多的医疗和机器人应用。