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

Ponte CastanedaDMS-1108847 该提案的主要目标是开发用于模拟形状记忆多晶体中的形状记忆效应和伪弹性的数学工具。 为此，研究人员利用适当的概括的线性比较均匀化方法，这些材料系统，将晶体结构和形态结构的影响（由两点相关的微观结构）。 主要的数学挑战是由于相关能量函数缺乏凸性（或者更准确地说，准凸性）。 众所周知，这种凸度的破坏导致在单晶水平上形成额外的微结构，这是形状记忆效应的核心。 正因为如此，线性比较估计也开发的“松弛”（或准凸）的单晶形状记忆合金（SMA），利用最近发现的连接均匀化在多晶水平和松弛在单晶水平。 该项目产生了新颖高效的多尺度建模技术，能够处理复杂的微观结构，耦合强非线性响应，微观结构演变以及可能的不稳定性发展。 在这项工作中开发的数学技术广泛应用于大类的聚合物，金属，生物和地质材料系统，包括多功能材料，并借给自己的数值实现本构子程序与标准数值包使用。 形状记忆效应是某些材料在加热时恢复的能力，在一定的临界温度以下维持明显的永久变形。 这种效应通常伴随着伪弹性，由此观察到这些材料的单晶样品在几乎恒定的应力下经历相当大的应变（大约10%），其在卸载时完全恢复。 这两个属性使得形状记忆合金（SMA）非常有吸引力的低频，鲁棒的致动器和传感设备的各种新的技术应用。 然而，SMA通常以多晶形式使用，通常作为线或薄条和膜，其中形状记忆效应大大降低。 因此，必须理解和模拟多晶体的单晶行为、微观结构和宏观行为之间的关系。 SMA多晶建模的改进应导致这些材料的性能提高，这些材料在工业中广泛使用，如致动器，传感器，耦合器和电连接器，以及越来越多的医疗和机器人应用。

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.