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

本提案的主要目标是开发数学工具来模拟形状记忆效应和形状记忆多晶体中的伪弹性。为此，研究者利用这些材料系统的线性比较均匀化方法的适当推广，结合晶体结构和形态结构的影响（由微观结构的两点相关决定）。主要的数学挑战是相关能量函数缺乏凸性（或者更准确地说，准凸性）的结果。众所周知，这种凹凸性的破坏导致了单晶水平上额外微结构的发展，这是形状记忆效应的核心。正因为如此，单晶形状记忆合金（sma）的“弛豫”（或准凸化）也得到了线性比较估计，利用了最近发现的多晶水平的均匀化和单晶水平的弛豫之间的联系。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高分子材料、金属材料、生物材料和地质材料系统，包括多功能材料，并有助于在本构子程序中与标准数值包一起使用的数值实现。形状记忆效应是指某些材料在加热后，在低于某一临界温度的情况下恢复明显永久变形的能力。这种效应通常伴随着伪弹性，即观察到这些材料的单晶样品在几乎恒定的应力下承受相当大的应变（大约10%），在卸载时完全恢复。这两种特性使得形状记忆合金（sma）在各种新技术应用中作为低频、鲁棒致动器和传感装置非常有吸引力。然而，sma通常以多晶形式使用，通常作为导线或薄条和薄膜，其形状记忆效应大大降低。因此，理解和模拟多晶的单晶行为、微观结构和宏观行为之间的关系至关重要。SMA多晶建模的改进应该会提高这些材料的性能，这些材料在工业中广泛使用，作为致动器、传感器、联轴器和电连接器，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.