均匀和非均匀功能梯度形状记忆合金样品中相变的解析研究

Shape memory alloys (SMAs) are characterized by two unusual properties: shape memory effect and pseudoelasticity. As a result, they are ideal materials in many engineering applications, such as sensors, actuators and various biomedical devices. To understand the properties of this type of materials, many experiments have been conducted on slender/thin uniform and functionally graded SMA structures under tension/extension. Among the findings, some important features are observed, including a hysteresis loop in a loading and unloading cycle, instabilities leading to stress drop/rise, the graded effect leading to a better controllability, etc. Despite of many works on SMAs, analytical results are relatively few. The major difficulty is due to the complexity of the governing nonlinear partial differential equations caused by the material nonlinearity and the high-dimensional effect. In this project, we shall carry out analytical studies through asymptotic methods on phase transitions in slender uniform and geometrically graded SMA cylinders under tension/extension. More specifically, we shall use the method of coupled series-asymptotic expansions by exploiting the slenderness of the specimen to derive the asymptotically approximate equations and then conduct analysis on them. It is aimed at deriving analytical solutions for the full loading and unloading cycle. The analytical results will be further examined in physical terms, which can then shed insights on the various phenomena mentioned above. The better understanding and mathematical interpretations of the experimental results can help in designing better devices made of SMAs.

形状记忆合金(SMA)具有形状记忆效应和拟弹性效应等特性，是许多工业应用的理想材料。 为了研究SMA材料的性能，人们开展了细长均匀和功能梯度SMA样品的拉伸实验，观察到许多实验特征，如循环加载的滞后回路、不稳定性导致的应力升高或降低、梯度效应对于可控性的改善等。尽管已有大量的实验和数值研究，相关的解析成果目前还非常少。解析研究的主要困难来自材料非线性和高维效应所导致的控制方程（非线性偏微分方程组）的复杂度。本项目拟采用渐近方法解析地研究细长均匀和功能梯度SMA样品在拉伸作用下的相变，特别地，基于样品的细长特征，将应用申请人提出的耦合级数-渐近展开法推导和研究渐进微分方程组，并采用正则摄动法、WKB法和均匀化方法等先进的渐近方法，得到描述完整循环加载过程的解析解。通过分析解析解的物理意义，揭示众多实验现象的内在机理。对实验结果的深入了解和数学诠释将有助于改进SMA材料的使用和相应设备的设计。

形状记忆合金(SMA)具有形状记忆效应和拟弹性效应等特殊的材料性能，在航空、医学等领域具有广阔的应用前景。已有的研究结果表明，SMA独特材料性能的内在机理是应力或温度诱导下的马氏体相变。在这一领域，研究人员已开展了大量的实验观测和数值模拟，但解析研究的结果还相对较少。本项目通过解析途径系统研究了均匀和非均匀SMA样品在不同加载条件下的力学行为特征，揭示其内在力学机理，从而为SMA材料的实际工程应用提供帮助。本项目开展的研究内容包括：(1) SMA管在扭转和预扭拉伸加载条件下所发生相变的解析研究；(2) 动态和冲击加载条件下SMA样品中相态边界与弹性波传播研究; (3) 具有几何梯度的SMA样品在轴向加载条件下所发生相变的解析研究; (4) 适用于有限（大）应变范围且满足一致性准则的新型板壳理论研究; (5) 基于Eringen模型的非局部欧拉-伯努利梁静态屈曲的解析研究。本项目的各项研究内容已顺利完成，研究计划和研究目标无变更。