Systematic study of instabilities in non-linear elasticity and martensitic phase transformations

格拉博夫斯基0707582 研究者专注于理解和系统研究可以用最小能量变分原理解释的不稳定性。 一种这样的不稳定性是屈曲。屈曲是工程力学中普遍存在的一个重要问题。 现代屈曲工程模型虽然非常成功，但与三维超弹性无关，三维超弹性应该包含所有弹性现象的描述，包括屈曲。 该项目启动了一个系统的研究所有的弹性不稳定性，把屈曲在其适当的settingwithin一个一般的理论。 该项目研究的另一个有趣而重要的不稳定性是马氏体相变中新相的成核。 这项研究还涉及到与心率无关的滞后现象--一种意想不到的历史依赖现象，目前还没有一个普遍认同的解释。 基于能量的马氏体相变模型在预测形状记忆合金行为的许多方面非常成功。 然而，滞后现象可以解释这个模型，只有当一个假设，该材料“卡住”在亚稳态，模拟为本地能量最小值。 该项目提供了一个迄今为止缺乏的通用工具，用于识别亚稳态，允许一个构建和分析连续介质力学模型与能量最小化原则的基础上定量的介电常数。 本项目旨在提高我们对非线性弹性材料的认识，如聚合物和橡胶以及具有形状记忆效应的材料。 当载荷施加到这些材料上时，它们以这样一种方式变形，即使存储在变形系统中的总能量最小化。 当载荷增加时，这些材料的非线性性质往往表现为出现新的降低能量的方法。当这种情况发生时，物理学家们就谈到了不稳定性。 最常见的不稳定性之一是屈曲，它发生在细长柱上的载荷超过某一临界值时。 研究人员与Lev Truskinovsky最近的合作产生了对屈曲的新理解，该项目用于探索计算具有复杂几何形状的结构（如弹性壳体和复合材料）的屈曲载荷的新方法。 在这个项目中进行的弹性不稳定性的系统研究也揭示了一些关于与速率无关的滞后现象--一种仍然很难理解的现象，即形状记忆合金在加载和卸载时遵循不同的变形路径。 该项目对更好地理解不稳定性的贡献允许对滞后现象的一个拟议解释进行定量分析，即材料“陷入”局部极小值。 该项目还为研究生和本科生提供了教育和培训机会。 该项目的一部分形成了研究者的研究生的博士论文的核心，而其他部分则告知了高级本科课程和独立研究的内容。

Grabovsky0707582 The investigator focuses on the understanding and systematicstudy of instabilities that can be explained by variationalprinciples of minimum energy. One such instability is buckling. Buckling is ubiquitous and very important in engineering andmechanics. Modern engineering models of buckling, though verysuccessful, are not related to 3D hyperelasticity, which issupposed to contain a description of all elastic phenomena,including buckling. The project initiates a systematic study ofall elastic instabilities, putting buckling in its proper settingwithin a general theory. Another interesting and importantinstability studied by the project is nucleation of a new phasein martensitic phase transitions. This study is also related torate-independent hysteresis -- an unexpected phenomenon ofhistory dependence, that does not yet have a universallyagreed-upon explanation. The energy-based model of martensiticphase transformations was very successful at predicting manyaspects of behavior of shape memory alloys. Hysteresis, however,could be explained by this model only if one assumes that thematerial "gets stuck" in metastable states, modeled as localenergy minima. The project provides a hitherto missing generaltool for identifying metastable states, allowing one to constructand analyze continuum mechanical models with hysteresisquantitatively on the basis of the energy-minimization principle. This project aims to advance our understanding ofnonlinearly elastic materials, such as polymers and rubbers andmaterials with shape memory effect. When loads are applied tothese materials, they deform in such a way as to minimize thetotal energy stored in the deformed system. When the loadsincrease, the nonlinear nature of these materials often manifestsitself in the appearance of new ways to decrease the energy. When this happens, physicists talk of instabilities. One of themost common instabilities is buckling, which occurs when the loadon a slender column exceeds a certain critical value. Recentwork of the investigator with Lev Truskinovsky produced a newunderstanding of buckling that is used in this project to explorenew methods of computing buckling loads for structures withcomplex geometries, such as elastic shells and compositematerials. The systematic study of elastic instabilities that isperformed in this project also sheds some light onrate-independent hysteresis -- a still poorly understoodphenomenon, whereby a shape memory alloy follows differentdeformation paths upon loading and unloading. The project'scontribution towards better understanding of instabilitiespermits a quantitative analysis of one of the proposedexplanations for hysteresis, that the material "gets stuck" inlocal minima. The project also presents educational and trainingopportunities for graduate and undergraduate students. Parts ofthis project form the cores of doctoral dissertations of theinvestigator's graduate students, while other parts inform thecontent of advanced undergraduate classes and independentstudies.