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

研究者着重于理解和系统地研究可以用最小能量变分原理来解释的不稳定性。其中一种不稳定性是屈曲。屈曲是工程力学中普遍存在的重要问题。屈曲的现代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。该项目启动了对所有弹性不稳定性的系统研究，将屈曲置于一般理论的适当设置中。该项目研究的另一个有趣和重要的不稳定性是一种新的相素马氏体相变的成核。这项研究还涉及到与历史无关的滞后——一种意想不到的历史依赖现象，目前还没有一个普遍认可的解释。基于能量的马氏体相变模型在预测形状记忆合金许多方面的行为方面是非常成功的。然而，只有假设材料“陷入”亚稳态，即局部能量最小值，这个模型才能解释迟滞现象。该项目提供了迄今为止缺少的识别亚稳态的通用工具，允许人们在能量最小化原则的基础上定量地构建和分析具有滞后的连续体力学模型。本项目旨在促进我们对非线性弹性材料的理解，如聚合物、橡胶和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，使变形系统中存储的总能量最小化。当载荷增加时，这些材料的非线性特性往往表现为降低能量的新方法的出现。当这种情况发生时，物理学家就会谈论不稳定性。最常见的失稳之一是屈曲，当细长柱的载荷超过某一临界值时，就会发生屈曲。研究人员与Lev Truskinovsky最近的工作产生了对屈曲的新理解，该项目用于探索计算复杂几何结构（如弹性壳和复合材料）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与速率无关的滞后-一种仍然知之甚少的现象，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是允许对迟滞的一种提出的解释进行定量分析，即材料“卡在”局部极小值中。该项目还为研究生和本科生提供了教育和培训机会。该项目的部分内容构成了研究者研究生博士论文的核心，而其他部分则为高级本科课程和独立研究的内容提供了信息。

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.