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

Grabovsky 0707582研究人员致力于理解和系统研究可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就是失稳。屈曲是工程和力学中普遍存在且非常重要的问题。现代屈曲工程模型虽然非常成功，但与三维超弹性无关，三维超弹性被认为包含了包括屈曲在内的所有弹性现象的描述。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而重要的不稳定性是马氏体相变中新相的形核。这项研究还涉及与主权无关的滞后--一种意想不到的历史依赖现象，目前还没有一个普遍同意的解释。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都是非常成功的。然而，只有假设材料处于亚稳状态，即被建模为局部能量极小时，磁滞才能用这个模型来解释。该项目提供了一个迄今缺乏的识别亚稳态的通用工具，使人们能够根据能量最小化原理来构建和定量分析具有滞后的连续介质力学模型。本项目旨在加深我们对非线性弹性材料，如聚合物、橡胶和具有形状记忆效应的材料的理解。当载荷作用于这些材料时，它们会发生变形，从而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新的降低能量的方法的出现。当这种情况发生时，物理学家会谈论不稳定性。最常见的失稳之一是屈曲，当细长柱上的载荷超过某一临界值时，就会发生屈曲。列夫·特鲁斯金诺夫斯基最近的研究工作对屈曲有了新的认识，该项目用于探索计算具有复杂几何形状的结构(如弹性壳和复合材料)的屈曲载荷的新方法。该项目对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现象--这是一种仍然很难被理解的现象，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使人们能够对迟滞的一种拟议解释进行定量分析，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教育和培训机会。这个项目的一部分形成了调查员研究生的博士论文的核心，而另一部分则提供了高级本科生班和独立研究的内容。

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.