Elasticity, Growth, and Stability

该项目的总体主题是通过使用和发展非线性弹性理论和相关的应用数学方法，研究物理和生物系统中的生长、结构和功能。这些项目分为三个相互关联的研究主线：(i)软材料（如生物组织）弹性生长的分析和建模；（ii）微生物系统的生物力学，包括力驱动渗透机制的研究，例如真菌中产生的破坏作物的机制；(3)具有重要工程应用的各种弹性杆、电缆和管道的动力学和稳定性。对于第一个主题，研究人员和他们的学生研究生长的机械后果及其通过几何和应力变化产生不稳定性的潜力，或者作为稳定和调节物理特性的机制。这些研究的目标是深入了解许多生物系统中生长和应激之间的基本耦合。微生物生物力学的第二个主题涉及细菌和真菌系统的数学建模。研究人员制定并分析了生长微生物的力学模型，以了解它们的整体结构（例如，在冰原真菌中形成附着胞）和它们通过渗透侵入宿主组织的能力（如在许多真菌中发现的），以及丝状细菌的生长和结构变化，如那些是抗生素的天然来源。这些生物的描述结合了膜生长的一般公式（基于第一个研究主题），经历大变形的细胞壁的弹性模型，以及使用塑性和断裂理论来描述渗透过程。弹性杆动力学的第三个主题涉及研究输送流体的弹性管，以及具有扭力和张力之间本构耦合的杆的动力学（半向杆）——这是与起重机缆索和其他编织结构设计相关的功能。细丝不稳定性的分析是通过使用和扩展研究人员在他们以前的工作中开发的非线性分析技术来进行的。该项目关注诸如管道和棒状纤维结构的动力学以及生物系统中出现的生长机制和入侵的物理分析。它在本质上是高度跨学科的，跨越了应用数学、机械工程、微生物学和生物力学等领域，并解决了具有实际重要性和数学意义的问题。这些主题在生物学（形态发生），生物医学工程（软组织的分析，它们的机械调节和功能），理解生物入侵的基本过程，如真菌对组织的渗透，以及经典的工程问题（管道输送流体，钻井中的不稳定性）中具有广泛的适用性。该项目还为来自不同背景的研究生和本科生提供了许多有吸引力的不同层次的培训经验。其中包括通过使用应用数学课程独特的实验设施，将数学建模与动手实验结合起来的机会。

GorielyDMS-0605029 The general theme of the project is the study of growth,structure, and function in physical and biological systemsthrough the use, and development of, nonlinear elasticity theoryand the associated methods of applied mathematics. The projectis divided into three main lines of interconnected research: (i)the analysis and modeling of elastic growth in soft materials,such as biological tissue; (ii) the biomechanics of microbialsystems including the study of force-driven penetrationmechanisms, such as those arising in fungi that destroy crops;and (iii) the dynamics and stability of various elastic rods,cables, and pipes having important engineering applications. Forthe first theme, the investigators and their students study themechanical consequences of growth and its potential to eithergenerate instabilities through changes in geometry and stresses,or to act as a mechanism to stabilize and regulate physicalproperties. The goal of these studies is to gain insight intothe fundamental coupling between growth and stress in manybiological systems. The second theme of microbial biomechanicsinvolves the mathematical modeling of both bacterial and fungalsystems. The investigators formulate and analyze mechanicalmodels of growing micro-organisms in order to understand theiroverall structure (e.g. the formation of appressoria in therice-blast fungus) and their ability to invade host tissues bypenetration (as found in many fungi), as well as the growth andstructural changes exhibited by filamentary bacteria, such asthose that are a natural source of antibiotics. The descriptionof these organisms combines a general formulation of membranegrowth (based on the first research theme), the elastic modelingof cell walls undergoing large deformations, and the use ofplasticity and fracture theory to describe penetration processes. The third theme of elastic rod dynamics involves the study ofelastic tubes conveying fluids, and the dynamics of rods withconstitutive coupling between twist and tension (hemitropic rods)-- a functionality that is relevant to the design of crane cableand other braided structures. The analysis of filamentinstabilities is carried out through the use and extension of thenonlinear analysis techniques developed by the investigators intheir previous work. The project concerns the dynamics of filamentary structuressuch as pipes and rods and the physical analysis of growthmechanisms and invasion appearing in biological systems. It ishighly interdisciplinary in nature, cutting across the fields ofapplied mathematics, mechanical engineering, microbiology, andbiomechanics, and addresses questions of both practicalimportance and mathematical interest. The themes have a broadrange of applicability in biology (morphogenesis), in biomedicalengineering (analysis of soft tissues, their mechanicalregulation and function), in understanding fundamental processesin biological invasion such as fungal penetration of tissues, andin classical engineering problems (pipes conveying fluids,instabilities in drilling). The project also provides manyattractive training experiences at different levels suitable forgraduate and undergraduate students from diverse backgrounds. These include opportunities to synthesize mathematical modelingwith hands-on experimentation through the use of the AppliedMathematics Program's unique experimental facilities.