The mathematics and mechanics of elastic growth; with biological and biomedical applications

GorielyDMS-0907773该项目的主题是通过使用和发展精确弹性理论和应用数学的相关方法来研究生理和生物系统的生长、结构和功能。该项目分为特定但相互关联的几条线：软材料弹性增长理论的数学发展；增长的长期动力学；增长弹性系统的线性和非线性稳定性分析及其在形态形成和图案形成中的应用；空化和空洞开放的内生问题；以及约化理论的研究。主要研究人员在精确理论的框架内研究弹性增长，其中增长是通过变形梯度的乘法分解来建模的。研究了增长的力学后果，它的长期动态，以及它通过改变几何尺寸和压力而产生不稳定的潜力，或者作为一种调节机制。此外，通过弱非线性分析技术的发展和应用，对生长组织进行了研究。还开展了用于生长结构的棒、膜、板和壳的简化理论的发展。本课程研究了生物学和生理学中的各种应用，特别是将生长的思想应用于细菌和真菌系统、茎、叶、动脉和血管。通过使用和推广首席研究员开发的非线性分析技术，进行了各种理论分析和具体的增长模型。生长是一个极其复杂的基本科学问题，涉及生物、化学和物理现象。该项目的目标是开发数学工具，以了解和模拟各种与生长有关的问题，包括植物和微生物的形态发生，动脉的稳定性，动脉瘤的发展，以及心脏、食道和气管等器官的正常和异常功能。尽管这些系统的生物多样性很大，但它们有共同的特征，可以通过使用类似的数学工具来建模。这些模型有助于科学界理解生长过程中力学和生物学之间的密切耦合，并更好地预测许多生物和生理系统的响应。

GorielyDMS-0907773 The theme of the project is the study of growth, structure,and function in physiological and biological systems through theuse and development of exact elasticity theory and the associatedmethods of applied mathematics. The project divides intospecific but interconnected lines: the mathematical developmentof a theory of elastic growth in soft materials; the long-timedynamics of growth; the linear and nonlinear stability analysisof growing elastic systems with application to morphogenesis andpattern formation; the problem of cavitation and void-opening ingrowth; and the study of reduced theories. The PrincipalInvestigators study elastic growth within the framework of exactelasticity theory, where growth is modeled through themultiplicative decomposition of the deformation gradient. Themechanical consequences of growth, its long-time dynamics, andits potential to either generate instabilities through changes ingeometry and stresses, or to act as a regulatory mechanism, areexamined. Furthermore, a study of growing tissues is carried outthrough the development and application of the techniques ofweakly nonlinear analysis. The development of reduced theoriesfor rods, membranes, plates, and shells for growing structures isalso undertaken. Various applications in biology and physiologyare studied; in particular the ideas of growth are applied tobacterial and fungal systems, stems, leaves, arteries, and bloodvessels. The various theoretical analyses and specific models ofgrowth are carried out through the use and extension of nonlinearanalysis techniques developed by the Principal Investigators. Growth is a fundamental scientific problem of greatcomplexity coupling biological, chemical, and physical phenomena. The goal of the project is to develop mathematical tools tounderstand and model a variety of problems related to growth,including plant and microbial morphogenesis, the stability ofarteries, the development of aneurysms, and the regular andabnormal functions of organs such as the heart, the esophagus,and the trachea. Despite the great biological diversity of thesesystems, they share common features that can be modeled throughthe use of similar mathematical tools. These models help thescientific community to understand the intimate coupling betweenmechanics and biology in growth processes and to predict betterthe response of many biological and physiological systems.