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。 该项目分为具体的，但相互关联的线：数学发展的理论弹性增长在软材料;增长的长期动力学;线性和非线性稳定性分析的增长弹性系统与应用形态和模式的形成;问题的空化和空洞开放向内生长;和研究减少理论。 PrincipalInvestigators研究弹性增长的框架内的exactelasticity理论，其中增长是通过模拟的变形梯度的乘法分解。 生长的力学后果，其长期动力学，其潜力，无论是产生不稳定性，通过变化ingeometry和应力，或作为一个监管机制，进行审查。 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。 发展约theoriesfor杆，膜，板，壳生长结构也进行。 在生物学和生理学的各种应用进行了研究，特别是生长的想法被应用到细菌和真菌系统，茎，叶，动脉和血管。 各种理论分析和具体模型ofgrowth进行通过使用和扩展的非线性分析技术开发的主要研究者。 生长是一个结合了生物、化学和物理现象的非常复杂的基本科学问题。该项目的目标是开发数学工具来理解和模拟与生长有关的各种问题，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。 尽管这些系统具有很大的生物多样性，但它们具有共同的特征，可以通过使用类似的数学工具进行建模。 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.