Collaborative Research: Mechanics of Growing Bodies: A Riemannian Geometric Approach

这笔赠款的研究目的是阐明生长物体的力学的微分/黎曼几何公式。所提出的工作是基于变化的物质流形的概念，其动力学预测物质物体的松弛状态的演变。这一理论适用于生长和重塑伴随着大变形的生物组织。为了实现这一建议的研究目标，引入了一种基于动态材料流形的连续介质力学理论，该理论将生物组织的生长/重塑与其大变形相耦合。这项拟议的研究将把增长和类似的非线性问题纳入具有动态物质流形的统一几何理论。这个项目的目标之一是通过展示几何技术的一些优点和应用来促进几何技术在力学领域的使用。该项目的结果将以工程师可理解的语言展示，并将展示几何方法带来的概念澄清和建模优势。这项研究的一个主要教育影响是建立了一个利用数学和力学的语言学习和教授生物学生长力学的一般框架，这是生物物理学目前所缺乏的一种背景。该项目将促进缅因州大学和佐治亚理工学院之间的合作研究机会，并促进本科生更广泛地参与生物力学领域的前沿研究。

The research objective of this grant is to elucidate a differential/Riemannian geometric formulation for the mechanics of growing bodies. The proposed work is based on the concept of a changing material manifold whose dynamics predicts the evolution of the relaxed state of a material body. This theory is applicable to biological tissues in which growth and remodeling are coupled with large deformations. To achieve the research objective of this proposal, a theory of continuum mechanics based on a dynamic material manifold is introduced that couples the growth/remodeling of biological tissues with their large deformations. The proposed research will put growth and similar nonlinear problems into a unified geometric theory that has a dynamic material manifold. One of the goals of this project is to promote the use of geometric techniques in the mechanics community by demonstrating some of their advantages and applications. The results of this project will be presented in a language accessible to engineers, and they will demonstrate the conceptual clarifications and modeling advantages brought by the geometric approach. A major educational impact of this research is the establishment of a general framework for learning and teaching growth mechanics in biology using the language of mathematics and mechanics, a context, which is currently missing in the biophysics. This project will foster opportunities for collaborative research between the University of Maine and Georgia Institute of Technology and facilitate broadening the participation of undergraduate students in the cutting edge research in the field of biomechanics.