Mechanics of surfaces: Continuum-based modeling and analysis for biomembranes and 2D fiber materials

拟议研究的目标是在表面力学领域建立一个研究计划，专注于开发用于设计和分析先进2D材料的连续介质模型和数值技术。尤其是，由于在生物力学和纳米力学领域的重要应用，拟议的研究计划将解决以下两个引起相当大关注的问题的主要分支。*1.受厚度膨胀和非均匀脂质分布影响的生物膜的力学*脂膜具有各种有用的医疗和工业应用，包括药物传输和输送，以及基于生物膜的传感器(例如，水毒性传感器)。对于这类应用，模拟和预测它们对囊泡形成、融合和膨胀涉及运动的力学响应是很重要的。这些分子将自己排列成具有相反方向的两层薄片，有效地将尾基与周围的水溶液隔绝。事实上，双层结构是所有生物膜(生物膜)的特征。这种排列发生在分子尺寸量级的长度尺度上。因此，一个脂类双层可以看作是一个闭合的膜，具有不同的微观取向的脂类分子最初排列在表面法线上。在这个项目中，我们将发展一个完整的连续介质模型和数值方法来描述存在厚度膨胀和非均匀分布的生物膜的形态。更准确地说，我们将通过计算膜相对于膜表面的平均曲率和高斯曲率的能量变化、膜表面的脂膨胀及其梯度来解析建模和检验脂双层。2.2D纤维复合材料的应变-梯度集成建模*本研究旨在开发能够适应纤维的拉伸、弯曲和扭转的通用2D连续介质模型。这将使二维纤维复合材料在各种外载荷作用下的应力场和位移场得到更准确的描述。特别是，通过提出的研究计划，我们的目标是为近年来引起相当大关注的纤维素纳米晶增强生物复合材料的设计和分析奠定坚实的理论和实践基础。通过计算纤维的应变梯度场，将纤维的运动积分到薄膜结构的变形场中。我们从单向纤维复合材料的情况开始，随后将结果扩展到更一般的场景，其中薄膜结构由多向纤维和最初的非直纤维支撑。**

The objective of the proposed research is to establish a research program in the area of the mechanics of surfaces focused on developing continuum models and numerical techniques for the design and analysis of advanced 2D materials. In particular, the proposed research program will address the following two major branches of problems that have drawn a considerable attention due to their important applications in the fields of biomechanics and nanomechanics.***1. Mechanics of biomembranes subjected to thickness distension and non-uniform lipid distribution***Lipid membranes have various useful medical and industrial applications including drug transport and delivery, and biomembrane-based sensors (e.g. water toxicity sensors). For such applications, it is important to simulate and predict their mechanical responses on vesicle formation, fusion and distension involved motions. Lipid bilayers are composed of transversely oriented lipid molecules containing hydrophilic head groups and hydrophobic tails. These molecules arrange themselves into a two layered sheet with opposing orientations that effectively shield the tail groups from the surrounding aqueous solution. In fact, the bilayer structure is characteristic of all biological membranes (biomembranes). The arrangement occurs over length scales on the order of molecular dimensions. Therefore, a lipid bilayer can be regarded as a closed membrane with distinct microscopic orientations of lipid molecules initially aligned with the surface normal. In this project, we will develop a complete continuum model and numerical method in the description of biomembrane morphology under the presence of thickness distension and non-uniform lipid distribution. More precisely, we will analytically model and examine a lipid bilayer by computing the energy variations of the membrane with respect to mean and Gaussian curvatures, lipid distension and its gradient on the membrane surface. ***2. Strain-gradient integrated modeling for 2D fiber composites***The research intends to develop general 2D continuum models that can accommodate fiber's resistance to extension, bending and twist. This will allow more accurate descriptions for stress and displacement fields of 2D fiber composites subjected to various external loadings. In particular, through the proposed research program, we aim to build a strong theoretical and practical foundation in design and analysis of biocomposites reinforced with cellulose nanocrystals (CNC) which has drawn a considerable attention in recent years. By calculating strain-gradient fields of fibers, we will integrate fiber's motions into the deformation fields of the film structure. We begin with the case of unidirectional fiber composites and subsequently extend the results to the more general scenarios where the film structure is supported by multi-directional fibers and initially non-straight fibers.**