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

拟议研究的目的是建立一个研究计划，在该地区的表面力学重点发展连续模型和先进的二维材料的设计和分析的数值技术。特别是，拟议的研究计划将解决以下两个主要分支的问题，由于其在生物力学和纳米力学领域的重要应用而引起了相当大的关注。 1.生物膜在厚度膨胀和非均匀脂质分布下的力学 脂质膜具有各种有用的医学和工业应用，包括药物运输和递送，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，模拟和预测其在囊泡形成、融合和膨胀运动中的力学响应是非常重要的。脂质双分子层是由横向取向的脂质分子组成的，这些脂质分子含有亲水性的头部基团和疏水性的尾部。这些分子排列成具有相反取向的两层片材，有效地将尾基与周围的水溶液屏蔽。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，脂质双层可以被认为是一个封闭的膜与不同的微观取向的脂质分子最初对齐的表面法线。在本计画中，我们将发展一套完整的连续统模型与数值方法，以描述在厚度膨胀与非均匀脂质分布下的生物膜形态。更确切地说，我们将分析模型和检查的脂质双层计算相对于平均和高斯曲率，脂质膨胀和膜表面上的梯度的膜的能量变化。 2.二维纤维复合材料的应变-梯度一体化建模 本研究旨在发展一种通用的二维连续体模型，该模型可以容纳纤维的抗拉伸、弯曲和扭曲的能力。这将允许更准确地描述的应力和位移场的二维纤维复合材料受到各种外部载荷。特别是，通过拟议的研究计划，我们的目标是建立一个强大的理论和实践基础，在设计和分析的生物复合材料与纤维素纳米晶体（CNC），近年来引起了相当大的关注。通过计算纤维的应变梯度场，将纤维的运动统一到薄膜结构的变形场中。我们开始与单向纤维复合材料的情况下，随后将结果扩展到更一般的情况下，膜结构是由多方向的纤维和最初的非直纤维。

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.