球形核-壳结构凝胶的非高斯行为及不稳定性的数学建模与分析

Polymer gels have important applications in the biomedical field and particularly, they are used as carriers of stem cells in the stem cell therapy. Experiments showed that the mechanical properties such as the modulus and morphology of the gels can determine the fate of stem cells. This project aims to explore theoretically how the mechanical properties of a spherical core-shell gel system can be controlled by parameters so that suggestions can be provided for designing a proper gel carrier. Specifically, we plan to study the non-Gaussian behavior of the modulus in the large inhomogeneous deformation. On the other hand, the mechanisms lying under the complex morphologies during the instability will also be explored. For the former aspect, there existed experimental observations but seldom did any theoretical works explore it systematically. We plan to define the modulus (or effective stiffness) of the swollen gel though derivation, then reveal its behaviors based on numerical as well as analytical methods. For the latter aspect, we aim to obtain the critical condition for the instability and influences of parameters on the critical state by virtue of linear bifurcation analysis. Based on above results, comparisons will be made between critical conditions for the non-Gaussian behavior and the instability so that relevant suggestions can be provided for the design of a proper gel carrier. After that, in the non-linear bifurcation analysis, we intend to derive the critical condition for crease and construct various pathways leading to crease patterns. The crease patterns are highly non-linear and we intend to use the idea of spline approximation as well as the consistent beam theory to tackle this problem. We expect to obtain results which will help to understand some complex biological morphologies (such as the sulcus of brain) and will also give advices on proper intervention.

凝胶材料在生物医学领域有重要的应用价值，特别地，凝胶在干细胞治疗中用作干细胞的载体。实验证明凝胶的力学性质，如模量及形态特征可以决定干细胞的分化方向。本项目试图从理论上探索球形核-壳结构凝胶的力学性质如何受参数影响，从而为设计合理的凝胶载体提供建议。具体地，我们打算研究在非均匀大变形中凝胶模量的非高斯行为，以及在不稳定性中复杂形态产生及相互转换的机理。对于前者，文献中存在实验观察的结果，但少有理论对其进行系统地探究。我们计划先通过推导定义出膨胀后凝胶的模量（或有效硬度），然后从数值以及解析的角度揭示该模量的行为。对于后者，我们打算先进行线形分叉分析得到发生不稳定的临界条件及参数对临界模态的影响。在此基础上，将模量的非高斯行为与材料的不稳定性发生条件进行对比，并给出设计合理凝胶载体的相关建议。接着，我们将在非线形分叉分析中得到crease皱褶发生的临界条件及形成路径，并考察参数的影响。对于crease皱褶这样高度非线形的变形，我们打算使用样条分析的思想以及一致性梁理论来进行处理，得到的结果将有助于理解一些复杂生物形态（比如大脑沟回）的形成机理，并有可能为合理干预提供建议。

凝胶的大变形和失稳及由此而带来的力学性能在医药、工程、能源、环境等领域均有巨大的实用价值和广泛的社会需求，因而具有重大的研究意义。本项目综合利用了数学及力学中各种分析工具（如级数展开、摄动展开、线性分岔分析、弱非线性分岔分析等）来研究凝胶的力学响应问题及相关课题。项目执行期间，研究工作进展顺利，在包括Journal of Mechanics and Physics of Solids, Physical Review E 等在内的国际权威期刊共发表SCIE论文5篇，基本完成预定目标。所取得的重要研究成果包括：使用板壳增量变形理论来研究板壳状凝胶的分岔及后分岔问题；该研究成果推导的板壳增量变形理论和有限元方法结合可以极大地提高数值计算的效率。半解析地研究了三层球形核壳状凝胶不均匀溶胀导致的凝胶内部硬度分布，并通过调整参数模拟了人体皮肤及甲状腺组织的硬度范围；该研究成果从力学角度提供了一个简单可控的方法来生产（干）细胞所需的组织环境。使用有限体积生长理论推导出二维及三维无应力生长模型，求得的解析解可用于描述多种植物及菌类的生长形貌演化；该研究成果为生物组织的生长机理提供了可能的解释，并且可以用于制作具有特定变形功能的凝胶。推导出适用于线性各向异性材料的渐进一致杆理论，该杆理论可以在将来被推广为新型有限应变杆理论，用来研究杆状软物质的大变形及不稳定性。

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