几何形态为曲面的连续介质有限变形理论（方法与应用研究）

The aim of the project is to develop and verify a kind of finite deformation theories of continuous mediums with surface geometrical configurations. As the factual thin layer flows or membrane motions are concerned, the related continuous mediums are modeled as surfaces and the surface densities are introduced to describe the factual thicknesses; the kinematics and kinetics of continuous mediums with surface geometrical configurations are developed based on the tensor field analysis on surfaces/Riemannian manifolds. The classified researches based on the general finite deformation theory are to be carried out: (1) the depiction of two dimensional flows on fixed surface in the view of vorticity dynamics with related numerical methodology; (2) the Lagrangian depiction of two dimensional flows on moving surfaces with related numerical methodology; (3) the Lagrangian depiction of finite deformations of elastic membranes with related numerical methodology. Correspondingly, all of the theoretical models with related numerical methodologies as mentioned above are to be modified and verified through the specified experiments of thin layer fluid flows on fixed and moving surfaces，finite deformation vibrations of elastic membranes. On the other hand, the following researches are concerned: (1) the relationships between the geometrical characteristics of the surfaces and finite deformations; (2) the ways through which the mathematical ideas and methodologies of modern geometry and so on can be adopted to mechanics. It seems that an original finite deformation theory will be established for thin enough continuous mediums.

本项目致力于发展并检验一种"几何形态为曲面的连续介质有限变形理论"。对于实际的薄层流动或者薄膜运动，我们将介质的几何形态模型化为曲面，并引入面密度刻画介质的实际厚度；基于曲面/Riemann流形上张量场场论建立曲面形态连续介质的有限变形运动学与动力学。基于所建立的有限变形一般理论，分类研究（1）固定曲面上二维流动的涡量动力学刻画方式及其数值求解方法；（2）运动曲面上二维流动的Lagrange观点刻画方式及其数值求解方法；（3）弹性膜有限变形运动的Lagrange观点刻画方式及其数值求解方法。对应地，研制固定及运动曲面上薄层流动、弹性膜有限变形振动实验装置，通过实验修正、检验上述理论模型及其数值求解方法。另一方面，注重研究：（1）曲面几何特征同其有限变形运动之间的关系；（2）现代几何学等数学思想及方法借鉴至力学研究的"桥梁"。通过本项目研究有望为薄层介质提供一种具有原创意义的有限变形理论。

本项目致力于发展并检验一种“几何形态为曲面的连续介质有限变形理论”。对于实际的薄层流动或者薄膜运动，我们将介质的几何形态模型化为曲面，并引入面密度刻画介质的实际厚度。本项目的主要研究成果，包括：（1）体积、曲面、曲线连续介质有限变形的统一理论及其应用，籍此不仅获得单独存在的曲面介质的运动学与动力学刻画，并且可以获得存在于体积介质之中的曲面介质运动学与动力学刻画。（2）基于曲面主方向的正交基的非完整基理论及其应用，籍此可为曲面及其邻域内的场论提供形式上最为简单，且揭示曲面几何特征与物理过程之间关系最为清晰的场论方法。（3）一般曲面（二维）运动的涡量动力学理论及其应用，包括固定与运动基面上不可压缩与可压缩流动的一般刻画方式。（4）可变形壁面上的涡量动力学及其应用，获得了可变形壁面上涡量、涡量法向梯度、变形率张量等的新的刻画形式，充分揭示了壁面几何特征与物理过程之间的关系。（5）张量场物质导数的变形分解及其应用，澄清了微分流形上的Lie导数与张量场物质导数之间的关系；并且提出基于加速度场的变形分解标示流场中的典型变形区域，包括剪切、旋转、拉伸等。值得指出，上述五方面的理论成果都基于严格的数学与力学分析，并且都为我们原创性研究成果。我们已经基于相关数值研究，在一定程度上检验了上述主要理论的适用性，相关数值结果与实验结果吻合。