超弹性细杆在大形变下的行为及其稳定性的分析研究

It has been known that the behavior and its stability of material specimen under load is an important subject in engineering and applied mathematics. It is very important in both theoretical research and security assessment. However, it is very difficult to analytically study the localized deformation states within a three-dimensional framework, especially when the deformation is large. This project intends to study the behavior and its stability of a hyperelastic cylinder subjected to axial forces at two ends when the deformation is large. We will improve the new method capable of processing large deformation problems published recently by the applicant and his co-author. Then we apply it to construct the 3D model equation for hyperelastic cylinder composed of generalized Blatz-Ko materials and Mooney-Rivlin polymerization nanometer fiber. Afterwards the model equation is used to deal with two types of boundary value problem (force-controlled problem and displacement-controlled problem) and discuss the stability of the solutions and the effect of the parameters. Finally, we will use finite element method to obtain the numerical solutions of the model equation, and do some comparative analysis with the analytical solutions. The results of this project will enrich the research method, and provide a mathematical explanation for the physical behavior of the material, and capture/predict the key features of the material behavior.

材料在荷载下的行为及其稳定性是工程和应用数学领域的一个重要课题，其无论是在理论研究还是在实际的安全评估中都有重要意义。但在三维背景下分析地研究局部变形状态非常困难，特别当形变很大的时候。本项目拟研究各向同性超弹性细杆在轴向拉力作用下发生大形变的行为及其稳定性。我们将改进申请人及其合作者最近发表的处理大形变问题的一种新方法，并应用它建立广义Blatz-Ko超弹性细杆和Mooney-Rivlin聚合纳米纤维在大形变下的三维模型方程，进而分别利用模型方程求两类边值问题（力控制问题和位移控制问题）的分析解并讨论解的稳定性和参数影响。最后用有限元求模型方程的数值解，并与分析解作比较分析。本项目所得结果将丰富现有的研究方法，对材料物理行为提供一种数学理论解释，并捕捉/预测材料行为的关键特性。