轴向载荷作用下半无限长超弹性圆柱壳轴对称运动的模型建立

作     者：赵冠充 王大卫 

作者机构：辽宁师范大学数学学院辽宁 大连 

出 版 物：《应用数学进展》 (Advances in Applied Mathematics)

年 卷 期：2024年第13卷第9期

页      面：4360-4369页

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

主　　题：超弹性圆柱壳 三项式Mooney-Rivlin模型 结构边界 模型建立 

摘      要：本文基于一类含有高阶项的三项式Mooney-Rivlin材料模型,计入边界条件、外部激励和结构阻尼,建立了描述半无限长超弹性圆柱壳非线性动力学行为的数学模型。假设圆柱壳产生径向和轴向对称变形,得到圆柱壳在变形过程中产生的动能和势能。考虑外部载荷作用,在轴向方向上得到外力对圆柱壳所做的功,进而得到圆柱壳的总势能。基于拉格朗日函数,利用变分原理,得到了描述圆柱壳非线性径向和轴向运动的非线性偏微分方程组。利用不可压缩约束条件,将圆柱壳的径向和轴向运动进行解耦,得到了一类描述圆柱壳径向运动的单一偏微分方程。为了方便对方程求解,利用行波变换,得到了一类非线性常微分方程,即行波方程,该方程可用于描述圆柱壳的非线性径向运动。Based on a class of ternary Moonie-Rivlin material models with higher-order terms, a mathematical model describing the nonlinear dynamic behavior of a semi-infinite superelastic cylindrical shell is established by considering boundary conditions, external excitations and structural damping. It is assumed that the cylindrical shell produces radial and axial symmetric deformation, and the kinetic energy and potential energy generated by the cylindrical shell during deformation are obtained. Considering the external load action, the work done by the external force on the cylindrical shell in the axial direction is obtained, and then the total potential energy of the cylindrical shell is obtained. Based on the Lagrangian function and the variational principle, the nonlinear partial differential equations describing the nonlinear radial and axial motion of the cylindrical shell are obtained. By using incompressible constraints, the radial and axial motions of a cylindrical shell are decoupled, and a single partial differential equation describing the radial motion of a cylindrical shell is obtained. In order to solve the equations conveniently, a class of nonlinear ordinary differential equations, namely traveling wave equations, is obtained by using traveling wave transform, which can be used to describe the non-linear radial motion of cylindrical shells.