摘要
将弹性细杆的“Kirchhoff动力学比拟”方法推广到弹性薄壳,使弹性薄壳的变形在物理概念上和刚体的运动对应,在数学表述上等同,从而可以用刚体动力学的理论和方法研究弹性薄壳的变形,为连续的弹性薄壳提供新的离散化方法.在直法线假设下,在弹性中面上构筑空间正交轴系,此轴系沿坐标线“运动”的角速度构成两自变量的弯扭度.沿两个坐标线的弯扭度表达了弹性薄壳的变形和位形,证明了弯扭度之间以及弯扭度与中面切矢间的相容关系.用Euler角和Lame系数表达了非完整约束和中面位形的微分方程,用弯扭度和Lame系数表达了应变和应力以及内力及其本构方程.导出了用分布内力集度表达的弹性薄壳在变形后位形上的平衡偏微分方程组,方程的形式与刚体动力学的Euler方程和弹性细杆的Kirchhoff方程具有相似性,实现了Kirchhoff动力学比拟对弹性薄壳的推广.总结了弹性薄壳静力学和刚体动力学以及弹性细杆静力学在概念上的比拟关系.最后给出了一个算例.为研究弹性薄壳的变形和运动提供新的建模方法和研究思路.也可进一步推广到弹性薄壳动力学.
The Kirchhoff kinetic analogy is generalized from thin elastic rods to thin elastic shells.The generalization makes thin shell deformations physically correspond and mathematically equivalent to rigid body motions.Hence theories and methods of rigid body dynamics can be applied to investigate deformations of thin elastic shell,and also provide a novel discretization for continuous thin elastic shells.An orthogonal spatial axis system is established along the coordinate lines of the middle surface under the straight normal assumption.The moving of the axis system along the coordinate lines in unit velocity forms its“angular velocity”,which is the curvature-twist vector with two independent variables.The curvature-twist vector along two coordinate lines expresses the deformation and the configuration of a thin elastic shell.It is demonstrated that curvature-twist vectors are compatible,and curvature-twist vectors and tangential vectors of middle surface are compatible.Nonholonomic constraints and differential equations of middle surfaces are established in the Euler angles and the Lame coefficient form.The strain,the stress and the internal forces are formulated in the curvaturetwist vectors and the Lame coefficients.The equilibrium partial differential equations are presented with distributed internal forces intensity of thin elastic shells.The forms of the equations are similar to the Euler equations of rigid body dynamics and Kirchhoff equations of thin elastic rods.The fact means that the Kirchhoff kinetic analogy of thin elastic rods is generalized to thin elastic shells.The analogy relations between thin elastic shells and dynamics of rigid body or thin elastic rods are concluded.Finally,an example is given to show the application of this method.The proposed analogy leads to novel views and approaches to model and to analyze deformation of thin elastic shells.It is possible to generalize further the analogy for dynamics of thin elastic shells.
作者
薛纭
陈立群
Xue Yun;Chen Liqun(School of Mechanical Engineering,Shanghai Institute of Technology,Shanghai 201418,China;School of Science,Harbin Institute of Technology,Shenzhen 518055,China)
出处
《力学学报》
EI
CAS
CSCD
北大核心
2021年第1期234-247,共14页
Chinese Journal of Theoretical and Applied Mechanics
基金
国家自然科学基金资助项目(11872159,11372195).
