非线性Donnell-Mushtari-Vlasov扁壳理论中的对称群和等价变换
Symmetry Groups and Equivalence Transformations in the Nonlinear Donnell-Mushtari-Vlasov Theory for Shallow Shells
摘要：
在只受横向载荷的扁壳情况下，各向同性弹性薄壳大挠度的非线性Donnell-Mushtari-Vlasov理论导出了一个由两个耦合的非线性四阶偏微分方程组成的系统，称为Marguerre方程。该系统包括两个任意元素--壳体中面的曲率张量和单位面积横向载荷函数。本文建立了Marguerre方程的点对称群，解决了相应的群分类问题。结果表明，Marguerre方程在与时间无关和与时间有关的情况下，都等价于von Karman方程。还观察到，同样适用于各向异性扁壳的场方程。
英文摘要：
In the case of transversely only loaded shallow shells, the nonlinear Donnell-Mushtari-Vlasov theory for large deflection of isotropic thin elastic shells leads to a system of two coupled nonlinear forth-order partial differential equations known as Marguerre's equations. This system involves two arbitrary elements -- the curvature tensor of the shell middle-surface and the function of transversal load per unit surface area. In the present note, the point symmetry groups of Marguerre's equations are established, the corresponding group classification problem being solved. It is shown that Marguerre's equations are equivalent to the von Karman equations for large deflection of plates in the time-independent case and in the time-dependent case as well. It is also observed that the same holds true in respect of the field equations for anisotropic shallow shells.
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