Symmetry Groups and Equivalence Transformations in the Nonlinear Donnell-Mushtari-Vlasov Theory for Shallow Shells

TL;DR

This paper analyzes the symmetry properties of Marguerre's equations in shallow shell theory, revealing their equivalence to von Karman equations and classifying their symmetry groups.

Contribution

It establishes the point symmetry groups of Marguerre's equations and demonstrates their equivalence to von Karman equations for large deflections.

Findings

Marguerre's equations are equivalent to von Karman equations.

Symmetry groups of Marguerre's equations are classified.

Equivalence holds for both time-independent and time-dependent cases.

Abstract

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.