On invariant integrals in the Marguerre-von Kármán shallow shell

Abstract

By employing the second-order Noether's theorem, several new invariant integrals have been derived for the non-linear shallow shell—the Marguerre-von Kármán shell. The dynamic effect is considered in the derivations. These invariant integrals are path-independent over the projection image of the middle surface of the shell in a Cartesian plane, in which the projection area of the middle surface of the shallow shell is maximum. The proposed invariant integrals can be used to evaluate the asymptotic field around a defect embedded in the shell. Unlike most other studies, the Lagrangian density of the invariant variational principle used here belongs to a mixed type variational principle.