A Dimensional Splitting Method for 3D Elastic Shell with Mixed Tensor Analysis on a 2D Manifold Embedded into a Higher Dimensional Riemannian Space (Journal of Mathematical Study, 2021)
In this paper, a mixed tensor analysis for a two-dimensional (2D) manifoldembedded into a three-dimensional (3D) Riemannian space is conducted and its applications to construct a dimensional splitting method for linear and nonlinear 3D elasticshells are provided. We establish a semi-geodesic coordinate system based on this2D manifold, providing the relations between metrics tensors, Christoffel symbols, covariant derivatives and differential operators on the 2D manifold and 3D space, andestablish the Gateaux derivatives of metric tensor, curvature tensor and normal vectorand so on, with respect to the surface $\vec{Θ}$ along any direction $\vec{η}$ when the deformation ofthe surface occurs. Under the assumption that the solution of 3D elastic equations canbe expressed in a Taylor expansion with respect to transverse variable, the boundaryvalue problems satisfied by the coefficients of the Taylor expansion are given.
Keywords: Dimensional splitting method; linear elastic shell; mixed tensor analysis; nonlinear elastic shell.