Variational formalism for generic shells in general relativity

We investigate the variational principle for the gravitational field in the presence of thin shells of completely unconstrained signature (generic shells). Such variational formulations have been given before for shells of timelike and null signatures separately, but so far no unified treatment exists. We identify the shell equation as the natural boundary condition associated with a broken extremal problem along a hypersurface where the metric tensor is allowed to be nondifferentiable. Since the second order nature of the Einstein-Hilbert action makes the boundary value problem associated with the variational formulation ill-defined, regularization schemes need to be introduced. We investigate several such regularization schemes and prove their equivalence. We show that the unified shell equation derived from this variational procedure reproduce past results obtained via distribution theory by Barrabes and Israel for hypersurfaces of fixed causal type and by Mars and Senovilla for generic shells. These results are expected to provide a useful guide to formulating thin shell equations and junction conditions along generic hypersurfaces in modified theories of gravity.

On a consistent finite-strain shell theory for incompressible hyperelastic materials

In this paper, a consistent finite-strain shell theory for incompressible hyperelastic materials is formulated. First, for a shell structure made of an incompressible material, the three-dimensional (3D) governing system is derived through the variational approach, which is composed of the mechanical field equation and the constraint equation. Then, series expansions of the independent variables are conducted about the bottom surface and along the thickness direction of the shell. The recursive relations of the coefficient functions in the series expansions can be derived from the original 3D governing system. Further from the top surface boundary condition, a 2D vector shell equation is obtained, which represents the local force-balance of a shell element. The associated edge boundary conditions are also proposed. It is verified that shell equation system is consistent with the 3D variational formulation. The weak formulation of the shell equation is established for future numerical calculations. To show the validity of the shell theory, the axisymmetric deformations of a spherical and a circular cylindrical shell made of incompressible neo-Hookean materials are studied. By comparing with the exact solutions, it is shown that the asymptotic solutions obtained from the shell theory attain the accuracy of O( h2).

Free vibration analysis of annular sector plates via conical shell equations

In this paper, free vibration analysis of annular sector plates has been presented via conical shell equations. By using the first-order shear deformation theory (FSDT) equation of motion of conical shell is obtained. The method of discrete singular convolution (DSC) and method differential quadrature (DQ) are used for solution of the vibration problem of annular plates for some special value of semi-vertex angle via conical shell equation. The obtained numerical results based on the two numerical approaches for annular sector plates compare well with the results available in the literature. The effects of some geometric parameters, grid numbers and types of the grid distribution have been discussed for curved plates.

Leslie Sydney Dennis Morley FREng FRAeS. 23 May 1924 — 16 June 2011

Leslie Morley's research focused on modelling structural behaviour, with particular emphasis on plates and shells. He developed the Morley shell equation, which has been acknowledged as the simplest equation consistent with first-order shell theory. As the finite element method rose to prominence he developed elements for both plates and shells. He then worked on developing a set of new finite elements able to handle complex shell behaviour in both the linear and nonlinear regimes. He also observed that it was possible to augment the finite element solution by using singular solutions to calculate the stress intensity factor at a crack tip in a thin-walled metal structure and thereby to compute crack propagation rates. In undertaking his research Morley probed into the mathematical and physical depths of the problems he confronted, and produced some outstanding and significant results.

Dynamic Model of Elastically Mounted Machinery with Cylindrical Shell

Mounting machinery by isolators can reduce vibration transmitted to the base and attenuated environmental noise. In this paper the machine having cylindrical shell such as electric motors is modeled by thin-wall cylindrical shell motion equation. The reaction force exerted by isolator is considered as point force and integrated in the shell equation. The typical vibration excitation of machinery is represented by point and line excitations. The forces transmitted to the base through isolators are then calculated under different excitations. Conclusions with respect to machinery and isolation system design are presented based on numerical results.