Spherical Sandwich Shells Research Articles

Free Damped Vibrations of Sandwich Shells of Revolution

Using a zig-zag model the free damped vibrations of sandwich shells of revolution are investigated. As special cases the vibration analysis under consideration of damping of cylindrical, conical and spherical sandwich shells is performed. A specific sandwich shell finite element with 54 degrees of freedom is employed. Starting from the energy method the damping model is developed. Numerical examples for the free vibration analysis with damping based on the proposed finite element approach are discussed. Results for sandwich shells show a satisfactory agreement with various reference solutions.

Axisymmetric Vibrations of Sandwich Spherical Shells having Cross-ply Laminated Faces.

This paper deals with axisymmetric vibrations of sandwich spherical shells having cross-ply laminated faces. Vibration analysis is carried out using the Ritz method. The expressions for the total strain and kinetic energies of the shells include the effect of transverse shear deformation of laminated faces and a core. Displacement functions satisfying boundary conditions are expressed as power series. The natural frequencies and corresponding mode shapes of the shells are computed. The accuracy and validity of the present results are confirmed by convergence study and comparison with published data. The results of parametric study are discussed for sandwich spherical shells having different orthotropy ratios, thickness ratios and opening angles.

On the non-linear buckling of circular shallow spherical sandwich shells under the action of uniform edge moments

In this paper, a non-linear axisymmetrical bending theory for a circular, planform, shallow, spherical, sandwich shell with a soft core under the action of uniform edge moments is constructed by means of calculus of variations. An analytic solution for the critical buckling edge moment is obtained by use of Yeh Kai-yuan and Liu Ren-huai's modified iteration method [ Bull. Sci. 2, 142 (1965) (in Chinese); Bull. Sci. 2, 145 (1965) (in Chinese); Bull. Sci. 3, 253 (1965) (in Chinese)].

Dynamic Response of Sandwich Shells of Revolution

Development of an annular finite element, for the linear elastic analysis of sandwich shells, is reported here. The derivation of stiffness and mass matrices is based on improved shell theory which takes into account the effects of rotary inertia and transverse shear deformation. Flexural rigidity of the faces is included in the formulation. The core of the sandwich shell is assumed to be incompressible in the radial direction. Numerical examples of spherical sandwich shells with two types of boundary conditions—(i) fixed and (ii) pinned along the outer periphery—have been presented. The results are generated for displacements and frequencies.

Curved Multilayered Element for Axisymmetric Shells

The study of axisymmetric multilayer sandwich shells with n stiff layers and n-1 weak cores by the finite element method is presented. A proposed curved multilayered element is used to investigate the elastic behavior of circular plates, cylindrical, conical, and spherical sandwich shells under various loading conditions. Numerical results obtained in these studies indicate excellent agreement with existing theoretical results.

Effect of Pulse Duration on Head Injury

The human head is modelled as a fluid-filled spherical sandwich shell. The fluid is inviscid, irrotational, and compressible. The axisymmetric loading function is a timewise Heaviside function with a variable duration. The effect of the pulse duration on skull fracture and brain injury is investigated numerically. Short loading pulses are found to be less damaging. Human tolerance curves are obtained using hypothetical skull fracture and brain injury criteria. The curves follow the some trend as that of the Wayne State Curve.

On non-symmetric vibration of deep spherical sandwich shells

This paper deals with the free non-symmetric vibration of deep spherical sandwich shells. The sandwich shell considered herein consists of three layers. A variational technique is utilized to obtain the equations of motion as well as the appropriate boundary conditions. The effects of transverse shear deformation and rotary inertia have been included in this analysis.

Dynamic analysis of a fluid-filled spherical sandwich shell-a model of the human head.

Abstract The human head is modelled as a fluid-filled three-layered spherical sandwich shell. The shell is considered thin and elastic. The fluid is inviscid, irrotational and compressible. The system is subjected to an axisymmetric but arbitrary time-dependent impact. A free vibration analysis of the model is obtained in the form of a frequency equation. The linear equations governing the transient axisymmetric response of the system are derived and they are solved by finite difference techniques. The stress distributions in the skull and the pressure variations in the brain are presented on the basis of the numerical computations.

Free vibration of deep spherical sandwich shells

This paper deals with axisymmetric vibrations of deep sandwich spherical shells. The shell consists of a thick core and two face sheets of the same isotropic material with equal thickness. The appropriate differential equations have been obtained in terms of two new arbitrary functions which replace the meridional displacement components. The solution is in terms of Legendre functions from which numerical results are computed by using an iterative technique. Effects of geometric and elastic properties of the core and the face sheets are presented in the form of graphs.

Classical and nonlinear buckling analyses of spherical sandwich shells

The buckling and initial postbuckling behavior of clamped shallow spherical sandwich shells with dissimilar face sheets under a uniform pressure is studied. The numerical results show that the buckling and initial post-buckling behavior of clamped shallow spherical sandwich shells with dissimilar face sheets is similar to that of the corresponding homogeneous shell.The classical buckling analysis for spherical sandwich shells under a uniform pressure is also presented. The results indicate that it is possible to obtain the buckling curves of spherical sandwich caps from those of the homogeneous cap using a magnification factor which is obtained via the classical buckling analysis, for large values of the sandwich shell parameter.

On the Buckling and Initial Postbuckling Behavior of Shallow Spherical and Conical Sandwich Shells

The boundary-value problems associated with the axisymmetrical, asymmetrical, and initial postbuckling behavior of the clamped shallow conical sandwich shell under a uniform pressure are developed for face sheets of the same material and equal thicknesses. The numerical results presented show that the buckling and initial postbuckling behavior of the sandwich cap is similar to that of the conical homogeneous cap. The effects of the thickness and material parameters on the axisymmetric snap-through buckling behavior of the clamped shallow spherical and conical sandwich shells are also studied. For one material parameter considered the axisymmetric buckling behavior of the sandwich shells is similar to that of the corresponding homogeneous shells; so it is suggested that, without any further analysis, the asymmetric buckling and the initial postbuckling behavior of the sandwich cap can be obtained by conjecture. For the other material parameters considered, the axisymmetric buckling behavior of the sandwich caps is different from the axisymmetric buckling behavior of the corresponding homogeneous caps.

Buckling and initial post-buckling behavior of clamped shallow spherical sandwich shells

This paper presents the development of a set of nonlinear differential equations that are suitable for the analysis of the buckling and initial post-buckling behavior of thin shallow spherical sandwich shells under axisymmetrical loads. The boundary value problems associated with the axisymmetrical, asymmetrical and initial post-buckling behaviors for clamped shallow spherical sandwich shells under certain axisymmetrical loads are developed also. Finally, numerical results of the buckling and initial post-buckling behavior of the clamped shallow spherical sandwich shell under pressures that are distributed uniformly over the entire reference surface of the shell are studied for face sheets of the same material and equal thicknesses. The numerical results show that the buckling and initial post-buckling behavior for the sandwich cap is very similar to that for the classical homogeneous cap.

The spherical sandwich shell under axisymmetric static and dynamic loading

A formulation and formal solution is presented for the dynamic response of a spherical sandwich shell under axisymmetric but otherwise arbitrarily distributed, time-dependent surface tractions, for arbitrary initial conditions and (admissible) homogeneous boundary conditions. The sandwich shell is composed of a core and two laminated covering facings. The core deforms in shear and bending. The thin facings deform by membrane action only. The solution is obtained in terms of the eigenfunctions associated with the free vibration of the shell, and appropriate orthogonality and normalization relations are formulated. The specific case of the free-vibration problem of a hemispherical sandwich shell is solved, and an example of shell response to a transient loading condition is presented. The corresponding static solution is also given in terms of an eigenfunction expansion.

Stress state around a circular cutout in a spherical sandwich shell

Stress state around a circular cutout in a spherical sandwich shell

Free Vibrations of Spherical Sandwich Shells

In the present study, the problem of axisymmetric vibrations of spherical sandwich shells is considered. In obtaining the governing differential equations, the longitudinal inertia terms u,it and v,it can be neglected. This makes possible the introduction of a stress function F similar to the one in elastostatic problems. The differential equations are written in terms of an arbitrary deformation function φ and the stress function F. For simplicity in the final results, the face sheets and core are considered to be isotropic elastic and of uniform thickness t and c, respectively. In some applications, the core material acts as a heat shield and an insulator. In this paper, the effect of the thermal gradient along the thickness of the sandwich is not considered. Since the core filler is of honeycomb thermoplastic or aluminum foam having low values of shearing modulus, an order analysis of various terms in the differential equations establishes that the problem can be studied in two parts. This results in two sets of differential equations, which are applied in accordance with the physical parameters of the core geometry and material. The final frequency equation |aij| = 0; i, j = 1, ⋯4 and the results are given for two types of boundary conditions (fixed and simply supported). The solution of the frequency equation degenerates to the classical solution for vibration of shallow shells and to that of two layered shells.

Buckling of spherical sandwich shells

An experimental investigation was carried out to determine the critical buckling loads of several shallow spherical sandwich shells. A cold-forming process simultaneously using pressure and vacuum was employed to manufacture the nearly perfect spherical facing layers from 5052 aluminum-alloy sheets of 0.006 and of 0.008-in. thicknesses. Eight shallow spherical-shell specimens of 20-in. base diameter and of 20 and 30-in. radii with 1/8 and 1/4-in. thickness of “Flexcore” have been tested in a 300-psi autoclave specifically designed for these experiments. The pressure on shells was developed by the differential pressure between the inner and the outer chambers separated by the shell being tested. When the inner chamber was maintained at atmospheric pressure and gas pressure was applied in the outer chamber, the testing procedure was termed “soft.” Alternatively, the inner chamber would be filled with fluid with the outer chamber remaining filled with gas. By initially pressurizing both chambers equally, a load on the shell could be developed by the differential pressure due to controlled bleeding of the fluid inside the inner chamber, while the gas in the outer chamber was maintained at the initial pressure. This is an accurate volume-control experiment and this testing procedure was termed “hard.” In the latter case, it was possible to monitor the displacements of the shell for each load increment with a nest of clip gages of an unique design. It was found that there is no substantial difference in the buckling loads between the hard and “soft” systems. All shells buckled in the plastic range. A reasonably good correlation is obtained with a linear theory using the double modulus for the sandwich segments.

Natural Frequencies of Closed Spherical Sandwich Shells

Solutions for the axially symmetric motion of an elastic spherical sandwich shell are obtained from a theory of shells that includes the effects of transverse shear deformation, transverse normal stress, and rotatory inertia. Frequency equations and mode shapes are derived for the free vibrations of a closed spherical shell. It is shown that three branches appear in the spectrum of torsionless modes, and two branches appear in the spectrum of torsional modes.

Note: The inertia effect in the tensile plastic instability of a thin spherical shell

Finite element simulations were conducted to investigate the dynamic responses of metallic sandwich spherical shells with graded aluminum foam cores under inner blast loading. The deformation of spherical shells, the energy absorption of each core layer, and the propagation characteristic of stress waves in the foam core layers were analyzed and discussed. The spherical shells exhibited an overall inflation–deformation mode as the foam cores were compressed gradually. The arrangement of the core layers with different relative densities had significant effects on the dynamic plastic responses of the spherical shells. The core layer arrangements of 15%–20%–10% and 20%–15%–10% (relative densities) from inside to outside demonstrate the optimal resistance to blast loading.

Effect of mass loss on the transient response of a shallow spherical sandwich shell

References 1 Huber, H and LeBihan, R , Emetteur capillaire d'ions de cesium: application a des convertisseurs thermioniques, Combustion and Propulsion Panel, AGARD-NATO, Athens, Greece (July 1963) 2 Dresser, D L , Dunlop, J D , Huber, H, LeBihan, R, Hill, V L , McMurray, N D , and Muszer, K E ,' 'Investigation of capillary emitter thermionic converters, Aeronaut Systems Div TDR63-865 (September 1963) 3 Dykman, I M , ' 'On the mechanism of activator vapours supply to the outer surface of a porous metal-film cathode,' Radiotekhn i Elektron 2,1525-1257(1957) 4 Clausing, P , ' 'Over de Stationmaire Strooming Van Zeer Verdunde Gassen, Physica 9, 65-80 (1929) 5 Langmuir, I , Phenomena, Atoms and Molecules (Philosophical Library, Inc , New York, 1950), pp 322-384 6 Houston, J M , ' 'Thermionic emission of refractory metals in cesium vapor, Proc Roundtable Discussion, Power Information Center, Rept PIC-ELE-T1 3/3, Appendix F-l, Univ of Penn (June 1961) ' 'Analysis of typical voltage-current Government-Industry Thermionic Power Information Center Rept 1-1-1-9, Univ of Penn (December 7 Nottingham, W B , curves, Proc Third Roundtable Discussions, PIC-ELE-T1 209/1, pp 1961) 8 Nottingham, W B , ' 'Sheath and plasma theory of an isothermal diode, Topical Rept for the Thermionic Emitter Materials Res Program, Contract NONR 3563 (00), Thermo Electron Eng Co (1962) 9 Dresser, D L , ' 'Analysis of capillary converter experimental results, Thermionic Conversion Specialists' Meeting, Gatlinburg, Tenn , October 7-9, 1963 (to be published) 10 Nottingham, W B , ' 'General theory of the plasma diode energy converter, TR TEE-7002-5, Thermo Electron Eng Co (1960) 11 Breitwieser, R , Cesium diode operation in three modes, Rept on 23rd Annual Conference Physical Electronics, Mass Inst Tech , Cambridge, Mass , pp 267-280 (1963) 12 Johnson, E O , ' 'Studies of externally heated hot cathode arcs, RCA Rev 16,498-532(1955) 13 Hernquist, K G , ' 'Analysis of the arc mode operation of the cesium vapor thermionic energy converter, Proc Inst Electrical Electron Engrs 51,748-754(1963) 14 Wilson, V C and Lawrence, J , ' 'Design and operation of laboratory type parallel plane cesium thermionic converters, Advanced Energy Conversion 2, 335-340 (1962) 15 Schneider, R T arid Dunlop, J D , Spectroscopic investigation of a thermionic capillary cesium energy converter, Appl Phys Letters 3, 73-75 (1963)