Three-Dimensional and Shell-Theory Analysis of Elastic Waves in a Hollow Sphere: Part 1—Analytical Foundation

1969 ◽  
Vol 36 (3) ◽  
pp. 431-439 ◽  
Author(s):  
A. H. Shah ◽  
C. V. Ramkrishnan ◽  
S. K. Datta
Keyword(s):  
Elastic Waves ◽  
Shell Theory ◽  
Three Dimensional ◽  
Frequency Equation ◽  
Wave Number ◽  
Normal Strain ◽  
Theory Analysis ◽  
Analytical Foundation ◽  
Transverse Normal Strain ◽  
Approximate Equations

In Part 1 of this paper an exact analysis of the nonaxisymmetric wave propagation in a hollow elastic sphere is presented. It is found that the characteristic frequency equation is independent of the longitudinal wave number. Approximate equations for thin shells and membranes are derived by way of asymptotic expansions. In general, the vibrations fall into two distinct classes, one of which is equivoluminal. Also included in the paper is a six-mode shell theory in which the effects of transverse normal strain are included. A technique due to van der Neut is used to separate the governing partial differential equations whereby two frequency equations corresponding to the two classes of vibrations are obtained.

On Free Vibrations of an Embedded Anisotropic Spherical Shell

1997 ◽  
Vol 119 (4) ◽  
pp. 481-487 ◽  
Author(s):  
W. Q. Chen ◽  
H. J. Ding
Keyword(s):  
Spherical Shell ◽  
Elastic Medium ◽  
Shell Theory ◽  
Three Dimensional ◽  
Homogeneous Solution ◽  
Free Vibrations ◽  
Normal Strain ◽  
Auxiliary Variables ◽  
Transverse Normal Strain ◽  
Isotropic Spherical Shell

In this paper, free vibrations of a spherically isotropic spherical shell embedded in an elastic medium of Pasternak type are studied by using a six-mode shell theory that includes effects of shear deformation, rotary inertia, and transverse normal strain. The separable homogeneous solution for displacements and stresses in a deep spherical shell is derived and two classes of vibrations are obtained by the introduction of five auxiliary variables. Numerical results are compared with those predicted by two simpler shell theories mentioned in the paper and those by three-dimensional elastic theory.

A Refined Theory for Laminated Anisotropic, Cylindrical Shells

1974 ◽  
Vol 41 (2) ◽  
pp. 471-476 ◽  
Author(s):  
J. M. Whitney ◽  
C.-T. Sun
Keyword(s):  
Shear Deformation ◽  
Shell Theory ◽  
Theory Of Elasticity ◽  
Small Radius ◽  
Refined Theory ◽  
Normal Strain ◽  
Governing Equations ◽  
Transverse Normal Strain ◽  
Anisotropic Cylindrical Shell ◽  
Good Agreement

A set of governing equations and boundary conditions are derived which describe the static deformation of a laminated anisotropic cylindrical shell. The theory includes both transverse shear deformation and transverse normal strain, as well as expansional strains. The validity of the theory is assessed by comparing solutions obtained from the shell theory to results obtained from exact theory of elasticity. Reasonably good agreement is observed and both shear deformation and transverse normal strain are shown to be of importance for shells having a relatively small radius-to-thickness ratio.

Impulsive Deformation of Mirsky-Herrmann’s Thick Cylindrical Shells by a Numerical Method

1973 ◽  
Vol 40 (4) ◽  
pp. 1009-1016 ◽  
Author(s):  
M. Ziv ◽  
M. Perl
Keyword(s):  
Shell Theory ◽  
Pulse Velocity ◽  
Normal Strain ◽  
Finite Difference Technique ◽  
Characteristics Method ◽  
Transverse Normal Stress ◽  
Infinite Cylindrical Shell ◽  
Shell Equations ◽  
Thickness Parameter ◽  
Transverse Normal Strain

The transient response of a thick elastic semi-infinite cylindrical shell subjected to impulsive step loads is obtained. This work presents solutions to two boundary-value problems. First, the shell is exposed to an axially step pulse velocity load while second, the pulse is applied radially to the shell. The influence of the transverse normal stress and the transverse normal strain on the deformation is being studied in great detail. The shell theory employed is based on the thick-shell equations derived by Mirsky and Herrmann, which comprise these transversal effects. These equations are solved by the characteristics method, while integration is carried out by the finite-difference technique. The results also present the influence of the thickness parameter (h/R) on the solution. Comparisons are made to solutions of other shell theories which neglect the transversal effects. Major conclusions show the existence of an important influence of the transverse normal strain on the deformation.

Three-Dimensional and Shell-Theory Analysis of Elastic Waves in a Hollow Sphere: Part 2—Numerical Results

1969 ◽  
Vol 36 (3) ◽  
pp. 440-444 ◽  
Author(s):  
A. H. Shah ◽  
C. V. Ramkrishnan ◽  
S. K. Datta
Keyword(s):  
Elastic Waves ◽  
Three Dimensional ◽  
Numerical Results ◽  
Mode Analysis ◽  
Theory Analysis ◽  
Frequency Spectra ◽  
The Past ◽  
Shell Analysis ◽  
Shell Equations ◽  
Thick Shells

In this Part 2 of the paper, numerical results obtained through an IBM 7044 digital computer are presented. Frequency values for the two classes of vibrations for different values of n are calculated from the exact frequency equations and also from the shell equations. Comparisons are made with the exact frequency spectra of those obtained from the present six-mode shell analysis and other available shell and membrane analyses. It is found that the lowest branches of the frequency spectra for the two classes of vibrations obtained by the six-mode analysis agree very well with the exact ones, even for very thick shells, whereas other shell analysis done in the past make erroneous predictions when the thickness-to-radius ratio of the shell is greater than 10 percent.

Axially Symmetric Motions of a Two-Layered Timoshenko-Type Cylindrical Shell

1966 ◽  
Vol 33 (4) ◽  
pp. 838-844 ◽  
Author(s):  
J. P. Jones ◽  
J. S. Whittier
Keyword(s):  
Cylindrical Shell ◽  
Shell Theory ◽  
Three Dimensional ◽  
Wave Number ◽  
Velocity Versus ◽  
Axially Symmetric ◽  
Low Frequencies ◽  
Timoshenko Type ◽  
Three Dimensional Elasticity ◽  
Rayleigh Wave Speed

A Timoshenko-type theory is presented for the dynamics of a cylindrical shell whose two layers are joined by a perfect bond. An assessment of the accuracy of the theory is obtained by solving the problem of axial propagation of an infinite train of axially symmetric waves and comparing the results with those obtained from the three-dimensional elasticity theory. Frequencies of the four lowest modes are accurately predicted by the shell theory for sufficiently long wavelengths and low frequencies. Preliminary comparisons of displacement distributions indicate that the shell-theory displacements are accurate in a more restricted frequency-wavelength regime. Timoshenko shear coefficients are determined by matching simple thickness-shear cutoff frequencies rather than by matching the lower Rayleigh wave speed. This is found preferable by consideration of the shapes of the first-mode phase velocity versus wave-number curves for the two theories.

Propagation of Harmonic Waves in Orthotropic Circular Cylindrical Shells

1973 ◽  
Vol 40 (1) ◽  
pp. 168-174 ◽  
Author(s):  
A. E. Armena`kas ◽  
E. S. Reitz
Keyword(s):  
Shell Theory ◽  
Cylindrical Shells ◽  
Three Dimensional ◽  
Longitudinal Shear ◽  
Theory Of Elasticity ◽  
Frequency Equation ◽  
Radial Coordinate ◽  
Harmonic Waves ◽  
Circular Cylindrical Shells ◽  
General Frequency

In this investigation, the general frequency equation for trains of harmonic waves having an arbitrary number of circumferential nodes, traveling in orthotropic, circular, cylindrical shells is established on the basis of the three-dimensional linear theory of elasticity, by expanding the displacement components in power series of the radial coordinate. Simpler forms of the frequency equation for axisymmetric nontorsional and torsional motion and for longitudinal-shear and plane-strain motion are established and discussed. The frequency equation has been evaluated numerically on an IBM 360/50 digital computer system and the numerical results are compared with those obtained on the basis of an approximate shell theory.

To the theory analysis of tightness of tank of pressure testing

2019 ◽  
Vol 5 (4) ◽  
pp. 129-142
Author(s):  
Vladislav Sh. Shagapov ◽  
Ismagilyan G. Khusainov ◽  
Emiliya V. Galiakbarova ◽  
Zulfya R. Khakimova
Keyword(s):  
Relaxation Time ◽  
Three Dimensional ◽  
Petroleum Products ◽  
Technical Condition ◽  
Excess Pressure ◽  
Soil Parameters ◽  
Theory Analysis ◽  
Pressure Testing ◽  
Tank Pressure ◽  
Over Time

This article studies the process of relaxation of the pressure in a tank with the damaged area of the wall after pressure-testing. The authors use different methods for the diagnosis of the technical condition of objects of petroleum products storage. Pressure testing is one of nondestructive methods. The rate of pressure decrease is characteristic of the system tightness. This article studies the cases of ground and underground location of the tank. Pressure testing involves excess pressure inside of a tank and observing its decrease. Over time, one can assess the integrity of the system. This has required creating mathematical models to account the filtration of the liquid depending on the location of the tank. The results include the analytical solution of the task and the formulas for describing the dependence of the relaxation time of pressure in the tank from the liquid and soil parameters, geometry of the tank, and the damaged portion of the wall. The two- and three-dimensional cases of liquids filtration for the case of underground location of the tank were considered. The results of some numerical calculations of the dependence of reduction time and the time of half-life pressure from the area of the damaged portion of the wall were shown. The obtained solutions allow assessing the extent of the damaged area by the pressure testing with known values of tank, liquid, and soil.

scholarly journals Alternative derivation of the higher-order constitutive model for six-parameter elastic shells

2021 ◽  
Vol 72 (2) ◽  
Author(s):  
Mircea Bîrsan
Keyword(s):  
Energy Density ◽  
Constitutive Model ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Shell Theory ◽  
Constitutive Relations ◽  
Three Dimensional ◽  
Classical Shell Theory ◽  
Three Dimensional Elasticity ◽  
General Method

AbstractIn this paper, we present a general method to derive the explicit constitutive relations for isotropic elastic 6-parameter shells made from a Cosserat material. The dimensional reduction procedure extends the methods of the classical shell theory to the case of Cosserat shells. Starting from the three-dimensional Cosserat parent model, we perform the integration over the thickness and obtain a consistent shell model of order $$ O(h^5) $$ O ( h 5 ) with respect to the shell thickness h. We derive the explicit form of the strain energy density for 6-parameter (Cosserat) shells, in which the constitutive coefficients are expressed in terms of the three-dimensional elasticity constants and depend on the initial curvature of the shell. The obtained form of the shell strain energy density is compared with other previous variants from the literature, and the advantages of our constitutive model are discussed.

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