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.

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.

Stresses Emanating From the Supports of a Cylindrical Shell Produced by a Lateral Pressure Pulse

1972 ◽  
Vol 39 (1) ◽  
pp. 124-128 ◽  
Author(s):  
M. J. Forrestal ◽  
G. E. Sliter ◽  
M. J. Sagartz
Keyword(s):  
Cylindrical Shell ◽  
Shell Theory ◽  
Pressure Pulse ◽  
Integral Transform ◽  
Circular Cylindrical Shell ◽  
Radial Pressure ◽  
Rectangular Pulse ◽  
Wave Fronts ◽  
Timoshenko Type ◽  
Stress Resultant

A semi-infinite, elastic, circular cylindrical shell is subjected to two uniform, radial pressure pulses, one a step pulse and the other a short-duration, rectangular pulse. Solutions for the stresses emanating from both a clamped support and a simple support are presented for a Timoshenko-type shell theory and a shell bending theory. Results from the Timoshenko-type theory are obtained using the method of characteristics, and results from the shell bending theory are obtained using integral transform techniques. Numerical results from both shell theories are presented for the bending stress and the shear stress resultant. Results show that the effects of rotary inertia and shear deformation are important only in the vicinity of the wave fronts. However, if the duration of the pressure pulse is short, maximum stresses can occur in the vicinity of the wave fronts where a Timoshenko-type shell theory is required for realistic response predictions.

Three-Dimensional and Shell-Theory Analysis of Axially Symmetric Motions of Cylinders

1956 ◽  
Vol 23 (4) ◽  
pp. 563-568
Author(s):  
George Herrmann ◽  
I. Mirsky
Keyword(s):  
Shear Deformation ◽  
Shell Theory ◽  
Plate Theory ◽  
Wave Length ◽  
Three Dimensional ◽  
Approximate Theory ◽  
Axially Symmetric ◽  
Rotatory Inertia ◽  
Phase Velocities ◽  
Transition Wave

Abstract The frequency (or phase velocity) of axially symmetric free vibrations in an elastic, isotropic, circular cylinder of medium thickness is studied on the basis of the three-dimensional linear theory of elasticity and several different shell theories. To be in good agreement with the solution of the three-dimensional equations for short wave lengths, an approximate theory has to include the influence of rotatory inertia and transverse shear deformation, for example, in a manner similar to Mindlin’s plate theory. A shell theory of this (Timoshenko) type is deduced from the three-dimensional elasticity theory. From a comparison of phase velocities it appears that, to a good approximation, membrane and curvature effects on one hand, and on the other hand, flexural, rotatory-inertia, and shear-deformation effects are mutually exclusive in two ranges of wave lengths, separated by a “transition” wave length. Thus, in the full range of wave lengths, the associated lowest phase velocities may be determined on the basis of the membrane shell theory (for wave lengths larger than the transition wave length) and on the basis of Mindlin’s plate theory (for wave lengths smaller than the transition wave length).

On a Variational Theorem in Elasticity and Its Application to Shell Theory

1964 ◽  
Vol 31 (4) ◽  
pp. 647-653 ◽  
Author(s):  
P. M. Naghdi
Keyword(s):  
Boundary Conditions ◽  
Shell Theory ◽  
Three Dimensional ◽  
Transverse Shear Deformation ◽  
Special Significance ◽  
Field Equations ◽  
Variational Theorem ◽  
Three Dimensional Elasticity ◽  
Strain Measures

After stating a variational theorem which is a further generalization of known variational theorems and which has as its Euler equations all of the field equations and the boundary conditions of classical linear three-dimensional elasticity, the remainder of the paper deals with its application to shell theory. A new characterization of the basic system of field equations and the boundary conditions of the linear theory of elastic shells is derived which includes the effect of transverse shear deformation and involves only symmetric resultants and symmetric shell-strain measures. These results are of special significance in relation to those of a number of recent investigations in shell theory under the Kirchhoff-Love hypothesis in which the boundary-value problem of shell theory is recast in terms of symmetric (but not necessarily the same) variables.

ELASTIC THIN SHELLS: ASYMPTOTIC THEORY IN THE ANISOTROPIC AND HETEROGENEOUS CASES

1995 ◽  
Vol 05 (04) ◽  
pp. 473-496 ◽  
Author(s):  
D. CAILLERIE ◽  
E. SANCHEZ-PALENCIA
Keyword(s):  
Shell Theory ◽  
Asymptotic Theory ◽  
Order Term ◽  
Three Dimensional ◽  
Leading Order ◽  
Three Dimensional Elasticity ◽  
Thin Shell Theory ◽  
Variational Formulations ◽  
Natural Way

Asymptotic (two-scale) methods are used to derive thin shell theory from three-dimensional elasticity. The asymptotic process is done directly for the variational formulations, and existence and uniqueness theorems are given for the shell problem. The asymptotic behavior is the same as that recently derived by the authors using classical hypotheses of shell theory. The role of the subspace G of pure bendings (inextensional motions) appears in a natural way. The asymptotic is basically described by a leading order term contained in G and a lower order term contained in the orthogonal to G. As in anisotropic heterogeneous plates, which exhibit a coupling between flexion and traction, in heterogeneous shells there is coupling between the terms in G and in its orthogonal.

scholarly journals Three-dimensional spectrum of temperature fluctuations in stably stratified atmosphere

2008 ◽  
Vol 26 (7) ◽  
pp. 2037-2042 ◽  
Author(s):  
A. S. Gurvich ◽  
I. P. Chunchuzov
Keyword(s):  
Internal Waves ◽  
Horizontal Plane ◽  
Additional Assumption ◽  
Three Dimensional ◽  
Temperature Fluctuations ◽  
Wave Number ◽  
Axially Symmetric ◽  
Dimensional Spectrum ◽  
Analytic Expressions ◽  
Good Agreement

Abstract. A phenomenological model is proposed for a 3-D spectrum of temperature inhomogeneities generated by internal waves in the atmosphere. This model is a development of the theory based on the assumption that a field of Lagrangian displacements of fluid particles, induced by an ensemble of internal waves with randomly independent amplitudes and phases, is statistically stationary, homogeneous, axially symmetric in horizontal plane and Gaussian. For consistency of this model with measured spectra of temperature fluctuations in the stratosphere and mesosphere the additional assumption was introduced in to the model about the anisotropy of inhomogeneities to be dependent on their vertical sizes. The analytic expressions for both the 3-D and 1-D spectra are obtained. A model vertical wave number spectrum follows a −3 power law, whereas a horizontal spectrum contains two regions with a −3 slope, and the intermediate region with the slope between −1 and −3 depending on the rate of anisotropy decrease as a function of increasing sizes of the inhomogeneities. In the range of a few decades the model showed a good agreement with the results of measurements of the spectra in the troposphere, stratosphere and mesosphere.

Three-Dimensional Analysis of an Elastic Plate-Cylinder Intersection by the Boundary-Point-Least-Squares Technique

1977 ◽  
Vol 99 (1) ◽  
pp. 17-25 ◽  
Author(s):  
D. Redekop
Keyword(s):  
Boundary Conditions ◽  
Least Squares ◽  
Boundary Point ◽  
Shell Theory ◽  
Three Dimensional ◽  
Middle Part ◽  
Elasticity Equations ◽  
Three Dimensional Elasticity ◽  
Tension Loading ◽  
Theoretical Results

The boundary-point-least-squares technique is applied to the axisymmetric three-dimensional elasticity problem of a hollow circular cylinder normally intersecting with a perforated flat plate. The geometry of the intersection is partitioned into three parts. Boundary conditions on the middle part and continuity conditions between adjacent parts are satisfied using the numerical boundary-point-least-squares technique while the governing elasticity equations and all other boundary conditions are satisfied exactly. Sample theoretical results are presented for the case of axisymmetric radial tension loading on the plate. The results compare favorably with previously published experimental data and provide supplementary data to theoretical results obtained using existing shell theory solutions.

On the Form of Variationally Derived Shell Equations

1964 ◽  
Vol 31 (2) ◽  
pp. 233-238 ◽  
Author(s):  
Eric Reissner
Keyword(s):  
Variational Problem ◽  
Shell Theory ◽  
Three Dimensional ◽  
Torsion Problem ◽  
Considerable Influence ◽  
Equilibrium Equations ◽  
Circular Cylindrical Shells ◽  
Shell Equations ◽  
Basic Equations ◽  
Three Dimensional Elasticity

In the first part of the paper the basic equations of three-dimensional elasticity are formulated as a system of four equilibrium equations and seven stress displacement relations, together with a variational problem which has these eleven equations as Euler equations. In the second part of the paper, the new variational problem is used for a derivation of shell theory which accounts particularly simply for the differences between the resultants N12 and N21 and the couples M12 and M21. In the third part of the paper a solution is given of the torsion problem for circumferentially nonhomogeneous circular cylindrical shells, as an explicit demonstration of the fact that certain terms in the shell equations which are often of negligible influence sometimes are of considerable influence.

Exact Elasticity Solution for Laminated Anisotropic Cylindrical Shells

1993 ◽  
Vol 60 (1) ◽  
pp. 41-47 ◽  
Author(s):  
K. Bhaskar ◽  
T. K. Varadan
Keyword(s):  
Shell Theory ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Elasticity Solution ◽  
Transverse Loading ◽  
Cylindrical Bending ◽  
Simply Supported ◽  
Classical Shell Theory ◽  
Three Dimensional Elasticity ◽  
Anisotropic Cylindrical Shell

An exact three-dimensional elasticity solution is obtained for cylindrical bending of simply-supported laminated anisotropic cylindrical shell strips subjected to transverse loading. Displacements and stresses are presented for different angle-ply layups and radius-to-thickness ratios, so as to serve as useful benchmark results for the assessment of various two-dimensional shell theories. Finally, in the light of these results, the accuracy of the Love-type classical shell theory is examined.

Sign in / Sign up

Export Citation Format

Share Document