Approximate Elasticity Solution for Orthotopic Cylinder Under Hydrostatic Pressure and Band Loads

1970 ◽  
Vol 37 (1) ◽  
pp. 101-108 ◽  
Author(s):  
A. P. Misovec ◽  
J. Kempner
Keyword(s):  
Approximate Solution ◽  
Shell Theory ◽  
Good Accuracy ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
External Pressure ◽  
Theory Of Elasticity ◽  
Numerical Comparison ◽  
Axial Loads ◽  
Special Case

An approximate solution to the Navier equations of the three-dimensional theory of elasticity for an axisymmetric orthotopic circular cylinder subjected to internal and external pressure, axial loads, and closely spaced periodic radial loads is developed. Numerical comparison with the exact solution for the special case of a transversely isotropic cylinder subjected to periodic band loads shows that very good accuracy is obtainable. When the results of the approximate solution are compared with previously obtained results of a Flu¨gge-type shell solution of a ring-reinforced orthotropic cylinder, it is found that the shell theory gives fairly accurate representations of the deformations and stresses except in the neighborhood of discontinuous loads. The addition of transverse shear deformations does not improve the accuracy of the shell solution.

Stability Loss in Thick Transversely Isotropic Cylindrical Shells Under Axial Compression

1993 ◽  
Vol 60 (2) ◽  
pp. 506-513 ◽  
Author(s):  
G. A. Kardomateas
Keyword(s):  
Axial Compression ◽  
Critical Loads ◽  
Shell Theory ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Stability Loss ◽  
Theory Of Elasticity ◽  
Dimensional Theory ◽  
Perfect Agreement ◽  
The Stability

The stability of equilibrium of a transversely isotropic thick cylindrical shell under axial compression is investigated. The problem is treated by making appropriate use of the three-dimensional theory of elasticity. The results are compared with the critical loads furnished by classical shell theories. For the isotropic material cases considered, the elasticity approach predicts a lower critical load than the shell theories, the percentage reduction being larger with increasing thickness. However, both the Flu¨gge and Danielson and Simmonds theories predict critical loads much closer to the elasticity value than the Donnell theory. Moreover, the values of n, m (number of circumferential waves and number of axial half-waves, respectively, at the critical point) for both the elasticity, and the Flu¨gge and the Danielson and Simmonds theories, show perfect agreement, unlike the Donnell shell theory.

Bifurcation of Equilibrium in Thick Orthotropic Cylindrical Shells Under Axial Compression

1995 ◽  
Vol 62 (1) ◽  
pp. 43-52 ◽  
Author(s):  
G. A. Kardomateas
Keyword(s):  
Axial Compression ◽  
Orthotropic Material ◽  
Shell Theory ◽  
Three Dimensional ◽  
Theory Of Elasticity ◽  
Elasticity Solution ◽  
Timoshenko Theory ◽  
Cylinder Axis ◽  
Dimensional Theory ◽  
Nonshallow Shell

The bifurcation of equilibrium of an orthotropic thick cylindrical shell under axial compression is studied by an appropriate formulation based on the three-dimensional theory of elasticity. The results from this elasticity solution are compared with the critical loads predicted by the orthotropic Donnell and Timoshenko nonshallow shell formulations. As an example, the cases of an orthotropic material with stiffness constants typical of glass/epoxy and the reinforcing direction along the periphery or along the cylinder axis are considered. The bifurcation points from the Timoshenko formulation are always found to be closer to the elasticity predictions than the ones from the Donnell formulation. For both the orthotropic material cases and the isotropic one, the Timoshenko bifurcation point is lower than the elasticity one, which means that the Timoshenko formulation is conservative. The opposite is true for the Donnell shell theory, i.e., it predicts a critical load higher than the elasticity solution and therefore it is nonconservative. The degree of conservatism of the Timoshenko theory generally increases for thicker shells. Likewise, the Donnell theory becomes in general more nonconservative with thicker construction.

Buckling of Thick Orthotropic Cylindrical Shells Under External Pressure

1993 ◽  
Vol 60 (1) ◽  
pp. 195-202 ◽  
Author(s):  
G. A. Kardomateas
Keyword(s):  
Critical Load ◽  
Shell Theory ◽  
Cylindrical Shells ◽  
Three Dimensional ◽  
Stability Loss ◽  
External Pressure ◽  
Elasticity Solution ◽  
Composite Shells

An elasticity solution to the problem of buckling of orthotropic cylindrical shells subjected to external pressure is presented. In this context, the structure is considered a three-dimensional body. The results show that the shell theory predictions can produce nonconservative results on the critical load of composite shells with moderately thick construction. The solution provides a means of accurately assessing the limitations of shell theories in predicting stability loss.

Three-dimensional stress distributions in hexagonal aeolotropic crystals

1948 ◽  
Vol 44 (4) ◽  
pp. 522-533 ◽  
Author(s):  
H. A. Elliott ◽  
N. F. Mott
Keyword(s):  
Harmonic Functions ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Quadratic Equation ◽  
Stress Distributions ◽  
Third Order ◽  
Isotropic Solid ◽  
Single Stress ◽  
Image Position ◽  
Special Case

The conditions for equilibrium in an elastically stressed hexagonal aeolotropic medium (transversely isotropic) are formulated, and solutions are found in terms of two ‘harmonic’ functions ø1, ø2, which are solutions ofν1, ν2 being the roots of a certain quadratic equation.It is also shown that in the case of axially symmetrical stress systems the solution may be expressed in terms of the third-order differential coefficients of a single stress function Φ.The solutions for an isotropic medium may be deduced as a special case.The problems of nuclei of strain in such a hexagonal solid are solved, and the results for zinc and magnesium contrasted with those for an isotropic solid.

scholarly journals ГРАНИЧНЫЕ УСЛОВИЯ КОНТАКТНОГО ТИПА В ЗАДАЧЕ О КРУГОВОЙ ЦИЛИНДРИЧЕСКОЙ ПОЛОСТИ В УПРУГОМ ПРОСТРАНСТВЕ

2018 ◽  
pp. 121-128
Author(s):  
В. Ю. Мирошников ◽  
Т. В. Денисова ◽  
В. С. Проценко
Keyword(s):  
Boundary Conditions ◽  
Three Dimensional ◽  
Fourier Method ◽  
Theory Of Elasticity ◽  
Numerical Comparison ◽  
Elastic Space ◽  
Dimensional Problem ◽  
Cylindrical Coordinates ◽  
Contact Type ◽  
Tangential Stresses

A three-dimensional problem of the theory of elasticity is considered, when contact-type conditions (normal displacements and tangential stresses) are given on a cylindrical cavity in elastic space. The solution is obtained on the basis of the Fourier method with respect to the Lame equations in cylindrical coordinates. The solvability and uniqueness of the problem for these boundary conditions is proved. Normal and tangential stresses are found in the elastic body. A numerical comparison is made of the influence of the boundary conditions in the form of displacements and boundary conditions of the contact type on the stressed state of the elastic space.

Elastic Waves in Generalized Thermo-Piezoelectric Transversely Isotropic Circular Bar Immersed in Fluid

2015 ◽  
Vol 8 (1) ◽  
pp. 82-103
Author(s):  
Palaniyandi Ponnusamy
Keyword(s):  
Classical Theory ◽  
Cross Sections ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Coupled System ◽  
Theory Of Elasticity ◽  
Inviscid Fluid ◽  
Displacement Potential ◽  
Modes Of Vibration

AbstractIn this paper, a mathematical model is developed to study the wave propagation in an infinite, homogeneous, transversely isotropic thermo-piezoelectric solid bar of circular cross-sections immersed in inviscid fluid. The present study is based on the use of the three-dimensional theory of elasticity. Three displacement potential functions are introduced to uncouple the equations of motion and the heat and electric conductions. The frequency equations are obtained for longitudinal and flexural modes of vibration and are studied based on Lord-Shulman, Green-Lindsay and Classical theory theories of thermo elasticity. The frequency equations of the coupled system consisting of cylinder and fluid are developed under the assumption of perfect-slip boundary conditions at the fluid-solid interfaces, which are obtained for longitudinal and flexural modes of vibration and are studied numerically for PZT-4 material bar immersed in fluid. The computed non-dimensional frequencies are compared with Lord-Shulman, Green-Lindsay and Classical theory theories of thermo elasticity for longitudinal and flexural modes of vibrations. The dispersion curves are drawn for longitudinal and flexural modes of vibrations. Moreover, the dispersion of specific loss and damping factors are also analyzed for longitudinal and flexural modes of vibrations.

scholarly journals Three-dimensional Green's functions for two-dimensional quasi-crystal bimaterials

2011 ◽  
Vol 467 (2133) ◽  
pp. 2622-2642 ◽  
Author(s):  
Yang Gao ◽  
Andreas Ricoeur
Keyword(s):  
Amorphous Materials ◽  
Green's Functions ◽  
Green’S Functions ◽  
Solid Phase ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Theory Of Elasticity ◽  
Two Dimensional ◽  
Special Cases ◽  
Quasi Crystals

Owing to their specific structure, which can neither be classified as crystalline nor amorphous, quasi-crystals (QCs) exhibit properties that are interesting to both material science and mathematical physics or continuum mechanics. Within the framework of a mathematical theory of elasticity, one major focus is on features evolving from the coupling of phonon and phason fields, which is not observed in classical crystalline or amorphous materials. This paper deals with the problems of combinations of point phonon forces and point phason forces, which are applied to the interior of infinite solids and bimaterial solids of two-dimensional hexagonal QCs. By using the general solution of QCs, a series of displacement functions is adopted to obtain the analytical results when the two half-spaces are supposed to be ideally bonded or to be in smooth contact. In the final expressions, we provide three-dimensional Green’s functions for infinite bimaterial QC solids in the closed form, which are very convenient to be used in the study of dislocations, cracks and inhomogeneities of the new solid phase. Furthermore, the paper is concluded by a discussion of some special cases, in which Green’s functions for infinite transversely isotropic solids and Green’s functions for a half-space with free or fixed boundary are given.

scholarly journals Extraction of consistent shell theory equations from 3D theory of elasticity

2019 ◽  
Vol 15 (2) ◽  
pp. 135-148
Author(s):  
Evgeny M Zveryaev
Keyword(s):  
Thin Shell ◽  
Shell Thickness ◽  
Shell Theory ◽  
Three Dimensional ◽  
Initial Approximation ◽  
Theory Of Elasticity ◽  
Curvilinear Coordinates ◽  
System Of Equations ◽  
Dimensional Problem ◽  
Output Data

Aims of research. Derivation of consistent equations of the theory of thin elastic shells without hypotheses and stress averaging over the shell thickness. Methods. Using the iterative method of Saint-Venant - Picard - Banach, the three-dimensional problem of the theory of elasticity is solved without any hypotheses. By the principle of compressed mappings, the solution converges asymptotically, regardless of the choice of the values of the initial approximation. Results. A method has been developed for integrating the spatial equations of the theory of elasticity in curvilinear coordinates for a thin shell. The presence of a small parameter allows the integration of the system of equations in such a way that the output data of the first operator is input to the next operator, etc., dividing the original complex operator into a sequence of simple integrable Picard type operators. Each equation contains terms of only one asymptotic order.

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