The Refined Theory of Axisymmetric Circular Cylinder in One-Dimensional Hexagonal Quasicrystals

2012 ◽  
Vol 217-219 ◽  
pp. 1421-1424 ◽  
Author(s):  
Bao Sheng Zhao ◽  
Di Wu
Keyword(s):  
General Solution ◽  
Circular Cylinder ◽  
Ad Hoc ◽  
Approximate Equation ◽  
Refined Theory ◽  
One Dimensional ◽  
Stress Components ◽  
Refined Equation ◽  
Independent Variable ◽  
1D Hexagonal Quasicrystals

A refined theory of axisymmetric cylinder in one-dimensional (1D) hexagonal quasicrystals (QCs) is analyzed. Based on elastic theory with 1D hexagonal QCs, the refined theory of axisymmetric cylinder is derived by using general solution of 1D hexagonal QCs and Lur’e method without ad hoc assumptions. At first, expressions were obtained for all the phonon and phason displacements and stress components in term of the three functions with single independent variable. Based on the boundary conditions, the refined equation for the cylinder is derived directly. And the approximate equation is accurate up to the second-order terms with respect to radius of circular cylinder.

A Refined Theory of Axisymmetric Thermoelastic Circular Cylinder with Transversely Isotropic

2012 ◽  
Vol 622-623 ◽  
pp. 1611-1615
Author(s):  
Di Wu ◽  
Bao Sheng Zhao
Keyword(s):  
Circular Cylinder ◽  
Radial Direction ◽  
Ad Hoc ◽  
Transversely Isotropic ◽  
Approximate Solutions ◽  
Refined Theory ◽  
Surface Loading ◽  
Refined Equation ◽  
Exact Temperature ◽  
Approximate Equations

For analyzing the exact stress field, the exact displacement field and the exact temperature field in axisymmetric thermoelastic circular cylinder with transversely isotropic, the refined theory of an axisymmetric circular cylinder was researched. Without ad hoc assumptions, the refined equation of an axisymmetric thermoelastic circular cylinder with transversely isotropic was obtained, which yields Bessel's function and the solution of the cylinder directly from the general solution. By dropping terms of high order, the approximate solutions are derived for a circular cylinder under radial direction surface loading. At last, we study the approximate equations with the temperature effect.

The General Solution for Torsional Circular Shaft of Cubic Quasicrystal

2011 ◽  
Vol 213 ◽  
pp. 206-210
Author(s):  
Bao Sheng Zhao ◽  
Jia Lian Shi ◽  
Ying Tao Zhao ◽  
Yang Gao
Keyword(s):  
General Solution ◽  
Boundary Conditions ◽  
Ad Hoc ◽  
Isotropic Plate ◽  
Refined Theory ◽  
Classical Elasticity ◽  
Circular Shaft ◽  
Cylindrical Coordinate ◽  
Stress Components ◽  
Classical Elasticity Theory

Gregory’s decomposed theorem of isotropic plate is extended to investigate torsional circular shaft of cubic quasicrystal with homogeneous boundary conditions, and the theory of equivalence that Cheng’s refined theory and Gregory’s decomposed theorem is extended to the cylindrical coordinate. The general solution of torsional circular shaft on cubic quasicrystal with homogeneous boundary conditions is proposed on the basis of the classical elasticity theory and stress-displacement relations of cubic quasicrystal without ad hoc assumptions. At first expressions are obtained for all the displacements and stress components in term of some 1D functions. Using Lur’e method, the exact equations were given. And the exact equations for the torsional circular shaft on cubic quasicrystal without surface loadings consist of four governing differential equations: two harmonic equations and two transcendental equations. Using basic mathematic method and the general solutions, an example is examined.

The Refined Theory of One-Dimensional Quasi-Crystals in Thick Plate Structures

2011 ◽  
Vol 78 (3) ◽  
Author(s):  
Yang Gao ◽  
Andreas Ricoeur
Keyword(s):  
Thick Plate ◽  
Ad Hoc ◽  
Plate Theory ◽  
Pure Shear ◽  
Refined Theory ◽  
Plate Structures ◽  
Thick Plates ◽  
Refined Plate Theory ◽  
One Dimensional ◽  
Quasi Crystals

For one-dimensional quasi-crystals, the refined theory of thick plates is explicitly established from the general solution of quasi-crystals and the Luré method without employing ad hoc stress or deformation assumptions. For a homogeneous plate, the exact equations and solutions are derived, which consist of three parts: the biharmonic part, the shear part, and the transcendental part. For a nonhomogeneous plate, the exact governing differential equations and solutions under pure normal loadings and pure shear loadings, respectively, are obtained directly from the refined plate theory. In an illustrative example, explicit expressions of analytical solutions are obtained for torsion of a rectangular quasi-crystal plate.

The Decomposed Theorem of Torsional Circular Shaft with Two-Dimensional Dodecagonal Quasicrystal

2011 ◽  
Vol 341-342 ◽  
pp. 1-5 ◽  
Author(s):  
Bao Sheng Zhao ◽  
Ying Tao Zhao ◽  
Yang Gao
Keyword(s):  
Boundary Conditions ◽  
Ad Hoc ◽  
Isotropic Plate ◽  
Refined Theory ◽  
Two Dimensional ◽  
Classical Elasticity ◽  
Circular Shaft ◽  
Cylindrical Coordinate ◽  
Stress Components ◽  
Classical Elasticity Theory

Gregory’s decomposed theorem of isotropic plate is extended to investigate torsional circular shaft for two-dimensional dodecagonal quasicrystal (2D dodecagonal QCs)with homogeneous boundary conditions, and the theory of equivalence that Cheng’s refined theory and Gregory’s decomposed theorem is extended to the cylindrical coordinate. The decomposed theorem of torsional circular shaft of 2D dodecagonal QCs with homogeneous boundary conditions is proposed on the basis of the classical elasticity theory and stress-displacement relations of 2D dodecagonal QCs without ad hoc assumptions. At first expressions are obtained for all the displacements and stress components in term of some 1D functions. Using Lur’e method, the exact equations were given. And the exact equations for the torsional circular shaft on 2D dodecagonal QCs without surface loadings consist of four governing differential equations: two harmonic equations and two transcendental equations.

The Refined Equations of Special Orthotropic Piezoelectric Plate-I: Anti-Symmetrical Transverse Surface Loadings

2012 ◽  
Vol 249-250 ◽  
pp. 348-351
Author(s):  
Bao Sheng Zhao ◽  
Di Wu
Keyword(s):  
General Solution ◽  
Boundary Conditions ◽  
Thick Plate ◽  
Ad Hoc ◽  
Piezoelectric Plate ◽  
Deformation Field ◽  
Elastic Theory ◽  
Surface Loading ◽  
Stress Components ◽  
Stress States

The deformation field and stress states of special orthotropic piezoelectric plate are analyzed. Based on elastic theory, the refined equations of bending thick plate are derived by using Elliott-Lodge’s general solution and Lur’e method without ad hoc assumptions. At first, expressions were obtained for all the displacements and stress components of a piezoelectric plate. Based on boundary conditions, the refined equations for the plate with anti-symmetrical transverse surface loading are obtained.

The Refined Theory of Plane Problems for One-Dimensional Quasicrystalline Bodies

2011 ◽  
Vol 79 (1) ◽  
Author(s):  
Yang Gao ◽  
Andreas Ricoeur
Keyword(s):  
Ad Hoc ◽  
Three Dimensional ◽  
Refined Theory ◽  
Shear Surface ◽  
Surface Loading ◽  
Thick Plates ◽  
One Dimensional ◽  
Nonhomogeneous Boundary ◽  
Deformation Regime ◽  
Nonhomogeneous Boundary Conditions

Without employing ad hoc assumptions, various equations and solutions for plane problems of one-dimensional quasicrystals are deduced systematically. A method for the exact solution of three-dimensional equations is presented under homogeneous and nonhomogeneous boundary conditions. The equations and solutions are used to construct the refined theory of thick plates for both an in-plane extensional deformation regime and a normal or shear surface loading. With this method, the refined theory can now be explicitly established from the general solution of quasicrystals and the Lur’e method. In two illustrative examples of infinite plates with a circular hole, it is shown that explicit expressions of analytical solutions can be obtained by using the refined theory.

Asymptotic derivation of a refined equation for an elastic beam resting on a Winkler foundation

2021 ◽  
pp. 108128652110238
Author(s):  
Barış Erbaş ◽  
Julius Kaplunov ◽  
Isaac Elishakoff
Keyword(s):  
Ad Hoc ◽  
Moving Load ◽  
Low Frequency ◽  
Elastic Beam ◽  
Winkler Foundation ◽  
One Dimensional ◽  
Beam Equations ◽  
Dimensional Equation ◽  
Phase And Group Velocities ◽  
Group Velocities

A two-dimensional mixed problem for a thin elastic strip resting on a Winkler foundation is considered within the framework of plane stress setup. The relative stiffness of the foundation is supposed to be small to ensure low-frequency vibrations. Asymptotic analysis at a higher order results in a one-dimensional equation of bending motion refining numerous ad hoc developments starting from Timoshenko-type beam equations. Two-term expansions through the foundation stiffness are presented for phase and group velocities, as well as for the critical velocity of a moving load. In addition, the formula for the longitudinal displacements of the beam due to its transverse compression is derived.

Solidification of an Aqueous Salt Solution in a Circular Cylinder

1992 ◽  
Vol 114 (1) ◽  
pp. 30-33 ◽  
Author(s):  
A. S. Burns ◽  
L. A. Stickler ◽  
W. E. Stewart
Keyword(s):  
Circular Cylinder ◽  
Binary Solution ◽  
Pure Water ◽  
Eutectic Point ◽  
Aqueous Salt Solution ◽  
Final Solution ◽  
One Dimensional ◽  
Boundary Temperature ◽  
Initial Melting ◽  
Diffusion And Convection

The situation of one-dimensional, transient inward solidification of a binary solution in a circular cylinder is studied numerically. The solution is assumed to be of a hypoeutectic initial concentration and to be initially at a superheated temperature above its initial melting point temperature. The boundary temperature of the cylinder is below that of its heterogeneous nucleation temperature and no supercooling occurs. The boundary temperatures and final solution concentrations are assumed to be above and below, respectively, the eutectic point of the solution. The finite difference numerical model predicts the time for the radial formation of the mush type of ice to reach the center of the cylinder and the time for the entire cylinder to reach the cylinder boundary temperature, based upon the assumptions of negligible diffusion and convection of solute during solidification. The results reveal that closure times are significantly increased for the solutions compared to pure water due to decreased conductivity of the mush compared to ice.

Modified Mixed Variational Principle and the State-Vector Equation for Elastic Bodies and Shells of Revolution

1992 ◽  
Vol 59 (3) ◽  
pp. 587-595 ◽  
Author(s):  
Charles R. Steele ◽  
Yoon Young Kim
Keyword(s):  
Variational Principle ◽  
Equations Of State ◽  
Classical Mechanics ◽  
Three Dimensional ◽  
Shells Of Revolution ◽  
Elastic Bodies ◽  
Mixed Variational Principle ◽  
Stress Components ◽  
Independent Variable ◽  
Nonsymmetric Deformation

A modified mixed variational principle is established for a class of problems with one spatial variable as the independent variable. The specific applications are on the three-dimensional deformation of elastic bodies and the nonsymmetric deformation of shells of revolution. The possibly novel feature is the elimination in the variational formulation of the stress components which cannot be prescribed on the boundaries. The result is a form exactly analogous to classical mechanics of a dynamic system, with the equations of state exactly in the form of the canonical equations of Hamilton. With the present approach, the correct scale factors of the field variables to make the system self-adjoint are readily identified, and anisotropic materials including composites can be handled effectively. The analysis for shells of revolution is given with and without the transverse shear deformation considered.

scholarly journals Revisiting Twomey's approximation for peak supersaturation

2014 ◽  
Vol 14 (19) ◽  
pp. 25901-25930
Author(s):  
B. J. Shipway
Keyword(s):  
Lookup Table ◽  
Accurate Calculation ◽  
Cloud Droplets ◽  
Closed Expression ◽  
One Dimensional ◽  
Additional Information ◽  
Droplet Nucleation ◽  
Physically Based ◽  
Thermodynamic Conditions ◽  
Independent Variable

Abstract. Twomey's seminal 1959 paper provided lower and upper bound approximations to the estimation of peak supersaturation within an updraft and thus provides the first closed expression for the number of nucleated cloud droplets. The form of this approximation is simple, but provides a surprisingly good estimate and has subsequently been employed in more sophisticated treatments of nucleation parametrization. In the current paper, we revisit the lower bound approximation of Twomey and make a small adjustment which can be used to obtain a more accurate calculation of peak supersaturation under all potential aerosol loadings and thermodynamic conditions. In order to make full use of this improved approximation, the underlying integro-differential equation for supersaturation evolution and the condition for calculating peak supersaturation are examined. A simple rearrangement of the algebra allows for an expression to be written down which can then be solved with a single lookup table with only one independent variable for an underlying lognormal aerosol population. Multimode aerosol with only N different dispersion characteristics require only N of these one-dimensional lookup tables. No additional information is required in the lookup table to deal with additional chemical, physical or thermodynamic properties. The resulting implementation provides a relatively simple, yet computationally cheap and very accurate physically-based parametrization of droplet nucleation for use in climate and NWP models.

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