Reflection of Longitudinal Micro-Rotational Wave at Viscoelastically Supported Boundary of Micropolar Half-Space

2016 ◽  
Vol 34 (3) ◽  
pp. 243-255 ◽  
Author(s):  
P. Zhang ◽  
P.-J. Wei ◽  
Y.-Q. Li
Keyword(s):  
Boundary Conditions ◽  
Half Space ◽  
Energy Flux ◽  
Incident Wave ◽  
Amplitude Ratio ◽  
Numerical Results ◽  
Elastic Parameters ◽  
Rotational Wave ◽  
Serial Connection ◽  
Delay Effects

AbstractThe reflection of longitudinal micro-rotational wave at the viscoelastically supported boundary of micropolar half-space is studied in this paper. The viscoelastic boundary is described by spring-dashpot model with parallel or serial connection. Both the spring and the dashpot contribute to the displacements and micro-rotation and the boundary conditions include the force stress and couple stress components. From the boundary conditions, the amplitude ratios and phase shifts of reflection waves with respect to the incident wave are obtained. Further, the energy flux ratios of the reflection waves to the incident wave are estimated. In order to validate the numerical results, the energy flux conservation with consideration of the energy dissipation of dashpot is used. Based on the numerical results, the influences of elastic parameters and viscous parameters are studied, respectively. It is found that the elastic parameters and the viscous parameters have evident influences on the amplitude ratio, the phase shift and the energy partition. The causes resulting in these deviations are related with the instantaneous elasticity of spring and the time-delay effects of dashpot.

Solution of Navier’s Equation in Cylindrical Curvilinear Coordinates

1978 ◽  
Vol 45 (4) ◽  
pp. 812-816 ◽  
Author(s):  
B. S. Berger ◽  
B. Alabi
Keyword(s):  
Fourier Transform ◽  
Finite Difference ◽  
Boundary Conditions ◽  
Half Space ◽  
Coordinate Transformation ◽  
Numerical Results ◽  
Three Dimensions ◽  
Curvilinear Coordinates ◽  
Rectangular Region

A solution has been derived for the Navier equations in orthogonal cylindrical curvilinear coordinates in which the axial variable, X3, is suppressed through a Fourier transform. The necessary coordinate transformation may be found either analytically or numerically for given geometries. The finite-difference forms of the mapped Navier equations and boundary conditions are solved in a rectangular region in the curvilinear coordinaties. Numerical results are given for the half space with various surface shapes and boundary conditions in two and three dimensions.

An Efficient Semi-Analytical Method to Compute Displacements and Stresses in an Elastic Half-Space with a Hemispherical Pit

2015 ◽  
Vol 7 (3) ◽  
pp. 295-322 ◽  
Author(s):  
Valeria Boccardo ◽  
Eduardo Godoy ◽  
Mario Durán
Keyword(s):  
Analytical Solution ◽  
Boundary Conditions ◽  
Half Space ◽  
Plane Surface ◽  
Elastic Half Space ◽  
Numerical Results ◽  
Quadratic Functional ◽  
Infinite Plane ◽  
Equilibrium Equations ◽  
Finite Element Software

AbstractThis paper presents an efficient method to calculate the displacement and stress fields in an isotropic elastic half-space having a hemispherical pit and being subject to gravity. The method is semi-analytical and takes advantage of the axisymmetry of the problem. The Boussinesq potentials are used to obtain an analytical solution in series form, which satisfies the equilibrium equations of elastostatics, traction-free boundary conditions on the infinite plane surface and decaying conditions at infinity. The boundary conditions on the free surface of the pit are then imposed numerically, by minimising a quadratic functional of surface elastic energy. The minimisation yields a symmetric and positive definite linear system of equations for the coefficients of the series, whose particular block structure allows its solution in an efficient and robust way. The convergence of the series is verified and the obtained semi-analytical solution is then evaluated, providing numerical results. The method is validated by comparing the semi-analytical solution with the numerical results obtained using a commercial finite element software.

Thermoelasticity of an Anisotropic Half Space

1974 ◽  
Vol 41 (4) ◽  
pp. 935-940 ◽  
Author(s):  
J. Padovan
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Half Space ◽  
Boundary Value ◽  
Integral Transforms ◽  
Numerical Results ◽  
Heat Sinks ◽  
Material Anisotropy ◽  
Body Forces

The thermoelasticity of an anisotropic Hookean half space is studied herein. A solution based on successive integral transforms is developed. The solution can handle arbitrary thermal and mechanical boundary conditions together with distributed body forces and heat sinks or sources. To illustrate the substantial effects of material anisotropy, a specific boundary-value problem is solved. Numerical results based on the solution are presented. These illustrate the significant effects of both thermal and mechanical material anisotropy.

The Bending, Buckling, and Flexural Vibration of Simply Supported Polygonal Plates by Point-Matching

1961 ◽  
Vol 28 (2) ◽  
pp. 288-291 ◽  
Author(s):  
H. D. Conway
Keyword(s):  
Hydrostatic Pressure ◽  
Boundary Conditions ◽  
Flexural Vibration ◽  
Frequency Equation ◽  
Numerical Results ◽  
Special Technique ◽  
Point Matching ◽  
Buckling Loads ◽  
Simply Supported ◽  
Flexural Vibrations

The bending by uniform lateral loading, buckling by two-dimensional hydrostatic pressure, and the flexural vibrations of simply supported polygonal plates are investigated. The method of meeting the boundary conditions at discrete points, together with the Marcus membrane analog [1], is found to be very advantageous. Numerical examples include the calculation of the deflections and moments, and buckling loads of triangular square, and hexagonal plates. A special technique is then given, whereby the boundary conditions are exactly satisfied along one edge, and an example of the buckling of an isosceles, right-angled triangle plate is analyzed. Finally, the frequency equation for the flexural vibrations of simply supported polygonal plates is shown to be the same as that for buckling under hydrostatic pressure, and numerical results can be written by analogy. All numerical results agree well with the exact solutions, where the latter are known.

scholarly journals Waves in Microstructured Conducting Sheath Helix Embedded Optical Guides with Chiral Nihility and Chiral Materials

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
N. Iqbal ◽  
M. A. Baqir ◽  
P. K. Choudhury
Keyword(s):  
Boundary Conditions ◽  
Flux Density ◽  
Energy Flux ◽  
Pitch Angle ◽  
Electromagnetic Waves ◽  
The Core ◽  
Dispersion Behavior ◽  
Chiral Materials ◽  
Energy Flux Density

The paper deals with the sustainment of electromagnetic waves in circularly cylindrical optical guide with chiral nihility and chiral materials in the core and the clad sections, respectively. A perfectly conducting tightly wound helix is introduced at the core-clad interface. The eigenvalue relation for such a complex optical microstructured guide is deduced by applying suitable boundary conditions at the core-clad interface, and the dispersion behavior is analyzed by varying the pitch angle of helix. The sustainment of energy flux density in such optical guides is estimated under various structural conditions, and the density patterns in core-clad sections are anatomized analytically.

A Variational Approach to Loaded Ply Structures

2003 ◽  
Vol 9 (1-2) ◽  
pp. 175-185 ◽  
Author(s):  
G. H.M. Van Der Heijden ◽  
J. M.T. Thompson ◽  
S. Neukirch
Keyword(s):  
Boundary Conditions ◽  
Energy Method ◽  
Variational Approach ◽  
Energy Analysis ◽  
Numerical Results ◽  
Equilibrium Equations

We show how an energy analysis can be used to derive the equilibrium equations and boundary conditions for an end-loaded variable ply much more efficiently than in previous works. Numerical results are then presented for a clamped balanced ply approaching lock-up. We also use the energy method to derive the equations for a more general ply made of imperfect anisotropic rods and we briefly consider their helical solutions.

Exact Elasticity Solutions for Stresses and Deflections in Curved Beams and Rings of Exponential and T-Sections

1991 ◽  
Author(s):  
Cemil Bagci
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Plane Stress ◽  
Numerical Results ◽  
Neutral Axis ◽  
Curved Beams ◽  
Equilibrium Stress ◽  
Different Boundary Conditions ◽  
Stress Functions ◽  
Elasticity Solutions

Abstract Exact elasticity solutions for stresses and deflections (displacements) in curved beams and rings of varying thicknesses are developed using polar elasticity and state of plane stress. Basic forms of differential equations of equilibrium, stress functions, and differential equations of compatibility are given. They are solved to develop expressions for radial, tangential, and shearing stresses for moment, force, and combined loadings. Neutral axis location for each type of loading is determined. Expressions for displacements are developed utilizing strain-displacement relationships of polar elasticity satisfying boundary conditions on displacements. In case of full rings stresses are as in curved beams with properly defined moment loading, but displacements differ satisfying different boundary conditions. The developments for constant thicknesses are used to develop solutions for curved beams and rings with T-sections. Comparative numerical results are given.

BENDING OF ORTHOTROPIC RECTANGULAR CANTILEVER PLATE

2021 ◽  
pp. 2-9
Author(s):  
M.V. Sukhoterin ◽  
◽  
A.M. Maslennikov ◽  
T.P. Knysh ◽  
I.V. Voytko ◽  
...  
Keyword(s):  
Iterative Method ◽  
Boundary Conditions ◽  
Trigonometric Series ◽  
Bending Moment ◽  
Partial Solution ◽  
Numerical Results ◽  
Cantilever Plate ◽  
Degree Polynomial ◽  
Fourth Degree ◽  
Beam Function

Abstract. An iterative method of superposition of correcting functions is proposed. The partial solution of the main differential bending equation is represented by a fourth-degree polynomial (the beam function), which gives a residual only with respect to the bending moment on parallel free faces. This discrepancy and the subsequent ones are mutually compensated by two types of correcting functions-hyperbolic-trigonometric series with indeterminate coefficients. Each function satisfies only a part of the boundary conditions. The solution of the problem is achieved by an infinite superposition of correcting functions. For the process to converge, all residuals must tend to zero. When the specified accuracy is reached, the process stops. Numerical results of the calculation of a square ribbed plate are presented.

Period equation for Rayleigh waves in a layer overlying a half space with a sinusoidal interface

1962 ◽  
Vol 52 (4) ◽  
pp. 807-822 ◽  
Author(s):  
John T. Kuo ◽  
John E. Nafe
Keyword(s):  
Wave Propagation ◽  
Boundary Conditions ◽  
Half Space ◽  
Rayleigh Waves ◽  
Wave Length ◽  
Linear Equations ◽  
Wave Number ◽  
Interface Plane ◽  
Period Equation ◽  
Phase And Group Velocities

abstract The problem of the Rayleigh wave propagation in a solid layer overlying a solid half space separated by a sinusoidal interface is investigated. The amplitude of the interface is assumed to be small in comparison to the average thickness of the layer or the wave length of the interface. Either by applying Rayleigh's approximate method or by perturbating the boundary conditions at the sinusoidal interface, plane wave solutions for the equations which satisfy the given boundary conditions are found to form a system of linear equations. These equations may be expressed in a determinant form. The period (or characteristic) equations for the first and second approximation of the wave number k are obtained. The phase and group velocities of Rayleigh waves in the present case depend upon both frequency and distance. At a given point on the surface, there is a local phase and local group velocity of Rayleigh waves that is independent of the direction of wave propagation.

Reflection of plane elastic waves from transition layers with arbitrary varration of velocity and density

1966 ◽  
Vol 56 (3) ◽  
pp. 633-642 ◽  
Author(s):  
Ravindra N. Gupta
Keyword(s):  
Transition Layer ◽  
Elastic Waves ◽  
Numerical Results ◽  
Elastic Parameters ◽  
Transition Layers

abstract The problem of reflection of plane elastic waves is generalized numerically to an arbitrary variation, with depth, of the elastic parameters inside a transition layer between two homogeneous half-spaces. Numerical results are given for some cases of interest.

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