scholarly journals Dynamic Stress and Displacement in an Elastic Half-Space with a Cylindrical Cavity

2011 ◽  
Vol 18 (6) ◽  
pp. 827-838 ◽  
Author(s):  
İ. Coşkun ◽  
H. Engin ◽  
A. Özmutlu
Keyword(s):  
Free Surface ◽  
Boundary Conditions ◽  
Half Space ◽  
Cylindrical Cavity ◽  
Circular Cross Section ◽  
Elastic Half Space ◽  
Least Square ◽  
Polar Coordinates ◽  
Wave Number ◽  
Forcing Function

The dynamic response of an elastic half-space with a cylindrical cavity in a circular cross-section is analyzed. The cavity is assumed to be infinitely long, lying parallel to the plane-free surface of the medium at a finite depth and subjected to a uniformly distributed harmonic pressure at the inner surface. The problem considered is one of plain strain, in which it is assumed that the geometry and material properties of the medium and the forcing function are constant along the axis of the cavity. The equations of motion are reduced to two wave equations in polar coordinates with the use of Helmholtz potentials. The method of wave function expansion is used to construct the displacement fields in terms of the potentials. The boundary conditions at the surface of the cavity are satisfied exactly, and they are satisfied approximately at the free surface of the half-space. Thus, the unknown coefficients in the expansions are obtained from the treatment of boundary conditions using a collocation least-square scheme. Numerical results, which are presented in the figures, show that the wave number (i.e., the frequency) and depth of the cavity significantly affect the displacement and stress.

scholarly journals Dynamic Response of a Circular Tunnel in an Elastic Half Space

2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
İrfan Coşkun ◽  
Demirhan Dolmaseven
Keyword(s):  
Boundary Conditions ◽  
Half Space ◽  
Dynamic Pressure ◽  
Equations Of Motion ◽  
Surrounding Medium ◽  
Elastic Half Space ◽  
Polar Coordinates ◽  
Wave Number ◽  
Circular Tunnel ◽  
Tunnel Wall

The vibration of a circular tunnel in an elastic half space subjected to uniformly distributed dynamic pressure at the inner boundary is studied in this paper. For comparison purposes, two different ground materials (soft and hard soil) are considered for the half space. Under the assumption of plane strain, the equations of motion for the tunnel and the surrounding medium are reduced to two wave equations in polar coordinates using Helmholtz potentials. The method of wave expansion is used to construct the displacement fields in terms of displacement potentials. The boundary conditions associated with the problem are satisfied exactly at the inner surface of the tunnel and at the interface between the tunnel and surrounding medium, and they are satisfied approximately at the free surface of the half space. A least-squares technique is used for satisfying the stress-free boundary conditions at the half space. It is shown by comparison that the stresses and displacements are significantly influenced by the properties of the surrounding soil, wave number (i.e., the frequency), depth of embedment, and thickness of the tunnel wall.

scholarly journals Refined boundary conditions on the free surface of an elastic half-space taking into account non-local effects

2016 ◽  
Vol 472 (2186) ◽  
pp. 20150800 ◽  
Author(s):  
R. Chebakov ◽  
J. Kaplunov ◽  
G. A. Rogerson
Keyword(s):  
Boundary Layer ◽  
Free Surface ◽  
Boundary Conditions ◽  
Half Space ◽  
Elastic Half Space ◽  
Interior Solution ◽  
Homogeneous Half Space ◽  
Classical Boundary ◽  
Non Local ◽  
Rayleigh Wave Speed

The dynamic response of a homogeneous half-space, with a traction-free surface, is considered within the framework of non-local elasticity. The focus is on the dominant effect of the boundary layer on overall behaviour. A typical wavelength is assumed to considerably exceed the associated internal lengthscale. The leading-order long-wave approximation is shown to coincide formally with the ‘local’ problem for a half-space with a vertical inhomogeneity localized near the surface. Subsequent asymptotic analysis of the inhomogeneity results in an explicit correction to the classical boundary conditions on the surface. The order of the correction is greater than the order of the better-known correction to the governing differential equations. The refined boundary conditions enable us to evaluate the interior solution outside a narrow boundary layer localized near the surface. As an illustration, the effect of non-local elastic phenomena on the Rayleigh wave speed is investigated.

Scattering of Subsurface Cylindrical Cavity near Multiple Semi-Cylindrical Alluvial Valleys under Incident SH-Waves

2011 ◽  
Vol 374-377 ◽  
pp. 1285-1290
Author(s):  
Hui Wen Wang ◽  
Xiao Juan Sun ◽  
Zai Lin Yang
Keyword(s):  
Cylindrical Cavity ◽  
Complex Function ◽  
Auxiliary Function ◽  
Elastic Half Space ◽  
Continuity Condition ◽  
Polar Coordinates ◽  
Wave Number ◽  
Algebraic Equations ◽  
Sh Waves ◽  
Incident Waves

The scattering of subsurface cylindrical cavity near multiple semi-cylindrical alluvial valleys under incident SH waves is studied in this paper by using methods of auxiliary function, complex function multi-polar coordinates. The model is divided into two parts, Domain I is multiple semi-cylindrical alluvial valleys, and Domain Ⅱ is an elastic half space with several subsurface circular cavities near multiple semi-cylindrical alluvial valleys. A series of infinite algebraic equations is then obtained based on the displacement and stress continuity condition on “common boundary” of two parts after constructing the associated displacement and stresses expressions of each part. Numerical examples illustrate that material parameters of semi-cylindrical alluvial valleys have great impact on DSCF around subsurface cavity and DSCF dose not always decrease as wave number increases especially under incident waves with high frequency when the alluvial valleys are “softer”.

Shear Horizontal Surface Waves in a Functionally Graded Magneto-Electro-Elastic Half-Space

2011 ◽  
Vol 304 ◽  
pp. 204-207
Author(s):  
Qian Yang ◽  
Yan Ping Kong ◽  
Jin Xi Liu
Keyword(s):  
Free Surface ◽  
Boundary Conditions ◽  
Surface Waves ◽  
Half Space ◽  
Material Properties ◽  
Elastic Half Space ◽  
Functionally Graded ◽  
Horizontal Surface ◽  
Propagation Characteristics ◽  
Shear Horizontal

We investigate the propagation of shear horizontal (SH) surface waves in a functionally graded magneto-electro-elastic half-space with hexagonal (6mm) symmetry. The material properties vary in the direction perpendicular to the free surface. The surface of the half-space is mechanically free, but subjected to four types of electromagnetic boundary conditions. These boundary conditions are electrically open/magnetically closed, electrically open/magnetically open, electrically closed/magnetically open and electrically closed/magnetically closed. The effects of the electromagnetic boundary conditions and the variation of material properties on the propagation characteristics of SH surface waves are analyzed. The results obtained show that except for the electrically open/magnetically closed condition, three sets of other electromagnetic boundary conditions sustain the propagation of SH surface waves. Different from the case in homogeneous materials, the existing SH surface waves in functionally graded half-spaces are dispersive.

Shrink fit of an elastic half-space with a cylindrical cavity

1995 ◽  
Vol 448 (1932) ◽  
pp. 81-88 ◽  
Keyword(s):  
Boundary Conditions ◽  
Half Space ◽  
Cylindrical Cavity ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Elastic Half Space ◽  
Series Method ◽  
Dual Integral Equations ◽  
Axisymmetric Contact Problem ◽  
Shrink Fit

This paper deals with the axisymmetric contact problem for an elastic half-space with a cylindrical cavity when mixed boundary conditions are prescribed on the surface of the cavity. The problem is simplified to that of finding the solution of dual integral equations arising from the mixed boundary conditions. The solution is obtained by the series method, and quantities of physical interest are calculated.

scholarly journals Transient Excitation of an Elastic Half Space by a Point Load Traveling on the Surface

1969 ◽  
Vol 36 (3) ◽  
pp. 505-515 ◽  
Author(s):  
D. C. Gakenheimer ◽  
J. Miklowitz
Keyword(s):  
Exact Solution ◽  
Free Surface ◽  
Steady State ◽  
Wave Front ◽  
Half Space ◽  
Displacement Field ◽  
Elastic Half Space ◽  
Point Load ◽  
Limit Case ◽  
Transient Waves

The propagation of transient waves in a homogeneous, isotropic, linearly elastic half space excited by a traveling normal point load is investigated. The load is suddenly applied and then it moves rectilinearly at a constant speed along the free surface. The displacements are derived for the interior of the half space and for all load speeds. Wave-front expansions are obtained from the exact solution, in addition to results pertaining to the steady-state displacement field. The limit case of zero load speed is considered, yielding new results for Lamb’s point load problem.

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.

Rayleigh-type surface wave on a rotating orthotropic elastic half-space with impedance boundary conditions

2020 ◽  
Vol 26 (21-22) ◽  
pp. 1980-1987
Author(s):  
Baljeet Singh ◽  
Baljinder Kaur
Keyword(s):  
Surface Wave ◽  
Boundary Conditions ◽  
Rayleigh Wave ◽  
Surface Waves ◽  
Half Space ◽  
Rotation Rate ◽  
Secular Equation ◽  
Elastic Half Space ◽  
Impedance Boundary ◽  
Impedance Boundary Conditions

The propagation of Rayleigh type surface waves in a rotating elastic half-space of orthotropic type is studied under impedance boundary conditions. The secular equation is obtained explicitly using traditional methodology. A program in MATLAB software is developed to obtain the numerical values of the nondimensional speed of Rayleigh wave. The speed of Rayleigh wave is illustrated graphically against rotation rate, nondimensional material constants, and impedance boundary parameters.

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