Surface Of Elastic Half-space Research Articles

The gradual removal of Hertz pressure from the surface of elastic half-space

Contact stress determination in non-stationary dynamic loading of elastic bodies is crucial for modelling structures at high speeds, but it presents mathematical challenges due to the time-dependent and often unknown contact area size and shape. The study aims to obtain an energy remainder estimation that forms waves during the contact interaction of elastic bodies, based on the exact solutions of non-stationary problems for an elastic half-space. For this purpose, the problem of the instantaneous loading half-space as an additional research problem was reconstructed using the Hankel transform concerning a radial coordinate and the Laplace transform concerning a time variable. The method of derivation of the displacements at an elastic half-space loaded (unloaded) gradually by Hertz's contact pressure has been proposed. Its availability made it possible to pass to the solution of the main problem – the problem of gradual loading of the half-space surface by Hertz pressure. The possibility of changing of the order of differentiation and integration operations in the obtained representation is substantiated based on the integrand properties. The cases when the speed of the indenter was constant when its motion was uniformly accelerated and when the motion corresponded to the law of the first quarter of the cosine period in the time were considered. It was concluded that the distribution of dynamic contact stresses is similar to the Hertz distribution. An estimation of the part of the energy spent on the formation of elastic waves was made for various laws of unloading. The practical significance of this study lies in its development of an effective method for calculating normal displacements on a loading area in dynamic contact interactions of elastic bodies, which can be valuable for modelling structures at high speeds

To Solution of Contact Problem for Rectangular Plate on Elastic Half-Space

Until the present time there is no exact solution to the contact problem for a rectangular plate on an elastic base with distribution properties. Practical analogues of this design are slab foundations widely used in construction. A lot of scientists have solved this problem in various ways. The methods of finite differences, B. N. Zhemochkin and power series do not distinguish a specific feature in contact stresses at the edges of the plate. The author of the paper has obtained an expansion of the Boussinesq solution for determining displacements of the elastic half-space surface in the form of a double series according to the Chebyshev polynomials of the first kind in a rectangular region. For the first time, such a representation for the symmetric part of the Boussinesq solution was obtained by V. I. Seimov and it has been applied to study symmetric vibrations of a rectangular stamp, taking into account inertial properties of the half-space. Using this expansion, the author gives a solution to the problem for a rectangular plate lying on an elastic half-space under the action of an arbitrarily applied concentrated force. In this case, the required displacements are specified in the form of a double row in the Chebyshev polynomials of the first kind. Contact stresses are also specified in the form of a double row according to the Chebyshev polynomials of the first kind with weight. In the integral equation of the contact problem integration over a rectangular region is performed while taking into account the orthogonality of the Chebyshev polynomials. In the resulting expression the coefficients are equal for the same products of the Chebyshev polynomials. The result is an infinite system of linear algebraic equations, which is solved by the amplification method. Thus the sought coefficients are found in the expansion for contact stresses.

The efficiency of application of virtual cross-sections method and hypotheses MELS in problems of wave signal propagation in elastic waveguides with rough surfaces

The efficiency of virtual cross sections method and MELS (Magneto Elastic Layered Systems) hypotheses application is shown on model problem about distribution of wave field in thin surface layers of waveguide when plane wave signal is propagating in it. The impact of surface non-smoothness on characteristics of propagation of high-frequency horizontally polarized wave signal in isotropic elastic half-space is studied. It is shown that the non-smoothness leads to strong distortion of the wave signal over the waveguide thickness and along wave signal propagation direction as well. Â Numerical comparative analysis of change in amplitude and phase characteristics of obtained wave fields against roughness of weakly inhomogeneous surface of homogeneous elastic half-space surface is done by classical method and by proposed approach for different kind of non-smoothness.

Нестационарная нагрузка на поверхности упругого полупространства

Нестационарная нагрузка на поверхности упругого полупространства

The flow of thin layer of plastic anisotropic material on the surface of elastic half-space

The paper presents the study of problem of a thin layer flow on the surface, limiting the elastic half-space, articulated in the framework of the generalized theory of the flow in a thin layer.

Wave propagation at the boundary surface of elastic and initially stressed viscothermoelastic diffusion with voids media

The present investigation is concerned with the wave propagation at the boundary surface of elastic half-space and initially stressed viscothermoelastic diffusion with voids half-space. The longitudinal and transverse waves are incident obliquely at the plane interface between uniform elastic half-space and initially stressed viscothermoelastic diffusion with voids half-space. It is found that the amplitude ratios of various reflected and transmitted waves are functions of angle of incidence, frequency of incident wave and are influenced by the initial stress, diffusion, voids, elastic and viscoelastic properties of media. The expressions of amplitude ratios and energy ratios are obtained in closed form and computed numerically for a specific model. The variations of energy ratios with angle of incidence are shown graphically. The conservation of energy at the interface is verified.

Excitation of Rayleigh and Stoneley surface acoustic waves by distributed seismic sources

Using the integral Fourier-transform technique, we obtain a solution in integral form to the problem of excitation of elastic waves in a homogeneous isotropic solid half-space and the bordering homogeneous gas by the time-dependent forces which are arbitrarily distributed in a solid over the plane parallel to the interface of the media. Different configurations of the force sources are analyzed from the viewpoint of excitation of different types of seismoacoustic waves. Expressions for the time-averaged radiated powers of the Stoneley wave at the gas–solid interface and the Rayleigh wave at the solid–vacuum interface as well as analytical expressions for the Rayleigh wave displacements, which are valid for large distances from the source, are obtained for the harmonic dependence of forces on time. Excitation of a Rayleigh wave by the point sources oriented vertically, i.e., along the normal to the surface of elastic half-space, and horizontally, i.e., parallel to this surface, is analyzed in detail. Analytical expressions for the Rayleigh-wave radiated power are obtained. The dependences of these powers on the source orientation and depth are derived. It is shown that the Rayleigh-wave radiated power decreases with distance between the point of the force application and the boundary and turns to zero for a source depth of about 17.5% of the wavelength of the transverse wave in the case of a horizontally oriented subsurface source and a medium with identical Lame parameters λ and μ. This power increases and reaches a relative maximum when the source depth becomes equal to about 42.4% of the wavelength of the transverse wave and then exponentially falls off as the source depth increases. This maximum is about 5.5% of the surface-source radiated power.

THREE-DIMENSIONAL SURFACE CRACK ANALYSIS OF ELASTIC HALF-SPACE UNDER ROLLING CONTACT LOAD

In this work a three-dimensional planar crack on the surface of elastic half-space was analyzed under rolling contact load. The stresses interior to an elastic half-space body under rolling contact load and those produced by an infinitesimal displacement jump loop for the elastic half-space body were used to reduce the planar crack problem to the solution of a system of two-dimensional hypersingular integral equations with unknown displacement jump. The ideas of finite element discretization were employed to construct numerical solution schemes for solving the integral equations. An appropriate treatment of the associated hypersingular integral in the numerical solution to the integral equations was proposed in Hadamard's finite-part integral sense. The numerical results showed that the present procedure yields solutions with high accuracies. The stress intensity factors near the crack front edge under rolling contact load were indicated in graphical form with varying the crack shape, the radius of rolling contact zone and the friction coefficients, respectively. In addition, the influence of the lubricant infiltrating the crack surfaces on the crack propagation was also discussed in the paper.

Excitation of the elastic half-space surface by normal rebounding impact of an indenter

A dynamic contact problem of the linear theory of elasticity regarding the response of the elastic half-space surface to the normal impact of an indenter is considered. If the indenter is spherical, the Hertzian theory serves to determine the impact parameters. The displacements of points on the surface remote from the contact spot are described by solving Lamb’s problem. In the case of impacts by a flexible plate indenter that transfers uniformly distributed load to the elastic half-space surface, a first order asymptotic model is applied, which takes into account the dissipative properties of the elastic base due to elastic energy transfer by waves to infinity. The main results of the work are obtained in closed form.

Interaction of slowly moving punches with an elastic half-space

The problem of slow dynamic contact interaction of a system of punches remote from each other with an elastic half-space surface in the absence of friction is studied under the assumptions that the diameters of the contact areas are smaller than the minimum distance between the punches and the time required for the shear wave to travel the distance equal to the punch diameter is comparable to the time scale of the process. A first-order asymptotic model is constructed. As an example, the case of steady-state vibrations of a system of two punches is considered.

A hybrid method for computing forces on curved dislocations intersecting free surfaces in three-dimensional dislocation dynamics

Dislocations intersecting free surfaces present a challenge for numerical implementation of traction-free boundary conditions in dislocation dynamics simulations. The difficulty arises when singular analytic expressions of dislocation stress fields need to be used in combination with numerical methods to calculate image stress fields due to the free surfaces. A new hybrid method is developed here in which the singular and non-singular parts of the image stress are dealt with separately. The analytic solution for a semi-infinite straight dislocation intersecting the surface of elastic half-space is used to account for the singular part of the image stress, while the remaining non-singular part is treated using the standard finite element method. The numerical advantages of this decomposition are demonstrated with examples.

Surface Green function for a soft elastic half-space: Influence of surface stress

Surface Green function for incompressible, elastically isotropic half-space coupled with surface stress is derived by using double Fourier transform technique. The result indicates that the surface displacement induced by a force tangential to the surface is the same as the usual solution for elastic half-spaces where the effect of surface stress is ignored. However, the displacement caused by a force normal to the surface involves an additional parameter, i.e. the ratio of specific surface stress to shear modulus. The parameter has the dimension of length, and may provide a means to introduce an intrinsic length scale for some related problems regarding the surface of an elastic half-space. This is extremely true for soft elastic media with very low shear modulus, because in that situation the magnitude of the parameter is relatively large. As an illustrative example, the proposed Green function is adopted to analyze the interaction between two molecules with circular section adsorbed on the surface of a soft elastic half-space. It is shown that surface stress remarkably affects the pair interaction potential when the distance between the molecules is not larger than several times of the intrinsic length scale.

On Taking into Account the Effect of the Critical Angle on the Parameters of Displacement of Transverse Elastic Waves Excited by Point Sources

Asymptotic expressions for displacements of transverse waves excited by sources like a point vertical and horizontal force on the surface of elastic half-space are obtained for the far zone. These expressions take into account the features of emission of these waves at angles close to the third critical angle. At a fairly large distance from the critical angle, these expressions go over into asymptotic estimates obtained earlier.

Simulation of Operation of Dry Point Contact Probes in the Emission Regime

Displacements of longitudinal and transverse waves driven by sources like point vertical and horizontal forces acting on the surface of elastic half-space are obtained in the form of Fourier–Bessel integral representations using the Smirnov–Sobolev method which makes it possible to transform two-dimensional solutions of wave equations into three-dimensional solutions. Asymptotic expressions of the displacements are obtained for the far zone. Results of numerical estimates are given.

Displacements of the Elastic Half-Space Surface Caused by Instantaneous Axisymmetric Loading

Displacements of the Elastic Half-Space Surface Caused by Instantaneous Axisymmetric Loading

Mechanism of Surface Crack Growth in Lubricated Rolling/Sliding Spherical Contact

A possible mechanism of the crack growth is described by calculating stress intensity factors for a semicircular surface crack under a spherical Hertzian contact moving over a cracked elastic half-space surface. In the analysis the crack is assumed to be inclined at an angle of 45 deg from the surface. In particular, the effects of surface traction, frictional force between crack faces, and oil hydraulic pressure caused by oil penetrated into the crack are discussed.

溝を通過するレイリー波群について (I)

溝を通過するレイリー波群について (I)

Response of Torsionally Excited Foundations

Investigated herein are two aspects of the response of a rigid massless foundation of circular plane which is supported at the surface of a homogeneous elastic half space and is excited by a torque about an axis normal to the foundation. A simple discrete model, characterized by frequency-independent parameters, has been presented for a torsionally-excited circular foundation supported at the surface of an elastic half space. In addition, closed form expressions have been presented for the impulse response functions of the system. The response functions considered include the rotation associated with a unit impulsive torque, and the torque associated with a unit impulsive rotation. With the aid of these functions, the response of the system to an arbitrary excitation and the force associated with a prescribed response may be evaluated by direct integration in the time domain.

The normal vibrations of a rigid spherical punch on the surface of an elastic half-space

SummaryA rigid spherical punch vibrates normally on the surface of a semi-infinite isotropic elastic half-space. The essential novelty of this problem, which is treated within the context of classical elasticity, is that of a changing boundary; the radius of the circle of contact on the free surface varies with time. The geometrical co-ordinates are modified to yield a boundary value problem with fixed boundaries. However the governing differential equations become more complicated. These equations are solved by a perturbation procedure for the case where the contact radius a(t) is of the formwhere a0 is constant and |ŋ(t)≪1. Finally the normal stress and the total load under the punch are evaluated in the form of series which are valid for sufficiently slowly varying ŋ(t).

Ground Shock Due to Rayleigh Waves from Sonic Booms

Steady state vertical displacements at surface of elastic half-space due to Rayleigh waves from sonic boom, obtaining shock amplification factors