Terms Of Displacement Potentials Research Articles

УЧЕТ ТРЕНИЯ НА НАЧАЛЬНОМ ЭТАПЕ ВЕРТИКАЛЬНОГО ВНЕДРЕНИЯ ВЫПУКЛОГО УДАРНИКА В УПРУГУЮ ПОЛУПЛОСКОСТЬ

The initial stage of indentation of a convex rigid punch into isotropic elastic half-plane with friction is considered. A closed mathematical formulation in Cartesian coordinates is presented by equations of motion in the absence of a body force for homogeneous isotropic elastic medium in terms of displacement potentials, Cauchy equations for deformations, Hooke law and translational equation for the punch. Initial conditions are homogeneous. Outside of contact region surface is free from stresses. Inside contact region normal displacements of the surface of a half-space and the surface of a punch are taken to be equal and the relation between tangential and normal stresses in a form of Coulomb friction law is given. Due to the short duration of the supersonic stage and smallness of the contact region radius the tangential stress direction is considered to be point independent. Resolving functional equations are given in a form of convolutions with influence functions. The latter is the solution the original problem for half-plane with a special boundary condition in a form of the Dirac delta function. This solution is obtained in the domain of Laplace transform by time and Fourier transform by spatial coordinate. The original is constructed with the combined Fourier–Laplace inversion method. Solution is given with consideration to the initial stage characteristic property of supersonic velocity of contact region expansion (no smaller than propagation velocity of expansion-compression waves). It is shown that kinematic parameters of the punch, resulting force and discontinuities of the first kind on boundaries of the contact region are independent of friction unlike contact pressure. Calculation examples for different values of friction coefficient are presented.

Modeling of wave reflection in gas hydrate-bearing sediments

The content and distribution pattern of the gas hydrate are the key factors affecting the free surface reflection of the gas hydrate-bearing sediment. In this study, a theoretical model of wave reflection in gas hydrate-bearing sediment is discussed for the incidence of P1-wave and S1-wave to analyze the influences of the gas hydrate. The gas hydrate-bearing sediment is modeled by a Biot-type three-phase theory and assumed to be composed of sediment grains with connected pore occupied by the mixture of fluid and hydrate. The incoming wave is split into three reflected compressional waves and two reflected shear waves at the free surface of a gas hydrate-bearing sediment half-space. The analytical solutions for various reflected waves at open-pore and sealed-pore boundaries are obtained in terms of displacement potentials. Then, two-phase type models based on Biot theory are proposed by assuming the gas hydrate and sediment frame are weakly coupled or essentially consolidated to verify the rationality of the theoretical model. A third two-phase type model is given by assuming there is no gas hydrate in the pore. At last, the influences of the content and distribution pattern of gas hydrate, the frequency of the incident wave, and the permeability condition on the horizontal displacements and displacement ratios are analyzed respectively through numerical calculation. It is revealed that the solutions based on the Biot-type three-phase theory are very consistent with the solutions based on Biot theory if the gas hydrate and sediment frame are weakly coupled or essentially consolidated. Furthermore, the model proposed in this study can be used to analyze the influence of the distribution pattern of the hydrate.

Dynamic Response of a Circular Tunnel in an Elastic Half Space

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.

Reflection and Transmission of Plane Waves at a Water–Porous Sediment Interface with a Double-Porosity Substrate

This paper investigates the wave propagation at the interface between the ocean and the ocean floor. The ocean floor is assumed to be composed of covered porous sediment with an underlying double-porosity substrate. For this purpose, plane wave reflection and transmission in the coupled water–porous sediment–double-porosity substrate system are analytically solved in terms of displacement potentials. Using numerical examples, the effects of the material properties of the underlying double-porosity substrate on the reflection coefficients are discussed in detail. Variations in pore and fracture fluid, fracture volume fraction, and permeability coefficients are considered. In addition, two cases of boundary conditions at the porous sediment–double-porosity substrate interface, i.e., sealed-pore boundary and open-pore boundary, are compared in the numerical calculations. Results show that material property variations in the double-porosity substrate may significantly affect the reflected wave in the overlying water if the sandwiched sediment depth is less than the critical value.

Reflection and transmission of plane waves at an interface of water/porous sediment with underlying solid substrate

Ocean floor is usually covered with porous marine sediment. The wave reflection and transmission at the interface separating semi-infinite ocean and porous sediment has been extensively investigated in the last three decades. This paper focuses on the wave propagation at the ocean–porous sediment interface with underlying solid substrate. The analytical solutions for the plane wave reflection and transmission in the coupled water–porous sediment–solid substrate system are obtained in terms of displacement potentials. Based on the derived theoretical formulation, the wave reflection and transmission coefficients at the water/porous sediment interface are examined numerically with various incident angles, excitation frequencies, and sandwiched sediment depths. In addition, the energy dissipation effects of the porous sediment and the flow condition at the porous/solid interface are investigated. The results show that the porous sediment layer has a significant effect on the reflected wave in the overlying water.

Analytical solutions for dynamic pressures of coupling fluid–porous medium–solid due to SV wave incidence

AbstractThis paper presents the results of theoretical investigation on the dynamic coupling of an ideal fluid‐porous medium‐elastic half‐space system subjected to SV waves to study the effect of sediment on the seismic response of dams for reservoirs that are deposited with a significant amount of sediment after a long period of operation. The effects of the porous medium and the incident wave angle on dynamic pressures in the overlying ideal fluid are analyzed, and the reflection and transmission coefficients of the wave at the material interfaces are derived using an analytical solution in terms of displacement potentials. The numerical test of modeling shows that the dynamic pressures significantly depend on the properties of porous medium. The fully saturated porous medium reduces the response peaks slightly, while the partially saturated porous medium causes a considerable increase in the resonance peaks. Copyright © 2009 John Wiley & Sons, Ltd.

Analytical solutions for dynamic pressures of coupling fluid-solid-porous medium due to P wave incidence

Wave reflection and refraction in layered media is a topic closely related to seismology, acoustics, geophysics and earthquake engineering. Analytical solutions for wave reflection and refraction coefficients in multi-layered media subjected to P wave incidence from the elastic half-space are derived in terms of displacement potentials. The system is composed of ideal fluid, porous medium, and underlying elastic solid. By numerical examples, the effects of porous medium and the incident wave angle on the dynamic pressures of ideal fluid are analyzed. The results show that the existence of the porous medium, especially in the partially saturated case, may significantly affect the dynamic pressures of the overlying fluid.

Hybrid volume scattering model: Application of the Biot theory to the fluid-bottom model

The volume backscattering model, which was derived by Yamamoto (1996) applying the Born approximation to the Helmholtz equation, is improved by replacing transmission coefficients at the fluid–fluid interface to those at the fluid–porous seabed interface. The spectra of velocity and density fluctuations in sediments are assumed to be the same. The sea bottom is hereby treated as an isotropic porous medium, in which sound speeds have frequency dependence. Frequency dependent reflection and transmission coefficients at the fluid–porous seabed interface for a compressional wave incident from fluid upon porous media and vice versa are easily calculated applying the mode’s decoupling method to Biot’s body waves without considering behaviors of inhomogeneous waves in porous media. In this problem the reflection and transmission coefficients in terms of displacement potentials are computed. The hybrid model is applied to medium to high frequency bottom backscattering from the sandy bottom. In soft sediments, however, the difference between the hybrid and the fluid–fluid bottom models is small. It is concluded that the hybrid model can be applied to volume backscattering from any kind of unconsolidated sediments throughout the frequency range. [Work supported by ONR Code 321OA.]

The use of displacement potentials in second order elasticity

This paper is concerned with the use of a representation in terms of displacement potentials in second order elasticity for equilibrium problems of homogeneous and isotropic materials. After justifying the adoption of an existing representation for linear elasticity for the purpose at hand, appropriate representations for solutions of second order elasticity problems in terms of displacement potentials (for both compressible and incompressible materials) are discussed. The use of the representations in obtaining complete solutions for equilibrium boundary-value problems is then illustrated by application to two examples of plane strain problems of compressible materials.

Computed waveforms in transversely isotropic media

Radiation of elastic waves from a point force or from a localized torque into a transversely isotropic medium has been formulated in terms of displacement potentials, and transient waveforms have been computed by numerical Fourier inversion. For isotropic sandstone, this procedure yields P‐ and S‐wave pulses whose arrival times and magnitudes agree with theory. For a range of anisotropic rocks, arrival times of quasi‐P‐waves and quasi‐S‐waves agree with asymptotic theory. For extreme anisotropy, some quasi‐S‐wave pulses arrive at times which are not predicted by asymptotic theory. Magnitudes have not been compared with results of asymptotic theory, but decrease with distance appears to be in agreement. This Fourier inversion method gives near‐source changes in waveform which are not obtainable from the asymptotic theory.