Wave-propagation in uniaxially-stressed elastic media

Summary The propagation of waves of small amplitude in an elastic medium, with a particular form of strain-energy function, and subjected to a large homogeneous uniaxial stress, is considered in detail. The main characteristics, including speed and polarisation, of these waves in an unbounded medium are calculated numerically, and the existence and general properties of Rayleigh waves on the surface of a semi-infinite medium are investigated.     

Scattering of plane P,SV or Rayleigh waves by a shallow lined tunnel in an elastic half space

Based on the plane complex variable theory and the image technique, an analytical solution is presented for scattering of plane harmonic P, SV or Rayleigh waves by a shallow lined circular tunnel in an elastic half space. The major contribution of this study is the treatment of the orthogonality of the boundary conditions along the half surface and the cavity wall. In terms of the image technique, the scattered waves by the half surface are simulated as transmitting from the image source of the origin of the tunnel. Using two different conformal mapping functions, we obtained the complex-valued stresses and displacements of the elastic medium and the liner in the image domain, respectively. The boundary value problem results in a set of infinite algebraic equations. The accuracy of the present approach is verified by comparing the present solution results with the available published data. Parametric study indicates that the embedment depth, the shear modulus and the thickness of the liner have significant influences on the dynamic response of the liner and the medium.     

TO RADIATE OR NOT TO RADIATE

Insight into radiation damping of an unbounded medium is developed by addressing the relative contributions of the elastic restoring force and the inertial force at infinity. When the inertial force dominates, radiation damping occurs. When the elastic restoring force dominates, no radiation damping arises. An unbounded medium with a cutoff frequency can also be identified.     

One-dimensional modelling of the non-linear far field in soil–structure-interaction analysis

In a bounded domain elasto-plastic wave propagation can be modelled accurately using the finite-element method. As is even the case for an elastic analysis, an unbounded domain, e.g. a semi-infinite soil or fluid, can, however, not be represented in this manner, as any spatial discretization has to be avoided. For one-dimensional wave propagation with a bi-linear elasto-plastic material law involving one stress component an analytical solution exists. The latter is used in modelling the non-linear far field of an unbounded medium using a rigorous bookkeeping procedure of the generated elastic and plastic waves propagating in both directions. The need for a non-linear model of the far field arises, as in a two-dimensional representation of soil-structure interaction the surface waves do not decay.     

Wave-propagation in biaxially-stressed elastic media

Summary The propagation of small-amplitude plane waves through a homogeneous, isotropic elastic medium, with a particular form of strain-energy function and subjected to a large primary deformation, produced by a homogeneous biaxial stress, is considered in detail. The properties of such waves in an unbounded medium, the conditions for their existence, their speeds and their polarisations are investigated and similar results are also obtained for Rayleigh surface waves.     

用弹性平板代替球壳计算板块弯曲形变的误差分析

在岩石圈板块弯曲问题的研究中常常忽略地壳曲率,用平板代替球壳计算．本文从弹性扁球壳和薄板形变的基本方程入手,通过对典型问题的解析解,推导出作简化处理所引起误差的表达式,着重分析了区域的几何尺度和地基模量对误差的影响．研究表明,用弹性平板模型取代球壳模型计算板块弯曲形变会引起较大的误差．但如果采用具有适当地基模量的弹性基础板,不但更符合地学实际,且能有效地改善误差状况,获得与用肩球壳计算弯曲问题接近的效果．     

Energy balance of simple elastodynamic sources

Complete relations are derived for energy and energy flux of elastic waves generated by an isotropic and double-couple source in a perfectly elastic, homogeneous, isotropic, and unbounded medium. In the energy balance of elastodynamic sources near-field waves play an essential role, transforming static energy into wave energy, andvice versa. For explosive and dislocation sources, the sources surface radiates a positive wave energy that is partially distributed to the medium transforming into static energy. For implosive and antidislocation sources, the source surface generates elastic waves, but it does not necessarily imply that it also radiates a positive wave energy. The energy transported by waves can originate in gradual transformation of the static-to-wave energy during propagation of waves through a stressed medium.On leave from Geophysical Institute, Czech Academy of Sciences, Boní II/1401, 41 31, Praha 4 Czech Republic     

Dynamic stiffness of unbounded medium based on damping-solvent extraction

In the damping-solvent extraction method, to calculate the dynamic-stiffness matrix of an unbounded medium, a finite region of the medium, adjacent to the structure is analysed in the first step, whereby hysteretic material damping is introduced artificially as a solvent. This leads to the dynamic-stiffness matrix of the damped bounded medium, which is assumed in the second step to be equal to that of the damped unbounded medium. In the third step, the effect of the material damping on the dynamic-stiffness matrix is eliminated, i.e. the damping solvent is extracted, resulting in the dynamic-stiffness matrix of the unbounded medium. The damping-solvent extraction method permits an efficient calculation of the dynamic-stiffness matrix of an unbounded medium by analysing the adjacent bounded medium only, which exhibits the same dynamic characteristics as the (bounded) structure. The familiar standard finite-element method is sufficient for the analysis and the hysteretic damping is introduced by multiplying the elastic moduli by 1 + 2i£. The introduced hysteretic material damping, the solvent, is extracted at the end of the analysis for each coefficient of the dynamic-stiffness matrix and for each frequency independently of the others by a very concise equation based on a Taylor expansion. The method is evaluated thoroughly for dynamic soil-structure interaction and for seismic reservoir-dam interaction using stringent simple cases with analytical solutions available and is also applied to practical examples, by calculating the dynamic-stiffness matrix of a semi-infinite wedge and an embedded foundation.     

Inferring the Orientation-distribution Function from Observed Seismic Anisotropy: General Considerations and an Inversion of Surface-wave Dispersion Curves

—A general relation linking the elasticity tensor of an anisotropic medium with that of the constituting single crystals and the function describing the orientation distribution of the crystals is derived. By expanding the orientation distribution function (ODF) into tensor spherical harmonics and using canonical components of the elasticity tensors, it is shown that the elastic tensor of the medium is completely determined by a finite number of expansion coefficients, namely those with harmonic degree l≤ 4. The number of expansion coefficients actually needed to determine the elastic constants of the medium depends on the symmetry of the single crystals. For hexagonal symmetry of the single crystals it is shown that only 8 real numbers are required to fix the 13 elastic constants which are for example needed to determine the azimuthal dependence of surface wave velocities. Thus, inversions of observations of seismic anisotropy are feasible which do not make any a priori assumptions on the orientation of the crystals. As a byproduct of the derivation, a formula is given which allows the easy calculation of the elastic constants of a medium composed of hexagonal crystals obeying an arbitrary ODF. An application of the theoretical results to the inversion of surface wave dispersion curves for an anisotropic 1D-mantle model is presented. For the S-wave velocities the results are similar to those of previous inversions but the new approach also yields P-wave velocities consistent with the assumption of oriented olivine. Moreover it provides a hint of the orientation distribution of the crystals.     

On the disturbance due to a spherical distortional pulse in an elastic medium

Summary Theoretical expressions are derived for the displacement, velocity and stress in the time domain induced by an axially symmetric shearing stress applied at the inner surface of a spherical cavity in a hornogeneous, isotropic, elastic medium of infinite extent. Theoretical seismograms are computed for a step source and for three sources with exponential decay in time. A satisfactory time-dependence of the source can be obtained by combining the step source with one or more exponentially decaying sources.     

垂直地震力作用下弹性半空间上扁壳基础的解析解

用布希涅斯克定义的弹性半空间内的垂直位移包括两项积分,除了积分号前面系数的差别之外,第一项积分是单层位势而第二项积分为双层位势。若扁壳基础是正高斯曲率的几何曲面,则壳底与半空间表面间的挤压强度就是半空间表面作用的分布垂直荷载。当越过边界时,双层势位的函数值和单层势位的法向导数值发生跳跃。利用这些性质,本文得出布希涅斯克积分的反演公式,从而避开要求解偏微分—积分方程组的巨大数学困难而易于得出解析解。以椭园抛物面扁壳为例说明本文方法的应用。     

Response of a circular cylindrical shell to disturbances in a half-space—numerical results

In this paper we present numerical results representing the response of an elastic cylindrical shell to disturbances in an elastic half-space. These numerical results show the radial displacement at the middle surface of the shell and the hoop stress in the shell. The incident disturbance is in the form of a plane dilatational wave or a Rayleigh wave.     

Dynamic response of a piecewise circular tunnel embedded in a poroelastic medium

Recently, considerable efforts have been devoted to evaluation of seismic dynamic response of a circular tunnel. Conventional approaches have considered integral liners embedded in an elastic medium. In this study, we re-examine the problem with piecewise liners embedded in a porous medium. Surrounding saturated porous medium of tunnels is described by Biot's poroelastic theory, while the liner pieces and the connecting joints are treated as curved beams and characterized by curved beam theories. The scattered wave field in the porous medium is obtained by the wave function expansion method. The differential equations governing the vibration of a curved beam is discretized by the General Differential Quadrature (GDQ) method. The domain decomposition method is used to establish the global discrete dynamic equations for the piecewise tunnel. The surrounding soil and the tunnel are coupled together via the stress and the displacement continuation conditions which are implemented by the boundary collocation method. Numerical results demonstrate that the stiffness difference between the liner piece and the connecting joints has a considerable influence on the internal forces of the liner piece.     

Finite element simulation of steady state SH wave motion

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Radial vibration of a composite spherical shell

Summary This paper deals with radial vibration of a composite spherical shell made up of orthotropic materials by the application of a periodic shearing stress on the outer surface. Two cases have been discussed. In the first the elastic constants of the material of the composite body are assumed to remain constant while in the second they are proportional to the distance from the centre of the composite body. Results were obtained in terms of Bessel functions.     

On disturbances produced in an elastic medium by twists applied on the inner surface of a spherical cavity

Summary In this note, disturbances generated in an isotropic elastic medium by some simple types of twisting forces applied on the inner surface of a spherical cavity have been investigated. Prof.S. K. Chakraborty, Surendranath College, 24 Harrison Road,Calcutta (India).     

Dynamic response of a submerged hemispherical shell to earthquake motions

The dynamic response to ground motion of hemispherical shells in a fluid, medium is studied numerically. In the analysis, linear thin shell theory is used and the fluid is assumed to be compressible and inviscid. The effect of the duration of the ground motion on the dynamic response is studied using two forcing functions, one with a very short duration and the other in the form of a Heaviside function. As special cases, dynamic responses of the shell in vacuo and of a rigid hemisphere in the fluid medium are investigated. The results are valid also for a ring-stiffened complete spherical shell accelerating in an acoustic medium.     

Propagation of plane waves in an unbounded elastic medium with Cauchy's initial stress

Summary Cauchy theory of initial stress has been applied to investigate the influence of uniform initial stress on the propagation of plane waves in an unbounded elastic medium. It is found that if the initial principal stresses are unequal, the velocity depends on the direction in which the wave propagates, whereas when the initial principal stresses are equal, the velocity is independent of the direction of wave propagation.     

On the effect of non-homogeneity

Summary Effects of non-homogeneity in respect of rigidity have been computed in the case of a gravitating sphere with a rigid core. For an isotropic spherical shell the effect of variation of rigidity has been calculated when the shell is under pressure or when the shell is subjected to some steady temperature on its internal surface.     

THE SUPPRESSION OF SURFACE MULTIPLES ON SEISMIC RECORDS*

A new approach is presented for the suppression of multiples reflected at the surface of a horizontally layered fluid or elastic medium, recorded at non-zero offsets from the source. The scheme used is to extract the effect of the free surface in the frequency-wavenumber domain and then to replace this surface by a non-reflecting boundary. The multiple suppression operator requires a detailed knowledge of the source time function and the elastic properties of the medium between the source and the surface. For a stratified fluid or a liquid layer overlying a stratified elastic medium, complete multiple suppression can be achieved with noise free data. If only the vertical component is available for an elastic medium an approximate approach may be used which removes most of the multiple energy. Good results may be achieved with this multiple suppression scheme in the presence of noise. The method is designed to be used before records are stacked in a CDP gather.