Torsional vibrations of elastic foundations embedded in an elastic half-space

This paper examines the axisymmetric torsional vibrations of an elastic pile and a hemispherical foundation embedded in a homogeneous elastic half-space. The embedded foundation–half-space system is decomposed into an extended half-space and a fictitious foundation. The deformations of the fictitious system are specified by an admissible function containing a set of generalized coordinates. The Lagrangian equations of motion are used to determine these coordinates associated with the assumed displacement function. Numerical results are presented for torsional impedance of an elastic pile and a hemisphere to illustrate the effects of relative flexibility and geometry. By employing certain simplifications on the pile–half-space system an approximate closed form solution is presented for the torsional impedance of an elastic pile.     

Transient flexural vibrations of an elastic column supported by an elastic half-space

An analysis is presented of the transient flexural vibrations of an elastic column supported by an elastic half-space under the condition that an arbitrarily shaped free-field lateral acceleration and displacement are given as inputs. Applying Laplace transformations with respect to time and numerical inverse Laplace transformations, the time histories of the column acceleration at the interface and free end, and the column and half-space displacement distributions are obtained. After the input free-field acceleration terminates, slightly damped and almost harmonically variable acceleration is observed. The acceleration frequency after the disappearance of the input acceleration nearly coincides with the resonant frequency of the system. The slight damping with the first resonant frequency, even if the half-space is soft compared with the column, is characteristic of the transient flexural vibrations of a column supported by a half-space. Such a phenomenon is not typical of the transient longitudinal vibration problem. Therefore, it may be concluded: when buildings and structures are subjected to an earthquake or an explosive force, their flexural vibrations will continue with their first resonant frequencies, even if their foundations are soft.     

Transmission of elastic waves through a second-order fluid layer sandwiched between elastic media

Summary The present paper is concerned with the transmission of elastic waves through a second-order fluid layer sandwiched between homogeneous elastic media having identical properties. It is concluded that the transmission ratio increases with the increasing absolute values of visco-elastic co-efficient and also with the wavelength of the wave transmitted.     

General energetic analysis of fracture instabilities in elastic media and the elastic field rupture

The singular integral equations method makes it possible to determine a general analytical solution to the problem of a crack subjected to any stresses, including singular ones. The singularity of stresses means that they tend to infinity in the concentration point. In exponential functions describing this relationship, the exponent characterizes the stress curvature growth. Also the energy released by crack opening can be described by a simple analytical formula. The problem is solvable for an exponent greater than −1. The class of all the cracks subject to stresses that exponentially grow to one of the crack ends is divided into three sub-classes. One of these embraces most of crack types, also Griffith’s. The remaining two are a source of microcracks in an elastic medium. The onset of such a stress concentration gives rise to a microcrack which cancels the stress singularity up to that with the exponent of −1/2, ensuring a strong stability of the medium. An analysis of the nucleation of such cracks brought about a concept of elastic field rupture without destruction of interatomic bonds, which has implications relating to the conductivity of metals. A general formula for the crack energy singles out a special crack of unit length, whose energy is constant and independent of stress concentration.     

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.     

Linearized elastic parameter sections1

Contrasts in elastic parameters can be estimated directly from seismic data using offset-dependent information in the PP reflection coefficient. A linear approximation to the PP reflection coefficient including three coefficients is fitted to the data, and relative contrasts in various elastic parameters are obtained from linear combinations of the estimated coefficients. Linearized elastic parameter sections for the contrasts in P-wave impedance, P-wave velocity, density, plane-wave modulus, and the change in bulk modulus and shear modulus normalized with the plane-wave modulus are estimated. If the average P- to S-wave velocity ratio is known, linearized parameter sections including the contrast in the average P- to S-wave velocity ratio and a fluid factor section can be computed. Applied to synthetic data, visual comparison of the estimated and true elastic parameter sections agree qualitatively, and the results are confirmed by an analysis of the standard deviation of the estimated parameters. The parameter sections obtained by inversion of a shallow seismic anomaly in the Barents Sea are promising, but the reliability is uncertain because neither well data nor regional trends are available.     

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.     

Decoupled elastic prestack depth migration

This paper presents a new decoupled form of the formula for common-shot or common-receiver amplitude-preserving elastic prestack depth migration (PreSDM), which can be used for estimating angle-dependent elastic reflection coefficients in laterally inhomogeneous anisotropic media. The multi-shot or multi-receiver extension of this formula is suitable for automated prestack amplitude-versus-angle (AVA) elastic inversion of ocean-bottom cable (OBC), walkaway VSP (WVSP) or standard towed-cable data at any subsurface location. The essence of the theory is a systematic application of the stationary-phase principle and high-frequency approximations to the basic elastic Green's theorem. This leads to nonheuristic explicit wave mode decoupling and scalarization of vector PreSDM. Used in combination, ray-trace and finite-difference (FD) eikonal solvers create a useful tool to calculate accurate Green's function travel time and amplitude maps. Examples of synthetic OBC data and applications to field WVSP data show that the new imaging technique can produce a clear multi-mode elastic image.     

Vibrations of composite elastic bodies

Summary The present paper deals with the vibrations of composite sphere and cylinder of different elastic materials characterised by particular types of non-homogeneity.     

Characterizing the effect of elastic interactions on the effective elastic properties of porous,cracked rocks

Elastic interactions between pores and cracks reflect how they are organized or spatially distributed in porous rocks. The principle goal of this paper is to understand and characterize the effect of elastic interactions on the effective elastic properties. We perform finite element modelling to quantitatively study how the spatial arrangement of inclusions affects stress distribution and the resulting overall elasticity. It is found that the stress field can be significantly altered by elastic interactions. Compared with a non‐interacting situation, stress shielding considerably stiffens the effective media, while stress amplification appreciably reduces the effective elasticity. We also demonstrate that the T‐matrix approach, which takes into account the ellipsoid distribution of pores or cracks, can successfully characterize the competing effects between stress shielding and stress amplification. Numerical results suggest that, when the concentrations of cracks increase beyond the dilute limit, the single parameter crack density is not sufficient to characterize the contribution of the cracks to the effective elasticity. In order to obtain more reliable and accurate predictions for the effective elastic responses and seismic anisotropies, the spatial distribution of pores and cracks should be included. Additionally, such elastic interaction effects are also dependent on both the pore shapes and the fluid infill.     

橄榄石矿物弹性研究进展

橄榄石是上地幔含量最丰富的矿物,对其弹性性质的研究对解释地球内部的物质组成和动力学过程至关重要.近年来,对橄榄石矿物弹性的研究,无论是实验测试还是模拟计算都取得了很大的进展.本文首先介绍了研究橄榄石弹性的主要方法,然后从铁、镁不同配分比例下橄榄石的弹性性质,水对橄榄石弹性的影响以及橄榄石各向异性等三个方面,对近年来橄榄石弹性研究所取得的主要进展做了介绍.实验方法方面主要介绍了1)超声共振谱法(Resonant Ultrasound Spectroscopy,RUS);2)X射线衍射法;3)X射线照相法;4)超声干涉法;5)布里渊散射法.     

SH waves in stressed elastic plates

Summary A theoretical study is made of the propagation ofSH waves of small amplitude in an infinite elastic plate subjected to a primary normal stress. It is shown that the effect of the stress may be represented by a re-scaling of the plate thickness, provided that one elastic constant is also redefined suitably.     

Dynamic stresses in an elastic half-space

This paper deals with the problem of time-varying point loads applied onto the surface of an elastic half-space and the stresses that such loads elicit within that medium. The emphasis is on the evaluation of the isobaric contours for all six of the stress components at various frequencies of engineering interest and for a full range of Poisson’s ratios. The extensive set of pressure bulbs presented herein may be of help in predicting the severity of dynamic effects in common practical situations in engineering—or even the lack thereof.     

Transmission of elastic pulses in bars

Summary The transmission of elastic pulses through solid bars has been investigated experimentally with particular reference to the shape of the bars. A single input pulse giving rise to a series of pulses at the receiver which help in the computation of dilatational and rotational velocities in the material has been exploited for the experimental work, and it has been found that the method is applicable to cylindrical and cubical (square) bars in an equal manner. Results of the experimental investigations carried out in cylindrical and square bars of brass are compared and found to be in good agreement. Also results in square bars of Aluminium, Steel, and a few rock samples are reported. The rock samples studied include limestone, sandstone and a crypto-crystalline rock. An attempt to study the applicability of this method to rectangular bars has also been made and the results obtained are consistent and in agreement with the results due to cylindrical and square bars.N.G.R.I.-68-117.     

Modelling cyclic elastic behaviour of sands

The cyclic stress-strain behaviour of cohesionless soils is considered to be composed of elastic and inelastic components, and the cyclic elastic stress-strain relationships for one type of uniform sand are investigated by conducting isotropic and laterally constrained compression tests under low frequency cyclic loads. The effects of relative density, stress history, and stress state on the cyclic elastic behaviour are studied. Simple and practically applicable elastic stress-strain models are developed based on Hertzian contact theory. It was observed in both types of tests that elastic strains can accurately be modelled by a power function of stresses.     

Mie elastic scattering generated by cracks

Some numerical experiments are designed to elucidate geometrical details of fractures by studying scattering of elastic waves. The response in the time domain does not seem to provide much information. In the frequency domain, geometrical features of diffractors stamp a signature on reflected and diffracted wave fields. Results show that radiation patterns in the Mie regime are a promising device to infer some aspects such as shape and connectivity of the fractures. A combination of the direct and the indirect boundary element methods is used to compute numerical results. The numerical method is validated with analytic solutions for canonical problems.     

Complex-parameter kelvin model for elastic foundations

A new rigorous approach in modelling mechanical behaviour is developed. The dynamic response of a rigid disc resting on an elastic half-space is approximated by a macroscopic force-displacement relationship, where not only the coefficients but also the order of time derivatives are complex valued. It is shown that comlex-parameter constitutive models are not simply an elegant alternative in modelling mechanical behaviour, but are the outcome of rigorous non-linear regression analysis on complex-valued functions, like the dynamic stiffness of dissipative systems. Complex-parameter constitutive models are very attractive since a minimum number of parameters is required to obtain a satisfactory fit of the ‘exact’ response. A two-parameter generalized Kelvin model reproduces closely the ‘exact’ dynamic stiffness of the disc for all three vertical, horizontal and rocking modes studied herein. Frequency- and time-domain algorithms to solve complex-order differential equations are developed, validated and used to calculate the foundation–response under earthquake excitation. It is the excellent agreement of results and the economy in number of parameters that make complex-parameter models so attractive in constitutive modelling.