Stresses produced by a normal load moving over a transversely isotropic layer of ice lying on a rigid foundation

Summary In this paper, the problem of a normal load moving over the surface of a layer of ice, lying on a rigid foundation, has been considered. FollowingHearmon, the ice is taken to be transversely isotropic. Two cases have been considered-in the first case, the rigid foundation is taken to be smooth, and in the second case, it is taken to be in welded contact with the layer of ice. Expressions have been obtained in both cases for the stresses at any point of the layer. From detailed numerical calculations, it has been found that the value of the normal stress at a point directly below the moving load falls of approximately as that of 1/z with increase in the depthz below the surface. On calculating the normal stress at the boundary between the layer of ice and the foundation, it is found that the normal stress falls off rapidly as we move away along the boundary from the point directly below the load.     

On stresses and displacements due to a moving line load over the plane boundary of a heterogeneous elastic half-space

Summary The problem of concentrated line load moving with supersonic speed along the boundary of an isotropic heterogeneous medium has been solved as a plane strain problem. The stresses and displacements in the heterogeneous case considered are found to decay exponentially with distance.     

Dynamic stress concentration in pre-stressed poroelastic media due to moving punch influenced by shear wave

During the occurrence of earthquake, the shear wave propagates in the rocks present inside/at the Earth’s crust. The propagation of shear wave may lead to the progression of punch present inside the rock medium. As a result of this, substantial stress accumulated at the vicinity of propagating punch inside rock medium which significantly affects the stability of various geological and human-made structure and, hence, may cause failure of structure. Therefore, the analysis of stress concentration at the vicinity of punch moving due to shear wave propagation has become prominent in the area of seismology. In the present paper, an analytical perspective has been employed to discuss the influence of velocity of moving punch associated with the propagation of shear wave on developed dynamic stress concentration (DSC) in three types of pre-stressed vertical transversely isotropic (VTI) poroelastic media viz. granite (an igneous rock); sandstone (a sedimentary rock); and marble (a metamorphic rock). The closed form expression of DSC for the force of constant intensity has been derived with the aid of Weiner-Hopf technique along with Galilean and two-sided Fourier integral transformations. The noticeable influence of different affecting parameters (viz. velocity of moving punch associated with the shear wave propagation, horizontal compressive/tensile initial stresses, vertical compressive/tensile initial stress, porosity, and anisotropy parameter) on dynamic stress concentration has also been reported. Numerical computation and graphical illustrations have been carried out for the aforementioned three different types of porous rocks to investigate the profound impact of affecting parameters on DSC. Moreover, some noteworthy peculiarities have also been derived from the obtained expression of dynamic stress concentration.     

Note on stresses produced by a shearing force moving over the boundary of a semi-infinite transversely isotropic solid

Summary Two dimensional problem of distribution of stresses produced by a pulse of shearing force moving uniformly on the boundary of a semi-infinite transversely isotropic solid has been studied here. It is observed that the shearing stress in the direction of propagation of pulse attains maximum value at a point directly below the point of application of the shearing force on the boundary.     

Ambiguous reflection coefficients for anelastic media

The calculation of reflection and transmission coefficients of plane waves at a plane interface between two homogeneous anelastic media may become ambiguous because it is not always obvious how to determine the sign of the vertical component of the slowness vector of the scattered waves. For elastic media, the sign is determined by applying so-called radiation condition when the slowness vector is complex-valued, but it has long been known that this approach does not work satisfactorily for anelastic media. Other approaches have been suggested, e.g., by requiring that the reflection and transmission coefficients should vary continuously with increasing incident angles, or by relating the sign to the direction of the energy flux. In the present paper, it is shown that these approaches may give different results, and that the results can be inconsistent with the elastic case even for weak attenuation. Instead, it is demonstrated that the ambiguity in the reflection coefficient can be resolved by expressing the seismic response of a point source over an interface as a superposition of plane waves and their reflection coefficients, and solving the resulting integral by the saddle point approximation. Although the saddle point itself (point of stationary phase) does not provide new insight, the ambiguity is removed by considering the steepest descent path through the point. Ray synthetic seismograms computed by this method compare well with synthetics computed by the reflectivity method, which does not suffer from the above-mentioned ambiguity since the integration path is taken along the real axis. This paper concentrates on the isotropic case, but it is discussed how the result may be extended to layered transversely isotropic media. The suggested approach, derived for a point source and plane layers, does not directly apply to 2-D or 3-D laterally inhomogeneous media, or to media of general anisotropy. A generalization of the result found is that the sign of the vertical slowness components should be chosen according to the energy flux direction for subcritical incidence and according to the radiation condition for supercritical incidence, even if this creates a discontinuity in the coefficients at the critical incidence angle. Such a discontinuity is sometimes necessary to get results which are consistent with the elastic case. It is discussed how the generalized result can be obtained by applying certain continuity criteria for the sub-and supercritical angle intervals, but the validity of this approach for general models remains to be proved.     

On the stresses produced in a liquid-saturated porous solid by normal and tangential forces on a moving strip of finite width

Summary This paper deals with the stresses produced in a semi-infinite liquid-saturated porous solid of the type considered byBiot [1]²), by moving normal and tangential forces acting simultaneously on a moving strip of finite width on the free surface. It has been found that the stresses can be obtained in a closed form. It has also been found that if the velocity of the strip exceeds a certain value, the nature of the stresses changes, and there are discontinuities of stress along certain lines. The results have been applied to study the changes in the stresses produced by moving normal and tangential forces with depth and distance.     

Microseisms: Its dependence on the movement of the source

Summary The displacements on the free surface of a homogeneous isotropic semi-infinite medium has been obtained assuming a moving normal stress on the surface. It has been studied how the displacements behave with (1) the direction of propagation of the source with respect to the recording station and (2) the velocity of the source.     

OnSH type of motion due to body forces and due to stress-discontinuity in a semi-infinite viscoelastic medium

Summary In this paper, surface displacements of theSH type of motion due to time-dependent body forces and due to sudden introduction of a discontinuity in the shearing stress in a semi-infinite, homogeneous, isotropic, viscoelastic medium have been worked out by the use of Laplace and Fourier transforms. The displacements due to a point force and due to a stress discontinuity over a fixed region have been calculated numerically.     

Constrains on stresses in isotropic homogeneous infinite half-spaces being in welded contact: 2D anti-plane and in-plane cases

Formulas are derived for two-dimensional problems relating stresses across a plane boundary that divides infinite homogeneous half-spaces being in welded contact. The calculations are made for both anti-plane and in-plane stress cases. The results obtained for the former case that involve only two stress components are useful in the analysis of fracture of strike-slip type. For the in-plane case, the relations that link stresses in one half-space with the corresponding homogeneous stresses in the other half-space are presented for arbitrarily oriented shear and normal stresses and for the center of compression (dilatation). The above relations provide a compete set of expressions that, among other things, make it possible to analyze stresses involved in faulting of deep-slip type in an inhomogeneous medium. The quantitative preliminary evaluations based on the results obtained demonstrate the great role of low rigidity media in fracture processes of all kinds within the Earth’s crust.     

Moving load on a porous,liquid-saturated solid

Summary The object of this paper is to study the stresses produced in a porous liquid-saturated solid of the type considered byBiot [1]²) by a normal load moving along the surface of such a solid. Two cases have been considered. In the first case, the porous solid is taken to be semi-infinite, while in the second case we consider a layer of the liquid-saturated porous solid resting on a smooth rigid foundation. Expressions have been obtained for the stresses at any point in the medium due to a normal load moving along the free surface of the porous solid. Detailed numerical calculations have been carried out for liquid-saturated sandstone, using the data given byFatt [9] for such a medium. These calculations show that the normal stress zz in the solid falls off approximately as 1/z with increase in the depthz below the free surface, and that the stresses in the solid and in the liquid fall off rapidly as we move away from the point directly below the moving load.     

Some problems of plane strain in a non-homogeneous isotropic cylinder

Summary The expressions for stresses and extension have been obtained in an isotropic circular cylinder in a state of plane strain, the elastic constants and the density of the material varying exponentially and the results have been compared with those in the homogeneous case.     

Generation of SH-type waves due to shearing stress discontinuity in a sandy layer overlying an isotropic and inhomogeneous elastic half-space

The generation of SH-type waves due to sudden application of a stress discontinuity which moves after creation at the sandy layer of finite thickness overlying an isotropic and inhomogeneous elastic half-space is considered. The displacements are obtained in exact form by the method due to Cagniard modified by De Hoop. The numerical calculations are obtained. Two cases of shearing stress discontinuities are considered for different sandiness parameters. The graphs are drawn to show the effect of sandiness in the displacement components.     

The offset‐midpoint travel‐time pyramid for P‐wave in 2D transversely isotropic media with a tilted symmetry axis

For pre‐stack phase‐shift migration in homogeneous isotropic media, the offset‐midpoint travel time is represented by the double‐square‐root equation. The travel time as a function of offset and midpoint resembles the shape of Cheops’ pyramid. This is also valid for transversely isotropic media with a vertical symmetry axis. In this study, we extend the offset‐midpoint travel‐time pyramid to the case of 2D transversely isotropic media with a tilted symmetry axis. The P‐wave analytical travel‐time pyramid is derived under the assumption of weak anelliptical property of the tilted transverse isotropy media. The travel‐time equation for the dip‐constrained transversely isotropic model is obtained from the depth‐domain travel‐time pyramid. The potential applications of the derived offset‐midpoint travel‐time equation include pre‐stack Kirchhoff migration, anisotropic parameter estimation, and travel‐time calculation in transversely isotropic media with a tilted symmetry axis.     

On the stability of a non-convolutional perfectly matched layer for isotropic elastic media

The multi-axial perfectly matched layer (M-PML) is a material boundary condition for wave propagation problems in unbounded domains. It is obtained by extending the formulation of the split-field perfectly matched layer to a more general absorptive medium, for which damping profiles are specified along all dimensions of the problem. Under the hypothesis of small damping, it has been demonstrated that the stability of the system of partial differential equations of the M-PML can be related to the ratio of the damping profiles, and stable M-PML terminations for isotropic and orthotropic elastic media have been constructed. In the present work, we use the Routh–Horwitz determinants to demonstrate that the conclusions regarding the stability of M-PML for isotropic media for small damping are in fact valid for the more general case of damping coefficients of any (positive) value. The effectiveness of the M-PML is demonstrated by constructing stable terminations for the abovementioned media. The stability analysis is presented for 2-D in-plane (P-SV) wave propagation in elastic isotropic continua.     

利用伪谱法模拟横向各向同性介质中的波

介质的弹性常数为三维四阶张量的分量,共有８１个,由于应力张量和应变张量的对称性及能量密度是应变的二次函数,一般各向异常性介质的独立弹性常数可减为２１个,如果介质具有较高的对称性,独立弹性常数的数目会更少。 对于地壳和上地幔,具有５个独立弹性常数的横向各向同性介质是一个非常好的近似,本研究中横向各向同性介质的对称轴方向可以是任意的（即对称轴可以不平等于铅直方向）,在此情况下,需要进行坐标变换,如果已知介质在某一坐标系（其坐标轴平行或垂直于介质的对称轴）中的弹性常数,我们能够容易地利用变换公式得到变换后新坐标系中的弹性常数。 本文提出了一种方案,利用伪谱法既能模拟横向各向同性介质中的平面波,也能模拟点源激发的波场。在勘探地球物理和地震学中,模拟横向各向同性介拮中传播的平面波及区域源产生的波是最重要的研究课题之一。然而在一般各向异性介质中,很难或不可能确定弹性波的相速度和偏振方向,但在横向各向同性介质中,则可以通过坐标变换来实现,这里我们所提出的方法可以用于横向各向同性介质中弹性波的模拟。     

Dynamic response of pipelines to moving loads

Vibration of buried pipelines induced by moving axial and radial loads has been studied in this paper. A thin shell model has been used for the pipeline, which is assumed to be lying in an infinite isotropic homogeneous elastic medium. In order to allow for the possible motion of the pipe out of phase with the surrounding ground a very thin layer of viscoelastic material is assumed to separate the pipe from the ground. Calculations indicate the presence of the interfacial viscoelastic layer does not influence the pipe response in a significant manner. So all the numerical results presented here are for the case when the pipe is assumed to be perfectly bonded. These show that the maximum pipewall displacements and stresses occur in a soft soil at long wavelengths.     

Determining maximum shear stress in confined substrate from elastic wave reflection coefficient

The relationship between the maximum shear stress in a substrate solid and the elastic wave reflection coefficient from the interface between the substrate solid and an overlying solid half-space is investigated. Both substrate and overlying solid media are assumed to be initially isotropic and stress-free. Then as the substrate is subjected to horizontal confined stresses it becomes anisotropic. It is shown that longitudinal and shear wave reflection coefficients are related to the degree of stress induced anisotropy in the substrate medium. From this relation the confined stress level and the maximum shear stress generated on the vertical planes of the substrate are estimated. Authors in their previous investigation computed plane wave reflection coefficient in a biaxially compressed solid substrate immersed in a fluid. This paper reports for the first time how the maximum shear stress in a biaxially compressed substrate medium can be measured from the plane wave reflection coefficients when the overlying medium is also a solid half-space.     

An efficient wave extrapolation method for anisotropic media with tilt

Wavefield extrapolation operators for elliptically anisotropic media offer significant cost reduction compared with that for the transversely isotropic case, particularly when the axis of symmetry exhibits tilt (from the vertical). However, elliptical anisotropy does not provide accurate wavefield representation or imaging for transversely isotropic media. Therefore, we propose effective elliptically anisotropic models that correctly capture the kinematic behaviour of wavefields for transversely isotropic media. Specifically, we compute source‐dependent effective velocities for the elliptic medium using kinematic high‐frequency representation of the transversely isotropic wavefield. The effective model allows us to use cheaper elliptic wave extrapolation operators. Despite the fact that the effective models are obtained by matching kinematics using high‐frequency asymptotic, the resulting wavefield contains most of the critical wavefield components, including frequency dependency and caustics, if present, with reasonable accuracy. The methodology developed here offers a much better cost versus accuracy trade‐off for wavefield computations in transversely isotropic media, particularly for media of low to moderate complexity. In addition, the wavefield solution is free from shear‐wave artefacts as opposed to the conventional finite‐difference‐based transversely isotropic wave extrapolation scheme. We demonstrate these assertions through numerical tests on synthetic tilted transversely isotropic models.     

Magneto-elasticSH-type of motion

Summary TheSH-type of motion in an isotropic elastic halfspace with infinite electrical conductivity subjected to an uniform magnetic field and disturbed by a buried source is investigated. Expressions for the surface displacements for various type of sources are obtained by an application of Cagniard technique. Numerical calculation has been performed for the case of a diapole source moving parallel to the surface of the halfspace with an uniform velocity.     

Evaluation of stresses in two geodynamically different areas: Stable foreland and extensional backarc

Areas which are geodynamically different have different behaviors both in their thermal regime and seismic activity. A stable area has a geotherm which can be considered as standard, extensional and compressional areas have, respectively, high and low temperature gradients. The Italian region includes different geodynamical areas and all such situations are present. We consider the Apulian platform as an example of a stable area and the Tuscany-Latium as an example of an extensional area. For both of them the present geotherms are calculated, taking into account, for the Tuscany-Latium, its thermal history. Assuming that each region is subject to a constant strain rate, the stresses are calculated as functions of depth and time. The rheological behavior is assumed to be linear viscoelastic, with viscosity dependent on temperature and elastic parameters dependent on lithology. The geothermal profile and the rheological structure of the lithosphere remarkably affect the processes of stress accumulation which control the distribution of seismic activity. The abrupt decrease of the temperature gradient at the Moho produces considerably higher stress values with respect to the case of uniform gradient, thus favoring subcrustal seismicity. In the case of a standard temperature gradient, subcrustal seismicity is predicted and a gap in seismicity, indicating a soft intracrustal layer, exists if there is a discontinuity in rheology. By contrast, in the case of a high-temperature gradient, subcrustal seismicity is not to be expected, even in the presence of a discontinuity in rheology, since subcrustal temperatures are already too high to permit a sufficient stress accumluation.