Showing papers on "Discontinuity (geotechnical engineering) published in 1969"

The age of discontinuity: guidelines to our changing society

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Elastic displacements in the near-field of a propagating fault

Abstract: Displacement, particle velocity, and acceleration wave forms in the near field of a propagating fault have been computed by numerical integration of the Green9s function integrals for an infinite medium. The displacement discontinuity (dislocation) on the fault plane is assumed to have the form of a unilaterally propagating finite ramp function in time. The calculated wave forms in the vicinity of the fault plane are quite similar to those observed at the strong motion station nearest the fault plane at the Parkfield earthquake. The comparison suggests that the propagating ramp time function is roughly representative of the main features of the dislocation motion on the fault plane, but that the actual motion has somewhat more high frequency complexity. Calculated amplitudes indicate that the average final dislocation on the fault at the Parkfield earthquake was more than an order of magnitude greater than the offsets observed on the visible surface trace. Computer generated wave form plots are presented for a variety of locations with respect to the fault plane and for two different assumptions on the relation between fault length and ramp function duration.

Wave propagation in an elastic solid with a line of discontinuity or finite crack

Abstract: With the aid of integral transforms, a method is presented for solving the problem of scattering of plane harmonic compression and shear waves by a line of discontinuity or crack of finite width embedded in an elastic medium of infinite extent. When the incoming waves are applied in an arbitrary direction, the scattered-wave field may be determined by separating the crack-surface boundary conditions into functions even and odd with respect to the variable along the line crack. The problem is reduced to the evaluation of a system of coupled Fredholm integral equations with special emphasis placed 011 finding the near-field solution which consists of a knowledge of the detailed structure of the displacements and stresses in a small region around the crack vertex. Dynamic stress-intensity factors, the critical values of which govern the condition of crack propagation, are defined and found to be dependent on the incident wave length and Poisson's ratio of the medium. At certain wave lengths, they are larger than those encountered under static loading. Such information is of particular importance in perdicting the fracture strength of structures subjected to oscillating loads. Introduction. Although the scattering of waves by obstacles of different shapes has been the subject of many past investigations in various branches of physics [1]—[3], to the authors' knowledge none of these investigations analyzed, in detail, the singular behavior of the stresses near a scatterer in the form of a line of discontinuity or finite crack. The main reason for this omission is the lack of an effective mathematical method for obtaining the near-field solution, which is of considerable theoretical interest and has innumerable applications in the field of fracture mechanics as well as in electromagnetic and acoustic theory. A popular approach to the diffraction of waves from obstacles has been that of separation of variables, where the formal solution of the wave equation is given by an infinite series of orthogonal functions. Such an approach, however, is effective only for obstacle shapes adapted to those coordinate systems in which the wave equation is separable. For this reason, the dynamic stress concentrations around circular and parabolic obstacles have received considerable attention in the past. A comprehensive survey of the literature in a field as wide and diversified as the propagation of elastic waves is clearly beyond the scope of this paper. In recent years, the Mow-Pao-Thau school [4]—[6] has * Received January 8, 1968; revised version received March 29, 1968. The research described in this paper was sponsored by the U. S. Navy under Contract Nonr-610(06) with the Office of Naval Research, Washington, D. C. 194 G. C. SIH AND J. F. LOEBER [Vol. XXVII, No. 2 published a number of papers on this subject. References to other work can be found in [4]-[6], It is well known that problems involving diffraction of plane harmonic, horizontally polarized shear waves (SH-waves) by a semi-infinite crack can be formulated in terms of integral equations, and solved by the Wiener-IIopf technique [7]. As pointed out by Sih [8], however, since the static limit of the semi-infinite crack solution is zero, it is not possible to estimate the precise magnification of the stresses due to dynamic effects. To overcome this shortcoming, Loeber and Sih [9] proposed to add another characteristic dimension into the problem, namely the crack width, and managed to obtain the exact behavior of the crack-front displacement and stress fields for the case of SH-waves diffracted by a finite or internal crack. Ang and Knopoff [10] have attempted to solve the internal crack problem earlier but their method yields results which are restricted to low frequencies and to distances far away from the crack. In elastodynamics, the farfield crack solution is not useful in the sense that it offers no information to the development of the theories of crack propagation. Generally speaking, the far-field solution can always be determined by the standard method of Wiener-Hopf [7] in a straightforward manner. On the other hand, considerable difficulty is encountered when the WienerHopf method is applied to find the near-field solution. One of the difficulties arises from the factorization of certain functions into functions analytic in the upper and lower half planes. The problem of the diffraction of electromagnetic waves' incident upon a slit has also been treated by Schmeltzer and Lewin [11] using the function-theoretic approach. Their results are left in terms of several complicated integrals the evaluation of which becomes a problem in itself, particularly in seeking the analytical form of the solution in the vicinity of the slit. Having discussed the previous work related to crack problems of SH-waves, it is natural to follow the discussion with a few remarks concerning the diffractions of plane harmonic compression waves (P-waves) and vertically polarized shear waves (SY-waves) by a line crack. Although both Miles [12] and Papadopoulos [13] have investigated crack problems of this type, their work discusses only the qualitative character of the displacement potentials without any explicit information given as to the nature of the local stress distribution. The mathematical description of these problems is somewhat complex because the scattered waves, caused by the line crack, are composed of both compression and shear waves even though the input wave may be of one type, either the Por SVwaves. For this and other reasons, the near-field solution of waves scattered by a crack with finite width is yet to be found. The purpose of this paper, aside from obtaining the stress solution close to the crack point, is to offer a method of solution for solving diffraction problems involving Pand SY-waves incident upon a line of discontinuity. The method can handle different types of boundary conditions2 on the line of discontinuity. For illustration, only the case of a traction-free crack will be considered. An important conclusion is that within certain ranges of wave lengths the dynamic stress distribution around the crack is quite sensitive to changes in the wave number. This is displayed graphically for different values of the irThe scattering of plane-polarized electromagnetic waves by a screen in a fluid medium is mathematically analogous to the SH-wave crack problem in elastodynamics. 2By following the steps outlined in this paper, it is clear that the problems of a rigid and rigidsmooth strip can be solved in the same way. 1969] WAVE PROPAGATION IN AN ELASTIC SOLID 195 Poisson's ratio. The knowledge gained in this investigation is believed to add further impetus to the understanding of the propagation of cracks under fluctuating loads. Field equations and input waves. Consider the propagation of elastic waves, produced by the action of oscillating compressional and shear forces, which vary harmonically in time and are applied in the .xy-plane containing a through crack. In the plane, there arise both compressional and shear waves, and the resulting displacements can be expressed in terms of two scalar functions and *p each of which depend upon x, y, and t. The rectangular components of the displacement vector are ux = d/dx + Sip/dy, uy = d/dy — d\\p/dx, (1) Substituting Eq. (1) into the equations of motion under the conditions of plane strain, the following wave equations on (/> and

d2 2(d~\\p d2\\p\\ d2 \\p .. C\\^? + W2) = ^• c*\\d? + w) = d?\" () In Eq. (2), cl and c2 stand, respectively, for the velocities of compression (irrotational) and shear (equivoluminal) waves in an infinitely extended elastic medium; they are given by c, = [(X + 2m)/p]1/2, c2 = (m/p)1/2 (3) with p being the mass density. As usual, in the case of generalized plane stress the Lame constant X in Eq. (3) is to be replaced by 2\\/i/(X + 2/x), while the shear modulus of elasticity u remains unchanged. From the stress and displacement relations, it is found that (d^ ay \\ \\5x'2 dx dy/ ' vxx = XV + 2fx +2\" @ £i) ■ « = (o _ ?Jk i

The Crush Zone of the Iranian Zagros Mountains, and its implications

Abstract: Structural deformity and stratigraphic discontinuity, basic intrusive rocks contemporaneous with the late Tertiary folding, crush zone related to Zagros orogeny (Miocene)

Instability of Three‐Layer Viscous Stratified Flow

Abstract: Instability of plane Couette flow of three superposed layers of fluids of different viscosity between two horizontal planes is investigated It is found that there are two modes of disturbance dominated, respectively, by the two interfaces The flow is found to be unstable in the zero‐order approximation of wavenumber α for certain values of the depth and viscosity ratios This is owing to a sort of resonance which prevails when a free wave at the lower interface forces a free wave at the upper interface As is known from results previously obtained, the existence of a single surface of viscosity discontinuity will cause instability The presence of an additional surface of discontinuity may or may not be stabilizing, depending upon the values of the depth and viscosity ratios at the additional interface

Crustal structure of Northwestern Ontario: Refraction seismology

Abstract: Deep seismic sounding of the earth's crust has been carried out between latitudes 49°30′ N and 51°30′ N, from longitude 93 °W to longitude 96 °W, by means of a refraction survey, using energy from underwater explosions. A two-layer crust was found, with velocities uniform both laterally and vertically within the layers. Velocities found were: Pg = 6.05 ± 0.05 km/s; Sg = 3.46 ± 0.05 km/s; P* = 6.85 ± 0.05 km/s; S* = 4.00 ± 0.05 km/s; Pn = 7.92 km/s; Sn = 4.60 ± 0.08 km/s. The discontinuity separating the crustal layers (called the Intermediate discontinuity) is believed to be similar to the Conrad discontinuity. Contour maps of depths to this discontinuity and the Mohorovicic discontinuity were produced. Average depths (below surface) are: Intermediate = 18.25 km; Mohorovicic = 34.28 km. Average surface elevation is 0.33 km. Velocity averaged vertically through the crust has a mean value over the area of 6.36 km/s. Structures on the discontinuities are related to at least one major surface geological featu...

Observations on small-scale structural discontinuities in the London Clay and their relationship to regional geology

Abstract: Summary The London Clay is well known as an engineering material and it is probably the most publicized example of a stiff fissured clay. Recently, growing interest has been shown in the small-scale structural discontinuities of this clay and their effects on the engineering behaviour of the material. The paper deals with observations of small discontinuities (or fissures) at sites in the London and Hampshire Basins. Orientation data from the various sites are plotted and analysed in the form of stereographic projections. The size and form of the discontinuity surfaces are considered and their distribution related to orientation and possible modes of origin. Both orientation and surface forms of discontinuities are related to the regional structure of the clay and adjacent sediments, and it is tentatively concluded that the general trends of discontinuity patterns in local areas, the local regional structure, and geological history can be correlated.

Plane strain necking of V-notched and un-notched tensile bars

Abstract: An analytical nonsteady solution for plane strain necking of V-notched, rigid/perfectly-plastic tensile bars was obtained by E.H. Lee in 1952. In his solution the plastic zone remains geometrically similar, a velocity discontinuity occurs on the rigid-plastic boundary, and when carried to the point of separation, the plastic flow results in a wedge-shaped neck. In this paper a new analytical solution is given for the same problem. However, the plastic zone does not remain geometrically similar, velocity discontinuities are absent, and plastic flow results in a convexshaped neck. Comparisons with measurements of neck profiles obtained under conditions approximating plane strain indicate that the new solution is a good representation of the physical situation n metals. Thus, it appears to be useful for estimating basic stress-strain relations from tensile tests after necking, and for estimating the stresses and strains required for fracture from notched bar tests.

The unloading problem for spherical elastic-plastic waves of small amplitude

Abstract: T his paper presents a closed-form solution for a uniform pressure applied instantaneously and smoothly released on the surface of a spherical cavity in an infinite elastic-plastic medium. The analysis is for a linearized theory, including plastic work-hardening, and complements the solution for the starting problem (P. Chadwick and L.W. Morland) when the applied pressure is maintained. The assumed wave-pattern is an attenuating elastic loading discontinuity front followed in turn by an expanding region of continuous elastic loading, an attenuating plastic loading discontinuity front, and a continuous elastic unloading region. After a finite time the plastic discontinuity is totally annulled, leaving purely elastic disturbances. An explicit solution is obtained by prescribing the attenuation of the plastic loading front and determining the required pressure release on the boundary over a short initial time; the subsequent release is prescribed. A variety of cases is investigated to determine the effects of varying the plastic front attenuation, the pressure amplitude, the final pressure release, and the difference between perfectly-plastic and work-hardening materials. Estimates of the maximum distance the plastic discontinuity front travels before annulment are deduced, and shown to be significantly lower than corresponding distances in the starting problem. Conditions for the onset of reverse yield are noted. A typical solution is illustrated.

Theoretical resistivity anomalies across a single vertical discontinuity

Abstract: Electrical resistivity anomalies of a symmetrical four-electrode co-linear system across a single vertical discontinuity are treated in relation to: a) the ratio of potential electrode separation to the current electrode separation that are employed in the system and b) the angle which the electrode alignment makes with the discontinuity. Several conclusions are extracted from this treatment and methods for obtaining an optimum sensitivity of the system, with respect to these parameters, are shown. Disadvantages of special arrangements, such as the Wenner configuration, are indicated. Methods are outlined to utilise variations in the apparent resistivity plot for determining the angle between the electrode alignment and the discontinuity, quantitatively or qualitatively. These variations include certain deviations from the standard curves obtained in longitudinal traverses made at right angles to the discontinuity. Also, a comparison is made between longitudinal and cross traverses, in relation to the discontinuity.

Flow to Wells in the Presence of Radial Discontinuities

Abstract: An analysis is presented for the nonsteady state problem of a well, pumping a constant discharge, and located at the center of a circular aquifer surrounded by a radial discontinuity. On either side of the discontinuity, the transmissibility and the storage coefficient are constant but may have different values from those on the other side of the discontinuity. The solution is obtained by using an approximate inversion formula for Laplace transforms. This method leads to a relatively simple set of equations which commonly requires only short computer time for solution.

On distinguishing between intrinsic and extrinsic faults in field-ion micrographs

Abstract: A simple inspection serves to distinguish between intrinsic and extrinsic faults in field-ion micrographs from f.c.c. and h.c.p. metals. The extrinsic fault gives rise to a double discontinuity in planes due to the insertion of an extra layer which is incorrectly stacked with respect to layers on both sides of the fault. The instrinsic fault gives rise to a single discontinuity. It is also shown that the distinction between the two varieties of intrinsic faults in the case of h.c.p. metals requires more detailed analysis of micrographs.

Water Wave Over a Rectangular Channel Through a Reef

Abstract: A practical method for predicting characteristics of waves propagating over a submarine channel, of rectangular cross section, is presented. The method was based on experimental results and theoretical models. The ripple tank proved to be a useful tool in wave refraction studies. In the case when waves propagate over an abrupt bottom discontinuity normal to their crests, the following were observed: (a) The adjacent wave fronts in both sides of the discontinuity were connected by smooth transitions, (b) two straight lines could be drawn, inclined to the discontinuity to define the transition zones, (c) the angles between the discontinuity and the two straight lines defining the transition zones in the deep and shallow sides were not constant. They vary with the ratio (h\D1\N/L\D1\N)/(h\D2\N/L\D2\N).

On the generation of double Kelvin waves

Abstract: This paper considers the linear response of a homogeneous uniformly rotating ocean of infinite horizontal extent with a discontinuity in depth to a variable horizontal wind stress. It is shown that, for either a transient or time-periodic wind stress which is suddenly applied to an initially calm sea surface, the asymptotic response far from the forcing region is dominated by an outgoing dispersive wave which is trapped along the depth discontinuity, i.e. a double Kelvin wave. Plots of the forced wave patterns in the neighbourhood of the depth discontinuity itself are also presented.

Effect of Mohorovicic topography on the amplitudes of seismic P Waves

Abstract: Recent crustal and upper mantle seismic experiments such as the 1963 Lake Superior experiment and the Project Early Rise experiment showed that the Mohorovicic discontinuity may in general not be a plane layer boundary but rather a very irregular surface. In this report a simple model is presented to illustrate the effect of large-scale undulations along this discontinuity on P-wave amplitude and travel-time curves. The results show that the discontinuity in acting as a lens may (1) focus the seismic rays, in which case small regions of relatively large amplitudes are produced, (2) diverge the rays to produce large areas of weaker amplitudes. For a continental crust, the magnitude of this effect is significant when the angle of incidence of P waves at the base of the crust are relatively large (greater than 40°) and when the radius of curvature of the M discontinuity is less than 100 km. Under certain conditions seismic rays may cross below the surface of the earth to produce triplications of the travel-time graph, in which case multiple arrivals with different apparent velocities, separated from each other by fractions of a second, are to be expected. A comparison of the theory with actual amplitudes observed along the Superior-Churchill line of the Project Early Rise experiment is made.

Transient stress concentration at an elliptical discontinuity

Abstract: : The purpose of the investigation was to determine experimentally the dynamic-stress concentration factors in a strut loaded by high-velocity impact and containing a symmetrically located elliptical discontinuity. The photoelastic technique was used and the conventional light source with its many inherent problems was replaced with a modulated ruby laser. (Author)

The electrical conductivity and internal temperature of the moon.

Abstract: Lunar dynamic response to discontinuities in interplanetary magnetic field to determine electrical conductivity and internal temperature

Development of Shock Waves in Atmospheres with Density and Temperature Variation

Abstract: The problem of the growth or decay of second‐order (acceleration) discontinuities using the theory of singular surfaces is investigated for the particular case of a medium consisting of a perfect gas exhibiting density and temperature fields which vary exponentially. Relations are derived for the amplitude of such waves as well as for the critical times for those cases where the acceleration discontinuities increase without bound, the criterion used for the development of a velocity or shock discontinuity.

Propagation of Shear‐Stress Waves in an Infinite Nonhomogeneous Elastic Medium with Cylindrical Cavity

Abstract: The problem of the radial propagation of shear waves in a nonhomogeneous elastic medium with a cylindrical cavity is solved by the theory of propagating surfaces of discontinuity. Thus, numerical integration that is required in using method of characteristics is avoided. An analytical solution, in the form of a Taylor series expanded about the time of arrival of the wavefront, is obtained. Two numerical examples, which are shown to agree with previous results, are also presented.

Zooplankton and the discontinuity layer in relation to echo traces in the Oslofjord

Abstract: During cruises with R/V trGunnar Knudsenn it was discovered that the echosounder nearly always recordecl echoes froin the depth of the thermocline. In accordance with the appearance of the traces (cf. Fig. 2) the term echo-hands was introduced. The echo-bancls might be caused by reflection from the border layer between two water masses (HASHIMOTO and MANIWA 1956, BANSE 1957, LENZ 1965) or from accunlulated particles in this layer

The paschen discontinuity in B stars

Abstract: Measurements of the Paschen discontinuity in stars withT eff≥104K leads to the conclusion that theD P/DB ratio increases with temperature faster than expected. The increase ofD P/DB with (logg)−1 is also steep.

Penetration of a density discontinuity by a turbulent vortex-pair.

Abstract: : Studies were carried out concerning the penetration of an isolated, rising fluid mass through a density discontinuity, such as may exist in the atmosphere (the tropopause) and in the oceans (the thermoclines). The rising mass was two-dimensional, turbulent, and had the characteristic shape and internal motions of a vortex-pair, or 'thermal'. An analysis of the pair's motion through a density discontinuity, based on the conservation laws for volume, buoyancy and energy, reveals that the maximum rise of the vortex-pair above the density interface is proportional to the square root of its characteristic densometric Froude number. Model experiments, carried out in a stratified, two-layer, saline-water tank, with 'thermals' impinging on the density interface, have clearly confirmed this conclusion. The densometric Froude number is defined in this case as the product of the vortex-pair's density and the square of its velocity, divided by the gravitational acceleration, the density difference of the two fluid layers and the pair's radius, all taken at the point where the pair first impinges on the density discontinuity. (Author)