Displacement Contours Research Articles

Geometric and kinematic coherence and scale effects in normal fault systems

Abstract Fault displacements derived from a seismic reflection survey of an offshore oilfield are projected onto a vertical plane parallel to the fault strike and the displacement values contoured. In addition to those for single faults, displacement diagrams are also constructed for fault arrays by aggregating the displacement values on selected faults. Displacement contours form regular and systematic patterns, even when there is no continuity between the fault surfaces in the array. A system in which linkage between the elements of a fault array is achieved by ductile strain of the intervening rock is referred to as soft-linked and is regarded as the general case. Geometrical coherence, with regular and systematic displacement patterns, exists at all stages in the growth of a fault array, implying a kinematic coherence requiring a high degree of synchronous movement, as opposed to sequential development, on individual elements in the array. Conventional maps and diagrams allow representation of only two orders of magnitude of fault displacement rather than the five or more which may occur. A soft-domino model is presented in which the role of ductile strains is acknowledged both in accommodating varying displacements on fault surfaces and in extension on structures too small to be represented individually, and in which the rotation of rigid fault blocks is of reduced importance when compared with rigid-domino models.

Automatic analysis of speckle photographs with extended range and improved accuracy

A new algorithm for automatic analysis of Young's fringes in speckle photography, applicable to fringe numbers >0.5, is developed. In the algorithm we first divide the diffraction pattern by the corresponding diffraction halo, then integrate the result over bands in various directions for detecting the fringe orientation, and use curve fitting for determining the fringe spacing. The algorithm has enabled fully automatic analysis of double-exposure specklegrams for a wide range of speckle displacement with improved accuracy. Contours of displacement and strain fields have also been plotted using the result of the analysis.

Fault displacement analysis in seismic exploration

In an important series of papers produced by geoscientists from British Coal and Liverpool University, it has been shown that the displacements on fault surfaces vary in a systematic and predictable manner and these variations can be mapped as displacement contour diagrams for any chosen component of the displacement vector (Rippon 1985, Barnett et al. 1987, Walsh and Watterson 1989). Furthermore, Barnett et al. (1987) have shown that the vertical component of displacement, measured in two-way time from seismic sections, when plotted on to a vertical fault projection can impose additional constraints on seismic interpretations.

Cosmetic evaluation of breast conserving treatment for mammary cancer. 1. Proposal of a quantitative scoring system.

In a population of 142 patients with stage I and II breast cancer, treated with tumor excision and external radiotherapy, using a wide range of radiation doses and fractionation schedules, an attempt was made to quantify the cosmetic outcome. Quantitative measurements of nipple displacement and breast contour retraction were compared and correlated with qualitative scoring by a panel. In the vast majority, the quantitative assessments correlate very well with subjective, qualitative scoring, making this method relevant for clinical use. There are a few exceptions, mainly cases where localized skin changes such as severe telangiectasia or skin necrosis affect strongly the cosmetic result but can go undetected in this measuring system. Also limited surgical deformations, which can detract seriously from cosmetic success, particularly when they occur in the medio inferior quadrants, can not always be assessed by this method. The routine use of these measurements in clinical practice is very simple. Only one photograph, taken in standard conditions is needed. Measurements can be carried out quickly, using the plotting device of a treatment planning system. This system may be of great use for follow-up of new treatment modalities and the study of the development of radiation fibrosis in breast cancer.

High resolution optical interference investigation of deformation due to thermal expansion mismatch around plated through holes in multilayer circuit boards

By adhering a thin and therefore flexible, microscope cover slip to the top surface of a multilayer circuit board, localised variations inz axis thermal expansion between the laminate and the copper sleeve on a plated through hole (PTH) have been observed and measured. Moire fringes have been generated from superimposed photographs of the pattern of optical interference fringes formed in the gap between an optical flat and the free surface of the cover slip; adjacent moire fringes are contours of normal displacement differing in magnitude by half a wavelength. Thin plate elasticity theory applied to the cover slip permits the distribution of normal stress to be estimated. Environmental cycling of soldered and unsoldered PTH's reveal that the copper sleeve experiences greater axial stress in the former and is never stress free at any point in the cycle. The maximum tensile stresses are found to be well in excess of the yield strength of annealed OFHC copper. The same optical interference fringe technique has been used to measure thez axis thermal expansion coefficient (a) over the range -40‡ C to 60‡ C. For the glass fibre/polyimide laminate boards studied here, there exists two linear regimes of expansion; α = 38.9 x 10−6 K−1 below and 68.2 x×10−6 K−1 above 29.7° C.

The Fault Analysis Projection System (FAPS) - A new seismic interpretation and structural geological interpretation tool.

Recent research has demonstrated that the displacement on faults varies in a systematic manner, both vertically and laterally. Contours of displacement on strike projections display ordered patterns for isolated normal faults, strike-slip faults and syn-sedimentary faults. Recognition of these ordered patterns forms the basis for a new seismic interpretation and structural/geological interpretation tool, enabling objective verification of fault correlation, identification of bad fault or horizon picks, and a better understanding of the overall kinematic framework. The Fault Analysis Projection System (FAPS) is a computerized implementation of this technique designed to run on UNIX-based workstations. The FAPS uses the X-windows graphics system which permits different elements of the interpretation/analysis to be viewed at the same time and is designed to make adjustments to interpretations a simple task. For example, an interpreter may work on a map in one window, a set of sections in another window, perform a graphics screen-edit on data in a third window and view the results of an analysis in a separate, fourth window.

Diffraction of plate waves at a fixed point

Rectilinear bending waves at a fixed frequency travel along an infinite thin plate in vacuo. One point on the plate is fixed: its displacement is zero, but the plate rotates about the point. Waves are scattered from the fixed point, and contours of the rms displacement near the fixed point are given. Next the fluid-loaded case is computed by evaluating numerically a complicated integral. The energy scattered to the farfield is calculated from the real part of the driving point impedance at various frequencies below coincidence for a steel plate, 1 cm thick, loaded with water on one side. The impedance in vacuo is real and independent of frequency, but, for the fluid-loaded plate, it is complex and frequency dependent. Charts of the pressure contours are given for the nearfield, both on the plate and in the water.

Stepper Motor Control in a Circulatory Model

A hydromechanical circulatory model consisting of a direct drive piston pump, driving mechanism and circulating tubings has been constracted for the aim of helping in cardiovascular studies. This paper describes the pump control mechanism in such a circulatory model. The pump is driven by a microcomputer controlled stepper motor to obtain a desirable pulsatile flow-rate contour in the proximal aorta of the model. Because stepping linearity of the motor is high enough, no feedback is required in the model. Since a direct drive piston pump is used in the model, the pump can be taken as a volume source, and the displacement versus time contour of the piston of the pump defines the shape of the flow-rate contour. Thus, the desired pulsatile flow-rate contour is obtained by controlling the movement of the piston by means of a microcomputer.

A Nonlinear Analysis of an Equilibrium Craze: Part I—Problem Formulation and Solution

This study investigates the effect of nonlinear cohesive forces on crack growth with the special problem of craze mechanics in mind. The work is presented in two parts. In the first and present one, we develop a numerical method for determining the equilibrium shape of a craze in an infinite elastic plane whose fibrils exhibit very general nonlinear force-displacement (P-V) behavior, including strain softening characteristics. The second part of this study deals with the numerical simulation of craze and crack growth (Ungsuwarungsri and Knauss, 1986).1 The problem formulation is based on the superposition of the relevant elasticity Green’s function and the solution for the resulting nonlinear problem is effected by using Picard’s successive approximation scheme. Both field equilibrium and the Barenblatt condition for vanishing stress and strain singularities are satisfied simultaneously, rendering the craze tip profile cusp-like. The formulation allows the stress distribution profile and the corresponding P-V relation to be computed from experimentally measured craze/crack displacement contours; it also allows the computation of the craze or crack/craze profile if the P-V relation, far-field load, and craze or crack size are specified. Numerical investigations indicate that only certain classes of the fibril P-V relations are consistent with measured craze profiles. In addition, it is found that for a given P-V relation, nontrivial solutions exist only for certain ranges of craze lengths.

FATIGUE CRACK TIP STRAINS IN 7075‐T6 ALUMINUM ALLOY

Abstract— The plastic enclave surrounding a fatigue crack in a 7075‐T6 aluminum alloy specimen is studied using moiré interferometry. Two distinct plastic zones are identified from inspection of moiré interferometric fringe patterns. The sizes and shapes of the monotonic and the cyclic plastic zones are correlated with previous theoretical and experimental results. Live load strain distributions in the plastic enclave are computed from high sensitivity (0.417 micron per fringe) full‐field in‐plane displacement contour maps. Comparison with theoretical and experimental strain distributions are included.

Displacement Geometry in the Volume Containing a Single Normal Fault

Fault displacements measured in coal mines and from seismic data are used to develop a model describing the near-field displacements associated with an ideal, single normal fault. Displacement on a fault surface ranges from a maximum at the center of the fault to zero at the edge or tip-line. The tip-line is elliptical, with the shorter axis of the ellipse parallel to the displacement direction. Contours of equal displacement form concentric ellipses centered on the point of maximum displacement. Displacement gradients vary with fault size and with mechanical properties of host rock; fault radius to maximum displacement ratios range from 5 to 500. Plotting of displacement contour diagrams and knowledge of displacement gradients are useful in interpreting seismic reflectio data, both for quality control of interpretations and for quantitative extrapolation of limited data. Displacements associated with faulting decrease systematically with increasing distance along the normal to the fault surface; this decrease is seen as reverse drag in both hanging wall and footwall. Hanging-wall rollover and tilting of the reflectors cannot be used to distinguish listric from planar normal faults; even where fault-block rotation can be demonstrated, neither listric fault geometry nor a flat detachment surface is geometrically necessary. Because faulting is accommodated by ductile deformation, rigid fault-bounded blocks cannot exist except in some special circumstances related to a free surface. The displacements within the rock volume affected by a single fault are not simply related to regional extension. Apparent horizontal extension by faulting varies from one layer to another, and a significant proportion of the extension in a basin may be due to ductile deformation.

Deformation due to a rectangular tension crack in an elastic half-space

AbstractClosed, analytic expressions are given for the displacement fields, their derivatives, and stresses from a rectangular crack in an elastic half-space having Burger's vector normal to its surface. Displacements are found by integration of Volterra's formula over the crack area using Mindlin's (1936) point force solutions for the elastic half-space. Differentiation of the displacement fields yields expressions for calculating stress and strain throughout the medium. Contours of stresses and displacements are given for selected geometries. The expressions for displacements give identical values to those calculated using angular dislocations (Comninou and Dundurs, 1975). All solutions are presented as a FORTRAN subroutine to facilitate practical implementation.

A shadow moire technique for the measurement of damage in composites

The purpose of this paper is to show that a simple form of shadow moire technique can be effective when applied in assessing the extent of damage in and stability of laminated composite materials. Typical shadow moire records of the out-of-plane displacement contours are shown for areas of damage in a carbon fibre/epoxy specimen under applied compressive loads, and for a composite repair patch on a cracked metal coupon under applied tension. The results are obtained using an optical system consisting of off-the-shelf components of low cost. The design of the system was restricted by the emphasis on simplicity and cheapness, nevertheless the performance is well suited to the application. It is hoped that simple out-of-plane field measurement techniques will be more widely used in the future to obtain the data needed to understand the sometimes complex behaviour of laminated composite materials.

Displacement and stress fields due to nonuniform slip along a strike slip fault

A closed-form analytic solution for the displacement and stress field due to a trapezoidal type of nonuniform slip along a strike slip fault is obtained by following Steketee’s method of integration. A set of displacement and stress contours are plotted and compared with the corresponding cases of uniform slip. A possible explanation of the equidistance distribution of strike slip faults and its relation to the amount of slip and the dimensions of the fault are shown.

Diffraction at a fixed point on a plate

Plane waves at a fixed frequency travel along an infinite thin plate, and are scattered from a fixed point on the plate. The point is constrained to have zero displacement, but the plate is free to rotate about the point. This boundary condition allows the expression for the nearfield diffraction pattern to be obtained in closed form: it contains the Bessel functions J0 and Y0 and the modified Bessel function K0. Computations of this expression have been made, giving contours of the rms displacement out to a distance of five wavelengths from the fixed point. The energy scattered to infinity is also considered.

Bimorph piezoelectric flexible mirror: graphical solution and comparison with experiment

The equation of state of a flexible piezoelectric mirror of rectangular cross section has been solved for arbitrary electrode voltages V(x, y) at a point x, y on the mirror. The corresponding displacement W(x, y) has been plotted in three dimensions for four forms of the voltage distribution. These surfaces give the actual shape of the distorted mirror. Contours of equal displacement W(x, y) = constant have also been plotted. It is found that these contours reproduce quite well the interferograms measured by Steinhaus Lipson [ J. Opt. Soc. Am.69, 478 ( 1979)] for comparable forms of the voltage distribution. The dynamic problem for a time-dependent voltage V(x, y; t) has also been solved. At certain frequencies resonant behavior is exhibited.

Plastic yielding on inclined slip-planes at a crack tip

Plastic yield at crack tips on singular slip-planes, inclined to the crack plane, has been studied under plane-strain conditions for combined tension, hydrostatic stress, and in-plane shear. The singular integral equation, which represents the equilibrium condition of edge dislocations on the slip-planes, is transformed into a Fredholm integral equation in order to avoid difficulties that occur with its numerical solution. Results are presented for the slip-band length, the plastic crack-tip opening displacement, stress fields, and crack-opening contours. A series expansion of the results obtained numerically confirms approximate analytical expressions given by J.R. Rice (1974), up to the third-order in the applied stresses. The results of finite element methods agree with values of the crack-tip opening displacement obtained here to within 10 per cent. Ahead of the crack tip, the principal tensile stresses exceed the principal shear stresses by a factor of 10, approximately.

Experimental analysis of the motion of a vibrating surface by speckle interferometry and optical spatial filtering

Optical spatial filtering [J. A. Clark and A.V. Mehta, J. Acoust. Soc. Amer. 54, 291 (1973)] and speckle interferometric techniques are employed to analyze the motion of vibrating surfaces for many cases (e.g., rectangular bar, circular disk, etc. ). Data, which is displayed in real time as contours of equal displacements over the image of vibrating surface, are recorded by time average, real time, and stroboscopic photographic methods using ordinary camera equipment. Calibration procedures are also described along with advantages and limitations of the method. The analysis compares well with the theoretical solutions.

Contours of equal in-plane displacement in holographic interferometry

By using twin apertures in the hologram plane contours of equal in-plane displacement can be obtained from a double exposure hologram. This makes the analysis of three-dimensional deformations of diffusely reflecting objects possible. Some of the possibilities of the method are discussed and a preliminary experiment is described.

Yield under plane-stress conditions

Characteristic jagged marks on moire interference photographs using diffraction gratings show clearly the growth of a yielding region in a mild-steel test specimen under plane-stress conditions. With more accurate measurements, analysis of the displacement contours in the elastic region and extrapolation to the elastic-plastic boundary should provide much needed experimental data for the theory of elasto-plasticity of metals.