SPHERICAL WAVE MOTION AND DYNAMIC STRAIN MEASUREMENTS by S* Norman Domenico ProQuest Number: 10795916 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. uest ProQuest 10795916 Published by ProQuest LLC (2018). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code Microform Edition © ProQuest LLC. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106 - 1346 A thesis submitted to the Faculty and Board of Trustees of the Colorado School of Mines in partial fulfillment of the requirements for. the degree of Doctor of Science# Signed: 5~ S» Norman Domenico Golden, Colorado February 1, 1951 Approved J. C. Hollister ACKNOWLEDGMENTS The writer wishes to acknowledge his indebtedness to Professor J. C. Hollister, Head of the Department of Geo­ physics, Colorado School of Mines, whose direct and sincere supervision made this investigation possible and also to Dr. G. T. Merideth for his assistance with many aspects of the theory. The writer further wishes to express his deep appre­ ciation to the Socony-Vacuum Oil Company, whose financial assistance aided materially in the completion of this work. CONTENTS Page INTRODUCTION ........................................ 1 THEORETICAL CONSIDERATIONS .......................... 5 General Wave Equation • • • • • • • • • • • • • • 5 Spherical Wave Equation ...... ............... 8 Minimum Oscillatory Requirement .............. 11 Change of Pulse Shape • • • • ................ .. 1® Energy Density. ........ ........... .. 2? Resumd. 33 EXPERIMENTAL RESULTS ................................ 3k Work Site ................ « 3^ Field Equipment and Procedure........ .. 35 Field Data and Computations............ .. 39 Résumé. ^9 CONCLUSION.......................................... 51 BIBLIOGRAPHY ................................ $k APPENDIX: _ INSTRUMENTATION....................... .............. la General . . . . . . . . Strain Gage . . . . . . . 5& Pre-Amplifier • • • • • • • • • • • • * • • • • • 1 ^ Central Control Unit. . . . . . . . . . . . 18a Tuning Fork ........ . . . . . . . . . . . . . . 23a Camera. . . . . . . . . . . ^5a Oscillographs . . . . . . . . . . . . . . 29a INTRODUCTION For quite some time geophysicists have employed the concepts of seismic wave propagation in many successful attempts to delineate subsurface structural features of the earth's outermost crust. This has been accomplished without specific knowledge of the form of the seismic pulse and of the manner in which this form changes with distance. The important factors in exploration geophysics are the deter­ mination of velocities of the seismic waves through various types of sediments, and the detection at the surface and accurate timing of the reflected or refracted waves. To this end the geophysicist has enjoyed considerable success and at present the seismic method of prospecting is perhaps the most effective known, although the most expensive. The physicist has attempted to keep pace with the very rapid de­ velopments in the field by explaining as best he can the various empirical observations of the field worker and, in many respects, he also has been successful. There are, how­ ever, numerous aspects of seismic wave propagation that are only partially understood and, undoubtedly, many others that remain to be discovered. 2 The problem presented here Is to examine the spherical seismic pulse resulting from an explosion in a small spheri­ cal cavity in an infinite, homogeneous, isotropic, and solid medium. The disturbance from the explosion does not cease when the pressure at the scene of the explosion becomes zero or negligible but continually progresses until all of its energy has been dissipated. We may, .at least theoretically, assign a definite length to the wave and properly term the wave a ntransient.*1 In all probability this is the simplest case of wave propagation, for we have not assigned any sur­ faces of discontinuity or free surfaces td the problem. Furthermore, the source of energy is spherical and the dis­ placement in the medium must be symmetric about this source. To attempt a theoretical solution of the problem is difficult enough, but to verify this solution by sound empirical obser­ vations is a matter of field or laboratory technique which, to this writer’s knowledge, has not been developed to the point that the results are free from serious extraneous fac­ tors. The measuring device, whether it be a geophone or some type of gage, must be placed at a considerable depth below the surface of the medium without appreciably changing the physical properties of the medium. This in itself is not a task easily accomplished. However, before we can fully under­ stand the effect of a free surface or of a discontinuous sur­ face on a seismic pulse, or before we can definitely verify the theoretical solutions concerning the wave motion at such surfaces, it does seem that we must first establish the form 3 of the seismic pulse in the absence of these surfaces. For the above reasons the writer has treated in theory only the case of a spherical seismic pulse in an infinite and homogeneous medium. Since suitable equipment for placing the gages at depth was not available, the experimental work con­ sisted of the measurement of strain at the surface of a rock outcrop and in the neighborhood of a rather shallow dynamite blast. Because the measurements were at the surface, the em­ pirical results are not a verification of the theoretical con­ clusions, but it is believed that the design of the equipment is such that dynamic strain in a solid medium may be measured accurately. The remaining problem is to measure this strain at depth and, when this is accomplished satisfactorily, many avenues of approach to the problem become apparent. For in­ stance, the exact form of the spherical pulse and the amount and type of energy loss may be determined. Also, the total energy and the maximum stress and strain at various distances from the source may be computed. The following dissertation has been divided into three general parts. The first of these is a theoretical discussion of spherical wave motion in which several physical features of a transient seismic pulse in a homogeneous and solid medium are treated. The spherical wave equation is developed from the general wave equation, and the dilatation and displacement functions in a spherical wave are shown to have a minimum os­ cillatory character. Also, it is demonstrated that the form of the displacement wave must alter in a continuous manner as the wave progresses and that the energy density at any point in the wave also must change in a continuous but quite com­ plicated way. The second portion is devoted to the experimental data and results. After the design and construction of suitable instruments, strain records were taken on the surface of a sandstone outcropping. Small charges of dynamite were placed in shallow bore holes, and the undistorted and unfiltered strain waves resulting from a dynamite blast were recorded. Most of the direct spherical wave was obscured by surface waves, but it was possible to observe how the maximum dis­ placement and maximum strain decrease with distance and also how the first compressional pulse in the direct wave changes in form as the wave progresses. The third and last part deals exclusively with the de­ sign and arrangement of the Instruments used in the measure­ ment of dynamic strain. Briefly, the components of the field equipment are (1) resistance-wire strain gages or the "pick­ ups," (2) a two-channel pre-amplifier, (3) two cathode-ray oscillographs, (4) a recording camera, (5) a central control unit, and (6) a 1000-cps tuning fork and drive amplifier. All of this equipment was mounted in a small panel truck and the power was supplied by a motor-generator and various dry cell batteries. The back of the truck was made light-tight and the records were developed immediately after they were taken. It was possible to record the strains occurring at two locations or in two directions at the same location. 5 THEORETICAL CONSIDERATIONS The following is a theoretical discussion of spherical wave motion in an infinite and homogeneous solid medium, , The differential equation of motion is developed from the familiar classical equation of wave motion and it is shown that the spherical wave equation is not as general as might be supposed in that the wave motion must fulfill certain basic require­ ments. It is shown further that the shape of the displace­ ment wave resulting from an explosion in a solid medium must change as the wave progresses. By assuming a particular form for the transient displacement wave, the writer has computed the corresponding dilatation, strain, and energy-density wave. The energy-density equation is developed for the case of spherical wave motion and it is shown how this energy density must vary with distance from the source. General Wave Equation For a medium under stress the relationship between the stress and strain may be expressed in vectorial form as f = >v?-y0 l + 2 /J$ (1) in which $ and $ are the stress and strain dyadics respec-