EFFEC‘I‘ OF SERESS 0N ULTRASONIC WAVE VELOCITIES‘ IN ROCK SALT Thesis for the Dam of Ph. D. MICHIGAN STATE UNIVERSITY thman M. Abu-Gheida 1964 O~169 Date This is to certify that the thesis entitled Effect of Stress On Ultrasonic Wave Velocities In Rock Salt. presented bg Othman M. Abu-Gheida has been accepted towards fulfillment of the requirements for Ph.D. Mechanics degree in I' - ,fi\ ' "’ , ; . \lvi,’ ._, J ,1; big) 5/: is ((1 .//( Mal V5311. . II roles or Lawrence E? Maivern July 1?, 196h ROOM USE ONLY ROOM USE ONLY. a I U ABSTRACT EFFECT OF STRESS ON ULTRASONIC WAVE VELOCITIES IN ROCK SALT by Othman M. Abu-Gheida The main objective of this investigation was to investigate the possible use of ultrasonic wave techniques for studying stresses in an undisturbed continuous medium. Present stress analysis methods employing strain gages or photoelastic and stress coat techniques cannot be readily used to analyze underground stress conditions. Recent tests by many investigators revealed that the velocity of ultrasonic waves propagating in rocks changed with changes in hydrostatic pressure. No mathematical explanation has been given for this change in rocks. Theoretical expressions for the wave velocities in an initially isotropic solid, subjected to homogeneous stresses similar to underground conditions, were developed in the present investigation. The theoretical development considered the superposition of small strains, due to waves, onto finite strains due to static stresses. The relevant elastic constants which determine the wave velocities were calculated in terms of the static stresses. Lamé's constants and the third-order elastic constants of Othman M. Abu—Gheida the finite theory of elasticity. Seven functions of the three third-order elastic constants were derived. Four of these functions were measured in rock salt. Preliminary investigations on uniaxially stressed salt and steel specimens were conducted to determine the changes that occurred in the velocity and attenuation of longitudinal waves. propagating in the lateral and axial directions. The basic information derived from the preliminary tests was used in designing tests to determine the third~ order elastic constants of rock salt. Longitudinal and shear wave velocities were measured in rock salt specimens subjected to hydrostatic compression. Similar measurements were made along the direction of uniaxial strain in tri— axially stressed’ specimens. The results revealed that the changes in velocity at high pressure are reproducible. This indicated an agreement with the theoretical predictions. Consequently, the third—order elastic constants of rock salt were determined from the changes in velocity at high pressures. It was concluded that the theoretical and experi— mental results of this investigation might have possibilities of being used to study stress conditions in underground salt formations. Other possible geophysical applications are also discussed. EFFECT OF STRESS ON ULTRASONIC WAVE VELOCITIES IN ROCK SALT BY Othman M. Abu-Gheida A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY IN MECHANICS Department of Metallurgy, Mechanics and Materials Science 1964 ACKNOWLEDGEMENTS The author wishes to acknowledge the National Science Foundation for providing financial assistance which made this investigation possible and Dr. S. Serata, thesis advisor and Project Director, for providing supervision and guidance throughout this challenging investigation and for assisting in the preparation of this manuscript. The author is also indebted to the members of his guidance committee for their continuing interest and support. Dr. L. Malvern, professor of Applied Mechanics and major professor. edited and provided invaluable sug- gestions in the theoretical analysis. Dr. T. Triffet. professor of Applied Mechanics, offered encouragement and supervision. Dr. R. H. Wasserman, Professor of Mathematics, for serving as minor professor. Others who were helpful in conducting this investi- gation were Dr. C. Tatro, former MSU professor of Applied Mechanics, who helped in the preliminary experimental investigation, and Dr. A. Chowdiah and Messrs. A. Dahir and S. Sakurai, who offered laboratory help whenever asked. AppreciatiOn is also extended to the author‘s Wife, Betty, for editing the manuscript. ii Chapter I. II. TABLE OF CONTENTS INTRODUCTION LITERATURE REVIEW . . . . . . . . . . . . . 2.1 2.2 2.4 Mechanical Properties of Rock Salt Theory of Stress Waves in Solids 2.2.1 2 2 2 2.2.6 2.2.7 Hooke's Law Wave velocities in an Infinite Linear Elastic Isotropic Medium Rayleigh Surface Waves Longitudinal Waves in Elastic Rods Limitations of the Rod velocity Equation Elastic Waves in Finite Cylinders Determination of Elastic Moduli From wave velocities Ultrasonic Techniques for the Measurement of Elastic Wave velocities and Attenuation in Solids 2.3.1 2.3.2 Ultrasonic Test Components Ultrasonic Test Methods Experimental Results on the Effect of Stresses on Ultrasonic wave velocities and Attenuation in Solids 2.4.1 Single Crystals 2.4.2 Polycrystalline Metals 2.4.3 Rocks 2.4.4 Attenuation iii Page 13 16 19 20 23 23 26 29 29 32 33 35 38 48 Page Chapter III. THE INFLUENCE OF HIGH STATIC STRESSES ON ELASTIC WAVE VELOCITIES . . . . . . . . . . 52 3.1 Previous Work 52 3.1.1 Finite Theory of Elasticity 55 3.1.2 Small Elastic Deformations Superposed on Finite Elastic Deformations 57 3.1.3 Determination of the Third- Order Elastic Constants From Wave velocities 58 3.2 Theoretical Derivation of Plane Wave velocities in an Isotropic Material Subjected to Homogeneous Deformation 60 3.2.1 Strain 61 3.2.2 The Strain Energy and the Relation Between Stress and Strain 65 3.2.3 Hydrostatic Pressure 69 3.2.4 Uniform Triaxial Stress with Uniax1al Strain 74 IV. EXPERIMENTAL INVESTIGATION . . . . . . . 82 4.1 Objectives 83 4.2 Apparatus 84 4.3 Uniaxial Compressive Stress 88 4.4 velocities in Unstressed Specimens 97 4.5 The Effect of Hydrostatic Pressure on velocity 100 4.6 Triaxial Compressive Stress With Uniaxial Strain 110 V. RESULTS AND DISCUSSION . . . . . . . . . 117 5.1 General Remarks 117 5.2 Uniaxial Compressive Stress Tests on Steel 118 5.3 Uniaxial Compressive Stress Tests on Rock Salt 119 5.4 Lamé's Constants 123 5.5 Hydrostatic Compression 128 5.6 Triaxial Stress with Uniaxial Strain 129 5 7 Determination of the Third-Order Elastic Constants of Rock Salt 130 154 5.8 Evaluation iv Chapter Page VI. GEOPHYSICAL APPLICATIONS . . . . . . . . . . 157 VII. CONCLUSIONS.................159 VIII. RECOMMENDATIONS FOR FUTURE STUDY . . . . . . 162 BIBLIOGRAPHY....................164 APPENDICES.....................172 LIST OF TABLES Table Page 2.1 Connecting identities for elastic constants of isotropic bodies (Birch) . . . . . . . . . . . . . . . . . 28 4.1 Reduced data for axial measurements in steel specimens . . . . . . . . . . . . . 96 4.2 Reduced data for the longitudinal wave velocity in the hydrostatic test . . . . 109 5.1 Dynamic elastic moduli of rock salt . . . . . 127 5.2 Calculation of (22 + 4m) for specimen TL2 from first cycle of loading . . . . . . . 134 5.3 Calculated values of (63 + 4m) and (3m - g) from hydrostatic tests . . . . . . . . . 135 5.4 Calculated values of (22 + 4m) and (m) from first cycle of triaxial tests . . . . . . 136 5.5 Third-order elastic constants of rock salt . 140 vi LIST OF FIGURES Figure Page 2.1 Effect of dimension and end friction on maximum stress (Serata) . . . . . . . . 11 2.2 Effect of end friction reducer on stress— strain relation (Serata) . . . . . . . . 11 2.3 Mohr's envelope representing stresses in a triaxial stress state (Serata) . . . 12 2.4 Serata's transition theory . . . . . . . 12 2.5 Relative velocity as a function of the radius—wave length ratio (Tu) . . . . 25 2.6 Regions of observation along the profile of x/L for longitudinal waves propa- gated with velocities VL and V in d SEa . 25 relation at a/L (Silaeva an mina) . 2-7 The pulse—echo method . . . . . . . . . . 34 2.8 Through transmission method . . . . . . . . 34 2.9 Variation of adiabatic elastic constants of KCl and NaCl with pressure (Lazarus) . 37 2.10 Variation of the shear moduli of KCl and NaCl with pressure (Lazarus) . . . . 37 2.11 Load—strain characteristics for the columns (Bergman) . . . . . . . . . . . 42 2.12 Experimental values of Poisson's ratio at different load levels (Bergman) . . . . . 42 2.13 Measured changes in the propagation velocities as functions of applied load (Bergman) . 42 2.14 Schematic diagram of Rollins experiment . . . 43 4.1 Arrangement of specimen and transducers for 1 tests J . . . . . . 90 the axial and latera vii Page Figure 4.2 Transducer attachment for lateral propagation in steel . . . . . . . . . . 91 4.3 Transducer attachment for lateral propagation in rock salt . . . . . . . 91 4.4 Steel disks and transducers for axial propagation . . . . . . . . . . . . . . . 91 4.5 Axial test on rock salt . . . . . . . . . 91 4.6 Testing machine, oscilloscope. camera. pulser and strain recorder used in uniaxial compression tests . . . . . . . . . . 92 4.7 Block diagram of ultrasonic wave apparatus . 93 4.8 Schematic diagram of two successive pulses as they appear on the oscilloscope screen . . . . . . . . . . . . . . . . 93 4.9 Typical traces from the uniaxial compression tests, showing the camera timing light. zero time and signals . . . . . . . . . 99 4.10 Typical longitudinal waves . . . . 99 4.11 Typical shear waves . . . . . . . . 99 4.12 Transducer and specimen coating in the hydrostatic test . . . . . . . . . . . 106 4.13 High pressure vessel and electrical . . . . . . 106 connections . . . . . . . . 4.14 Various components of the hydrostatic test’ . 106 4-15 A block diagram of the comparison method for measuring small changes in velocity in the hydrostatic and triaxial compression tests . . . . . . . . . . 107 4.16 Delay time adjustment in comparison method . 108 4.17 Typical traces from a longitudinal test at 2/5 p sec/scope div. . . . . . . . . 108 4.18 Typical traces from a shear test at 2” sec/scope div. . . . . . . . . . . . . 108 viii Page Figure 4.19 Thick walled steel cylinder . . . . . . . . . 115 4.20 Longitudinal transducer and specimen assembly in triaxial test . . . . . . . . 115 4.21 Shear transducers . . . . . . . . . . . . . . 115 4.22 Assembly for shear wave measurements . . . 115 4.23 Various components of triaxial compression - test . . . . . . . . . . . . . . . . 116 5.1a Stress-strain relationship . . . . . . . . . 120 5.1b Energy attenuation and velocity changes versus axial stress in steel . . . . . . 120 5.2 velocity change of ultrasonic longitudinal waves propagating through five identical 3.5—inch cubic specimens with increase . . . 121 of uniaxial compression . . . . . . 5.3 Energy attenuation of ultrasonic waves propagating through six identical 3.5-inch cubic specimens with increase of uniaxial compression . . . . . . . . . 122 n time of longitudinal and shear 5.4 Propagatio waves in rock salt specimens of various lengths . . . . . . . . . . . . . . . . 126 5.5 Change of longitudinal velocity with . . 143 hydrostatic pressure . . . . . 5-6 Change of shear velocity with hydrostatic pressure . . . . . . . . . . 144 O O O O O I 5.7 Longitudinal velocity versus axial stress in triaxial tests . . . . . . 145 O O C 0 O o axial stress in 5.8 Shear velocity versus . . . . . . . 146 triaxial tests . . . . . . axial stress in 5.9 Lateral stress versus 0 O O O O 147 triaxial tests . . . . . . 5.10 Change of (63 + 4m) with hydrostatic pressure . . . . . . . . . . . . . . . . 148 5.11 Change of (3m — g) with hydrostatic pressure . . . . . . . . . . . 149 ix Figure Page 5.12 Change of (24 + 4m) and (m) in the triaxial test—~first cycle . . . . . . . . . . . . 150 5.13 Change of (22 + 4m) and (m) in the triaxial test--second cycle . . . . . . . . . . 151 5.14 Typical axial strain versus axial stress in triaxial tests . . . . . . . . . . 152 5.15 Determination of the third-order elastic 153 constants of rock salt . . . . . . . . . Appendix I. II. III. LIST OF APPENDICES xi Page 173 175 182 B.. 1J NOTATION Unstressed, unstrained state of an isotropic body. State of finite strain State of finite plus infinitesimal strain Coordinates in state A0 Coordinates in state A' Coordinates in state A Jacobian matrix of the transformation x: ~ xi Jacobian matrix of the transformation x! + xi X . 1 1 Jacobian matrix of the transformation x: * Particle displacementsfrom state A0 to A' Particle displacements from state A' to A Strain, or infinitesimal strain from state A' to A . 0 Strain, or finite strain from state A to A Stress, or total stress in state A xii iJ’ cij ijr kij k1 kcs mcs u in/in Increment of stress from state A' to A Identity square matrix of dimension 3 Second-order elastic moduli Effective elastic moduli Third-order elastic constants Third-order elastic constants velocity: lst subscript refers to type of wave. 3 for shear and L for longitudinal, 2nd subscript refers to direction of particle motion, 3rd subscript refers to direction of propagation. velocity; lst subscript refers to type of wave, 2nd subscript refers to type of stress, h for hydrostatic and t for triaxial stress with uniaxial strain Young's modulus Poisson's ratio Bulk modulus Density Hydrostatic Pressure Linear hydrostatic strain Uhiaxial strain Kilocycles per second ‘ Megacycles per second Micro-inches per inch xiii I. INTRODUCTION The use of ultrasonic pulse techniques to study stress waves in solids advanced after world War II when fast pulsed circuits were developed. During the past decade, extensive literature has been published on the use of ultrasonic propagation methods for studying dynamic elastic moduli, propagation velocity and internal friction in unstressed solids. Most of the investigations were restricted to single crystals and unstressed specimens. The factor of stress was recently introduced in some experiments to study dislocations,27 residual stresses.58 changes in elastic moduli and propagation characteristics in single crystals,39 polycrystalline metals30 and rocks.13 Such studies are still in the early stages of development due to limited available data and absence of well defined theories. To date, characteristic changes in velocity versus hydrostatic pressure in rocks have been interpreted as due to pore closure at low pressure,14 and to undetermined intrinsic changes at high pressures.53 A study of the basic principles involved in wave propagation is required to explain the intrinsic changes in velocity. With the growing use of radioactive materials, there is an increased demand for safe and economic disposal of radioactive wastes. Recently, rock salt cavities have been suggested for this purpose.66 Serata6O concluded that knowledge of the existing underground stressvfield was needed for the structural design of such cavities. The objectives of the present research were to investigate the possibility of using ultrasonic-wave methods to study stress conditions in underground salt formations and to determine the third-order elastic constants of rock salt. The theoretical development included a derivation of wave velocities in an initially isotropic material when subjected to homogeneous deformations due to hydrostatic pressure or to triaxial stress with uniaxial strain. Preliminary tests were conducted on uniaxially stressed steel and salt specimens to determine the sensiti- vity of the ultrasonic pulse method for measuring relative attenuation and absolute velocity at different stress levels. The results of these tests were used to design a more sensitive circuit for measuring the changes in velocity with an accuracy of 0.05%. This circuit was used to measure changes in longitudinal and shear wave velocities in rock salt due to hydrostatic pressure and to triaxial stress with uniaxial strain. The maximum axial and lateral stresses in the triaxial tests were 13,800 and 11,000 psi respectively. The maximum hydrostatic pressure was 9,000 psi. The dynamic elastic moduli and the third—order elastic constants 0f rock salt were determined from the collected data. ‘II II) I... -n: .l. v... u... "u v.‘ v. II. LITERATURE REVIEW 2.1 Mechanical Properties of Rock Salt The ultimate and safe disposal of radioactive waste has become one of the important problems of this age. The feasibility of using underground salt cavities for this purpose has initiated extensive exploratory research on the geophysical, radiological and economic aspects of using rock salt cavities for reactor waste disposal. In 1955 the cumulative results of this research led the Committee on Waste Disposal of the Division of Earth Sciences. National Academy of Science, to endorse the disposal of radioactive waste in salt cavities as a most promising and practical solution.66 In 1959, Serata6O developed design principles for the disposal of reactor fuel waste in underground salt cavities. To arrive at his results, he conducted an experimental and theoretical investigation of the chemical, radiological, thermal and structural factors which affect the salt cavity. Of these factors, he found that the structural Stability of the cavity is of prime importance. Consequently, 61'62 and later Morrison.4S RamaHSl and ChOWdiahzo Serata studied the mechanical properties of rock salt such as strength, Young's modulus, creep behavior, etc. A summary of the results obtained by the above investigators which are related to this investigation will be presented below. For a summary and an account of previous investigations that were done on rock salt the reader is referred to Serata.6O Uniaxial Compression Serata6O analyzed the results obtained from uniaxial compression tests on rock salt. He suggested that the wide variations observed in the shape of the stress strain curve. yield strength, Young's modulus (E). and Poisson's ratio (V) were due to the variation of certain factors in the testing procedure. Of these factors. he analyzed the follow- 1. Use of a friction reducer on the loading surface of the salt. 2. Ratio of height to cross sectional area of the salt. 3. Method of measuring the strain by use of strain gages or dial gages. The results obtained by Serata indicated the following: 1. The friction developed on the loading surface of the salt increased the yield strength through the formation of a triaxial stress zone in the central region of the specimen. This effect became more pronounced when the ratio of cross sectional area to the height was increased. Figs. 2.1 and 2.2. 2. The specimen size effeCL could be eliminated by the use of a friction reducer. 3. The mechanical properties obtained after eliminating the end friction were as follows: (a) Mean maximum stress was 2,300 psi with a standard deviation of 200 psi. (b) Mean value of E was 0.14 million psi with a standard deviation of 0.03 million psi. (c) Poisson's ratio was more than 0.5 for stress beyond 3,000 psi. Chowdiah20 used the friction reduction technique developed by Serata. The results he obtained from the uni- axial compression of 5 inch cubic specimens, indicated the following: 1. The stress strain curve did not exhibit any linearity and as such the conventional methods of calculating E and v could not be adapted. 2. The average value of E as calculated from the slope of a straight line connecting two points on the stress strain curve (chord modulus) was as follows: E E Stress range. (SR—4 Gages) (Dlal Gages) _ 981'. Million psi Million pSi 0 to 1.000 1.408 0.4323 1.000 to 2,000 0.1913 0— 3- The value of v, when dial gages were used. varied linearly from 0 to 0.5 as the stress increased from 0 to 1,500 psi, and increased very slightly for higher stresses. The value of v when SR-4 gages were used was consistently higher than the values obtained from dial gages. The high values of v, 0.5 and above, were explained as due either to errors in the measurement or to brittle fracture between the crystal grains resulting in a volume increase of the specimen. Biaxial Compression Chowdiah20 used strain gages, dial gages and photo- stress technique to determine the behavior of rock salt due to biaxial compression. Five inch cubic specimens were subjected to vertical and lateral stresses of equal or different magnitudes. His results indicated the following: 1. Rock salt in a biaxial state of stress fails by plastic flow. 2. Yielding begins when the octahedral shear stress reaches a value of 1,885 psi. The octahedral shear stress is defined by: 1 To =‘% [01-02)2 + (oz-o3)2 + (o3-ol)2]2 Where: To = dCtahedral shear stress O1'02'03 = principal stresses For condition of equal lateral and vertical stresses 0=G=O O = 0 this equation reduces to: =~L3 o 3 U '1’ Triaxial Compression Handin26 studied the triaxial behavior of cylindrical salt specimens subjected to a confining liquid pressure and compressed in the third direction. His results indicated that rock salt exhibits an increased ductility and plastic behavior with an increase in confining pressure. A specimen, at 1,200 atmosphere confining pressure, was shortened 75% be- fore fracture. Serata61 used the experimental results of Handin to plot Mohr's envelope for rock salt. Mohr's envelope is represented by the line A B C D E, Fig. 2.3. This envelope is tangent to Mohr's circles for a number of triaxial tests in a large range of the mean principal stress. Serata Suggested that the envelope could be considered to be com— posed of three portions: 1. A B corresponds to brittle behavior and can be best described by the Coulomb-Mohr theory of triaxial failure in the form: T = t + 6 o (2.2) c where: T = shear stress on the plane of failure 9 = coefficient of internal friction of the material o ‘d I ~ 4... o = normal stress on the plane of failure T = the constant part of the shear strength which depends on the material. 2. B D represents a transition state. 3. The horizontal line D E represents a plastic behavior which could be explained by the octahedral shear strength theory of failure in the form of Eq. 2.1. 1 _.l 2 2 _ 2 2 = = To - 3 [01-02) .+ (02-03) + (03 01) ] const. ko when 01 = 02 = 03 this condition reduces to J) To - if (02’03) _ ko If the octahedral shearing strength of salt is defined by k0. k ) (2.3) =[2—_ o 3 (02‘03 then the maximum shearing stress corresponding to the line D B will be: O2‘03 = -—3-— k max. _ 2 2J2 0 (2’4) By an extension of Mohr's envelope method, a theory 0f underground stress field was developed by Serata61 and investigated by Serata,61 Morrison45 and Raman.51 This theory predicted that the underground stress field is elastic at small depths and abruptly changes to plastic at a certain depth depending upon the material. This behavior was represented by Mohr's envelope FDE, Fig. 2.3- FD represents the elastic state and DE the plastic state. The theory and the results are given below: The relation between the principal stresses and strains is: - _.1 ex - B [OX V(OY + 02)] e =-; [o -v(o + o )] (2.5) y E y z x e = l-[o —v(o + o )] z E z x y where: compressive stress and strain are considered to be positive. ex'€y = strains in horizontal directions £2 = strain in vertical direction OX'OY = stresses in horizontal directions Oz = stress in vertical direction E = Young's modulus v = Poisson's ratio The underground stress condition is essentially a triaxial stress state with uniaxial strain. The horizontal strains ex.€ are zero since no lateral strain is possible when the medium extends to infinity. The vertical stress is equal to the overburden pressure. The lateral stresses are equal and are developed due to the restriction of lateral deformation. Substituting these conditions in Eq. 2.5 leads to: v L 1-v 0z where: O = O = O 10 This equation represents the relationship between the lateral and vertical pressures at small depths. Eq. (2.6) remains valid for increasing depths until the octahedral shearing stress To reaches the value of the octahedral shearing strength k0. At this point there is a sharp transition from elastic to plastic behavior and the relation between UL and Oz could be derived from the yield condition: 1 1 2 2 2 2 = =._ _ - + - 2.7 To ko 3 [(0x oy) + (CY 02) (oz OX) ] ( ) When 0 = oy = 0L. this equation reduces to: 3 = _. 2.8 0L Oz-i 2 ko ( -.? Using Eqs. 2.6 and 2.8, Serata's transition theory was represented in a plot of C vs. oz.(Fig. 2.4). Line AB L . V represents an elastic state with a slope defined bY l-v° Point B is a transition point. Line BC represents a plastic condition with a slope of unity. Line CD represents elastic unloading and DE plastic unloading. AB is equal to the residual lateral stress after a complete cycle. This transition theory was experimentally investi- gated using different rocks.51 and salt.45 A uniaxial state Of strain was created by applying axial loads to cylindrical rOCk Specimens which were tightly fitted in thick-walled Steel cylinders. Lateral stresses were calculated from the measured strains in SR-4 gages on the outer surface of the steel cylinders. For the tests on rock salt the steel stress (psi) maximum stress (1,000 psi) 11 2300 psi 1 _ (true maximum stress) without friction reducer friction reducer l 2 3 )1? ; ratio of width to height Fig. 2. 1. Effect of dimension and end friction on maximum stress. (Serata) 6000 composition of friction reducer 5000 _. Without ‘\ aluminum [steel I friction \\ £011 «mg-m \ m reas W1 \ .1..:§§—_—_m ‘1 h t 4000 _. 8 es WIT-I \ ~ \ with friction p . . 2000 ._ friction reducer reducer \ standard salt \\ specimen 100° ' 1. 75"xl- 75"xl. 75" \ o ‘ . . . . '. . . . 0 Z 4 6 8 10 12 l4 16 18 ' strain (%) Fig. 2. 2. Effect of end friction reducer on stress-strain relation. (Se rata) shear stress 12 brittle C D plastic E B A 'r max. elastic F I (TL 0' i principal stresses Fig. 2. 3. Mohr's envelope representing stresses in a lateral stress (UL) ['1 triaxial stress state. (Serata) plastic / // ’C - +3—k “13‘s.” 0/ axial stress (0'2) Fig. 2. 4. Serata's transition theory. l3 cylinders had an inside diameter of 3.25 inches, an outside diameter of 4 inches and a height of 3.25 inches. A friction reducer consisting of a plastic sheet coated with a grease— graphite mixture was applied to all sides of the salt to prevent the development of shear stresses. The experimental results obtained from the tests on rock salt showed remarkable agreement with the theory as outlined by Serata. The value of v, obtained from line AB, was 0.17 and k0 1,750 psi. Fig. 5.9 shows a typical plot of 0 vs. oz as obtained by the present author during this L investigation. 2.2. Theory of Stress Waves in Solids 2.2.1. Hooke's Law A generalization of Hooke's linear law may be stated a3: Each of the six components of stress is at any point a linear function of the six components of strain. This may be expressed as: r - ._. - r' -‘ Oxx C11 C12 C13 C14 C15 C16 Exx ny C21 C22 C23 C24 C25 C26 6yy 022 = C3i C32 C33 C34 C35 C36 Ezz (2.9) Oyz C41 C42 C43 C44 C45 C46 yyz Ozx C51 C52 C53 C54 C55 C56 sz LonyL :61 C62 C63 C64 C65 C6: b'YXYJ 14 where: o o o -- r 1 xx' yy' 22 no ma stresses 0 O O —— sheer s r YX’ ZX’ XY t esses C , C , ... C..-~ elastic constants of the material 11 12 13 E . € . e -- normal strains xx yy 22 ny. sz. sz -- are two times the shear strains 40 . . . It may be shown that the matrix Cij lS symmetric i.e., C, , = C 1) ji' For a completely aeolotropic (non symmetri- cal, non isotropic) material, the elastic constants Cij reduce to twenty-one constants. Where the material has axes or planes of symmetry, relations may be established between the elastic constants. For a cubic crystal there are only three independent constants, and the Cij matrix becomes: - _ —y C11 C12 c12 0 0 0 C12 C11 C12 0 O 0 C12 C12 C11 0 0 O 0 0 0 C44 0 0 0 0 0 0 C44 0 _ 0 0 0 0 0 C44 For an isotropic solid. C11 = C12 + 2 C44 C12 = A C = u 44 15 where A and H are called Lamé's constants. The matrix can be written as: R... I. . . . .' A A + 2a A O A A l + 2n 0 0 0 O O O u 0 O O O O O u 0 0 0 0 O O u _ 1 The generalized Hooke's law (Eq. 2.9) for an isotropic solid in the elastic state may now be written as: _ _ _ du 5V 0xx — A A + 2” Exx ny — M WXY _ u (5V'+ 5;) du 5w OYY -— A A + 2p. €YY ze — LL 'YXZ — LJ. (5'2— + 3;) (2.10) 5v 5w = = = + O... I A + 2.. 0y. M 'sz .. (a; 37) where: u, v. w = displacement components in x. y, 2 directions. A = e + s + 5 xx yy 22 =-§E +-§! +-§! = dilatation = change in volume 0X By dz per unit volume Lame's constants can be expressed in terms of E.V. G. and K. where: E = YOung's modulus Poisson's ratio .P _ Applied hydrostatic pressure Bulk modulus = TZT'- Change in volume u V K II G = Shear modulus = 16 The relations are: E ___ L(3?\ + 20.) ‘ A + u V = )\ 2 (x + u) (2.11) K = A + 3H 3 _ _ E u — G - 2 (1 + V) 2.2.2 Wave velocities in an Infinite Linear Elastic Isotropic Medium The equations of motion for an isotropic linear elastic medium are: 52u " doxx Box 50x2 Pg‘Pu=PX+T§;—+ y+—'az— do do 50 2 p6 = pY + —5£§-+ 0y + z (2.12) asz 502 6022 pw=pZ+—5X—+ Y+ z where: X. Y, Z are body forces in x, y, 2 directions p = density Substituting for the stresses in Eq.(2.10L.and neglecting the body forces X. Y. Z, in Eq.(2.12 yields: Pfi = (A + u)-%% + uvzu (2.13s) PV = (A + 0) %$ + uvzv (2.13b) pfi = (i + u) 37f:- + LLVZW (2.13c) 17 These equations may be shown to correspond to the propagation of two types of waves through a medium.36'41 Differentiating (2.13a) with respect to x,(2.l3b) with respect to y and (2.13c) with respect to z, and adding the three equations yields: 2 5 A _ ~ 21‘. 36'41 which shows that the This is a wave equation dilatation A is propagated through the medium with a velocity (Vi) equal to: LLB—E , (2.15) On the other hand. A can be eliminated by differentiating (2.13b) with respect to z, (2.13c) with respect to y and subtracting the results. The resulting equation is: 2 5 0w 0v _ 2 Qw _.§y p597“??? ””7 (.y 52’ or' 52; P 22‘ = )1: (72;, (2.16) atz yz where: W92 is the rotation around the x axis. This is another form of the wave equation and shows that the rotation‘Wyz is propagated with a velocity VS: 18 (2.17) Similar soultions for the rotation about the y and z axes may be obtained. For a plane wave component v, into the form: independent (1 + 2H) with a single particle displacement of x and 2. Eq. (2.14) can be put ———'i u = w = 0 (2.18) It can be verified by direct differentiation and . 41 substitution that this equation describes a wave of the type, V where: (.0 V cos w (t — AL) (2.19) 0 V L angular frequency, which travels with the velocity VL 0f Eq- (2-15) and has a particle displacement v in the same direction as the wave Prepagation y. This type of wave is called a longitudinal Plane wave, and may be designated as: where: the first subscript, L, indicates a longitudinal wave ' the second subscript, y, indicates the direction of particle displacement. and the third subscript, y, indicates the direction of wave propagation. 19 By similar procedure, it may be shown that two other plane waves, VLxx' VLzz’ could be propagated in an isotropic medium with the same velocity Vi. For a plane wave. with a single particle displacement v independent of x and y. Eq. (2.16) can be put into the form: 2 pa—-}-’=u at u .-.-.. w = O (2.20) This equation describes a wave of the type, 2 V — vO cos w (t - Vs) (2.21) which travels with the velocity VS of Eq. (2.17) and has a particle displacement v perpendicular to the direction of propagation z. This type of wave is called a transverse or shear wave, and may be designated as Vsyz' Similar solutions ‘ . V . can be obtained for stz’ sty' Vsyx' Vszx and szy 2.2.3. Rayleigh Surface Waves In an unbounded isotropic elastic solid only longitudinal and shear waves can be propagated. However. when there is a bounding surface, elastic surface waves may also occur. These waves are similar to gravitational surface Waves in liquids and were first investigated by Lord Rayleigh in 1887, who showed that their effect decreases rapidly With depth, and that their velocity of propagation is smaller than that of shear waves. Kolsky36 gave a discussion of these waves and showed that, 20 If Vn R ='V— . s where: Vh = velocity of Rayleigh wave, VS = velocity of shear wave, then R is found to satisfy the equation. R8 - s R4 + (24 - 16 A2) R2 + (16 A2 — 16) = 0 (2.22) where: 1 - 2v A = 2 - 2v v = Poisson's ratio. Equation (2.22)shows that R depends on Poisson's ratio. For V = 7 R = 0.9194; for V = 0051 R = 009554! 9H4 for steel v = 0.29. and R = 0.9258. 2.2.4. Longitudinal Waves in Elastic Rods There are three types of vibrations which occur in rods. They are classified as longitudinal. torsional and 36 flexural. The following discussion will be limited to longitudinal waves. An approximate solution for the longitudinal wave velocity in an elastic rod can be obtained by assuming that plane transverse sections of the bar remain plane during the passage of the stress wave.36 Consider a small section. RS: 0f the bar with a cross-sectional area, A, as shown below: 21 R S o + §§XX 6 XX“ r—'—"XX X X Ex Newton's second law of motion yields: 2u doxx PAOX-SZ— = A'—3;— 5X where: A = area of cross section u = particle displacement or, 2 do 0 u _ xx patz— 5X But, Oxx E: u 5; Therefore. 2 dzu d On _ 512 . 2 gp‘wflEa‘gl-Exz (2'3) This is another form ofthe wave equation; its solution may be written as:36 - - 2.24) u — f (VRt x) + F (VRt + x) ( where: VR = \/% = propagation velocity in rods. (2.25) and F. f are arbitrary functions depending on the initial s t A: 22 conditions; f corresponds to a wave traveling in the direction of increasing x while F corresponds to a wave in the opposite direction. For a wave traveling in the decreasing x direction, u = F (VRt + x). (2.26) Differentiating Eq. (2.26) with respect to t yields, 3E'= VRF' (VRt + x). Here F' denotes differentiation with respect to the argument (th + x). Thus: 0 d du _ Egg 3% = vR—afi But. X — E E du _ du Oxx _ (V;)'dt — pVR'dt o _ V du xx — P R 6‘.“ (2.27) This equation shows that there is a linear relation between the stresses at any point and the particle velocity. When a wave reaches the end of a bar it will be reflected. The nature of the reflected wave depends on the boundary conditions at the end. It may be shown that36 for a free end (no stress) the reflected wave will be oPPosite in sign to the original wave. Thus, a compression PU1Se'will be reflected as tension. Applying the condition that the displacement is zero at the end of the bar, leads to the conclusion that the reflected wave is of the same Sign as the incident pulse. 23 2.2.5. Limitations of the Rod Velocity Equation In deriving Eq. (2.25) it was assumed that plane transverse sections remain plane. However, there is a lateral expansion and contraction in the rod that results in a non-uniform distribution of stress in the sections. This causes the sections to be distorted. This effect was first investigated by Pochhammer50 in 1876. An account of Pochhammer's treatment is given by Love,40 Kolsky36 and Mason.41 An exact solution is given for cylindrical bars whose length is much greater than their diameter. The equation derived is valid for wave lengths which are much longer than the diameter. They lead to the result that harmonic waves cannot be propagated in bars at a velocity larger than VR. 2.2.6. Elastic Waves in Finite Cylinders Kolsky36 pointed out that "exact solutions have not been obtained for vibration of cylinders of finite length." The Pochhammer treatment leads to the result that no energy can be propagated in a bar at a velocity larger than VR. However, Kolsky, Silaeva,64 Tu and his colleagues,70 have stated that a bar could be assumed to be an infinite medium, when the ratio of the bar radius to the wave length is large. This statement is especially important when elastic wave velocities are measured by ultrasonic pulse methods using rock specimens of small dimensions. In such experiments it is desirable, in many cases. to 24 measure Vi directly so that the velocities measured can be compared to velocities observed in nature. The following experimental results may be considered as design criteria for the selection of specimen dimensions and frequencies to be used to measure V in small specimens. L Tu and his colleagues70 measured the velocity of longitudinal pulses, with different carrier frequencies, in metal bars using quartz and barium titanate transducers. Their results are presented in Figure 2.5. vg in the figure indicates the group velocity. This is the velocity with which the energy is transmitted in the bar41 and is equal to the measured pulse velocity.7O They indicated that their results agree with the theoretical calculation of Pochhammer for values of-% less than 0.8, where a is the bar radius and L is the wave length. At large % the group velocity becomes equal to VL- Silaeva and Shamina64 used ultrasonic pulse methods to measure distribution of elastic waves in cylindrical brass rods of different radii. Their results are presented in Figure 2.6. This figure shows the regions in which VL and VR are detected easily, depending upon the relative dimensions of the specimen. The relative dimensions Of the specimen are presented by the ratios of the length (X) (distance between transducers) and radius (a), to the wave length (L). Vg/VL 25 1.0 0.80\\ 0. 6 Vg = group velocity ‘ a = bar radius 0. 4 / L = Iwave length 0. 2 , ‘0 0.8 1.2 1.6 2.0 2.4 2.8 3.2 a/L Fig. 2. 5. Relative velocity as a function of the radius- wave length ratio. (Tu) S I... I 5'." I H I I 8 / I 5 / s I '3 , / a .9. I :1 / .3 n-l an H / no \ o I m o X H I :3 / :4 m / // .-l / / => //// ’ / / 1 l l 1 i I O. 5 l. 0 l. 5 Z. 0 a/L Fig. 2. 6. Regions of observation along the profile of .x/L for longitudinal waves ropagated with velocities VL and VR in relation to a L. (Silaeva and Shamma) 26 Figure 2.6 may be interpreted to state that the velocity measured by ultrasonic methods in rods will be VR or v depending upon the ratio of 2-as well as the length of L L the specimen. For small ratios of-%. the measured velocity will be VL for short specimens, and VR for longer specimens. As the ratio-% increases, the length of the specimen for which vL is observed becomes longer. 2.2.7. Determination of Elatic Moduli from Wave Velocities In general, the elastic moduli (Cij9Eq’ 2.9))1 of a solid are determined by measuring the velocities of longitudinal and shear waves in different directions. The types of waves and directions to be chosen depend upon the number of independent elastic moduli of the material. For an isotropic solid, the elastic moduli A and u are determined by measuring V and V3 in any direction. L The value of VR’ if known, provides an additional check. _ A + 2g = E(1-V) YL _ p . fiV/p (l+V)(1-2V) {—21— i 27 _ 2 u - P vS A _ 2 — p VI - 2a (2.28) 2 ._i- Vs ‘ 2 2 2 2 (VI - vS ) Additional relations are shown in Tabka 2.1. For a cubic crystal the elastic constants C11. C12 and C44 can be calculated from the measurement of longitu- dinal and shear wave velocities in a (110) crystal.* If the velocities are measured in the [110]direction, then:39 ll 12 44 _ 2 2 ‘ P V1 C - C 11 12 2 = 2.29 2 pv2 ( ) 2 C44 ‘ p-V3 where: Vi = velocity of longitudinal wave. V2 = Velocity of shear wave having a particle motion in the [llO] direction. V3 = Velocity of shear wave having a particle motion in the [001] direction. Additional checks are furnished from the measurement of longitudinal and shear wave velocities in a (100) crystal. *The notation (110)or [110]refers to standard Miller indices of crystals. The ( ) means plane. the [ ] refers to direction.35 . 28 Table 2.1. Connecting identies for elastic constants of isotropic bodies (Birch). K = bulk modulus; E -= Young's modulus: E = shear modulus: B = compressibility = l/K; A = Lame's constant7V = Poisson's ratio; p = density: R1 = VL/Vs' 2 2 2_ K L pVR E A v pVL AVS -H 3A+2 ._A___ + . 1+2u/3 -7;fifi 2(i+u) 1 2n K—A A 3K-2A 3(K-A)/2 ° 9K 3K—i 3K—i 9K8. _ ————i£——3K"2 K+4 3 . . 31““ K Zu/3 2(3K+p.) IV E E—ZE _ 4u-E 3K"E §§:§. éfiifi _§§§ o o o 0 3K 9K‘E 6K 3K 9K-E 9K“E 1+V (1+v)(1-2v) A 1:1 A 1—2v A 3v A v ° .' V 2V 2 1+v 2v 2—2v . )‘L 3(1‘2V) 2u(l+V) LL l-ZV LL l-ZV ’ 3K(1'2V) 3K 1+v 3K 1+v 3K 2+2v E Ev 3(1'V) E 3(1-2v) ° (1+v)(l—2v) 1+.)(1-2v) 2+2V 2 (R - 2) 2 4 2 2 l P(V --v ) .. p(v -2V ) ° ' 29 C11 — P v42 (2.30) C44 = 9 V52 where: v4 = velocity of longitudinal wave in [100] direction, < ll velocity of shear wave propagating in [100] direction. 2.3. Ultrasonic Techniques for the Measurement of Elastic Wave velocities and Attenuation in Solids 2.3.1. Ultrasonic Test Components The basic components of an ultrasonic test system 1. Electrical signal generator and indicator, 2. Transmitting and receiving transducers. 3. specimen and couplant. 1. Electrical signal generator and indicator: The function of the signal generator is to provide an electrical signal in the form of a pulse or harmonic wave. This pulse is applied to the transmitting transducer. The electrical indicator is usually the CRO screen Of an oscilloscope. Its function is to display the signal from the receiving transducer. 2. Transmitting and receiving transducers: The transducers used in ultrasonic testing can be best described by the term "high frequency electromechanical .7- ~ . Ali 74: 3O transducers" i.e., elements that have the ability to transform high frequency electrical energy into high frequency mechani- cal energy and vice versa. The transmitting transducer vibrates when an electrical pulse is applied to it. These mechanical vibrationsare transmitted to the specimen through a couplant. When the receiving transducer "feels" the vibra- tions in the specimen it vibrates and produces electrical signals which are displayed on the oscilloscope. A conductive material such as silver paint is needed on the two opposite faces of both transducers, for connection to the electrical system. Transducers are made from piezoelectric materials, such as quartz crystals, or from polarized ferroelectric ceramics such as barium titanate. A description of the Properties of these materials and their electrical and acoustic constants, can be found in any of the standard books 17,13,41 on this subject. For the purpose of this investi— gation, it is sufficient to note the following: Quartz piezoelectric transducers can be cut in various orientations. to generate the type of wave desired. An X-cut is used to generate longitudinal waves; a Y-cut is used to generate shear waves. A ferroelectric ceramic is polarized by applying a large electrical field, usually at a temperature above the Curie Point, and cooling the ceramic with the field applied. After cooling, the ceramic acts like a piezoelectric crystal. 31 The three most common transducers are quartz, barium titanate and lithium sulfate.43 Quartz transducers have the advantage of being stable over a wide temperature range and the disadvantage of being the least efficient transmitters. Barium titanate transducers have the advantage of being the most efficient transmitters and the disadvantages of low Curie point and a tendency to age. Lithium sulfate transducers are the most efficient receivers. However. their use is limited because lithium sulfate is soluble in water. The frequency of a transducer varies inversely, with its thickness. However, the relation is not linear and depends on the type of transducer, X or Y cut, and on the material from which it is made. 3. Specimen and Couplant: The surface of the specimen to which a transducer is applied must be smooth to minimize the loss of acoustic energy due to reflection and refraction at the interface. A thin layer of a couplant is applied between the transducer and the specimen.' The couplant serves as a means by which the acoustical energy is transferred between the transducers and the specimen. For longitudinal waves, a thin film of grease or oil serves this purpose. For shear transducers the couplant should possess the ability to transfer shear waves. Satisfactory results can be achieved by using Phenyl Salicylate or regular sealing wax. These materials 32 are applied to the heated surfaces of the transducer and the specimen to be joined. The transducer and the specimen are then clamped together and allowed to cool. 2.3.2. Ultrasonic Test Methods Several methods have been used to measure the velocities and attenuation of ultrasonic waves in solids. The most favored method is the pulse method in which a short train of high frequency waves are propagated in the specimen. The frequency of the waves is equal to the vibration frequency of the transmitting transducer. When a rectangular voltage pulse is used, the transducer is assumed to vibrate at its natural frequency which depends on its thickness. waever, a transducer can be driven to vibrate over a wide frequency range by using a pulse which consists of a short train of harmonic waves. The efficiency of the transducer is best when driven at a frequency which is equal to its natural frequency. Pulse Echo Method (Figure 2.7) This method uses only one transducer which acts as both a transmitter and receiver. The original pulse is reflected back and forth between two opposite parallel faces of the specimen. Both velocity and attenuation can be measured from the received signal. velocity is determined by measuring the time delay between two echoes. Attenuation is measured from the relative amplitude of successive echoes. 33 Through Transmission (Figure 2.8) In coarse grained materials, such as rock, the echo method cannot be used because of scattering and loss of acoustic energy both in the specimen and on the reflecting surface. In such materials, two transducers are used on the opposite faces of the specimen. The velocity is deter- mined by measuring the delay time in the specimen. Relative attenuation can be determined by comparing the amplitude of the signals received from specimens of different lengths. 2.4. Experimental Results on the Effect of Stresses on Ultrasonic Wave velocities and Attenuation in Solids* During the past two decades, ultrasonic waves have assumed an increasingly important role as an industrial and research tool to study the physical properties of matter. In industry, ultrasonic inspectroscopes and thickness gauges are used to determine the integrity and dimensions of metal castings or other solids.41 In research ultrasonic waves, and their variations with temperature, stress and frequency are used to study the composition of materials. elastic moduli, mechanisms of internal friction, atomic structure, imperfections in a crystal lattice and "even the interaction of lattice vibrations with free electrons and phonons."41 *The term "ultrasonic waves" refers to periodic disturbances in a medium above the audible range, i.e., above 20.000 cycles per second. 34 ape cimen "‘_—_—;| transduce :- ,______.3. a. 'U ‘H attenuation F—L 4 3 \ f :1 \ a. E \ . 2L ‘6 \\ veloc1ty = T [[\\ _ Tl echos L_t .4 Fig. 2. 7. The pulse-echo method. scope '___L___,‘ p—t pulserv- “188 -——-> " 4” transducers velocity = 1;- Fig. 2. 8. Through transmission method. 35 The purpose of this section is to review the use of ultrasonic methods to measure the changes of elastic moduli and attenuation in solids, with special emphasis on rocks. Comprehensive bibliographies and discussion of additional uses can be found in any of the standard books on this subject.18'31'41 2.4.1. Single Crystals Experimental results of the dependence of the elastic constants of single crystals on pressure, are aimed at investigating the atomic structure of crystals. Measured changes in the elastic constants are compared with theoretical changes that can be computed from Fuch's theoretical calcu- lation of the cohesive energy contributions.59'75 For example, the compressibility, B, for a cubic crystal can be expressed as59 E15; =% (C11 + 2 C12) = 1:32 (2-31) where: 5¢ = change in elastic energy per unit volume. -%¥ = relative change in volume. From quantum-mechanical considerations-Ex is V expressed in terms of the change in the radius of a sphere having the same volume as the atomic cell. (¢) is expressed in terms of the cohesive energy per atom of the crystal.59 The compressibility, B, at zero absolute temperature can then be expressed as?9 ... .. u -\~ 36 l_.__l__ 9.23;) B—lZTrr 52 (2°32) where: r = radius of a sphere having the same volume as the atomic cell when €(r) is minimum. €(r) = cohesive energy per atom Lazarus39 made the first ultrasonic measurements of the changes of elastic constants with pressure. He measured the changes in the elastic constants of NaCl, KCl, CuZn. Cu and A1 with pressures up to 10 kilobars.* The elastic constants were calculated from the measured velocities of equations (2.29) and (2.30). The results for KCl and NaCl are presented in Figures 2.9 and 2.10. Lazarus pointed out that the increase in anisotrophy, C44 compared to l . . §.(Cll ' C12). of NaCl and KCl With pressure agrees With 75 predictions which are based on Fuch's calculations Zener's Concerning the effect of short range exchange between closed- shell ions on shear moduli. Daniels and Smith21 reported similar measurements on Cu, Ag and Au. The values found for the pressure deriv- atives of the elastic constants were as follows: Cu Ag Au @ 6.18 6.43 dP 5.59 dC -— . 5 2.31 1.79 61, 2 3 dC' - . 80 0.639 0.438 dP o 5 2 ~ . *1 bar = 106 dynes/cm2 = 1.01972 kg/cm w 14 p31. 37 NaCl C11 (312 Z 4 6 8 10 12 pressure (1, 000 bars) Fig. 2. 9 Variation of the adiabatic elastic constants of KCl and NaCl with pressure. (Lazarus) ‘ as 11-clz)-K01 -;—(c 11 -c 1 Z)NaC1 O 12 Q U l. 1. C44 - NaCl 0. C44- KCI 1 L 1 O 4 8 10 pressure (1 , 000 bars) Fig. 2. 10 Variation of the shear moduli of KCl and NaCl with pressure. (Lazarus) 38 where: B = C11 + 2C12 3 C = C44 C._ C11 ’ C12 2 The main conclusion of the authors was that the conventional assumption that short—range repulsions are considered to act along lines joining nearest neighbor atoms and depend only on the distance between them, does not account for the shear constants and their pressure derivatives in Cu, Ag and Au. 2.4.2. Polycrystalline Metals While extensive literature has been published about the effect of stresses on ultrasonic wave velocities in Single crystals, very little work has been done with poly— crystalline metals. A summary of this work follows: Bergman and Shahbender5 measured the velocity changes Of ultrasonic waves propagating in transverse direction to the applied stress in (4 x 4 x 20 inch) aluminum columns under uniaxial compressive stress. The ultrasonic modes considered were longitudinal waves, shear waves with particle motion along the direction of applied stress, and shear Waves with particle motion transverse to the direction of applied stress. The authors used a through transmission method (Section 2.3.2). X and Y—cut quartz crystals were used. All transducers were driven at 4 mcs. A similar set ' ide a 0f transducers was placed on an unstressed spec1men to PrOV .Au :5 39 delay line and a comparison circuit for more accurate measurement of velocity changes. The results obtained are presented in Figures 2.11-2.13. Figure 2.11 shows the load—strain curve. Figure 2.12 repre- sents the experimental values Ll, Al) A2 For the rocks tested, a peak attenuation was observed at a frequency which corresponded to the resonance frequency Of the largest grains of the rocks. This frequency is defined as: ."i 2L Where: VL = velocity of longitudinal wave L = maximum macroscopic size of the grains. O 1(d M b\ :5 III. THE INFLUENCE OF HIGH STATIC STRESSES 0N ELASTIC WAVE VELOCITIES 3.1. Previous WOrk Elastic wave velocities in a medium are derived from the constitutive equations and the basic principles of conservation of mass and balance of momenta. The wave velocity V can be expressed as: C P C = relevant elastic constant p = density The relevant elastic constant C depends on the type Of wave, direction of propagation, temperature and the material properties of the medium. Stress produces a change in temperature, density, material properties, and structural symmetry. The change in p can be determined from the principle of conservation of mass. p0 d v = pl d Vi density '0 ll d V = unit volume 52 53 To find the changes in C it is necessary to examine the constitutive equations, stress-strain relationships. in which C appears. These equations are derived from energy considerations. For isothermal processes it is assumed that the work done on a medium, by the stresses applied to it, is stored in the form of strain energy and that the density of this energy is a function of the strain components. In the classical theory of elasticity the strains are assumed to be infinitesimal and the strain energy function is expressed up to the 2nd power of the strain components. Each component of stress is then the derivative of the strain energy with respect to the corresponding strain com- ponent. When the strains are measured from a state of zero stress. the strain energy and the stresses are expressed as: ¢=lC.. 6.6. (3'2) 2 ij i j G. = 5¢ = C.. e. (3.3) i 86. ij j i where: ¢ = strain energy/unit volume ilj =1, 2’ 000 6 repetition of a suffic implies summation with respect to that suffix. Oi = six components of stress €i = six components of strain Cij = 2nd order elastic constants Cij are Called 2nd order elastic constants because d . . they are multiplied by 2n powers of strain in the strain energy function (¢)- 54 Equation (3.3) is the well known Hooke's Law. For an isotropic material this equation reduces to Eq. (2.10) with only two independent elastic constants A and u. In this case the elastic wave velocities for an infinite medium are: Vi = l_%_2& (3.4) v8 = '\/—%%- (3.5) where: V: = longitudinal wave velocity VS = shear wave velocity When‘the strains are measured from a stressed state. EqS- (3.2) and (3.3) become: 0 1' = ._i . . , (3.6) ¢ Oi ei + 2 cij eleJ 0. = 09 + c2; 5. (3.7) i i ij j where: O: = initial stresses In this case the Cij depend on the state from which the strain 6 is measured, i.e., they depend on the initial stresses 0:. For an isotropic material under hydrostatic pressure P. the Cij reduces to two indepdent constants l' and M'- Brillouinl6 used the infinitesimal strain theory, Eqs. (3-6) and (3.7), and derived the following expressions for A' and u': )\' A + P 55 The wave velocities Vi and vS are then: Vi 2 -\/H + 2 _—\/%p + 2n - P (3.8) ' = -(/13.'_p=-(/Li_:__P_ V8 p' p' (3-9) density in the stressed state '0 II These results were later rejected by Biot10 and Birch12 because they are contrary to experimental data and lead to the result that at sufficiently high pressures the velocities will be zero. A more accurate derivation of the changes in the relevant elastic constants, C, was derived from the finite theory of elasticity by expanding the strain energy function ¢. Eq. (3.6), to include higher powers of strain. 3.1.1. Finite Theory of Elasticity In the finite theory of elasticity the assumption 0f infinitesimal strains is removed and the strain energy function is expanded to include higher powers of strain. Thus, the energy stored in the body in its stressed state is: 0 =-% c..n. n. + C . n. nj nr + higher order terms (3.10) 0i = finite strain C- are called 3rd order elastic constants because they are multiplied by 3rd powers of strain. 56 The 3rd order constant Cijr form a 36 x 6 matrix. Bhagavantam6 has shown that for a body with no structural symmetry there are only 56 independent Cijr' For cubic cyrstals the number of independent Cijr is 6 and for iso- tropic bodies only 3. The stress is given by: Oi = Cij nj + Cimn nm nn + higher order terms (3.11) Thus, stresses are not linear functions of the strains. This is why the finite theory of elasticity is also called a non-linear theory of elasticity. Another consequence of the finite theory is that the final coordinates of a point are not interchangeable With the original coordinates as is assumed in the infinitesi— mal theory. This is because the strains are large. Thus. in the finite theory, stresses and strains can be expressed in terms of either the initial or the final coordinates. The choice of coordinates wouhfl depend on the type of problem and the given boundary values. The initial coordinates are Called "Lagrangian Coordinates“ and the final coordinates are called "Eurlerian Coordinates." The formulation of the finite theory of elasticity in a completely general tensor notation is given by Murnaghan,46-Eringen,23 Brillouin16 and Green and Zerna. Murnaghan47 also gave the formulation using matrix notation. BiOt9 and Rivlin54'55 discussed some of the fundamental Concepts of the theory using rectangular cartesian coordi- nates. 57 3.1.2. Small Elastic Deformations Superposed on Finite Elastic Deformations Green, Rivlin and Shield24 developed a general theory for small elastic deformations superposed on known finite deformations. Their theory was later specialized by Rivlin and Hayes,56 Truesdall69 and Toupin and Bernstein68 to the case where the small elastic deformations are due to wave motion in a material originally subjected to pure homogeneous deformations. The theory is developed by the above authors in general tensor notation and without assuming any special form for the strain energy function. Their treatment does not explicitly bring out the third—order elastic constants. Bhagavantam8 used Murnaghan's theory of finite strain to derive the "effective elastic constants" of cubic crystals under hydrostatic compression. The "effective elastic constants“ are defined as the constants relating additional stresses to the additional infinitesimal strains. Bhagavantam assumed that: ¢1 = “’2 - 4’3 (3.12) Where: 91 = strain energy of the infinitesimal strains 92 = strain energy of the infinitesimal strains plus the finite strains considered as a single composite finite state of strain 93 = strain energy of the initial finite strain The effective elastic constants appear in @1' The original 2nd and 3rd order elastic constants appear in 58 ¢2 and ¢3. The effective elastic constants are then derived by equating the coefficients of the infinitesimal strains appearing in both sides of the equation. The results are as follows: ' _ C11 ‘ C11 + n (zcll + 2C12 + 6C111 + 4C112) 0 _ _ _ C12 ‘ C12 + n (C123 + 4C112 C11 C12) (3'12) c . _ 144 C44 ‘ C44 + n (C44 + Cl1 + 2C12 + 2 + C155) Birchll obtained similar results for C11 but slightly different expressions for C12 and CA4 by equating stresses instead of energies. He assumed that: l 2 where: T1 = stresses due to infinitesimal strains T2 = stresses due to infinitesimal plus the finite strains P = hydrostatic pressure 3.1.3. Determination of the Third-Order Elastic Constants From Wave velocities Examination of the formulas in (3.12) indicates that certain functions of the third-order elastic constants can be calculated when the effective elastic constants are measured from wave velocities in the stressed state. The second-order elastic constants can be calculated from wave velocities at zero stress. Hydrostatic strain can be measured for any pressure. However, there are only three 59 equations for five unknown third-order elastic constants. Thus, an explicit solution cannot be obtained. Hughes and Kelly:0 were the first to obtain a sufficient number of independent relations to determine the third-order elastic constants from wave velocities for an initially isotropic material. They used Murnaghan's form of the strain energy function, Eq. (3.26), and considered two types of stresses; hydrostatic pressure,P, and uniaxial compressive stress, T. The velocities were given by: Hydrostatic pressure: 2- _—3— 2+ + i+10 p0 VL - l + 2n 3K0 [6 4m 7 u] (3.14) 2 P n = -—*— -—+ 1+6 Uniaxial compressive stress: 2 T 1+9 = —————- ++ 4m+4)\+10) p0 szz A+2u 3K0 [22 A u ( u ] p V 2 = A+2u — -1L-[2£- 2—)5-(m+)\+2u)] (3.15) o Lyy 3KO 8 p0 V 2 = u --—g- [m +All + 4% + 4p] T An Po V 2 = u - [m +-—— + x + 2n} syx 3Ko 4p 2 _ _._E_ - A_i_;E n - 2x] po Vszx - u 3K0 [m 20 where: Ko = compressibility = densit at zero stress Po Y P = hydrostatic pressure 60 T = uniaxial compressive stress along 2 direction E,m,n = 3rd order elastic constants or Murnaghan constants. Hughes and Kelly used ultrasonic pulse methods to measure the velocities of Eqs. (3.14) and (3.15). The values of £,m and n were calculated from the known stresses. These values are presented in the table below. They are the only experimental values of £.m and n that the present author found in the literature. Units 1011 dynes/cm2 Material ‘ "A ‘ ' u fl m n Polystyrene 0.2889 0.1381 —1.89 -1.33 -1.00 .:0.001 (:0.001 (:0.32 .10.29 .10.14 Armco Iron 11.00 8.20 -34.8 -103.0 +110.0 10.04 10.10 :6-5 .17.0 .:110 PYrex 1.353 2.75 1.4 +9.2 +42 .:0.003 .:0.03 (:4.0 ‘15 .135 3.2. Theoretical Derivation of Plane Wave Velocities in an Isotropic Material Subjected to Homogeneous Deformation Expressions for wave velocities in an isotropic. material subjected to hydrostatic compression or triaxial Stress with uniaxial strain, will be developed by the Present author in the following section. The equations mine the derived will be used in Chapters IV and V to deter third-order elastic constants of rock salt. 61 3.2.1. Strain Consider an isotropic body which is initially un— stressed and unstrained. In this initial state the body will be called in state A0. When this body is subjected to known uniform stresses of the type Oil # 032 ¢ 033 # 0 with all shear stresses = O and with no rotation of the body as a whole it will be in state A'. Infinitesimal strains (due to plane waves) carry the body from state A' to the final state A. Throughout the development ea rectangular cartesian coordinate system will be used. The coordinates of a particle in each state are as follows: State A0 : X? (i = l, 2, 3) State A' : x; (i = l, 2. 3) State A : xi (i = l, 2. 3) . o . xi and x. can be expressed in terms of xi as follows. i i xi = x: + U; (xi) 'x5 = x; + U: (x?) x. = x: + v: we where: U: are the displacements due to the stresses 0 o o 011' O22' O33 . 0 I The Jacobian matrix of the transformation dxi » dXi is given by Jl 62 I + f—ax! l Bll O O — l — Jl - 5X0 0 l+B22 O (3.17) J — J 0 0 1+833 _. 4. where: 0 \ O \_ O B BUi = oUZ B _ 0U3 11 5x9 22 5x0 33 5x0 i 2 3 Thus: 0 (3.18) dx' = J dx Eq. (3.18) is a matrix equation between the dif— and dxi. If the additional (1 = 11213): , o ferential elements dxl infinitesimal displacements are denoted bY ui’ then the final coordinates of a particle that was originally o . at xi are given by: , = ' ' I' = l, 2, 3 3.19) x1 xi + ui(xj) l J ( ) ( Here ui is a function of all three coordinates xi. xé and The Jacobian of the transformation dxi -—* dxi is I X3. given by J2. V _ l+b11 b12 b13 = (3.20) J2 b21 l+b22 b23 b31 b32 1+b33 _ where: dul 2 3 =2 .1 . = 1! bij sq l J ' J2 can be written as the sum of a symmetric strain matrix and a skew symmetric rotation matrix as follows: 63 l oui do 1 oul ou -E3+ ‘2‘ (3? + XJ + 5‘s...- x') J 1 J J J2 = E3 + Eij + 1] where: 1 aui du. Eij = 2'(5;§ + X.i) = strain matrix 1 dul Bu. wij — 5-(5;§-— 5;?) = rotation matrix E3 = identity square matrix of dimension 3 The matrix equation for the transformation dx'—*-dx can now be written as: dx = J2 dx' (3.21) but dx' = J1 dxo (Eq. 3.18) Therefore, dx = J2 J1 dxO or 0 dx = J dx (3-22) where: ’ ‘1 (1+bll)(1+Bll) b12(l+B22) bl3(l+B33) = 3.23 J J2Jl= b21(1+Bll) (1+b22)(1+322) b23(l+B33) ( ) + 1+B wb3l(l+Bll) b32(1+322) (1 b33)( 33L The total strain from state A0 to state A will be designated by n. This strain is composed of two parts: In '1 (n 64 the large initial strain, which may be expressed in terms of 311' B22 and B33. and the additional infinitesimal strain ei., which may be expressed in terms of bij' J Throughout the development the initial strains will be treated as finite strains of order less than 0.1. The value of 0.1 represents 10% strain which is much higher than the maximum strain of 2.5% as measured in the tests during this investigation. To obtain a consistent order of accuracy the following rule of approximation will be adopted: In those terms which do not include bii' . nd . the first and 2 powers of B11, B22 and B33 Will be retained. In coefficients of biionly the first ' t ' ed. No powers of B11. B22 and B33 Will be re ain terms involving bij to a higher power than the first will be retained. . . . 47 The total strain n is given by: n =% [J* J - E3] (3'24) where: J* = transpose of J The elements of this strain are developed in Appendix 1 using the approximation rule above. They are: 2 B11 '11 “ b11 + B11 + 2b11 B11 + 2 B22 T'2 = b22 + B22"2b22 B22 +"§“ B332 n3 = b33 + B33 + 2b33 B33 + 2 b + b _ 23 32 n4 - 2 (l + B22 + B33) (3.25) b + b _ 13 31 T‘5 ' 2 (1 + B11 + B33) b + b _ 12 21 1‘6 ‘ 2 (l + B11 + 322) where: n1 = n11 ' T‘2 = ”22 ' n3 = T‘33 n4 = T‘23 = U32 ”5 = T‘13 = T‘31 Tl6 = 1]12 = 1'‘21 or [W] = 3.2.2. The Strain Energy and the Relation Between Stress and Strain Since the total strain n is considered to be finite. the strain energy can be represented by Eq. (3.10): = l— + hi her order terms ¢ 2 Cij ”i ”j + Cijr "i ”j T1r 9 66 where: ¢ = strain energy per unit volume in A0 state. All higher order terms in the eXpression of ¢ will be omitted throughout the development. For an iso- tropic material.¢ can be expressed more conveniently as a function of the strain invariants as follows:47 0 =-A—§—gE 112 - 24 12 + £_§_2m 113 - 2m 1112 + n I3 (3.26) where: I1' 12, 13 are first, second and third invariants of strain as given below A, u = Lame's constants £.m.n = Third—order elastic constants or Murnaghan constants. = (3.27) I1 T‘1 + n2 + n3 n2 T14 n3 T‘5 ”5 I2 = det + det + det Tl4 ”3 _F5 T‘1 n2 (3.29) I3 = det [n] where: det means determinant of the matrix. The total stress 0 in the A state is related to the 47 total strain n by the formula: 67 O = E; J'%% J* (3.30) where: p = density in A state p0 = density in A0 state -9— ~ 1 - I p0 ( 1) 50 In this equation 53-is a symmetric 3 x 3 matrix. The elements of this matrix can be developed using the following basic formulas from Murnaghan: 61 1 = E 53—" 3 612 .56— = I]. E3 _ T] (3.31) 513 Bfi—-= cof n where: cof n means cofactor of the matrix n. 511 512 513 3 . ____ .___ ' - 3 5 metric Each of the an . 50 and Sfi—-ls a X Ym matrix. Combining (3.31) and (3.26) leads to — 2 + f - A I1 E3 + 2n n + (211 — 2mI2) E3 + 2mIl n n co n (3.32) 010/ .3 e The elements of g% are developed in terms of A. H: 3. m. n. bij and Bij in Appendix 2. EqS- (A 19:20)- 68 The stress matrix 0 is then obtained from Eq. (3.30). The procedure for obtaining the stresses is outlined in Appendix 3. The results are: B 2 B 2 O1 — 011 [A (B +'% Bll2 ' "52"" ‘5;- ' 2B22 333’ + 2n (B +-§ B 2 - B B - B B ) 11 2 11 11 22 11 33 + z 82 + 2m(B 2 - B B ) + n(B B )1 11 22 33 22 33 + bll [A (1 + 4 Bil) + 2n (1+4 Bll - 322-333) + 22 B + 4m311] ((3.34) + b22 [A(1-2 B33) + 2n ('Bll) + 22 B + m(-ZB33) + nB33] + b33 [1(1-2 822) + 2n (-Bll) + 22 B + m (-2B22) + nB22] where: B=B +B22+B33 O are written from (3.34) by cyclic 22 = 02 and 033 = 03 Permulation of the numbers 1, 2 and 3. 23 32 - + [24 (1+822 + B33 B11) + (2m 2A)B ‘ “311] + 2” (b32322+b23333) (3.35) 0 of the 3.2.3. or: 4 and 69 05 are obtained from (3.35) by cyclic permutation of numbers 1, 2 and 3. Hydrostatic Pressure In this case B11 B22 = B = - a O _ O _ O _ O11 ‘ O22 ‘ O ‘ The stresses of Eqs. (3.34) and (3.35) become: {-d (3A + 210-02 (g-A + u - 92 _ n)] l + b11 [1 + Zu - a (41 + 4p + 62 + 4m)] _ ‘2 + _ + (b22 + b33) [1 + a (21 + Zn 6 2m n)] 2 3 2 02 = [—a (31 + 20) — a '2 1 + u - 9 - n)] + b22 [1 + 2n — a (41 + 4n + 62 + 4m)] + - 62 +2 - + (b33 + bll)[1 + a (21 24 m n)] 2 3 g 03 = [-d (3A + 2n) — a 5-A+ u — 9 -n)] + b33 [1 + 2u - a (41 + 48 + 62 + 4m)] (3.36) + 2 — 62 + 2m-n)] + (bll + b22)[1 + o (21 u b + b 23 32 . _ = = =——* 2 -Q. 6A+6I~L+6mn)] O4 O23 G32 2 [ u ' b + b 13 31 _ = = = 2 -a (6A+6(J.+6m 11)] 05 O13 031 2 I u b + b 12 21 _ ) = _ — 2 —a (6A + 62 + 6m n 1 06 O12 021 2 ' u 7O ’ ‘E '- " " ‘1"' fl 0 -P A'+2 ' 1' 1' 1 u 0 0 0 bll _ . | - I I | 02 P 1 1 +20 1 0 0 0 b22 03 -P A A A +2u 0 0 O b33 = 1. . (3.37) 04 0 0 0 0 u 0 0 b23+b32 05 0 , 0 0 O 0 u 0 l3+b31 06 0 [ , 0 0 0 0 0 u L 12+b21 __ _. L. _ “F 7_ ...-1 .— where: -P = -a(31 +211.) - 012(3- 1+ LL - 92 - n) u' = u - a (3A+3u+3m --§) (3.38) A' = 1+ a (2A+2u - 6E +2m-n) 1' + 2u' = 1+2u — a (41+4u+62+4m) where: A' and u' are the effective elastic constants for the state A'. The stresses 01 :to C6 are the stresses in the A state in the x, y, 2 directions: therefore they satisfy the equation of motion 50. azu. X. p i P a 2 J t where: Oij = total stress in A state Xi = body force. If the body forces Xi are neglected. the x component Of the equation of motion becomes: 71 0 anx any Oxz dzu This equation can be transformed to the independent coordinates xi of the stressed state A' by the chain rule. a For example, the first term -5§§ becomes: aoxx BO dx' do ' BO 02' XX XX 5x = dx' dx + By' dx + dz' BE’ (3°41) The partial derivatives of X". y”. z' with respect to x.y.z are given by the Jacobian transformations of the type: 5 (f, g, h) det I I ax. = 3 (xi: Y I Z ) 8x. D i where: f = -x + x' + u 9=-y+Y'+V h = -z + z' + w D = det J2 — Bu Bu T -1 7| 2' d det 0 l + 31;: 5§| 5w 8 (f. g. h) 5". l + . 5X. = _ det 54X, 3]., X.) = - _‘=_0—_ y 52—- x D . D 8 8w 5v 5w _ 5v dw .1 “3‘;- +'o'z'- +33;- B‘z" 52' 5?; D 72 Neglecting products of the bi' in the numerator yields: ox' = 1 + b22 + b33 5x D Similarly. 5 (fr 9: h) dy' = _ det d (x', x, z') = _ b21 x D D 5(f. g. h) 52' _ _ det 5(x', y', x) _ _ b31 5x - D — D Thus ooxx _ doxx (1+b22+b33) + ooxx (_ b21) + aoxx (_ b31) 5x _ Bx' D dy' D 52' D '50 do -5§¥'and-§§§ of Eq. (3.40) can be transformed by similar procedure. With the above transformations and the additional relation47 _B_'_ p D Eq. (3.40) becomes: Bo l+b +b do b do b xx 22 33 xx _21_ XX _.__1 W" w‘o “‘6?" D)+—SE'_( 0’ 00 b 50 1+b +b do b32 ltd-“til” 3’3“ 11.1 ””313“? 50 b 60 b 00 l+b +b xz 13 xz 23 X2 11 22 +W1-TH‘6—YT “'33—“an D ' .2163: (3.42) D at2 73 Oxx' Oxy' Oxz are given by Eq. (3.37). They can be written as follows: 0xx = 011 = -P + Sxx Oxy = 06 = 012 = 021 = Sxy (3.43) Oxz = 05 = 013 = 031 = sz where: -P = hydrostatic pressure = 03x S = additional stresses which are related to the ij infinitesimal strains eij by the effective elastic constants A', u'. With the above notation Eq. (3.42) becomes: asxx dSX BSXZ ' 52u ‘ 5X: + —5Y| + 52! = P at2 (3'44) Eq. (3.44) was derived using'gg. = 0, since P is homogeneous . asi. and neglecting the higher order terms 5;:1‘(bij). By similar procedure, the equations of motion for the y and z directions can be transferred to the xi coordinates, and Eq. (3.39) becomes: 3—11 — ' ' i (3.45) x! — p 2 i at This equation is similar to the classical equation (2-12) from which the velocities Vi and Vé were derived. Thus, hydrostatic pressure does not produce any change 74 in the laws of propagation and the velocities are given by: I I Longitudinal velocity = Vi = A—-%7;E- I Shear velocity = VS = '%T Substituting for A', u' from Eq- (3.38) yields: v =\/1+211-a (41.+411+62+4m (3.46) L P u-a (31+3u+3m—52'-) v = . (3.47) s P 3.2.4. Uniform Triaxial Stress withiUniaxial Strain In this case. 311 = B22 = 0 B33 = -e (3.48) 033 = -Oz 0:1 = 022 = ’0L The stresses of Eqs. (3.34) and (3.35) become: ll 22 or, 33 2 A [ eA e (2 + b22 [A+e 2 [-eA -e (g + bll [A+e +b + 11 2 (b b33 [1+2u + b 32 2 [2n + b 31 [2LL + b 21 [2” + C' O 1 L + C' O 1 L + c' O 3 2 b 23 + I C44 b + 0 C44 13 i . 2 (C11 where the CE. lJ [-e (A+2u) + e 75 - 2)] + bll [1 + Zu + 22)] (2A-22+2m-n)] + b [A -2£e] 33 _ 3)] + b22 [1 + 2n + e (Zu - 22)] (2A —2£+2m-n)]+ b A - 23e] 33 [ 2 (g-A + 3n + 2 + 2m)] (3.49) 2) [A + e (2H - 22)] -e (4A+8u + 22 + 4m)] -e ( 2A + 2u + 2m)].- 2 b23 u e —e (2A + 2u + 2m)] - 2 b 13 u 9 (2A - 2n + 2m-n)] + c' b b 13 + C' b 12 1 ll 22 33 I i I + C b22 + Cl3 b b 11 2 11 33 . I 1 b11 + C31 b22 + C33 b33 (3.50) C' b 55 32 C' b 55 31 ' C12) (b12 + b21) are the effective elastic constants. 76 C11 = 1 + Zu + e (Zu - 22) C'12 = A + e (2A — 22 + 2m-n) ' - - C13 - A 22e C51 = A + e (Zu - 22) ' _ (3.51) C33 = 1 + 2n - e (41 + Bu + 22 + 4m) CA4 = u - e (A + Bu + m) C55 = u - e (1 + u + m) .1. I I _ n 2 (Cli'clz) ‘ u ‘ e (A ' u + ”‘2' The initial stresses CL and Oz are given by 2 A —0 = - _ ._ _ L eA e (2 2) (3.52) _ 2 3 -Oz - -e(A + Zu) + e (5-1 + Bu + 2 + 2m) Thus the stresses Si' are related to the infinitesimal strains b.. by: l) = I I I S1 Sxx Cll bll + C12 b22 + C13 b33 S2 Syy C12 b11 + C11 b22 + C13 b33 = = I I S3 Szz C31 (bll + b22) + C33 b33 (3.53) 34 = syz = Szy ‘ C44 b23 + C55 b32 s = s = s I I 5 xz zx C44 b + C b31 13 55 = — i I _ ' + S6 Sxy - Syx 2 (C11 C12Hb12 b21) 77 These are the stress-strain relationships for the material in the A' state. It is interesting to note the differences between these equations and the equations which are derived from the classical theory for an unstressed body which has the same symmetry as A'. In the classical theory, an unstressed body which has the same symmetry as A', i.e., symmetric about the z 40 axis, would have only five independent elastic constants. The stress-strain relationships are given by: —' — - (— 7 _ S1 611 d12 d13 0 0 0 b11 0 b S2 d12 d11 d13 0 0 22 b S3 dl3 dl3 d33 0 O 0 33 84 = o 0 0 d44 0 0 b23+b32 s5 0 0 0 0 d44 0 bl3+b31 1 $6 0 0 0 0 0 -2--(d11 - d123LLf12+b21_ Thus, in the classical theory the elastic constant matrix is symmetric while in (3.53) Ci3 # C51. Furthermore. in the classical theory d44 = d55 and the shear stresses S4 and S5 are related to twice the strains (b23+b32) and whereas in (3.53) CA4 # C55 and the (b13+b31) by 644’ shear stresses S4 and S5 are related to the deformation ' I I . gradients b23, b32, bl3 andb31 by C44 and C55 These differences are due to the presence of initial stresses in A'. Similar relations to Eq. (3.53) ’w‘é .C‘ 78 were obtained by Green24 and Biot.9 Both of them also obtained the following relations between the effective elastic constants: I _ I = I _ I = _ C31 C13 C55 C44 02 0L (3.54) These relations can be obtained from Eqs. (3.51) and (3.52) by approximating oz and 0L to the first powers of e. The equations of motion for the state A are: 501. dzui 'g—l+PX.=P This is the same as Eq. (3.39) and it can be transformed to the independent coordinate of body A' by the same procedure leading to Eq; (3.45). When the body forces are neglected the equations of motion become: aSxx aSx asxz 52“ (3 55 ) = - ___ . a WJ'BTX”? P at2 as as ' as 2 x = z = .§__\_' (3.55b) 31‘- Tifi-l’r—s“? P at2 58 a .QEE (3.55c) s as zx _ z 22 _ . Bx'—‘5y_'¥+—B?"p atz Where: S._ are the stresses given in Eq. (3.53) 1J u, v, w are particle displacements For a plane longitudinal wave propagating in the z direction, the particle motion can be described as: _. . __ I w — f (V'Lzz t z ) 79 (3.55c) (3.66) The velocity Vlzz can be found from Eq. aszx as 6322 a2w 4+ _I____ 52x + aZ'—p52 t Substituting for the stresses from (3.53) and noting that oui bij = 3;? yields: J s - b +C'b _ c' zx 44 13 55 31 SW I C44 g—‘ + CSS ox' _ C. I Szy _ 44 5—3 + C 55 5_' . ou Szz = C31 (§§‘+ '5§') + C33'32' and zx _ z _ 22 = 5x' ‘ 0 5y' “ 0 52' C33 Thus, (3.66) reduces to Cl égj'lz =p|.a_2_w 33 522 dtz This is a wave equation and the velocity is given by Lzz p' p' C33 A + 2p - e (4x + an + 21 + 4mL (3.67) (3.68) By similar procedure the velocities of plane longi— tudinal waves in the x and y directions are: V _1/ C11 _q /A + zu + e (2“ - 23) LXX (3-691 80 For a plane shear wave propagating in the z direction with a particle motion (u) in the x direction, the particle motion can be described as: f (stz t - z) u v = w = 0 The wave velocity stz can be obtained from (3.55a). as as as 2 xx x xz - u __E I—3;T + y' + z' p dt2 I all . 8V . 5W Sxx _ C11 8§'+ C12 5§'+ C13 5;“ BSXX = 0 5X. _.l . . du 5v xy ‘ 2 (C11 C12) (oy' +'5§') BS 3’4 = 0 Y _ I an I aw sz _ 44 52' + CSS 5;“ asxz CI @32 52' 44 a .2 Thus, (3.71) reduces to the wave equation: and (C44 fiVfiI- e (A + Bu +1ml V = "T’ = I sxz p P (3.70) (3.71) (3.72) 81 By similar procedure, the wave velocity of a plane wave propagating in z direction with a single particle motion (v) in y direction is given by Vsyz' = V (3.73) syz sxz Similarly, 1 ~ n - (C' - c' ) u —e (A —u. +m --) v = v = 2 11. i2—= . 2 (3.74) sxy syx P P v _V =j/C'55=W-e0+u+m1 szx — szy p' p' (3.75) IV. EXPERIMENTAL INVESTIGATION The experimental investigation may be divided into two major parts; preliminary and major investigations. In the preliminary investigation, the changes in ultrasonic attenuation and velocity of longitudinal waves were measured in steel and rock salt specimens subjected to uniaxial compressive stress. Measurements made in a direction parallel to the applied load will be designated as "axial" measurements. Measurements made in a direction perpendicular to the applied load will be called "lateral" measurements. In the major investigation the following four velocities were measured: . . ,/ —a47\+4+61+4m v =1/7.+2g = 3+2u. ( Ll Lh ph Pb (4.1) .- u-G(37\+3H+3m-%) V =-(/£L = ‘ 7 (42) Sh ph ph ° - + 8 + 22 + 413) v = x + 211 e (4% P (4.3) Lt Pt V = u-e(x+3u+ml (4.4) st pt where: 82 83 The first subscript of V indicates type of wave: L for longitudinal and s for shear, and the 2nd subscript of V indicates stress condition: h for hydrostatic pressure, t for triaxial stress with uniaxial strain in z direction. Vlt = VLzz of Eq. (3.68), i.e., longitudinal velocity in z direction. VSt = Vsxz = Vsyz of Eqs. (3.72) and (3.73), i.e., shear wave velocities in z direction. Lame's constants for unstressed rock salt. .>" s p ll pt, ph = density at a certain stress level. a = measured linear hydrostatic strain. e = measured strain in the triaxial test. 4.1. Objectives The main objective of the tests on steel was to Gain experience in ultrasonic systems. The main objective of the preliminary investigation of rock salt was to. determine if changes in wave velocity and attenuation could be used to detect anisotropy due to compression or Predict yield before it occurs. The main objectives of the major investigations were: 3, 1. To determine the third—order elastic constants, m and n of rock salt. 84 2. To investigate the feasibility of using ultrasonic pulse methods to determine the initial stresses (hydrostatic pressure or triaxial stress with uniaxial strain cases onlnyrom wave velocities as predicted by the equations above. 3. To determine if the velocity measurements would detect the transition from elastic to plastic states|ofstress in the triaxial test. 4.2. Apparatus The principal components of the apparatus may be divided into six major groups; specimens, loading systems, strain equipment, electronic equipment, transducers and optical equipment. Specimens Rock salt: The rock salt used in all the tests was mined from the International Salt Company mine in Avery Island, Louisiana. Large specimens, about one foot cubes or larger, were cut from the mine floor without the use of explosives. These specimens were cut in the laboratory to the desired dimensions for each test. The rock salt was compact, white in color and 99.9% pure NaCl. The macroscopic grain size of the crystals ranged from 0.4 to 1.2 cm. Steel: Hot rolled, mild steel "C1018" columns. The ends and two sides were polished to final dimensions of 1.97 x 1.98 x 6 inches. 85 Loading Systems Loading machines: An Olson loading machine, with a loading capacity of 300,000 lbs was used in the uniaxial stress tests (Fig. 4.6). A Forney, Jobsite Press tester model FT-ZO, was used in the triaxial tests. This tester is designed for testing specimens in compression. It has a manually operated pump and an accuracy of 500 lbs (Fig. 4.23). Thick walled steel cylinder: A thick-walled stainless steel cylinder, 3.25 inches long with inside and outside diameters of 3.25 and 4 inches, was used in the triaxial tests (Fig. 4.19). This cylinder provided the lateral stresses by restricting the lateral expansion of the salt. High-pressure vessel: The pressure vessel used in the hydrostatic test had a capacity of over 10,000 psi. It consisted of an 11 inch long steel cylinder with a 5 inch inside diameter and 7 inch outside diameter, circular end plates and high strength bolts to fasten the plates. Sealing was accomplished by teflon and oil rings placed between the cylinder and the end plates. Insulated wires were threaded through bolts in the top plate to pro- Vide electrical connection through the vessel (Fig. 4.13). Leakage was prevented by using Hyso cement to fill the holes in the bolts. 86 Pressure pump: A Blackhawk model P—85 hand pump was used to develop the oil pressure in the vessel. Strain Equipment SR—4 strain gages: Post-yield strain gages, type PA—3, were used in the uniaxial stress tests. Two— element rosette gages, type FABX—SO—lz, were used on the outer surface of the thick walled cylinder in the triaxial test. Type A—l gages were used to measure linear strain in salt in the hydrostatic test. Dial gages: Half-inch displacement Ames dial gages, with an accuracy of 0.0001 inch, were used to measure the strain in the salt. Strain indicators: A Sanborn Strain Recorder (Fig. 4.6) and Baldwin Strain Indicator (Fig. 4.23) were used in the uniaxial and triaxial compression tests respectively. Electronic Equipment Oscilloscopes: Tektronic types 541 and 551 with type D or CA plug-in units. Pulsers: A Hewlett-Packard type 212A and a Rutherford pulse generator type B7 were used. These pulsers give rectangular 70 volt pulses. The pulse repetition rate and width are adjustable in both pulsers. Both pulsers are between the trigger-in e(flipped with an adjustable delay time or trigger-out and the main output pulse. 87 Transducers One inch diameter barium titanate disks vibrating in thidkness mode were used to produce and detect longitudinal waves. The disks used for the uniaxial stress tests were 0.5 inch thick with a fundamental frequency of 180 kcs. The disks used for the triaxial and hydrostatic tests were 0.2 inch thick with a fundamental frequency of 500 kcs. Barium titanate shear plates were used for shear velocity measurements. The plates were 0.8 x 0.8 x 0.13 inches and had a fundamental frequency of 350 kcs. The two opposite faces of each transducer were coated with a layer of conductive silver paint or silver epoxy cement. The paint on the face to be attached to the specimen was extended to the edge of the back face. The silver epoxy cement was used to attach thin phonographic wire leads to the transducers. Optical Equipment A Dumont Oscillograph Record Camera, type 321A, was used in the uniaxial stress tests (Fig. 4.6). This camera accommodates a 50 foot roll of 35 mm film. The film could be driven at different speeds. A projector equipped with a mirror that reflects the image to a gridded paper was used to analyze the film. Different enlargements were obtained by changing the distance between the mirror and the paper. 88 A Polaroid Oscillograph Camera was used in the hydrostatic and triaxial tests. The films were analyzed using a two-dimensional measuring microscope which is manufactured by PYE Company of England. This device has an accuracy of‘: 0.01 mm per 20 cm and a magnification of 5 and 25. 4.3. Uniaxial Compressive Stress Changes of attenuation and velocity of longitudinal waves were measured in two directions; parallel to the applied load (axial tests) and perpendicular to the applied load (lateral tests). The same experimental procedure was used for the rock salt and steel tests. Specimen dimensions and transducers were as follows: Steel: 1.97 x 1.96 x 6 inch steel columns Salt: 2.5 and 3.5 inch cubic specimens Transducers: 180 kcs barium titanate disks placed in a holder. Experimental Setup A schematic diagram of the transducer attachment in the axial and lateral tests is shown in Figure 4.1. In the lateral tests two identical transducers were placed on opposite sides of the specimen and held tightly by rubber bands. High—vacuum Dow—Corning grease was used as a couplant. Figures 4.2 and 4.3 show the transducer attachment for steel and rock salt specimens. 89 In the axial tests the transducers were cemented, using plybond rubber cement, to one-inch steel disks placed on both ends of the specimen. Bellow thick—walled steel cylinders were placed around the transducers in order to transmit the load without compressing the transducer (Fig. 4.4). The steel disks provided uniform stress distribution in the sample. The interfaces between the steel disks and the specimen were coated with vacuum grease. Figure 4.5 shows the specimen assembly. Figure 4.6 shows the various test components. A schematic diagram of the electrical circuit is shown in Figure 4.7. The basic principles of this circuit are the same as those of the through transmission test discussed in Section 2.3.2. Figure 4.8 shows a schematic diagram of two successive pulses as they appear on the scope screen when a low sweep rate is used. The pulse is applied to the transmitting transducer at point A, which is the point at which the sweep starts. Point B represents the instant when the wave reaches the receiving transducer. The output of the receiving transducer, after point B, shows a complicated damped oscillatory motion, D, which disappears before point A' (the instant when the second pulse from the pulser is applied to the transmitter). The received signal D consists of many wave components that travel with different velocities in addition to waves that are reflected from the sides of the sample. 90 Load Input T ransduc er and Load Pulse holder 1 l l i Transducer ’ Holder Rosk Dial gages ____ R°°k ____ Shielded Salt —--- Salt “‘- Cable . 5 ‘ Spec1men ‘ 7 peeimen _ Rubber backing Received 1 inch ' I I I—1 4- . Signal steel disk Load Thick walled steel cylinder Load Axial propagation Lateral Propagation Fig. 4. 1 Arrangement of specimen and transducers for the axial and lateral tests. 91 Figure 4. 2. Transducer attachment Figure 4. 3. Transducer attachment for lateral propagation in steel. for lateral propagation in rock salt. Figure 4. 4. Steel disks and transducers Figure 4. 5. Axial test on for axial propagation. I‘OCk salt. Figure 4. 6. Testing machine, oscilloscope, camera, pulser and strain recorder used in uniaxial compression tests. 93 Z Pulser H Rock 5 1 - > Salt 5‘” Specimen 24 1 3 7 8 +_ - [__fiq/ " V - I I I I L__J 1. Pulse Generator 2. Output pulse: Amplitude 70 volts 3 Driving Barium Titanate transducer: 1 inch dia. , §inch thick, 180 kc frequency, silver plated on both faces. 4 Receiving transducer; same as the driving transducer 5. Received signal 6. Tektronix type 541 oscilloscope 7. Dumont oscillograph record camera 8. Sweep trigger signal Fig. 4. 7 Block diagram of ultrasonic wave apparatus DI B .\\ . t~4 Fig. 4. 8 Schematic diagram of two successive pulses as they appear on the oscilloscope screen. 94 An analysis of the complicated pattern D is quite difficult and for all practical purposes all measurements of this kind are confined to the measurement of the time, t, which will be the propagation time of the fastest wave between the two transducers. Relative attenuation was determined from the changes in amplitude of the first received signal (C in Figure 4.8). Procedure -- Sweep speed and amplifier gain were adjusted to give a trace similar to that of Figure 4.8. -- Load was raised slowly to 500 lbs. -- Pictures of the trace were taken with a Polaroid camera to determine the best screen adjustments (illumina- tion, focus, astigmatism and intensity). -— The Dumont camera was attached to the oscilloscope. -- The oscilloscope screen scale was illuminated and pictures of the grid were taken. -- Scale illumination was turned off. -- The scope intensity was adjusted so that the point of zero time (sweep start) would show as a continuous line on the film (Fig. 4.9). -- The oscilloscope single sweep switch was turned on and the film was driven by the camera motor. -- Loading was started and pictures of the trace were taken at different loads by pressing the single sweep buttom of the oscilloscope. 95 -- The camera timing light served as a record to determine the load at which a picture was taken. When this light is on, it leaves a trace on the left side of the film (Fig. 4.9). -- Strain and dial gage readings were reccrded. -- Steel was loaded up to 45,000 psi. Yield occurred at about 30,000 psi. -- Salt was loaded to 4,000 psi which is about 80% of the crushing strength. Data Reduction As mentioned in the apparatus, the film was analyzed by projecting the trace on a gridded paper. The grids on this paper were drawn by projecting a picture of the oscilloscope grid. _ In the axial test, the wave propagates through the steel disks and the salt. Therefore, it was necessary to measure the changes in propagation time and amplitude of the first arrival in the steel disks alone. The results of this test were combined with the results of the axial test (steel disks and salt) in order to determine the propagation time and amplitude of the first arrival for the salt alone. Table 4.1 shows typical reduced data for the axial test on steel specimens. velocity is obtained by diVlding the path length by the propagation time. The propagation . , . . \ time was measured from the straight line portion (Fig. 4.9; 96 ms.a emmo mm.mm roam.m 0w.mm 0 n.0HH ~.0m mn.a mmmm mm.m~ mvem.m 0H.0N o ¢.Hmm n.-a an.a memo mm.mm mmem.m 0m.0m NH 0.mnm N.0ma mm.a memo oh.mm omem.m 00.0w 0N 0.mmm 0.¢ma Hm.a demo ob.mm moem.m 0m.®~ mm h.mmm ¢.mma 00.~ mmmo Nb.mm hmmm.m 00.nm Hm 0.00m m.mma hm.m Nome mm.m~ mmmm.m 0N.>m hm pecans III III II II III momm.m 0m.hm me 0.ma¢ ¢.0ma nh.m comm mm.m~ mmem.m 00.0w me ¢.mHe «.mma m0.m 00mm mm.mm Nnmm.m 00.¢~ Ne h.HHv ¢.0ma em.m momo Nb.mm 0mm.m 00.0H 0e e.HHe H.0ma Hm.m memo mm.mm m0a0.m 00.ma hm N.maw 0.0ma m0.H meme ma.vm H¢m0.m 00.HH mm III m.mma II ,II N¢.¢m IIII III mm h.aH¢ m.0ma 0¢.H eamw 0m.¢m oemm.m 00.H 0m n.mmm m.¢ma m0.H mmam 00.¢m mm00.m, hm.0 mm 0.0mm m.mma m0.0 mwam 00.¢N wmmm.m 0b.0 0m H.0mm m.0ma 00.0 00am 00.0m momm.m mm.0 ma m.mmm m.mma No.0 mmam em.¢m mhmm.m mm.0 0H «.00 m.am 0 mmao hm.¢m mmmm.m hH.0 m 0 m.hm 0 mmam Ha.mm 0000.0 0 0 o o o O . d n 4 muao>HHHE R > u > omm\E omij mesons sfl\cH i Hmm 5 04L... «a .Z o>..> >. > u A oooa oooa ......e. .. en“... e... ”um. CH mmcmsu CH mmsmno . . . . ucmoumm unmoumm .mcmEHome ammum CH mucmEmHDmmmE amaxm How sump Umosbmm .H.v magma 97 with an accuracy of 1%. Percent change in velocity was calculated as: V - V 4!? _._____2 V 96-— V x 100 o where: V = velocity at a certain stress V6 = velocity at zero stress Relative attenuation was calculated from the changes in the amplitude of the first received signal. For steel specimens, percent change in amplitude was calculated as: A - A AA _ o o where: A0 = amplitude in mv at zero load A = amplitude at any load For rock salt, relative attenuation was calculated in decibels as follows: db = 20 loglo -§— 0 The results are presented in Chapter V. 4.4. Velocities in Unstressed Specimens The circuit shown in Fig. 4.7 was used to measure the longitudinal and shear wave velocities of different size specimens. Longitudinal and shear wave veloc1ties were measured using different size specimens. The transducers were attached to the specimens by Phenyl Salicylate. This F o . comPound was heated to its melting pOint (40 C) and a few 98 drops were applied between the transducers and the salt. The bond obtained after cooling was found to be satis— factory for both longitudinal and shear waves. Care was taken to have the two opposite faces of the specimens parallel to each other and to align the shear transducers so that their particle motions (directions of polarization) were in the same direction. Figure 4.10 shows typical signals from the longi- tudinal transducers. The straight line portions of the trace indicate the propagation time in the specimens. Figure 4.11 shows typical signals from the shear transducers. The first small signals on the straight line portion of the trace are early arrivals of some longitudinal components. The arrival of the shear wave is indicated by the first large signal. Figure 5.4 indicates the propagation time in different length specimens. The longitudinal and shear wave velocities are the slopes of the lines in the figure. Lamé's constants were calculated as follows: 2 LL-PVS pV 2 - 2H )‘ L where: p = average density 99 Figure 4. 9. Typical traces from the uniaxial compression tests showing the camera timing lights, zero time and signals. .3" lflllmfllllllllllul ll Ill Illll lllll lllllll lllfllllllillllllllll “WWW Figure 4. 10. Typical Figure 4. 11. Typical longitudinal waves. shear waves. 100 4.5. The Effect of Hydrostatic Pressure 0n velocity The purpose of this test was to measure the longi— tudinal and shear wave velocities given by Eqs. (4.1) and (4.2). V =—\/7\+2u-a(47\+4u+6£+4m) ph Lh ,u — a.(3x + 3n + 3m --§) v h = * s ph In the above equation, the hydrostatic linear strain (a) and the density ph depend on the hydrostatic pressure. Thus, it was necessary to determine d. Then: = a ph Po(l+3) where: p0 = density at zero pressure a = linear hydrostatic strain Strain Measurement The following procedure was used to determine the strain a. -- A thin coat of SR-4 strain gage cement was applied to specimens and allowed to dry. This coat served the Purpose of filling any holes or cracks on the surface of the Specimen. -- The coat was polished with sand paper and SR-4 gages of type A-l were attached. 101 -— The strain gages and salt specimens were coated with a Dow-Corning silicone rubber caulk (commercially known as Dow-Corning Bathtub Caulk) to prevent oil leakage to the strain gages or the specimen. -- After drying, the whole assembly was coated with approximately a 1/4 inch layer of Gulf Microwax 75. Coat- ing was accomplished by dipping the specimen in the melted wax. Figure 4.12 shows the layers of silicone, rubber and wax. -- A similar procedure was followed for coating and attaching a similar strain gage to a steel specimen. This gage was used as a compensating gage. -- The steel and rock salt specimens were placed in the high-pressure vessel. The electrical connections to the strain gages were the same as discussed in the apparatus. -- The gages from the steel and salt were connected to form adjacent arms of a wheatstone bridge. This arrangement provided for temperature and pressure compen- sation. The strain output was the difference between the strain in salt and steel. -- a was calculated by adding recorded strain to the strain in steel which was calculated from the known compres— sibility of steel. -- The results indicated a linear relation between pressure and strain (a). The curve was very close to the theoretical value of a: 102 "0 ll pressure l,u = Lame's constants calculated from Section 4.4. Experimental Setup for velocity Measurements Longitudinal transducers were attached to the speci- mens with a thin coat of high vacuum grease. The transducers and the salt were then covered with coats of Dow-Corning silicone rubber and Gulf Microwax 75.(Fig. 4.12). The electrical leads from the transducers were soldered to the electrical wires on the end plate of the vessel (Fig. 4.13). Shear transducers were cemented to the specimens with regular office sealing wax. This was accomplished by warming the surfaces of the transducers and the salt and applying a thin coat of the melted wax (melting point 120°C) to the two surfaces. The transducers were then clamped to the salt and allowed to cool slowly. It was mentioned in Section 4.3 that the accuracy obtained by using the circuit of Fig. 4.7 to measure the delay time in the specimen was of the order of.i 1%. Better accuracy was needed in the hydrostatic and triaxial compression tests. This was obtained by using a modified comparison method to measure the changes in the delay time due to stress, rather than the absolute time at each stress level. A block diagram of this method is 103 shown in Fig. 4.15. A description of this method as developed by the present author, is included in the following procedure. Procedure —- The l—inch steel disks with the transducers attached (Fig. 4.4) were used to represent the unstressed specimen in the circuit of Fig. 4.15. The delay time in the steel disks was about 9 0 sec. -- The test specimens were placed in the pressure vessel and the circuit was connected as shown in Fig. 4.15. Figure 4.14, shows the various components used in the test. The unstressed specimen assembly is shown to the right of the pressure gage. -- The straight line portions of the two signals were superimposed to appear as a single straight line. -- The oscilloscope sweep speed was increased to l or-% u sec/cm scope division. At these speeds, only a small portion of the straight line appeared. -- The amplitude of the signal from the test specimen was adjusted to coincide with the top horizontal grid line of the oscilloscope screen. This adjustment was accomplished by using the oscilloscope gain. -- Loading was started. At each load level the ampli- tude of the signal from the stressed specimen was adjusted bask to its original level. Figure 4.17 shows three pictures from a longitudinal test. Each picture was taken at several 104 load levels. For example, the top picture was taken at 5 load levels. The traces of the received signal from the test specimen are the thin approximately parallel traces. The thick slanting trace represents the signals from the unstressed Specimen at different load levels. A single curved trace from the unstressed specimen indicates that the zero time (scope trigger), which is not shown in the picture, did not change. Thus, the horizontal distance between any two signals represents a true change in propa- gation time. -- Measurements were taken for two cycles of loading and unloading. Maximum pressure was 9,000 psi. In the unloading of the first cycle and during the second cycle, the changes in propagation time were small as shown in the bottom picture of Fig. 4.17. Thus, measurements were taken at larger load intervals. -- Four specimens were used; two for longitudinal tests and two for shear tests. -- The same procedure was used for the shear and longitudinal measurements. Typical traces from a shear test are shown in Fig. 4.18. This figure shows the early arrival of the longitudinal components. Data Reduction The polaroid pictures were analyzed using a two dimensional traveling microscope. The change in propagation time between any two pressure levels was calculated from 105 the horizontal distance between the two traces. This distance was measured along the center of the scope grid. The accuracy of the time measurements was of the order of .1 0.005 n sec. Table 4.2 shows typical data for a hydrostatic test. The quantities were calculated as follows: -- Hydrostatic strain a was determined from the results described in the hydrostatic strain measurements: _ 3% + 2H where: P = pressure in psi 3A+2u = 9.238 x 106 psi;determined from measurement of Vb and VS at zero pressure. -- The path length at any pressure is given by LC + AL where: L0 = path length at zero pressure AL = - a LO -- At step: represents the change in propagation time between two consecutive loads. -- tO + 2 At: represents the propagation time at any pressure. ~"- 2 At = total change in propagation time. 106 Figure 4. 12. Transducer and Figure 4. 13. High pressure specimen coating in the vessel and electrical hydrostatic test. connections. Figure 4. 14. Various components of the hydrostatic test. 107 main pulser stressed HP 212A ._ / specimen F bar um titanate tra 37ducers variable delay t uns re 3 as J spec1men .— Ruthe rfo rd pul se r scope trigger + 44 I f :£ Fig. 4. 15. A block diagram of the comparison method for measuring small changes in velocity in the hydrostatic and triamal compression tests. 4. l6. Delay time adjustment in comparison method. III-Illit‘ ill IIIIIIIWIIII III-i ”Ill/Ila! -"~- I\ 4. 17. Typical traces from a 4. 18. Typical traces from a longitudinal test at 2/5 shear test at 2 p. sec/ p sec/scope div. scope div. 109 . . .m mm oom . mmNoo+ Nomfi 0mm Md mmN 0+ m¢hv. mmmw www.wm NHH.o+ hams Nos.ms mHH.o+ smsv.~ mos ooom mmov mmo.ms mmo.o+ mmov www.ms Hwo.o+ mosv.m mam ooo mow smm.ma omo.o+ meme mvm.ma mmo.o+ mass m smm ooom H- - - owes mmm.ma No.o+ mmsv.m mms wmwm . . . + News Hom.mH mvo.o+ owes.m osm swms Sm-mH «mm-o - - - - was oooo . ovo.o+ sass Ho¢.ms smo.o+ amps.m mm» ooos mmms mmm-ms - - - - - wow ooom pmoHcs some www.ma «mo.o- wmwv www.ma smo.o- onev.m mum ooom It I: II II II it mhh¢.N vow 0000 mm®¢ Ham.ma 0V0.0I mmov va.ma mh0.0I Hmh¢.N 0mm 0005 II :1 II II II II flmbw.m mvm 0000 mwmw Hmm.ma ¢m0.01 mwmv mmm.MH ®m0.01 0mh¢.N 0¢m 000m II II II vmmv mH0.ma 000.0! mmh¢.N wa 000% mmmd m00.ma 0m0.0l 0mm¢ H05.ma mma.01 N0b¢.N fimm 000m NH®¢ mm©.ma 0HH.0| Hmmv 0mm.mH Hm.0I mmh¢.N QHN 000m whmv mmb.ma mNN.0l Hmfifi mfia.¢a HN.0I hmb¢.N 00H 000a Nomv omm.ma II hmmfl 0mm.va mm.01 005%.N mm 00m - - - mmms mom.ss o ooms.m o- o .Umm\E .omm d .omm 1 omm\E .00m 1 .Umm 1 some :a\:a 1 and huflooam> pd W + on mwum >ufloon> u< N + on mmuw o. dflmwum snowmoum U4 U4 1H<+ 1H .U m mauww pcoomm wHwNU umuwm mam cmEHUme .ummu oaumumouphn mnu CH >UHUOHm> m>m3 HmsflmsuflmCOH on» How sumo nmosomm .N.v manna 110 t - propagation time at zero pressure. This was calculated from the average velocities in unstressed specimens. The actual calculation is given in Section 5.4. VL = 4225 meters/sec. V = 2450 meters/sec. Lo t - vaverage -- The velocity at any pressure is given by V: L + AL . (meter) 0 (inch ) 106 V sec = t0 + At u sec (9 sec)(0.0254)(meter) sec sec The results are presented in Chapter V. 4.6. Triaxial Compressive Stress With Uniaxial Strains The purpose of this test was to measure the longi- tudinal and shear wave velocities of Eqs. 4.3 and 4.4, _-\/l + Zu - e L4k + Bu + 22 + 4m) VLt _ Pt _jvfp - e ix + 3p + m) v _ st pt where: V and V t are the longitudinal and shear wave Lt S velocities in the direction of uniaxial strain e. lll Experimental Setup A plastic sheet was coated on both sides with a grease-graphite mixture. The sheet was wrapped around the solid cylindrical salt Specimens. The diameter of the speci— men was 3.24 inches and the height was about 3.15 inches. The specimen was then tightly fitted into the thick-walled steel cylinder described in Section 4.2. Figure 4.19 shows the thick-walled steel cylinder and the salt specimen. The assembly for the longitudinal wave measurements was the same as for the axial test of Fig. 4.1. Figure 4.20 shows the transducer attached to the one-inch steel disk, the dial gages used to measure the strain in the salt- and the thick—walled cylinder. Shear transducers were embedded in Armstrong NH adhesive cement, type C-4, which filled the center of-— inch thick steel rings (Fig. 4.21). Electrical wires from the transducers were embedded in two thin grooves in the back of the rings (left ring of Fig. 4.21). The front faces of the rings were placed in direct contact with the salt. Loading was applied to the back faces of the rings. This arrangement provided uniform stress on the salt and at the same time direct contact between the transducers and the specimens. Figure 4.22 shows the assembly for a shear test. 112 Procedure Figure 4.23 shows the experimental setup. Load was applied to the specimens by the hand pump of the press tester. The electrical circuit was the same as the circuit used for the hydrostatic tests (Fig. 4.15). The same procedure was used for measuring the change in propagation time. At each load the following readings were taken: (1) picture of trace, (2) dial gage reading, and (3) tangential strain from SR-4 gages attached on the thick-walled cylinden Measurements were taken for two cyCles of loading and unloading. Maximum axial stress was 13,800 psi. Four specimens were used; two for longitudinal waves and two for shear waves. Data Reduction The films were analyzed by the same procedure des- cribed in the hydrostatic test. Uniaxial strain e was calculated from the average reading, AL, of the two dial gages: 8:91. L o where: AL = average change in path length L0 = path length at zero stress. Velocity was calculated by the same procedure as used in the hydrostatic test: 113 L +AL O V=t +At 0 Lateral stress on the salt was calculated from the tangential strain (et ) on the outer surface of the thick- 4 s walled steel cylinder as follows.45 0 = - b2 - a2 o L 2a ts where: CL = lateral stress in the salt. ct = tangential stress on the surface of the s thick-walled cylinder. b = 2 inches, outside radius of steel cylinder. a = 1.625 inches, inside radius of steel cylinder. 0 = E e + v (a + o ) t3 t8 r5 25 where: v = Poisson's ratio of steel. oz , or = axial and radial stress on the outer S‘ s . surface of the steel. or = oz = 0 on the outer surface s s E = 30 x 106 psi. Substituting for a, b and 0t yields: 5 b2 - a2 o = - ———————- E e =-7.72 e L 2a2 t3 t8 114 where: 6t = measured tesile strain in u in/in s o = compressive lateral stress on rock salt, psi. 115 Figure 4. 19. Thick walled Figure 4. 20. Longitudinal steel cylinder. transducer and specimen assembly in triaxial test. Figure 4. 21. Shear transducers. Figure 4. 22. Assembly for shear wave measurements. 116 Java couscoumfiou its?» mo 3:23.530 unamhd> .MN .w ohflmmh [Ill-lull I'll! V. RESULTS AND DISCUSSION 5.1. General Remarks Before discussing the results, it might be desirable to state that the velocities measured in this investigation are assumed to be equal to the wave velocities in an infinite medium. Furthermore, it was assumed that the waves in the hydrostatic and triaxial tests are plane. The assumption that the velocities measured are equal to the velocities in an infinite medium can be verified from the experimental results of Sileava (Fig. 2.6) as follows: The lowest frequency used was 180 kcs. The longitudinal wave velocity in rock salt was about 4225 meters/second. Thus, the maximum wave length was: 422,500 cm/sec 180,000 cycle/sec = 2'35 cm L: - - . a The minimum value of the ratio Ev Where a = minimum radius of specimen, was 3.2 cm 2.35 cm = 1'36° The maximum length of the specimens, x, was 3.5 inches = 8.9 cm. Therefore, the maximum ratio % was: 8.9 _ 2.35 ‘ 3'8 117 118 From Figure 2.6 it can be seen that the velocities measured for these extreme values of E and.% are the wave velocities in infinite media. The plane wave region (sketch, Sect. 2.4.4) in the hydrostatic and triaxial tests extended to 2.3 cm, while specimen length ranged from 2 to 3.1 inches. It was pointed out that this would cause large errors in absolute attenuation measurements. However, attenuation was not measured in these tests. The assumption of plane waves was justified by the sharp rise time in the received signals and by the fact that velocity was measured along the main beam. 5.2. Uniaxial Compressive Stress Tests on Steel The results obtained from these tests are presented in Fig. 5.1. Fig. 5.1a shows the stress-strain relationship for the steel used. Fig. 5.lb shows the results of the axial and lateral measurements. For the axial test, attenuation is most probably due to improvement of the coupling between the transducers, end blocks, and the specimen interfaces. For the lateral test, no improvement of coupling is expected since the holders are fixed in position by rubber bands across them and the change in the lateral dimension is too small to affect the force with which the rubber bands are stretched. 119 In both the axial and the lateral test, the velocity was found to increase with increasing stress with a steeper slop near the yield region. The maximum velocity is reached at the maximum load. The maximum velocities are l.4%-and 2.8% higher than the zero stress velocity for the lateral and axial tests respectively. This is higher than the resolution of the method: therefore, the change is assumed to be the true behavior of steel. During unloading, the velocity and attenuation showed a small change in both tests. 5.3. Uniaxial Compressive Stress Tests on Rock Salt Figures 5.2 and 5.3 show the results obtained from uniaxial compressive stress tests of six identical 3.5 inch cubic rock salt specimens. Similar results were obtained for six identical 2.5 inch cubic specimens. ~Relative attenuation is calculated in decibels as discussed in Section 4.3. Negative values indicate an increase in attenuation and vice versa. The behavior indicated by the results may be explained by Serata's6O observations on the development of fracture or cracks at the lateral surfaces and the development of a triaxially stressed zone at the loading surfaces of the specimen. A schematic diagram of these two zones is 'presented below. voad ‘\ 41" i, I ‘ 4——Triaxial Zone Fracture————— \ \ {[3100 axial stress (1,000 psi) energy attenuation ('1. 120 .0- 30 20 10 ' _l l o 5 10 15 20 25 axial strain (1.000 p in/in) Fig. 5-.~1a Stress-strain relationship. loading unloading 3 0 400 - 300 - - 2.. 0 200 1- ... axial test 1 031); . ‘5 \. . . m L4 1 L l - _1_ L l m n a 800’ 1 '3 L 1. S 8 60 - l. 0 T, > 40 L e 0. 5 lateral test 20 - e 0. 0 o e \ d: 1 1 l —L i 0 '10 20 30 40 45 40 30 2.0 10 axial stress (1,000 psi) Fig. 5. 1b Energy attenuation and velocity changes versus axial stress in steel. velocity change (%) 121 .. 0 1 propagation in axial direction +5 - l I I 0 3 4 5 axial stress (1,000 psi) -5 ,_ propagation in lateral direction -10 __ -15 _ A O O -20 T . G) . (D c O A -25 _ . A Fig. 5. 2 Velocity change of ultrasonic longitidunal waves propagating through five identical 3. 5-inch cubic Specimens with increase of uniaxial compression 122 +15 T? ’) "'- c’r. 0'3" O o 0 +10 propagationin axial direction +5 3 f . « :3 0 | 1 1 5 a 0 h 1 2 3 ' 4 5 .8 .- ‘ V 2' ' axial stress (1, 000 psi) 0 g g -S 0 :3 «I a propagation inlateral :3 direction I: '3 ~10 , A -15 A -20 l 0 C -25 F' 0 Fig. 5. 3 Energy attenuation of Ultrasonic wave propagating through six identical 3. S-inch cubic specimens with increase of uniaxial compression. ' 123 In the triaxially stressed zone, the salt becomes more compact and better transmission is expected as indi— cated by the increase in velocity and decrease in attenuation in the axial measurements. Opposite behavior is observed in the lateral direction due to propagation in a fractured zone. As the load increases, the triaxial zone extends to the center of the specimen. From 1,000 to a 3,000 psi, lateral waves are partially propagating in the triaxial zone; this explains the relatively constant values of attenuation and velocity in this stress range. At higher loads, the specimen starts to fracture and attenu- ation in the lateral direction increases very fast. Attenuation measurements are further complicated by changes in coupling between the transducers and specimens. The effect of this coupling cannot be separated from the true changes in attenuation in the salt. The difference between the velocities measured in the axial and lateral directions.increases with increasing axial stress. The velocity difference reaches 25% at 3,000 pSi (about 75% of the crushing strength). 5.4. Lame's Constants Velocities were measured in different-length speci- mens and in three mutually perpendicular directions in cubic specimens. The purpose of using different—length Specimens was to determine the effect of the couplant on 124 the propagation time. The purpose of the measurements made in three directions was to determine if the assumption of isotropy is valid for rock salt in its unstressed state. Figure 5.4 shows the propagation time for longitudi- nal and shear waves versus specimen length. The straight lines connecting the points pass through the origin indi- cating that the couplant effect is negligible. velocities were calculated from the slopes of the straight lines. longitudinal velocity = 4255 meters/sec i 1% shear velocity = 2450 meters/sec 1.2% The accuracies indicated are determined from the accuracy with which the propagation time (straight line portions, Figs. 4.10, 4.11) was measured. The cluster of six points at the end of the longi- tudinal line (Fig. 5.4) was determined from measurements in 3 perpendicular directions in two approximately S-inch cubic specimens. They indicate that the velocity is the same in all directions and thus the assumption of isotropy is valid. Lame's constants were calculated as follows: _ 2 IJ.--pvS (50].) where: p = average density of all samples = 2.157 gm/cm3 (2.157 gm/cm3) x (245,000 cm/sec)2 1.2947 x 1011 dynes/cm2 3: ll 1: ll 125 _ 2 _ A - p VL 2% A = 1.261 x 1011 dynes/cm2 (5.2) A + 20 = 3.85 x 1011 dynes/cm2 1 dyne/cm2 = 1.45 x 10.5 psi Thus, 6 . u = 1.877 x 10 p51 i = 1.828 x 106 psi i + 20 = 5.58 x 106 spi Poisson's ratio, V, was calculated as: V 1 - 20—842 VL v = = 0.247 Vs 2 2 — 2 (E70 L The dynamic value of YOung's modulus 'was calculated as: = 0 (3K + 2H) 20 (1 + V) k + u D! II 4.68 x 106 psi The bulk modulus, K = (3A + 2H) = 3.08 X 106 psi. uuH The values of VL. Vs, A, 0 obtained from velocity measurements in situ (Section 2.4) are slightly higher than the values obtained by the present author. The value of v compares very closely (Table 5.1). The differences are due to the fact that rock salt in situ is in a triaxially com— pressed state while the specimens used by the author are in an unstressed state. Compression causes an increase in VL- Vs‘ A and u as will be discussed in the following sections. 126 lZ _ 10"— longitudinal E u v 8 .... O I.- o 8 "3 shear a A g e a ..— G o 0 E u .o .s in 4 " G 3 z .- l I I l J 10 ’ 20 30 4o 50 propagation time (in. sec. ) Fig. 5. 4 PrOpagation time of longitudinal and shear waves in rock salt specimens of various lengths 127 Table 5.1. Dynamic elastic moduli of rock salt. Present author In situ VL (meter/sec) 4225 4370 V3 (meter/sec) 2450 2550 V = Poisson's ratio 0.247 0.241 0 = G (106 psi) 1.877 2.05 i (106 psi) 1.828 1.91 E (106 psi) 4.68 5.09 x (106 psi) 3.08 3.28 It is also interesting to note that the values of A and u are very close to each other. This is in line with the quantum mechanical assumption regarding the existence of central forces between the particles of single crystals of NaCl.68a This assumption leads to the conclusions that C44 = C12 in single crystals of NaCl and l — u in isotropic bodies.68a For NaCl:68a C = 1.25 x 1011 dynes/cm2 12 1.26 x 1011 dynes/cm2 C44 22 and C12 C44 128 For the rock salt tested: 11 i 1.261 x 10 dynes/cm2 a C12 of NaCl 0 1.295 x 1011 dynes/cm2 a C44 of NaCl and A m u. 5.5. Hydrostatic Compression Figure 5.5 shows the variation of longitudinal velocity due to hydrostatic pressure in two specimens. Figure 5.6 shows the variation of shear wave velocities. In both figures, velocity increases very fast with an increase of hydrostatic pressure from zero to about 1,000 psi. velocity increases at a progressively slower rate with an increase of hydrostatic pressure from 1,000 to 4,000 psi. velocity shows a slow but steady increase with increasing hydrostatic pressure from 4,000 to 9,000 psi. velocities measured during unloading of the first cycle are slightly higher than the corresponding velocities measured during loading. velocities measured during the second loading and unloading cycles are very close to the velocities measured during unloading of the first cycle. Similar results for different types of rocks were obtained by many investigators and reported in Section 2.4. The characteristic shape of velocity versus pressure curves was interpreted by the hypothesis of pore closure. Birchl4 explained this hypothesis as follows: "Pressure affects the elastic constants of rocks first by reducing porosity and eventually, as pressure is increased, by an 129 intrinsic effect upon the crystalline components." For example, "in igneous rocks, porosity is of the order of one tenth of one percent and under pressure of order of one kilobar solid contact is restored and above this point the pressure effect is close to the intrinsic one." The higher velocities obtained during unloading are interpreted as due to the fact that some pores remain closed while unloading. The present author agrees with the above descriptive hypothesis. The fast increase of velocity at low pressures is not reproducible and is most probably due to difference in porosity between the specimens. 'However, the "intrinsic" increase of velocity at higher pressures is reproducible. Rinehart53 has stated that no satisfactory mathematical explanation has been given for this intrinsic change in rocks. The present author will give a mathematical explan- ation for this effect in the following sections based on the equations derived in Chapter III. 5.6. Triaxial Stress with Uniaxial Strain Figures 5.7 and 5.8 show the variation of longi- tudinal and shear wave velocities, measured in the direction of uniaxial strain, as a function of the axial stress. The shape of these curves is similar to the one obtained in the hydrostatic tests and the same discussion applies here. One of the objectives of this test was to determine if ultrasonic methods would detect transition from elastic 130 to plastic states of stress as discussed in Serata's transi- tion theory (Fig. 2.4, Section 2.1). Figure 5.9 shows a typical experimental plot of lateral stress 0L versus axial stress oz. This figure shows a close agreement with Serata's transition theory. However, no correlation is observed between the transition points of this figure and the velocity curves of Figs. 5.7 and 5.8. The absence of correlation may possibly be explained by Sternglass and Stuart's65 experimental results which indicated that the wave front of longitudinal waves propagating in metal bars, which are prestressed to the plastic region, travels with the elastic wave velocity. 5.7 Determination of the Third-Order Elastic Constants of Rock Salt The following theoretical equations, for the measured longitudinal and shear wave velocities during the first loading cycle in the hydrostatic and triaxial compression tests, were derived in Chapter III and redefined in Chapter Iv. Hydrostatic Compression. =_V/x + 2n - a (4x + 4p + 63 + 4m) (5 3) ph ' \¢/u.- a (3% + 3n + 3m 'g) v = (5.4) sh ph Lh 131 Triaxial Stressf _ ‘ 2 v =W+2g e(4k+@1+2 +4m) (5.5) Lt pt S Pt where: A = 1.261 x 10ll dynes/cm2 u = 1.295 x 1011 dynes/cm2 -a = hydrostatic linear strain -e = uniaxial strain in triaxial tests ph = density at a given strain a pt = density at a given strain e 3 po (1 + 30;) 3 pt p0 (1 + e) p = 2.157 gm/cm3 3. m. n = unknown third-order elastic constants V Lh' V sh’ VLt and VSt are the measured velocities in the first loading cycle of Figures 5.5 through 5.8 respectively. The unknown third—order elastic constants can be calculated from the above equations. For example, the constant function (62 + 4m) can be calculated from Eq. 5.3. 2 th ph = k + 20 — a (4% + 40 + 62 + 4m) 2 )- + 211 - V p (5.7) Lh h (4% + 40) 62 + 4m a 132 Similar procedures can be used to calculate the remaining functions (3m —-§), (22 + 4m) and (m). These four functions of 2, m, and n should be constant for any strain a or e and the corresponding velocity. However, due to pore closure effects and non-reproducibility of velocities at low pressures- the four functions of 2, m, and n were calculated by a dif— ferential approach. Table 5.2 shows a typical calculation for the constant function (22 + 4m) by the following procedure: 2 = — + + 2 + 4 th pt 1 + 2n el (41 Bu 2 m) l 1 2 = - 4x + 8 + 22 + 4m) th pt A + Zu e2 ( u 2 2 Thus:. 2 2 VLtl Ptl ' VLtz pt2 22 + 4m = — (4i + 80) (5.8) e - e 1 where: el = strain at a stress 01 e2 = strain at a stress 02 > 01 e > e . e is positive 2 1 V j) v Lt2 Ltl e2 — el = Ae A similar procedure was used to calculate the other functions (62 + 4m), (3m —-%) and (m). The results are tabulated in Tables 5.3 and 5.4. 133 Since rock salt remains isotropic in the hydrostatic test, the same procedure was used to calculate (62 + 4m) and (3m -'%) for the second loading cycle. The only change in the seconc cycle is that A and u are different from their values in the first loading cycle. The new values of A and u were calculated from the average longitudinal and shear wave velocities at the beginning of the second loading cycle. VLh = 4450 meters/sec. v = 2640 meters/sec. sh 2.157 gm/cm3 .0 ll Using Eqs. 5.1 and 5.2 yields: 2 1.51 x 1011 dynes/cm u A 2 1.24 x 1011 dynes/cm These values were substituted in the velocity equations. The results are included in Table 5.3. A similar procedure for the second loading cycle in the triaxial tests would not be valid because of the induced anisotropy. -M0C.~n0004 H0 .040th Una-H EON“ NITH. EOE-momvam HON «EV + WNV M0 COHUNHDUHMU .N-m .WN-nNm-UL... 134 mm m.ma NH~.0 www.mw mom.m subs osmo. mmH mm H.HN mhm.o emo.mv mom.~ have ammo. oma mm H.ma OHN.o mom.mv mo~.~ hows oamo. ¢HH mm m.ma Nm~.o mom.m¢ Hom.m mace mono. moa we H.0m om¢.o hem.m¢ mma.m come Nmao. om we N.om mm¢.o Ham.>¢ mma.m meme undo. we we o.mm «mm.o mm¢.h¢ mma.m mmoe mmao. 00 no m.am mm¢.H wom.m¢ mma.m Omoe mdao. em on ¢.¢m emm.a Hm¢.m¢ NmH.N some haao. me am m.m¢ oem.a www.me ona.m move mmoo. cm 00 m.v¢ oma.a bmo.m¢ boa.m hove heoo. ma vm N.mh Hmm.a bmm.o¢ NOH.N mmmfi NNoo. NH ham o.~o~ NHN.H was.mm mma.m omme 0000. 0 III III III mom.mm bma.m mmme o o I I I + Eo\Em oom\Eo GH\CA amm Haoa Haas oaoa oaoa 0a mmH sq m ooa muamcoa > cwmnum N Nau\mmc>p mufloon> mmomum ma as + em m4 ma mew-Ha H> QN> a .02 as ~>-aa H> m m N N .msapmoH no masks uma Eoum N09 seafloomm How AEv + wmv mo :oHumasono .N.m manna 135 Table 5.3. Calculated values of (62 + 4m) and (3m - g) from hydrostatic tests. -(62 + 4m) -(3m - g9 11 dynes/cm2 10ll dynes/cm2 Priggure _giggt Cycle Second Cycle First_§ycle Second Cygle psi HL2 HL3 HL2 HL3 H51 H82 H81 H52 0 _- -_ -_ _- __ __ -- _- 5 6330 5555 -- -- 2762 3442 -- -- 10 1524 1107 842 2798 1580 1982 799 615 20 -- 1811 227 677 303 568 245 -- 30 261 836 194 274 210 292 91 161 40 -- 530 —- -- 117 120 —- —- 50 -- 366 140 146 63 76 45 77 60 127 -- -- -— —- ~- —— _- 7O -- 256 130 127 31 32 34 27 80 -- —— -— -- -- —— __ __ 90 121 192 105 109 20 21 23 21 Selected values: 107 .i 16 22 .i 5 Table 5.4. Calculated values of (22 + 4m) and (m) from 136 the first cycle of triaxial tests. Axial -(23 + 4m) —m 8:3:ss 10ll dynes/cm2 loll dynes/cmz psi TLl TL2 T81 T82 o --- ___ --- --- 6 i 218 217 '-* "‘ 12 85 94 17 20 18 72 60 54 57 30 59 59 63 —-- 42 63 7O 75 47 54 53 67 63 -~- 66 6O 48 52 41 78 55 46 51 —-- 90 46 44 --- 30 102 40 35 34 —-— 114 43 35 --- 17 126 35 36 --- --- 138 19 35 16 9 Selected values: 35 < 9 ’1 ..4. f) (D ’J ’4 ii If f: 137 Figures 5.10—5.12 show a plot of the four functions of 2, m, and n, tabulated in Tables 5.2 and 5.3, with respect to the stress. The horizontal straight line segments indicate the stress ranges for which the functions were calculated by the procedure illustrated in Eq. (5.8). The results indicate the following: -— The four functions of 2, m and n increase with an increase in stress. This increase is attributed to porosity effects. At high stresses, porosity effects become less and the four functions of 2, m and n approach constant values. -- Differences between two replications are also due to differences in initial porosity. Lower values of any of the functions were obtained from specimens that showed higher changes in velocity (Figs. 5.5-5 8). Thus, it is reasonable to reject low values of the functions obtained from specimens which showed large changes in velocity. For example, in the first cycle of the hydrostatic longitu- dinal test, the results of Specimen HL3 are rejected and the results of Specimen HL2 and accepted. -- Figure 5.10 shows that the difference between the two Specimens HL2 and HL3 becomes small during the second CYcle. Furthermore, the function (62 + 4m) seems to con- verge faster at higher loads. Therefore, the value of $3 + 4m) is taken as: _ (62 + 4m) = (107 i 16) x 1011 dynes/cm2 (5.9) 138 where: 107 x 1011 is the average value from 7,000 to 91000 pSi. ‘: 16 x 1011 represents 67% of the difference between 107 x 1011 and the average value in the load range of 5,000 - 7,000 psi. -- By similar argument, the average value °f(3m -'§) is taken as: 11 — (3m - g) — (22 i 5) 10 dynes/cm2 (5.10) .— —- The results of specimen TL2 (Fig. 5.12) incidated that (22 + 4m) converged to a constant value of -35 x 1011 dynes/cm2 over a wide pressure range (9,000 - 13,800 psi axial stress). The average value of (2E + 4m) from specimen TLl, over the same stress range, is also —35 x 1011 dynes/cmz. Therefore: (22 + 4m) = - 35 x 1011 dynes/cm2 (5.11) -- The value of (m) as determined from the triaxial shear test (Fig. 5.12) did not indicate good convergence. The results of specimen T82 indicate that (m) can be assumed as: - (m) < 9 (5.12) -- The four equations above do not yield a unique solution for E, m and n. This behavior can be explained as due to differences in testing procedures: the functions (62 + 4m) _ , .-, .-a.a-do-..*‘.‘I.’ 1". -. . 139 and (3m —-§) were determined from a hydrostatic test while the functions (22 + 4m) and (m) were determined from triaxial tests. —- Large errors can be expected in the measurement of uniaxial strain, in the triaxial test, due to creep and plastic behavior of the salt. In fact, the strain e at each stress level was measured after a waiting period of 3 minutes during which some creep did occur. Thus, it is reasonable to assume that the true values of Ae are less than the values used in the calculation of the functions (22 + 4m) and (m). Careful examination of Eq. (5.8) would indicate that the true values of - (22 + 4m) and (-m) must then be higher than those of Table 5.4. -- It can be assumed that the unknown creep effects are the same in the longitudinal and shear velocity measurements of the triaxial test. Therefore, the results of the test can be represented by the following single formula: N(ZB + 4m) N(m) = constant (5.13) N = unknown factor -N(2£ + 4m) = 35 -N(m) < 9 -- The third order elastic constants can be calculated from Eqs. (5.9, 5.10 and 5.13) by assuming different _(gi + 4m) (N) m(N) ratios. 140 Table 5.5. Third-order elastic constants of rock salt. lOll dynes/cm N(2§m+ 4m) + Nm m 2 n 4 38 -8 -2o.5 : 3 -39 i 0.5 -79 i 21 5 -7 -15.3 i 2.3 -7.6 i 1.1 -48 i 17 5.84 —6 -1o.0 : 1.6 -9.4 i 1.5 -12 i 14 A sample calculation is given below: Assume —(N)m = 7 x 1011 then: N(ZB + 4m)= §§_= 5 N(m) 7 Therefore, 22 = m From Eq. (5.9): 6£ + 4m = (—107 i 16) x 1011 7m = (-107 i 16) x 1011 Therefore: 11 (-15.3 i 2.3) x 10 dynes/cm2 m 2 = (—7.6 i 1.1) x l0ll dynes/cm2 From Eq. (5.12): 3m _.§ = (-22.: 5) x 1011 n = 6m + (44 i 10) x 1011 = [(-92 i 13.7) + (44 : 10)]10ll n = (~48 : 17) 1011 dynes/cm2 141 where: .i 17 = WVQ10)2 + (13.7)2 -- Figure 5.13 shows the average values of (23 + 4m) and (m) for the second loading cycle in the triaxial tests. The salt was assumed to be isotropic at the beginning of the second loading cycle, and the functions (22 + 4m) and (m) were calculated by the same procedure used to calculate (63 + 4m) and (3m -'§) in the second loading cycle of the hydrostatic test. It was pointed out that the assumption of isotropy is not valid in the triaxial test. However, it is assumed that the errors introduced in this assumption are less than the advantages obtained in the calculation of (23 + 4m) and (m), due to less creep effect in the second loading cycle. Figure 5.14 shows the strain behavior for two typical specimens. The strain in the second loading cycle shows a linear behavior up to about 12,000 psi. very little creep was observed in this range. After 12,000 psi the strain increments become large and start to show creep similar to that observed in the first loading cycle. This behavior justifies the elimination of the values of (22 + 4m) and (m) in Figure 5.13, at loads higher than 12,000 psi. -- It can be concluded that the average values of (22 + 4m) and (m) for the second cycle of the triaxial test (Fig. 5.13) are: 142 23 + 4m (-55 i_3) x 1011 dynes/cm2 (-12 + l) x 1011 dynes/cm2 m -- These values suggest the following choice of 2, m and n from Table 5.4: m = (—12 i 1) 1011 dynes/cm2 z = (—9 i 1) 1011 dynes/cm2 n = (—26 i 6) 1011 dynes/cm2 -- This choice was made by plotting the values of E, m and n from Table 5.4 as shown in Fig. 5.15. The value of (m) was chosen as (-12 i l) 1011. Three horizontal lines were drawn to intersect the (m) curve at the selected 11 dynes/cmg. Three vertical values of (-11, -12, -13) X 10 lines were then drawn from these points. The values of E and n were obtained from the points at which the three vertical lines intersected the Z and n curves. -— The measured values of (6£ + 4m) and (3m -‘%) in the first and second cycles of the hydrostatic test (Figs. 5.10, 5.11) and the average values of (22 + 4m) and (m) in the second cycle of the triaxial test, are all in close agreement with the final selected values of E, m and n. Similarly, the measured value of (m) in the first cycle of the triaxial test was -9 x 1011 for specimen T82 and -17 x 1011 for specimen TSl. The average of these two values is also in close agreement with the final choice of m. longitudinal velocity (meters/sec. ) 143 4700 '- 4600 ' 4500 -4 HL2 , 1 4 "' ‘ - 40° ‘ f A 7 lst cycle loading ,‘ , ° lst cycle, unloading x ~2nd cycle loading . 2 ' 4300 ‘ . nd cycle unloading 4200 , J ' l L i ‘ 2 4 6 8 . 10 hydrostatic pressure (1, OOOpsi) Fig. 5. 5 Change of longitudinal velocity with hydrostatic pressure shear velocity (meters/sec. ) 2800 2700 ‘ Z6 00 2500 2400 144 HSZ _ ,,;..——-—-—--—”“" t 4 J 0 / HSl ;‘ . ‘ ‘ g A /" o _- ./ A 1st ‘cycle - loading ° lst cycle - unloading X an’ cycle - loading 0 2nd cycle - unloading I A l * I I I 2 4 6 8' 10 hydrostatic pressure (1, 000 psi) Fig. 5. 6 Change of shear velocity with hydrostatic pressure 145 a; momma 3323.3 5 awoken #308 m5mno> .3339, fimcmofifimsod h .m .mrm 1 3mm ooo .3 mmouum H.208 om w o v o _ _ _ . 4 I. _ oo~¢ \I oomw MATH Cu .13 \ -25.. 3 1:. 393 m t» mcfivdoa umufl umumm \\ £325. - £0 0 EN . .39»... 3.43 Sn. L mg no - ... 96 a x N -83. o H a o N NAB mcwododfia .. 393 a: o mswvdoa u 393 a: 4 1.83. 1,83. cogs ( 'aas/sxaiamhinsopn [eutpnufium 146 mumou 33873 5 mmouum #308 mama»; >fioo~v> umvsm m .m .mrm 3mm coo .3 much—m H.303 14 cl) w o N o _ _ _ r . oo¢~ I comm 4 was moan: oo>o G . .o H H o N 4 I ooem mcwomofl u 393 com x mcwomoficn I 3930 a: o I corn mcfiumg u o~u>o and 4 I cowN a W p I‘lall‘ 1 NE. ... 83 l 000m 0 “m8 I ooum ... ) - I” q .)|I||I|I|‘.. . comm ( 'oas/sxaism) .Aitootan .I'eaqs NmrH. CUEwUWQm ~147- .mumou H.203?» 5 mmouum aflxm mdmuo> mmmuum Hmumudq o .m .wfih 3mm ooo .3 mmouum Hmwxm a; Na 0H m o v N _ _ _ q _ _ _4 \ < \ Q \ mcwomoHcs n 30.3 HEN . \ mcflomofi u 393 35 x \ o wcfiomofic: . 393 «ma 0 wcflosofi n 30.3 “3 NmB nonhuman oa (lad 000 ‘1) seems 1919191 -(61 + 4m) in units of 1011 dynes/cmZ 148 lst cycle H HL2 400 — H HL3 0——o 300 - 200 c (L 4) 100 "' J ‘ I I 1 ~ 1 ~ 2nd c cle 300 )— v I ._. HL2 o—oHL3 O-——-o 200 _- k 9 3 g 5 Q 43 100 ' l I I I I o 2‘ 4 6 8 10 hydrostatic pressure (1, 000 psi) Fig. 5.10 Change of (61 + 4m) with hydrostatic pressure 149 lst cycle O——o HSl 0‘0 H52 300 -- "—'—°: N F—d g 200 '- \ G} d.) C >~ o " ‘2:9 "‘3 100 — ‘04 o____|o o .___. .‘E.’ '8 q=====9 5 Q===9 .5 o I I I I - I ... 2 4 6 8 10 cm 2nd cycle ' .... HSl E H H82 1". ' 300 - .———-O 200 - e— fl 100 "" ,____. e— —e_ é: 3'. O I I l I I ' 0 2 4 6 8 10 Fig. hydrostatic pressure (1, 000 psi) 5. 11 Change of (3m - n/ 2) with hydrostatic pressure VFF-U\.flrvfi~>~rv NAOH Ho WHMCJ Cw Acfihvl VEU\m~mv:\n~U HHOH No @ch3 Cw. «SW: + NNVI -(m) in units of 1011 dynes /cm2 -(21 + 4m) in units of 1011 dynes/cm2 150 100"- l'st cycle ._. TSl 0‘0 T52 80“ 0—0 60"- fi-———O 0 at c:. o 40- ° 0 O--———o 0——-—O 20 ‘I- H °——-u—-—_=LL 1 O-—-——O I 1 I I L I I Q 2 4 6 8 10 12 14 _q )- lst cycle . -- TLl ' o—O TL2 ¢——-e 80 .. o—-o O-—-—O o——o O—-——o 6O _ 0—8 .__.. o———o .....3—8—0 623 ___. 40 _- .___. 0—._o—8=‘b—"_° 20 - I I I I I I I 0 2 4 6 8 10 12 14 axial stress (1, 000 psi) Fig. 5.12 Change of (21 + 4m) and (m)in the triaxial test-first cycle c c C 0 0 0 0 6 4 2 8 6 4 2 VEU\3UC>—Ul~ad°~iko I.u.~C§...C« AEIV NEU\00C\AHV HNQH H0 owns...- nuw AEWV 4r \N»I 151 2nd loading cycle ' F- N _- TSl S °—°TSZ $60 - E p .m- km. .9 ‘5' 40 "' k 385 ”“333 .S’ 8 20 I— c - ' o c? gg 4 0 l 1 J l I ' Z 8 10 12 14 “— an loading cycle 0—0 TLl NE o—o TL2 m 0 I: >. 'u o——o :2 60 .. ,_____, _ # ° F a J" H ___. W ‘H e c o '8 5 40 ‘ .S E- 5 =5, ‘1' + 20 '- E l I I l - 0 2 8 10 12 14 axial stress (1, 000 psi) Fig. 5. 13 Change of (21 +g4m) and (m) in the triaxial test-second cycle g. 152 mummy H308?» 5 mmonum #308 moms“; comp? #303 303.».H ¢~ .m .wflh 3mm ooo .3 mmouum H.303 3 NH oa , m o v _ _ _ _ i 0 wcfiomomcs n 30.3 95 . wswomoa u 0H0>0 95. X mswomodfio n 30.0 and O wcwoNOH I 30.0 “ma Q I NAB moo .o o~o.o mdoo o~o .o mNo .o (nu/Una 'uyens 121x12 2 153 ll \ a \ 3 \ n \ 8 \ >~ . 'U 2 a. .fi fl :3 \ . \\ \\ m =' '12 I 1\ m = -12 + 1 ~"" -—-§. If / ______ ___—.....——— __— 6” I 4. 38 2 _ ratio of. g ( 14’4"!) from first cycle of triaxial test Fig. 5.15 Determination of the third order elastic constants of rock salt 154 Therefore, the final choice of E, m and n is in close agreement with the measured values in seven tests but disagrees with (22 + 4m) in the first loading cycle of the triaxial test. Therefore, the probability of the final choice of the values of 2, m and n is %° 5.8 Evaluation The literature review did not reveal any work on the third-order elastic constants of rocks. However, the experimental values of 2, m and n obtained in this investi—o gation are in agreement with Brillouin's predictions and Hughes'30 experimental results which indicated that the third-order elastic constants are negative and larger than the second-order elastic constants A and u. Bergman and Shahbender5 measured the changes in longitudinal and shear wave velocities propagating in a direction perpendicular tothe applied uniaxial stress in aluminum columns. They concluded that changes in shear wave velocities could be explained by changes in density and shear modulus while changes in longitudinal velocities could the 10 be explained by changes in density alone. However, 30 . . 56 . theoretical development of Hughes, Rivlin, Biot, Bhagvantum,6 and the present author indicate that changes in both longitudinal and shear waves are due to changes in both density and effective elastic moduli. Furthermore, the experimental results of Hughes and the present author 155 were in close agreement with the theory. Therefore, the present author cannot find any theoretical justification for Bergman and Shahbender's assumption that changes in longitudinal velocity are due to changes in density alone. Hughes' experimental results provided the third- order elastic constants of Polystyrene, Armco iron and Pyrex. In the present investigation, the third-order elastic constants of rock salt were calculated from the changes in velocities at high stresses where the porosity effect is small. This suggests the possibility of measuring the third-order elastic constants of other solids. Consequently, it is reasonable to assume that for compact isotropic materials the changes in wave velocities due to known applied stresses can be predicted from theoretical considerations provided that p0, 0: u. 3, m and n are known. The question arises whether a reverse procedure is possible, i.e., given po, 1, u, E, m and n for a compact material which is under the influence of unknown forces, is it possible to calculate the stresses from absolute velocity measurements in the material? This question is difficult to answer because of the limited available data on this subject. Furthermore, the theoretical development is restricted to the special cases of uniaxial stress (Hughes), hydrostatic compression (Hughes and present author), and homogeneous triaxial stress (Rivlin and present author). Essenaitlly, there are two things that must be considered. 156 First an assumption must be made regarding the stress distribution in the material to be tested, and second the stress-strain relationship and the history of the deformation must be known. The stress-strain relationship and history of the deformation are needed because the velocity equations include some functions of the strains and not stresses. Therefore, the answer to the question may be affirmative if the stresses can be uniquely determined from the strains and if an intelligent assumption can be made regarding the stress distribution, provided that all necessary velocities can be measured. VI . GEOPHYSICAL APPLICATIONS The theoretical and experimental investigations in this study may be extended to certain geophysical appli- cations. Ultrasonic transducers might be placed at the ends of drill holes, which are filled with liquid pressure, for detection and transmission of waves. The data obtained in the uniaxial compressive stress tests suggest that measurement of longitudinal velocities in the vertical and horizontal directions of mine pillars may indicatesstructural stability of the pillars. Currently there are two theories regarding the nature of the underground stress field: the common hydrostatic theory and Serata's61 triaxial theory which was reviewed in Section 2. Measurement of velocity in three mutually perpendicular directions could provide an experimental verification of either theory. Equal velocities indicate that the rock is still isotropic and thus the underground stress field is hydrostatic. If two velocities in the horizontal direction are equal but different from the velocity in the vertical direction, then the underground stress field is triaxial. The data obtained in this investigation revealed that the changes in velocity at high pressure are reproducible. 157 158 The third-order elastic constants of rock salt were calcu— lated from the intrinsic change in velocity. This suggests the possibility of extending the experimental data to determine the changes in overburden pressure with increasing depth by measuring the velocity of longitudinal or shear waves. Wave velocity logging around underground cavities might indicate some correlation with the stress and strain distribution. 1. VII . CONCLUSIONS Unstressed rock salt is nearly isotropic. The velocities measured in three mutually perpendicular directions in five—inch cubic specimens were rela- tively consistent with a maximum variation of less than 3%. The average longitudinal velocity in rock salt is 4225 : 1%.meters/sec. The average shear wave velocity is 2450 i 2%.meters/sec. The dynamic elastic moduli of rock salt are: I 1 = (1.83 i 0.05) x 106 psi = (1.26 _t 0.05) x 1011 dynes/cm2 Lame's constants) 6 p. = (1.88 _t 0.05) x 10 psi I (1.29 i 0.05) x 1011 dynes/cm2 Poisson's ratio, V (0.247 :_0.01) Young's modulus, E = (4.68 i 0.10) x 106 psi Bulk modulus, K = (3.08 :_0.07) x 106 psi The dynamic elastic moduli of rock salt are much higher than the corresponding static moduli. The dynamic value of the bulk modulus is in close agreement with the static bulk modulus as measured from a hydrostatic test. 159 5. 160 Uniaxial compressive stress in rock salt produces an increase in velocity of longitudinal waves propagating along flie axis of compression and a decrease in the velocity of longitudinal waves propagating in the lateral direction. The dif— ference between the two velocities reaches 25% at 3,000 psi (about 75% of the salt crushing strength). Longitudinal and shear wave velocities increase with an increase in hydrostatic pressure or tri— axial stress with uniaxial strain. The rate of velocity increase is very fast at early stages of loading and approaches a steady slow rate at high loads. The initial rapid increase in velocity is not reproducible and can be explained as due to porosity effects. The small but steady increase in velocity at higher loads is reproducible and can be explained as due to intrinsic changes in the effective elastic moduli of rock salt. The third-order elastic constants of rock salt were determined from the changes in velocity at high loads assuming that the porosity effects are negligible at the highest loads used. The calculated values are: 2 (~13.0 i 1.4) x 106 psi 2 (—9 i 1) x 1011 dynes/cm 10. 11. 161 3 II ("17'4.i 1.4) x 106 psi (‘12 i.1) X 1011 dynes/cm2 (37.7 i 8.7) x 106 psi :3 || (’26.i 6) X 1011 dynes/cm2 The experimental results did not show any cor- relation between the measured absolute velocities and magnitude of stress. Furthermore, there was no correlation with the transition from elastic to plastic states of stress. Velocity changes at high stress levels were reproducible and indicated a correlation with the change in stress. This behavior might possibly be used to determine changes in over— burden pressure with increasing depth in salt formations. The data obtained from the uniaxial compressive stress bestssuggest that measurement of longitudinal velocities in the vertical and horizontal directions of mine pillars may provide information about the development of fracture zones in the pillar. Since ultrasonic waves are sensitive for the detection of the degree of anisotropy in a stressed medium, ultrasonic wave methods might be used to determine if the underground stress field is hydrostatic. VIII. RECOMMENDATIONS FOR FUTURE STUDY The following aspects of the effect of stress on wave propagation in solids are recommended for future study: 1. The literature review revealed a lack of experimental data on the third—order elastic constants of solids. A study of these constants is essential for any investigation of material behavior under high static stresses. A general theoretical development to determine the necessary and sufficient velocities to be measured in order to determine unknown stresses. A theoretical and experimental investigation to correlate the third-order elastic constants of single crystals with the non—linearity of intermolecular interactions as predicted by quantum mechanical analysis such as Porn—Mayer theory of ionic crystals. Determine the best experimental procedure to measure attenuation in stressed solids. An experimental investigation, using wave reflection and refraction techniques, to detect the development of a plastic region in stressed solids. 162 163 Certain difficulties were encountered during the course of this study. Based on these difficulties, the following laboratory investigations on rock salt or other rocks are recommended: a. Repeat the same experiments with higher hydrostatic and triaxial compressions. Measure the longitudinal and shear wave velocities in the lateral direction in the triaxial test. Measure the changes of longitudinal and shear waves in the same specimen. Measure the wave velocities in biaxially com- pressed salt blocks. Measure the wave velocity in triaxially or biaxially stressed hollow salt blocks. 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Chicago: The University of Chicago Press, 1960. APPENDICES APPENDIX I _ 1. * - n — 2 [J J - E31 r(1+bll)(1+Bll) b21(l+Bll) b3l(1+Bll) J* = [J:J.] = b12(l+B22) (1+b22)(l+B22) b32(1+B22) (A-l) _bl3(l+B33) b23(l+B33) (1+b33)(1+B33j _(1+bll)(1+811) b12(1+B22) bl3(1+B33) i J = [Jij] = b21(1+Bll) (1+b22)(1+B22) b23(1+B33) (A-2) b31(1+811) b32(1+B22) (1+b33)(1+B33) Fl 0 0‘ E3 = 0 1 O 10 0 1_ where: i indicates row and j column. For example (J32) b23(1+833). By regular matrix multiplication, J J is a square matrix of dimension 3 whose elements are:' 3 (J*J). = 2 J7. J. 1k j=l ij jk * 3 * (J J)ll = z Jlj le 174 _ 2 2 2 — (1+b ) (1+B ) + b 1+ 2 2 2 11 11 21 ( 311) + b31 (1+311) _ '2 . - 1 + 2Bll + Bll + 2bll + 4bllBll + higher order terms. Thus: 1 2 = b + B + 2b Ell— (A-B) 11 11 11 B11 + 2 By similar procedure, 2 n = n = b + B + 2b B +-E;;— 2 22 22 22 22 22 2 B332 ”3 ‘ T‘33 = b33 + B33 + 2b33 E33 + 2 (J*J) = 8 J* - J* J + J* J + J 12 i=1 13 32 11 12 12 22 13 J32 (1+bll)(1+811) b12(1+B22) + b21(1+Bll)(1+b22)(1+822) + b31(1+Bll) b32(1+B22) (b + B22) + higher order terms. + b21)(l+B 12 11 . . * * Similarly (J J)21 - (J J)12 Therefore, 1r‘6 = 1112 = 1121 = 2'[(J*J)12 ‘ 0] = biz : bZl (1+8ll + B22) By similar procedure. T‘5 = 1113 = 1]31 = blB 2 b3l (1+311 + 333’ 04 = n23 = 032 - b23 : b32 (1+B22 + B33) APPENDIX II l + 2 2 - 3 0 = (——§——E) Il - 21112 + ( 3 ) 11 — 2mIlIZ + n13 (A-5) I1, 12 and I3 are functions of the symmetric matrix H. Therefore, 51 BI 01 ¢ 2 83': (1+2u) 11(53l) -Zu(5;;) + (2+ 2m)Il (53;) 812 811 813 -2mIl (_339 - 2m12(dfi_) + n(sfi—) (A-6) 00 . . . . . '5; IS a symmetric matrix according to the follow1ng theorem from Murnaghan.47 "If f(A) is a function, written symmetrically, of the symmetric matrix A, then g% is, like A, a symmetric matrix." By the same theorem, the gradients of 11’ 12 and I3 with respect to n, are symmetric, given by:47 .511“. an 3 512 ( 7) ___. = - A- dn I1 E3 n 51 175 176 cof n means cofactor* of n r —-1 cof n= n6 n5 n1 n5 n1 ”6 _ _ (A-B) 4 n3 1‘5 n3 n5 T‘4 It is important to note that in the process of differentiation. the symmetry of the strain matrix is neglected and the nine elements of n are regarded as independent elements. For example, I2 is given by equation (3.28) as, _ 2 2 2 I2“‘2"3"‘4 +“3nl n5 +1‘1"2 ”6 81 In this form 3;; = -2n4 which is wrong. To get the right 4 answer the elements of n42 must be considered to be independent. Thus, 2 -114 n23 T‘32 and-gig- = - n = - n n23 32 4 *If A is any square matric of dimension n, the cofactor matrix of A (denoted by cof A) is the matrix obtained by replacing each element of A by its cofactor, the cofactor of qu being the product of the determinant of the (n-l) dimen- sional matrix. obtained by erasing the p column and q row of A; by (-1)p+q. 177 812 B-fiz—n =_n “32 23 4 and B Bia=gi=alz =_n T‘4 T‘23 an32 4 Combining Eqs. (A-6) and (A-7) yields: d¢ 2 fi=xIlE3+2un+MIl -2mIZ)E3+2mIln+ncofn (A-9) or, 5¢ = x I + Zu + z I 2 2 2 Efifi_ 1 n1 :1 — m 12 + 2m Il n1 + (n2n3 - n4 )n 5¢ = x I + 2 + 2 I 2 2 I 2 2 535 .1 u ”2 1 I“ 2 + m 11 T‘2 + (”1"3 ‘ T‘5 )n ace 2 a" = — ' 2 n3 A II + 2” T‘3 + 2 I1 2m I2 + 2m 11 n3 + (”lnz T‘6 )n a¢ = 2 + 2 I + ( )“ 371-4 u M m 1 ’14 ”6 T‘5 ' n1 1‘4 a¢ '335 = 2n n5 + 2m.I1 n5 +'(fl6 n4 - n2 n5)n (A—lO) a¢ = 2 + 2 I +( - ) 3716 u 1"6 m 1 1‘6 T‘4 T‘5 1r‘6 113 n The strains in the formulas above can be expressed in terms of b.. and B.. as follows: 1) lJ Il=nl+n2+n3 Substituting for n1, n2 and n3 from Appendix I, yields: 11 = B + b + 2 (bll B11 + b22 B22 + b33 B33) 2 2 2 + B11 + B22 + B33 2 2 2 (A-11) 178 where: 2 2 . . I = B + 2 Bb + higher order terms. (A—12) = 2 2 2 From Appendix I; n4, n5, me have terms of b 1J appearing alone. Therefore, the squares of these strains are neglected and Iz.is written as: 2 n2T‘3'H‘37‘1’Hhr'2 Hz U3 = b22333 + b33B22 + 322833 + higher order terms (A—13) n3 n1 = b33811 + bllB33 + B11B33 + higher order terms n1 n2 = bllB22 + bZZBll + 811822 + higher order terms Therefore: I2 = b11(322+B33) + b22(Bll+B33) + b33(322+311) + 322333 + 311333 + 311322 (A—14) = B nlIl B Bll + bBll + bll (A-ls) ”212 = B B22 + b822 + bZZB U3I3 = B E33 + bB33 + b3BB Iln4 = [B+b+2 (bllBll + b22322 + b33333) l. 2 2 2 1 + 2 (Ell + B22 + B33 )‘ b + b 23 32 [ 2 (1 + 322 + B33)] 179 _ .3. I1 ”4 2 (b23 + b32) Similarly, I n =-5 (b + b ) l 5 2 13 31 (A—l6) I n = g-(b + b ) 1 6 2 12 21 2 _ 2 _ _ T‘4 " T‘5 ‘ ”6 "6 ”5 ”6 "4 T‘4 1r‘5 = 0 (A47) B _ 11 n1 n4 ‘ 2 (b23 + b32) n2 T‘5 2 13 31 (A'ls) B 33 T‘6 T‘3 2 (b12 + b21) From Eq. (A-lO), a¢ _ a¢ _ 2 2 Substituting for the strains and strain invariants yields: 5¢ 1 2 2 2 Efii = A[B + b+2(bllBll+b22822+b33B33) +I§(Bll + B22 + B33 )1 . 2 + 2 B +b +2b B + Ell—1 + z [B2+2bB] u l 11 11 11 11 2 + 2m ElBBll+bBll+bllB] - [bll(322+B33)+b22(311+333) + b33(322+Bll)+B22333+BllB33+Bllezl + n“9221333 + b33322 + 322333] 180 a¢ = 1 B+b+ '531 [ 2(b11311 + b22322 + b33333) 1, 2 2 2 + 2(311 + E22 + B33 )3 2 B11 2 + 2n [311 +bll+2bllBll+ 2 } + £[B +2bB] 2 + .. ... - .. 2” [311 +2b11311 b22333 b33322 322333] (A 19) + “ [b22333 + b33322 + 322333] “$2 = d¢ and 6§$_ =-%$- are written from g2-by cyclic 1‘22 6:2 1‘33 T‘3 1‘1 permutation of the numbers 1, 2, 3. a¢ a¢ a¢ , = = = 2n n + 2mI n + n(n n - n n ) 534 5n23 5n32 4 1 4 6 5 1 4 b + b 23 32 + 2H [ 2 (1+322+B33)] b23 + b32 (B) + 2m I 2 1 b + b 23 32 _ 54) b +13 23 32 _ —————————— [zu (1+32 +333) + 2mB nBll] 53; = 2 2 Similarly, a¢ _ b13 + b31 _ A-ZO Iafis - 2 [2u(1+Bll+B33)+2mB “322] ( ) a¢ b12 + b21 _ .33 = 2 [2u(1+Bll+322)+2mB nB33] where: 181 APPENDIX III a¢ * °=%;J5-5J (A—Zl) 9' - l - (1 21 l p - Det J _ - 1 + 412 + 813) 2 (A-22) O The elements of J, (Jij) are given in Appendix I, Eq. (A-Z). The elements of J*, (Jij) are given in Appendix I, Eq. (A-l). 5¢ '53 is a symmetric square matrix of dimension 3, and can be written as: ¢ ¢ ¢ 11 12 13 ¢ ¢ ¢ 31 32 33 where: a¢ ¢,,= d¢..=¢.. 11 5nij an 11 31 The elements of (¢ij) are given in Appendix II, Eqs. (A-l9) and (A—20). ¢ _ Let N = J'gfi J* (A 23) Then N is a symmetric square matrix of dimension 3 whose elements are given by: 182 183 3 3 * N. = z z . J ir j=l k= 1] Jk kr 3 3 * N = z 2 J ¢. J 11 J=l k=1 1) JR kl =§J ¢ J* +J J* +J * j=l 13 jl 11 lj 32 21 lj J3 J31 N - J ¢ J* + ¢ * ¢ * 11 ‘ 11 11 11 J11 12 J21 + J11 13 J31 * 12 21 11 12 22 J21 + J12 ¢23 J31 +J¢J*+J¢J*+J¢* 13 31 11 13 32 21 13 33 J31 All terms of ¢.., i # j contain terms of b or B . All 1) rn rn terms of J.. or J*., i # j contain terms of b . All 1] 1) rn * . . . terms of Ji' or Jij' 1 8 J contain terms of order 1. Thus, all terms of ¢i., i # j multiplied by Jr J n or J:n' r # n, are higher order terms. Similarly all ' * terms of ¢ij' i # j, multiplied by the product Jrn ka, r # n and k # m, are higher order terms. Therefore, _ * - N11 - Jll ¢ll Jll + higher order terms 2 2 = (1+b11) (1+Bll) $11 ' t ms (1 + 2 b11 + 2 B11) $11 + higher order er or (1 + 2 b + 2 B A—24) 1 11 11) ¢ , ( 1 2 II 184 By similar procedure, it can be shown that: N2=N22=(1+2b22+2822)<192 N3 = N33 = (1 + 2 b33 + 2 B33) $3 and N4 = N23 = N32 = b32 ¢2 + (l + B22 + 333’ ¢4 + 323 ¢ N5 = N3l = N13 = b13_¢3 + (1 + B33 + 311) ¢5 + b31 ¢ N6 = N12 = N21 = b21 4’1 + (1 + B11 + 322) (D6 + blZ ¢ 5— can now be approximated as: 0 -g— = l - Il + higher order terms 0 = 1 - B - b where: B = B11 + E22 + B33 b = bll + b22 + b33 The higher order terms of %-are neglected because their 0 products with N are all higher order terms. The stress 011 is now given by: - = £- 01 " 011 p N1 0 substituting for ¢1 from (A—l9) yields 185 o = A B + b + 2 + 1, 2 2 2 l [ (b11811 b22B22 + b33333) + 2 (311 + E22 + B33 ) + B (2 bll - b) + (2 B11 - B) (B + b)] 2 B11 + ___. .- 2u[Bll+bll+2 bllBll + 2 + 311(2 bll b) + (2 B11 - B)(Bll + bll)] 2 +£[B +2bB] 2 + 2” [311 + 2 b11311 ” b22533 " b33322 ' 322333] + n [b22333 + b33322 + 322333] Collecting terms of bll' b22 and b33 yields; 2 2 B B — ._ 2 _22_._ _§é_._ G1 ‘ [1‘3 + 2 B11 ‘ 2 2 2 322333) + 2 (B +-5 B 2 - B B - B B ) u 11 2 11 11 22 11 33 +£(B2)+2m(B 2—B B )+n(B B )1 11 22 33 22 33 + bll [1(1+4 B11) + 2u(1 + 4 B11 - B22 — B33) + 2(2B) + m (4 B11)] (A-26) + b22 [1(1—2 B33) + 2u(—Bll) + £(2B) + m(—ZB33) + n (B33)] _ + _ + b33 [1(1—2 322) + 2p ( B11) + 2(23) m( 2322) + n (822)] 186 Similarly 02 and 03 are written by cyclic permutation of the numbers 1, 2, 3 2 3 + 2“ (322 + 2 B22 ' B22333 ’ 322811) 2 2 + £(B ) + 2m (B22 - B33B ) + n(B 11 33311)] + b 2 [x (1 + 4B22) + 22 (1-4 B - B - B 2 22 33 11) + 223 + 4mB (A—27) 22] [1(1-2311) + 2B ('322) + 223 — ZmB + "311] + b 11 33 + bll [3(1-2B33) + Zu (-B22) + 2£B—2mB33 + nB33] o = [%(B +-— B — ———— - —§——-- 2 811322) + 2 (B + 3-B — B B - B B ) u 33 2 33 33 11 33 22 2 +2 B + 2m (B — B 33 11 322) + n (B11 322)] + b i 1+4B ) + Zu (1-4B - B - B ) 33 I ( 33 33 11 22 (A-28) + 22B + 4mB33] [1(1-2B ) + 2% (-B33) + 223 - ZmBzz + “322] + b 11 22 + b [3(1-311) + 2n (~333) + 22B — 2mBll + n 311] 22 _ ¢ 04 ‘ 'fi; (b32 (D2 + (1+322 + B33) $4 + b23 3) 2. 4, $3 are given in Appendix 2, Eq. (A-l9) and 187 Therefore, -9—- = = p0 b32 $2 b32 ¢2 b32 (%(Bll + B22 + B33) + Zn 322] R— = = p0 b23 $3 b23 $3 b23 [3(311 + B22 + B33) + 2” B33] b + b LL. = _ = 23 32 _ Po (1+B22+B33) ¢4 (1 B11) ¢4 2 [2u(l+B22+B33 B11) +2m(Bil + B22 + B33)‘nB11] Thus: a — b23 + b32 [2u(1+B + B -B ) + (2m+21)(B +B +B ) 4 ‘ 2 22 33 11 11 22 33 ‘n 311] + 2” [b32 B22 + b23 B33] Similarly, — bl3 + b31 2 (1+B + B ‘- B )+(2m+21)(B +B +B ) O5 ' 2 [ u 33 11 22 11 22 33 _ A—29) n 322] + 2”[b31311 + b13 B33] ( b + b _ 12 21 - +B +B o6 _ 2 [2u(1+Bll+B22 B33)+(2m+2?\)(Bll 22 33) 'n B33] + 2”“012 B22 + b21 B11]