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Determination of viscosity of rocks by a dynamic resonance technique Janakiraman, Coimbatore Subrahmanyam 1967

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DETERMINATION OF VISCOSITY OF ROCKS BY A DYNAMIC RESONANCE TECHNIQUE by COIMBATORE SUBRAHMANYAM JANAKIRAMAN B.E., University of Madras, 1961  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF M.A. Sc. in the Department of , MINERAL ENGINEERING  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA  August, 1967  In p r e s e n t i n g  for  thesis  an a d v a n c e d d e g r e e  that  the  Study.  thesis  Library  for  agree  scholarly  or  publication  without  shall  I further  Department  or  this  at  in p a r t i a l  the U n i v e r s i t y of  make i t  that  freely  purposes  my w r i t t e n  this  thesis  may be g r a n t e d  for  permission.  Department The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  for  Columbia  It  is  financial  of  British  available  permission  by h.i.<s r e p r e s e n t a t i v e s .  of  fulfilment  for  the  Columbia,  I  reference  and  extensive  by  the  requirements  copying  Head o f  understood  gain  shall  this  my  that  not  of  agree  be  copying  allowed  ABSTRACT Results of forced vibration tests in longitudinal and transverse modes on three different rocks are presented.  The experimental investigation leads  to evaluation of solid viscosity parameters as a function of frequency and logarithmic decrement. Linear viscoelasticity theory is applied to the test results within the frequency range studied.  The test results indicate that the rheological  parameter viscosity of the rocks tested, quartzite, granodiorite and an 8 9 a r g i l i t e , is of the order of 10 to 10 poises. Methods of predicting solid viscosity parameters from forced vibration resonance tests by linear viscoelastic theories are derived.  The  correspondence principle, which is based on the solution to steady state sinusoidal o s c i l l a t i o n is not s t r i c t l y applicable, but does yield results which are of the same order as the measured relationships.  The behaviour  of rocks to idealized rheological models has been examined.  Measurements of  the viscosity and complex moduli are described and from a consideration of the results obtained the type of mechanical model most respresentative of the behaviour of rocks is suggested. A method for predicting directional properties of rocks using photoelastic studies for different loading conditions is examined.  This enables  allowances to be made for the wide variation in dynamic test results on rocks. However, i t is f e l t that the dynamic method of determining viscosity may have more application in the examination of rock structure, since a comparison of the laboratory and f i e l d test results at the same temperature and pressure yields a method for structural design involving rocks. i  TABLE OF CONTENTS Page CHAPTER 1 INTRODUCTION  1  CHAPTER 2 REVIEW OF LITERATURE  5  Historical Background Terminology Methods of Measurement Discussion  5 6 9 12  T  CHAPTER 3 VISCOELASTICITY OF ROCKS Linear Viscoelasticity Dynamic Techniques r  T  17 17 24  CHAPTER 4 TESTING PROCEDURE Description of Rocks Tested Procedure for Collecting Samples from the Field Preparation and Description of Samples Description of Test Equipment Testing Technique  29 29 29 32 35 41  CHAPTER 5 DISCUSSION OF TESTING TECHNIQUE  47  %  Introduction Phase Displacement Between Stress and Strain The Effect of Environment and Supports in Forced Vibration Tests Effect of Size of Specimens Internal Friction Measuring Methods Rate of Testing Grain Boundary Effects in Forced Vibration Tests Frequency Considerations Considerations of the Anelastic Behaviour of Rocks CHAPTER 6 DETERMINATION OF VISCOSITY  47 48 52 54 56 59 61 62 63 66  From Forced Vibration Tests Introduction Method 1 Method 2 Method 3  66 68 72 78  CHAPTER 7 TEST RESULTS Introduction  80 80 ii  Page Characteristics of Rocks Tested Comparison of Resonance Curves for Different Rocks Viscoelastic Properties of Rocks Frequency Response of Rocks CHAPTER 8 CONCLUSIONS AND SUGGESTIONS FOR FURTHER RESEARCH Conclusions Suggestions for Further Research  80 91 99 103 113 113 115  LIST OF SYMBOLS  116  LIST OF REFERENCES  119  APPENDIX I  123  APPENDIX II  127  iii  LIST OF TABLES Page TABLE  I  PETROGRAPHY OF ROCKS TESTED  31  TABLE  II  RESULTS FROM FORCED VIBRATION TEST  75  TABLE III  RESULTS FROM FORCED VIBRATION TEST (.Torsional .Mode).  §5  TABLE ; H  RESULTS FROM FORCED VIBRATION TEST  iv  102"  LIST OF FIGURES Figure  Page  1  Resonance Curve for Forced Vibrations  11  2  Relationship Between Specific Damping Capacity and Logarithmic Decrement for Decrements Exceeding 0.01  15  3  Rheological Models  19  4.  Elastic Modulus E and Internal Friction Q as a Function of Frequency etc  23  5  Sinusoidally Varying Stress and Strain, Relationship Between Complex Moduli and Complex Compliance  25  6  Dependence of Storage Modulus (G'), Loss Modulus (G"), Resultant Modulus, etc 27  7  Road Cut Looking East on Highway 7 1/2-8 Miles South of Brittania Beach  30  8  Photo micrographs of Thin Sections  33  9  Photograph of Equipment Used and Prepared Core Samples  34  10  Circuit Diagram of Apparatus  36  11  Sonic Mounting Clamp  38  12  Position of Transmitter and Receiver for Longitudinal Velocity and Transverse Velocity  40  13  Schematic Diagram of Sonic Apparatus  43  14  Shear-Strain Time Curves for Three Faces of an Argil l i t e Specimesn  45  15  Shear Strain-Time Curves for a Quartzite Specimen  46  16  Graphical Representation of Internal Friction  51  17  Internal Friction as Measured by Bennewitz and Rotger  51  18  Piezoelectric Excitation of Rocks  57  19  Resonance Curve for a Quartzite (fQ Rock v  76  Figure  Page  20  Plot of Frequency Versus Log Decrement  82  21  Graph of Magnification Factor Versus Frequency Ratio for a Quartzite Rock 84  22a, b, c Graphs of Magnification Factor Versus Frequency Ratio for Three Specimens of Rocks 86,87,88 23  Graph of Attenuation Versus Frequency  89  24  Graph of Attenuation Versus Frequency for a Granodiorite Specimen  90  25  Variation of Retardation Time with Frequency  92  26  Resonance Curves for Three Specimens of Rocks  93  27  Resonance Curve for Torsional Mode  97  28  Comparison of Longitudinal and Torsional Resonance Curves  98  29  Plot of Log u> Versus Log J-j  106  30  Plot of Log w Versus Log J  31  Plot of J  32  Comparison of Variation of J-j with Frequency  33  Comparison of Variation of J  1  107  2  Versus co  109  ?  with Frequency  vi  110 Ill  ACKNOWLEDGEMENT The investigation reported herein has been supported by funds provided by the Mines Branch at Ottawa. support for the writer.  These funds also included financial  Grateful appreciation is expressed for the assistance  without which the graduate studies and this thesis could not have been accomplished. The writer wishes to express his thanks to Dr. C. L. Emery for his guidance and c r i t i c i s m during the preparation of this thesis. The writer also wishes to thank Professor L. G. R. Crouch and Professor J . Leja for their help in the preparation of this thesis. The technical assistance and help supplied by the graduate students of the Mineral Engineering Department and my friends is gratefully acknowledged.  vi i"'  CHAPTER 1 INTRODUCTION In the design, construction and operation of functional structures in or on rock, a considerable technology has developed with respect to the structure but not to those rocks which form i t s base.  In s t a b i l i t y analysis,  the structure is analyzed to insure that the sum of the resisting forces on any potential failure surface is greater than the sum of the driving forces. No attempt is made to determine the magnitude of the deformations. resisting forces are determined from strength tests on rocks.  The  In mines where  large forces are common, the study of the mechanical properties of rock and the application of this knowledge to engineering problems dealing with rock has resulted in the establishment of a specialized technology called rock mechanics. When designing an underground structure in rock or evaluating the s t a b i l i t y of an existing structure, we must determine (1) the stresses and/or deformation in the structure resulting from external or body loads, and (2) the a b i l i t y of the structure to withstand these stresses or deformations. A l i m i t on this a b i l i t y is generally evaluated in terms of the stress necessary to cause a structural f a i l u r e , which is manifested usually by a sudden collapse of one or more structural components, although sometimes excessive deformation may also be a limiting factor.  Designing a structure in rock is in a number  of ways a more d i f f i c u l t problem than designing or evaluating the s t a b i l i t y of surface structures made of s t e e l , concrete or other conventional construction materials.  Subsurface rock is known to be under a pre-existing and unknown  2  state of stress due to both the overburden and possible tectonic forces. Although previous investigators have estimated the subsurface state of stress by assuming that i t is due only to the wieght of the super-incumbent rock, stress measurements made in existing structures indicate that estimates made on this basis may not be entirely v a l i d . The absence of specific information on the mechanical properties of in situ rock before underground access is possible creates another problem. The mechanical properties of most conventional structural materials can be obtained to a given specification and a structure can be designed to u t i l i z e these materials; however, the design of underground structures is usually limited because of a lack of information about the properties and behaviour of the in situ rock.  It was with this as one of the objects in view that a  testing programme was undertaken in the laboratory.  Also dynamic methods of  determining the mechanical properties of rocks are also of practical significance when fragmentation by explosives or seismic prospecting or comminution studies are to be conducted. Emery (1961) indicated the d i f f i c u l t i e s involved in the design of rock structures based on the use of Young's modulus and Poisson's ratio alone. Furthermore, the directional properties of rocks and the pehnomenon of relaxation was explained by using a rheological equation for a Kelvin S o l i d , e l a s t i c i t y plus solid viscosity and the term solid viscosity was used to describe the viscoelastic nature of rocks.  Rana (1963, 1967) has investigated  the viscosity of rocks using different rates of loading in static tests and under no load conditions in dynamic tests.  3  Kelvin (1875), and Maxwell (1890) have shown that a unique relationship exists between rate of loading, effective stresses and strains for solids. Since then, Jeffrey's (1929) and Reiner (1960) have done considerable work on this problem.  These and other investigators have emphasized that viscosity  is an important parameter in the rheological behaviour of solids.  The prime  purpose of this experimental investigation was to determine i f the rheological parameter viscosity could be evaluated for rocks by a forced vibration resonance method.  A secondary purpose was to crampare the behaviour of rocks  to rheological models suggested by Emery (1963) and Hardy (1959) for an idealized "rock" and in particular to compare the stress-strain relations according to the linear viscoelasticity theory. The methods used in the laboratory for determining the mechanical properties of rocks have been a r b i t r a r i l y divided into static and dynamic methods, this division being based primarily on the rate of loading. is rapid rate of loading/vas in vibration tests, i t is called dynamic.  If the  Although there is  a wide variety of test results reported in the l i t e r a t u r e , there is very l i t t l e information available on viscoelastic parameters, particularly from dynamic tests.  This testing program was undertaken by the writer in an attempt to  determine the viscoelastic parameters of rocks from dynamic tests. A review of pertinent literature is presented in Chapter 2. A discussion of viscoelasticity theory applied to forced vibration test is presented in Chapter 3.  The rocks tested, test equipment and testing techniques  are discussed in Chapters 4 and 5.  A method for predicting the viscoelastic  parameters of rocks is presented in Chapter 6.  The results from forced  4  vibration tests on three different rocks are presented in Chapter 7. and suggestions for further research are presented in Chapter 8.  Conclusions  6  where  Y = displacement of c e n t r e of sphere a = t h i c k n e s s of band b = b r e a d t h of band g = a c c e l e r a t i o n due to g r a v i t y  Cady (1922) proposed a theory of longitudinal vibrations of viscous rods and used the term "viscosity" which is equivalent to j(l+u)n/p.  Wegel  and..Waliner (1935) used the term longitudinal viscosity, n^: n  a  = |(l+y)n  2  "and observed internal dissipation in solids for small cyclic strains-  Getfmant  and Jackson (1937) studied this later and found that the logari-thmie decrement* was constant over a wide frequency band. When choosing a proper nomenclature for expressing damping capacity a discussion of the present practice is considered necessary.  The actual  behaviour of the material is not determined solely by the energy losses occuring in i t but rather by the relation between the energy losses and the e l a s t i c energy.  Hence, i t is necessary to consider the wide variation in  terminology in the past two decades for measuring damping capacity. Terminology: Potter (1958) found eight quantitative expressions for measuring damping capacity in use, which were:  viscosity, damping capacity, constant  of internal f r i c t i o n , hysteretic constant, specific damping capacity, log decrement, elastic phase constant and coefficient of internal f r i c t i o n . The following and perhaps other terms have been added since that time. *Note:  Log decrement stands for Logarithmic decrement in this thesis. definition of log decrement is given on Page "'7.  The  Damping modulus, resonance-amplification factor, specific damping energy, stress-strain phase angle, mobility, and log viscosity.  This variety in  terminology arises, in part, from the fact that the damping capacity in a material may manifest i t s e l f and be evaluated in different ways. At present, v i r t u a l l y the same definition is used for the terms "damping capacity" and "internal f r i c t i o n " , which Nowick (1956) states, is "the capacity of a vibrating solid to convert i t s mechanical energy of vibration into heat, even when so well insulated that energy losses to i t s surroundings are negligible". The term "specific damping capacity" has been used consistently in the U.S. Bureau of Mines to indicate the ratio of the energy absorbed in one cycle of vibration to the potential energy at maximum displacement in that cycle, expressed in percent.  The value is frequently expressed as a decimal  fraction, rather than in percent. The designation "internal f r i c t i o n " is widely favoured in physical metallurgy and metal physics and although essentially synonymous with damping capacity, is usually reported in more abstract units. "Logarithmic decrement" 6, can be defined as the natural logarithm of the ratio of amplitude of two successive cycles of a freely decaying vibration on opposite sides of equilibrium.  In practice as the difference  in amplitude between two successive cycles is very small and d i f f i c u l t to measure with precision, the decrease in amplitude over a number of cycles "n" is defined as follows:  8  <5 = J m (  I  J L  n+1  -)  A  where f = frequency of vibration ?. 1  A and A -| represent the amplitudes of two successive cycles of a n  n+  freely decaying vibration The "attenuation" of a plane stress wave is another way of defining and measuring internal f r i c t i o n .  For plane sinusoidal progressive waves,  the amplitude decays exponentially with distance according to equation: Ax = Ao exp (-ax)  3  Where Ax is the amplitude at the distance x and Ao is the amplitude at x = 0. The quantity a is defined as amplitude attenuation (or absorption coefficient). According to Maxwell "the viscosity of a substance is measured  by  the tangential force on a unit area of either of two horizontal planes of indefinite extent at unit distance apart, one of which is fixed while the other moves, the space between being f i l l e d with the viscous substance". 2 The unit of viscosity is "poise", l b . - s e c . / s q . i n . in F.P.S. system or 2 dyne-sec./sq. cm.  in C.G.S. units.  Although there is some uncertainity about  applying the term viscosity, to rocks, i t is widely used in rheological studies.  Hence, i t has been decided to retain this term in this thesis to  analyse the viscoelastic behaviour of rocks. At the present, the study of viscosity or internal f r i c t i o n is of considerable importance in three distinct f i e l d s , v i z . , metallurgy, solid-state research and rock mechanics.  A brief mention is made of the experimental  investigations with materials other than rocks helpful to interpretation of behaviour of rocks.  9  Methods of Measurement: Although there are numerous methods of measuring damping capacity or internal f r i c t i o n only two relevent methods w i l l be discussed here.  The  two methods to be discussed are as follows: 1) resonance methods; 2) ultrasonic pulse methods; The resonance methods are based on the theory of vibrations while the pulse methods are based on the theory of wave propagation. Experimental investigations have proved that whenever a solid is mechanically strained, the imparted elastic energy is never f u l l y recovered. Some energy is always converted from mechanical to heat energy, therefore, a perfectly elastic solid is only a theoretical concept.  This non-elastic  behaviour within Hooke's law region has been called "anelasticity" by Zener (1948) and the various mechanisms which bring about the non-elastic properties have been grouped under the catchcall term, "Internal Friction" (Kolsky, 1953). The study of internal f r i c t i o n in rocks is of more than academic interest because the decrease in the intenrsnty of a seismic wave with distance is caused by internal f r i c t i o n in earth materials, and the change in shape of the seismic wave with distance is also, in part, a result of internal f r i c t i o n , s p e c i f i c a l l y , i t s dependence on the frequency of the stress wave (Born, 1941). Hopkinson and William (1912) and Fdppl (1936) measured the internal of steel frictionAby determining the ratio of the heat developed per cycle to the maximum strain energy of that cycle, the ratio being defined as the specific damping capacity or specific loss.  Birch and Borncroft (1938) measured internal  f r i c t i o n in a long column of granite 8 feet long by 9 inches diameter, and i t s dependence on frequency.  Born (1941) using bars of shale, limestone and  10  sandstone in lengths up to 6 feet and subjected to frequencies in the range 100-10,000 Hz found the attenuation to be proportional to frequency. Most investigators because of experimental convenience have measured internal f r i c t i o n indirectly by means of resonance techniques.  It is possible  to define the measurement of internal f r i c t i o n for a resonant system by the "logarithmic decrement" provided the following assumptions are made: (1) the e l a s t i c restoring force of the specimen is proportional to amplitude (Hooke's law) and, (2) the dissipative force is proportional to the rate of s t r a i n . Assumption (1) is valid when the resonance experiments are performed at s u f f i c i e n t l y small amplitude or s t r a i n .  Assumption (2), that the dissipative  force is proportional to the rate o f . s t r a i n , is allowable as a f i r s t approximation i f the loss per cycle is very much smaller than the e l a s t i c energy per cycle (Gemant, 1950).  The log decrement for a freely o s c i l l a t i n g resonant  system is then defined as the natural logarithm of the ratio of two successive maximum deflections in the same direction.  Loga^ttfrrc decrement can also be  obtained from the measurement of the bandwidths and the resonant frequency of the system, Figure 1. 6 = Af ~f )/f z  where f-| and  }  r  4  are the frequencies on either side of the resonant frequency  at which the amplitude is reduced to 4=-of i t s value at resonance (Potter, 1948). \/2 An often used measure of internal f r i c t i o n is the Q factor, which is related to log decrement, 6, and specific damping capacity, S, by the following equations:  11  1  \  A f  t, J f  f  1  Frequency  Figure  1 -  Resonance  Carve  jor  Forced  Vibrations  12  Q6  =  TT  5  OS =  2ir  Some of the principal investigators were Birch and Borncroft (1938), Born (1941), Bruckshaw and Mahanta (1954), Krishnamurthi and Balakris'hna (1957). Excepting for the last mentioned investigators, the afore mentioned i n v e s t i gators determined the internal f r i c t i o n of rocks in the frequency range from 10-1000 Hz.  Birch and Borncroft using a resonance method, found that the  exponential absorption coefficient ( a ) for granite was proportional to the frequency in the range 140-1600 Hz; however, their experimental data were not sufficiently precise to establish a definite correlation.  Born, using a  resonance method for shale, limestone, sandstone and caprock Cores, found that the absorption coefficient was proportional to frequency for dry cores, but that the absorption coefficient could be represented by the sum of a linear and frequency squared term when water was added to the sandstone samples. Discussion: It is seen that there is a considerable difference of opinion both with regard to the terminology and the testing methods.  It would appear,  from the work done by those mentioned here, to obtain any useful information, A.  investigation should be carefully &g carried out under controlled conditions to give reasonably reproducible data, regardless of the l o c a l i t y from which the samples were collected.  The test results to have any validity and i f i t  has to be of some use in design work involving rocks,the above c r i t e r i a must be s t r i c t l y adhered to. When a specimen is caused to vibrate by an external source and the  15  frequency of the driving force is varied, at the resonant frequency the amplitude would pass through a maximum value, and a plot of amplitude versus frequency would have the appearance of Figure 1.  Increased damping tends to  reduce the amplitude and broaden the base of this resonance curve. decrement  Based on  this phenomenon, expressions relating log/, resonance frequency, viscosity and elastic constant have been derived by Potter (1948) and Kolsky (1963). 6 =  (Maxwell  Liquid)  p G 6 =  where  n  (Kelvin  Solid)  6 = Log decrement E, G = Elastic constants p-| = A factor which depends on frequency n = Coefficient of viscosity One of the main differences between resonance and ultrasonic pulse  methods is the frequency range they cover. is approximately proportional to i t s length.  Resonance frequency for a specimen Hence, i f extremely small or  embarrassingly large specimens are to be avoided, the frequency range for the resonance method must range from about 1 KHz to at most 200 KHz.  The  resonance method chosen should be able to handde specimens at least 6 i n . in length.  Another striking difference between resonance and ultrasonic pulse  methods l i e ? in the stress and strain distribution.  Both methods operate  under extremely low stress l e v e l .  Resonance methods have less precision  than the ultrasonic pulse methods.  Either method, however, is l i k e l y to  give satisfactory results with rocks.  It appears that i f adequate care is  14  taken then resonance method is l i k e l y to y i e l d reasonably accurate results.  If  corrections are applied to the experimental results obtained with care, then the test results from resonance method and that from ultrasonic pulse methods could be compared. When log decrement,"6 is less than 0.01, the specific damping capacity S = 200 6 within normal experimental error.  Relationship between  specific damping capacity and log decrement for decrements exceeding 0.01 are shown in Figure 2. Evidence for the unique relationship between viscosity and log decrement is rather interesting, but suggests that for the frequency range of 10 Hz to 20 KHz, to be covered in this investigation, the relationship is approximately true.  No data on torsional vibration of rock is available, but  i t has been suggested that a similar relationship might hold for rocks tested by resonance methods.  For torsional modes, the relationship would appear to  be more complex. However, the literature also suggests that the relationship between viscosity and 1 oga-rithm-i-e-decrement may not be identical for the following reasons: (1) Method of testing not identical for determining the resonance frequency and specific damping capacity; (2) The actual method of wave propagation is different for longitudinal and torsional modes; (3) Non-uniform distribution of stresses and strains due to location of transmitter and pick up; (4) Porosity, moisture content, texture, grain size and orientation  15  16  (4) Continued of samples in test w i l l affect the final results; (5) D i f f i c u l t i e s in mathematical analysis arising from different ways in which the material dissipates energy in the two modes of vibration. The effects of 1 and 2 may be eliminated by checking the results obtained with an oscilloscope to obtain correct wave shapes in both cases. If tests are repeated varying the position of transmitter and receiver 3 could be eliminated.  If sufficient number of samples were carefully chosen  whose porosity is negligible, whose texture and grain size are uniform, theoretically i t is possible to minimize error due to 4. be sampled according to aeolotropy of rocks.  Also, rocks must  A method for estimating the  viscosity parameter from test results is discussed in Chapters 3, and 6. The viscoelastic behaviour of rocks under forced vibration test is:considered Chapter 3.  in  1:7  .CHAPTER 3 VISCOELASTICITY OF ROCKS Linear Viscoelasticity The theory of Linear Viscoelasticity provides a theoretical basis for analyzing data of time-dependent strain and for extrapolation beyond the range of experiment.  The mechanical behaviour of many glasses, rubbers and high  polymers have been described by a linear viscoelastic law in which the stressstrain relationships can be written down as a linear differential equation which simply involves stress, strain and their time derivatives.  In particular,  analysis of problems of creep and flow of rocks in the earth requires some such basis, and.iri.ltht role viscoelasticity can be helpful, as long as i t s limitations are understood. Considering single components of stress and s t r a i n , the viscoelastic law is usually expressed as: P a = Qe  6 n  where P and Q are linear operators of the form E A D , D being the time N  °  9  derivative —r. ot  In order to define completely the mechanical behaviour of  a linear viscoelastic s o l i d , derivatives of a l l orders are required but i t is usual to adopt only a limited number of terms in the expansion and ignore'.higher orders. Rheological models perform a useful function in acting as the f i r s t step in the development of rational thought as to the inherent causes and processes o.ffpTastic deformation in solids.  Sometimes tamdem and parallel  arrangements of dashpots and springs, simulating viscous and elastic elements  1(8  are used in which springs deform according to Hooke's law and viscous dashpots obey Newton's law of viscosity.  The equations from these-,/.; and other phenomeno-  logical models in one dimension are treated at length by Nadai (1950) and by several authors in Eirich (1958) and in a clear exposition by Reiner (1960). Both Maxwell and Kelvin elements contain f i r s t - o r d e r terms and therefore describe the dependence upon the rate of loading.  The Maxwell  element allows for no delayed recovery while the Kelvin element describes no purely elastic s t r a i n .  The standard linear solid shown in Figure 3 -  originally suggested by Boltzmann (1876) although limited is more representative of deformation processes in solids since i t incorporates the elements of instantaneous response, delayed response, and flow.  The writer w i l l discuss  further the behaviour and usefulness of these models later in this thesis. Emery (1964) suggested a better approach to consider, that rocks, and other technical substances, combine the properties of e l a s t i c i t y , l i q u i d i t y and p l a s t i c i t y .  This introduces variations in rock behaviour with  time and with magnitudes of loads. ° where  =  E e  +  n  dt  a = Unit stress e = Unit strain E = Young's modulus n = Coefficient of viscosity  and the rate of strain would be  7  19  60Q  Iridioa  Dr^  U  1  model iWutrating ^  befiaviour of rocks  1,  t StaacLard.  Butters  Figure  3 -  m.od.20  Rfteol!ojiicai  Mo<del?s  Linear  Sokd.  Mode.!  20  The standard linear solid could be interpreted by relating the e l a s t i c modulus under high frequency conditions to the modulus when the relaxation process is allowed to terminate.  The former can be termed the  dynamic or unrelaxed modulus and the l a t t e r , the static or relaxed modulus. By equating stress and i t s f i r s t derivative with respect to time to the strain and strain rate, we have that: a^o  + a^a = b^e  Now letting T and x  b^e  +  9  be the times of relaxation of stress for constant strain  and of strain for constant stress, and denoting the relaxed and unrelaxed moduli by E^ and Ey respectively, i t can be shown that: ^  =—  10  Nowick (1953) introduced a dimensionless proportionality constant A  M  called the relaxation strength which related "equilibrium non-elastic  strain" to the perfectly e l a s t i c strain which obeys Hooke's law.  It provides  a measure of the total relaxation, v i z . :  ( E  uV -  Now'ithe-singlerelaxation time, x , associated with the standard linear solid •iis equal to half the sum of the two particular relaxation times (Attewe], 1964) and the internal f r i c t i o n can be expressed by the equation: r.~l ^ "  A  -j „  COX  /  2  M <  1  +  CO  2 X  )  1  2  where co = 2irf  If a further Newtonian dashpot is inserted in series with the standard linear solid model, the resulting Burgers model-widely used by Terry and Morgan (1958) to describe the mechanical response of coal-allows for a degree of permanent s t r a i n .  The simplified form of equation for a Burgers model in the  more easily manipulated form of the standard linear solid would be: a + t a = M (e-'H- t, e) m s k  The coefficient t  m  13  is the relaxation (Maxwell) time at constant s t r a i n , t  k  is the  retardation (Kelvin) time at constant stress, and M is the static (or relaxed) g  e l a s t i c modulus. The relaxation time for a s o l i d , expressed in viscoelasticity theory as the ratio of the viscosity n , to some e l a s t i c modulus M, is a parameter often used in viscoelasticity for the time of decay to I  o f  s t r e s s  o r  strain.  Relaxation under constant stress or constant strain is an important property of rocks, but only for small strains, before irreversible effects occur. The relaxation and retardation times are not constants; they vary with temperature and pressure and probably are not significant for large strains.  These  concepts are useful in creep, flow and high-strain rate rock deformation studies.  S i m i l a r l y , the parameters of viscosity and e l a s t i c i t y vary over a  wide strain-rate-frequency range, but nevertheless are indispensable concepts, as operational tools. To gain a better understanding of viscoelastic properties of rocks, in general, -anelastic dynamic tests should be considered.  This w i l l be the  22-  prime purpose of this investigation on experimental studies on rocks, although certain results of experiments on metals and other non-metallic materials, w i l l be considered wherever found necessary. studies of internal f r i c t i o n  The usefulness in metallurgy of  is apparent from the very great interest in the  subject. The viscosity, determination for rocks has been mostly confined to monotonic creep tests, made at room temperature in a narrow stress range. 15 17 The averaged values of n in poises range from 10 to 10 for h a l i t e , 10 16 to 1017 for coal and 1022 for limestone. From his experiments Heard (1960) 18 22 -10 gives n = 10 to 10 poises for Yule marble in the.ranges e = 10" to 10"  Rana (1963, 1967) from his experimental 1g results gi,ve§coefficient of structural viscosity n - 1.35 x 10 and 1 fi 7 rij = 8.10 x 10 at o = 1500 l b . / i n . and under no load conditions 9 14  sec" and T = 25 C to 400°C. 1  Q  s  n = 10 poises. It is apparent that n is a greatly changing parameter over the range of a ande of interest.  Robertson (1963) defines instead — - — as a new log e  parameter, and calls i t logarithmic viscosity n * ; i t has the advantage that i t s value w i l l reflect the signoidal shape of the curve of o-loge, as in Figure 4.  However^ n* is only an empirical parameter, and is not very useful  in the theory of v i s c o e l a s t i c i t y . The present analysis leads to the conclusion that n is an important rheological parameter whose real value is not known and further experimental investigation is necessary before this could be applied in general for rock deformation.  23  t  1  Log j  are  4 -  Elastic c^s  a  ( loq_  Modlu/s junction  Logarttfemic  5rrairz  ,  rate  I  I  C.\JS)  £  or  and  of  Vi^ctu £  (Ajirer  Log  I S  internal.  jrequencj . ^  Oog  Qfi  a  Sec~*)  Friction Q <r  and  junctron  C-J  5t>e^  Zencr , 1948)  Dynamic Techniques When a linear viscoelastic body is subjected to a stress varying sinusoidally with time, i t s mechanical behaviour at a single frequency must be specified by two independent quantities.  These may be chosen in various ways.  The phase relationship between stress and strain are illustrated in Figure 5;. For deformation in shear, the component in phase divided by the strain is the real part of a complex modulus of r i g i d i t y , G'; the out of phase stress component divided by the strain is the imaginary part G".  The moduli G and G" may 1  be added vectorially on a complex plane to give G* = G' + i G " . The absolute modulus is |G*| =(,(G') + ( G " ) ) ' : 2  peak s t r a i n .  2  1/  2  this is the ratio of peak stress to  The phase angle between strain and stress is 0.  The two  independent quantities used to specify dynamic behaviour may be chosen as |G*| and 0 instead of G' and G".  For a single cycle of deformation at a  given amplitude, for energy stored and recovered is proportional to G , and 1  the energy dissipated is proportional to G".  In extension, the corresponding  quantities are E and E", the real and imaginary parts of a complex Young's E" 1  modulus E", that i s , p - = tan 0. In order to compare data obtained by different dynamic methods, i t is necessary to convert a l l data to a common basis.  The imaginary parts of  tfe modulus are damping terms which determine the dissipation of energy when a material is deformed. by "6 = |rr— . factor by  The logarithmic decrement 6, is given approximately  The half width of a resonance peak i s related to the dissipation  25  At  Helahonrti} Compliance  between  Comply  Module  and  Cornel  26  Dynamic mechanical properties may be expressed in terms of complex viscosities rather than by complex moduli,complex viscosities are defined by: n* =n ' :  -  14  TI"  The complex viscosities are related to complex moduli by such equations as: G =  can"  G" =  con'  1  15  Where co is the frequency of oscillations in radians per second, that i s , co = 2 i r f .  The real part of the viscosity  n'  is related to the imaginary or  loss modulus G". Dynamic properties are often expressed in terms of complex compliances, J* = J ' - i J " , rather than as moduli.  The moduli and compliances are related  by the equations:  a I so where  J' =  K  G"  J"  G' = ^ -  16  E"  = j-r = p G = (G') + (G")  1/  Q-r 2  2  Conildiirabli  2  and J  2  = (J ) 1  2  + (J")  2  theoretical and experimental work has been carried out  by dynamic tests on polymers, glass and rubber in Eirich (1958) (see Figure 6) and wave propagation in solids by Kolsky (1963).  The literature suggests  methods of f i t t i n g experimental behaviour of a material under investigation. A great role is played by viscoelasticity theories in the solution of experimental investigations.  For example, the functions of creep and  relaxation constitute a measure of the mechanical properties of the body. The solution to many viscoelastic problems have been simplified by the use of  27  Frequency  Low Or  Hiafi  Higft  Temperature  Or  Rubbery  Fre agency  Low Temp  Glas5y  o  77  Locj  Decreasing Fiqure  '—  Jemperahire  Defence  of  Restaur  Modulu,  On  Scale  Tem\>erahire  Rubbery •  Constant  at  Storaqe  Moduli*  C&r) , and  and  frequency  Frequency (G')  toe jor  ,  Low  Lo?A  (6"),  Modulus  factor  P«a*tic*  ^ <^d  28  the (Correspondence principle.  The Alfrey-Lee (1944, 1955) 'Correspondence  principle states that the Laplace transformed viscoelastic solution is obtained directly from the solution to an associated elastic problem by merely replacing the e l a s t i c constants by certain functions of the Laplace transform parameter depending on the nature of stress-strain relations.  The  character of the associated elastic problem is established by Laplace transforming the differential equation, and boundary and i n i t i a l conditions defining the viscoelastic problem.  Berry (1958) stated that for a steady  state sinusoidal stimulus, the solution to a viscoelastic problem is obtained from that of the corresponding elastic problem by the substitution of complex moduli or constants; for example,  where E is replaced by  E*(co)  and C^ and C^^ are elastic and viscoelastic con-  stants. Although i t is of considerable interest from theoretical considerations to try to analyse experimental results by using viscoelasticy theory, in practice, i t is generallythe measured values of modulus and viscosity are required.  Hence, one must be careful in applying viscoelasticity theory to  problems of rock mechanics and the limitations must be clearly understood before any attempts are made to use experimental results.  It appears that the theory  is useful in relaxation mechanisms and in dynamic measurements of internal f o c t i o n but the theory may not apply equally well to creep and flow of rocks. It may be necessary to try to isolate the coefficient of viscosity i f the laboratory tests do not duplicate the f i e l d conditions with regard to confining pressure and temperature effects.  29,  CHAPTER 4  i  TESTING PROCEDURE Description of Rocks Tested: The rocks used in this testing program were taken from three different locations in B r i t i s h Columbia. 1.  The igneous rock, granodiorite Ex-cores were taken from Highland  Valley, Skeena Silver Mining property 25 miles east of Achcroft in British Columbia.  A l l cores were taken approximately normal to the shear zone. 2.  The metamorphosed quartzites were taken from a pile of core  samples obtained from Pine Point area. 3.  The a r g i l l i t e s were from massive outcrops near Brittania Beach.  These samples were kindly provided by Dr. I. Bain of Mineral Engineering Department.  A general outline of the area from which the samples were  collected is shown in Figure (Z. Procedure for Collecting Samples from the Fieilid: Using a Brunton compass, the bearing was taken, the dip and strike of the face from which the sample was to be taken was marked. size of sample for sonic test was 6" x 6" x 12".  The required minimum  Samples of rock were brought  from the site (Brittania Beach) to the laboratory, suitably marked so that the o  orientation relative to the geometry of their surroundings was known.  The  preliminary attempts to obtain samples from a quartz mine had to be dropped because of the d i f f i c u l t y in d r i l l i n g core samples and due to the presense of too many cleavage planes in pure massive quartz rock.  It was possible to  L E G E N D -  Master  Joints:  $  Strong  Joints:  Strike?  t n k e  l  7  0  °-  Vertical  150° -  45 5W s  Figure  7 -  Sample 1  Oriented  Sample  Oriented  2  Road  Scum  Cut  of  PPane  Uferen.ce Reference  Loafcina  Brittania  East  Beacfi  PPane  Oft  Strike  190  5 trite  J60  Wcjftwa^  fcf>  7  Dib  7- - 8  5  ^  85* E  Mites  C O  o  31  TABLE I PETROGRAPHY OF ROCKS TESTED  Grain Size  Quartzite (Fine grained)  Mineral  Amount Present  Quartz Feldspar Mica  Large Large 10 per cent  Quartz Microcline Albite-01igocla.se Biotite-' Muscovite Calcite  Large Large Moderate Moderate Small Small  Silicates  Large  10u  Granodiorite (Medium to coarse grained) 0.5 mm Argillites (Fine grained) lOu  obtain cori;@fited samples of argy'llites only while the core samples kindly made available by a mining company in B r i t i s h Columbia were made use of for other tests.  The cores were carefully preserved and identified during the duration  of the testing program, "thin sections of core samples were prepared for petrographic studies.  Photo micrographs of these sections are shown in  Figure 8. Preparation and Description of Sample: Core samples 9.5 to 11 inches long and diameter from 7/8 inch (Ex Core) to 1 5/8 inches (Bx Core) were prepared in the laboratory using a diamond saw and polishing machine. Figure ;9.  Prepared core samples and equipment used are shown in  End pieces were used for specific gravity determinations and for  preparing thin sections.  An averaged value of specific gravity was calculated  and used in a l l test results.  Precautions had to be taken in preparing  specimens of a r g j l l i t e s for test purposes due to non a v a i l a b i l i t y of core samples. The prepared samples were weighed in air-circumferential and length measurements were determined. >The specific gravity was determined according to the procedure described by obert et al (1946).  The prepared samples were  a i r dried before testing in the sonic equipment.  Some of the highly hetero-  geneous samples and those with prominent cracks were rejected from the stock pile.  From the side trimmings of a r g j l l i t e rocks, samples for relaxation  tests were prepared.  Each of the samples were cut to form prisms with three  mutually orthogonal faces oriented on the prisms to agree with the orientation of the original sample collected from the f i e l d .  The a r g i l l i t e was cut int!o  FIGURE 8 Photomicrographs of Thin Sections of Rocks A - Quartzite B - Granodiorite x 160 x 150  33a  FIGURE 8 a Photo Micrographs of Thin Sections x 120  34  FIGURE 9 Photograph of Equipment Used and Prepared Core Samples  four parallel slabs retaining the original orientation of the sample.  From  these slabs, prisms of 1" x 1" x 10" were cut in another smaller diamond saw. The faces were made parallel with a polishing equipment.  The prepared samples  of a r g i l l i t e were tested and shear-time curves plotted. Description of Test Equipment: The testing program was carried out in a sonic equipment designed originally by Mr. D. W. Mckinley and modified suitably by the writer for the present experimental investigations. aluminium and brass.  Sonic tests were performed on s t e e l ,  After completion of these-tests, certain modifications  were made in the equipment, principally the checking of the performance of beat-frequency o s c i l l a t o r , preamplifier and voltmeter. change the receiver cartridge. patterns during the test.  It was necessary to  An oscilloscope was used to test the wave  At the end of the series of preliminary tests with  metals, a further change was made in the shielding procedure.  An amplifier with  higher output capacity was tried to improve the output while testing rocks for comparison purposes.  A number of t r i a l tests were then run with different  lengths and of different rock types.  But the noise level was too high and  made i t d i f f i c u l t to take any useful reading. output had to be dropped.  Hence, the idea of improving the  Shielded wires in place of ordinary wires were used.  The c i r c u i t diagram of the test equipment used is shown in Figure HO. The technique used was i n i t i a t e d by Wegel and Walther and developed to i t s present form by various investigators.  A beat-frequency audio generator type  1304-B was used, capable of producing 20 Hz to 40 KHz., in two ranges.  The  main control is engraved from 20 Hz to 20 KHz and has a true logarithmic scale.  Oscillator  Transmitter  Figure  10  -  Circuit  Diagram  oj  Receiver  Preamplifier  Apparatus CO  37 the of For^use^higher range throwing a panel switch adds 20 KHz to the scale frequency The frequency increment dial was calibrated from +50 to -50 cycles.  This was  used to obtain bandwidth measurements. -The accuracy of the frequency calibrati had been set to +(1% +0.5 cycles) when the o s c i l l a t o r had been set to the line frequency.  The output of the amplifier was amplified by a preamplifier with  a power output of 20 watts and frequency response 20 to 20,000 decibels.  <HZ:."!:G,.  +2  The amplifier has controls for eliminating noise in the input s i g -  nal and a master control to increase the amplitude of input signal. No significant distortion of the signal was noticed from the output of the amplifier. The vibrational energy was applied by means of a magnetic record cutting head.  The transmitter needle when brought into contact with the core  transmitted vibrational energy through the specimen.  The specimen was held at  i t s modal plane by a special clamp designed according to the specifications of U.S.B.M. Report of Investigations No 3891. The clamp made of aluminium had a V-groove cut to balance the specimen at i t s centre shown in Figure l-eK  If  necessary, the specimen could be clamped in the gjig with the clamp l i g h t l y touching the specimen.  The entrir.eframe of the [J% to support the specimen  was mounted on an ebonite board.  The energy transmitted through the core was  detected at the other end by means of a variable reluctance pick up cartridge. The output signal amplitude from the pick up (receiver) was indicated by the vacuum tube voltmeter. . A preamplifier before the voltmeter cuts out low frequency noise and increases the output signal.  From the readings of the  valve voltmeter corresponding to the different feequencies about a natural  38  Ftqure 0 -  5ORLC  Mounting  clamp  resonance frequency, the mechanical resonance curve for the specimen under test could be deduced. In Figure 12, is indicated the manner in which the cartridges (transmitter and receiver) were placed depending on whether longitudinal or torsional velocity is desired.  An adjustable regulated power supply type  1205-B, voltage output 0 to 300 volts was used in the c i r c u i t to supply power to preamplifier.  A 90 volt battery is connected to this c i r c u i t .  Frequency calibration was achieved by following the procedure recommended by the manufacturers.  The audio generator was connected to a  power source and the parallel ground connection was connected to an external ground in order to keep hum and noise to a minimum.. TIhe.:;frequency calibration was standardized usually at the power-line frequency or occasionally as a check at zero.  Since maximum s t a b i l i t y of calibration was desired, during the f i r s t  few minutes of operation the standardization was checked frequently.  The  frequency d r i f t was found to be less than ten cycles in the f i r s t hour of opeaation and was substantially stable after two hours.  The audio generator  was thoroughly checked at Electrical engineering workshop, using a different standard o s c i l l a t o r for comparison as well as a Tektronix type 561A dual trace oscilloscope. The line frequency standardization was done as follows: 1.  The frequency range switch was set to normal.  2.  The output control was adjusted to give a midscale meter indication.  3.  The output attenuator switch was set to the line frequency  calibration position.  40  Transmitter  p  Figure 12 -  Portion  of  Transmitter  lon^nai  Velocity  Tran/vew  Velocity  above  and and  —  Receiver below  for for  4.  The cycles increment dial was set to zero and the main dial to the  line frequency. 5.  The frequency was varied with the zero adjust control until the  beat between the o s c i l l a t o r and the power-line frequencies, as i n d i cated by the fluctuations of the meter pointer, was as slow as possible.  When the o s c i l l a t o r feequency was near the line frequency  there was a large amplitude fluctuation.  At multiples of line f r e -  quency small-amplitude fluctuations were obtained. 6.  To determine that the adjustments were made at the correct beat',,  the output attentuator was turned to a decibeT"position, and a rough check with zero beat was made. After carrying out the routine checks, the following alterations were made.  The pick up cartridge was changed, and shielded wires were used were for a l l connections. Resistances of 3300 and 120 ohms connected in parallel A  across the audio generator and amplifier.  Proper earthing and shielding  reduced experimental errors. Trial tests were run with s t e e l , aluminium and brass samples to check the performance of the equipment.  When i t was found that i t was performing  s a t i s f a c t o r i l y , tests were conducted with rock samples and the results wchen found reliable w2ine recorded. ;  Also, tests on rock samples of average lengths  6-12 inches were tested to study the effect of different lengths.  These t r i a l  runs were discontinued when standard lengths of 10 inches cares tested sonically gave more reliable restl-lts. Testing Technique: Core samples approximately 10.0 inches long and 0.85 inches to 1.10 inches  42  diameter were prepared (from cores broughtrfrom the mines) in the laboratory using a diamond saw and a polishing equipment.  Prepared coressamples were k r c -  air-dried for a week before testing them in the sonic apparatus. of rock were tested as follows:  The specimens  a schematic diagram of sonic equipment used is  shown in Figure 13. The power was turned on. preamplifier in the c i r c u i t .  The 90 volts battery was connected to the  Sufficient time was allowed for the tube compo-  nents to warm up and then a t r i a l test was run with either a steel or aluminium sample to test the experimental set up.  A d r i l l core sample was then balanced  at i t s centre on the j i g and the clamp l i g h t l y held the specimen in position. The vibrational energy was applied at one end of the specimen by bringing in the transmitter needle directly beneath and l i g h t l y touching the specimen.  The receiver was then kept at the proper position at the other end of  the specimen.  The positions of transmitter and receiver are shown in Figure 11.  The small increment dial was set to zero and the amplitude was adjusted to be of.a fixed value throughout the test run.  The frequency was  increased with the main dial and i n i t i a l l y the whole range was covered noting the amplitudes at different frequencies until the resonant frequency was found. This was indicated by the maximum deflection on the tube voltmeter.  Then, the  experiment was repeated and at every stage whenever a significant deflection of the voltmeter needle was observed, the main dial was set at this frequency and the small increment dial was used to get the correct amplitude. quency setting bandwidth was measured.  At this f r e -  This procedure was carried out throughout  Driver Osci itator  Figure  Poller  fti^R  Specimen  Amplifier  J3 -  Peck- Uf>  Schematic  Diagram  Cain  Hmf>dijCer  OJ"  Sonic  /J.C.  Voltmeter  Apparatus  CO  the frequency range on the main d i a l .  The maximum amplitude observed and the  corresponding value of frequency and bandwidth was used in calculating the modulus and specific damping capacity. The bandwidth measurements were carried out according to the following procedure.  The main increment dial was set at the resonant f r e -  quency, the frequency was decreased with the increment dial until the voltmeter reading was reduced to —\=r A , . where A .,„ was the maximum voltmeter reading in max max m  at that resonant frequency.  m  Next, the frequency was increased until the volt-  meter reading was again reduced  to —7= A . in max  The total change in frequency  required to produce these amplitude reductions was taken as Af, the bandwidth measurements.  These measurements were repeated at other frequencies when ampli-  tude readings were significant.  In the preliminary investigations, an o s c i l -  loscope was used to determine the fundamental resonant frequency and the corresponding voltmeter readings were observed.  The lengths of the core  samples tested sonically were determined accurately. The samples prepared for photoelastic work were observed under polarized light after allowing an adequate setting and curing time for the cement. Those rocks which showed isochromatics, a point at the highest observable shear strain (E-J - z~) was selected for measurement, an each photoelastic piece. Three rocks of radically different characteristics were compared.  They com-  prised a quartzite (K) from Pine Point, granodiorite (J) from Skeena and an argjillite (0) from Brittania Beach.  In each case, the pattern of s t r a i n , both  in magnitude and direction, appeared to be d i s t i n c t i v e .  The shear strain-time  curves for the orthogonal faces of the specimen of rocks, K, J* and 0 are shown in Figures 14 and 15.  45  70  60  to  3  0  Tune fin are l'4-r Sfieax  5trc\in  - Time  iz  '  Curve Jor  Hie  Vnree.  46  90  Figure  15 -  Sftear  Strain -Time  Orthogonal  Face*  Curve  of a  for  the  Quartette  Kree Specimen  CHAPTER 5 DISCUSSION OF TESTING TECHNIQUE Introduction The main purpose of the testing program was to try to determine the solid viscosity parameters of rocks using a forced vibration test and i f so to establish the relationship between this rheological parameter and i t s relationship to other structural properties of rocks.  It was important that the v a r i -  ables be kept to a minimum in these tests and so the other variables, such as temperature and confining pressure were therefore kept constant.  However, due  to the complex nature of forced vibration tests errors arise which may not affect static or monostonic creep tests in the same manner. In a forced vibration test, the aim is to subject a sample to a vibrational energy at one end and to measure the amplitude of the output at the other end.  It is often inconvenient to set a low "internal f r i c t i o n " , Q,  material such as rocks into free vibration.  More often, a condition of forced  vibration is set up in which the test specimen is driven at a constant amplitude; under these conditions, the fractional decrease in vibrational energy per cycle gives a measure of the internal f r i c t i o n , the response of this system being a maximum at the resonant frequency.  The energy of the vibration is  proportional to the amplitude squared, the log ' ' '  :  decrement being given  by: r  6  _-  TT  (Band j Width) _ uAf j— r  r  ,(io) \ nQ  Now since Q = ^ > then: -1  Af  (19)  Q  where 6 = Logarithmic decrement Af = Band Width f  = Resonant frequency Q = Internal f r i c t i o n  The d i f f i c u l t i e s in the measurement of internal f r i c t i o n by the forced vibration method ar.e' loss of energy at the supports, extraneous damping and the disadvantage of improper coupling between the driving system (transmitter) and the specimen.  Although adequate precautions were taken in these tests, certain  errors arise in measuring the amplitude and band widths at resonance frequencies. Phase Displacement Between Stress and Strain: In a forced vibration test dissipation of mechanical energy would cause phase displacement between stress and s t r a i n .  The phase angle or log decrement  is generally calculated from the band width which is determined by assuming that the rate of dissipation of energy in the material is not too high.  Leconte (1963),  and Attewell (1964) have indicated that the phase displacement  log  "(j)",  decrement 6, internal f r i c t i o n Q , and specific damping capacity S are simply related by the expression: S -  26  ?  TT/Q = ntan $  (20)  Kolsky (1963) has suggested that the relation between the log decrement and the coefficient of viscosity for longitudinal vibrations of a specimen is given by:  m  (21) where 6 = Log decrement E = Young's Modulus Pfj>== A factor depending on frequency and dimensions of specimen n = Coefficient of viscosity So that 2 percent error in the bandwidth measurements would result in an error of 5 percent in the determination of log.  .'  c decrement but s t i l l the relative  error in the determination of viscosity w i l l be within reasonable l i m i t s . The specific damping capacity is generally calculated by dividing the bandwidth at resonant frequency by resonance frequency and i t is assumed that the log decrement varies inversely both with the frequency and with the effective viscosity of the material.  Kolsky (1963) showed that for forced vibration tests  on s o l i d s , the log decrement does not behave as predicted for Maxwell and Kelvin elements and is often independent of frequency.  In general, decrement (log)  was found to be higher for coarse grained than for fine grained rocks. could be in agreement with Zeners  1  This  (1940) suggestion that the maximum damping  occured when the grain size was close to that predicted by equation: 3TTK'  where  a  average grain size Specific heat at constant pressure  p  Densi ty  K  Thermal conductivity  N o  Relaxation frequency  (22)  Zener s (theoretical relation between log decrement, internal f r i c t i o n 1  and frequency is i l l u s t r a t e d in Figures 16 and 17. These curves i l l u s t r a t e that experimental values of internal f r i c t i o n are higher than those predicted by theory and this indicates that other effects are becoming relatively more important. For polycrystalline materials, Zener has measured internal f r i c t i o n and concludes that the neighbouring grains due to different crystallographic directions with respect to the direction of strain w i l l be stressed by different amounts when the specimen is deformed.  Ke (1947) has investigated internal  f r i c t i o n by "viscous s l i p " at the crystal boundaries.  For rocks Emery (1960)  has explained grain boundary effect and comes to a similar conclusion.  It  was observed that considerable variation in test results for polycrystalline grano'diorite rock, in these tests could be attributed to the grain boundary effect.  The previous discussion serves to i l l u s t r a t e the complicated nature  of damping in heterogeneous materials and hence a discussion of testing technique is considered important to analyze the usefulness of the present investigation. The change in damping capacity occurs at the resonant frequency and slowly decreases in value on either side of this frequency.  In forced vibration  tests on specimens at different temperatures, pressures and loads the behaviour of rocks would be different from that of metals.  However, a similar trend  could be expected, with maximum strain occuring near the higher frequencies and different zones at lower and intermediate frequencies. It is apparent, therefore, that stresses and strains are not uniform throughout a sample.  The ratio between the peak stress and peak strain could  x I0" (jl :  Mi  11 1  1  T T"T~I 1 r  i  1 1 1-1—  */  4  x/^  */ */  1  \ X  1  p m L  i r e  1  1 1  J6 _  *  Exftrimerttul Points — Theoretical Curve  \  x  i  1  1  1 i • • • •  1,  Frequency ( HiXv:-:. )  Graphical  1  Representation  C /liter  l  1  oj  1  \_  L1  l 1 o o  Internal?  Friction  >9«-0  Uner,  .06 0 004 0003  350  84 ^  1  a 1000  100 0 Frecy-iercc y  Internal and  1  loo C Hz^>-; .:.,>  Friction.  RStger  Silver  (After  as  Aluminium. steeE: Brass 1-0  Measured lener  ,  by  BemteWi^ «  5®  be calculated in the manner explained before: E" tan  (j> =  | r r  since the actual peak stress and peak strain could not be reliably calculated for any element of the sample, the usual method of taking the damping at resonant frequency  was adopted.  It was hoped that the errors would be similar in  both direct and indirect tests and that calculated average values would provide reliable comparisons. The Effect of Environment and Supports in Forced Vibration Tests: It is common practice in materials testing to select a set of conditions so that the combined effects of the extraneous factors are relatively small and hence the measured results are, to as large a degree as possible indicative of the material properties. be eliminated.  Although these effects can be minimized, they can not  It was mentioned earlier that the temperature and confining  pressure were kept constant during the duration of the test.  A l l tests were •  conducted at room temperature and pressure and thus eliminating the effect of these two important variables. In the past, in order to reduce loss at supports, piano wire suspensions were used.  This eliminated the effect of supports in the investigation  of mechanical properties of soft materials such as rubber and polymers.  In .  the present investigation, to minimize the loss at supports, the V-groove support as suggested by Obert et al (1946) for hard solids such as rocks was adopted.  It was found that the losses due to supports were negligible since  the specimen could be held in position during the test run without using the -  clamp.  Recent developments in Rock Mechanics have generated considerable interest in the determination'of energy storage and energy dissipation properties of rocks.  Direct determination of such dynamic properties of rocks using  vibration methods can be quite d i f f i c u l t experimentally and requires considerable complex electronic instrumentation.  Hence, because of the complicated nature  of the material under investigation and due to the complexities involved in electronic instrumentation, the present investigation was confined to testing rocks at room temperature and pressure.  Furthermore, indirect methods using  transformation techniques of viscoelasticity are limited because of the general d i f f i c u l t y in obtaining analytic functions to adequately represent the experimental response obtained from quasi-static experiments such as forced vibration tests.  Because of these d i f f i c u l t i e s , approximation techniques take on  considerable importance and significance in evaluating the integral transforms in order to obtain rock response properties over the desired time (or frequency) scale of interest.  Such approximation techniques are presented in  the thesis to determine energy storage and energy dissipation properties of rocks using the results obtained from dynamic tests. Tests on rocks at room temperature and pressure requires a standardized procedure before i t could be easily adopted for high temperature and confining pressure work.  In the present investigation, a series of tests  were conducted at normal atmospheric pressure and temperature.  Although  the internal f r i c t i o n or viscosity measurements, would be expected to be more temperature sensitive; however, for these measurements, the order of magnitude of the parameters was more important than a measure of their temperature  5$  dependence, and hence temperatures were not controlled. Preliminary tests were conducted to determine i f the use of high temperature would allow measurement of damping. out with a group of granodiorites.  The experiments were carried  As the high temperature values of elastic  modulus E, shear modulus G and specific damping capacity S, were erratic and since high temperature produces a permanent change in the physical properties of the specimen, testing in this state gave no useful information and therefore had to be discontinued. Tests on 10 i n . long 0.886 i n . (Ex-Core) specimens disclosed that the moisture content in rock affects the values of elastic constants and the specific damping capacity,-necessitating a different test procedure that would take this into effect.  In the present investigation, i t was observed that for  most types of rock when the moisture content exceded 50 percent saturation, the value of the specific damping capacity was greater than the upper l i m i t of measurement of the apparatus.  Since the value of the damping capacity for the  saturated state could be made only with considerable d i f f i c u l t y , the testing was confined to air-dried samples. Effect of Size of Specimens: In the theoretical development forthe longitudinal and torsional velocities of sound wave given by equation: V  2fJ  (23) (24)  where, V  Longitudinal velocity of sound  55  = Torsional velocity of sound f-| = Fundamental longitudinal frequency f  t  = Fundamental torsional frequency  1 = Length of specimen i t has been assumed that the ratio of the radius to length of the specimen is small.  A correction factor for this effect is given by the equation:  where, V-j = True longitudinal velocity V^ = Longitudinal velocity calculated by Equation (23) m  ii = Poisson's ratio •r = Radius of specimen In the present investigation, the radius to length ratio suggested by Obert et al (1946) 1:10 was followed; for this ratio the correction factor produced less than 0.25 percent and therefore was neglected. The writer was concerned about the value of the specific damping capacity in sonic tests because of the inhomogeneous nature of the rock specimens.  During testing large variations in damping would take place due  to structural change and difference in grain size.  Hence, i t was thought  possible that log decrement might be more or less than usual.  A method was  derived for calculating difference in the value of specific damping capacity from the bandwidth measurements.  This was done by running tests at a slower  - rate, but allowing sufficient time for the experimental conditions to become  se  normal.  The method and results are discussed in detail in Chapter 6.  Internal Friction Measuring Methods: Although much of the theory on forced vibration tests was developed in the past 30 years, i t was only comparatively recently that experimental techniques have been available for testing many of the results of this theory. Electronic methods have greatly f a c i l i t a t e d both the production and detection of high-frequency e l a s t i c waves.  In the pre-electronic era, experimental work  on the propagation of e l a s t i c waves in solids was largely confined to the detection of seismic waves, and to the investigation of vibration at audiable frequencies in acoustical experiments.  Of the reports written on the sonic  methods of measuring the elastic and damping properties of rock, concrete, building materials, e t c . , the paper by Wegel and Walther (1935) described a method most easily adopted to the testing of diamond d r i l l cores.  In this  method, a cylindrical specimen supported at centre, is vibrated, f i r s t , in the fundamental longitudinal mode, which gives the fundamental longitudinal frequency, and, second, in the fundamental torsional mode.  From these experi-  ments, i t is possible to determine the longitudinal and torsional velocities of sound.  It is also possible to determine the specific damping capacity when  this method is suitably modified.  More recently, wave attenuation techniques  have been used to measure internal f r i c t i o n in rocks.  A mechanical pulse is  transmitted (through rock specimens sandwiched between two identical discs) allowing calculations of attenuation from the strength of the signal projected by specimen (see Figure 18). If the internal f r i c t i o n within the sample is changing, and i f i t is  Generator  Figure  )8 -  Piezoelectric  Excitation  °i  R  °  c k s  58  assumed that the change would be independent of the amplitude of vibration, the graph of amplitude against frequency on log linear paper would be a straight line.  In the freee vibration method, a movement of some observab.le amount must  f i r s t occur before the 1 ogar-ithmie decrement can be measured, and hence, i t is often inconvenient to set a low G)-factor material into free vibrations.  In the  same way for such materials, the resonant peak is too sharp for accurate work in the forced vibration tests.  Hence, considerable d i f f i c u l t y was  encountered in the measurement of damping capacity. A mathematical equation was derived by Quimby (1925) for viscosity for a given value of band width A f and resonant frequency f .  This expression  also depends on density, mass, shape of specimen and the nature of material tested. 3pft2A * A  9  (26)  J-  2irm  where p = density i = length of specimen A f = band width  2nf r m = an integer at resonant frequency^ = — — Terry and Morgans (1959) defined damping by a mechanical quality  factor given by equation: f "'ST For rocks tested between the ranges of band width A f , 20 to 120 Hz, and resonant  frequencies f , 1-10 KHz, the error was less than +5 percent.  Had the c a l i b r a -  tion of the small increment dial with which the band width was measured had a vernier scale, then the measurement of bandwidth (with an osci 1 loscope) could have been very much closer to actual value. Certain discrepancy occurs with the vacuum tube voltmeter due to i t s extreme s e n s i t i v i t y .  It was found that for rocks vibrating in longitudinal  mode, the range was in between .001 to .01 volts scale.  The deflection in this  case is a function of input energy from transmitter, the nature of sample material, and the output of the pick up needle.  Had a l l the connections been  made of shielded wires, a very much more accurate reading would have been obtained. In general, the values of damping is very much closer to the actual value for predicting the values of viscosity and consequently errors less than 5 percent have not been considered.  However, i f internal f r i c t i o n is measured  for metals and polymers, sensitivity would be an important factor. Rate of Testing: Since the frequency was to be the only variable, i t was necessary to have the frequency readings as accurate as possible in a l l tests.  In addition,  i t was rqquired that the oscillations had to be slowly built up as would give reliable values of amplitude for the frequency range covered.  The approximate  time for testing one specimen of rock in longitudinal and transverse mode was 20 minutes.  If the tests were run at a faster rate then i t became d i f f i c u l t to  measure resonant frequency and the corresponding maximum amplitude. Some^ preliminary tests were conducted at higher amplitude input but the results did  603  not j u s t i f y further attempts. two hours.  At best, three to four samples could be tested in  The readings were erratic after two hours operation, no readings were  taken after this period.  Since the testing was non-destructive in character, i t  was possible to repeat the tests as many times as required.  In forced vibration  resonance tests, since the amplitude at resonance rose very rapidly with maximum amplitude occuring after adjusting the small increment d i a l , a considerable portion;; ©f the investigation was sentred around t h i s . of the readings were taken during these operations.  In f a c t , two thirds  Therefore, the measured  values of damping at other harmonics might be considered unreliable.  However,  Lecomte (1963) points out that errors in the measured values of damping caused by resonance methods w i l l be less than 1 percent and is a function of the material.  For rocks, errors are l i k e l y to be small and consequently readings  at less than 5 percent accuracy may be quite acceptable.  (Vide U.S.B.M.R.I.  3891) Many investigators believe that under a wide variety of conditions of confinement, temperature and rate of loading untrasonic pulse technique could be used and reliable internal f r i c t i o n measurements obtained as a basis in the study of rock deformation.  Terry and Morgans (1959) studied the  rheological properties of coal, based on the Burgers model, using an-acpustic puslse technique^;.  They measured the internal f r i c t i o n and retardation time for coal as  a function of frequency and no attempt to determine the rheological parameter viscosity was made.  The experimental results proved that the simple Burgers  model does not completely describe the behaviour of coal under stress and suggested a suitable model,  With a large number of retardation elements  covering a wide range of retardation times.  61  Grain Boundary Effects in Forced Vibration Tests: In rocks, the intergranular and intragranular strains could be explained in a large part by the dislocation theory applied to polycrystals. Most rocks contain more than one kind of polycrystal and, then there w i l l be differential reaction between polycrystals because of their different physical properties across interfaces.  Stress relaxation along grain boundaries is  therefore an important source of internal f r i c t i o n . strated  Ke (1948) f i r s t demon-  the strong internal f r i c t i o n peak due to grain-boundary relaxation by  experiments on high purity aluminium wires.  The physics of granular media as  suggested by Emery (1961) was involved in this testing programme.  Emery (1963)  also observed that the cementing material between grains have characteristics akin to the cement in concrete.  It was observed that this cementing material  tends to be amorphous or cryptocrystalline to some extent and exhibits some degree of e l a s t i c i t y over short time intervals, but during f i n i t e times flow l i k e viscous liquids.  This is a characteristic of Maxwell liquids.  This  behaviour is consistent with the assumption that grain boundaries behave to a certain extent in a viscous manner and at elevated temperatures or frequency this effect is more pronounced. It was observed that attenuation in rocks tested was generally higher than those reported for amorphous substances of negligible grain size or in metals and this could be ascribed to the relatively inhomogeneous nature of rocks.  Attenuation increased with frequency to some l i m i t and that for  quartzites (fine grained, homogeneous, close-packed), the attenuation was moderate but linear with frequency (absence of scattering).  (Dn the other  62  hand, attenuation in a r g i l l i t e s was rather sharp and dependent on frequency (fine grained and close packing) while the larger grain sizes in granodiorites caused the attenuation to become independent of frequency.  There was some  scatter in the results observed with granodioirite samples.  The role of grain  boundaries and porosity effects seem to be important in this context. Frequency Considerations; In most resonance test equipment, the frequency applied is measured outside in the o s c i l l a t o r included in the c i r c u i t .  The vibrational energy is  applied through a pole piece attached to the specimen or through a magnetic record cutting head as in the present case.  In any case, this transmitter  coupled to the ends of the specimen reduce both the longitudinal and torsional frequency, thereby producing an error.  This small error could be corrected by  using the following equations, Obert et al (1946): .  where  f  b  -  f  s  .  % (1 •  ,  (, • &L)  (27) (28)  f^ = longitudinal frequency without pole-pieces = longitudinal frequency with pole-pieces f  = torsional frequency without pole-pieces  f  = torsional frequency with pole-pieces  w = weight of both pole-pieces 1  w = weight of specimen k = ratio of radius of gyration of pole-piece to that of specimen The correction for rock specimens amounted to less than 5 percent; hence, this  6:3,  correction is applied to a l l longitudinal and torsional frequency measurements. For torsional frequency corrections, Equation (28) was used. To minimize this correction factor, light weight transmitter and pick up cartridges have been used.  The ball bearing type with counterweights was  found to be the most suitable of these because the weight of the mountings would not produce any additional force.  A proper mounting frame was necessary  to have direct coupling during the duration of the test.  A check was made by  moving the transmitter and receiver to see i t s effect on resonant frequency and amplitude measurements.  It was thought that significant effects might  develop i f the cartridges were placed improperly.  Sutherland (1963) suggested  the position for placing transmitter and pick up for longitudinal and torsional oscillations.  Since the frequency range of interest was rather narrow a l l  measurements were confined to 10 Hz to lOKHz.  The frequency range of 0 to  1 KHz was found to be rather less amenable to experimental investigation. Losses (introduced in the driving and recording apparatus, losses from a i r f r i c t i o n and acoustic radiation vary with the type of vibration.  For  longitudinal vibrations they are less, acoustical radiation is negligible except on the end surfaces and air f r i c t i o n is also less.  In torsional vibrations,  acoustic radiation losses are low and a i r f r i c t i o n losses are small. Available evidence indicated that the errors are not l i k e l y to be more than .01 percent and the results of resonance frequencies measurements are reproducible within 5 percent. Considerations of the Anelastic Behaviour of Rocks: Rocks are known to be crystalline and granular heterogeneous media and adjust by intergranular movements and by crystal deformation.  Emery (1963)  described rock as a heterogeneous, aelotropic medium, a product of i t s own history from photoelastic experiments.  The static and dynamic moduli and  Poisson's ratio measurements have indicated that rocks exhibit directional properties.  Hence, any measurement of viscosity of rocks has to take note of  the directional properties of rocks.  From the previous discussions, i t is  obvious that comparable specimens must be comparatively-oriented.  One would  expect different results from a specimen oriented by d r i l l i n g perpendicular to the bedding and from one d r i l l e d parallel to the bedding, or in any other direction.  Since there was no way of checking how the d r i l l cores were  originally obtained, i t was not possible to analyze the directional properties of the granodiorites and quartzites.  But the following procedure was adopted  for chekcing the anelastic properties of a r g i l l i t e specimens. Samples of a r g i l l i t e specimens were cut in a diamond saw into prisms 1 i n . x 1 in.xlO i n . and polished in a small diamond saw.  Oriented samples were  cut for relaxation studies out of the original rock and the specimens were allowed to relax for 24 hours after which time, testing was commenced immediately. In dynamic tests, i t was assumed that a l l stresses applied was a result of the previous strain history of the sample.  The tests using photoelastic coatings  on three orthogonal faces of a r g i l l i t e specimens indicated shear stress. shear strains (e-| - z^) were high and are shown in Figure  The  15'.  The difference in viscosity values is considered to be due to the nature of rocks. rocks.  It is closely related with the inherent or residual stress present in  If further relaxation is allowed by creating a new face, the rock relaxes  due to structural rearrangement, while i f i t is prevented, there is a change in  6|  the magnitude of s t r a i n .  Therefore, some of the damping values measured could  be attributed to the residual stresses present.  Perhaps this influences the  directional properties of rocks under dynamic tests.  However, the test results  were f i n a l l y analysedAfor viscoelastic materials since thri.se was one of the purposes of this investigation.  66  CHAPTER 6 DETERMINATION OF VISCOSITY FROM FORCED VIBRATION TESTS Introduction Phase displacements between strain and stress is always present in some magnitude in dynamic sinusoidal tests.  Change in internal f r i c t i o n in  solids is caused by difference in phase displacements within the sample and the resulting changes are referred to as analastic effects. The method of measurement was chosen such that the phase displacement could be expressed in terms of specific damping capacity or log decrement.  If  displacements along different axes had been observed as in the case of slowly applied angular frequencies to rods fixed at one end whose displacements could be observed at the other end, i t would have been possible to relate the measurement of displacements to viscoelastic coefficients.  In the present case, however,  i t is the ratio of output to input amplitude that was measured.  Also, the  internal f r i c t i o n was measured from band width and resonant frequency measurements.  The viscosity coefficient is related to log decrement, resonant f r e -  quency and elastic modulus according to the formula: irE  Thev:above formula was derived from theoretical considerations by Kolsky (1963).  However, i t does not allow the calculation of average solid viscosity  coefficients.  An expression for determining coefficient of viscosity was  derived by Potter (1948) based on the assumption that the viscous stress at any point in the material is proportional to the strain velocity at that point.  6>7)  This was then considered to determine the upper bound for the average degree of dissipation in the material.  Since the rate of dissipation in forced vibration  test is known to be a constant, the expression for estimating viscosity would not be suitable for estimating the rheological parameter viscosity and was not intended to be so. Alternate methods for estimating viscosities were therefore considered'. Since test data was to be analyzed on a computer, numerical methods of relatively complex form could be tolerated.  Three methods were considered and w i l l be  referred to as Method 1, Method 2 and Method 3.  The following common assumptions  were made: 1.  The rocks chosen is assumed to be reasonably homogeneous. .  2.  Elastic restoring force is proportional to the displacements.  3.  Dissipation force is proportional to velocity.  4.  Internal f r i c t i o n depends on frequency i f the dissipative forces  are of a purely viscous nature. 5.  The material obeys linear viscoelastic laws within the frequency  range considered. 6.  The behaviour of the rocks can be considered by a combination of  Maxwell and Kelvin elements. 7.  Sample deformations are small.  8.  The stresses and strains applied are of a negligible magnitude.  9.  The radius to length ratio of specimen is small.  10.  Only harmonic wave propagation is considered.  It is assumed for both methods that the material is linearly elastic  ©8  and inertia effects are negligible.  However, inertia effects can not be neglected,  without introducing an error in a l l calculations, particularly in wave propagation methods.  The average viscosity coefficient rather than the maximum value  has been calculated.  The reason for this is that a relationship between  viscosities and internal f r i c t i o n was being examined and i t was thought that the viscosities should be the average v i s c o s i t i e s .  It w i l l be shown that when  the degree of dissipation in a material is high, as i t should be in a forced vibration test, the maximum viscosity is 3 1/2 times the average.  So that the  maximum value of viscosity is readily obtained from the average and vice versa. METHOD 1: The determination of internal f r i c t i o n or indirectly viscosity is complicated because of the heterogeneous nature of rocks.  Variation in test  results under forced vibration tests are caused by the : anisotropic behaviour of rocks. tions.  The choiee=of testing methods is dictated by the aim of the investiga-  Here, the prime purpose was to analyse the viscoelastic nature of rocks.  The internal f r i c t i o n is given by the formula:  Q = 2TT(^)  (29)  1  where Q = Internal f r i c t i o n AW = Energy lost when the body is unloaded W = Total elastic energy stored in the stressed body The above formula was derived from theoretical consideration by Kolsky (1963). However, i t does not allow the calculations of viscosity parameters.  An  expression for damping was derived by Wegel and Walther (1935) by considering the viscosity of a vibrating medium. The mathematical theory underlying forced vibrations could be developed for longitudinal vibration of a solid with support at one node, the driving force applied at one end, and the amplitude of vibrations measured at the other end. 1.  The basic assumptions made were as follows:  The viscous stress at any point in the material is proportional  to the strain velocity at that point. 2.  The viscosity is inseparately associated with the shearing  motion, and the coefficient of viscosity, n , of a homogeneous material dx dx is defined by the equation a = n  where  is the time of shearing  strain and a is the contribution of the viscous forces to the corresponding shearing stress. When the specimen is caused to vibrate by an external source and the frequency of the driver is varied, at the resonant frequency, the amplitude w i l l pass through a maximum value, and a plot of amplitude versus frequency was shown in Figure 1.  Increased damping tends to reduce the amplitude and broaden the  base of this resonance curve.  Based on this phenomena, an expression relating  log decrement, the band width of the peak, and the resonant frequency has been derived.  Potter (1948) derived an expression by considering the viscosity of a  vibrating medium.  -  :  .  ;  The log decrement based on the peak width measured at any fraction of the maximum amplitude reads:  (30)  where A f is the width of the resonance peak at amplitude A , in cycles per A  second, and A  i s the maximum amplitude at resonance frequency f .  m a x  Potter measured A f from the width of the resonance curve at an ampli tude A  x  = 0  -  7 0 7 A  max  <> 31  (that i s , 2 2 = A ^ ) A  and the equation reduces to 6 =  *f  (32)  r  This compared favourably with that of Zener (1948)  " h r  (33)  r  From Figure 1, Amax „ i s the maximum value of amplitude and occurs at the resonant m  3  frequency f  r  r  when A f = 0. A i s some other value of amplitude (U-jj less than A  which occurs at the frequencies f-| and f^.  m a x  For materials in which, n is small,  the value of f - j ^ i s small compared to f , and the frequency can be considered to be equal to f r  Then the variation in the absolute value of the amplitude U-j  is determined, according to Potter, by the term /  (  16Tr^co^m^,  / , v2 , „ 2 2/Af^v\ 1  2~4"—Mn )  + V m (-E—;)  p c where co = 2prf, f i s frequency  r  ^  (34)  ,  71 2  m = An integer given by c = jE/p  V  - - Velocity of propagation of wave  p = Density of material Term ( 3 4 ) describes the motion of the end of the rod in the neighbourhood of the resonant frequencies in i t s simplified form.  From this term, i t could be shown  that: A1  2  1  ) max-'  p~A  =  (JT~  ,  »9p c  ,Afx  4to ( n f  f  r  (35)  9"  )  r  2  where A f = —^— " n could then be calculated from an experimental resonance l curve by choosing any convenient ratio for ^— and using the corresponding A-, max value of, At. A convenient ratio is ^ = 0...707 . (36) max Therefore, 1  A  r  9 P  2  4u>  C 4  (n'T  2 ( 22 Tf - >  and, , _ 3p r  n  -  {  A  r  2  =  1  < > 37  Afx  p  2um  where A f is the width of the resonance curve at an amplitude A^ = 0.707 A  m a x  .  Although the above method is of considerable interest from theoretical considerations for determining damping, in practice, i t is generally the rheological parameter, viscosity, is of particular interest in rock mechanics work.  The damping value calculated from above has no significance in visco-  elastic calculations.  Hence, i t would be necessary to try to determine  viscosity by laboratory tests which could be duplicated in the f i e l d to obtain design data.  72  Method 2: This method is based on the equation of motion of a vibrating rod: (Reiner, 1960) P = M^'+ n£*+ Ec  (38)  where P = Impressed force M = A factor which depends on mass and shape of specimen n = Coefficient of viscosity E = Elastic modulus The damping term is assumed to depend on the nature of dissipative forces, this quantity involving the dimensions of the specimen. forces is discussed in detail elsewhere.  The nature of the dissipative  According to linear viscoelastic theory,  the damping forces are of purely viscous nature and the phase angle between stress and strain depends on frequency. Method 2 was found to predict viscosity values for longitudinal vibrations of rocks assumed to behave as Maxwell element.  However, Kelvin solid  element behaviour was predominant in the final adjustments for times long compared to the duration of the time of dynamic tests.  It was f e l t that the values  of viscosity thus derived, could be used to predict the viscoelastic behaviour of rocks.  An examination of the work of Kolsky (1963) and Reiner (1960) i n d i -  cated that the basic equation of motion could be used to yield a much better expression for viscosity parameters i f certain assumptions were made.  This  alternative approach is discussed in this section and is considered to have more merit than Method 1.  The mathematical derivation is given in Appendix II-I.  7J8  The equation of motion of the rod is given below: |^x dt  +  n  dx  +  E  x  =  P  (  t  )  ( 3 9 )  J  where  w = Weight of the material g = Acceleration due to gravity x = Displacement E = Elastic Modulus n = Coefficient of viscosity  This equation could be solved either by using the usual methods for solving second order linear differential equations or by using Laplace transform. two methods are discussed in greater detail in Appendix I.  The  The solution for  this differential equation.is given by: M  =  —  1  /  [1 - ( a > / u ) ] 2  2  2  + [2U/ui )(c/c )] n  c  (40) J  where M = Magnification factor co = Angular frequency = 2irf co = Natural frequency of the system c/c  c  = Damping ratio It is seen from expression (Si) that for a Maxwell element, the log  decrement varies inversely both with the frequency and with the effective viscosity of the material and directly proportional to frequency for a Kelvin element.  Consequently, the viscosity is a changing parameter and could be  obtained only by an indirect method such as the one discussed here.  7/4  Expression (40) is very readily programmed for the computer. much simpler than the expression involved in Method 1.  It is  Since i t involves no  material analysis, and in f a c t , could be easily calculated without a computer. Only one unknown, the damping ratio c/c appears in the equation instead of c  more than one occuring in Method 1.  However, an assumption was made that the  log decrement was a function of frequency.  This necessarily implies that #re  log decrement should be a function of frequency throughout the frequency range of interest, which would probably be s t r i c t l y true only i f the material studied were homogeneous and isotropic.  For frequency range of 100 Hz to lOKHz and at  room temperature, this expression is considered to give good approximation of viscosity values. The variables in Equations (21) are; the factor B-J which includes the frequency f , the dimensions of the specimen, area and length, and the elastic modulus, E, and log decrement, 6. The length and area of cross section are easily found.  The elastic modulus is calculated from resonant frequency  measurements.  Since the log decrement calculated from band width measurements  were not very accurate, a method based on measuring magnification r a t i o , M, and frequency r a t i o ,  was developed and a theoretical resonance curve was  compared with the experimental resonance curve.  The correct damping ratio  assuming the theoretical resonance curve to be reliable was used.  This  improved the accuracy of a l l calculations and proved as a double check on the measurement of experimental damping values.  The calculated values of viscosities  for different rock specimens are shown in Table II.  The relation between  viscosity and frequency plotted on a log-linear paper w i l l be useful in  TABLE II RESULTS FROM FORCED VIBRATION TEST 1  GROUP  —  i  i  SAMPLE NO.  LENGTH CM.  K .  J  mice  [  r  .  —  " •  1  LOG.DECREMENT 6 •  .  :—i  YOUNG'S  VISCOSITY nxlO POISE DYNES DYNES/CM " SEC CM"  .. MODULUS.  Ix-loM  ?  2  1  25.60  2.75  42.0  6250.0  0.0191  2.776  7.307  2  25.20  2.70  38.0  6400.0  0.0196  2.893  7.234  3  24.90  2.74  54.0  6300.0  0.0209  2.804  6.675  4  25.00  2.68  62.0  6100.0  0.0213  2.739  6.226  5  25.80  2.62  32.0  • 6400.0  0.0206  3.176  7.561  6  25.30  2.80  44.0  6200.0  0.0274  3.071  5.688  . 1  25.60  2.58  . 120.0  4680.0  0.0806  1.496  1.247  2 .  25.50  2.54 ••  115.0  5020.0  0.0720  1.654  1.439  3  25.60  2.54  105.0  4820.0  0.0684  1.638  1.559  24.88  2.62  130.0  4630.0  0.0882  1.417  1.090  25.60  2.52  .95.0  4920.0  0.0607  1.693  1.781  6  25.80  2.65  125.0  4650.0  0.0844  1.627  1.302  1  25.44  2.54  _65:o  5800.0  •0.0352  2.316  3.563  2  25.48  2.46  . 75.0  5860.0  0.0402  2.454  3.273  3  25.54  2.62  68.0  6100.0  0.0350  2.705  3.978  4  25.68  2.56  72.0  5960.0  0.0379  2.523  3.505  .  4  i  5  0  DENSITY  .. RESONANCE. BAND WIDTH FREQUENCY Af . f C/6EC  -fi _  —  —  •  9  76  30  to —  T f e o r e t c c a t Carve  X  Experimental*. Pointjs  DaiupLf^  Factor  0-4  0-6  = O025  10 9  vj  6 ^  5  0  Fiqure  19  ReSONAMCE  0-8 Frequency CURVL  '-0  1-2  1 - 4  Ratco FOR  A  QLWRTZITE  ROCK  predicting viscoelastic behaviour of rocks.  It was not possible to study the  effect of amplitude of vibration on damping.  The magnification ratio versus  frequency ratio co/<^ calculated from Equation (40) are shown in Figure 19. Since the tests to determine the directional properties were run with the a r g i l l i t e samples only, the effect of size and shape of specimens are not considered, the. samples in the shape of a rectangular prism is'assumed to behave similar to the core samples.  The average values of viscosity as a function of  log decrement and resonant frequency are shown in Table II.  The coefficient  of viscosities calculated from Method 2 are seen to be different from the measured values of viscosities calculated by assuming retardation time (x=n/E), where x is inversely proportional to frequency (Terry and Morgans, 1958). Thes.e: discrepancies are explained in detail in Chapter 7. It is f e l t that Method 2 gives a reliable measure of coefficient of viscosity from forced vibration tests.  However, since the test results on  rocks are not very accurate without a s t a t i s t i c a l average being computed, a considerable number of tests have to be carried out with a number of samples before these results could be used in practice.  This investigation is of a  preliminary nature to indicate the usefulness of forced vibration tests in the laboratory.  It is possible to determine a relationship between coefficient of  viscosities or damping and frequency in rocks. Method 2 also has interesting alternative of predicting viscoelastic behaviour of rocks under varying viscosity conditions when phase angles are measured in vibration tests.  If the viscosity is small, the curve has a sharp  peak and i f viscosity is high, the curve is f l a t .  Again, by Method 2, i t is  '78  possible to find a viscoelastic solution depending on the model assumed.  For  any rock type for a given phase angle and creep or complex compliance, i t is possible to reduce test data to model parameters, so that a measure of the actual behaviour of the rock under test could be obtained by comparing with standard viscoelastic models. Measurement of phase angle, frequency and subsequent calculation of viscosity and complex compliances in dynamic tests would allow a very simple check on the concept of viscoelastic behaviour of rocks.  If rocks having  identical mineral content ,, porosity, texture e t c . , are to have the same 1  structural behaviour then their viscosity parameters under identical conditions should be checked.  Since the change in structural behaviour depends on i  viscosity, i t appears to be a suitable method for predicting rock deformation. The structural viscosity, i f i t could be accurately predicted from such experimental methods, could be used in predicting the behaviour of rocks in mine design in the future. Method 3.: Viscoelastic Solution to a Forced Vibration Problem;  Hardy (1959)  presented a method for predicting stress-strain relationships from the results of creep tests.  This method enables prediction of viscoelastic parameters  without using indirect methods of transformation techniques of v i s c o e l a s t i c i t y . This method also appeared to predict that i f the relationships between stresses and strains were unique, then the modified Burger's rheological model (with N different retardation times) would be more useful in analysing the viscoelastic nature of rocks, which is in general agreement with that of Terry and Morgan. However, since i t was found that the stress-strain relationship from dynamic  09  tests is not"!easily obtained s a another viscoelastic method was further examined.  Although i t s usefulness in the present investigation was rather  limited but i t affords ample scope for future investigations i f the mathematical basis underlying forced vibrations are to be studied. This method assumes ithat the deformations are small and that the complete stress-strain relationship of a linear viscoelastic solid copld be described from the complex modulus and the phase angle measurements in a dynamic test.  Data obtained from dynamic test measurements, under sinusoidal  loading conditions, can be carried over to any of the other description of stress-strain behaviour and used in the solution of various problems. Details of the connecting transformation may be found in literature. method to be described was mentioned briefly in Chapter 5.  This  A detailed derivation  of the mathematical basis underlying forced vibration test is examined in Appendix III.  However, the derivation indicates that i t w i l l not be that  easy to program this method.  The number of variables involved are not known  directly from experimental measurements.  Therefore, the viscosity values  determined by Method II were^sed in determining the shape of the frequency response curve and a solution to the forced vibration test on rocks assumed as a viscoelastic material was examined.  It can be inferred that this approach  is in reasonable agreement with the ones previously discussed.  80  CHAPTER 7 TEST RESULTS Introduction: The main purpose of the testing program was to determine, i f for rocks, a unique relationship existed between viscosity parameters and log decrement, which is independent of frequency. The main program of testing consisted of six samples of rock type.quartzite (k), 6 samples of rock type granodiorite (J), and 4 samples of rock type (0) a r g i l l i t e , a l l rocks were obtained from British Columbia and tested in the laboratory by forced vibration methods.  In addition, relaxation tests  were performed on rocks tested sonically to determine their directional properties.  At least three tests were performed for each sample of rock  tested so that the reproducibility of results could be checked.  Test data  was analyzed on the I.B.M. 7040 computer at the University of British Columbia. Results from a l l the tests are shown graphically and in tables in the following pages. Relaxation characteristics of a r g i l l i t e s are shown in Figure 15. Resonance curves are useful in interpreting the results discussed in subsequent sections.  Resonance curves for the three types of rocks from forced vibration  tests are compared in this chapter.  Log decrement corrections, the p o s s i b i l i t y  of predicting theoretical resonance curve relationships and the effect of damping ratio on the shape of this curve are also discussed. Characteristics of Rocks Tested the The most d i f f i c u l t problem in attempting to determine relationship  between viscosity and damping ratio is to find a rock of sufficiently uniform composition that test results could be compared for different samples. frequency versus log decrement relation is shown in Figure 20.  The  Although!the  previous investigators indicated that the log decrement from a l l tests on aluminium l i e on a single curve, whereas decrements from tests on rocks show some difference.  There was some d i f f i c u l t y encountered with granodiorite and  a r g i l l i t e samples. The results were not that consistent with;the theory. The six samples of specimen quartzite (k) and the six samples of granodiorite were obtained from core samples in boxes and in fact had to be picked out at random.  It is unfortunate, therefore, that the specimens could not be selected  carefully and analyzed.  A r g i l l i t e s were carefully prepared samples so the  difference in shape of specimen is the^only factor involved. •j  When comparing resonance curves from different rocks, i t is necessary to consider the locations from which the samples were collected, grain s i z e , and directional properties of specimens selected for the test.  The difference  in damping ratio for forced vibration tests (Figure 20) appears to have l i t t l e effect on the viscosity values calculated from these tests as may be seen in Table II.  The effective viscosity of the quartzite rock are very similar to  that of granodiorites tested.  It appears that for the samples of quartzites  tested, the log decrement values determination affect variation in viscosity values.  It was concluded from this evidence that for the purpose of  comparing viscosity values, i t would be reasonable to assume that the resonance curves of the specimens of rocks and their damping ratios should be obtained. Magnification and frequency ratios were plotted for different damping  82  xio  Figure  ZO -  *ot  of  Fr^ufl^ c  Versus  Log  decrement  83  ratios rather than amplitude versus frequency.  The magnification ratios or  more correctly amplitude r a t i o , M, is given by M =  In forced vibration tests, the input amplitude is small and the ratio of input to output is therefore considered in finding the resonance curve.  Since the  effects of gain in the system are being examined, i t appears more reasonable to compare the ratios of amplitudes rather than individual amplitudes.  Also,  the frequency ratios form a common basis on which to compare the test results for different specimens. The maximum magnification versus frequency ratio obtained from test results is shown in Figure 21.  It is seen that while samples of damping ratio  0.02 had a magnification ratio of 26.0 samples with low damping show maximum value of magnification ratio 55.0, thus indicating the influence of damping ratio ori the shape of the resonance curve.  This is in agreement with the general  observation that the material with low damping ratio has a sharp peak as compared to material with high-damping.  It w i l l be shown later that materials  with different damping capacities behave differently.  Magnification versus  frequency ratio is plotted for three different samples from forced vibration tests.  Each test is shown separately so that the reproducibility of results  can be examined.  It is seen that the damping ratio has a remarkable effect on  the shape of these curves as would be expected.  For quartzite specimens with a  damping ratio of 0.021, the curve is sharper than for granodiorites with a damping ratio of 0.06.  There is similarity between a l l the resonance curves  84  <  /\ -  Theoretical  Carve  Q  Expe rime ntafi  Ci  <  ) -  Points Jjarnbiria  Factor =r  O02 \  \  \  \ -  F rec^aency  Fi a ure  21 -  Gmbfl  0/  Frequency  Ratio  Magnification Ratio  for  Factor a  Versus  Quart^te  R  o  c  k  85  for a l l tests and are represented in Figures 22a, 22b and 22c.  The magnification  ratio or amplitude ratio continues to rise with frequency, the rise is pronounced at resonance frequency and decreases gradually after t h i s . is symmetrical about this peak.  The resonance curve  There is some d i f f i c u l t y in obtaining the  i n i t i a l and after resonance points in the curve since the amplitude is very low and i t was d i f f i c u l t to obtain accurate readings from the voltmeter.  At  maximum damping ratio of 0.009, the magnification ratio is 55.6. The corresponding 9 value of viscosity is 1.09 x 10 .  Maximum viscosity values were computed  corresponding to damping ratio of 0.02 and this appears to be a characteristic of the rock tested.  This could be due to the grain size and texture of  rocks.- Attewel1'arid Brentnal1 found similar results from resonance tests by using a magnitostrictive method on calcareous sandstone and Darley Dale sandstone.  Although they did not calculate absolute values of v i s c o s i t i e s ,  the values suggested for quality factor, Q, are in reasonable agreement.  They  f e l t that the graph of amplitude against frequency on log-linear paper w i l l be of a straight line form, provided that the internal f r i c t i o n is not dependent upon the amplitude of vibration.  On the other hand, i f damping is a function  of vibration amplitude, then the slope of the resultant curve w i l l define log decrement for a given amplitude. Attenuation against frequency for quartzite and a r g i l l i t e specimens from forced vibration tests are shown in Figure: 23, and that for a granodiorite in Figure 24.  It may be seen that the attenuation exponent appears to vary  approximately linearly with frequency over  a range  of 0-6000 Hz and  generally conforms with the results of pervious workers.  This is  86 30  A 2.0  TfteoreXLcol  x  curve  Experimental  Dambing  Points  Factor-OOZI  10  3 .a  8  7 y  6  VJ  S  4  0-2  0 4  0-8  0-6  Frequency Figure  '9  22 a  Graf>8 Frequency  of  10  16 -  1-4  Ratio  Maj^ni/tcfliton Ratio  1-2  jot a  Factor  Versus  cjuart^ite  Rock  1-3  87  Ratio  for  a  amnodiorite  Rock  88  ZO-  F^urezzc-  Gra|>ft  0  Magai/icaKoiz  Ratio  for  cm  T  AgiLute  ^dor Roc*  Verju*  Fr^umy  Freo^uenc^  FLflure  2.3  -  A Li emiat ion  C KH*/second  Versus  )  Fre<juencjj  Figure  24 -  Grc^G for  a  of  Frec^uenc^  Qranodlorite  Versus S\>zcwen  Alien  further evidence of attenuation at the lower porosity ranges and is due to scattering at grain boundaries.  The attenuation of longitudinal waves appear  to be proportional to frequency at frequencies in the range of 10 to 6000 Hz. Maximum attenuation occurs at a frequency of 5950 Hz for a quartzite specimen. Maximum viscosity values occur at a damping rate of 0.02 and varies from 5.0 x 10 to '7.0 x 10 for quartzite samples (k), 1.5 x 10 Poise for (J) 9  9  9  granodiorites (J) and a r g i l l i t e s 3 x 10 Pbjse: 9  negligible and are not shown.  T h e  Porosity effects were  The grain size and the particular packing  pattern undoubtedly play a major role in the behaviour of rocks. A comparison of retardation time against frequency is shown in Figure 25.  It may be seen that the retardation time is a strong function of  frequency.  The maximum retardation is at resonant frequency(frl'/w) while the  different retardation times at lower and higher frequencies contribute in different proportions to the mechanical losses. Comparison of Resonance Curves for Different Rocks Resonance curves determined from three different rock specimens are -  shown in Figure 26.  It may be seen that the curves of amplitude ratio versus  frequency rattos for the three rocks have different shapes near resonance frequencies.  Thus, for rocks,there does not seem to be a simple unique  relationship between log decrement and other parameters.  The data shown in  Figure 26 was prepared from the amplitude to frequency ratios for different damping ratios.  The other variables such as grain s i z e , directional properties  porosity and texture have been neglected in arriving at this relationship. Since rock is a granular, aelotropic and heterogeneous material th'e-re; can not be a simple unique relationship.  Only a general relationship such as the one  Ficjure  25 -  Variation Frequ.en.cy  Oj  Retardation  Time ( T )  ^  L  93  30  20  -  Tfteoretiaafi  Curve  ®  Quartzite  Rock  Aranodiorile  Rock  10  u  o -L->  a  9 8 7  04  oz are 26 -  06  Resoaance  of  Roc J<5  0-6  /-0 Fredas ricjj Ratio Curves  /or  Itircc?  Specimens  9/4  discussed in Chapter 5 can be derived.  It is f e l t necessary to infer conclusions  based on the experimental results available so far. It was stated earlier that longitudinal and torsional waves are  test propagated in different ways thus necessitating a different procedure for analysis.  In longitudinal vibrations, the stress is uniform throughout the  section of the specimen, but varies along i t s length, being zero at a loop and maximum at a node.  In transverse or torsional vibrations, the stress varies  over the cross-section of the specimen as well as  along i t s length.  If the  damping loss is due to the relative motion of particles in the material, the stresses are greater.  The effect of micro cracks and orientation of specimens  in studying the damping capacity measurements is different in different modes of vibrations. Due to practical d i f f i c u l t i e s only, a few tests were performed in torsional vibrations.  The comparison between longitudinal and torsional  vibrations and analysis for latter is not considered very satisfactory for checking any hypothesis.  The relationships between solid viscosities and log  decrement is assumed to be identical to that for longitudinal vibrations. Nowacki (1963) developed a method by which the forced vibration method could be analyzed by numerical methods.  Attewell (1964) suggested a simpler  experimental method for studying the dynamic properties of rocks using shear waves.  This method is basically identical to that proposed by Jamieson and  Hoskins (1963) in transforming waves i n i t i a l l y generated in longitudinal mode into the shear mode and then after passing through the test specimen can be converted back in the longitudinal sense for ease of measurement.  Shear waves  TABLE III RESULTS FROM FORCED VIBRATION TEST (TORSIONAL MODE) _  - 1  : ,  1 i  FREQUENCY-—eftoup  -NO.  V  --  CYCLES SEC.  K  J  0  BAND WIDTH  :  .LOG DECREMENT 5HEAR MODULUS  Af  - -G-x 10  8  "  x 10 \ 6  11  " DYNES/CM  2  VISCOSITY •9 n x .10  1  4500.0  14.0  0.980  1.44  10.28  2  4850.0  16.0  1.040  1.66 •  10.38  3  4200.0  12.0  0.900  1.25  10.40  4  5200.0  24.0  1.450  1.75  7.30  5  ' 5250.0  28.0  1.670  1.87  6.66  ••6  4620.0  21.0  1.430  .1,71  8.12  1  2380.0  28.0  3.700  .376 .  1.34  2  2360.0  38.0  5.060  .380  1.00  3  2060.0  60.0  9.150  .280  0.47  4 '  2560.0  72.0  8.830  .458 '  0.64  5  2420.0  54.0  7.040  .441  0.82  6  2380.0  46.0  6.840  .402  0.88  1  3200.0  22.0  2.160  .728  3.31  2  3450.0  26.0  2.370  .889  3.42  3  3640.0  34.0  2.940  .936  2.75-  4  3420.0  28.0  2.570  .893  3.19  ;  i  .  *96  have received rather less experimental attention than longitudinal waves.  In  the present analysis, the measurements and calculations for torsional vibrations were made similar to that for longitudinal vibrations.  For torsional  vibration, the damping values in general, were found to be much less and are shown in TaBlTe IV.  A typical resonanee curve obtained for torsional mode is  shown in Figure 27.  In Figure 28, resonance curves from longitudinal and  torsional vibrations from forced vibration methods are compared. on quartzite (K) are shown.  Only tests  In finding resonance curves, the values of shear  modulus and the corresponding damping ratios (different in longitudinal and torsional vibrations unless otherwise stated)(were taken into account.  It is  seen that the resonance curves for the two modes do not l i e on the same curve and that for other rock types, there is even poorer agreement.  Since the  torsional vibration tests were conducted in a similar way, the values of viscosities (shear) show general agreement with that obtained from longitudinal vibrations.  This substantiates the findings that for rocks there is not a  relationship (simple) between viscosity, thus computed and other parameters or that the viscosity values cannot be computed directly from forced vibration tests.  It may be of interest to note that had forced vibration  test results been used for f i t t i n g to a model such as suggested by Terry and Morgan (1958), the tests would have to be more refined thannat present and the mathematical analysis w i l l be moee complicated.  The concept of applying only  e l a s t i c i t y theory to the study of rock behaviour under such conditions is no longer considered v a l i d .  97 200  100 90  80  10 60 50 40 _  30  x  20  Theoretical  Experimental  Cur\/e  Pointy  c o Damfnita u  Ratio  =  I  00041  V 7 6  /  k  \  Frequency Figure  27  -  Grab/?  0}  Ratio jor  Ratio  Maj^rujication a  Quartzite  Factor Versus  Rock  (  Frequency  Torsional Mode)  98 200  100 30 80 70 60 50  _  40  u  Theoretical  Curve  Experimental  VoLr&f,  Torsional  Mode Experimt t&aL ftx/ths Lort^KudircaL Mods  30  o u.  Z0  o  <d io 8  6 5  02.  0-4  0-6  0-8  10  Frequency Figure  28 -  Comparison  oJ  Resonan.ce  curve for  Darrc|>in^  Ratio  Torsional  Factors  12.  a * °  -  and  Lon^ituAina.1  Quartette Rock 00047 O0Z5  14  9,9  Viscoelastic Properties of Rocks Viscoelastic properties of rocks were discussed in Chapters 2 and 6 and i t appears that the elastico-viscous method as set out by Emery  (1962)  that  rocks behave as a combination of Kelvin and Maxwell elements is the most logical approach.  These rheological concepts have been applied to the rocks tested and  the results are in general agreement with the predictions of the theory. If a Newtonian dashpot is inserted in series with the standard linear model, the resulting Burgees model (fluid) allows for a degree of permanent strain.  The equations for Burgers model can therefore be written in the following  form: a + l|{}p = M ( + t £ ) §  where  e  k  T~ = r ^ / ^  This equation implies that a , the attenuation constant, is defined such that the amplitude of an elastic wave f a l l s to 1/eth of i t s i n i t i a l value after travelling a distance 1/e in the rock.  The viscoelastic constants E^, E~, n-j and  ri2~  a r e  the elastic and viscous components associated with the Maxwell and Kelvin elements respectively.  To evaluate the significance of these viscoelastic  coefficients in rocks, time-dependent strain experiments, i . e . creep or dynamic tests w i l l be necessary. -This was indicated in Chapter 3 . The retardation time versus frequency for the rocks tested was shown in Figure 2 5 .  The coefficient of attenuation, a , was calculated to be  -4  4 . 1 2 x 10 " for a quartzite rock and the corresponding retardation time, the r a t i o , T , between viscosity and elastic modulus was found to be, therefore, 3 . 8 x 10 .  Hand calculations of a l l the test results shown would have been  IQO  time consuming and the accuracy would have been less.  In addition, certain  results are not the direct result of measurements, but rather those evaluated from mathematical relationships.  From the retardation time versus frequency  curve, i t is seen that the former is a maximum at resonant frequency since i t is assumed to be inversely proportional to frequency^  Hence, the material exhibits  an internal f r i c t i o n peak at a frequency equal to x ~ \  If we assume that the  Burgers body closely approximates the rock behaviour, the attenuation constant .of sound wave travelling through the medium could be calculated.  The relation-  ship between coefficient of attenuation and the retardation time, x , associated with the dashpot, has been shown by Terry and Morgans 2co  xq C  -  (a^-co^/c ) - p w^/E 2  s  (1958)  to be:  = 0  (42)  where co = 2 i r f ,  Angular Frequency  c = Velocity of propagation of wave p = Density of material = Young's Modulus resulting from the combination of springs ii Maxwell and Kelvin elements in the model In the aforementioned assumption that rocks could be reasonably represented by a Burgers model, the attenuation in the solid is given by: a  sinjTl .(gjqjjjr).  (43)  length  where n  number of harmonics  A simpler approximate expression was derived by Born  (1941)  viz.:  101  w  a=  From the values obtained for a , E , and x , the viscosity coefficient can be s  calculated: n„  s  =  E s"T  Some results derived by this writer using the resonance technique are given in Table IV. X2~l/cu  The values for retardation time are calculated using the expression  instead of the expression given in Equation (43).  This was necessary  to obviate the d i f f i c u l t y encountered in the expression 42 where really two equal quantities are to be subtracted. inaccurate results of retardation time.  Hence, the arithmetic expression gave The difference in actual values  calculated assuming the simpler expression was found to be negligible. a l l results are expressed using this method.  Hence,  In determining E , f i r s t E was s  calculated by using resonance frequency, length and density measurements. Subsequently, E was calculated assuming the relationship given by Terry: g  1/B  S  = 1/E  ;1  +  1/E  2  Considering that rock is a relatively complicated material for experimental analysis and that the relationships presented so far were developed for idealized materials, the results especially for forced vibration tests appear to be in reasonable agreement with the theory. In terms of rheological models, Kolsky (1963) using the data on high polymers derived by Lethersich (1950) demonstrated that both the Maxwell and Kelvin models were inadequate for describing the frequency response of  TABLE IV TEST RESULTS FROM FORCED VIBRATION TEST  RESONANCE • NO. • 'FREQUENCY '  GROUP  BAND WIDTH  INTERNAL  WAVE VELOCITY  Af  FRICTION  C x IO  •  CYCLES SEC CYCLES SEC" 1  1  K  Q  FREQUENCY .  5  ATTENUATION  u'x 10  CONSTANT  4  Cm. SEC"  RADIANS SEC"  1  1  a  X  10. . .  YOUNG'S MODULUS E xlO  RETARDATION TIME  VISCOSITY  n x 1.Q DYNES SEC CM" 8  1 1  DYNES CM"  ;  T 2  x 10"  . SEC  3  6250.0. • '  42.0  148.81  3.20  3.9Z7  4.12  2.82  | 3.789  2.67  . 2  6400.0  38.0  168.42  3.26  1 4.021  3.70  2.81  4.188  2.94  3  6300.0  54.0  116.67  3.14  3.958  5.41  2.70  2.947  1.98  4  6<b00.0  62.0  98.39  3.05  3.833  6.39  2.49'  2.567  1.60  5  6400.0  32.0  200.00  ' 3.30  4.02T  3.04  2.86  4.973  3.55  6  6200.0  44.0  140.91  3.14  3.895  4.41  2.76  3.617  2.49  1  4680.0  120.0  39.00  2.40  2.941  15.70  1.48  1.326  4.91  2  5020.0  115.0  43.65  2.56  3.154  14.10:  1.66 '  1.383  .5.76,  3  4820.0  105.0  45.90  2.47  3.029  13.40"  1.55  1.516  5.86  4  4630.0  130.0  35.62  2.30  2.909  17.70  1.39  1.224  4.26  5  4920.0  95.0  51.79  2.52  3.091  11.'80*  1.60  1.675  6.70  6  4650.0  125.0  37.20"  2.40  2.922  16.40  1.53  1.273  4.86  "'. "1  5800.0  65.0  89.23  .2.95  3.644  6; 92  2.21  2.448  1.35  .  5860.0  75.0  89.71  3.12  .3.833  6.86  2.54  2.340'  1.49  6100.0  68.0  78.13  2.99  3.682  7.89  2.19  2.122  1.16  72.0  82.78  3.06  3.745  7.39  . 2.40  2.210  1.33  .  J  0  1  ANGULAR  2 ' 3 4  ;  5960.0 '  :  103  materials but that the standard linear solid gave reasonable agreement over about a decade of frequency. ITerry measured the frequency response of coal and found even a four parameter model inadequate to represent the mechanical response of coal.  Hardy (1959) from his experimental work on creep found that the  behaviour of rocks could be reasonably represented by a Burgers model.  The  Burgers model assumed appears to answer many questions regarding the behaviour of rocks.  The present work although of a preliminary nature, indicates that  the behaviour of rocks could be approximated to by a combination of Maxwell and Kelvin elements in series.  But the different relaxation mechanisms operating  in a relatively complicated material such as rocks cannot be described completely by a simple relationship such as that given by a four parameter model. Frequency Response of Rocks: At low stresses, rubbers and plastics follow a linear viscoelastic law.  The stress limits of linearity are of the same order of magnitude for  both rubbers and plastics but the limits within which the strains are linear become smaller in p l a s t i c s .  If the viscoelastic behaviour is linear the shear  modulus can be expressed in terms of real and imaginary parts, the former representing or describing the stiffness of the material and the latter describing the damping capacity, (Chapter 3).  Since the dynamic behaviour  of these materials is strongly temperature and frequency dependent, one would expect to find a c r i t i c a l frequency band or bands within which the internal f r i c t i o n becomes more pronounced. modulus and the  IIQSS  The dependence of storage modulus, loss  factor n , on temperature and frequency for plastics and  rubbers was shown in Figure 6 in Chapter 3.  m Frequency effects on rocks have a particular significance in seismic prospecting where the frequency composition of a pulse w i l l determine i t s behaviour in rocks of different mineral content and grain size.  Some of the  important research work carried out by different investigators have been discussed in greater detail in Chapters 2 and 3. Methods of predicting viscoelastic behaviour are examined in this section.  Terry (1958), Hardy (1959) and Attewell (1964) have presented methods  for predicting stress-strain relationships. in detail in Chapter 3.  These methods have  been discussed  An attempt has been made to apply these methods to rocks  but the results are in satisfactory agreement with the measured relations only in limited cases. Hardy (1959) and Robertson (1963) presented methods and results from monotonic creep and dynamic tests on rocks and other materials by which stress-strain relationships could be predicted.  In the results of creep and  dynamic tests conducted, the stress and strain levels were low and the effect of confining pressure and temperature were taken into account in the final analysis.  The method presuppeses a unique relationship between stress level  and material propertis of rocks.  Since for rocks, the relationships could  hardly be considered unique, the method is not s t r i c t l y applicable.  However,  i f i t is assumed that the stress levels are negligible and the testing is done at room temperature and pressure, the viscosity values and i t s relationship to other rheological parameters can be predicted according to linear viscoe l a s t i c i c i t y theories.  In the previous discussions, i t was observed that the  viscosity values and modulus values from forced vibration tests were found to  be reasonably r e l i a b l e . From the phase angle or internal f r i c t i o n values storage modulus (G ) 1  and loss modulus (G") were calculated for different frequencies.  Complex  compliances were extrapolated and graphs of log.frequency versus log compliance (J')  and that of J " are i l l u s t r a t e d in Figure 29i. The method of calculating  viscosity values from phase angle measurements was indicated in Chapter 3. The frequency response for a Maxwell element is idealized and shown in Figure 30.  The increments of frequency and complex compliance values were determined  from assumed model relationships.  No direct tests were performed to determine  the stress-strain relationships.  The following stress-strain relationship for  a Burgers model was used: (l-e' 2) + E + t/r,-,] t / T  e - a[l/E  1  x  2  2  (44)  where 2  = g/E n  e = Strain o = Applied stress t = Time after the application of the stress T = Retardation time E.| and E = Elastic elements 2  n-| and n = Viscous elements 2  The values of n and E have already been determined and substituted in equations  106  and  J ( ) 2  w  = Vn  2  + l/n  3  —  j^T  2  E + a; n 3  where u> is the frequency at any instant.  3  It was assumed that the Burgers model  with i t s single retardation time, gave only an approximate description of the mechanical behaviour of rocks.  The fact that a number of short retardation  times are present makes the observation and interpretation of the complete time-strain curve d i f f i c u l t .  Therefore, the forced vibration method with i t s  relative advantage of a rapid loading rate is assumed to be more useful in f i t t i n g test data to model parameters.  By these phenomenological approach, the  complex compliance relationships were calculated and shown in Figure 321.  It  is seen that although the predicted shape of the curve is approximately the same as measured relations, the correlation could "inot be considered as s a t i s factory.  Since strains could not be measured directly for the frequency range  covered, streas-strain relationships cannot be predicted easily from forced vibration tests.  Moreover, many such time-strain tests at various frequency  ranges w i l l be required. Bland and Lee (1953) presented a method for matching experimental results to phenomenological models.  However, this method is based on the  validity of the experimental results and the assumed model behaviour.  Figure  3c2 shows the measured values satisfactory representation by a four-element model over a frequency range of 100-10,000 Hz. J  1  Comparison of variation of  with frequency measured experimentally and corresponding to four element  model and similarly for J " are shown in Figure 32.  It should be emphasized  that when such a representation of a viscoelastic material by a simple model,  60  Figure 31- Grapft.  Versus J .  no  Frequency  in  (4-0  izo  100  ®  Experimental  1  Computed  Points  |rom  the moded  « 80 <==J  e o u  60  X  \ \  \  \  \  o u  40  \ \  \  \ N  20  \  \  N  \ \ : \ s\  o  io  10  30 Freo^Liencj^  fiaure A  33 -  Comparison  0J  5.0  A-.O  j  Variation  of J With z  1?3  valid over a narrow frequency range, is used, i t pertains to the material and the particular loading program studied and that the same material would be represented by a different viscoelastic law for a different loading program. The foregoing description..of f i t t i n g to a model is by no means theonly procedure.  It is indicated here to show the usefulness of such analysis in the  study of dynamic properties of rocks.  Actual f i t t i n g to a model w i l l be more  complicated and a fourier analysis w i l l be required to f i t to any physical model.  113  CHAPTER 8 CONCLUSIONS AND SUGGESTIONS FOR FURTHER RESEARCH CONCLUSIONS: Test results presented in the previous chapter lead to the following conclusions: 1.  The results indicate that the coefficient of solid viscosity of rocks tested 9 10 v i z . quartzite, granodiorite and a r g i l l i t e , is of the order of 10 to 10 Poises in longitudinal mode. order of lO ^ poises. 1  In torsional mode, the viscosities were of the  The lowest recorded viscosities were that of grano-  q  diorites (0.47 x 10 poises). 2.  For rocks there is no simple unique relationship between viscosity parameter and log decrement measured, which is independent of frequency and modulus or alternatively there is not a unique method for determining viscosity for rocks.  The viscosity parameters and their relationship to other parameters  can be determined by postulating rheological models.  The viscosity to mass  ratio suggested by Emery (1963) needs further examination. 3.  The linear viscoelastic solution appears to apply to the study of rock behaviour within experimental limitations.  But this depends very much on  the frequency range chosen i f an attempt is made to f i t to a model.  The  determination of viscosity parameter enables the formulation of a model to predict the time-strain behaviour of rocks.  This was found to be approximately  true for Maxwell and Kelvin elements and the Burgers model formed by combining the two elements in series.  This is true only in the frequency  range of 100-6000 Hz.  This suggests the p o s s i b i l i t y that the internal f r i c t i o n ,  log decrement, and viscosity a l l arise from the way in which solids dissipate energy and that, in f a c t , there is only one fundamental damping parameter viscosity, n , which corresponds to attenuation of waves in layered material, and this is dependent on strain and strain rate.  The test results further  implies that viscosity, n , depends on s t r a i n , i t is also dependent on grain s i z e , grain packing, and the number of contact points in a sample.  However, the  directional nature of viscosity i s of particular interest in practical design problems, since i t is this property that determines the strength of rock structures. 4.  The Burgers model assumed for predicting viscoelastic behaviour of rocks needs to be further examined with more experimental results.  It may be  necessary to have a considerably modified approach since rocks do no have a unique stress-strain relationship. 5.  The elastic theory alone for predicting stress-strain relations does not apply to mine rocks.  6.  Viscosity values for rock samples tested by forced vibration method are less than that quoted from creep testsr  The difference could be attributed  to the test being conducted under no load conditions. 7.  Measurements taken with resonance apparatus shows that the viscoelastic properties of rocks are sensitive to temperature, rate of deformation, (internal energy changes) and amplitude of deformation.  The desired  environment must be produced by designing a suitable equipment preferably methods using ultrasonic pulser\for making dynamic measurement. •:..  115  Suggestions for Further Research: During the course of this testing program and the subsequent preparation of this thesis, interest developed in the following topics: 1.  It was suggested in this thesis that viscosity is a fundamental rheological parameter and is directional in character.  This could be checked by  dynamic tests on oriented core samples. 2.  In the discussion of test results, i t was suggested that the viscosity of a rock is dependent on load and the rate of application of load.  This could  be examined by designing a suitable apparatus where measurements could be taken at different rates of loading. 3.  Since viscosity of rocks is dependent on temperature and confining pressure, i t is f e l t that very useful information with regard;'- to rock structure could be obtained from laboratory and in situ tests conducted in different areas in British Columbia.  The viscosity concept of the earthls- mantle could be  checked in this manner.  LIST OF SYMBOLS A--Area of cross-section A--A constant a--Average grain size a--Viscoelastic constant (stress) b--Viscoelastic constant (strain) c--Velocity of propagation of wave Cp--Specific heat at constant pressure C/C --Damping ratio c  D--Constant, time derivative E--Young's Modulus Ei--Real part of elastic modulus E"--Imaginary part of Elastic Modulus E*--Absolute value of Elastic modulus f--frequency f^—Resonant frequency f^--Fundamental longitudinal frequency ^--Fundamental torsional frequency g--Acceleration due to gravity G--Shear modulus G'--Real part of shear modulus G"--Imaginary part of shear modulus G*--Absolute value of shear modulus  13117  i~/T  •  J ' - - R e a l part of complex compliance J"—Imaginary part of complex compliance J*--Absolute value of complex compliance k--Ratio of radius of gyration of the pole pieces to the radius of gyration of the specimen K=-A constant K --Thermal conductivity 1  ^--Length m--A constant, mass M--Mass of specimen M--Magnification Factor M - - S t a t i c (or relaxed) elastic modulus s M'--Weight of transmitter and receiver n--Number of harmonics N --Relaxation frequency Q  ,-P-|--A factor depending on frequency and shape of specimen p--Impressed force P, Q--Differential Operators Q--A Constant, internal f r i c t i o n r--Radius of specimen ^ - - R e l a x a t i o n (Maxwell) time at constant strain ^--Retardation (Kelvin) time at constant stress t--Time U--Displacement  U.8  U^—Amplitude ratio ^ - - L o n g i t u d i n a l velocity of sound ^ - - T o r s i o n a l velocity o f sound W--Weight of specimen W--Total energy stored in a body W—Weight o f both the pole-pieces x--Distance x--Displacement a--Attenuation constant 6--Logarithmic decrement Af--Bandwidth e--Strain e - - S t r a i n rate Aa)--Energy lost when the body is unloaded n--Coefficient o f viscosity x--Wave length p--Densi ty a--Stress T--Retardation time 4>--Phase angle between stress and strain ^--Angular frequency u> --Natural n  frequency  LIST OF REFERENCES 1.  ALFREY, T., 1944, "Non Homogeneous stresses in viscoelastic media", Quarterly Applied Mathematics, Vol. 2, pp. 113-119.  2.  ATTEWELL, P . B . , and BRENTNALL, B., 1964, "Internal f r i c t i o n : some considerations of the frequency response of rocks and other metallic and non-metallic materials", Int. Jour. Rock Mech. Mining S c i . , Vol. 1, pp. 231-254.  3.  BENNEWITZ, V.K., and ROTGER, H., 1936, "On the internal f r i c t i o n of solid bodies: absorption frequencies of metals in the acoustic range", Physikal Ztschr., Vol. 37, pp. 578.  4.  BENNEWITZ, V.K., and ROTGER, H., 1938, "Internal f r i c t i o n in s o l i d s " , Part Ztschr. Tech. Phys., Vol. 19, pp. 521-526.  5.  BERRY, D.S., 1958, "Stess propagation in viscoelastic bodies", Jour, of Mechanics and Physics of Solids, Vol. 6, pp. 177-185.  6.  BIRCH, F., 1943, "The e l a s t i c i t y of igneous rocks at high temperature and pressure", Bulletin Geological Society of America, Vol. 54, pp. 263.  7.  BLAND, D.R., 1960, "The theory of linear v i s c o e l a s t i c i t y " , Pergamon Press, N.Y., 1960.  8.  BOLTZMANN, L., 1876, Pogg. Ann. Erg., 7, 624.  9.  HOPKINSON, B., And Williams, G.T., 1912, "The elastic hysteresis of s t e e l " , Proc. Roy. S o c , A. Vol. 87, p. 5(52. BORN, W.T., 1941, "The attenuation constant of earth materials", Geophysics, Vol. 6, pp. 132-148.  10.  II,  11.  BRUCKSHAW, J . M . , and MAHANTA, P.C., 1954, "The variation of elastic constants of rocks with frequency", Petroleum, Vol. 17, pp. 14-18.  12.  CADY, W.G., 1922, "Theory of longitudinal vibrations of viscous rods", The Physical Review, Vol. 16, pp. 1-6.  13.  DICKSON, E.W., and STRAUCH, H., 1959, "Apparatus for the measurement of internal f r i c t i o n and dynamic young's modulus kilocycles frequencies", Jour, of S c i e n t i f i c Instruments, Vol. 36, pp. 425-428.  14.  EMERY, C.L., 1961, "The measurement of strains in mine rocks", Int. Symp. on Mining Research, Roll a, Missouri.  15.  EMERY, C.L., 1962, "The photoelastic technique for studying rock strains", Trans. Can. Inst. Mining Mett.  m  16.  EMERY, C.L., 1964, "Strain energy in rocks", State of Stress in the Earth's Crust, edited by W.R. Judd, pp. 235-279.  17.  EIRICH, F.R., 1958, "Rheology Theory and Applications", Vol. 1 and 2, Academic Press Inc. Publishers, N.Y.  18.  GEMANT, A . , 1935, "Compressional waves in media with complex viscosity", Phys. Vol. 6,pp. 363-365.  19.  GEMANT, A . , and JACKSON, W., 1937, "The measurement of internal f r i c t i o n in some solid d i e l e c t r i c materials", P h i l . Mag., Vol. 23, pp. 960-983.  20.  GRIGGS, D., and HANDIN, J . , 1960, "Rock deformation", Geological society of America Memoirs.  21.  HARDY, H.R., 1959, "Time-dependent deformation and failure of geologic materials", Quart. Col. School of Mines, Vol. 54, No. 3, pp. 134-175.  22.  HONDA, K., and KONNO, S . , 1921, "On the determination of the coefficient of normal viscosity of metals", P h i l . Mag., Vol. 42, pp. 115-123.  23.  JAEGER, J . C . , 1962, " E l a s t i c i t y , fracture and flow", Methuen's Monographs on Physical Stubjects, p. 208.  24.  JEFFREYS, H., 1952, "The earth, i t s o r i g i n , history and physical constitution", Cambridge Univ. Press, N.Y.  25.  JENSEN, J.W., 1959, "Damping capacity-its measurement and significance", U.S. Bureau of Mines, R.I. No. 5441, p. 46.  26.  KE, T . S . , 1947, "Experimental evidence of the viscous behaviour of grain boundaries in metals", Physical Review, Vol. 71(8), pp. 546-553.  27.  KELVIN (Sir. W. THOMSEN), 1875, " E l a s t i c i t y " , Encyclopaedia B r i t t a n i c a , 9th Edition.  28.  KIMBALL, A . L . , 1941, "Vibration problems, Part IV-Friction and damping in vibrations", Jour, of Applied Mechanics, Vol. 8, A37 and A135, Trans. ASME.  29.  KOLSKY, H., 1963, "Stress waves in s o l i d s " , Dover Publications, Inc., p. 213.  30.  KRISHNAMURTHI, M., and BALAKRISHNA, S . , 1957, "Attenuation of sound in Rocks", Geophysics, Vol. 22, pp. 268-274.  31.  LECOMTE, P . L . , 1963, "Methods for measuring the dynamic properties of rock", Third Symp. on Rock Mechanics, Queen's Univ., Kingston, Ont., p. 11.  N.Y.,  121  32.  LEE, E.H., 1955, "Stress analysis in viscoelastic bodies", Quart. Appl. Math., Vol. 13, No. 2, pp. 183-190.  33.  LETHERSICK, W. and PELZER, H., 1950, "The measurement of the coefficient of internal f r i c t i o n of solid rodssby a resonance method", British Jour. Appl. Phys., Vol. 1, pp. 18-22.  34.  MASON, W.P., and MCSKIMIN, H.J., 1947, "Attenuation and scattering of high frequency sound waves in metals and glasses", Jour. Acous. Soc. Am., Vol. 19, p. 464.  35.  NADAI, A . , 1963, "Theory of flow and fracture of s o l i d s " , McGraw-Hill Book Co., Inc., N.Y.  36.  NOLLE, A.W., 1948, "Methods for measuring dynamic mechanical properties of rubberlike materials," Jour. Appl. Phys. Vol. 19, p. 753.  37.  NAWACKI, W., 1963, "Dynamics of e l a s t i c systems", Chapman and Hall L t d . , London, p. 396.  38.  NOWICK, A . S . , 1953, "Internal f r i c t i o n in metals: progress in metal physics", Vol. 4, 15, Pergamon Press, London, pp. 1-70.  39.  OBERT, L., WINDES, S . L . , aod.DUVALL, W.I., 1946, "standardized tests for determining the physical properties of mine rock", U.S. Bureau of Mines, R.I. 3891, p. 67.  40.  OBERT, L., and DUVALL, W.I., 1967, "Rock mechanics and the design of structures in rock", John Wiley and Sons, Inc., N.Y.  41.  PESELNICK, L., and OUTERBRIDGE, W.F., 1961, "Internal f r i c t i o n in shear and shear modulus of solenhofen limestone over a frequency of range of 10 Hz", Jour. Geophys. Res., Vol. 66, pp. 581-88. 7  42.  PESELNICK, L., and ZIETZ, I., 1959, "Internal f r i c t i o n of fine-grained limestone at ultrasonic frequencies", Geophysics, Vol. 24, pp. 285-296.  43.  POTTER, E.V., 1948, "Damping capacity of metals", U.S. Bureau of Mines, R.I. No. 4194, p. 48.  44.  QUIMBY, S . L . , 1925, "On the experimental determination of the viscosity of vibrating s o l i d s " , Phys. Rev., Vol. 25, p. 559.  45.  RANA, M.H., 1963, "Experimental determination of viscosity of rocks", M.Sc. Thesis, Department of Mining Engineering, Queen's Univ., Kingston, Ontario.  1:22  46.  RANA, M.H., and MCKINLAY, D.W., 1967, "Experimental determination of viscosity of rocks by a sonic method", Can. Inst, of Mining and Metallurgy Annual Meeting held in Ottawa, p. 19.  47.  REINER, M., 1960, "Deformation, strain and flow", 2nd Ed., H.K. Lewis and Co. L t d . , London.  48.  ROBERTSON, E.C., 1963, "Viscoelasticity of rocks", State of stress in the Earth's Crust, Edited by W.R. Judd, American Elsevier Publishing Co., Inc., N.Y., p.181.  49.  SPINNER, S . , REICHARD, T.W., and TEFFT, W.E., 1960, "A comparison of experimental and theoretical relations between Young's modulus and the flexural and longitudinal resonance frequencies of uniform bars", Jour. Research. Nat. Bur. Standards, Vol. 64A, pp. 147-55.  50.  SUTHERLAND, R.B., 1963, "Some dynamic and static properties of rock", Rock Mechanics, edited by C. Fairhurst, The MacMillan Co., N.Y. pp. 473-491.  51.  TERRY, N.B., and MORGANS, W.T.A., 1958, "Studies of the Rheological Behaviour of coal", Mechanical Properties of Non-Metallic B r i t t l e Materials, Butterworths S c i e n t i f i c Publications, London, pp. 239-258.  52.  WEGEL, R.L., and WALTHER, H., 1935, "Internal dissipation in solids for small cyclic s t r a i n s " , Physics, Vol. 6, pp. 141-157.  53.  ZENER, C , 1948, " E l a s t i c i t y and Anelasticity of metals", Univ. of Chicago Press, Chicago, 111., p. 170.  APPENDIX I METHOD II Measurement of Dynamic Viscoelastic Properties of Rock; Forced Vibration—KELVIN SOLID Consider the Kelvin Solid Model shown.for equilibrium of such a system under the influence of a periodic external impressed force 2 g  d t  + n -JT +  2  "dt  Ex = F^ cos  wt Mass Mr  i.e. x  +  1  £m + m = „ ocos x  ut  F  kelvm  Substituting for ^ = c E x + cx + - x m  = F cos  cot  Model  (1)  o  •-  Assuming as usual a solution of the form Y = A cos cot + Bsin cot" and substituting in equation (1), collecting terms, and equating to zero the coefficients of cos cot and sin cot, we obtain the two conditions (E  -  co m)A + cocB = F 2  Q  - cocA + B (E - com) = 0 2  from which we find immediately  1243  OlC  [E - A ]  2  cos  j m]  x =F  p  + [oic]  2  [E  -  0  sin  ait + [oic]  oj m] 2  +  2  [we]  hit  2  E -  o [E  -  A]  2  +  (oic)  LJ  2  aiC  IE  J  ai m] +  -  2  2  [E  sin [aiC]  -  ai m  ai m] 2  2  +  COS ait [aiC]  2  ait]  2  Now referring to the triangle shown in Figure 1, i t is evident that x can be written in either of the equivalent forms: F [E  -  (cos  oi m] 2  2  +  [oic]  /  [E  -  oi m]  +  [oic]  2  [E  -  ai m] +  (aiC)  2  2  a  + sin  ait  sin  a)  2  COS (ait 2  cos  ait  -  a)  ooc  X =  /  (cos  2  2  aifc sin  6 + sin  u>t  F.  i  [E  =  sin  ai m] + 2  2  [aiC]  (3)  COSB)  (ait  +#)  2  The f i r s t of these equations is the more convenient because i t involves the same function (the cosine) as the excitation term in the Diff. equation. Hence, the phase relation between the response of the system and the d i s turbing force can be easily inferred. f i r s t expression for x.  Accordingly we shall continue with the  If we divide the Numerator and Denominator by E and rearrange s l i g h t l y , we obtain F  X = • [1  -  o  =  co m/E] 2  +  2  [coc/E]  F  [1  -  co (E/m) 2  +  2  6  (1  -  co /co 2  ,  / E  n  2  )  +  2  o  t  COS (cot  -  ,  a]  2  / E  (co/  COS  E/m  (cot  -  a)  ' 2c/ 4E/m)  2  st (2(co/co )(c/c )) n  c  2  When 6 = F /E is the static deflection which a constant force of st o +  2  2  magnitude F would produce in a spring of modulus E, and, co = E/m and c Q  n  c  =  4 E/m. The quantity M=  J  i  (1  -  co /^ ) 2  2  2  +  [2(co/o) )(c/c )] n  is called the magnification r a t i o .  c  2  It is the factor by which the static  deflection produced in a spring of modulus E by a constant force F must be Q  multiplied in order to give the amplitude of the vibrations which result when the same force acts dynamically with frequency co c.-(u=2Trf).  Kelvin Solid co n G  A when  1  (log decrement) -  co^ =  2irf  G = Shear modulus n = Shear viscosity  n  Determination of Solid Viscosity from Experiment For a Maxwell Liquid the equation of motion is  when t_ = C£  X  where  1  M£  2T  cos (p,t +e)  # t  2  The logarithmic decrement A ' , t h e natural logarithm of the ratio of two successive amplitudes on the same side of the equilibrium position.  Since  the amplitude is reduced by a factor exp (l/2x) per unit time, i t w i l l be reduced by exp (ir/xp^) in one o s c i l l a t i o n , hence, A  Since for a Maxwell Liquid, the time of relaxation T can be regarded as the ratio between i t s "effective viscosity" n and i t s e l a s t i c modulus E, so that  Thus for a Maxwell l i q u i d , the logarithmic decrement varies both with the frequency and with the effective viscosity of the material.  12-)7| APPENDIX II METHOD III Expected Viscoelastic Response for a Forced Vibration Test Forced Vibration Test To find u(ft, t) u(o, t)  Mass Unit Vol  u ( o , t ) - U . Cos cot  Elastic solution a ( x , t) = E (x, t) £  But, 9 a  3X  (x, t) = p' ^ 4 (x, t) E ^ (x, t) 3X  o(x, t)  Consider the wave equation E_ 32u _ 92 u 8X  p  2  3  t  2  where C =  JE/P  Steady State Solution Here we assume u(x, t) = R[0(x) e  i w t  ]  then the differential equation becomes c R [0 2  c  2  X X  e* ] = R [-0 J  [u ] + xx  wt  U  U)  2  e  t B t  ]  u(x) = A cos ^ x + B sin I x Apply boundary conditions u(0, t) = u cos t = R[A e Q  u  1 w t  ]  A = u_  r(£, t) = 0 = E |£ U , t) = ER[|(-A sin ^  + B cos | i )  e  1 a , t  ]  Hence, -A sin ^ + B cos ^= 0 c c B = A tan ^ C  t)  c  0  + tan 0)1, — sin c  u(x) = u"J c o s c u(x,  co&  = u tan  = u(xj_  COS  cut  .._2  u ( i , t) = u [cos  / c  COS  COS  Jl,  t)  u(o,  t)  U<  o)£/C  cot  c 1  COS  Viscoelastic Response: J*(w)  ^ ]  s i n  o  COS  COXj  1  co£  0  Maxwell Model 1 _ i  E  nco  129  -  „  /?> = JLU/PJI/E - i/nw = b + id  1/E  -  i / n w = + (1/E  w£  o V 4 -i<t>/2  k  « r~~> / i / c  j. i / /  2  b = w£V P (1/E d = -ai£/?  (1/E  + l/(nu)  N \ 2  + l/(nw)  C  O  «.  9/  S  2  )  + l/(n«F)  2  1 / 4  )  sin  e/2  Eriiii -  /  /  i>/2  /v  2  P  E_  . +  2 2  ,  n a)  + no).  •E :TO I) u ( £ , t) = Re [  e  "V  VE  c  o  s  i w t  ]  ^E*(co)/P  cos (b + id) = cos b cosh d - sin b sinh d / a  .\  UwcU> t) v t  = u  v  (cos b cosh d cos ut - sin b sinh d sin mt) 9  cos 0  . Note:  ( u i t +<j>) _ _  (cos b cosh siri b sinh d * " cos b cosh d 2  t a n  n  ;  -  (cos^ b cost/ d - sin^ b sinh^ d) '  0  u  n  7)  2  d  - sin  2  b sinh d ) 2  1 / 2  The above solution is an expected viscoelastic response of the material to forced vibration.  

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