AB ST RAC T THE TRANSITION TEST AS A METHOD FOR DETERMINING THE MECHANICAL PROPERTIES OF MATERIALS IN A CONTINUOUS MEDIUM by Donald MacKenzie Morrison Rock tends to behave as a brittle material when differentially stressed under a low confining pressure. As the confining pressure on the material is increased, however, it is found to behave more plasti- cally. The triaxial technique most commonly used for this type of study employs a liquid confining medium. The inherent characteristics of this technique are such that the mechanical properties of a material in a continuous medium cannot be determined with accuracy. A testing technique developed by Serata has a cylindrical speci- men compressed axially while enclosed in a steel cylinder. The speci- men is thus exposed to triaxial strain restrictions as well as triaxial pressures. Employing this technique, named the transition test, Serata found that dolomite, paraffin, and rock salt behaved elastically up to a certain stress, peculiar to the sample, beyond which the material changed abruptly to the plastic state. From the relation of the lateral stress to the axial stress in the plastic state, Serata obtained values for the octahedral shear strength of the material. Donald MacKenzie Morrison In this paper the method of obtaining the octahedral shear strength is reviewed. Relations and the additional instrumentation necessary for determining the elastic properties of the material are developed. Improvements in the testing method which permit accurate measure— ments of the lateral strains and which resulted in the reduction of friction between the specimen and the enclosing cylinder are described. Several cycles of loading were conducted on each of two samples of rock salt. Values of octahedral shear strength, Poisson's ratio, modulus of elasticity, and compressibility were obtained for each cycle of loading. The experimental results agree very closely with the theory upon which the testing principle is based. The transition test is shown to be a useful method for determining the elastic and plastic properties of a material in a continuous medium. Because of the similarity of boundary conditions, the mechanical properties of a material as determined by this test are directly appli- cable to the analysis of underground structures. THE TRANSITION TEST AS A METHOD FOR DETERMINING THE MECHANICAL PROPERTIES OF MATERIALS IN A CONTINUOUS MEDIUM By Donald MacKenzie Morrison A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Civil and Sanitary Engineering 1962 ACKNOW LEDGEMENTS The author would like to express his sincere thanks to his major professor Dr. Shosei Serata, Assistant Professor of Civil Engineering, Michigan State University; for his guidance in the study and for his careful checking of the manuscript. Gratitude is also extended to the International Mineral and Chemical Company and the International Salt Company for supplying the samples used in this study, the National Science Foundation for their support of this project, and to the Depart- ment of Civil Engineering for the use of its facilities. ii TABLE OF CONTENTS Page ACKNOWLEDGEMENTS ................................. ii LIST OF FIGURES ....................................... iv LIST OF TABLES ....................................... vi LIST OF SYMBOLS ..................................... vii Chapter I. INTRODUCTION ................................ 1 II. SUMMARY OF TRIAXIAL STUDIES ON ROCK SALT ......................................... 3 III. STRESS CONDITIONS IN A CONFINED SPECIMEN . . 7 IV. TESTING APPARATUS AND INSTRUMENTATION. . . 15 V. RESULTS AND EVALUATION .................... 21 VI. SUMMARY AND CONCLUSIONS .................. 26 Appendix- -Figures ..................................... 30 Tables ....................................... 51 BIBLIOGRAPHY ........................................ 53 Figure 10. ll. 12. LIST OF FIGURES Stress-strain curves of rock salt deformed in compression with specimens exposed to confining liquid ........................................... Cut-away view of the thick-walled cylinder with specimens and pistons in place ..................... Mohr diagram showing transition from the elastic to plastic stress condition in a material restrained from lateral expansion ................................. Lateral stress-axial stress diagram illustrating the stress relations in a specimen restrained from lateral expansion .................................. Thick-walled cylinder with strain gages and wiring attached ......................................... Device for measuring the coefficient of static friction of friction reducer material ........................ Section view of device for measuring the coefficient of static friction of friction reducer material ......... Determination of the coefficient of friction of six friction reducer materials ......................... Illustration of the location of strain gages on the thick- walled cylinder ................................... Apparatus enabling internal pressure to be developed in the thick-walled cylinder ........................ Internal hydraulic pressure in cell vs. pressure observed from strain gage readings ................ Reihle testing machine with specimen in place ....... iv Page 30 31 32 33 34 35 36 37 38 39 4O 41 Figure l3. 14. 15. 16. l7. l8. 19. 20. 21. Specimens 88-3 and SL-Z .......................... Lateral stress-axial stress diagram showing loading cycles 2, 4, and 5 on specimen SS-3 ................ Lateral stress—axial stress diagram showing loading cycles 3 and 5 on specimen SL-Z ................... Octahedral strength vs. cycles of loading: specimen SS— 3 and SL- 2 .................................... Poisson's ratio vs. cycles of loading: specimen 88-3 and SL-Z ................................ . ........ Modulus of elasticity vs. cycle of loading: specimen 88-3 and SL-Z .................................... Compressibility curves for specimen SS—3 .......... Compressibility curves for specimen SL-Z .......... Compressibility parameters b and 11 vs. cycle of loading ........................................... Page 43 44 45 46 47 48 49 50 LIST OF TABLES Table Page I. Coefficients of static friction of 6 friction reducer materials tested ................................. . 51 II. Mechanical properties of specimens 88-3 and SL-2 . . . 52 vi LIST OF SYMBOLS ratio of lateral stress to axial stress in the elastic state internal radius of the thick-walled cylinder external radius of the thick-walled cylinder strain in direction of major principal stress strain in direction of intermediate principal stress strain in direction of minor principal stress axial strain in the external surface of the thick-walled cylinder tangential strain in the external surface of the thick-walled cylinder strain in the specimen in axial direction modulus of elasticity of rock salt modulus of elasticity of steel octahedral shear strength the uniform internal pressure on a thick-walled cylinder the uniform external pressure on a thick-walled cylinder stress on specimen in lateral direction axial stress on specimen Poisson's ratio of rock salt Poisson's ratio of steel volume of sample at the beginning of each cycle of loading vii apparent coefficient of internal friction observed in a material restrained from lateral expansion ratio of axial strain to lateral strain in elastic state major principal stress axial stress in the external surface of the thick-walled cylinder minor principal stress normal force on a shear surface mean principal stress tangential stress in the external surface of the thick-walled cylinder tangential stress at radius r in the thick-walled cylinder shear stress acting in the plane of failure component of shear resistance which is independent of normal stress on the shear surface coefficient of internal friction viii I. INTRODUC TION Rock, when submitted to high confining pressures, exhibits many of the characteristics of a plastic material. Bridgeman1 and others, 2' 3 employing pressures of from 3000 to 75, 000 p.s. i. , have produced plastic flow in marble, limestone, and rock salt. To develop these high pressures, most investigators have used a liquid confining medium. The liquid exerts a uniform hydrostatic pressure on the specimen. A differential stress is then applied axially through pistons acting on the ends of the specimen. Being surrounded by a liquid of negligible shear strength the specimen may or maynot fail by rupture after developing large lateral strains. Serata4 devised a triaxial technique in which the cylindrical sample is fitted closely inside a steel cylinder. Axial pressure is developed through pistons which act axially on the specimen. Lateral pressure results from lateral expansion of the rock under axial stress being restricted by the confining cylinder. Using this technique, Serata created plastic states of stress in dolomite, paraffin and rock salt and obtained values for the octahedral shearing strength of these materials. He also demonstrated that, under these conditions of restraint of lateral strain, the transition from elastic to plastic state in rock is abrupt. This testing method will be referred to as the "transition test. " In this study, additional relations and instrumentation are devel- oped for the transition test. This development permits values of Poisson's ratio, modulus of elasticity, and compressibility, to be obtained for the material. Rock salt from two source areas is tested. Values for Poisson's ratio, modulus of elasticity, compressibility, and the octahedral shear strength are obtained. In addition, the effect of repeated cycles of loading on the mechanical properties is studied. II. SUMMARY OF TRIAXIAL S TUDlES ON ROCK SALT l . . Bridgeman, in a comprehensive study of the phy51cochem1cal properties of the elements and some common compounds, measured the compressibility of salt. The results of his investigation can be ex- AV - - 2 , 2 pressed by the relation: - -V— = a x10 7 p - b x10 12 p (p in kg/cm 3 o where: at 300C; a = 4-1. 82 and b = 50.4 at 75°C; a = 43.44 and b = 51.9 p = hydrostatic confining pressure In 1936 Griggs2 published the first results of a comprehensive series of tests on Solenhofen limestone, marble, and quartz. He employed a triaxial technique with which he obtained confining pres- sures as high as 13, 000 atmospheres. Tests were made on unjacketed 1/2 inch diameter cylinders. The purpose of Griggs' investigation was to study the relationship between strength and confining pressure. Griggs also investigated strength as a function of time. ‘Even at the highest confining pressures that he employed, Griggs was unable to produce continuous flow in any of the materials. He states however, ”as the confining pressure is increased, the physical character changes gradually from dominantly brittle to dominantly plas - tic. No sharp line can be drawn between the two. " Griggs found that: 1. For each material, the value of the elastic limit obtained with the highest confining pressures was only 10% greater than that observed at the lowest pressures. 2. The ultimate compressive strength of marble increased about 1, 400 percent in the same series of tests. Handin 3developed an elaborate high pressure triaxial apparatus for the Shell Oil Company Laboratory. A number of tests were made on rock salt with confining pressures as high as 5100 atmospheres. As a result of this investigation involving both extension and compression tests, Handin made the following observations. 1. The salt has remarkable ductility in compression at very low confining pressures. At 1200 atmospheres a specimen was shor- tened nearly 75 percent before fracture. 2. Initially, there is either no elastic limit or yield point, or it is too low to be detected with the measuring devices employed. 3. Deformation results in work-hardening and the appearance of an elastic limit obtained by unloading and reloading the specimen. 4. The increase of strength and ductility with confining pres- sure, so striking in many rocks is much less pronounced in the case of salt. Much of the effect is observed in the first few hundred atmospheres. In 1957 Handin performed triaxial tests on halite single crystals under confining pressures from 0 to 2000 atmospheres. Handin noted again that the yield stress is not affected greatly by variation in confin- ing pressure alone. He also noted, from the straight-line character- istic of the stress-strain curve beyond the elastic limit, that the work- hardening function is nearly linear at both 1000 and 2000 atmospheres confining pressure. Handin found yield strength of the crystal halite to be 1400 psi. The results of Handin's work are shown in figure 1. This review would not be complete without mention of the work done by Adams5 at the turn of the century. The confining pressure on Adams' specimens was developed by enclosing the specimen in a tightly fitting steel jacket. In that feature his method anticipated the work done in this study. Due to a lack of measurement of axial and lateral strains, the results of Adams' experiments were essentially qualitative. He demonstrated clearly, however, as was his intention, that rocks could be made to flow under high confining pressure at room temperature. This survey is not in any sense complete. The purpose in citing this material is to illustrate the characteristics of the triaxial test and particularly to show some of the results of tests on rock salt. Figure 1 from Handin's work is illustrative of the results obtained with the tri- axial method. It is seen that due to the rather inelastic nature of the salt there is no well defined yield stress evident. In addition, the value of E as well as the ultimate stress is largely a function of the confining pressure. The boundary conditions of the transition test are quite different from those of the conventional triaxial tests. This study investigates the properties of material subjected to a condition of lateral restraint of strain as well as a lateral confining pressure. III. STRESS CONDITIONS IN A CONFINED SPECIMEN An illustration of the testing apparatus with a specimen in place is shown in figure 2. The thick-walled cylinder encloses and confines the cylindrical sample against lateral expansion. Lateral pressure is developed on the specimen through restraint of the surrounding steel as the specimen is compressed axially. The steel pistons through which the axial pressure is applied are the same diameter as the specimen. Axial stress is thus imposed on the specimen with no axial load being applied by the pistons to the thick- walled cylinder. A theory, describing the triaxial behavior of rocks under conditions of controlled triaxial strain, has been developed by Serata. 4 For com- pleteness of presentation an outline of the development is given below. Most rock materials show a brittle type failure under uniaxial loading. Under triaxial loading these same materials fail in a manner which becomes increasingly plastic in nature as the confining pressure is increased. This change of character which occurs can be illustrated with a Mohr diagram. The Mohr diagram of a material is the locus of stress conditions which cause failure of the material. In figure 3, the portion of the Mohr diagram from A to B illustrates the Coulomb-Mohr theory of failure. The shear resistance developed at failure on the failure sur- face is expressed as: T = T + 490’ (1) c n where 'r = shear stress acting in the plane of failure. TC = component of the shear resistance which is independent of the normal stress. <1) = coefficient of internal friction of the material. O'n = normal force on the failure plane. The Coulomb-Mohr failure criterion applies strictly only when the coefficient of internal friction (4)) remains independent of the normal stress. The horizontal portion of the Mohr diagram (C to D) represents the Henchy-Von Mises theory of plastic yielding. This theory states that a material becomes plastic when the shear stress on an octahedral shearing surface reaches a maximum limiting value. The octahedral shear stress (To) is a function of the three principal stresses. 2 2 2 70—1/3~\/(crl-0’2) +(0’2-CT3) +(U3‘Ul) (2) If the minor principal stresses are equal, crz 2 0'3 -‘-= (TL and equation (2) reduces to: 1' =£2 (CI-U) {2'8” The maximum octahedral shear stress which can exist in a particular material is its octahedral shear strength (KO). When the octahedral shearing stress is equal to K0 the maximum shear stress in the material is: T _(°1‘ UL) _ 3 K (3) max 2 z—v’ 2 o Equation 3 is a plot of the horizontal portion of the Mohr diagram. Referring again to the elastic state of stress, stress-strain relations in a material restrained from lateral expansion are derived below. The relationships of the principal stresses to the principal strains at any point in an elastic material are: l e1-—-—E [01- u (02 + 03)] ‘;1 u (o + cr )1 ' A (4) 6.2 E “2 1 3 e =—1—-[0' -u (0' +0 )] 3 E 3 l 2 where e1 = axial strain e2, e3 = lateral strain 01 = axial stress 0' ,0“ = lateral stress 2 3 E = modulus of elasticity Poisson's ratio C. II 10 If a condition of no lateral strain is assumed, in a cylindrical specimen, the following boundary conditions prevail. 0-1 :Sz “2 :03 :SL e =e (5) 1 z e2=e3=0 Substitution of equations 5 into equations _4 gives the following equations for a laterally restrained cylindrical specimen. [S - ZuSL] (6-3) - u(S +5 )] (6-1)) Z L From equation 6-a, the relation of lateral stress to axial stress is: U) L _ _ u Sz - tan a —_l-u (7) Equation 7 is represented in the Mohr diagram (figure 3) by the line OE passing through the origin and tangent to the Mohr circle with the prin- L' The slope of this line in the Mohr diagram is cipal stresses Sz and 8 given as: sin w = l - 2u (8) As is shown by the relationship of equation 8 to the Mohr diagram, the stress conditions in a laterally confined material are stable until a plastic state of stress is reached. The relation of angle 4) to angle w in ll figure 3 is a characteristic of each material. For most materials how- ever w is smaller than <1). Thus as is shown in figure 3, the elastic condition of equation 8 intersects the Mohr failure envelope in the plastic state. Serata postulated from this that the transition from the elastic to the plastic state in a material restrained from lateral expan- tion is abrupt. A material restrained from lateral expansion, as in the transi- tion test, does not fail in the elastic state. The relation of shear stress to the principal stresses in a confined specimen is such that a condition of shear failure cannot occur. The relationship of lateral'stress to axial stress in a cylindrical specimen with lateral expansion restrained can be more clearly shown on a diagram on which the lateral stress is plotted as the ordinate and the axial stress as the abscissa. A diagram of this type is represented by figure 4. The elastic condition of equation 7 is represented by the line OE. The plastic state of stress is represented by the line EF which has a slope of 450. The plastic condition is derived from equation 2 by substituting in the boundary conditions of a restrained cylindrical specimen, equations 5, giving: 12 Differentiating 9 with respect to SL to obtain the slope of the line we obtain dS E's—z-Ztanfizl (10) L The sequence of stress conditions, which a loading cycle of the transition test produces in a material can also be illustrated by figure 4. Under the initial axial loading up to the stress corresponding to point E on the diagram, the material behaves elastically. As the axial stress is further increased, the material changes rather abruptly to a plastic state of stress represented by line EF. The maximum axial stress to which the specimen may be subjected is a function of the yield strength of the steel cell enclosing the specimen. The stress conditions which the material experiences as the axial load is gradually reduced are illus- trated by the line of elastic stress FG, and the line of plastic stress GH. The residual stress H which remains after the axial stress has been removed is a result of creep and plastic flow which has occurred in the specimen. The dashed curve in figure 4 represents the locus of brittle failure obtained when one of the principal stresses is zero or tensile. As is seen from figure 4, the octahedral shear strength of a material is equal to 1/3 the perpendicular distance between the lines of plastic stress. In the development above, it was assumed that no lateral strain occurs in the specimen. It is obvious however that some lateral 13 expansion must occur if the steel cell is to exert any confining pressure on the sample. The small amount of lateral strain which occurs does not invalidate the theory or the description above. If accurate values for the elastic constants are to be obtained, however, the effect of lateral expansion must be considered. Relationships for determining values of the elastic constants from the data of the transition test are developed below. If the effect of lateral expansion is included, the boundary condi- tions for equations 4 become: 0Fl : Sz 0' = 0' = S 3 L 2 (11) e1 : e2 e2 : e3. : eL With these boundary conditions equations 4 become: e =-—1- [S - 2u S ] z E z R L R e =-1—[S - u (S +S )] (12) L ER L R L z From equations 12, the following values for Poisson's ratio (u) and the modulus of elasticity (E), are obtained. _ 14?». R 2A-1“(A+1) u (13) Where: S A‘t a——_L - an S Z e I‘=tanY:'e_E L 52- 5L + 25 (SL 52) ER: e e z L 25 (sL+Sz) 14 (14) IV. TESTING APPARATUS AND INSTRUMENTATION Thick-walled cylinder and pistons The cylindrical steel cell was made of stainless steel tubing. The tubing, with an outside diameter of 4. 00 inches, was cut to a length of 3. 25 inches and bored to an inside diameter of 3. 250 inches. Figure 5 shows the cell with strain gages and wiring attached. The mechanical properties of the stainless steel are given below. . 3 . yield strength 60 x 10 p51 . . 6 . modulus of elast1c1ty 29 x 10 p51 3 tensile strength 100 x 10 psi The pistons used for compressing the specimen axially are 1. 5 inches thick and were machined to a diameter of 3. 240 inches. They are made from structural carbon steel. Friction reducer In developing the stress-strain relations for the transition test, it was assumed that no shearing stresses existed between the specimen and the surrounding cylinder. That a shearing stress does exist is evident from consideration of normal stresses between the cell and specimen and from the relative movement which occurs as the specimen is com- pressed axially. The axial stress on any cross section of the specimen is less than that applied by the testing machine due to this friction. The 15 l6 lateral stress exerted by the specimen is a function of the actual axial stress in the specimen. If lateral stress is plotted versus the axial stress exerted by the testing machine a slope of less than 450 for the "plastic line” will result if friction is significant. In his earlier experimental work with the transition test, Serata found that the ”plastic line" in the lateral stress-axial stress diagram (see figure 4) had a slope which varied from 21 to 35 degrees rather than 450 as expected from the theory. A slope of 450 was obtained in a test on paraffin however. On the basis of these results, Serata pro- posed that friction between the specimen and the surrounding cell caused the divergence in the slope of the plastic line from that expected. In this study, therefore, an effort was made to reduce the friction between the cell and specimen as much as possible. As a means of reducing this friction, a thin layer of friction reducing material was introduced between them. The device shown in figures 6 and 7 was employed to measure directly the coefficients of static friction of several materials considered for use as friction reducers. 'The friction reducer was placed on either side of the central sliding plate. Steel blocks were then set in place and the device was placed in the testing machine as illustrated in figure 7. The normal stress on the friction reducer was generated by the Reihle testing machine. The shearing stress was applied to the sliding plate by a Blackhawk hydraulic \ ram. 17 In conducting these tests, a normal force was first imposed on the friction reducer with the testing machine. With the normal stress held constant, the force exerted on the sliding plate by the hydraulic ram was gradually increased until movement occurred between the sliding plate and the loading blocks. With the shearing stress required to cause movement and the corresponding normal stress known, the coefficient of static friction of the material was obtained. Figure 8 is a plot of shear stress vs. normal stress on the friction reducer. Five tests at different normal stresses were run on each material. Table I describes the six friction reducers tested and gives the coefficient of friction obtained for each. Type D which reduces the friction force to less than 0. 3% was used almost exclusively in this study. Instrumentation Six type AR-l, rosette SR-4 strain gages were attached to the thick-walled cylinder as is shown in figure 9. The gages were located around the cell 1200 apart in two rows. The rosette gages were oriented with one of the three gage axes parallel to the axis of the cylinder and one perpendicular to the cylinder axis. The lateral stress in the specimen was determined from the tangential stress in the external surface of the cell. The axial and tangential stresses on the surface of the cell at each gage location were Calculated from the following equations. 18 E — s - (re —————2 (ee+ uS e2) (15 a) (l-u ) 5 Es cr =————(e +u 8) (15'b) z 2 z s 9 (l-us) The distribution of tangential stress in a hollow cylinder sub- mitted to uniform pressure on the inner and outer surface is given by the equation: c2d2(PO-Pi) 1 Picz- Pod =- - ——-+ “e d2_C2 r2 d2_C2 ”6) where 0'e = the tangential stress at radius r c = internal radius of the cylinder d 2 external radius of the cylinder Pi = the uniform internal pressure on the cylinder P0 = the uniform external pressure on the cylinder According to equation 16, the conditions of the transition test can be expressed as follows: a = 1. 625 in. b = 2. 000 in. Pi = SL (17) P = 0 o r = 2. 000 in. 19 Substituting the conditions of 17 into equation 16 and solving for SL: 0' = 0. 258 0' (18) z z Equations 15-a and 18 were used to determine the lateral stress in the specimen from the strain gage readings. The apparatus shown in figure 10 was devised to determine the accuracy of the above method of finding the lateral pressure in the specimen. The device permitted the cylinder to be submitted to a known hydraulic pressure Pi. Strain gage readings were taken and using equations 15-a and 18, the apparent internal pressure was calcu- lated. The results of this investigation are shown in figure 11. To measure axial strains in the specimen, an Ames dial gage was mounted between the platens of the testing machine. This permitted axial strains to be measured to 1/10, 000 of an inch. Lateral strains were obtained from the SR-4 readings. The Reihle testing machine has two load ranges. In the lower range, which has a capacity of 60, 000 lbs. , the total applied force can be measured to the nearest 100 lbs. The machine has a capacity of 300, 000 lbs. in the high load range. In this range readings can be made to the nearest 500 lbs. which is equivalent to 60 psi of axial pres- sure on the specimen. Figure 12 shows the testing machine with a specimen in place. 20 Specimen Preparation The rock salt specimens were machined to a diameter of 3. 240 inches and a length of 3. 250 inches. The ends of the specimen were made exactly perpendicular to the axis of the cylinder to insure uniform loading. Any imperfections existing in the surface after machining were filled with paraffin. The imperfections were usually the result of small crystals chipping out of the surface in the maching operation. The friction reducer was applied to the rock surface after these imperfections had been filled. A thin film of the grease-graphite mix- ture was applied to the rock surface and to both sides of the plastic sheet. The plastic was then placed on the cylindrical surface with care being taken to remove all wrinkles and excess grease from the plastic sheet. A similar procedure was followed in applying the friction reducer to the ends of the specimen. V. RESULTS AND EVALUATION Rock salt specimen SS-3 The rock salt used in this study was obtained from two sources. The specimen designated SS-3, came from the Yorkton Saskatchewan mine of the International Minerals and Chemical Company. The salt was obtained from a depth of 3020 feet below the ground surface. Sample 55-3 is pink to red in color and contains small inclusions of foreign materials. The salt crystals range in mean diameter from 0.15 to 0. 75 inches with the average crystal diameter of the specimen about 0. 45 inches. Five cycles of loading were run on specimen SS-3. Rock salt specimen SL-2 Specimen SL-2 was mined from the Avery Island, Louisiana, mine of the International Salt Company. The sample was obtained from a drill core at a depth of 600 feet below the ground surface. This salt is white in color and contains much less impurities than specimen SS-3. Specimen SL-2 is somewhat finer grained than specimen SS_3. Speci- men SL-2 has a maximum crystal diameter of 0. 50 inches and an average diameter of 0. 31 inches. Four cycles of loading were run on this specimen. Samples SS-3 and SL-2 are shown in figure 13. 21 22 Octahedral shearing strength Figures 14 and 15 show the result of tests on samples 85-3 and SL-2 respectively. Not all of the cycles of loading are shown in these figures due to the congestion which would occur. It may be readily seen that the perpendicular distance between the lines of plastic stress in- creases as the specimen undergoes successive cycles of loading. This indicates that the octahedral shear strength of the material increases as it experiences successive loadings. Table 11 gives the values of Ko obtained for each cycle of loading on both specimens. Figure 16 is a plot of K0 versus the cycles of loading. It is seen that the octahedral shear strength of specimen 88-3 is greater than that of sample SL-2. Both samples obtain close to their maximum strengths after five cycles of loading. These values are 2100 psi and 1700 psi for samples 55-3 and SL-2 respectively. Poisson's ratio Table II also gives the values of Poisson's ratio (u) which were obtained for the two samples. Figure 17 shows Poisson's ratio plotted against cycles of loading. The values for specimen SS-3 are slightly greater than those of SL-2. Modulus of elasticity Table II presents the data obtained for the modulus of elasticity. 23 The only value for sample SS-3 was obtained on the fifth cycle of load- ing. Figure 18 shows the values obtained in this study along with results 7 3 obtained by Serata, Handin, and Wuerker. Compressibilig The compressibility of the salt was obtained as a function of the mean principal stress 0' am =1/3(sZ + 25L) (19) To obtain expressions for the compressibility of the specimens, the unit change in volume V_ was plotted versus the mean principal 0 stress 0- on logarithmic paper. The equation for the straight line m obtained is of the form: AV _ n v— ' ‘b‘Tm (20’ o where: VO = the volume of the sample at the beginning of each cycle of loading b and n = compressibility parameters The volume change AV was obtained from the axial and lateral strains in the specimen. Figures 19 and 20 show the compressibility curves for specimens 58-3 and SL-Z respectively. Table II gives the values for b and n obtained from the several tests. Bridgeman‘s com- pressibility results are of the same order of magnitude as those obtained 24 in this study. Figure 21 shows the parameters b and n of equation 20 plotted versus the cycles of loading. A diagram similar to figure 20 was not plotted for specimen SS-3 since only two compressibility measurements were made on that sample. Evaluation In comparing the experimental results with the theory, two points of verification are noted. 1. The transition from the elastic to plastic states and from plastic to elastic are generally abrupt. 2. The slope of the ”plastic line" closely approximates 450. The maximum deviation is 2. 5° with an average of 0. 9°. The deviation which does occur is attributed to friction and strain hardening. The effect of friction was discussed earlier. Strain hardening is accom- panied by an increase in K0 in the material. From the expression, S = S - —— KO, (figure 4), it is seen that an increase in KO causes S and consequently the slope of the "plastic line” in a lateral stress— axial stress diagram to decrease. Figure 18 and Table II compare the values for the modulus of elasticity obtained in this study with those of other investigators. Handin employed a conventional triaxial apparatus and used kerosene as a confining liquid. His results are shown in figure 1. The range of values he obtained for the modulus of elasticity are shown by two 25 points in figure 18. Similarly, the range of values of E obtained by Serata using a uniaxial technique are shown by two points in figure 18. Wuerker's triaxial test data are also shown in figure 18. The values found in this study are larger than those obtained by other investigators. This discrepancy can be explained by the difference in boundary conditions of the transition test compared to those of the other testing methods. In the uniaxial test and the conventional tri- axial methods, employing a liquid confining medium, slip on the crystal interfaces is relatively free to occur as the material adjusts to the differential stress. In the transition test, the lateral restraint of strain prevents much of this intercrystalline movement. The result is a higher modulus of elasticity. In a continuous medium such as a homogeneous rock material, beneath the earth's surface, the conditions of triaxial pressure and re- straint of strain are similar to those of the transition test. The several mechanical properties obtained are thus seen to be directly applicable to the analysis of stress conditions underground. VI. SUMMARY AND CONCLUSIONS The work done in this study includes improvements in the transi- tion test as well as determination of the mechanical properties of two samples of rock salt. Development of a more efficient friction reducer and replacement of single axis strain gages with rosette gages are the principal modifications made in the testing technique. Reduction of the friction between cell and specimen permits the experimental boundary conditions on the specimen to more closely ap- proach the assumed condition of no friction. Rosette gages were employed so that the tangential strain reading can be corrected for the effect of vertical strain in the cell. The effect these modifications have had in increasing the accuracy of the testing method is shown in figure 11, and the slopes of the "plastic lines” in figures 14 and 15. Figure 11 shows that the internal pressure in the cell as indicated by the strain gages agrees very closely with the true internal pressure. The slopes of the "plastic lines” in the lateral stress-axial stress diagrams have been shown to deviate not more than 2. 50 from the theoretical of 450. This is in comparison to deviations as large as 200 obtained prior to this study. Relationships were developed which permit the elastic properties of the sample to be determined. These relationships consider the effect 26 27 of lateral expansion of the cell but assume no friction exists between the cell and the specimen. Several cycles of loading were conducted on each of two rock salt specimens. Values of Poisson's ratio, the modulus of elasticity, compressibility, and the octahedral shear strength were obtained for each loading cycle. On the basis of the observations enumerated above, the following conclusions may be drawn. 1. With the use of the improved friction reducer, the effect of friction on the test results is negligible. 2. The technique of measuring the internal pressure on the thick-walled cylinder, through the tangential strain on the external sur- face of the cylinder, gives accurate results if rosette strain gages are used. 3. In the plastic state of stress, the ratio of lateral to axial stress experimentally obtained differs less than 5% from the theoretical relation. 4. The boundary conditions of the transition test are seen to approximate those existing in a continuous medium. The mechanical properties obtained by this testing method are thus applicable to the analysis of structures existing in a continuous medium such as underground. 5. Figure 16 shows that the octahedral shear strength of speci- men SS-3 ranges from 1100 psi to 2000 psi for the first and fifth cycles 28 of loading respectively. In comparison the octahedral shear strength of the relatively pure specimen SL-2 increases from 1067 psi to 1567 for the first and fourth cycles of loading respectively. 6. Values of Poisson's ratio of 0. 057 and O. 043 were obtained on the first cycle of loading for specimens SS-3 and SL-2 respectively. Figure 17 illustrates the large increase in Poisson's ratio up to O. 20, which occurs in successive cycles of loading. 7. Figure 18 shows that the modulus of elasticity for rock salt, obtained in this study, is greater than the values obtained by three other investigators. Variations in materials tested by the several in- vestigators, and the rate of loading employed by them, are probably partially responsible for the difference. The most significant factor causing the variation, however, is the difference in boundary conditions among the triaxial, uniaxial, and transition test. 8. The change in volume of the specimens is related to the mean principal stress according to the following exponential relation: AV _ n T - - “m o where: cr = mean principal stress m b and n = compressibility parameters 9. Figures 14 through 21 show that repeated cycles of loading result in a general increase in the strength of the material. The increase 29 in strength is partially evident in the increase of octahedral shear strength and modulus of elasticity. 10. 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Ohm .o on: .o mmmfi m mmv .o oomoooo . mmo .o 02 .o mmmfi N man .0 womoooo . com .o mvo .o noofi H Nsdm 0mm .0 03.000 . cod x one .H 13.0 ooom m uuuuuuuuuuuuuuuuuu cox: .o nmmd w uuuuuuuuuuuuuuuuuu oom~.o Neda m uuuuuuuuuuuuuuuuuu 0:; .o oovfi N uuuuuuuuuuuuuuuuuu nmo .o 00: H Mnmm c Q (mm (d 0M WM ”HUANG oHQEmm Nudm paw Mumm mcoawoomm mo mofluomoum 138.9302 u E 3an BIBLIOGRAPHY Bridgeman, P. W. , The Physics of High Pressure, London: Bell and Sons, 1949. Griggs, David T. , ”Deformation of Rocks Under High Confining Pressure," Journal of Geolgy, Vol. XLIV, No. 5, July-August, 1936. Handin, J. , ”An Application of High Pressure in Geophysics, " A. S. M. E. Transactions, Vol. 75, pp. 315-324, 1953. Serata, S. , ”Transition from Elastic to Plastic States of Rocks Under Triaxial Compression, ” Proceedings of the 4th Symposium on Rock Mechanics, Proceedings of ASCE, Vol. 86, No. SA 3, February 1960. Adams, F. D. , and Nicholson, J. T. , "An Experimental Inves- tigation into the Flow of Marble, ” Proceedings of the Royal Society of London, Series A, Vol. 195, pp. 363-401., [90/ Timoshenko, S. and Goodier, J. N., Theory of Elasticity, McGraw-Hill Book Company, Inc. , New York, p. 59., I75" Serata, S. , Gloyna, E. F. , ”Principles of Structural Stability of Underground Salt Cavities, " Journal of Geophysical Research, Vol. 65, No. 9, September 1960. Wuerker, R. G. , “Annotated Tables of Strength and Elastic Properties of Rocks, " A. I. M. E. , Petroleum Branch, December 1958. 53 31293 02808 0665