#l ‘IHIIIIM W! I 122 653 THS ELASTEQPLASTEC TRANSITION TESTS ON VARIOUS ROCK TYPES Thesis for ”12 Degree 0* M. S. MICHEGAN STATE UNIVERSETY Athipet Bashyam Raman 1962 THESIS 0-169 This is to certify that the thesis entitled ELASTIC-PLASTIC TRANSITION TESTS 0N VARIOUS ROCK TYPES presented 'bg Ath ipet Bashyam Raman has been accepted towards fulfillment of the requirements for L degree in m «W 7w Major Mess“ Date Q’QV ’3\ \(\(° 2/ LIBRARY Michigan State University ABSTRACT ELASTIC-PLASTIC TRANSITION TESTS ON VARIOUS ROCK TYPES By Athipet Bashyam Raman Rocks in general behave as a brittle material when subject to low confining pressure. But as the confining pressure is increased, they are found to exhibit plastic behavior. In the conventional methods, the confining pressure is applied by subjecting the rocks to a liquid confining medium. A new technique invented by Serata develops the confining pressure in a cylindrical sample of rock by enclosing the same in a tightly fitted steel cylinder and com- pressing the specimen axially. Thus triaxial pressure is developed in the specimen as there is strain restriction in all directions. By em- ploying this method, called the transition test, Serata obtained plastic state of deformation in paraffin and rock salt, and the change from elas- tic to plastic state was found to be abrupt. From the relationship between the lateral stress and axial stress, he was able to obtain the octahedral shearing strength of the materials tested. Morrison improved this technique in his studies on rock salt by applying better friction reducers and also obtained values for elastic constants. Athipet Bashyam Ram an In this work the earlier triaxial tests on rock types are summa- rized and the new technique of transition test is reviewed along with the method of obtaining the octahedral shear strength and elastic constants. Successive cycles of loading were conducted on limestone, sand- stone, granite, marble, shale, anhydrite, and rock salt reaching high pressures up to a maximum of 110, 000 psi and the relationships between lateral and axial stresses were studied. Evaluation of the test results show that there is a general trend in transition from elastic to plastic state, though it may not be very abrupt. All rocks except salt show a gradual transition. The slopes of the plas- tic lines are much smaller than the theoretical prediction. This dif- ference may be attributed to the nature of the strength of grains and of their mutual bonding. ELASTIC-PLASTIC TRANSITION TESTS ON VARIOUS ROCK TYPES BY Athipet Bashyam Raman A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Geology 1962 AC KNOW LEDGEMENTS The author desires to express his sincere thanks to his major professor Dr. Justin Zinn, Professor, Department of Geology, for his valuable suggestions and careful scrutiny of the manuscript. Special thanks are also due to the National Science Foundation under whose research project the present work was carried out. The author is particularly indebted to the director of the project, Dr. Shosei Serata, Assistant Professor of Civil Engineering, for his constant and clear sighted guidance. Grateful thanks are also due to Mr. Shunsuke Sakurai, Graduate Assistant, Civil Engineering Department, for his assistance in conducting the experiments and analyzing the results. Gratitude is also extended to Dr. Leonard Obert, U. S. Bureau of Mines, College Park, Maryland, and to the Michigan State Geological Survey, for supplying the core samples from which the specimens were machined for the tests. ii CONTENTS Page Acknowledgements ....................................... ii List of Figures ........................................ iv List of Symbols ........................................ vi Chapter I. Introduction ................................... 1 II. Summary of Triaxial Tests on Rocks ............. 3 III. Theory of the Experiments ..................... 8 IV. Testing Method ............................... 19 V. Specimen Description and Testing Results ........ 30 VI. Evaluation of Experimental Results and Discussion . 44 VII. Conclusions .................................. 48 VIII. Suggestions for Additional Research ............. 50 Bibliography ........................................... 51 iii Figure 10. 11. 12. l3. 14. LIST OF FIGURES Mohr's diagram representing underground stress conditions ....................................... Relation of axial and lateral stresses in transition test ............................................ Cutaway view of the thick-walled cylinder with the enclosed specimen ................................ Photograph of the testing machine in operation ....... Photograph of the testing cell with the strain gages attached and of the specimen with the plungers ........ Illustration of the strain gage locations on the thick- walled cylinder ................................... Strain gage wire diagram .......................... Photograph of the specimens tested ................. Lateral stress — axial stress diagram showing loading cycles I, III, and V on limestone ................... Lateral stress - axial stress diagram showing loading cycles 1, III, and V on sandstone .................... Lateral stress - axial stress diagram showing loading cycles I and III on granite .......................... Lateral stress - axial stress diagram showing loading cycles I, II, and III on marble ...................... Poisson's ratio versus loading; cycles I to III; marble .......................................... Modulus of elasticity versus loading; cycles I to 111; marble ........................................... iv 15 20 22 23 24 25 31 32 33 35 36 37 38 Figure 15. 16. 17. 18. Lateral stress - axial stress diagram cycles I and II on shale ............. Lateral stress - axial stress diagram cycles 11 and III on anhydrite ........ Lateral stress - axial stress diagram cycles I and II on rock salt 'A' ...... Lateral stress - axial stress diagram cycles I and II on rock salt 'B' ....... showing loading showing loading Page 39 40 42 43 LIST OF SYMBOLS ratio of lateral stress to axial stress in the elastic state angle of elastic line angle of plastic line internal radius of thick-walled cylinder external radius of thick—walled cylinder modulus of elasticity modulus of elasticity of rock modulus of elasticity of steel strain in axial direction strain in lateral direction strain in lateral direction tangential strain in the external surface of the thick-walled cylinder axial strain in specimen ratio of axial strain to lateral strain in the elastic state maximum octahedral shearing strength uniform internal pressure on the cylinder uniform external pressure on the cylinder coefficient of internal friction of the material lateral stress in specimen vi axial stress in specimen major principal stress intermediate principal stress minor principal stress mean principal stress normal stress perpendicular to failure plane tangential stress developed in the external surface of the thick-walled cylinder tangential stress at radius r shear stress acting in the plane of failure true shearing strength which is constant for a given material octahedral shear stress Poisson's ratio Poisson's ratio of rock Poisson's ratio of steel apparent coefficient of internal friction of a material under conditions of restraint in lateral expansion Vii INTRODUCTION Most rocks are exceedingly brittle when deformed under ordinary atmospheric pressure conditions. But these very same rocks exhibit a large amount of flow deformation as revealed in the highly contorted sections of the earth's crust. This increase in ductility is considered to be due to the environmental conditions existing deep in the crust, the conditions being high confining pressure, high temperature, stress differences and the presence of hydrothermal solutions. That confining “I “I“! 'A‘I'a_ Min!" pressure plays an important role in bringing about the transition from brittle to ductile flow was first proved by Adams and Nicolson [l] (1901), Adams [2, 3] (1910, 1912), and Adams and Bancroft [4] (1917). Later, Griggs [8] (1936), Balsley [5] (1941), Bridgeman [6] (19493.), Robertson [16] (1955), Handin and Hager [11] (1957), and Paterson [15] (1958a) have all illustrated the same quantitatively with improved techniques. To subject the rocks under high pressure conditions, the above investigators used a liquid confining medium and the specimen was put to differential stress by means of pistons acting axially on the ends of specimens. Instead of confining the specimen to a liquid confining medium, Serata [7] (1961) has devised a very simple triaxial technique in which the cylindrically cut specimen is exactly fitted in a cylindrical steel cell. The specimen is then subjected to axial pressure by means of pistons. Under this set up, a triaxial pressure condition is attained as the lateral expansion of the specimen under axial stress is restrained by the confining steel cell. By applying this technique, Serata obtained plastic deformation in paraffin and rock salt and also showed that under such conditions of restraint of lateral strain, the transition from elastic to plastic state in the rock is abrupt. This testing method is called by him the "Transition Test." From this method he also derived values for octahedral shearing strength for the materials tested. This octahedral shearing strength measures the intensity of stress which is needed to bring a solid substance into a plastic state. By applying this technique on rock salt and with additional relations and instrumentation, Morrison [14] (1962) obtained values of Poisson's ratio, modulus of elasticity, and compressibility for the materials tested. The object of the present study is to test a number of rock samples of varying strength under this new technique, to evaluate the relationship between the lateral and axial stresses in the specimens and to compare the trend in transition from elastic to plastic state in the different rock types. The specimens tested are limestone, sandstone, granite, marble, shale, anhydrite and rock salt. II. SUMMARY OF TRIAXIAL TESTS ON ROCKS Since the beginning of this century, extensive experiments have been carried out on the deformation of rocks. It has been known for some time that the rocks become stronger and more ductile with the application of confining pressure. The earliest pioneers in the field were Adams and his collabo- - ‘r 1—53- ‘3.“- .— rators [1] (1901), [Z] (1910), [3] (1912), [4] (1917) who had established at least qualitatively that both the ultimate strength and the ductility of iv some rocks increase with the confining pressure. This was established in experiments using constraining steel tubes. Von Karman [20] (1911) was the first to perform the triaxial tests under a truly hydrostatic confining pressure. He tested Carrara mar- ble and red sandstone in a device which permitted compressive loading of samples subjected externally to liquid pressures of several thousand atmospheres. He attained data for quantitative stress-strain curves and found that at high pressures the curves were similar to those of ductile metals which show poorly defined yield points, work hardening and large permanent deformations. Notable contributions to rock deformation were made by Griggs [8] in 1936. He subjected solid cylinders of rocks, half inch in diameter, to triaxial compression under a liquid confining medium. The liquid used mostly was kerosene. The specimens tested were Solenhofen limestone, marble, and quartz. In his experiments he reached pres- sures up to 13, 000 atmospheres. The stress-strain curves obtained by him for limestone and marble show the well established effects of pres- sure on strength and ductility. The earlier specimens that he tested were unjacketed, but when jacketed they showed only a 40 per cent in- crease in strength. Even at the highest confining pressures that Griggs was able to attain, he was not able to produce continuous flow in any of the materials tested. However, he states, "as the confining pressure is increased the physical character changes gradually from dominantly brittle to dominantly plastic. No sharp line can be drawn between the two.” Further, he found that (l) for each material, the value of the elastic limit obtained with the highest confining pressures was only 10 percent greater than that observed at the lowest pressures; and that (2) the ultimate compressive strength of marble increased about 1, 400 per cent in the same series of tests. Balsley [5] (1941) made extension tests on marble under pressures up to 1000 atmospheres. The stress-strain relations were suggestive of ductile behavior. Griggs and his collaborators [9] (1951), [12] (1951), [19] (1951) at the University of California have made a detailed study on the deforma- tion of Yule marble. Their earlier studies were conducted at room temperature under a 10, 000 atmosphere confining pressure. Their analysis of data, together with the observed micro-fabric changes, proved to be an aid in establishing the deformation mechanism involved, twinning planes, gliding planes etc. Their later investigations (Griggs, Turner, and Heard [10] 1960) in triaxial tests were performed under high temperatures ranging from 500°C to 8000C. Varieties of rocks like pyroxenite, peridotite, basalt, granite, dolomite, marble, quartz crystals and other aggregates were tested. They have reported the strength of these as a function of temperature. Bridgman [7] (1952) has made some extensions tests on rocks under extreme pressures up to 30, 000 atmospheres. He obtained a 20 per cent reduction of area in his extension experiment on a rock salt at a load of 510 kg/cmz. For Solonhofen limestone he measured a 53 per cent reduction of area at a load of 14, 000 kg/cmz. Robertson [16] (1955) has made deformation tests on limestone, sandstone, marble, slate, quartzite, granite, diabase and other rocks. He also tested minerals like pyrite, quartz, microcline and fluorite. He reached pressures up to 4, 000 atmospheres. His tests included com- pression of solid cylinders, crushing of hollow cylinders, and punching of disks. He found all rocks exhibited a range of elastic linearity of stress with strain. He showed that limestones and marbles could be made to flow plastically to large deformations. His heating experiments 9 on limestones demonstrated an increase in plasticity with a rise in temperature. But he found no plastic behavior for the silicate rocks and minerals tested. Handin and Hager [11] (1957) in connection with their work with the Shell Development Company have performed a number of triaxial tests on dry sedimentary rocks at room temperature and at various con-~ fining pressures ranging from 0 to 2, 000 atmospheres. The rock types tested were siltstone, shale, sandstone, limestone, dolomite, and an- hydrite. Their tests reveal small increases in elasticity and yield stress and large increases in ultimate strength under pressure. Most rocks also show some increase in ductility. Serata [17] (1961) and his collaborators in the Engineering Re- search Division of Michigan State“ University have studied the triaxial behavior of paraffin, rock salt, and dolomite under the new triaxial technique of confining the specimen in a closely fitted steel cell and then applying axial pressure. The new technique is definitely more advantageous than the conventional triaxial tests conducted under liquid medium. This new method is very simple, inexpensive, accurate, and allows a very high pressure capacity. By applying this technique and from the study of the materials tested, Serata has postulated a new theory regarding the transition from elastic to plastic state in rocks. He arrives at the following conclusions on the static stress conditions: l. The static stress field in underground formations can be calcu- lated if the octahedral shearing strength and the Poisson's ratio of the formations are known. 2. In underground formation, the state of stress should be either elastic or plastic, and no intermediate state should exist. There- fore, the plastic state is distinguished from the elastic state by a definite boundary of transition between the two states. 3. In a homogeneous formation, there should be a plane of the tran- sition boundary from elastic to plastic with increasing depth. 4. No hydrostatic state of stress exists in an underground formation, although the state of stress approaches the hydrostatic condition with increasing depth beyond the transition boundary. 5. The transition of the states between plastic and elastic is caused by an abrupt change in the mechanism of yielding in underground formations. Recently in Salzburg, Austria, John [13] (1962) has developed a new approach to the study of triaxial compression of rocks. This new technique involves the compression of rocks of standard dimensions in situ. The detailed results of his experiments are awaited. He postu- lates that the analysis of his test results will help in practical design problems and in the estimation of overbreaks in rock excavations for tunnelling. III. THEORY OF THE EXPERIMENTS It is a well known fact that rocks generally exhibit varying degrees of plastic nature under progressive increase of triaxial compression. This phenomenon is explained by solid state mechanics as molecular (or ionic) displacement taking place in solids when the shearing stress ex- ceeds a definite limit of the intermolecular attraction. From studies of triaxial behavior of materials under conditions of controlled triaxial strain, a new theory of transition from elastic to plastic state has been proposed by Serata. The development of this theory is briefly outlined below. Rocks, though they exhibit a brittle behavior under uniaxial load- ing, become increasingly plastic under triaxial loading. This change of character can be visualized by means of a Mohr diagram. The Mohr diagram of a material represents the locus of stress conditions which bring about the failure of the material. The results of triaxial compression on a material can be shown in a Mohr diagram as shown in figure 1. The principal stresses are plotted in the abscissa and the shearing stresses on the ordinates. The semicircles indicate the yielding conditions of the material, and the envelope obtained is the result of a number of triaxial yielding tests in a large range of mean principal stress. The mean principal stress is the average of the three principal stres ses: ‘3' _ I. ‘ ‘ 3 .‘r 113.“. H mcomfipcoo mmoflm Unsoywnopcd wcfiqomoumop Emnwmmp 9.3on A madwrm 3mm: mommohm Hmmwocfanfi N N 1H m b m o a 3 :N I H n 3 Em < o W\/.N V38 VH i H m ofimgo \\ m l l‘\ I] owummauoomflo Q osmfla \ o wagon coflfimcmhfl A N «Coaoaflcoo 35d: 5L3 “mob finger: enumfipho cw mommouum 6.33.3 n . . b w 1H N Goflwpcoo padou Hops: paw «mob coflwmcmfi. E mommobm u . m (isd) sas sans Buixeaqg 10 : + + . . 1 (Tm 01 0'2 0'3 where 0'1, 02, and 0'3 are pr1nc1pal stresses ( ) A study of the Mohr envelope reveals that it consists of three parts. The first part AB has a definite slope, i. e. , it has a definite value of tan 9 within a certain limit on the mean principal stress. This part illustrates the Coulomb-Mohr theory of triaxial failure and applies to brittle state only. Here the coefficient of internal friction remains independent of normal stress. The state of stress can be represented as: T - <1>° U (2) where Tc = true shearing strength which is a constant for a given material; 7 = shear stress acting in the plane of failure; 4) = coefficient of internal friction of the material; 0' = normal stress perpendicular to the failure plane. n In the above equation > sign indicates a stable state and the = sign represents a failure or yielding state. The second portion BC of the envelope shows a gradual decrease in the slope, finally leading into the third part where the envelope reaches horizontality. The second portion represents an intermediate stage called the elasto—plastic or plasto-elastic state. up: «Him 11 In the third portion, CD in the figure, the envelope, as aforesaid is horizontal. This portion represents the condition of plasticity defined by Henky and von Misis, based on the energy of distortion theory. According to this theory, every material has a maximum limit in strain energy of distortion beyond which it becomes plastic. The strain energy is obtained by subtracting the elastic energy of volume expansion from the total elastic energy stored in a material. This r leads to the octahedral shearing stress theory of failure which makes 3 . 2'11; - 1'; \ it possible to apply the energy of distortion theory of failure by dealing only with stresses instead of dealing with energy directly. This theory states that a material becomes plastic when the shear stress on an octahedral shearing surface reaches a maximum limit Ko which is called the octahedral shearing strength of the material and which is specific to the material concerned. The octahedral shear stress To can be expressed as a function of the three principal stresses: 1 Z 2 2 To : 3J