ABSTRACT CONTINUOUS MEDIUM ANALYSIS OF ELASTIC, PLASTIC AND VISCOELASTIC BEHAVIOR OF A MODEL SALT CAVITY by Abdallah G. Dahir The main objective of this investigation was to study creep motion and the distribution of stress and strain in the medium around a cylindrical cavity. Rock behaves as a brittle material when differentially stressed under a low confining pressure. It was observed in nature that under- ground rocks show a tendency to flow under a triaxial state of stress. Accordingly, the possibility of using the information derivable from laboratory triaxial tests to predict the behavior of natural underground formations was investigated. Theoretical analysis was made of stress, strain and boundary motion of a cylindrical salt model containing a coaxial cylindrical cavity. This analysis was based on the mathematical theories of elasticity, plasticity and viscoelasticity of a continuous medium. The constitutive equations assumed to describe the behavior of the rock salt were derived from a mechanical model consisting of elastic, viscoelastic and viscoplastic elements. The elastic, viscoelastic and viscoplastic coefficients of rock salt were determined from experimental results of a triaxial Ab dallah G . Dahir transition test simulating underground stress field in situ. The be- havior of a model cylindrical salt cavity was studied in the laboratory by using a high pressure vessel, employing a liquid confining medium. The material constants obtained from the triaxial tests were used to calculate the theoretical cavity closure of the model salt cavity. The experimental results were in close agreement with the theoretical predictions, indicating the validity of both the mechanical model and the theories of stress, strain and creep motion. The agreement between the theory and experimental creep results suggests that this study may be extended to underground rock formations. CONTINUOUS MEDIUM ANALYSIS OF ELASTIC, PLASTIC AND VISCOELASTIC BEHAVIOR OF A MODE L SALT CAVIT Y BY Ab dallah Geor ge Dahir A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOC TOR OF PHILOSOPHY in Me chanic 5 Department of Metallurgy, Mechanics and Mate rial Science 1964 ACKNOWLEDGEMENTS The author would like to express his sincere thanks to Dr. S. Serata, Thesis Advisor and Project Director, for his guidance in this study and careful checking of the manuscript. The author is especially indebted to Dr. L. Malvern, Profes- sor of Mechanics and Major Professor, for his valuable suggestions, help and encouragement for the theoretical development. Gratitude is also extended to Dr. G. Mase, Professor of Mechanics, for his interest and help whenever requested, Dr. G. Martin, Professor of Mechanical Engineering, for serving in his graduate committee, and to the National Science Foundation for their support of this project. The author wishes to thank Mr. S. Sukurai for his valuable suggestions in the development of the mechanical model of rock salt. Appreciation is also extended to Messrs. O. Abu-Gheida and E. Fiskers for providing. laboratory help whenever requested, and to my wife Nancy for her encouragement and editing of this manuscript. ii TABLE OF CONTENTS Chapter I. INTRODUCTION .................................. 1.1 General Remarks ........................... 1.2 Objectives ............................. l. 3 Experimental Techniques ..................... PART ONE: THEORETICAL ANALYSIS II. PRINCIPLES OF STRESS AND STRAIN FIELDS AROUND A CYLINDRICAL CAVITY ................ 2. 1 Summary of Previous Work Done on Rock Salt . . . 2. la Uniaxial stress tests 2. lb Biaxial stress tests 2. 1c Triaxial stress tests 2. 2 Elastic Analysis of Stress and Strain Distribution Around a Cylindrical Opening ...... 2. 2a Stress and strain analysis in completely elastic cylinder in plane strain 2. 3 Elastic Plastic Analysis of a Hollow Cylinder in Plane Strain .............................. 2.. 3a Theoretical consideration of yield criteria 2. 3b Stress analysis in elastic -plastic domain 2. 3c Stress-strain relation in combined elastic and plastic domain 2. 3d Small strain plastic boundary motion solution in plane strain with maximum shear stress yield condition and plastic potential theory 2. 3e Determination of the inner radius of a hollow thick walled cylinder for large- strain solution and plane strain iv DJ 00m l3 17 21 21 25 3O 35 39 Chapter III . 2. 4 Analysis of Completely Plastic Cylinder with Plane Strain ................................ 2. 4a Stress distribution in completely plastic cylinder DETERMINATION OF THE MATERIAL PROPERTIES 3. 1 Linear Viscoelastic Behavior of a Solid Cylinder with a Laterally Constrained Motion 3. la General remarks 3. lb General viscoplastic behavior of rock salts 3. 1c Plastic stress relaxation equation and the determination of the plastic constant, n1 3. 1d Viscoelastic stress relaxation equation and the determination of viscosity constant, 772 IV. THEORETICAL BEHAVIOR OF A CYLINDRICAL V. CAVITY UNDER CONSTANT TRIAXIAL STRESS ..... 4. 1 Strain Rate Equations in a Hollow Thick Walled Cylinder ..... . ............................ .. 4.2 Comparison of Theoretical and Empirical Equations Describing the Creep Behavior of the Model Cavity ................................ PART TWO: EXPERIMENTAL INVESTIGATION APPARATUS AND EXPERIMENTAL PROCEDURE 5.1 General Remarks ............................ 5. 1a Testing techniques and their objectives 5. lb Sources of salt 5.2 Transition Test ....... . ...................... 5. 2a Testing devices 5. 2b Specimen preparation 5. 2c Testing procedure 47 47 51 51 51 52 54 61 65 65 70 74 74 74 77 78 78 81 81 Chapter Page 5. 3 Hollow Cylinder Test ......................... 84 5. 3a Testing devices 84 5. 3b Specimen preparation 84 5. 3c Jacketing of a hollow cylindrical specimen 9O 5. 3d Closure measuring system 91 5. 3e Testing procedure 94 PART THREE: EVALUATION VI. EXPERIMENTAL RESULTS AND DISCUSSION ....... 96 6. l Fundamental Structural Properties of the Material .................................... 96 6. la Young's modulus 97 6. 1b Strength of salt 102 6. 1c Cyclic stress tests and the effect of strain hardening 106 6.2 General Behavior of a Cylindrical Cavity with Loading ..................................... 111 6. 2a Experimental verification of the theoretical cavity behavior . 111 6. 2b Stability conditions 1 l9 6. 3 Transition test .............................. 120 6.3a Axial stress-lateral stress relationship 120 6. 3b Stress relaxation equations 123 6. 3c Plastic and viscoelastic states 128 6. 3d Determination of time -dependent constants from stress relaxation equations 133 6. 4 Creep of Model Cavity ............... . ........ 136 6. 4a Factors affecting creep 136 6. 4b Mechanism of creep 137 vi Chapter VII. VIII . IX. 6. 4c Creep analysis of a cylindrical model cavity and verification of theoretical creep equations OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 141 148 153 158 161 LIST OF TABLES Table Page 5.1 Components of high pressure vessel ............... 86 6. 1 Values of Young's modulus as obtained from hollow cylinder (first cycle) ........................... 99 6.2 Values of Young's modulus obtained from transition tests ................................ . .......... 101 tests ........................................... 106 6. 4 Experimental analysis of strain hardening characteristics of rock salt ....................... 111 6. 5 Theoretical analysis of the behavior of a thick walled cylinder. (External diameter of 4 9/16" and internal diameter of 1 inch) ..................... 115 . G . . . 6. 6 Values of the ratio, 71—5, as obtained from transnion 2 tests ........................................... 135 6. 7 Comparison of the values of p and $.13}. as obtained 1 from two different testing techniques .............. 147 iriii Figure 5.3 LIST OF FIGURES Lateral stress - axial stress diagram illustrating the stress relations in a specimen restrained from lateral expansion .......................... Lateral stress -axia1 stress diagram showing loading cycles 1, III, and V on limestone .......... Lateral stress - axial stress diagram showing loading cycles I and III on granite ............ . . . . Theoretical analysis of radial displacement in thick walled cylinder with plane strain and triaxial compression ................................... Mathematical concept of model salt cavity ........ Plastic viscoelastic model of rock salt ............ Theoretical prediction of the change of lateral stress in viscoplastic region in transition test ..... High pressure testing system providing versatile triaxial loads up to 200, 000 pounds with automatic loading control ................................. Top view of high pressure vessel showing ring and cap -plate .................................. General setup of transition test showing 250, 000 pounds press tester, shock absorber, pressure gages, strain indicator and automatic loading control ....................................... Thick walled cylinder cell with enclosed specimen, plungers and dial gages used for triaxial transition tests ........... O 0000000000000 00.0.0 .......... Cross-sectional view of high pressure vessel of triaxial testing assembly, showing various com- ponents and arrangement of test specimen ........ ix 12 13 14 38 40 53 54 75 75 76 76 85 Schematic diagram of triaxial testing assembly . . . Schematic diagram of pulse reduction and cut- away view of shock absorber .................... Plastic jackets and form ....................... Various wall thickness of hollow cylinders ........ Spe cimen cutting auge r ........................ Experimental analysis of elastic constants from triaxially compressed hollow cylinder .......... Analysis of Young's modulus from experimental results of triaxially compressed hollow cylinders . . Experimental determination of K0 from large- deformation theory of hollow cylinder ........... Experimental analysis of cycling effect on strain of triaxially compressed hollow cylinders ........ Closure of a model salt cavity .................. Theoretical stress-strain relation of a model cylindrical cavity compared to experimental results ...................................... Variation of the modified inner radius, a , of a cylindrical cavity compared to experimental results ...................................... Lateral stress - axial stress diagram illustrating typical results of transition tests, in which solid cylinder is restrained from lateral expansion Plot of strain readings illustrating the technique of determining zero drift by electrically revers- ing active and dummy gages .................... Stress relaxation behavior in transition test ...... Page 88 89 92 92 92 98 100 104 108 112 116 118 121 124 125 Figure Page 6.11 Stress relaxation behavior in transition test illustrating contribution of viscoelastic and viscoplastic flow and determination of their coefficients ............... 127 6.12 Lateral stress - axial stress diagram illustrating the stress relations in viscoplastic region in transition test ................................. 130 6. 13 Experimental analysis of constant stress region compared with the general theory of stress relaxation .................................... 131 6.14a Creep curve showing different stages of creep . . . . 138 6.14b Creep curve showing basic components ......... 138 6.15 Closure rate of a model salt cavity ............. 142 6. 16 Reduction of a model cavity radius as a function of viscoplastic deformation . 144 6. 17 Experimental analysis of triaxial creep in hollow cylinders illustrating determination of viscoelastic coefficient ..... ..... 147 xi 0'1, 0'2, ‘73 9 3 U+cre+o 3 O'+O'9+O' 3 E 2 =6 z e 6e 6' z P 6P 9' z NOTATION principal stresses radial stress tangential stress axial stress stresses in rectangular coordinates lateral stress generalized or effective stress deviatoric stresses octahedral shear stre ss 8 octahedral shear stress at t II C octahedral shear stress at t octahedral shear strength principal strain in rectangular coordinates lateral strain radial strain tangential strain total elastic strains in 3 principal directions total plastic strains in 3 principal directions xii ’71 T12 effective plastic strains in 3 principal directions principal strain rates 2 r 0' z in 3 directions slope of the equivalent stress/plastic strain curve scalar or proportionality factor inner and outer initial radii of a model cylindrical salt cavity, respectively reduced cavity radius radial coordinate radius of the elastic -p1astic boundary cylindrical coordinates internal pressure of cavity external pressure of cavity pressure at the elastic plastic boundary Young's modulus Poisson's ratio octahedral shear modulus retarded shear modulus viscosity coefficient in visc0plastic region viscosity coefficient in viscoelastic region material constant time xiii CHAPTER I INTRODUCTION 1. 1 General Remarks Rock is a heterogeneous mixture of grains of polycrystalline. materials which are randomly oriented and held together by bonding forces. It generally exhibits isotropic and homogeneous qualities under different kinds of loading. Geophysical explorations have demonstrated that rocks change their character and property and flow in varying degrees depending on the environmental conditions. Under uniaxial stress, they behave like brittle material. However, under a triaxial stress condition, which is normally the stress in an under-I ground formation, it exhibits many of the characteristics of plastic material. Adams,l Bridgman,9 GriggsZ—I and others have employed pressures that varied from 3000 to 75, 000 psi and produced plastic flow in marble, rocks and limestone. This phenomenon can be ex- plained by solid state mechanics since molecular (or ionic) displace- ment in solids takes place whenever the shearing stress exceeds a definite limit of the intermolecular attraction. Rock salt has been chosen as a model in the present study. It is encountered in large quantities as rock salt formations in many parts of the world. It flows more readily than other rocks as demonstrated by many natural salt domes. It is sufficiently homogeneous and isotropic for theoretical studies and easy to machine any desired form of testing specimens. Recently, a great attention has been given to the triaxial behavior of rock salts because underground cavities can be used for radioactive waste disposal in quantity when the nuclear energy becomes necessity in the near future. To date, the study of failure of rock salt has been restricted to conventional uniaxial compression tests or limited triaxial tests which cannot be used to distinguish the deformation mechanism of an underground rock salt. Because of these limitations, a transition test technique and a hollow cylinder technique were used to create testing conditions similar to that of underground formation. Numerous observations have been made on stress and strain and creep motion in underground formations. However no basic principles describing this behavior or the mechanical properties have been formulated and verified. Some of the basic reasons are listed by Serata70 as follows: 1. An underground formation is always subjected to triaxial stress, under which the behavior of the materials is quite dif- ferent from the uniaxial compression. 2. The laboratory study of triaxial compression exceeds a certain value which is specific to the individual materials of the formation. 3. The laboratory study of triaxial compression requires 1.2 extreme high pressure testing instruments which make the study rather difficult. 4. In addition to the static overburden pressure, the ground formulations may be subjected to tectonic pressures . 5. Usually, an underground medium consists of a variety of formations, whose structural properties are different from each other. 6. Strain confinement in the underground formation, different from the conventional triaxial test. Objective 5 The main objectives of this investigation were to: 1. Establish basic mathematical relations describing the distribution of stress, strain and creep motion in the medium of a thick walled cylinder. 2. Verify that the mechanical model, consisting of elastic, viscoelastic and viscoplastic elements, describes the overall behavior of rock salt as a function of stress and time. 3. Determine the mechanical constants of the material such as, Young's modulus, Poisson's ratio, octahedral shear strength, retarded shear modulus and viscosity coefficients. 4. Verify from laboratory data the basic mathematical equa- tions of stress, strain and strain rate distribution around a cylindrical cavity and the mechanical model of rock salt. 1. 3 Experimental Techniques By the very nature of the restraint offered by the massiveness of rock salt, underground openings are normally in a triaxial stress state. A thorough understanding of the distribution of stress, strain and creep motion in the medium around such openings requires a study of their behavior under multiaxial stress conditions. Elastic, viscoelastic, elastic ~plastic, completely plastic and other stress con— ditions might prevail in underground formations. Laboratory study of these stress states necessitated the devel- opment of two basically different testing techniques. First, a triaxial transition test was used to verify the applicability of the proposed mechanical model and to determine the elastic, viscoelastic and viscoplastic constants. These constants were used to calculate the theoretical cavity closure of the model salt cavity. Second, a high pressure, automatically controlled, vessel was developed to simulate underground cavity. Three different stress states of elastic, elastic- plastic and completely plastic states were obtained from this testing technique. The data obtained was used to verify the various maths.- matical equations describing the distribution of stress, strain and creep motion of the medium around a cylindrical salt cavity. PART ONE: THEORETICAL ANALYSIS CHAPTER II PRINCIPLES OF STRESS AND STRAIN FIELDS AROUND A CYLINDRICAL CAVITY 2. 1 Summary of Previous Work Done on Rock Salt Rock salt tested in different laboratories indicates isotropic and homogeneous qualities under static and dynamic loading: while at the same time, it exhibits elastic, plastic, elasto -plastic and brittle properties within a practical range of testing pressures. Most rocks are exceedingly brittle when deformed under ordinary atmospheric pressure conditions. However, they in general exhibit a plastic nature in varying degrees with the increase of triaxial compression. This increase in ductility depends on environmental conditions such as high confining pressure, high temperature, differential stress and exposure to liquid. The confining pressure plays an important role in bringing about the transition from brittle to ductile flow as demonstrated by 1’ 9’ 27’ 3° Accordingly, it is desirable to review the many investigators. properties of rock salt under uniaxial, biaxial and triaxial stress states . 2. la Uniaxial stress tests The uniaxial test is a common method extensively used to deter- mine structural behavior of rocks under stress. The behavior of a rock, with a consistent structural property, varies depending upon the testing method employed. The structural properties are usually defined by certain coefficients such as Young's modulus, Poisson's ratio and yielding stress. The value of the coefficients showed a wide variation depending on the investigator and method employed. The maximum strength of salt varies between 2300 psi and 5000 psi, while Young's modulus varies between 0.1x 106 psi to 1.0 x 106 psi. Serata70 attributed this wide variation to several factors, some of which are as follows: 1. Friction developed upon the loading surface. 2. Strain measuring devices, dial gages or strain gages. 3. Dimension of test specimen. The friction developed on the loading surfaces of the salt increases the ultimate strength and modulus of elasticity by the formation of a triaxial stress zone in the central region of the specimen. The strain measured by a dial gage is quite different from the same strain measured simultaneously by SR—4 strain gages. The size of specimen has an important influence upon the stress- strain curves, which can be eliminated by use of the friction reducer and proper proportion of height-to-width ratio. After these effects had been eliminated Serata 70 found the fol- lowing mechanical properties: 1. The maximum strength of salt is 2300 psi with a standard deviation of 200 psi. The yield strength of salt is found to be in the range of 1800 psi to 2200 psi. Mean value of Young's modulus was 0. 14 x 106 psi with a standard deviation of 0.03 x 106 psi. Poisson's ratio of value 1 has been found in granular aggregate, which is much higher than the theoretical limit of 0. 5. The Poisson's ratio taken on a single crystal grain in the same ag- gregate reveals a value which is not greater than 0. 5. Chowdiah,ll following the same procedure developed by Serata, reached the following conclusions: 1. The stress strain curve of rock salt does not exhibit linearity at any stage. The approximate value of chord modulus of elasticity E is: Stress range Average E (SR-4 gages) Average E (dial gages) 0 to 100 1.408 x 106 psi 0.4559 x 106 psi 1000 to 1000 0.1913 x 106 psi 0.1757 x 106 psi The value of the Poisson's ratio obtained from SR-—4 gages was greater than 0. 5, whereas with dial gages the value increased linearly from O to 0. 5 in the stress range 0 to 1500 psi. and was larger than 0. 5 beyond the stress 1500 psi. The average failure strength of the material is 3, 800 psi. 2. 1b Biaxial stress tests The failure of the material in a biaxial state of stress is differ- ent from that of uniaxial state. The load is applied in two directions and the material flows from the unconfined side. Chowdiah,ll using dial gages, strain gages and photo-stress techniques, reached the following conclusions: 1. The material has the tendency to flow by yielding at equal hori- zontal and vertical loads of 4000 psi (corresponding to an octa- hedral shear stress value of 1885 psi). 2. Under equal biaxial compressive stresses of 4000 psi the circular opening tended to flow without collapsing, while an oval opening collapsed by undergoing large deformation under the same biaxial loads. 2 3. Photo-stress results justified the assumption of statistical homogeneity and isotropy of rock salt material. 2.1c Triaxial stress tests Some of the earlier triaxial tests on rock salt were done by 1'4 on the increase in ductility and ul- Adams‘2 and his collaborators timate strength of rock salt under confining pressure. Von Karman78 performed triaxial tests under hydrostatic pres— sure. Samples of white marble and red sandstone were subjected to compressive external liquid pressures. The results show that the stress-strain curves were similar to those of ductile metals which Show work hardening and permanent deformations . Griggs27 continued the work started by Adams to study the relation of the triaxial characteristics of ultimate strength to the confining pressure and time under load. A liquid kerosene medium with a confining pressure of 13, 000 atmospheres was applied to small solid cylinders of limestone, marble and quartz. Some of the main results obtained are: l. A gradual change between brittle and ductile state is attained as confining pressure increases, and no sharp transition was noticed. 2. Continuous flow was not obtained in any of the materials used. 3. The ultimate strength of rocks decreases asymptotically to a certain value of strength at infinite time under load. 4. The rate of plastic deformation varied in proportion to the applied force. 5. The rate of plastic deformation at constant high loads decreases rapidly with time . Bridgman9 made some extension tests under hydrostatic pressures on rock salt crystals at an extension pressure of 420, 000 psi. He obtained a reduction of 20% in sample area in rock salt, while on solenhofen limestone under hydrostatic pressure of 400, 000 psi, he obtained an area reduction of 53%. 10 Handin30 studied the triaxial behavior of cylindrical salt speci- mens (1" x 1/2") under a confining pressure of 5200 atmospheres. He indicated that salt has a remarkable ductility in comparison. At 1200 atmospheres a specimen was shortened 75 per cent before fracture. He also indicated that the increase of strength and ductility with con- fining pressure, so striking with many rocks, is much less pronounced in salt, and much of the effect is observed in the first few hundred atmospheres. 30 work on the triaxial strength of salt was utilized by Handin's Serata70 to construct Mohr's envelopes. The envelopes at low con- fining pressures gave smaller ultimate shear strengths than the envelopes at higher confining pressures. The minimum required mean principal stress for the creation of a plastic state was found to be 5500 psi. Beyond this value, the Mohr's envelope became horizontal. Serata70 created a large mean stress by subjecting cylindrical salt specimen (6" O.D. , 2" I.D. , 2" h), confined in a steel jacket, to an axial load of 16, 000 psi. The important conclusion of his study is that the yield condition based on octahedral shear theory is appli- cable to rock salt whenever the mean stresses exceed 5500 psi. Serata70 and his collaborators55’ 65 developed a laboratory method by which the octahedral shear strength was determined and the transition region from elastic to plastic in continuous medium was illustrated. In their laboratory experiments, a cylindrical salt ll specimen was closely fitted in a thick steel cylinder. The lateral pressure is developed on the specimen through restraint by the sur» rounding steel as the specimen is compressed axially. The lateral stress was calculated from the strain on the outer surface of the steel cylinder, measured by SR-4 strain gages. The stress condition of the test specimen may be illustrated in the diagram of axial stress vs. lateral stress as in Fig. 2.1. The elastic state of stress is given as a straight line from the origin at the angle a defined by tan a 2 Tit—u , where u is Poisson's ratio, . “L p , . . Since U—- : I—t-i by Hooke 3 law under the condition of zero lateral z - strain. This elastic region extends between the two elastic lines OE and FG. The plastic state of stress is represented by another straight line, EF or HG, with the angle of tangent, B = 450 and expressed by The elastic and plastic lines can be interpreted as follows: the stress condition of the specimen remains elastic until the overburden pres- sure, 02, reaches the yield point. When this point of transition is exceeded, the stress condition becomes plastic, and remains plastic regardless of the initial overburden load. (The elastic line FG and the plastic line HG represents the unloading stress state.) Young's modulus and Poisson's ratio of the triaxially stressed medium can be obtained from measurement of the stress and strain 12 acmmcmmxo 1:32 Scum vocmunuaou .5532: a 5 25.330.” much.» as» mnmuauuuaflm Swans; anon: Hamxdunuonum “ascend TN .mwh 32: noon: 333 \ \ \. sud—«mun u. \ \ ms 2 u a :3 4 s c 0 v3.3 , \ \IIII'I. UMH.“.—o (19d) 999118 [2.19121 l3 mv ov mm > o~o>o i4 iii E 396 l #1.?! 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How- ever, Raman's65 results on limestone and granite, Figs. 2.2 and 2. 3 respectively, showed an unexpected behavior that is explained in the analysis to follow. 2.2 Elastic Analfiis of Stress and Strain Distribution Around a Cylindrical Opening An exact solution to the stress strain relation around a cavity opening using Hooke's law requires a simple geometrical form of the l6 cavity such as a circle, sphere or an ellipse. In an ideal form of cavity a theoretical solution to the problem can be obtained, which may provide a close approximation to a real cavity. Most of the under- ground openings do not have ideal geometrical forms and their exact solution is difficult to obtain. However, for practical purposes, the results of the analysis of idealized forms approximates that for the real forms. Thus the theory of elasticity can be applied to the stress and strain analysis provided the maximum shear stress around these geometrical openings never exceeds its critical value. According to the mathematical theory of elasticity, when radially symmetrical stress is applied around a cylindrical cavity in plane strain the maximum shear stress always occurs on the boundary. By increasing the applied stress, the maximum shear stress at the boundary reaches a certain critical value. The magnitude of these critical stresses depends upon the forms of the cavity and the initial stress conditions. When these critical stresses exceed the maximum shear strength of the material, it causes failure at the boundary. The high stress concentration is then relieved by the development of the plastic zone in the highly concentrated stress area. In this area, the salt undergoes plastic flow, which causes a gradual reduction in the stress until an equlibrium condition is reached. This theoretical analysis considers only a cavity in an ideally elastic and plastic medium. The cavity is assumed as a thick walled l7 tube loaded with a uniform external pressure P0 and a small internal pressure Pi subjected to plane strain conditions and axial symmetry. From this imposed condition the cylindrical coordinate directions, r, 0, and z are the principal axis and the stresses crr, 0 and 0 are 0 z the principal stresses. 2.2a Stress and strain analysis in completely elastic cylinder in plane strain The general stress solution can be obtained from equilibrium equations and is generally represented for the case of radially sym- metric plane strain with a superposed uniform axial strain €z as follows:77 r r 0’ —-‘£+ZC (2-2-1) 9 1.2 dz = E €z,+ 4pc where A and C are constants. When a0 and b0 are the inner and outer radii of the cylindrical cavity and Pi and Po the uniform internal and external pressures respectively, the two constants are determined by the given boundary conditions, and the equations become l8 bOZ 8'02 2 (P - po 2) 2 1 a0 0 0 0- : (Po’p.) _ Z '1' 2 r 1 r2 1— 3L2 1- €22 b0 b0 2 2 aoz (9—130 '27) 39.. (P P) i- + 1 a0‘ bOZ (2 2 2) 00 7 O i r3 a02 1 a02 ' 130‘2 1302 bo‘2 ao?‘ (P-P — — 0 02 b02 = E “z 6s + 2 H 1 — (aOZ/boz) Thus: a02 A = P —P ' ° 1' 1 - (ax/boa) and b02 3'02 P -P 2.2- < i o :0.) £72 ( 3) 2c = 1 " (302/1302) Substituting or, 06 and 02 of Eq. 2. 2-2 in the following Hooke's law 6 :31in - u(0'e+0'z)] r r €e=%[06-u(0r+az)] (2.2-4) _l 62 — E [O'Z - (.L (O'r+0'e)] the relations between strain and applied pressure are obtained, such that (for plane strain) 19 z ') 1 [3° 2 a0“ 6 = z 2 (l+u)(PO-P,)+(l-u-2u)(P,—2—‘P0)] r E(1-3_°_) r 1 1b. be?- (2.2 1 a 2 2 - -01 P-P 1——22P°-P)1 60" a‘2[ (+p)(o i)H P P)(- o E(1-_.°_. r b,2 is,2 b 2 210‘2 (p -po 42—) b 2 1 O 0 O 6z - E Hz.H 1- a: '— be?- The principal strains are related to the displacement by du 6 = r r du ur : _ 2.2-«6 69 r i ) du 6 = z = 0 in lane strain z dz p ' whereu =f(r); u =u =0 r 9 z The radial displacement is then expressed as 1 a02 5'02 ur= -"——;—z'[-;(1+u)(Po-Pi) -r(1—+L-2+L“)(Pi g"; - 130)] E(1-._°_) ° 2 b0 (2.2-7) For P, = 0: 1 (1+ p.) Po a02 = _ _— -3 2.2- ur a z [r '1' (1 H) r] ( 8) E(1-.._°_) b,z 20 which is a linear relation between the external load and the radial displacement. From Eq. 2.2-8, the modulus of elasticity may be expressed as, (H11) . E=— 2 2s0(1—p) (2.2-9) £311 (1_ 5:12.) APO 2 b0 where P0 = [1(a0) . By assuming a value of the Poisson's ratio and small strains where a E a0 the modulus of elasticity may be determined from laboratory measurement of Auo/AP0 by using specimens of thick walled cylinder. If plane strain is assumed and internal pressure is set equal to zero, the stress distribution of Eq. 2.2-2 reduces to 3'0 a = -———— (— +1) (2.2-10) 0' = - -———2— (1-31) b07- It is important to notice that in case of plane strain and external pressure only, the axial stress is not always the intermediate prin- cipal stress. The necessary conditions for 0'z not to be intermediate 21 area 2 0' or 0 S0 . From Eq. 2.2»10 this leads to z r z 0 2 < 30 -2|-'~--(—'2‘ 1) r , 2 21139-9?- +1 2 a _0_ _ ZH'I This proves that 0 is always the intermediate principal stress so z long as r :30 and 11 #1/2. 2.3 Elastic-Plastic Analysis of a Hollow Cylinder in Plane Strain 2. 3a Theoretical consideration of yield criteria Strength theories of a solid are usually described in terms of state of stress, strain and energy of distortion. Some of the theories that describe the yielding behavior of the material and relevant to this study are briefly summarized here. 1. Mohr's theory of rupture The state of stress at any point may be represented graphically by a plot known as the Mohr diagram. The Mohr theory of rupture may be expressed by the statement that for a material there exists a boundary called Mohr envelope such that a Mohr circle within the envelope represents a stable condition, whereas a circle tangent to the envelope represents failure on the plane denoted by the point of 22 tangency. The hypothesis formulated by Mohr may be presented as follows: 1. The line of rupture is independent of the means by which it is obtained. 2. The line of rupture is independent of the intermediate principal stress. 3. The angle of rupture in the material is equal to the angle of the line connecting the points of the minimum stress and the tangent in the Mohr's circle. In this theory the failure occurs when the maximum shear at a point ' ' - 0 - 0 reaches a critical value. The maXimum shear stress, T = ——1 3:: max 2 . . 01 + 0'3 , where k is a function of ———Z , the abSCissa of the center of Mohr circle. Triaxial study of rock salt70 showed that the Mohr's envelope theory is applicable only when the temperature surpasses 5000F or the mean principal stress exceeds 5, 500 psi. At this temperature and pressure the material becomes extremely ductile. 2. Coulomb's yield criteria In principal stress space, the yield surface is a right hexagonal pyramid equally inclined to the deviatoric stress axes (01', 02', 03'), with its apex on the line 01' = 02' = 03' . The hexagonal pyramid is irregular since the yield stress in tension differs from that in com- 72 pression. The stress equation on any failure plane is given by: Z3 - :: T 0rmax N4) crmin ZCV I\ct) N = tanz (450+cp/2) C = is the cohesion II (b is the angle of internal friction. It can be seen from this equation that when d) = 0, N4) = l and Coulomb's criterion becomes equivalent to the maximum shear theory, which is then a special case of Coulomb's criterion. 3. Maximum shear theory (Tresca yield condition) This criterion is usually stated as "yielding occurs when the greatest absolute value of any one of the three principal shear stresses "37 If crl and 03 are the maxi- in the material reaches a certain value. mum and minimum principal stresses, and k is a constant for the material, then where k is the yield stress in pure shear. . . . . Y In case of un1ax1al condition 01 = Y, 02 c 03 = 0 then 2 = k; hence the yield criterion can be stated as This theory may be considered as a special version of Mohr's theory. Both theories are independent of the intermediate principal stress and 24 consider only the maximum and minimum principal stresses to in~ fluence failure. In Mohr's theory the critical value, k, is a function “9.1.151 , the abscissa of the center of Mohr's circle, while in the maximum shear theory, k is a constant radius to any Mohr's circle representing yield conditions . 4. Energy of distortion theory (Mises yield condition) This takes into account the intermediate principal stress, while the maximum shear stress theory and Mohr's theory assume yield independent of the intermediate principal stress. According to this 33 "yielding begins when the (recoverable) €18.5th energy Of theory, distortion reaches a critical value. Thus a hydrostatic pressure does not cause yielding since it produces only elastic energy of compres— sion in an isotropic solid. " This criterion can be written in alterna— tive form as 2 J2' = Clio-1": 0'1'z+ 02‘2 + 03'2 = 2 k‘2 J (UX-UY)Z + (0y+ crz)‘Z + (02 - 0X)?‘ + 6(Tyzz + szz + TXYZ) = 6 k2 where k is a parameter depending on the amount of prestrain. The yield locus of this criterion is a circle of radius J? k as indicated from the above equation. The significant difference between Mises and Tresca's yield criteria is that "Mises criterion predicts that the maximum shear 25 stress in pure torsion is greater by a factor of 1.155 than in pure tension. Tresca's criterion, on the other hand, predicts that they are equal. However, both imply the same yield stress in uniaxial tension and compression, and both are independent of mean stress, 0m: l/3(01+02+0'3). 2.3b Stress analysis in elastic-plastic domain Plasticity is an important factor for structural stability of salt cavities. The creation of a cavity results in large concentration of stresses that will be reduced to certain tolerable values by the plastic deformation of salt around the cavity. This deformation starts when the differential pressure acting around a cavity reaches a certain critical value. If this value is exceeded, the cavity deforms plastically and never fracture in brittle fashion under increasing pressure but merely widens the plastic zone surrounding the cavity. When the external pressure is gradually increased from zero, the yield limit will be first reached at the inner surface. On further increase of the outside pressure the plastic domain extends outward into the elastic domain. If the cavity is considered as a thick walled tube, then the plastic domain may extend until the entire tube becomes plastic. The radius at the boundary between elastic and plastic domains is denoted by a "plastic radius, " p. For a symmetrical thick-walled cylinder where the external load is always larger than internal load, the main axis may be 26 considered as principal axis and 0 2 0 2 09. The general stress in r z the elastic region is governed by Eq. 2.2-l. From maximum shear theory 0 -cr=Y>O (2.3-1) 0 r The stresses from Eq. 2.2-l become U _U =___ (2.3-2) From Eq. 2.2-2, it is seen that Eq. 2. 3-1 can be satisfied only when Pi is larger than PO. This contradicts the earlier assumption that PC is always greater than Pi’ and hence the maximum yield theory may be written as 0 - 0 = constant = (2.3-3) 3 —-——K r 0 21f; o where K0 is the octahedral shearing strength of rock salt determined from the octahedral shear stress theory of failure. 2 0'1 "73 To 2 KO = 3(01 - 03) for the condition 02 = 03 . Since Tmax = 2 , therefore 'r = i-Ko' max 2V2 From Eqs. 2.2-l and 2.2-3 _ 34 or 09 — r2 at first yield, r = a0 3 z A = a0 K0 (2.3-4) 2 2 Substituting this value into Eq. 2.2-2 gives the value of the pressure at first yield 27 2 a . 3 .2. (PO-P.) at first yield: K0(l— 2) (2.3-5) 1 b0 2 2 At the elastic -plastic boundary (r = p) the internal. radial pi”r:.%.*.>;11‘(i is defined by Pp and A = -§—- pz K0 215 The pressure necessary to cause yielding at r = p is then defined as 2 Po-P =—§—K0(1-f3) (2.3—5) p 2"2 0 From this equation the unknown value of the radial pressure, Pp, at r = p is given by P =P0 - 3 Ko (17%) (2.3-7) p 25 O and from Eq. 2.2-3 atr=p 2 2 (P -POPL)£§ P 7- b p 0 2C : (2.3-8) [1' (Pa/1302)] Substituting the value of P into above equation yields, 2 Zc=-Po - 3 K0 'P— (2'3”9) 2 2 b0?- The general stress distribution in the elastic region under the condi— tion or 2 0'2 2 09, is then defined as a function of plastic radius 9 by substituting the value of A and 2 c into Eq. 2. 2-1 such that r 2 2 2 2 1302 l 2 2 O. :--—2—K0(E—+-B—) _p0 (2.3—10) 0 2 1.2. b0?— 2 2 2 Uz=-H(—3-T_Kop—+ 2P0) 2\/2 b0?- this stress distribution equation holds in the elastic region prescribed by p S r S be . The stress distribution equations in the plastic region prescribed by a S r S p can be determined by using the equilibrium equation written in total derivative form since there is no variation with respect to z or 0. (10 0-0' r+_____:0 (2.3-1].) The rest of the equilibrium equations vanish because of axial symmetry. Assuming ideal plasticity with no strain hardening, the yield condition is satisfied throughout the plastic region with the same constant K0 . From maximum shear theory of yield 3 0 -0' =——Ko rah/7 therefore do-r_ 3 E3 dr ZI—Z r or 3 or = - K0 1n r + cl (2.3-12) N—z 29 The constant of integration is determined from the continuity condition at the elastic plastic boundary. At r = p 0] = 0 ] r . r . elastic plastic Equating Eq. 2. 3-10 and Eq. 2. 3-12, the constant of integration is 2 C1 21- KO(l--p—2-)+ 22$K0 boZ [VF—O Substitute the value of cl back into Eq. 2. 3-12 and use the maximum KO lnp -PO (2.3-13) shear theory to obtain the following stress distribution. 2 sr = —:-5-K01n(f7) +Zl—1—KO (1- if)- P0 2V2 21/2 bo 2 3 P 0' 1n(P—) --——-—K (1+—) - Po (2.3-14) e 2V2 2 2V2 0 ho For plane strain, E = 0 and 0 = p(0' + 0‘ ), therefore: z z r 0 :2 0' =2p.[--—KO ln(E-) -—K — ZO—P] 7‘ 2W0 I 2% ° These equations describe the stress distribution in plastic region (a s r s p). The radius to the front of the plastic zone can be ob- tained as a function of the loading conditions of the cavity. By sub- stituting the condition at the inner boundary (r = a), P, = - or, into 1 Eq. 2.3-l4 it may be expressed as: 30 l l p=aexp.[—3-——(PO-Pi)-:Z- (1--‘E-2) (2.3-15) -—K o If the ratio, p/bo is sufficiently smaller than unity, Eq. 2. 3-15 be- comes with good approximation p:aexp. [—3—]:— (Po-P) -l] (2.3-1.6) K 1 2 2x5 0 which exhibits a linear relation between p and Po - Pi on semi—log diagram. These two equations (2.3-15 and 2.3-l6) determine the advance of the plastic radius into the elastic region, if it is assumed that the change in internal radius is so small with respect to the mean radius that “a0" instead of "a" can be used in the calculations. How- ever, if the change in "a" is significant compared with the "ac" value, this cannot be neglected and an expression for "a" in terms of p is to be solved simultaneously. It is interesting to note that the plastic radius is independent of the axial strain, 62, so long as or > oz > 09. 2. 3c Stress strain relation in combined elastic plastic domain. When a specimen reaches the yield point at the inner boundary, a permanent plastic deformation will occur. Then the total deforma— tion becomes a sum of a recoverable elastic deformation and a permanent plastic deformation. As the load increases the plastic region spreads outward until the whole cylinder becomes completely plastic, reaching an equilibrium with the outside loads. Whenever a 31 plastic condition is present, the general strain increment equation can be written as d6..=d€.. + de.. (2.3-l7) Where the superscripts e and P denote the elastic and plastic com- ponents respectively. The elastic strain increments are given by Hooke' 5 law as: deez r [dar- pd(cre+0‘z)] NIH e 1 (166 - E [doe-ud(crr+crz)] (166 = -1-[d0' -pd(0' +0')] z E z r 0 by knowing the changes in stresses, the elastic strain rate can be determined. If the plastic potential theory is used, the plastic strain increment may be determined as follows: The strain increment vector has to be normal to the yield surface in stress space such that (2.3-19) If the Mises yield condition is assumed, then f=l-[(0'-0‘)2+(0'—0')Z+(0'—0)Z]+£[T2 +T2 +T2 +T2 +72 ] 6 x y y z z x 2 xy yx zy zx xz (2.3-20) 32 and iii _ . 601j ij P Eq. 2.3-l9 may then be expressed as deij = dx 0"ij (2.3-21) or P P d€i. %- dyi ,3 = J = d). 0'. . T. . 1.1 1.1 This may be expressed as P P P derp - deep deep - dez dez - dsr : 2 = CIA (2.3-22) - 0' 0 - 0 0 - 0 r 0 0 z z r If Eq. 2. 3-21 is expanded the total stress-strain relation which is 33 known as the Reuss equation may be written as: l l dEr—E[d0'r-pd(06+crz)]+§d)\(20'r-06-0'z) d6 =-]3-[d0'-pd(0'+0")]+ldk(20 -0" ~0) (2 3-23) 0 E 0 r z 3 0 r z ' l l dEZ —E[d0'z-pd(06+0"r)]+-3-d)\(20'z—0'r—(Ie) The strain increment equations may be determined by relating dlx with the octahedral shear stress To and the increment of plastic octahedral , P shear strain dvo as follows: The octahedral shear stress and strain are defined as 1/2 -1; Z Z Z 2 2 Z To-3[(‘[((rr 00) +(UG-Uz) +(UzuO-r) +6lllllmimum uHN nW+ddadiUN OM1>|HS \\ m m Nit: ms 2 a 3-: m \ c .. \ \ M \\ m \ d \ \ a s \\ \ IIII‘ I‘\ \ 111 II .\ 1 \\ o l o \ use m ... M \ \ \ \ mm 0 H H o l m use ooom u M 39 (where u is a negative quantity). A mathematical concept of a model salt cavity is presented in Fig. 2. 5. 2. 3e Determination of the inner radius of a hollow thick- walled cylinder for large --strain solution and plane strain. If the compressibility equation is used and the movement of the plastic boundary radius, p, is taken as scale of "time" (see Hill33, p. 100), the relation of inner radius to plastic radius is calculated as 51 follows: _ l-Zu d dr+de+dz-( E )dp (O’r + 06+0'z) (2.3-39) d . . . . . . where —- is a material derivative applied to a function of p and r such do that d 8 6 a; - 8 + V( 8r) V = radial velocity = 37:1, where r is the coordinate of a moving particle. dV dr-a; d =35 6 r It is known that crz = u ( or + 0‘9) , in plane strain with maximum shear yield condition and plastic potential theory, where z __ P___E_. _ Ko(21nr b2) 2P0 0'+0‘- r 9 2V; 0 fiTTTTTOv "U o n q H Displacement 3 2 .1 _ (l - 2") L23 K ( 1' In D + 1‘) Por Z‘rzKo p r] u - ———-— -—:- 0 - - — - - --——-—-— G [I 3 Z r 4 Z 4 b 2. —--- K 2 0 Fig. 2. 5 Mathematical concept of model salt cavity 41 From above expression and using Eq. 2. 3-39 for plane strain with d = 0, gives ——+—+o=(1—'—-— 2HL>—[2 —-3-——K (1+u)1nP—~2<1+u.)P r d 2 2 o o z - (1+ H) -—3- K E-] (2.3-40) 2\/—2- O boa Now let 2G =—_?— 1 P _. 3. a l ”Pi. .. 2\/_2_Ko[ln a + 2 (1 boa” (2.3—41) dP _9. __L) :2“.- K0 (p boa Substitute the value of P0 back into Eq. 2.3-40 to eliminate PO, then Taking the derivative with respect to "time, " p, and collecting terms, itbecomes 1 d LEE 3 K (_-_?_&,[_1._s._1_ 11;] (23.42) da . . . . . where -— is the rate of change in internal radius With respect to dp 42 "time, " p, which is the velocity expressed as V(a), and 3% = V. Hence V dV 3 1-2 V V F+T‘—Ko( GP” (am-:1 ‘2'3’43’ zs/"é 1- Let M = —3—KO( G2“) 2 2 so that Eq. 2.3-43 reduces to fl+ (1+M)‘—’=MY—(—§-) (2.3-44) dr r a Assuming M to be very small compared to unity, this is closely appr oximate d by d_V + X = M V(_a) dr r a or dV _ d _ V(a) rdr+V-dr(Vr)—M a r whence z Vr = M V(aa) £2- + c(p) (2.3-45) Since V is a function of both r and p, c(p) is the constant of integra- tion, to be determined from the continuity conditions at the elastic- plastic boundary. Recalling that r = p at the elastic —plastic boundary VezVPatr=p Assume 6 is zero in plane strain and replaces Po by its value as a z 4.3 function of p. Then the elastic region radial displacement of Eq. 2.3-31 reduces, after collecting terms, to 3 —--—K _ £_____‘/ZO[&Z_ 20 2 ue - r +(l-'2p) rung-+3?” (3.546) Let du=Vdp -i‘i=§3 21.1 dp 8p 3r or 33”.. 3.5., hence, vzaL/gE (2.3.47) 1'5: From Eqs. 2.3-46 and 2.3-47, the value V at the elastic-plastic aue aue boundary may be determined by taking derivatives E and -<-i—p_ and substituting into Eq. 2. 3-47 to get for r = p v ] = (2.3--48) since 3 —K 2J3 o E 1 2c; (1'2“)(1na+2)<<1 44 we may approximate Ve] r=p 3 3 __ 1_ __ 2 5K0 ( 2P) 2 2 K v =--———— (iii/(an r=P 2G 2G a or 3 ---—---K O V :_2\/-..E__M(1_Bv(a)) (2.3-49) r=p 2G 2 a Now apply the condition that Ve = V at the elastic-plastic boundary P which implies that Eq. 2.3-45 is equal to Eq. 2. 3-49 at r = p. Therefore do) =- p 3 2 5 O M 2G 2 if this value is substituted back into Eq. 2. 3-45, the solution for plastic velocity is then __3_._K _MV(a)r_(2V/—Z— ° M)p_ 2G r _ _ ,.0 V 2a +2 (235) which describes the velocity at any point in the plastic region. . . , a From the preceding results, an expressmn for the ratio (a—) of the 0 current inside radius to the original inside radius can be obtained in terms of the elastic plastic boundary radius. Evaluating Eq. 2. 3-50 at r = a and substituting da V(a) = 21-;- yields 45 3 - K da Z/E O P. ivl ;: —(1-—2‘)=-——'—— “7'11”“ G a .. (‘1. or a —-—-—— K p 1 2‘12 0 M add. : ——'—"" ..._.........._....___.-- .. .2-) priD M 26 z. (1- 7)( , where the limit on a is from initial yield internal radius al to cur— rent internal radius a and the integral on p is fron'i same point a1 to current position p. Integrating, and neglecting M in comparison to unity, yields 3 —-—K 25° (612 -a1"') = -K—-—ZG + %)(92 “12% or 3 —-—- -—-——K 2V—z—O M ,_(2\/—° 2- -—- — = 2 2.3-l (1+ 2G + )a1 +-2-)p ( 5) Now 0 0 where uo is the boundary displacement at first yield. aIZ=a2+2au +uz, o oo 0 Since uO << a , a good approximation is 0 a12 - ao‘2 = 2a 11 (3.3-52) and u as a function of a0 may be obtained from the elastic solution 0 46 radial displacement at p = a0 when 3 ——K 2 2V2 0 21o P =———— (l--—-—? the boundary conditions for the elastic solution being imposed as usual in the undeformed configuration with a = a . Substituting the 0 value of P and collecting terms reduces the displacement given 0 by Eq. 2.3-46 to 3 ——K a. O u = - —°- (5— “2 + M) (2.3-53) 0 2 2c; Substitute this back into Eq. 2.3-52 to obtain Hence an expression for current radius a in terms of initial radius a0 may be obtained by substituting the value of "a1" into Eq. 2.3-51 to get 3 K 3 K az : az[1- (LE—O— + M.)2] __ (ii-L: + M) pz 0 2G 2 2G Since 3 --——K 2V2 0 M2 —— — << ( 20 + ) 1 2 The above equation is closely approximated by 3 K az=az_(_fl_9.+_1;£)p2 2 o ZG 47 01' __§_K Z O 2 [a2 =1-E—Q—(1-H)L],3?l (25-54.) a G a‘2 a 0 o o This equation describes the behavior of inside radius as a function of plastic radius and structural properties of the material. It is clear that as the plastic radius p, which is a function of loading, increases, the ratio of ;a_ decreases. For p = a0 at the initial yield, the ratio of O 50- is equal to the ratio of elastic radius a to initial radius a0. This relation between the inside radius and plastic radius defines too, the stability of the cavity. As p increases, the outside pressure increases accordingly and hence the inside radius, a, decreases and thus the cavity remains stable. It should be remarked here that the second term on the right is small in comparison to unity until p becomes large in comparison to a0. 2.4 Analysis of Comletely Plastic Cylinder with Plane Strain 2.4a Stress distribution in completely plastic cylinder. From equilibrium equation, we obtain, as in Eq. 2. 3—12 0' =—-KO lnr+B (Z.4--1) erF—0 The constant of integration B is determined from the internal boundary condition that at r = a, or = - Pi' Hence B=-—-K Olna-Pi 2F° 48 and 0' -' 3 K(lr -) P r 2f; 0 3 a (fez—Kofln; -1) -Pi (2.4-2) 2 2 1 UZ=Zu(-—3——Koln-r--E-Pi) Z 2 Eq. 2.4-2 describes the plastic stress distribution in the domain where a S r 5 b. The outside pressure necessary to create a com- plete plastic state is determined by considering the boundary condi- tionatr=b, (r =-P. Hence r o -PO=—3——Ko(ln g) -P 2 2 1 or 3 a P =—--K ln(-) +P, (2.4-3) 0 2 2 o b i The behavior of the inner radius in the plastic region may be expressed as .. _..__._ (p0 - Pi) (2.4-4) It is very convenient to obtain the octahedral shearing strength from the slope of this curve. The theoretical expression for the ratio a/b when plotted against the outside pressure on a semi-log paper gives a linear relation whose slope is - 3/2V 2 K0, a function of the 49 octahedral shearing strength. The above equation will be used later to determine the shear strength of the material. If incompressibility is assumed, the plastic strains and dis- placements may be calculated as follows: r 9 z or du u ——+—+€ :0 dr r z On integration this equation yields u— 2+1 —2 r , Ez - constant (Z.4~5) The constant of integration c1 is determined as a function of radial displacement at r = a where, 63- C 2 l u=u =- +— a r=a 2 a or eaz C Z + a = u l 2 a Substituting this into Eq. 2.4-5) yields 5 2 u-l(i—-r)+u -2 r a The displacement at the elastic plastic boundary is obtained by substituting r = p into above equation to give 62 ‘57“ m N -p) +u (2.4-7) P a N '0'” Neglecting the first term on the right, which is zero in plane strain, the displacement at the innerface of the cavity may be expressed as _ E u .. ( ) u (2.4-8) e Because of the continuity at the elastic plastic boundary, up =ur_P Hence by Eq. 2.3-31 _§_K l 2 2 ° 3 uazg' E —J——Z—_[(ZH-1)ELZ ‘P]+(2P"1)ppo (2'4-9) This result was derived under the assumption of compressibility in the elastic region and incompressibility in the plastic region. Finally, the tangential and radial strains in the plastic region are also obtained from Eq. 2.4-6, such that 6Z a2 a : — : — — .. 1 + _ 69 r 2 (r7- ) ua. r2 (2.4”10) € du z a2 a : -— : — — — 1 — — 61' dr 2 (r2 + ) ua. r2 where ua is given by Eq. 2.4-9. CHAPTER III DETERMINATION OF THE MATE RIAL PROPERTIES 3.1 Linear Viscoelastic Behavior of a Solid Cylinder with a Laterally Constrained Motion 3.1a General remarks Elastic and plastic analysis of rock salt has been treated solely in the previous chapters with no reference to the viscous behavior of the material. The triaxial study of rock salt in the laboratory showed that rock salt exhibits certain plastic and viscoelastic properties. These properties are needed to describe the time dependent behavior of the hollow cylinder with mathematical formulas. Solid materials that possess the characteristics to flow are called viscoelastic or viscoplastic materials. Such materials possess rigidity and at the same time flow and dissipate energy by internal friction. Under different external conditions, these materials may exhibit both solid and fluid characteristics simultaneously. This study is mainly concerned with those materials which possess charm- acteristics of both the elastic (Hookean) solid and the viscous New"— tonian fluid. The ideal linear elastic element is represented by a "spring" and a linear viscous element by a ”dashpot. ” Since it is desired to look at the material behavior in between the spring and dashpot, different mechanical models were arranged to fit different 51 52 materials. There are several different possibilities for combining elasticity and viscosity, but two approaches have received special attention. Simplified forms of these are the elements in parallel (Voigt, Kelvin body) and the elements in series (Maxwell). The constitutive equations of the above two models are indicated respec- tively as or = (E + 71D) 6 (Kelvin) and ' D l 6: (—+ —)o (Maxwell) E n . . . . . d where n is the Viscos1ty coeffiCient and D = Tit . Both of the above equations contain the material time derivative of small strain and the Maxwell contains the material time derivative of stress. Both 52 and hence cannot be applied for of these equations are not objective unrestricted motions. However for symmetrical loading, where there is no rotation, the principle of objectivity is not involved. 3.1b General visco-plastic behavior of rock salt. The physical behavior of a rock salt specimen subjected to homogeneous stresses of the type oz i ox = (Ty is investigated. This investigation intends to describe the stress and strain behavior of rock salt by a mechanical model. Based on the triaxial behavior of a laterally confined specimen the following observations have been made: 5 .3 Rapid plastic deformation occurs during loading and shortly afterwards and reaches its maximum value in :3 few hours period. The amount of deformation in Part 1 is a relatively large part of the total deformation. A slow deformation process continues to take place for a. long period of time extending in some experiments up to a period of 40 days . From the above observations, it was concluded that A rapid plastic deformation takes place almost instantaneously during loading, and rapid deformation continues for a short period after load is kept constant, indicating a relatively small viscosity coefficient. A slow viscoelastic deformation, whose effect is negliglble in Part 1, continues to take place for a. long period of time ind1=- cating a relatively larger viscosity coefficient than the previous part. Such behavior suggested the following modelzf’g E2 771 * —-—-/\/\/WN——. fl J—J E1 W“ >- 0' 4:} . ”2' . K ' 0 Fig. 3.1 Plastic. viscoelastic: model of. rock. salt {one dimensional) 54 where (I - stress, psi K' = maximum yielding strength El - Young‘s modulus, psi [Tl II retarded Young‘s modulus, psi 11'1 = linear plastic viscosity coefficient,poises in one dimension 77'; = viscoelastic viscosity coefficient, poises in one dimension 3.1c Plastic. stress relaxation equation and the determination of the plastic constant: 171 The elastic and plastic. model, which describes the rapid change during a short period after loading is represented as follows ”1 T1 '_.l—_ G] T ———J ~—-——>- T O A—mwix O K o where _ E , . _. G1 - --.-<'---- elastic: shear modulus 2 (1+ p.) T = octahedral shear stress 0 K = octahedral shearing strength 0 n, = viscosity coefficient in viscoplastic region The nonhomogeneous constitutive equation of the above model may then be presented as follows: i =_9..+ 3.--_.9. (3.1-1) 55 where Yo = octahedral shear strain. The Levy-Mises equation presented before may be written in tensor notation s 0 that where with the understanding that if the deformation is small, the natural strain increment dei, may be represented by small strain increments deij, where del, deg and d€3 are the components of small strain increment. The meaning of the scalar function dk was explained in the previous chapter as a function of octahedral shear stress and the increment of the plastic octahedral shear strain is presented as follows: Substituting this value into the Levy-Mises equation and dividing by dt yields the strain rate equations (16! dy 1j_ l O , _ dt " 27 dt orij (3'1 2’ whe re The rate of the plastic OCtahedr a1. shear strain is obtained from the plastic model such that P dyo T .. KO dt ‘ n, Substituting this value into the deviatoric strain rate equations gives deij 1 K0 .._...- : __.._g'1-.—— ' 3.1-3 dt 2m ‘ To) “‘11 ( ) This describes the three dimensional flow where incompressibility is assumed in the plastic state. The deviatoric stresses may be obtained as a function of the deviatoric strain rate, such that 1 de'ij . : ______. .1-4 Uij K 2 n1 dt (3 ) ll - '53) T where the coefficients of the strain rates contain the viscosity term 2r)l and plasticity term KO/TO Including the effect of the elastic strain rate in the deviatoric strainmrate equation, the result is as follows (3.1.5) 1 _ —— (1 2771 where the last term on the right is contributed by the elastic strain rate. From this constitutive equation, it is desired to obtain the stress relaxation equation in the viscoplastic region. Assuming incompressibility, defl. = deifreduces Eq. 31-5 to delJ 1 KO 1 do'lJ -.- —— 1 - —— ' .— ' ,1- dt 2m ( To) “ij ZG, dt (5 6) The axial elastic plastic stress-strain relation is written from the above e quation a s d6z 1 K0 1 dcr'z = —— l- —- ' + —— —— .l-» ’ dt 2771 ( To )Uz 2G1 dt (3 6) From boundary condition of solid cylinder, the following substitution can be made: E. 1.0: T(UL-Uz) Z l=_ _ 02 3(6 UL) I (102‘ E—dUL' cr‘cnt t dt“3dt’ z‘08am Experimental results for the tests described in sec. 6.3b indicate dcz dt that is approximately zero. Hence, substituting the above quantities into Eq. 3.1-6' we obtain, 0’ e =(0' .;——3 K )enl +C (3'1”?) 58 From initial boundary conditions, at t = 0 after the initial purely elastic deformation, since EL = 0, then _.__L*_ UL lop OPz Thus, the integration constant, C, in Eq. 3.1-7 becomes _ 41..-. -~__§_ C-O'z(1- 1)+ K u f; 0 Hence, the lateral stress relaxation equation in the viscoplastic region is described as follows: K)+[(—*—*—- -1)a;-———K]e ”1 (3.1~8) o z o For the loading cycle equation 3.l~-8 takes the form of ~01 _t 3 "1 .L’L. __ __ .. “H 1) crz+ ZK0]e (3.19) (r =(cr-—3—-K)+[( Lzfio "I As time approaches infinity + 3 0- :cr -—-—-K (3'1"10) L z \/ 2 0 Therefore, the tangent of the ”plastic line“ is do L: tanB =1 (3.1-4.11) dcrZ From Eq. 3.1—8, it is observed that the last term is a function of time and the plastic viscosity coefficient n1. Hence, the one to one 59 relationship between lateral stress and axial stress demonstrated in Eq. 3.1—ll may be obtained by either letting t approach infinity or by assuming that 171 is a very small number. The change of lateral stress as a function of axial stress and at constant interval of time may be demonstrated from Eq. 3.1-9 as follows: 1. define t0 as the time measured after each increment of loading, i.e., to = 0 at points A, B, andC of Fig. 3.2.. 2 0’ > (r z z 2 1 3. t = constant Then from Eq. 3.1-9 u 3 Acr =0 -a =[(—-1)cr +— 1D (3-1-12) L1 th1 tho l-H Z1 (2 o and AUL=ULt-0'Lt=[(——1P-l)cr +—-3 K]D 2 22 20 -H 22 2. O -E.lt whereD—e m , t-tl-t =tZ-t O 0 Therefore, from the above equation AU > A0 L L 2 1 Since 0" > 0' z z 2 1 This indicates that AUL is a linear function of crz and may be repre- sented graphically by Fig. 3. 2. 60 In the laboratory, an experimental stress relaxation equation equivalent to that of Eq. 3.1-,9 was determined as A0 3 ‘ Hydrosta . t = 111 L Line C / / ‘. Measu | | AOLZ t =~‘t . a l/ 0 / B AoLl I ,/ ' A Theoretical l / *i” / / . / t = O / / ' o o’ 0' F'— 21 22 23 Fig. 3. 2 Theoretical prediction of the change of lateral stress in viscoplasticregion _ 3 3 ”alt UL—O’z-an-J—ZZ—(T,-Tf)e (3.1-l3) where, Tf - octahedral shear stress at t = 00 Ti - octahedral shear stress at t = 0 al - material constant Comparison between Eq. 3.1-9 and Eq. 3.1-l3 gives ’71 = Sf (3.1-l4) where (11 is determined experimentally from the slope of plastic deformation. 3.1d Viscoelastic stress relaxation equation and the determination of viscosity constant, n2 For a Kelvin body and spring in series, Fig. 3.1, the one dimensional constitutive equation, is presented as follows: dcr E, + E2 de E, E2 An equivalent three-dimensional constitutive equation may be written as, d‘r dy 0 61+ G2 0 GI G2 — + ._________ :: —— + _ .1-1 dt ( n2 )To GI dt nz' Yo (3 6) where G1 = shear modulus retarded shear modulus .9 I viscosity coefficient in viscoelastic region :5 N II 62 Eq. 3.1:.16 may be solved either for creep or stress relaxation response depending on the condition prescribed. For a laterally confined solid cylinder with homogeneous stress distribution, it is as sumed that. = (- YO viuxt) u(t) = unit step function t l,t=O Mt): 6(t) where[ 6(t)dt={ o 0,t>0 Hence, d‘r O GI+GZ GI G2 __.._ + __ T =_______ , dt ( n2 ) 0 n2 yiU(t)TG1yi6(t) integrating G +G G +G (—‘—n——?:)t (3le (Jr-in T e Z = (--——) y. e 2 u(t)+Gly.u(t)+C 0 GI +GZ 1 1 From initial conditions att = 0, To = Gly'l, then (3,02 “*m’n Substitute this back into the above equation and arrange terms to get To: -(Elfitgi)t -(E}._I_T.7.+_ciz.)t GIGZ (t) (3117) : 2 - Z -————-— v.11 O - To GIYI e u(t) + [1 e ]G1+GZ 1 As time approaches infinity 63 1- : 1- : 212’; y, = constant (3.148) G1+G2 where 'rf is the minimum value of To below which yielding never occurs 0 The lateral stress relaxation is obtained from the relation _ (Mn -(G1+Gz)t G G + 3 ’72 1 2 1 2 O'L—O‘z _V—_—: [Glyi e u(t) +[ -e ]G1+Gz yiuU‘.) (3.1-l9) which indicates that on the lateral vs. axial stress diagram, stress . . . . . 0 . relaxation in Viscoelastic region starts from 45 line to reach a con- stant value at t = 00, indicated by (TL: Uzti-(fi) Yo u(t) (3.1-20) V 2 G1 + G; Eq. 3.1-20 will be used to obtain the retarded shear modulus G2 and to verify the constant stress relaxation region as indicated in Fig. 6.13. In the laboratory, a stress relaxation equation equivalent to Eq. 3.1-l7 was determined such that -azt T=T+(-r,--r)e o i f f 64 where a2 is a material constant defined by the slope of the stress relaxation curve in the viscoelastic region. A comparison of Eq. 3.1-17 and Eq. 3.1-21 gives + n; =E—’——93 (3.1-22) “2 Eq. 3.1-14 and Eq. 3.1-22. define the viscosity in the plastic and viscoelastic region re spectively. CHAPTER IV THEORETICAL BEHAVIOR OF A CYLINDRICAL CAVITY UNDER CONSTANT TRIAXIAL STRESS The creep formulas, for the same thick walled cylinder treated in Chapter II, are derived for the purpose of measuring the strain rate of a model salt cavity in the laboratory. The derivation is based on the following assumptions: 1. Deformation is uniform and sufficiently small to produce no appreciable change in geometry of the cavity. 2. At a constant external stress, the deformation occurs in such a way that the stress distribution across the cylindrical sur- face remains constant with time and'that the octahedral shear stress may be expressed as T = To u(t) . 3. The material has a different creep rate depending on the state of stress and the temperature employed. The temperature is kept constant across the thickness of the cylinder. 4. The mechanical model presented in Chapter III describes the viscous behavior of rock salt. 4.1 Strain Rate Equations in a Hollow Thick Walled Cylinder. Several theories have been proposed for predicting creep strains in terms of stresses for the combined states of stress. Most of these theories are based on an assumed tensile creep stressn-strain relation. 65 66 A generalized stress-strain relation based on Hooke's law and the basic laws of plastic flow is described in Eq. 2. 3—27 where the elastic stresses and the octahedral shear strain are functions of time. The characteristic feature for large deformation in a thick walled cylinder is that for every increment of time there is not only a set of increments of plastic strains but also a set of increments of stresses. By assuming small deformation with stress increment E 0, the octa- hedral creep strain function, y0(t), for rock salts may be determined from the mechanical model as follows: The three dimensional constitutive equation for the viscoelastic part of the model is given by dT d 0 “913)? = G, 133+ G‘G" v12 (4.1-1) dt T): 0 dt 772 where Yiz = Y1 + Y2 y, = octahedral shear strain in elastic element y; = octahedral shear strain in viscoelastic element Taking 'r = To u(t), then lez G2 To 6(t) GI+GZ ——+— =————+—————Tut dt 17; Y” G, G2 272 o () solving this equation to get, G2 GZ —t T G,+G2 1?; -—t O __ vlze"2=-—u(t)+ '-e"2 ut +C G, Gm. G. H 67 Att=0 T . 0 “2:5: Then Gl+Gz Thus, the viscoelastic creep response function is derived as, GZ G2 - —t - --t To ”2 G1 +G2 G1+Gz n2 z __ ._______ _ _ r. Y12 G, e u(t)+ GIGZ To u(t) G102 e Tou.t) Collecting terms, it reduces to C32 - —t G1+G2 1 n2 : __ _ _ 4.1- YIZ [ GIGZ 62 6 ]TO 11(t) ( 2) Similarly, the octahedral shear strain from the viscoplastic part, Y3: can be derived as follows: To -Ko y3 = ( ”1 )dt for To > K0 0 1b -Ko v3 = ( )t (4.1-3) 771 Hence, the total creep response for the model in plastic and viscoelastic regions is derived as, 62 - —-t v 2 [._—————..-l- e ]1"u(t) + (-£L——i1)t (4.1-4) 0 GIGZ G2 0 n1 68 By taking time derivative, this function yields G2 G2- -—t -——t dYO G1+ G2 1 n2 n2 TO- K —— : —— - ——- T + —- T ' + 4.1- dt GIGZ G2 6 ] 06H) 772 e on”) 771 ( 5) Fort> 0 G2 --—-—t dYo 1.0-K0 l T12 ——+- = ———-- + -— e T u t 4.1-6 d, < m ) n2 0 < ) ( > where the first term on the right is constant with time and contributed from plastic flow just after load is applied. When stresses are constant with time, the general stress— strain relation is de dy r l o 1 dt ‘ 270 dt 3 (ZUr'Ue'O'z) d6 y l o l ___ . _ _ _ .1- dt 270 dt 3 (2% or 02.) (4 7) d d€z-_i__YE._(ZO. U 0.) dt ‘z-ro dt z . 0 . . . . Substitute the value of dt into the above equation to obtain de dt 69 G2 -—t T -K 772 l o o l 2 1 z +-—— — -- ZTO[ ’71 ’72 e 'r u(t)] 3 [a 2(0'r+ oz)] G2 -—t l 1 Ko l 772 -_-.—[——(1-__)+__e ]cr' 2 771 T 712 r o 4.1-8 G2 ( ) -—t K 172 _l l o l , -2[nl (l-T)+Ze ]Ue deviatoric stresses are: 2 l l___ __ 0' — 3 [or 2 (09+ 02)] 2 l .3—[09- -2- (Ur+qz)] For the case where To = K0' the first term on the right of Eq. 4.1-8 vanishe s and de dt dose dt. 02 -—-t ._. ,. . _Le "2 r Zn; Ga (4.1-9) -—t __ I o 1 n2 — (re ane As time approaches infinity, the strain rates approach zero. This 70 is true only in the viscoelastic region. The strain rate equation for any cylindrical cavity may then be defined by substituting the values of the deviatoric stresses for any particular cavity. These stresses are usually defined as a function of the boundary conditions, octa- hedral shear strength and applied external loads . From Eqs. 4.1.5 and 4.1-8, it is observed that the plastic deformation at t = 0 consists of an instantaneous elastic part and a plastic part described by der 1 G, + G2 1 K I o _ _ _ + __ 1.. __ ' - 4.1-10 dt 21-0 [GIGZ G, Th ( 1- )0r T0u(t)] ( ) 0 At t > O, the elastic deformation vanishes and the above equation reduces to the triaxial plastic flow where the plastic strain rates are linearly proportional to the deviatoric stresses, such that der 1 o = _ m — l .1” dt 2271 (1 To) “r (4 11) 4. 2 Comparison of Theoretical and Empirical Equations Describing the Creep Behavior of the Model Cavity From the viscoelastic state, the tangential strain rate equation is defined as 71 The total amount of tangential strain in the opening can be calculated by integrating the above equation such that Gz t -—-t 2 n2 By integrating between t, = t and t2 = 00, gives 02 , - —t o—e nZ (4.2-1) Eet’eeooz'zcze where 6 —20-I_L_ 900‘4 szn2 The radial displacement, u, at r = a0 is obtained from the above equation such that, e (4.2-2) where u is defined as a - a0, the above equation reduces to 62 a a o" - T1: ’ f e 7- : _ 4.2- a 26,6 ( 3) o where, af = reduced radius a at t = 00. By comparing Eq. 4. 2—3 with the following empirical relation obtained in the laboratory, 72 =Ce (4.2—4) The following relationship is found: 0. I 6 C ‘ ' 20, and G2 P = —- ’72 Since the values of the above relationships are constant for a given material and geometry, the empirical and the theoretical equations are essentially the same. This has been proven by a different method as discussed in the following chapters. The plastic deformation is assumed to occur at the early stage of the creep since the viscosity coefficient of the plastic flow, 77,, is very much smaller than that of the viscoelastic flow. Just after loading, the plastic flow dominates the creep and the strain rate may be described by 5:9- _i. (1 {(2) . 0.: dt - 217, 'r 9 o The total strain is then 1 t Ko —- _ _ _ I 69—277, (1 T)0”9 dt 0 o where To and (79' are not constant with time. Assuming t is very small, then the change in To and 09' is insignificant, hence 73 K e zszi ,.(,__9_,, 9 r 2,716 T0 at r = a0, this reduces to a-a 1 KO :— .— l 2771(1 o)tUG a O (4.2-5) (4.2-6) Eq. 4. 2-6 holds true only in a short period after the load is applied. PART TWO EX PERIIVIENTAL INVESTIGATION CHAPTER V APPARATUS AND EXPERIMENTAL PROCEDURE 5.1 General Remarks 5.1a Te sting techniques and their objectives Two basically different testing techniques were utilized for testing the theories of rock salt behavior. The first one used a hollow thick-walled cylindrical specimen triaxially compressed in a high pressure vessel developed by Serata. The confining pressure of the vessel may reach up to 10, 000 psi. The vessel was used for a long term triaxial creep test with confining pressure of up to 8, 000 psi. This high pressure vessel was used to study the distribution of stress and strain in a hollow thick—walled cylinder; the general setup is shown in Fig. 5.1. The second technique is the confining cylinder of the "transition test technique" developed by Serata. The axial load was applied by a Forney's press tester whose capacity is 250, 000 pounds. This tech- nique was developed to study the triaxial behavior of rocks for the purpose of simulating underground stress field in relation to time, material property and loading conditions; a general setup of this apparatus is shown in Fig. 5. 2. 74 75 Figure 5. I. High pressure testing system providing versatile triaxial loads up to 200, 000 pounds with automatic loading control. Figure 5. l. a. Top view of high pressure vessel showing ring and cap—plate. Figure 5. 2. General set up of transition test showing 250, 000 pounds press tester, shock absorber, pressure gage, strain indicator and automatic loading control. -'-.'-‘.,. . Figure 5. 3. Thick walled cylinder cell with enclosed specimen, plunge rs and dial gages used for triaxial transition tests. 77 5. lb Sources of salt The rock salt used was obtained from three different locations: Saskatchewan, Louisiana and Michigan. The specimens designated by SS came from the Yorkton Saskatchewan mine of the International Minerals and Chemical Company. The salt was obtained from a depth of 3020 feet. These samples were pink in color and contained small inclusions of foreign materials. The salt crystals ranged in mean diameter from 0. 15 to O. 75 inches with an average of about 0.45 inches. The specimens designated by LS were mined from the Inter- national Salt Company mine in Avery Island, Louisiana. The samples were obtained from a drill core at a depth of 600 feet. This salt was white in color and contained much less impurities than the SS salt. The maximum crystal diameter of LS salt was about 0. 50 inches with an average diameter of 0.31 inches. The MS salt was obtained from the Detroit Salt Mines. The samples were obtained from a depth of 1000 feet. The salt was white in color with varying degrees of impurities. It was more homogene- ous and clearer than the SS salt. The average crystal diameter was about 0. 50 inches. Michigan and Louisiana salt were used for most of these tests. The first number following the lettering SS, MS or LS indicates the sample block from which the specimen was cut and the second num-- ber indicates the test number. 78 5. 2 Transition Test 5. 2a Testing devices 1. Thick -walled cylinder The cylinder is a stainless steel tubing with a length of 3.250 inches, inside diameter of 3. 250 inches and outside diameter of 4.00 inches. The cylinder assembly is shown in Fig. 5.3. The two Ames dial gages were used to obtain average values of the axial deformations. Z. Creep measuring devices The strains of the cylinder were measured by SR-4 strain gages with the Baldwin SR-4 type M indicator. An A. C. voltage regulator was also used to eliminate fluctuation of the supply voltage to the strain indicator. The use of a regulated power supply reduced the instrumental drift in the extended creep measurement to an insignifi- cant value. The usual indication of instability is a shift of the null balance point, which is known as instrumental drift or zero drift. This kind of drift was eliminated by studying its possible sources 60 as follows: which are suggested by Perry and Lissner 1. Incomplete temperature compensation of the active strain gage. Z. Instability of the wheatstone bridge, power supply and amplifier. 3. Improperly bonded strain gages. 4. Creep of one or more strain gages. 5. Insufficient protection from humidity, or reduction in the impedance between the gage wires and ground. 79 6. Variation in the impedance of the lead wires. It is necessary in the long-term creep tests to obtain continuous records based on an initial zero reference point. Three rosette SR-4 strain gages, type FABX--50-12, were attached symmetrically (i.e. , 1200 apart) to the external surface of the cylinder. The rosette gages were oriented with their two direc- tions of the wire resistance parallel and perpendicular to the axis of the cylinder. To prevent creep in the gage cement, the following cur- ing cycle was adopted (Boldwin-Lima-Hamilton) . "The absolute minimum curing cycle is 1 hour at 1750 F followed by 2 hours of 2500 F with temperature brought up slowly to each baking point over a 1 hour period. Performance is improved considerably by an additional 2 hours at 300 to 3500 F either before or after removing the pressure clamp." The instrumental drift for long-term creep tests was detected by a switch, which alternates the positions of the active and dummy gages to reverse the apparent sign of strain. An average of the two abso- lute values was taken as the true creep strain (Fig. 6. 9) . The lateral stress in the specimen was determined from the tangential strain (€ts) measurement on the external surface of the cylinder. The distribution of a tangential stress in a hollow cylinder subjected to uniform pressure on the internal and external surfaces is given by 8O aZbZ(P -P,) 13,312-sz OO O 1 10 OO cres = + r?‘(b2-az) bZ-a2 o o o o where, a0 = internal radius of cylinder = l. 625 inches b0 = external radius of cylinder = 2. 00 inches Pi = radial internal pressure = (TL P0 = radial external pressure = 0 (TBS = tangential stress on the surface of thick-walled cylinder Substituting P0 = 0 and P.1 = - O'L at r = bO into the above equation, gives b 2 - a 2 o o 0' = - ( ) (r L 0 2a 2 S o b t =1E e l— + u 00 ts H (a s 025) where, us = Poisson's ratio of steel Urs' 625 = radial and axial stresses on the outer surface of steel 0’ = cr : 0 rs 25 b2 -a2 o o ULz- _Es Ets 2a2 0 Substitute the values of be, a0 and ES to get 2 - 7.722 oFL ets 81 This is used to determine lateral stress from strain gage readings as will be shown in s'everal diagrams later. 5. 2b Specimen preparation The specimens used for the transition tests were solid cylinders with a diameter of 3.240 inches. The ends of all specimens were made perpendicular to the axis of the cylinder to insure uniform loading. All imperfections existing in the surface after machining were filled with plaster of paris to insure an ideal model shape. The im- perfections were usually caused by small crystals chipping out of the surface in the machining operation. Thin films of the friction reducer (grease-graphite mixture) were sandwiched between two layers of thin plastic film and one layer of tin-foil. Each set of these layers was placed on the top and bottom ends of the cylindrical specimen to reduce lateral friction. A thin film of friction reducer was placed on both sides of a thin plastic sheet. The plastic was then wrapped around the cylindrical surface with great caution to eliminate any air bubbles or wrinkles and any excess grease between the plastic sheets and the cylinder. 5.2c Testing procedure The general objective of this series of tests was to define, then determine the necessary material constants to describe time- and stress-dependent behavior of the material. The test involved the following four procedures: 82 1. Lateral stress-axial stress relationship Axial pressure to the solid cylinder specimen placed in a con- fined cell was gradually increased at 600 psi intervals up to 13, 400 psi and lowered gradually to zero. Strain-gage measurements, 6 s' and dial-gage measurements, ez, were recorded at each stress level. The process was repeated for several cycles in each test. The purposes of these tests were to study the instantaneous transition behavior between the elastic and plastic states of stress and determine the mechanical properties of rock salt. 2. Stress relaxation tests The axial load was gradually increased from zero in 600 psi intervals up to 10, 400 psi. The axial load was maintained at that level for about 20 days by an automatic pressure control system. After that period, the load was increased gradually from 10, 400 psi up to 13, 400 psi and kept at that level for another 20-day period. The purposes of these tests were to: (a) study the lateral stress relaxation behavior as a function of time, (b) determine the slopes of the visco—plastic and visco -elastic stress relaxation curves and (c) verify that the state of stress after 20 days is elastic, so that further loadings always start from an elastic region. 83 3. Constant stress region The axial load was gradually increased at 600 psi intervals up to 13, 400 psi with immediate strain-gage readings taken at each stress level. The load was then decreased at 1200 psi intervals down to 6000 psi and maintained at each of these stress levels for a period of 90 minutes. The purpose was to study the lateral stress relaxation around the hydrostatic line and to verify the existence of a constant lateral stress region defined by Eq. 3.1—20. 4. Visco-plastic stress relaxation tests The axial load was increased gradually at 600 psi intervals up to 12, 000 psi and lowered back to zero. At each stress level, the load was maintained constant for a period of 10 minutes. Strain gage readings were taken as soon as the stress level was reached at the end of this waiting period. The purpose of this test was to verify the theoretical Eq. 3.1-8 which indicates that the slope of the line of lateral waxial stress rela- tion is not 450 if instantaneous readings were recorded. However, because n1 is very small, this one to one relationship will be attained in a short period of time. 84 5. 3 Hollow Cylinder Test 5. 3a Testing devices 1. High pressure vessel and assembly The components of the high pressure vessel are shown in de- tail in Fig. 5.4 and Table 5.1. The vessel is pressure sealed to withstand an internal operational pressure up to 10, 000 psi. The cylinder is made of stainless steel, which was quenched at an air temperature of 18000 F and drawn at 11000 F to develop a Brinell 310-Rockwell C-34 hardness with a yield strength of 1.2 x 105 psi. Provisions for hydraulic pressure input and deformation measure- ments are illustrated in the previous figure. A cylindrical rock specimen is held in the vessel (Fig. 5.4) by means of a tightly fitted plastic jacket. The peripheral ends of the plastic jacket are held tightly between the circumference of the cylinder top and the cap plate. The O-rings and the teflon rings, shown in the figure, serve as an oil seal. As pressure increases, they flatten out and more sealing effect is produced. The specimen is subjected to a lateral confining pressure of up to 10, 000 psi. The load is applied by an automatic pump through the pressure control system which is illustrated in Fig. 5.1. an.) 85 dial gage for measuring cavity closure gage axial strain \\\>\\l '3. r \ Q) ' .§— _ \ - §.__- § plastic _ E— \ . __ _. __ b to hydraulic speci- _ __ __ — a pump men __ _ _ _ cover W/// /////// J \ l I 2 a 3 4 L_JI_I _______________ 449/ Fig. 5. 4 Cross-sectional view of high pressure vessel, showing various components and arrangement of specimen 86 6% :v: o>onw “350.30 2 m .:N x :N "3.3m ommofion ondmmoum N w Mommas» mafia Momma oudmmoum 3w“: mo mucoaomEoU Tm 3an 87 2. Pressure control system A schematic diagram of the triaxial testing assembly, Fig. 5. 5, shows the pressure control system and the testing vessel. The auto- matic pump "Vanguard" operates in a range of 100 to 10, 000 psi. A hand pump is connected to the pressure system to provide a slow loading and a fine control. The shock absorber system is devised in order to reduce the surging wave produced by the automatic pump during a constant load- ing in the creep tests. The system consists of a shock absorber unit and several damping chambers with needle size orifices shown in Fig. 5.6. This arrangement reduced the shock wave amplitude from approximately 700 psi to 150 psi. This system is particularly impor- tant for creep experiments. 5. 3b Specimen preparation The same process of specimen preparation discussed in sec. 5.2b was followed here. The only difference was that the friction reducers were placed at top and bottom edges of the cylindrical specimens. The specimens used for the cavity closure tests were hollow cylinders with a diameter of 4. 625 inches and a length of 4. 625 inches. The bore of the cylinders ranged from 0. 75 inches to 1. 50 inches for different tests. A few specimens of diameter 3.25 inches, length of 3.25 inches and a bore of 0. 65 inches were used. .annEoumd HdeMHAu mo EaudeHu ofldfionow m .m enamHh 88 mean _ @559 Houudou Hobs—cu 33535.6 . 3353:“ _ H quHOmnd xoonm pagan was: |U \ VVF V7 4‘) .\ “Had. . N a. a i f: N .0: H .0: Hoflfioon Hommo> Hence? HEM shammuum unannoum 3mm”; :3: gm} amazon : .3 83. :a 25.2 \ swam swam :Hdnum Hmed n3 mommm Hump uusmmmum ousmmoum @3430 .39m H _ «dugousmmofi 0.2530 3366 you mowmw _ 89 _'__l_T shock 1 absorber ’ ‘ Pump _ A if ""—" ,-_,‘. '1 damping chambers ‘ Testing Stand l3. 0" Fig. 5. 6. Schematic diagram of pulse reduction and cut-:aways view of shock absorber. 9O 5. 3c Jacketing of a hollow cylindrical specimen Considerable time and energy were spent in order to develop a jacket that will transmit the liquid pressure to the specimen and yet will not permit the liquid to enter its pores. Such a. jacket should be impermeable, ductile, strong enough to resist shear and not be affected by liquid medium. The following materials were tried unsuccessfully: 1. Several layers of thin wrapping plastic were found not strong enough to prevent the hydraulic oil from having access to the specimen. 2. A thick, 1/4 inch, rubber jacket was made to fit tightly on the specimen, but it was soluble in mineral oil if kept in the oil for long-term creep tests. It was also permeable at high pressures. 3. Thin copper tubings were made to fit 4 9/16 x 4 9/16 inches specimens. These were found unsuccessful, because of the difficulty of obtaining a. strong weld at the top and bottom ends of the jackets. Besides, a large amount of air, which is unde- sirable for accurate measurements, was found to be trapped between the specimen and the jacket. Finally, a successful jacket was made of 7~mil thick plastic material, known by the trade name of Plexiglass. This material proved to be impermeable, strong enough to resist shear, and flexim ble at low confining pressure. The jacket is also very easy to make. 91 Fig. 5. 7 shows the plastic jackets and the cylindrical form used to make them. The form was of the same dimensions as the specimens. The plastic was cut and wrapped around the circumference and bot- tom of this frame. All seams were cemented with plastic cement. Steel rings were then tightened around all seams. Final sealing was accomplished by heating the rings to the melting point of the plastic. A. tight fit was secured by moderately heating the plastic jacket and placing it around the specimen. Thus, most of the air usually trapped between specimen and jacket is eliminated before specimen is immersed in the confining liquid. Several tests were performed with these jackets at an axial load of up to 200, 000 pounds and lateral liquid pressure of 10, 000 psi. These jackets are reusable, if handled properly during installation and removal of the specimen. 5.3d Closure measuring system The closure of the salt cavity under triaxial compression was measured by mercury displacements into graduated tubes. The tubes are of 4 mm. inside diameter and calibrated to read to one -tenth of a millimeter. A zero reading was recorded at a zero liquid pressure. As liquid pressure increases, deformation at the inner face of the cavity occurs and the corresponding mercury displacement was measured. 92 Figure 5. 7. Plastic jackets with form. Figure 5. 7. a. Various wall thickness of hollow cylinders. Figure 5. 7. b. Specimen cutting auger. 93 Mercury was chosen as the measuring medium because of its inert chemical characteristics and its heavy density that might occupy any present pores or crevices in the bores of the salt cylinders. Thin rubber tubes were efficiently used to line the surfaces of the cavity and prevent mercury leakage between the specimen and the jacket. Any air trapped inside the cavity, when mercury was poured in, was eliminated by an air relief mechanism indicated by components 8 and 9 in Fig. 5. 4. The reduced radius of the cavity, a, was measured from the displacement of mercury, AV, by the following relation. W O a= ——- 1711 where a = reduced cavity radius V = original volume of the cavity AV = change in volume h 2: height of cylindrical cavity Also, the radial displacement, ur, was calculated from the relation 1133-3 1‘ O whe re p u reduced cavity radius a: u initial radius of cavity 94 The axial deformation measurements were performed by a dial gage that fits on the top of a perpendicular steel rod connected to the bottom of specimen, Fig. 5.4. 5. 3e Testing procedure The procedure followed in studying the behavior of a model cavity under triaxial compression was divided into three main categories: Elastic -plastic deformation tests: The objectives of these tests were to determine: (a) the structural properties of rock salt such as Young's modulus E, and octahedral shear strength KO, (b) the development of the plastic zone as a function of cavity depth and strength of salt, and (c) to verify the mathe- matical equations describing the variation of cavity closure based on theories of elasticity and plasticity. The pressure was raised uniformly in steps of 500 to 1000 psi. Immediate mercury displacements were recorded over a pressure range of 500 to 9000 psi. Cyclic tests: In this part, the interest was to study the effect of triaxial compression tests on the properties of rock salt. The pressure was raised uniformly in steps of 200 psi up to 2000 psi and unloaded uniformly to zero. The process was repeated for about: 15 cycles; in a few of these the load was 95 maintained at that level for 5 minutes till the plastic flow effect was decreased and then unloaded to zero. The cyclic effect on Young's modulus and octahedral shear strength were observed and compared with the first cycle. Immediate mercury displacements were recorded for the loading and unloading cycles. Results were plotted in several diagrams and the cyclic effect on Young's modulus and octa- hedral shear strength were observed and compared with the values of Young's modulus from the first cycle. Creep test: The objectives in this series of tests were to study (a) the creep behavior of a hollow cylinder under multiaxial compression, (b) creep rate equations and material constants, and (c) the effect of geometry, external pressure and time on these behaviors. The outside dimensions of the specimen were all the same, but the inside diameters varied between 0. 50 inches and 1.50 inches for different tests. The pressure was raised at a uniform rate from zero up to 3000, 5000 or 7000 psi. It was maintained at these stress levels by the automatic control system. Creep measurements were calculated from mercury displacement over a period of 10 to 40 days . PART THREE: EVALUATION CHAPTER VI EXPERIMENTAL RESULTS AND DISCUSSION 6.1 Fundamental Structural Properties of the Material The structural properties of most materials are usually defined by fundamental coefficients, such as Young's modulus, E, Poisson's ratio, [4, octahedral shear strength, KO, viscosity coefficient, 11, and retarded shear modulus. These coefficients vary from one material to the other. They also vary within the same material if subjected to various factors such as strain hardening, physio-chemical interaction, radiation, heat . . . etc. However, under normal laboratory condi— tions these coefficients are constant and essential to describe the physical behavior of most engineering materials. The reaction of different materials to different stress conditions depends entirely on their mechanical properties. These properties are divided into two groups: first, time wdepencence constants such as viscosity coeffic- cients and retarded shear modulus that are discussed in sec. 6.3 and second, time -independent property constants such as Young's modulus, Poisson's ratio and octahedral shear strength. The two main con- stants that have been investigated in this section under triaxial com- pression are Young's modulus and octahedral shear strength, also the strain hardening was investigated. 96 97 6.1a Young's modulus For a material that obeys Hooke's law, Young‘s modulus is defined as the proportionality constant between the stress and strain in uniaxial loading, or it may be defined as the slope of a straight line which includes most of the linear part of the stress-strain curve below its yield limit. The values of Young's modulus varied depending upon the testing procedures used. Values of Young's modulus determined by different test procedures are listed by Serata.70 Young’s modulus under triaxial compression was determined in the laboratory by using equation of a thick walled hollow cylinder, Eq.2.2-8, Au 1 (I'll) a0 ——=— —+(l-2)r APO E a0, [ H l (1-—) b2 0 u . . where is the slopw of the curve of displacement versus external po pressure. This equation describes the radial displacement in the elastic region as a function of cavity geometry and external stress, p0. If the geometry and Poisson's ratio are known, Young's modulus may be obtained by determining the slope of the displacement «stress curve at r = a. The displacement—stress curves for different speci- mens of hollow thickowalled cylinder subjected to triaxial compression and plane strain are shown in Fig. 6.1. This figure is characterized by the following features: 98 .umvnHH>o BoHHoHH HuommoumEoo >HHmHNdHuH Eonw mucmumnou oHummHo .Ho $93.95 HHGQEHHomxm .H .o .erH San: ondmmoum comm cowm ooon oooH oomH cow oo¢ o _ H a H H H i d o l m endomoum can Hm maps." u .m I manna HdHuHGH u on on: .o u 0‘ GOHHMEHOHHvHV USmMHv I an s3 x 02:. .o n H~-H-m2m .. n Ilv I an 2 x ~23 .o u 3 H 92m Irimlu. ix 0 IV _ _I _ m |+ " _ l V \ HNIHumSH \ I \ .IJ.H;|. l "H .ww \ \d : 0 V I. \ \ o .m u o GOHHMEHOHeHu uHummHm\ \ TH Aims: + .H HH~ Hv Wm... n “4 \O t mom 1 u H H Dd IA ON 0* oo ow ooH CNH owH on: owH n 2l (‘11-) 5-01 (B ‘- 99 1. There is a definite elastic straight line. 2. p0 = 1600 to 2000 psi when yield first occurred. 3. Rapid changes in slope occur approximately after these limits, which may indicate the beginning of the plastic deformation. Values for Young‘s modulus obtained from this and similar curves by using the above mentioned equation are tabulated as follows: Table 6.1 Values of Young's modulus as obtained from hollow cylinder (first cycle) A. V‘— Test Young's modulus . . Size of spec1men No. Ex 106 psi MS-l-15 0.987 Di = 1.0"; D0 = 4 9/16"; ho = 4 9/16" MS-1-16 0.749 Di = 1.0”; D0 = 4 9/16"; ho = 4 9/16" MS-l-18 0.40 Di =1 15/16”; D0 = 4 9/16"; ho = 4 9/16" MS-l-Zl 0. 964 Di = 1.0"; Do = 4 8/16"; ho = 4 9/16" MS-l-24 1.137 DI :1 16/64"; Do = 4 9/16"; ho=4 8/16" These results are based on data obtained from the first cycle. The above values compare closely with the results reported by Serata.70 Higher values of Young's modulus were observed in the subsequent cycles as indicated in Fig. 6.2. Another testing method, the transition test technique, was used to determine Young's modulus. In this method, a triaxially stressed solid cylinder was used to determine the lateral stress axial stress 100 m .HoHoGHHcHo BoHHoHH commonmgoo >HHmemHnu Ho 33?: 3.30830me 80.3 35.358 mimssow. Ho mHmanmsaaa No .me HHmmv enummoum oowH oooH oowH ooNH 2:: com 000 ooH... oom H _ H H H H H . an 02 x .24. ... m .4 mmHoH - m - o mu .3 an 83 ma u.|ll.r d o H . H I 4 .l :H enoH Xmawm o CH 73 on Him 54 D 0H 0 0 Au MW .1 em - I»... l. .4 an s2 x 2.5 “H m 0 one .H 1 Mm. H. Hmm x .lnlllu .4. eoH cmo .H m. \ O . . 4» NH 07H 3.0 U \ QIOH um ONO.H H llblMqul H “WM I OW \ 72 on .4 w .02 oHo>U 0 Wm H on: om HV 0 m .02 oHohU m N u H TEN- T. : mulmq _. NdzmHSG H o.m H1 + Hv H 54 x l ww N x H .02 oHo>U n ‘iuawao'etdstp o '9) (In) 5.01 (P - .101 relation. Using these results and the equation of Young's modulus (Eq. 2.1—2), indicated on page 13, values of Young's modulus were calculated as shown in Table 6.2. Table 6. 2 Values of Young's modulus obtained from transition tests Young's modulus (106 psi) Test Cycle Cycle Cycle Cycle Aver- No. I II III IV age MS-l-26 ---- 2.0 2.68 2.45 2.378 LS-l-l ---- 2.71 2.83 ---- 2.77 LS-1-3 ---- 2.23 2.83 ---- 2.53 LS-l-4 0.4x 106 3.0 an an ---- These values are higher than those obtained from the first cycle of the hollow cylinder tests, Table 6.1, but they are smaller than the values obtained from second and subsequent cycles (Fig. 6.2) . This discrepancy can be explained by the difference in the boundary con- ditions and in the strain measuring technique of the transition test compared to that of hollow cylinder test. In the latter, slip on the crystal interfaces is relatively free to occur as the material adjusts to the differential stress. On the other hand, the lateral restraint of strain in the transition test prevents most of this behavior and creates a more compact crystal arrangement. Therefore, this wide variation of results is mainly attributed to the difference between the two 102 testing procedures. Similar observation has been reported by various investigators as reported by Serata.7o A certain amount of variation is expected to occur in a material like rock salt due to the following reasons: 1. Different stress history of the specimen 2. Deviation from isotropy and homogeneity of the specimen tested 3. Various grain size of the specimen 4. Degree of impurity concentration 6.lb Strength of salt The strength theories of solids may be described in terms of state of stress, state of strain and energy of distortion. Some of the theories which are relevant to this work were used to define the strength of rock salt. The octahedral shear strength was chosen to describe the strength of salt under triaxial compression. However, in the uniaxial test the maximum strength is usually used. Two dif- ferent procedures were used to determine the octahedral shear strength. These are described as follows: 1. Hollow cylinder test In thick-walled cylinder the maximum shear theory was used to evaluate the octahedral shear strength, KO. The ratio of the inner radius of the cylinder "a" to that of the external radius "b” is related to the external pressure Po and the octahedral shear strength K0 as follows: 103 - (P-P.) 3 K o 1 a 2V2 ° Eze 01' a 1 ln e (g) — - 3 (PO— Pi) (2.4-4) -——— K 2V? 0 This indicates that the logarithm of the plastic front ratio, a/b, is directly proportional to the stress difference between the external and internal pressures. If this relation is plotted on a semi-log paper, the slope would yield the octahedral shear strength of rock salt for completely plastic state. Experimental data obtained from three identical specimens are shown in Fig. 6. 3. The results indicate that the elastic limit is about 2000 psi for MS-l-15 and MS-l-l6 and about 2500 psi for MS-l-Zl. An elastic -plastic state extends from these two limits up to approxi- mately 7500 psi. A large deformation region is expected to exist beyond this value. The octahedral shear strength is calculated from the slopes of the curves of the plastic region which extend beyond 7500 psi. For MS~1~15 and MS--l-l6, the octahedral shear strength K0 is approxi- mately 2300 psi. This value is based on the asymptote to the last few points of the curve. More accurate data could have been obtained if the capacity of the high pressure vessel is larger than 9000 psi, so 10310 (a/b) lU4 -O. 65 l I -0.66 O 1‘ fl '0061 ‘ l lst cycle MS-l-lS “ -O.68_ 9 lat cycle MS-l-Z6 ‘ [3 13th cycle MS-l-Zl 3 - —— 6 0 O I ¢¢ ON 0 O O O O O HumN omN flaw NHVN HVNN CNN on NHN mom HQ H d < 7 n H o _ _ _ cHdnum ,. _ uHHmmHm _ .. o3. A 5.93m . oHummHa _ s 4 com _ a m _ x w _ - 83 I mHmou0am>£ W _ l e C p _ . . . um _ Amy \ \ .. . - 8.: u .. um oHoxro . \ . \ m\ oHo>o VA 0Ho>o HHH oHo>o \ _ \ i 88 @0090 90m BoHHd o» 053 CH ommmHo .55 m - HNIHanZ 109 minutes and then unloaded. The results are shown in Fig. 6.4, where elastic recovery responded immediately since the rapid plastic flow had clearly diminished and only the elastic recovery was being measured. The deformation of the compressed cylinder beyond the proportionality limits is partially elastic or recoverable and par- tially plastic or permanent. This is shown in Fig. 6.4, where the region AC represents the total. deformation, 6 = 68 + 6P. Region AB . P . e is the unrecoverable part, 6 , and BC is the recoverable part, 6 4. Transition test The effects of cyclic loading and strain hardening are shown in Tables 6. 2 and 6.3. From these tables, it is clearly seen that the values of K0 and E increased with the number of loading cycles. Most of the increase occurs between the first and second cycles and approaches a small value after a certain. number of cycles. The in— crease in Ko for MS--l-26 between the first and second cycles was about 17 percent, while only 5 percent between the second and the fourth cycles. Similarly, a large increase in E value was observed between the first and second cycles and was almost the same for the subsequent cycles (Fig. 6. 2). Theoretically, the strain hardening of rock salt in a triaxially compressed state may be described as illustrated in Eq. 2. 3~3O by 110 using the Levy-Mises relation and the work hardening hypothesis suggested by Hill.33 From these relations, an expression for the slope of work hardening material is expressed as follows: . _ 2 _di. ‘ 2 (fax where H‘ = slope of the effective stress ~effective strain curve c} = effective stress 1 1/2 = —-[0' -0')2+(0' -cr)2+(0' -0’)2] x y y z z x 2 dc? = incremental change in the effective stress -P . . . 3 dX = proportionality function = 2 T .. V 1/2 dep= -—2Kde -de )Z+(d€ -d€ )2+(de -de )2] 3 x y y z z x BY using the experimental results from test MS-l-26, the strain hardening characteristics of rock salt were obtained as presented in Table 6.4. As the effective stress increases beyond the proportionality lirriit, the material hardens with increasing effective strains. The S10pe of the strain hardening portion H' decreases rapidly until it approaches zero. This behavior is demonstrated in Fig. 6.8, Chap- ter VI. The slope H‘ decreases rapidly to zero as the axial stress inGreases approximately between 8000 to 11, 000 psi. Fig. 6.8 shows that this is the transition region between the elastic and plastic states. 111 Table 6.4 Experimental. analysis of strain hardening characteristics of rock salt Cycle 11 O'z psi 0' psi At; psi dx 10"6 1.5 -&—.G H' 7839 5828 742 2652 0.190 72 9045 6470 642 2908 0.149 51 10251 6988 518 3280 0.111 33 11457 7359 0 0 Cycle III 8442 6315 757 1570 0.175 114 9648 6980 665 1720 0.142 83 10251 7274 712 2276 0.138 61 11457 7691 O 0 Beyond this value the "plastic line" reaches 450 which describes the perfectly plastic state with A; = H' = 0. 6.2 General Behavior of a Cylindrical Cavity with Loading 6. 2a Experimental verification of theoretical cavity behavior. The strength of an elastic ~p1astic deformation of a cavity created in a salt medium has been studied in the laboratory by a model cylindrical cavity. Closure of a cavity with respect to loading is illustrated in Fig. 6. 5. The pressure was increased uniformly at 112 affirms «Ham Havoc» .Ho oHSmOHU m .o .mHh .GoHuofiHuou 05.39, mo 03d." "kw: >4 mm 3... 8.. 2. 2. mo. 4o. o H H H H H H H i HE S .Ho>.mEo..eH-H-m2 + \.\ HES .Ho>.m€o-mH-H-m2 o . I \ \ \\\\ 2-792 . .\\ I a ...\ .. + + +\ \f \ m \ \ \\ \\ nCH..H1.w.H>H \ tsd 0001 ‘ssans I'euxsixa 113 equal intervals of stress from zero up to 9000 psi and the corre- sponding cavity closure was recorded. A maximum closure of approximately 30 percent was registered at an external pressure of 9000 psi, and less than 5 percent at a stress of 3000 psi. This closure is an important factor in the design of salt cavities when these are created at a depth of more than 3000 ft. Theoretical stress distribution around a cylindrical cavity has been developed in the elastic, elasticuplastic and completely plastic states as indicated by Eqs. 2.3-10, 2.3-14 and 2.4-2, respectively. Based on these equations, the theoretical radial displacement at r = a is calculated and compared with experimental results. The conditions imposed on the displacement equations are either elastic compressibility in plastic regions or compressibility in elastic region and incompressibility in plastic region as indicated by the following equations. For elastic compressibility in elastic region, the radial displacement is: 0 2 2V—Z—_B_ (2.3-38) For compressibility in elastic region and incompressibility in plastic 114 region, the radial displacement is: _3___K i.12____V2"[ a 2 3 20 (24-1) L-pluzu-nppo (2.4-9) b 2 o The theoretical behavior of the radial displacement as a function of applied stresses and plastic radius is worked out for values of p that varied between 0. 50 inches to 2.20 inches. The steps involved for computation of the radial displacement are as follows: 1. The material properties are obtained from the previous labo- ratory data where, E =1.2 x106 psi and p = 0.16. 2. For assumed values of p, values of "a" are calculated from Eq. 2. 3-54. 3. By using the values of p and “a" obtained from step 2, values of external stresses necessary to create the given plastic radius, p, are calculated from Eq. 2.3-15. 4. By substituting the values of p, a, and Po into Eq. 2.3-37 and Eq. 2.4-9, the theoretical radial displacements at each stress point are calculated. A sample of the calculation is presented in Table 6. 5. Three different values of K0 of 1500 psi, 1750 psi and 2000 psi were used in Eq. 2. 3-38. The stress—displacement curves thus obtained were presented in Fig. 2.4. In the laboratory, radial displacements were measured for different specimens at equal intervals of pressure between 0 and 115 O . . 3.0 + ~£N\~damcH:dG:m* wamov: owmfih. hmom 0¢hN.0H H0m0¢.0 NNNN thh.0l 0¢wwb.0 m00m0.0 00Hh¢.0 OHMN0.0 $0Nm0.0 ¢0.v 0N.N NehNMME woomh. Hmom N000M.0 Hm¢wm.0 ¢hmh H>m>.03 mHm00.0 N0005.0 oowhv.0 000m0.0 N>>h0.0 00.v 00.N Noomoms MNNNF. mcmm H000.0 ONHHM.0 0mm> 0¢N>.0! 0hhwm.0 wm-0.0 hmwm¢.0 0H0¢0.0 m0000.0 ¢N.m 00.H wNNNONS mHvOF. 0v¢m vmmNN.m m0m¢N.0 H¢0h NMH>.0l 005$.0 HOHOV.0 moowv.0 00000.0 VNHN0.0 0m.N 00.H vmvhmHs mhmww. whNn 0HH00.M Hmme.0 Nmow vm0h.0I fivemm.0 Noohmoo 00¢0v.0 N0000.0 000¢0.0 00.H 0¢.H mmovHHl NHmHo. wNom vm000.N mmme.0 0N00 m00.0i NMNwH.0 0>0hN.0 0000*.0 mohh000 00mm0.0 v¢.H 0N.H oHooh! Nwwvm. mmom 0N000.H #0000. 0mmm 000.0! 000.0 mHme.0 vVNOm.0 omvwo.0 m¢¢00.0 00.H 00.H ohmom! 00¢m¢. V¢NN N000N.H wvHoc. Hv¢¢ mwo.0l MNN.01 NONNH.0 mwnom.0 0H000.0 m¢000.0 $0.0 00.0 VHHwNE *00Nm. NfimH mmoON. mmfim0. 000m www.0l HHmo0l FH000.0 mwhom.0 0w¢00.0 00wmo.0 wmoo 00.0 mmthl mmmm,. MVHH HhHOW. HCVND. 0¢NN 0h0.0l m00.05 momvooo Nwmomno M00000 mmm00.0 mmoo 0m.0 H o: o a. 0.3 0 on o... Na. #35. * .0 mm .Im NQ. DH NH: QCH INIQI M “I . .. NQ Q N M N wrrmo OEH noUCHH>u Homing vHoHHHH .m Ho MOH>mHHoQ 9H» 00 mHm>HMCd HmuHumnooeHH 365 H Ho ~308me HmcuoHcH Huse 23:. H. Ho uoHcEsHp Hmcuouxmv m .0 anmH. 116 mqum on Hangman: omens on. ponmnHEoo twat/mo HmoHuccHH>o H0008 .m Ho coHumHou chnHmummouHm HmoHHmnomaHH. c .c .mHuH 2: .03 x .90 unsavomHnHmHHo HdHHumu comm coom comm cch comH cooH com o H a _ _ _ 1 H c; N v. 3-792 4 i m mHHdmon 330830me a mHuH...mH>H . w“. a o 1 cm: u H w .383 33.30 CH. tfiHHHnHHmmoumEoo l H. T. 03930 .HOH mHm>Hmca HmoHHmnooeHH m. m 0 S in om: n M m .383 oHHmmHnH CH >HHHHQHmmoum§ooaH L m - oHonEoo .HOH 393.93 HauHumnooHHH m. m d m. H J 0 117 9000 psi. At each interval, the load was maintained at that level for a period of one minute before the data were recorded. A sample of the experimental results was compared with the theory of Eqs. 2. 3-38 and 2.4-9 (for Ko = 1750 psi) as shown in Fig. 6.6. A general close agreement has been observed between the experimental results and the curve of complete incompressibility in the plastic region up to approximately 5500 psi. Above this stress level, the theoretical curve flattens out quickly and the degree of variation between the two curves increases. If sufficient time is allowed for salt to flow, the experimental results might have approached the theoretical curve. On the other hand, a good agreement was observed between the theory of Eq. 2. 3-38 and experimental results up to approximately 4000 psi. Above this stress level, the degree of variation between the experi- mental and theoretical curves increases due to rapid plastic deforma- tion. Whenever the plastic stress state is reached, the discrepancy between theoretical and experimental results was found especially when the experimental results were taken one minute after the load was reached. Similarly, the relationship between the reduced cavity radius, a, and the external pressure was calculated simultaneously from Eq. 2.3-54 and Eq. 2. 3-15. A comparison between the theoretical reduced radius, a, and its value obtained from experimental results is illustrated in Fig. 6. 7. Close agreement between experimental and a: inside radius, inches 118 \elastic - x I! ' v p I/ theoretical = 2000 psi 0.50 0.49 --+- MS-1-15, a0 = 0.51141" K 0.48 -— ---°"""' MS-1-16, a = 0.51014" \ o \o —-—-‘a——— Theoretical results 0.47 — MS-l-15 MS-l-16 I. 1 l l l J 0.4 6 flr3 4 5 6 7 8 external stresses, 1000 psi Fig. 6. 7 Variation of the modified inner radius, a, of a cylindrical cavity compared to experimental results 119 theoretical values was observed up to a pressure of approximately 4000 psi, where the variation of the inside reduced radius is very small. At higher pressures the experimental values seemed to be lower than the theoretical ones. This might be explained by the fact that the experimental values were recorded one minute after the pressure was reached. It is reasonable to assume then that the ex- perimental values would have been closer to the theoretical values had the readings been taken instantaneously. 6. 2b Stability conditions These analyses have demonstrated the important fact that a salt cavity will eventually reach a state of equilibrium between the advancing plastic zone and the elastic part. The advancement of the plastic zone is defined by Eq. 2.3-15 where 1 1 p2 p = aexp. “:37"? [(130 -Pi) ~2'(1-- 371% 2V3— ° If an infinite medium is considered where bo>>p, then 1 If”? (Poupi) '2' 3 —— K 2‘\/ 2 0 It should be noted that the natural log of (p/a) is directly propor- tional to the stress difference and inversely proportional to octahedral shear strength. Hence, the development of the plastic zone is 120 controlled by: (1) increasing the external load or (2) decreasing the internal pressure. With the increase of external pressure, the plastic radius increases and accordingly the internal radius decreases as shown in Eq. 2.3-54, where a2 2V2 ° PZ —‘1'—E_“1"”’_ a2 a2 o o and thus a state of stable equilibrium exists. In the case of completely plastic state and for 0r 2 oz 2 06, the relation between external pressure and the reduced cavity radius is P=——Ko(ln—) O ZVZ—O where for incompressibility conditions, +bz _az 1/2 0 ZV—o a2 From which it is clearly observed that as the inner cavity radius "a'l decreases, the cavity gain ability to withstand a greater external pressure and thus the system is stable. 6. 3 Transition Test 6.3a Axial stress-lateral stress relationship In the laboratory, the lateral stress-axial stress relationship has been determined by Fig. 6.8. The stress conditions produced by doHacdnxu HanoumH scum cough—con 3 .3657: 33¢ #03? :H .mumou coHuHmcdnu mo 33mm; Hugh? 0:» mcHudnumaHHH 5.0.5ch amouum Hmemnmmonum Hagen—min w .c .erH Hum cooH .unonum Hanna u as 4H NH oH a v N H H - H H Q E 296 © 8-7m? o E 396 D 2.5 HH oHu>U o oHHdeoqu>m\.H \ x H uHu>D 4 11.. .IL ‘\ 85 03.20 .. 83 03.20 Q ted 0001 ‘sssns [219121 V‘ '10 a loading cycle can be illustrated as follows: during the initial load- ing up to point A on the stress diagram, the material behaves elasti- cally. As the axial stress increases, the material changes rather abruptly to a plastic state of stress represented by line AB. The stress condition which the material experiences as the axial load is gradually reduced is illustrated by the elastic stress line, BC, and the plastic stress line, CD. The line DO represents the residual stress caused by creep and plastic deformation during the cycles of loading and unloading. The change in the residual stress becomes very small after the first cycle as indicated in the figure. However, the distance between the "hydrostatic line" and the ”plastic line, " which is related to the octahedral shear strength of the material, in-- creases with the number of cycles. It was observed from laboratory results that the transition from the elastic state CA to the plastic state AB is rather abrupt. Experimental results obtained by Raman‘63 from limestone and granite as indicated in Figs. 2. 2 and 2. 3 show the gradual transition between the elastic and plastic states. This difference in the transition state between salt and the previously mentioned materials may be attributed to the viscosity constant of the materials. In case of rock salt, the transition was abrupt as indicated by line AB and this reflects a low viscosity constant. On the other hand, dolomite and lime stone show a gradual change between the elastic and plastic states and this indi- cates a large viscosity value. 12.3 The discrepancy observed at the beginning of the unloading line BC may be due to viscous plastic deformation. As the axial stress is gradually reduced, for the first few readings, no strain recovery was observed. This was explained by the hypothesis that at high axial stress levels, plastic deformation is still taking place and thus can- celling the strain recovery produced by unloading. 6. 3b Stress relaxation equations The triaxial stress relaxation test involves the application of a constant load to the test specimens and the measurement of the de- formation over a long period of time. Instrument creep was elimi- nated by taking the mean indicator reading illustrated in Fig. 6. 9. Fig. 6.10 illustrates the general stress relaxation behavior for a triaxially compressed and laterally confined rock salt specimen. It is observed that the octahedral shear stress decreases rapidly at the beginning of the creep period and relaxes asymptotically to a constant value as time increases. The experimental equation describing this behavior is presented as follows: ~ at 1- :T +(T_ --T)e (3‘1"-13) o f i f Rearranging this equation, it reads 124 mowmm >EESHV Head 0330 agape/0." >HH00H500H0 >3 «HHHHV on0u mchHEn0H0p Ho 053G300» 05 mcHudnHmsHHH amends?“ 50.30 mo uoHnH c .c .mHh ccc .HH 9 £an ou0u coo .NH. coo .cH cccw ‘- mcHHomou no? 305. an 08 1‘ as: is. coco nudge?” 5.0 .30 09.3 I. cmHu () .4 a l OOH!- cm: ccc¢ cocN I | ‘ 9.. lcmH LomH ensue; nu; .iad saqout-ozotin titans -sAtssa.tduioo 125 33 55335.3 5 nogmson 553.3339 mmouum o“ .o .mfim A933 033 v.2 ex: on: an: hww .m..m Nun _ . _ _ q . J . _ o _ Amousamev $53 ooowm ooo¢~ oooom oooofi ooomd ooow ooo¢ o A _ a 7 a _ _ ll® 0” 0 lallallTfilldILlfiii e O o O m 1:.m w P 1 m. S u. a a . 1 1mm m w m -m u 0 n... 1N \N 4.0 8 ad: wN... ON .9 r: a; m, ..M m .. P. l. Sneak \Nv + N- ‘6 W. 0 w omnaumz U o m' 12.6 which indicates a linear relationship between octahedral shear stress and time on semi-log paper. Laboratory results from stress relaxation tests were plotted as shown in Fig. 6.11. This showed that there are two independent stress relaxation equations represented in the same stress relaxation figure. A straight line was drawn asymptotically to the curve at large values of time. This represents the linear relationship of the viscoelastic stress relaxation. The difference between the values of the total stress relaxation curve and the line of the viscoelastic relaxation measured from the abscissa. represents the stress relaxation curve of the viscoplastic state. This difference was found to be another straight line. The tangents of the angles of inclinations of these two lines, denoted by (11 and a2, are related to the Viscosity coefficients of the material. Theoretically, the stress relaxation behavior in the viscoplastic and viscoelastic regions is described by Eqs. 3.1-~8 and 3.1-l9 re spectively, s 0 that C'1 -Tt + 3 H - 1 0‘-‘-'(0 9 o». + A... _ 9 9 \ é . \ . /\/. + \. \ a. O «(o + We \ w \ NIHImJ $111k '10 ~D w 19“ 0001 ‘883119 [2.19321 : od Na 131 cowudxmgon muonum .«o .3023 fiduocmvm on» sumspfimvummacoo common macho acdumcou mo 39:35 aficogmnooknm are .mwh mum ooofi .mmouum “MUS "no m.: mg m.- o~ mg. m m.~ o _ d _ . . _ . 7 , c 63555 oo Hon—ma @093 x \ \ m . . \ \. I N \ \ x < H HHH 393 O \x us. .. m ~ a Q. g" an 0%E IWI+ Nb Han—”.0 ~0 H0 1 mg. common 3330.609? i S \\\||\ o~ cm @093 ll \ . - 5. .\\\\\x m 2 \\ .. m common cwumdauuoomg k \ “IT-m4 Gods. . m much—u mauucoU\ ted 0001 ‘999139 I'ezaret :10 132 relation for the third cycle. The axial load was raised to 15, 000 psi and maintained at that level till the lateral stress reaches a constant value. Through this point, a line was drawn parallel to GH intersecting IG and EH at B and C respectively. This line, BC, represents the constant stress (TL, such that, Gle + 3 ( L 2V? o,+o,Z 2.l. )yo I) A similar line AD was drawn parallel to BC and at equal distance from the hydrostatic line representing the change in lateral stress during unloading cycle. Accordingly, if. lines BC and AD represent the constant lateral stress as time approaches infinity, then region ABCD is a constant stress region and no change in lateral stress with respect to time is expected along AB and DC. To prove this constant stress region, the load was raised uniformly from 15, 000 psi up to approximately 18, 000 psi as indicated in Fig. 6.13. At that stress level, the axial stress was maintained constant for a period of 90 minutes. Then it was reduced to zero. At each stress level, during this unloading cycle, the load was main-- tained constant for a period of 90 minutes to allow for any stress relaxation. The results showed that stress relaxation occurs only at 18,000 psi with a lateral. stress change of 500 psi. As unloading proceeds 133 through the region indicated by line CD, no appreciable variation occurred. At point D, stress relaxation in an opposite direction to that of the loading cycle starts to increase as the axial stress ap- proaches zero. 6. 3d Determination of time -dependence constants from stress relaxation equations Assuming rock salt has standard structural properties, the time-dependence constants are evaluated as indicated in the following example: Test No. LS-l-l Young's modulus, E = 2.83 x106 psi (3rd cycle) Poisson's ratio, u = 0.16 E Shear modulus, G1 = 2(m) = 1.219, 827 psi As time approaches infinity, the stress relaxation is described by Eq. 3.1-20, where 3 GIGZ + (r :0" _ ( I u(t) L z VE- (31+G2 Yo The retarded shear modulus G2 is the only unknown quantity, since 0' L’ oz, (31 and Y0 may be calculated. From Fig. 6.13, as time approaches infinity q II 15,000 psi q H 12,500 psi 134 for laterally confined strain, E = 6 = E . Therefore, x y L =lg-g-V—2—(6 L- 6-z) t 8439 x 10‘6in/in 11 Yo on loading cycle Yo is a negative quantity if both (TL and oz are positive. The retarded shear modulus may be expressed as G1(UL—Uz) G2 = 3 -—-—- Gly - (0' - 0') 2%- o L z G l, 219, 827 (psi) (-2500 psi) 2 _ — —-3— (1, 219, 827)(psi)(8439 x10"6) + 2500 (psi) 2 G2 = 148, 000 psi The plasticity constant 711 was described by Eq. 3.1-l4, such that G’1 T713-— (11 where (11 = 995 x 10‘5/m1n. Therefore, 06 mln "' lbs 171 =122.6x1 Z in Using conversion factors from min —lbs/in2 to dynes-sec/cm2 to get: . — 6 1 01‘ 122 6 min lbs x 60 sec 0 10 dynes x 2 inz mm 2.248 lbs cm 6. 45 . Z in m = 5.0 x 1014 poises 135 Similarly, the Viscoelasticity coefficient was described by Eq. 3.1-22, such as G1+G2 772 ‘12 where a2 = 5.577 x low/min Therefore 772 23.2 x108 min—lbs/in2 or 772 96 x1014 poises This indicates that 172 18 almost 20 times larger than 111. G2 — , n2 equation of a cylindrical cavity, is presented as follows: The ratio, that will be used in sec. 6. 4 to verify the creep Table 6. 6 Values of the ratio 57323 , as obtained from transition tests T st 772 x 104 G2 er mi 1: Ex106psi Glxioépsi szlo6psi min-lbs 32’? n °' in2 10’4 LS-l-l 2. 83 1,220,000 148,000 232,000 0.63 MS-1~26 2.45 1, 060,000 166, 900 224,000 0.74 LS-l-3 2. 83 1,220,000 128,000 239,000 0.54 Accuracy of the time --dependence constants determined by the above procedure are affected by the accuracy of the time -independent property constant G,- . This does not minimize the applicability of this method to determine the viscosity constants 771 and n2, particu- larly if the structural properties are well defined. 136 6. 4 Creep of Model Cavity When a constant stress is applied to a material the atomic or molecular structures of the material readjust themselves with sus— tained loading. This readjustment of the internal structure of the material produces deformation as time passes on. The deformation produced is called creep. In other words, creep is the property of solids to change their shape with time under constant loads. ASTM defines creep as “the time -dependent part of the strain resulting from stress. " The phenomenon of creep is the most complex of all mechan- ical behavior of materials. Freudenthal‘Z4 refers to the complexity of creep as being in direct relation with the complexity of the internal structure of matter itself. This phenomenon is observed in metals, ionic and covalent crystals and in amorphous materials such as glasses and high polymers . 6. 4a Factors affecting creep Temperature plays an important role for creep responses of metal, rocks and amorphous material. Those having low melting points tend to creep extensively at elevated temperatures. Heat is the most critical factor affecting the structural property of rock salt. High temperatures in salt cause thermal stresses, increased creep rate and reduced strength. Seratam reported that at a temperature of 5000 F, rock salt exhibits a uniaxial yielding strength around 1000 137 1500 psi instead of 2300 psi. He. further reported that, under triaxial compression, the differential stress reduced from 1700 to 1000 psi, and remained nearly constant for temperatures varying from 5000 F up to 14750 F (melting point of rock salt). The extent of creep is also affected by the grain size, micro- structure, previous strain history and boundary characteristics. The latter are usually influenced by impurities and, with a large concen- tration of them, the cohesive force of the aggregate material might increase. 6. 4b Mechanism of creep l. Uniaxial On the basis of the mechanism of creep, the total creep of any instant is divided into three groups by Richard. 67 First, is the elastic plus plastic creep that occurs immediately after applying the load. Second, is the transient or cold creep. Finally a combination of transient and steady state creep, usually referred to as a viscous creep or hot creep. The components are illustrated for comparison in Fig. 6.14a, b. The last two components of creep are of prime importance and represent basically two different phenomena. They are treated separately as follows. 2. Transient creep Decreasing rate is the principal characteristic of transient / / / ,/ , 4,1" L .«4 L4 .”-”" . ‘ .1 Transient / {D / ___...1/ / ./ /’ Steady state 6 / o / / llc‘f_____._.._._____._.._._.- ___. __.._.___...... ___”-.. ___—___... Elastic plus plastic L [Linc . _ Fig. 6.1-4a Creep curve show1ng d1fterent "stages of creep first second third «___—aunt :w-Hu—m-nfi stage stage stag; I l I . { curve B - h1gh stress / / / 1: -r4 (‘6 u .5.) co / \_. curve A - low stress ___._./ first sta e ,, ,q $_ third m... g [-4 sesond stage stage time Fig. 6.14b Creep curve showing basic components 139 creep. The deformation rapidly increases but the flow rate gradu- ally slows down and finally the creep deformation approaches its maximum value. In non-crystalline material, as in amorphous materials, elastic after-effects often constitute a major part of the total creep. In crystalline material, however, elastic after-effect is small and insignificant compared with the other creep deformation mechanisms, especially at high stress levels. In this range, the transient creep consists largely of yielding produced by plastic stresses and thermal activation. The plastic deformation is accom- panied by initial plastic strain which decreases as the stresses are balanced by strain~hardening. 3. Viscous creep Ideal viscous flow is characterized by steady increase of deformation at constant stress. In nonstrain-hardening materials like thermoplastic or amorphous polymers, viscous flow is the natural form of inelastic deformation. It is produced by permanent change in the molecular structure of the material. The changed molecules slip past each other, constantly breaking and regrouping with no strengthening results. In strain hardening material, like most crystalline materials, viscous flow takes place when the strain hardening effect is just balanced by the softening effect of heat. This softening is produced by atoms of the crystals which migrate 140 or diffuse to positions of lower energy. Thus, dislocations are made more mobile and can detour around obstacles or take them along, given a little time. Also, in polycrystalline materials, viscous creep is produced by what is called grain-boundary shearing which is due to flow of grains themselves as a semi-rigid body. 4. Triaxial In triaxial stress state such as creep of a thick walled cylinder subjected to uniform external pressure, the transient state is defined "as that portion of the deformation history of the tube when the stresses 1112 This transient state may throughout the cylinder vary with time. start when loads are just applied and is considered completed when the complex creep-relaxation process in the thick walled cylinder adjusts itself such that all the stress distribution is constant with time. Eventually, a steady state of stress will be reached and from then on the stress distribution will remain constant with time. The creep rate characteristics of a region such as a thick walled cylinder with external pressure are that the transient creep rate decreases from large to a small value in a short period of time. However, after stress reaches a constant value, the creep rate con- tinues to decrease slowly until it reaches an asymptotic value at t=oo. 141 6. 4c Creep analysis of a cylindrical model and verification of theoretical creep equations The development of the mathematical theory of creep has been rather slow. This is due to the difficulty of obtaining experimental data for the entire life of the material concerned. Most of the re- search so far has been in the direction of developing empirical rela- tions that fit the available data. A sound mathematical theory based on logical assumptions and guided by practical experience to describe ' the complex physical phenomena of creep is yet to be developed. 1. General creep behavior of a salt cavity In the laboratory, as discussed earlier, a creep testing device has been setup to determine the creep characteristics of salt. This triaxial creep test involves the application of constant load to the test specimen and the measurement of deformation at a long period of time. A sample of creep behavior of salt is presented in Fig. 6.15. The creep in these samples was divided mainly into two stages. The first stage is the one during which most of the transient creep takes place. This type of creep is usually called transient creep. When the transient creep has reached a substantially constant value, the strain continues to increase at a more or less decreasing creep rate, under the action of the viscoelastic component. This type of creep Continues at a decreasing rate until eventually it reaches its minimum 142 >3>du 3mm Hobos mo 3m.” 9:530 mg .0 .mrm 93p . 053 o.~.~ o.vN oAN o.m~ o.m~ o.- o6 06 0.». c q q A . n _ _ — _ 0.7.732 n + 1' l: ' | coflnfihgov 3»de pad 03930 2525»:ng OTTmz a MNIHImz mwudumz soflmfiuaop omuuma was 33.30 msoocmucmumfi .r .1 11-1-1.-.th : H\® ¢ :©~\O V bl m..m 04V m.¢ o.m ‘uogonpax amnIOA was .tad A AV (001) 143 creep rate at a large value of time. Because of the nature of the loading, the third stage of creep discussed in uniaxial creep is difficult to observe in this study. 2. Transient and plastic deformation In the laboratory, the deformation rate at the early stages of creep decreases rapidly from some large value to a small value (Fig. 6.15) . To analyze this type of deformation, the change in cavity radius, a, was plotted as a function of time as indicated in Fig. 6.16. This figure shows that the cavity radius decreases rapidly to a small value within a short period of time. At this state of creep, where the deformation rates are large, the stresses in the salt are not con- stant even though the external pressure was maintained at the same level. For each increment of time, there is not only an increment of plastic strains but also an increment of stress which eventually reaches zero with time. The relation of cavity radius as a function of time and stresses in plastic region is demonstrated theoretically by Eq. 4.2-6, where aO-a 1 K0 :— 1..— ' O O The relation between a and t is difficult to obtain without determining the variation in the deviatoric stresses. However, for small values of t, the changes in (76' and T0 are small and a linear relation may be 144 .970 a/a o S-l-IZ .960 M .950 0.98— a/a MS-l-l4 0.97 b o 0.96- a/a O 0. 93 0.92 0.91 100 200 300 time - minute 3 Fig. 6. 16 Reduction of a model cavity reduces as afunction of viscoplastic deformation 145 approximated only by drawing a tangent to the curve at time ap- proximately equal to zero. The slope of this linear relation is described by 1 0 tan Zr” (1 To) 0'e if these variables are known the viscoplasticity constant r7l may be approximated from the tangent of these curves. 3. Viscoelastic deformation It has been observed from several test results that the creep curves of salt have some common features. Thus, they may be ex- pressed by a common creep curve. Although no mathematical expression as yet describes the exact process of creep, many em- pirical relations have been suggested and used to fit the creep data obtained. An empirical creep function that fits very closely the creep data for a cylindrical. cavity was proposed. The proposed function may be written as: - Pt a - a = a e f o where a = reduced cavity radius at time t a = reduced cavity radius as time approaches infinity P = material constant 146 a more general formulation of this equation is n «Pt where n is equal to unity in a linear material. The advantage of this empirical equation is that the coefficients can be determined from simple graphical methods as illustrated in Fig. 6.17. This figure indicates that the empirical equation fits very closely the viscoelastic. creep data of the salt cavity. The discrepancy ob- served at the early stages of creep may be attributed mostly to the plastic deformation. At the end of the plastic deformation period, salt exhibits a linear behavior as demonstrated in the above figure. The material constant P, which is related to the viscosity of the mate- rial, was determined from the slopes of the curves. It has been noted that the proposed creep function developed as a result of laboratory creep measurements is essentially the same equation obtained from the mathematical analysis of a cylindrical cavity in a linear viscoelastic material. In particular, a conclusion G2 . was reached such that P = 77—- . To prove that these equations are the 2 same, the value of P, as determined from creep analysis of a cylin- . . G2 . drical salt caVity, should be equal to the value of .1“)— as determined 2 from stress relaxation equations of the transition tests. . . Gz From Fig. 6.17 and Table 6.6, the values of P and —are '72 calculated and presented in Table 6. 7. 147 Table 6.7 Comparison of the values of P and %i as obtained 2 from two different testing techniques P from cavity creep tests :3 from transition tests 2 0.34 x10.4 per min(MS-1-12) 0.63 x10"4 per min (LS-l—l) 0.67 x 10'4 per min (MS-l—l4) 0.74 x10”4 per min (MS-1-26) 0.72 x10.4 per min (MS-1-19) 0.54 x10"4 per min (LS-l-3) The above comparison of the two different experimental results G2 . . shows a close agreement between the values of P and 77—, verifying 2 that Eqs. 4,2~3 and 4.2-4 are essentially the same. This further supports the validity of the general assumptions made in developing the theory of creep behavior of a salt cavity. Therefore, by using these coefficients the creep rate of underground cylindrical cavity can be described, provided the conditions of the cavity agree with the basic assumptions of the study. 300— 200 —J 100-—-1 90 80 70 60 50 40 30 20 U1 oxiooxoo 115 147A MS-l-l7 P3 =tan 03 = 0.34 x10-4/min. tan0=P n=1 7 l n r '- P2 =tan 02:0.67x10'4/min. ’4 02 ‘- P1: tan 9, = 0.72 x10-4/min. .4 MS-l-l4 1 1 1 1 0 4 8 12 M5449 16 20 1 l 4 l 1 0 8 16 24 MS-l-IZ 32 40 time, days Fig. 6.17 Experimental analysis of triaxial creep in hollow cylin- ders illustrating determination of viscoelastic coefficients CHAPTER VII SUMMARY Theories of stress, strain and boundary motion were developed to study the stress distribution in. a cylindrical model cavity. Distri— bution of stress and strain and creep motion of the medium around the cylindrical cavity were described based on the mathematical theories of elasticity, plasticity and viscoelasticity of a continuous medium. 2' > U -- 0 were assumed. The conditions of plane strain with Ur z 0 In this analysis six mechanical constants of the material were included, which have to be determined for the experimental verifica- tion of the theories. A proposed mechanical model was verified by the triaxial transition test technique, which was used to determine the mechanical constants of the material. These constants are: Young’s modulus, Poisson's ratio, octahedral shear strength, viscosity coefficients, and retarded shear modulus. The mechanical model consists of elastic, viscoelastic and viscoplastic elements that describe the over- all behavior of rock salt with respect to time and stress conditions. Behavior of the model cavity was described by substituting some of these constants into the theories of stress and strain and boundary motion equations. The validity of the theoretical calcula- tions was confirmed in the laboratory by using the technique of liquid 148 149 confining medium for a model cavity. A summary of this theoretical and experimental investigation is as follows: From experimental model cavity analysis and theoretical Eq. 2.2-9 (1 ) 2 '1’“ o E=- a2 [—+(l-2u)r] Au (1 o) APO b2 o Young's modulus was determined to be in the range of 0. 7 x 106 psi to 1x 106 psi for first cycle. Higher experimental values (4 x106 psi) were found for second cycle and thereafter. Transition test results show that Young's modulus for second cycle and beyond range between 2. 4 x106 psi and 3 x106 psi. Plasticity solution for large boundary motion such as Eq. 2.2-4 1 where --—--—- (P - P.) o 1 _ .3 K o a - 2V2 b — e were utilized to determine the octahedral shear strength of rock salt which is around 2300 psi. Experimental data from the transition test were found to agree generally with these values. Cavity reduction as a function of elastic-plastic deformation is described by Eqs. 2. 3-37, 2. 3-54 and 2. 4~9, such that 150 Ko 2 Pa u (l-‘2u)[ 3 K (ilnp— a) 2V2- f_ o ] r=a— G o 2 4 4 b2 2 2 2 0 ——?—K 2 (1 M2 2 0 E- 2G ___}... 2 a2 2V2 O _ ~1- G (1-111— a2 a?" o 0 based on elastic compressibility; and 3 . 1/ 0 3 ZE._1_ ___—.2 2;... 7 - L- + —l P urza a 2G 2 [(‘lJ' 1.) b2 p] (2” )p O 0 based on total incompressibility in plastic region. The radial displacement and the reduced cavity radius were expressed as functions of geometry and external pressure. Close agreement is observed between theory and experimental results, indicating rock salt may be considered as ductile, isotropic and homogeneous under triaxial compression. From the transition test, the stress relaxation equation in the plastic and viscoelastic regions are defined respectively as: 3 11 3 G‘t .—. t -—‘—— —-— -- '— '717 (TL (0'2 fiK0)+[(1‘H 1)Uz+V-EKO]6 3 ” n t -—n t GIGZ _ + __ _ ~ 1 -. 2 O'L— 02-. 2 [Glyouhy e + (1 e )G1+Gz yout] 151 from which the viscosity coefficients n1 and 172 are determined such that GI n1: — = 5 x1014 poises “1 and (31+ G2 17, = —— z 96 x1014 poises “z where 11, and (12 are experimental values determined from the stress relaxation figures. The creep behavior of a model salt cavity is determined by considering the viscous behavior of rock salt beside its elastic and plastic deformations. The creep rate equations are as follows: G2 -—t d6r l 1 Ko l 772 =-[—(l-—)+—-e ](I' dt 2 721 To n2 r G2 d K ”_t l:l[_1_(1__°_)+_1_e 772101 dt 2 1 To Z 8 a-a -Pt that fits very closely the experimental data in the viscoelastic region, is compared with the theoretical equation developed from the above creep equations such that 152 G2 a a -—t f:. (79' e 772, a 2G; Both equations have exactly the same creep form. To verify the . . . . G2 validity of this comparison the value of P and 77-;- were calculated from experimental results of model salt cavities and those of transi- tion tests respectively. Close agreement between these values were observed which led to the verification of the mathematical equation and the conclusions that and NI... N O N CHAPTER VIII CONCLUSIONS Based upon the theoretical and experimental analysis discussed in the previous chapters, the following conclusions have been drawn. Me chanical Prope rtie s The following fundamental properties of rock salt were deter- mined by two different triaxial testing procedures. 1. Young's modulus from the first cycle of the hollow cylinder tests varied between 0.7 x 106 psi and 1.2 x106 psi with an average of about 0. 85 x 106 psi. While, from second and sub- sequent cycles, it was about 4 x 106 psi. Value of Young's modulus from first cycles of the transition test was about 0. 4 x 106 psi. From the second and subsequent cycles, it varied between 2 x 106 psi and 3 x 106 psi with an average of 2.6 x106 psi. Variations of Young's modulus within the same test technique were relatively small compared to the variation between the two different testing procedures. The mean octahedral shear strength obtained from the large deformation theory of thick-walled cylinder was around 2,300 psi. The octahedral shear strength of the LS-specimens ranges 153 154 from 1300 psi to 2000 psi for the first and third cycles of load- ing respectively as shown in Table 6. 3. In comparison, the octahedral shear strength for the relatively less pure MS- specimens ranges from 1700 psi to 2100 psi for the first and fourth cycles of loading respectively. Rock salt exhibits strain-hardening characteristics. The slope of work hardening part, H', decreases rapidly and approaches zero within a short range of stress. Repeated cycles of loading result in. a general increase in the strength of the material. The increase in strength is partially evident in the increase of octahedral shear strength value. Structural Behavior l. The elastic theory can be applied for analysis of the stress~ strain distribution around a salt cavity so long as the octahedral shear stress is less than its maximum elastic value. The elasticity theory has been used effectively to determine Young's modulus of the material as a continuous medium under triaxial compression. The maximum shear theory and the energy of distortion theory are directly applicable for analyzing the strength of rock salt under multiaxial compression only if the material tested is assumed to be ductile, isotropic and homogeneous. 155 Theoretical analysis has been completed on the stress, strain and boundary motion of a cylindrical cavity. Theoretical re- sults compared reasonably well with the experimental results. Fig. 6. 6 shows the theoretical stress-displacement relation in comparison with the experimental results. The experimental behavior of the reduced cavity radius '‘a" as a function of applied stresses compares reasonably well with the theoretical results. The volume reduction of a salt cavity is a function of the applied loads. The higher the loads, the larger the volume reduction. An instantaneous stress equilibrium always exists between the advancing plastic zone and the surrounding elastic part. As the applied stresses increase the plastic radius increases and the modified inner radius decreases, resulting in a stable cavity. The salt cavity is unstable and can be fractured if the internal pressure exceeds the external pressure. Plastic deformation increases with the increase in applied loads. However, contrary to uniaxial tests on salts, plastic deforma- tion rate of the cavity under constant external loads decreases rapidly with time . A mechanical model describing the structural behavior of rock salt was proposed and verified in the laboratory. The model consists of elastic, viscoelastic and viscoplastic elements whose coefficients were successfully determined from laboratory tests. 10. The elas 156 ticity and plasticity constants were determined from the transition tests. The viscoplasticity coefficient, 77], is relatively much smaller than the viscoelastic one. A simple graphical method was developed to determine these coefficients. They are calculated as follows: 771 and 772 where ’71 772 GI C'z = viscosity constant in the plastic region = viscosity constant in the viscoelastic region = slope of the octahedral stress relaxation curve in the plastic region = slope of the octahedral stress relaxation curve in the viscoelastic region 2 octahedral shear modulus retarded shear modulus 11. The empirical creep equation of a hollow cylinder a-a --Pt f =ce a 0 established in the laboratory, agrees with the theoretical 157 equation derived from mathematical theory of viscoelasticity, Gz a a (r ' - _ t f _ i .6. e "2 a. 2 G2 0 where af = modified inner radius at t equals infinity a = modified inner radius at any time t a0 = initial inner radius C = constant The P value of the empirical equation of the cavity was found C . . . to agree very closely to the value of ;1—2- obtained from tranSition 2. tests. CHAPTER IX FUTURE RESEARCH A phenomenological approach to rock mechanics has been studied based on simple isotropic, homogeneous and linear visco- elastic characteristics which provided a satisfactory agreement be- tween theoretical calculations and experimental results. However, during the course of the present investigation, certain complications were encountered from which the following recommendations for future research can be made. 1. The experimental results for the determination of Young's modulus under triaxial compression, from the hollow cylinder and transition tests failed to agree. Part of this may be con- tributed by experimental errors in both techniques. However, a considerable amount of this error was observed to be due to the use of dial gages in measuring the axial strains. Hence, a more precise device to measure this strain would eliminate this potential error and improve the experimental results. 2. An attempt was made to determine the octahedral shear strength, ' Ko' from the boundary motion of a hollow thick -wa11ed cylinder. The values obtained were based on a few readings in the com- pletely plastic state. It is suggested to use a higher capacity pressure vessel to obtain more readings and improve the value of K . O 158 159 The analytical results in the three-dimensional analysis of model cavity deformation did not show complete agreement with the experimental results at high external loads. It would be more correct to consider instantaneous cavity measurements to eliminate the plastic flow effects at such high stresses. Theoretical analyses have shown that the plastic strain rate is time «dependent and a function of the deviatoric stresses. Therefore, it is inadequate from the present experimental re- sults to determine the viscosity constant, 171. Hence, a plastic stress-strain rates diagram might be used to evaluate m. In the present investigation, an effort was made to determine the material properties and verify the creep equations. No effort was made to determine the creep rate equations for any particular opening such as tunnels, mines or oil wells. Such analysis might be of practical importance for different mining or drilling operations. It is suggested then, to apply the ex- perimental and theoretical analysis of the model salt cavity developed in laboratory, to actual field measurements. 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