THE BEHAVIOR OP ROCKS MD ROCK MASSES
IB RELATION TO MILITARY GEOLOGY
By
Wilmot R. MoCutohon ProQuest Number: 10781375
All rights reserved
INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted.
In the unlikely event that the author did not send a com plete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. uest
ProQuest 10781375
Published by ProQuest LLC(2018). Copyright of the Dissertation is held by the Author.
All rights reserved. This work is protected against unauthorized copying under Title 17, United States C ode Microform Edition © ProQuest LLC.
ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106- 1346 $EGti3
A thesis submitted to the Faculty and the Board of Trustees of
the Colorado School of Hines in partial fulfillment of the requirements
for the degree of Master of Mining Engineering*
Signed
a** Wilmot H* MeCutohen
Golden, Colorado
Date # 18d8
4
V Approvedi r C* W, Livingston I . Golden, Colorado Date /<£- , 1948 Abstract
Following a brief introduction giving a general classification of rooks in the earth's crust which are of Interest to the military geologist,
Table 1 i s presented to summarise the e ffe c t of various factors on the physical properties of rooks. Accompanying d efin ition s f a c ilit a t e the interpretation of the table and provide a reference of terms used in sub sequent discussions.
The properties of elasticity and plasticity in rocks are treated in
Chapter 2m Some other characteristic phenomena of importance in the study of rock strengths, such as creep, fatigue, and endurance, are also mentioned.
Failure of rock specimens under stress is given a detailed study.
First, the accepted classical theories of failure are stated brieflyj then the Mohr Stress Diagram is developed for a number of types of load ings, ending with a general state of stress. Using the Mohr Stress Dia gram, a carefully controlled laboratory experiment on rook specimens is analysed to deduce the manner of failure of rooks and the form of the envelope of rupture.
Chapter 4 discusses several examples of static and dynamio loadings on rock masses, including stress distribution around a tunnel opening, propagation of elastic strain, and crater blasting. The last part of the chapter is devoted to a discussion of the principles of similitude as they may be used in the study of military geology. A speolfic example involving the detonation of an atomic bomb above an underground tunnel is * presented as. an illustration of the application of these similarity principles. TABLE OPCOBTENTS
Page
Introduction I Hooke in the Earth's Crust X Definitions 4
Elasticity and Plasticity of Rooks 8 E la s tic ity 8 P la s tic ity 10 Stages of Deformation X 0 Fatigue and Endurance 12
Failure of Hook Specimens Under Stress IS
Classical Theories of Failure 13 Mohr's Stress Diagram. 14 Simple tension 14 Triaxial loading 17 Pure shear 18 Pure shear and compression 19 General state of stress 21 Further discussion of failure of rocks 27 Envelope of rupture 28 The form of the envelope of rupture 31 Influence of the intermediate principal stress 34 Summary of methods o f fa ilu re 34
Physical Behavior of Rook Masses Under Statio and Dynamic Loadings 36
S ta tic Stresses Around a Tunnel Opening 37 Hanna1s solu tion 40 Character of stress distribution 42 Dynamic Loadings ymA A General considerations 44 State of stress on a body 43 Relations between stresses and strains 47 Propagation of elastic deformations 48 Repeated deformations 33 Wave motions 53 Dynamic loadings involving rupture 58 The Concept of Similitude as an Aid to the Study of Military Geology 61
Appendix 71
Bibliography 73 ILLUSTRATIONS O pposite Figure No* Subject Page No*
X Stress-Strain Diagram 9
2 Deformat ion-Time Diagram 9
3 Simple te n sio n 15
4 Mohr Stress Diagram for Simple tension 15
5 Triaxial loading 18
6 Mohr Stress Diagram for, triaxial loading 18
7 Pure Shear 19
8 Mohr S tre s s Diagram fo r pure shear 19
9 Pure shear and compression 20
10 General S ta te of S tre s s 21
11 Equilibrium oonditions on a body in general state of stress 22
12 Mohr Stress Diagram for general state of s tr e s s (Method A) 23
13 Resolution of Stress Components 25
14 Derivation of Mohr Stress Diagram for general state of stress (Method B) 25
15 Mohr Stress Diagram for general state of stress (Method B) 27
16 Specimens of rock tested to failure under various loadings 28
17 Mohr S tre s s Diagrams fo r specimens of > Figure 16 showing envelope of rupture 29
18 Influence of moisture and porosity on Mohr S tre s s diagram 32
19 Stress distribution around a tunnel a* Radial Stress pattern 37 b. Tangential stress pattern 37 o* Shear s tr e s s p a tte rn 37 d* Pattern of shear planes of weakness 43
20 State of stress on a body 46 O pposite Figure No* S ub ject Page Ho»
21 Blastio deformations from a dynamic load 50
22 Elaatio deformations at a distant point from the impulse 53
23 Pattern of shear weakness around a orater charge and outline of typical o ra te rs 60 TABLES
O pposite Table Page No»
I The In flu en ce of V arious F ao to rs on the Physical Properties of Rooks 4
II Values of the Principal Stresses and of - the Shear and Normal Components on the Planes of Weakness for Various Points Around a Circular Tunnol Opening 44 AC KNCMLB DGXENTS
I wish to express my appreciation for the courtesy extended me by the Structural Research Laboratory of the United States Bureau of Reclamation, Denver, Colorado, in making available the results of valuable experiments pertinent to the subject of this thesis*
Thanks are also due to Professors C* W# Livingston, Henry
Baboook, and W* H. Jurney of the Faculty of the Colorado School of
Mines for their helpful criticism s and suggestions* 1
THE "BEHAVIOR OF ROCKS AND ROCK MASSES
IN RELATION TO MILITARY GEOLOGY
Chapter I
Introduction
Rocks In tha Earth* s Crust
The earth’s crust is generally defined as the siliceous zone forming the outermost layer of the earth* The depth of the crust is more or less arbitrarily taken at about ten miles* At present* the military geologist is concerned with only the upper portion of the ©rust* down to a depth of several hundred feet, which falls within the rang© of moderate depths as present-day mining operations go* This upper layer of the earth consists of "rocks"—hard, cohesive material composed of minerals or aggregates of minerals bound together into a definite unit.
Rocks are in turn generally covered with a mantle of decomposed* uncon s o lid a te d m in eral and organic m a te ria l known as s o il* The boundary between the lower lim it of the mantle and the upper lim it of the "bed rock" is usually not well-defined inasmuch as the upper zone of the bed-rock has usually undergone alteration and decay from its original state through the action of the agents of weathering and erosion* This discussion w ill be concerned mainly with the physical properties of the bed-rock as it is found in its more or less unaltered, cohesive, and consolidated state*
Approximately twenty minerals, called "rook-forming minerals , 0 make up the great majority of the mineral aggregates of which rocks are composed* Notable among the rook-forming minerals are the t feldspars, micas, pyroxenes, amphibolos, quartz and caleite.
Rooks are generally classified according to their modes of origin as ( 1) igneous rocks, or those directly derived from a magma* ( 2 ) sedimentary rocks, or those secondarily derived rocks formed through the disintegration, transportation, and subsequent induration of pre viously existing rocks; and (3) metamorphic rooks, or rocks which were originally igneous or sedimentary and which have been partly or wholly reconstituted as a result of the action of heat and pressure at depth*
Each of the Inroad classifications of rocks—igneous, sedimentary, and metaraorphie—may in turn be subdivided into further genetic categories.
The igneous rocks are, for example, classified as extrusive (formed through the cooling of the molten magma on the earth’s surface) and intrusive (formed through the cooling of the magma in a relatively deep- seated environment)* Examples of extrusive igneous rooks are rhyolites, basalts, scoria, obsidian, and other volcanic glasses* The intrusive igneous rocks of note are granites, syenites, monzonites, gabbros, and peridotites* The intrusives are, in the order given, progressively more basic in composition* Granite stands as the most acidic of the intrusive igneous rocks, while gabbro and diorite contain a much larger proportion of basic minerals*
Sedimentary rocks also fall into two main subdivisions: namely,
( 1) those formed through the consolidation of clastic fragments
(sandstones and shales) and ( 2) those formed through the precipitation of solid material from solution (limestones)* Metamorphic rocks, on the other hand, are usually classified according to the degree of di as trophism to which they have been subjected* Low degree or low
”rankn metamorphies consist of such rocks as slates, while the higher 5 rank metamorphics ar© such rocks as gneisses and schists#
A further detailed classification of rocks according to mode of origin or mineral composition is of no significant interest to this discussion. However, any particular rook may exhibit characteristic features of texture and structure which may affect its physical proper ties to a marked degree. Extrusive igneous rocks are characteristically fine grained and quite often either glassy or scoriaceous. An important structural feature of intrusive igneous rocks is the presence of inter locking crystalline minerals. Clastic sedimentary rocks are usually characterised by mineral particles held together by a cementing agent*
Metamorphic rocks may exhibit the structural characteristics of both sedimentary and igneous rocks and, in addition, have such unique features as banding, foliation, schistosity, and gneissos© structure*
Rocks at ordinary conditions of temperature and pressure, and even under the environment of moderate depths, are inherently brittle sub stances, Individual rock specimens are able to withstand a relatively large compressive load before rupture, but are noticeably weak in shear and even weaker in tension* The rook masses which constitute the earth 1 s crust are weaker still when considered in relation to the tectonic forces to which they are subjected. As a consequence of the action of these earth forces, certain well-known features as folds, faults, joint ing, cleavage, and foliation are present to produce planes of discon tinuity or zones of structural weakness which serve to further reduce the strength of the rock mass. Other structural characteristics of rock masses such as bedding, degree of consolidation, degree of alteration, gradation from one type of rock to another, porosity, and moisture content affect the physical and mechanical properties of rock masses*
The accompanying table (Table I), together with the definitions given TABLS I (C ont’d) s,Suno^ sntnpoj? L^Susj^g gJUOSSTO^ i r sepjdoioo a r f v ^ C^xpox©A •I'T^TT C^TPT^TH -*swTd *ofq!i8H /'VTOf 6 u M 3 M H fft HP *H S ft © C ft O © 09 © P« © «T° o O P ( *—-> 31 *H §# A •H •P •P P •H n3 M ©
p T3 H <—1 •s W ft R 25 G 5© T5 r-i fl t -P ft © © ft © a ft © O O G • P • rP O «j p +3 H ♦ •H o P "©C © « © ^O tH »PO d o ©© ,M p © G i t a © O 4“ 4 I
o P ft rP 3 O 23 P >P rP P P ft rP f «O P« ft © •P • © P © > 05>> O © 5 O, rP , bO 05 u M rP o r-i •H tJ p + X? ft p 05 o o o o © O © « o o « © £ O X ♦
id « a? *P p 3 3 p •H «w P ft O <3 bO'd ft © © © 3 O a p .ft •Hi •P © ft G S o ft ***» p r-i rP A p p t—i P 'ft P l r • r-t p p •P d T O p t tp © o © d u S3 (0 h © •p © ft © s ft bO t—r d bO © § o bJ3 u ft 05 © ft* & o © d s © © © © o © bO © © £ o ft © 3 A £ § U) • h “ Indicates increases, while— indicates decreases Influence of Various Factors on Physical Properties of Rocks O D M55 * —S3 I ^ — - P P •HO^« ft *Hft >»e U © CO 3 Vi3 (0 © O © © © H ft *4 ft O P WO © ft © ft t © ft © M © r-i Q •Hft 03 P ft © © P P*O © P t O ft fl ft © o ca t f © t f HO© OP© r-i > Capable of more plastic or "elastico-viscous" deformations, § If coarser grained below, will serve to illustrate the effect of various factors on the physical behavior of rock masses# It should be remembered that any number of such factors may be operating at any one time to produce a resultant effect which may be different from the behavior under any one of the component factors. Definitions (See Appendix for lis t of symbols used herein) Elasticity. The ability of a}body, when deformed to a certain ! extent by a system of applied forces, to regain its original shape upon removal of the forces. The degree of elasticity of materials is measured by the amount of deformation (strain), expressed as a fraction of the original linear dimension, which a body will undergo under a given stress (force per unit area) within the range of elastic action of the material. Steel, for example, is a much less elastic substance than rubber, because a given stress in steel will produce much less elastic deformation than will the same stress in rubber# Inasmuch as in any study of elasticity, Hooke’s law is assumed to hold below the elastic lim it—that is, for any material the applied stress is proportional to the resulting strain— Young’s modulus has been used to describe the relative elasticity of two materials# Since Young’s modulus, E Z will have a much larger s t r a i n value for substances such as steel than it will for a material like rubber, it is readily seen that, the lower the value of the elastic modulus, the more elastic a substance must be. Characteristics of elasticity w ill be discussed further In the next chapter. Plasticity. The property of a body, whereby, when it is subjected to a deformation by the application of a system of forces, it remains deformed upon removal of the forces* In other words, an ideally plastic body should exhibit no elastic recovery* All solids exhibit both elas ticity and plasticity to varying degrees* Steel under ordinary conditions of temperature and pressure exhibits elastic properties under stresses below the so-called "elastic lim it" (beyond which Hooke’s law no longer holds), then with increasing stresses it displays plastic behavior until rupture* Rocks under the same conditions show very little if any plastic deformation beyond the elastic lim it and before rupture* However, with increasing depth and confining pressure, rocks may exhibit a considerable amount of plastic deformation* * * Viscosity* The resistance offered by a fluid to the relative motion of its particles. A viscous deformation is characterized by a continuous displacement of the particles of the fluid body at a rate proportionate to the rate of application and magnitude of the applied shearing load. Such dependence on the rat© of application of the shearing load dis tinguishes viscous flow from plastic flow, for in the latter type of deformation the shear stress necessary to produce plastic shear is very nearly independent of the rate of shear* Elastioo-visoosity* A substance is called "elastico-viscous" when it exhibits elastie properties (and perhaps plastic properties as well) under loads of short duration, yet under relatively small loads of long duration deforms viscously* In such materials the coefficient of viscosity is quite high* An example of elastic-viscous material is seal ing wax* Rooks are said by some authorities to be elastic-viscous* Creep* A slow deformation of a material when subjected to relatively small stresses acting over a relatively long period of time. Brittleness* A relative term denoting the amount of plasticity which a body fails to exhibit before failure under a given condition of loading. Rooks at ordinary conditions of temperature and pressure are said to be brittle. Hardness. As used herein, hardness refers to the ability of a body to stand abrasion. Such a descriptive term can only be applied with reference to a generally accepted standard, or scale of hardness,, to which a given material is compared* In the testing of hardness of mineral specimens, Moh’s scale is the usually accepted criterion. In this scale, talc is assigned a hardness of 1, gypsum 2, caloite 3, fluorite 4, apatite 5, orthoclase 6 , quartz 7, topaz 8, corundum, 9 and diamond 10. Toughness. The p ro p e rty o f a body by v ir tu e o f which v/ork may be done on the body when stressed beyond its elastic lim it. Strength. The particular combination of stresses which a material can undergo before failure either by rupture or plastic flow under a given set of conditions of time and temperature. Fundamental strength is the strength of a material regardless of the time period over which the stresses my be applied. Porosity. The amount of voids in a given volume of material in relation to the amount of solids in the samevolume. Elastic Constants. a. Modulus of E lasticity (Young's Modulus) u> — stress s ** strain V b. Poisson's ratio* The ratio of the lateral unit strain to the longitudinal strain such as found in a test specimen subjected to a simple axial load. Poisson's ratio - m - € l a t e r a l I longitudinal 7 c* skQar modulus or modulus of rigidity# The ratio of the applied shearing stress to the resulting shearing strain# fl r Ss r S -77- 2(T+'iIJ d. Modulus of Flexibility# The reciprocal of the modulus of rigidity, or A # G e. Modulus of volume elasticity# (Lama’s constant) \ mE - -(yt n)'(r--s) f • Compressibility factor# A proportionality factor which may be used to determine the compressibility of a material under a given confining pressui*©. For hydrostatic pressure 5 K I 5(l-2m) E g# Incompressibility faotor (k)# Also called the wBulk Modulus , 11 is the reciprocal of the compressibility factor# k* Modulus of Resilience# The energy absorbed per unit volume of a material when stressed to the elastic lim it. For specimens tested in simple tension or compression the modulus of resilience, U, is: 2 TT 1 1 “ • t ' l c e t ^ L . Velocities of Wave propagation. a# Compressional, longitudinal, or primary waves: *» = b# Shear, or transverse waves: vs = V -y-nSe— 5— : -J-SL V 2w(l + m) Y w C hapter 2 Elasticity and Plasticity of Rocks E l a s tic ity General. An elastic deformation is one which disappears completely upon release of the stresd causing the deformation. As already indicated under the definition of elasticity in Chapter 1, there is a definite relation, established by experiment, between elastic stress and elastic strain—that is, they are proportional to one another within a certain range. This elastic property of proportionality is known as Hooke’s Law and is one of the funda mental precepts on which investigations of elastic behavior are based. A hard, rigid substance such as steel will behave in a truly elastic manner under a small stress according to the defini tion of elasticity, but a highly elastic substance such as rubber w ill not behave in a truly elastic manner. When stretched slowly, rubber does not immediately return to its original shape after the release of stress. It would seem, therefore, that the harder and more rigid a material is, the more truly elastic is its behavior under a small stress. Rocks as encountered near the earth’s surface ally themselves more nearly to the harder substances and behave in a more or less truly elastic manner under small stresses* Stress-Straln Diagram. An idealized ”stress-strain” diagram for a rock specimen is shown in Fig. 1. The specimen has been ' subjected to a slowly applied, simple, axially compressive load under ordinary conditions of temperature and atmospheric pressure. If the rock specimen behaved in a truly elastic manner, the line QA fcUpTUft-E. po in j SjR. A I N Figure 1 Stresa-Strain Diagram for Rook, pLASjlC OGjroa^NAfiON ELAS-pC AfJSft*EffECj I LA4JIC R.E COV E tQ • LoAOlKCi COAM£UC£D t, : Loaoim ^ ft e leased : U LjIMAfE <2.£COVEft-Y Figure 2 in the diagram would be straight* Actual specimens tested, however, indicate that the stress-strain relationship is not entirely in accordance with Hooke’s Law, so that the line QA established by experiments on rooks is slightly curved* Inasmuch as the area under the curved line OA represents the actual work done on the system, it is evident that less potential energy is stored up in the body than would be possible if the rock had behaved in a truly elastic manner* The horizontally shaded portion of Fig* 1 indicates the amount of possible elastic potential energy which, through a phenomenon called Relaxation", is not stored in the body during deformation* Relaxa tion may be described as the process whereby certain portions of the rock make use of some of the potential strain energy as heat energy of particle vibration* Moreover, upon gradual release of the load, the rock specimen does not return to its original shape along the same line as in loading the specimen* for a certain stress on the loading curve OA is associated with a greater strain on the unloading curve* The area between the loading curve and the unloading curve is the amount of stored potential energy lost as heat, which is dissipated in pro ducing a change in the internal structure of the rock in loading it to a condition corresponding to point A* This loss of energy is called "hysteresis of strain*" If at point A the specimen is subjected to further load, it would finally fail by rupture at a point B, corresponding to a stress value Bf• The entire deformation before rupture would be approximately elastic, as already indicated* Creep* The deformation illustrated in Fig* 1 is brought about on a specimen during a relatively short time experiment* In addition to this type of deformation, it may be observed that, if the specimen 10 is deformed to a point such as A by the corresponding stress A*, and then if the stress is kept constant over a period of time, the rook w ill continue to deform in ever decreasing amounts over equal time intervals until the deformation virtually ceases* This phenomenon, called creep, is thought to be another manifestation of relaxation* If the stress A* is removed after the creep deforma.tion occurs, the specimen will in time usually regain its original shape, or nearly so* Thus, the rock may still be considered as behaving more or less elas tically under "creep deformations*" P l a s t i c i t y If the phenomenon of relaxation is extended to a consideration of the behavior of rocks under much higher confining pressures than ordinary atmospheric pressure—that is, at pressures to be found in the earth at depth—it will be found that the rock specimen w ill behave in an elastic manner under much higher stresses and in addition w ill under go a type of flow deformation from which it will not recover its original shape* The type of flow deformation which produces a permanent "set" in the rock specimen is called "plastic flow*" If the confining pressure is high enough, the rock specimen may undergo a considerable amount of plastic deformation* Stages of Deformation Rocks are made up of aggregates of mineral matter, bonded together by various means. It is evident that each of the mineral constituents may exhibit individual elastic and plastic properties which make it conceivable that under a given loading some mineral crystals may be undergoing plastic deformation while other crystals are still in their elastic range. This heterogeneity of action under stress has been y advanced by Burgers to explain the deviation of rocks and other VHouwink, "Elasticity, Plasticity, and Structure of Matter", Cambridge Univ. Press, 1940. polycrystalline materials from a truly elastic behavior. The inner tensions arising between the elastically and plastically deformed portions may, in addition to the deformations already described,give rise to an "elastic after-effect," or slow recovery from deformation after release of the load. We may then summarize (see Pig. 2) 'the deformations produced on a typical rock specimen under suitable conditions of temperature and confining pressure as follows: 1. Small loads producing truly elastic action of all crystalline matter and bonding m aterial. 2. Increasing loads to produce a plastic deformation of a few of the crystals (or bonding m aterial), so that the rock as a whole behaves as essentially elastic but with potential elastic after effect and hysteresis produced. 3. Static loading over a period of time (creep), giving rise to a sufficient relaxation and internal adjustment of the rock structure to produce a more favorable equilibrium with the external loads* 4* Increasing loads resumed. Deformation of the crystalline matter and bonding agents is so preponderant that, if the load were released, the elastic potential energy of those crystals still under going elastic deformation is insufficient to restore the rock to its original shape. A "permanent set" is produced. 12 5. Further increasing of loading. Plastic deformations occur to such a degree that the elastic elements no longer are able to form any concerted resistance to the deformation, hence appreciable yielding is evidenced with small increase of load. Plastic flow is dominant. If, before the state of deformation in 5 above begins, the load were gradually released, the following stages of recovery of the rock would be observed: 1. An elastic recovery (accompanied by hysteresis) from a considerable proportion of the deformation produced. 2. A recovery from creep, combined with the elastic after-effect mentioned previously, to further reduce the amount of deformation produced. A portion of the deformation w ill s till remain which was brought about by the plastic flow. The rock will not recover from this plastic deformation. Fatigue and Endurance Under repeated loading cycles of the nature ^ust described it is probable that the changes brought about upon the internal structure of the rock by hysteresis, plastic deformations of some of the crystals or bonding material, and localized strains will bring about the eventual failure of the material under loadings below those nec essary to rupture the rock during a single test. Yfhether these factors completely explain the reduction in strength of a rock under repeated loadings is problematical* However, it is sufficient to know that such repeated stresses actually do reduce the strengths of rocks* Chapter 3 Failure of Rook Specimens Under Stress Many varieties of rocks have been subjected to standard laboratory tests to determine their ability to withstand mechanically induced stresses. Unfortunately, rocks have not been tested as exhaustively as have some of the more common structural materials such as steel or concrete; nevertheless, the data available from rock tests is suffi cient to justify a theoretical discussion of the manner of failure of rocks under stress. A. Classical Theories of Failure Several theories of failure have been advanced to explain the cause of the commencement of inelastic action or the manner of failure of a material. Failure of elastic action is tantamount to rupture in a brittle material such as rock under ordinary conditions of tempera ture and atmospheric pressure. Briefly summarized, these classical theories of failure are: 1. Maximum Stress Theory. This theory states that inelastic action begins at a point only when the maximum normal stress (tensile) across any plane through the point exceeds the stress corresponding to the proportional limit of the material in a simple tension test. 2. Maximum Strain Theory, This theory postulates that the strain associated with the failure of a material in a simple tensile test must not be exceeded at any point in a body if the material is not to fail elastically. 3. Maximum Strain-Bnergy Theory. The "modulus of resilience" (see definitions) is used as the criterion of elastic failure in this / 14 theory. The maximum strain-onergy stored per unit of volume by the body under any complex system of stresses must not exceed that which can be stored per unit volume by the body as determined by a simple tension test. 4. Maximum Stress-Difference Theory or Maximum Shear Theory* (Guest*s Law). This theory states that the factor producing failure of elasticity is the greatest shear stress in the material (or the greatest difference between the principal stresses). j / The maximum stress theory has been modified by 0. Mohr. Instead ^ Mohr, 0., “Technische Mechanik”, pp. 192-234, W. Ernst u, S.# 3rd Ed., Berlin, 1928. of placing reliance on the maximum shear stress alone as a cause of failure, Mohr states that of all planes having the same normal component, that which sustains the greatest shear stress is the plane on which inelastic strain will occur. As far as rocks are concerned, it seems most plausible to accept Mohr's hypothesis, with several important amendments, in order to explain the failure of their inelastic action. Before discussing failure further, it is advisable to explain in detail Mohr’s Stress Diagram, which should be understood in order to fully appreciate the manner of rook failures. Inasmuch as most references do not include a thorough discussion of this valuable graphical means of representing a state of stress, a comprehensive analysis is included in the following paragraphs* Mohr’s Stress Diagram. 1. Simple Tension. Figure 3a illustrates a rock body subjected to an axial tensile load, F, distributed uniformly over the cross V <*r dx a. F igu re 3 JSfiS ION ^ COAAfR.es SI OVI - S H £ * f c . F ig u re 4 Mohr S tr e s s Diagram fo r F ig . 3 section of the body taken at point P perpendicular to the direction of F* The "force intensity*” or "stress,” across such a plane is p^ * F <• A.# where A is the cross sectional area* This particular plane is known as a "principal stress plane” for the point P, such a plane being defined as one on which there is no shearing or tangen tial stress# It can be shown that for the states of stress on the many planes which may be passed through point P under any system of external forces, there w ill be three and only three such principal stress planes* Furthermore^ the three principal stress planes are mutually perpendicular. Thus, the directions of the three principal stresses—acting always normal to the principal planes of stress— are also mutually perpendicular* In & general case, none of the principal stresses will be equal, so that, if we designate these stresses as p^, p£, and p^, and if P i > P2 > * V then p^ is the maximum principal stress* pg is the intermediate principal stress, and pg is the minimum or least principal stress* In the particular type of loading shown in Fig* 3a, pgS pg - 0* Through the point P, of Fig* 3a, a random plane such as a-a may be passed, making an angle / with the plane of the maximum principal stress* Upon the small body, dV, of unit thickness shown in Figs* 3a and 3b the following equilibrium of forces exists; ZFX * 0 XFy^Q SgCos/ dl* snsin/ dl sncos/ dx + sssin/ dx = p^dx c o s / c o s / Solving the above equations for ss and S&, the shoar and normal components of a resultant stress, B, on the plane a-a, we obtaint 16 c ss 8 J p i s in 2/ ( 1 ) 8n * i P i + "k Px ©os2/ ( 2 ) The relationship between ss and sn may be represented geo metrically by the construction shown in Pig* 4* The horizontal axis in the drawing measures tensile stress from the origin, 0 , to th e left, and the vertical axis represents shear stress (the sign of the shear stress is arbitrary). A length 02, equal to the magnitude of pi, is laid out along the horizontal axis and with C as a center a circle of radius -g p^ is described through points 0 and 2* An angle 2/ is measured off from the horizontal axis, its radius CA intersecting the circle at A. Thus the abscissa and ordinate of point A represent the magnitude of the normal and shear components of stress on plane a-a since equations ( 1) and ( 2 ) are satisfied by the geometrical con struction. Length OA is the resultant stress, R. The circle, therefore, represents the loci of all points, such as A, whose coordinates determine the shear and normal stress components on planes passed through point P. The random orientation of the specimen in Fig. 3a and hence the random direction of with respect to the axes of the Eohr diagram in Fig# 4 illustrate the method of using a “pole," P, to determine the point A by drawing parallel lines to those of Fig. 3a on the stress circle instead of laying off the angle 2$ in the procedure just stated. A line is drawn from point 2 parallel to the direction of the maximum principal stress plane in Fig# 3a# Its point of intersection with the stress plane is called the "pole" for the circle. Any line, such as PA, drawn from the pole parallel to a random plane through point P in the specimen w ill give the required intersection (e.g. point A) with the stress circle for that plane. It is more convenient, generally, to orient the horizontal axis itself parallel to the maximum stress plane so that the pole P will coincide with the intersection of the circle with the horizontal axis. The pole in the Kohr diagram is of particular use in locating the directions of the principal stresses and principal stress planes in case the principal stress directions are the unknown quantities. 2. Triaxlal Loading. Consider now anothor condition of load- ing such as is obtained in a "triaxial compression" testing machine. If an impervious jacket surrounds the specimen as it is immersed in a liquid under a confining pressure, p, and then the specimen Is sub jected to an axial pressure, Ap, in addition to the pressure p, we have for a unit volume of the specimen cut out by planes parallel to the directions of the principal stresses the loading shown in Fig. 5a. In this sketch = p + A p, and pg s - p. The stress pg is acting in a direction perpendicular to the plane of the paper. Following the procedure of resolving the forces necessary for equilibrium of a small volume dV (Fig. 5b) in a horizontal and vertical direction and solving for sfi and sg in terms of p^, p^, and we o b ta in : «n * I (P i + P3) + i (P i - Pj)oos 2/ ( 3 ) ss = i (pt - P3) sin 2/ (4) If we plot the Mohr stress circle to represent the relation given in equations (7) and ( 8), it will appear as in Pig. 6 . Kota that the sign of the shearing stress is merely conventional, depending upon « - P. i a. b Figure 5 IsK&tOK 2— CO*sp«.£$$iOW Figure 6 18 whether the angle / is greater or less than 90°, Compressive stresses are taken in the positive direction of the horizontal axis, while tensile stresses are considered negative. QZ represents again the magnitude of p^, laid off in a direction to the right of the origin (to signify compression). OX represents the value of p , also a 3 compressive stress. An identical stress circle would have been ob tained if the elemental volume dV and the plane a-a had been disposed in the same attitude with respect to p as it is with respect to p 2 3 in Fig. 5a, inasmuch as p - p in triaxial loading. 2 3 In F ig. 6 the orientation of the horizontal axis is parallel to the plane of the maximum principal stress, so that the pole P, described in Fig. 4, coincides with point X in Fig. 6 * The lin e XA is parallel to the plane a-a of Fig. 5. The line OA represents the magnitude of the resultant stress, R, on the plane a-a, and 9 i s th e angle between the resultant and the normal to the plane a-a* 3* Pure Shear. Another type of special loading which may be imposed upon a body is that of ’’pure shear.** If the specimen in Fig. 7 is subjected to a couple as shown, the resistance to deformation is set up as a shearing stress along the plane a-a as it is influenced to move in relation to a parallel plane a*-a*. The shearing stress is I * P J A, assuming that the shearing stress is distributed uniformly over the cross-sectional area A cut out by the plane a-&.* * This assumption is probably not Justified for the stress condition at any point throughout the specimen, for it is probable that a complex stress distribution is evident near the boundaries of the specimen. The same complex stress distribution probably exists in the simple tensile test specimen already described, but in each case, if the specimen is sufficiently large, the fundamental definition for stress^Le. s - Force divided by area, should be sufficiently valid for points well in the interior of the body. ------CL d * till F ig u re 7 - X - - F ig u r e 8 An elemental cube in the body, shown in Fig* 7b is acted upon by the tangential or shearing stresses, T, on its surfaces* A random plane b-b, passed through the element w ill, from the requirements of equilibrium of the free body, have a normal tensile stress equal in magnitude to T acting across the plane (with no shear component) if & - 450, and a normal compressive stress equal to T in value if & - 135°. Then the compressive streas (= T, & *135°) may be considered the maximum principal stress and the tensile stress Pg(s T» /3 = 45°) the minimum p rin c ip a l s tre s s * Under th e co n d itio n s of loading specified, the intermediate principal stress is s Q, which algebraically is actually intermediate in value between p^ and p^* The equilibrium condition on a small volume dV in Fig* 7o may be expressed in terms of sn, sg, T, and for any assigned value of Q • From such conditions we obtain the relations; sn • T sin 2 & (5) ss - T cos 2j3 ( 6 ) On the other hand, if s„ and s„ were determined in terms of their JX 5 relation to the principal stresses, p^ and p^, and the angle (s 135° - /3 ), the resulting equations would be identical with equations (3) and (4). ’The stress circle for !lpure shear" is shown in Fig, 8, Note the position of the pole P for this diagram* If the tensile stress p^ is considered the maximum principal stress, then the pole P would be thrown into position P* as shown* 4# Pure Shear and Compression* If a specimen such as the one shown in Fig* 7a were "first subjected to a uniform hydrostatic pressure. b V C. 4 a F ig u r e 9 p, before the imposition of the couple F, the question arises as to the effect of this confining pressure upon the Mohr Diagram of Fig* 8* In conformity with the discussion under Par. 3, let the couple F aot in the plane of the paper as in Fig. 9a* Thus, the intermediate principal stress pg is equal to the confining pressure, p* Also, let the magnitude of p be less than that of T. Then the magnitudes of the other principal stresses will bet p-^ » T + p (compressive stress) j v. Pg = T - p ( te n s ile s tr e s s ) As b e fo re , the plane of th e minimum p rin c ip a l s tr e s s w ill c o rre s pond to 0 - 45°, and the plane of the maximum principal stress w ill occur when 0 ~ 135°. Solving for sn and ss (Fig* 9a) in terns of T, 0 , and p, we obtaint en = T sin 20 - p (7) j ss - T cos 20 ( 8) Figure 9c illustrates the Mohr Diagram for planes parallel to the direction of pg, while Fig. 9d shows the stress relations across planes parallel to the direction of pg. As might be surmised, both the stress circles of Figs. 9c and d could result in a specimen from an axial compressive stress • p 4- T, a lateral compressive stress of p^ s p, and a lateral tensile stress of pg = T - p at right angles to the direction of p^. Practical difficulty would perhaps be encountered in producing such stresses, however, so that it would seem, from an experimental point of view, that the "indirect approach" utilizing triaxial compression and a superimposed couple would furnish the means of securing such a condition of loading* z M — -^P_ - /> o ^ y s > . o F igu re 1 0 , It should be noted from the discussions of pure shear and pure shear combined with confining pressure, that, whenever a plane in a body under a state of stress has only a shear or tangential stress acting upon it with no normal component, then one of the principal stresses has a different sense in relation to the other two principal s tre s s e s * 5* General State of Stress* Thus far, several special condi tions of loading have been considered* In each case, the planes passed through a certain point in a body in a state of stress have been parallel to one of the three principal stress directions* It is possible to examine thf state of stress on other planes at random directions to the principal stress axes as well. Two methods of constructing Mohr's Stress Diagram will be presented. The first method is based upon the same “equilibrium of forces” approach used heretofore, while the second method utilizes the principle of resolution of stress components* a* Method A* Figure 10 represents a very small body at point P which is bounded by planes normal to the three principal stresses and by a fourth plane, ABC, whose normal in turn makes direction angles and Y with the principal stress directions* Denote the direction of the stress p^ by ZO* p^ by YG, p^ by XQ* For convenience of illus tration, all the principal stresses in this derivation are assumed compressive, although the proof is applicable to any signs of the principal stresses. Further let EMS7 0X1 P ° Y = P* a* b . Figure II fLH$. Figure 12. Mohr Stress Diagram for General State of Stress ( Method A) 22 L et R signify the magnitude of the resultant stress acting across the plane ABC* The problem is to find the value of this resultant stress, R* The stress R may be resolved into two components, H and T, normal and ta n g e n tia l re s p e c tiv e ly to the plane ABC* Once th e mag nitudes of the components of S are determined, then the magnitude of R may be determined by adding the component stresses vectorially* The angles which the normal, OP, to the plane ABC makes with the axes X, Y, and Z are at , 0 , and 7 ♦ The ta n g e n tia l s t r e s s T may be resolved into two components, and Tg, so that T^ is perpendicular to trace AB on the XY plane and T£ is parallel to AB* A sm all, wedge-shaped volume may be delineated by the plane ABC extended, by the XY plane, and by planes A*B*DE, AA’E, and BB*D, all erected perpendicular to plane XY* Planes AA*S and BB*D are in turn perpendicular to A*B’DE, whose trace A*B* on the XY plane is parallel to AB* The state of stress across A’B’DE, BB*D, and AA*E can be determined in the manner cited in Par* 2, inasmuch as they are parallel to the direction of p^* The angles which these planes make with the direction of p£ are / and 90°“ / as shown in Fig# 10* The angle / is determined through the relation: The Mohr Stress Circle for planes A*B*DE and BB*D is shown in Fig* 11a# The loading condition on the wedge-shaped volume is shown in Fig# lib* Resolving the forces on this free body in the direction of BB* and B’D, we have: XF 0 B*B an db + cosy db - E sin 7 db - 0 sin y s in 7 23 £ F. I 0 B*D p da - T sin*/ da - H o o s'7 da 0 X 1 Cm y co sy And solving for H and we have; T1 ~ i (p^ - «n ) s in 2 7 (9) 11 = S (Pi + sn) + 4 (px- sn) ooe 2 7 ( 10 ) For equilibrium in the AB direction: s_s5 db - T<>6 dbMM s m ^ o r T2= s s 8in T ( i d The Mohr S tre s s Diagram f o r plane ABC i s shows in Fig* 12* The procedure for geometrical construction of this stress circle (Method A) may be summarized as follows: ’ (1) Determine the angle / which the trace of plane ABC on the maximum principal stress plane makes with the direction of the plane of the intermediate principal stress (or with the direction of the least principal stress)* This angle may be determined analytically or by g eo m etrical c o n s tru c tio n through th e r e la tio n s ta n * oosot # OOSjS (2) Lay off on the horizontal axis OX = p3j 0Y s pgj and 0Z z p^# Describe circle XT* From X, lay off angle YXA s to intersect circle XY a t A* (3) 0A* is the abscissa of point A* With C* as a center and with radius * ^ A*Z, describe circle A* Z through points A* and 2* (4) Prom A* lay off angle ZA’B ST to intersect circle A*Z at B* (5) With A* as a center and with AAf as a radius, describe an aro to interseot line AfB at *Bf* ( 6 ) The abscissa of point B is the value of the normal stress, N, on the plane ABC* The ordinates of points B and B* represent the values of the shear stresses, T- and T , respectively* The resultant X 6 stress, R, acting on the plane ABC,-may be found by adding vectorially the three stress components N, T-^, and T£* b* Method B* Another graphical construction to represent the state of stress across any plane through a point in a body is the one originally proposed by Mohr* Before proceeding with the graphical representation, it is advisable to consider a principle connected with the resolution of stress components* Pig* 13 is a small block of material whose faces ar 8 perpendicular to any two random directions. Ox and Ox*, with the other two faces parallel to the plane of the paper* Let a resultant stress, R*x* act on the planes normal to Ox, and resolve this stress into two components, Xf anc| Y*, parallel to the directions Ox* and Oy*. Similarly, a resultant stress, B*,, acting upon planes normal to Qxf can be resolved into components and Yyt* If the cosine of the angle between vectors S^i and X x, be denoted by 1 and the cosine of the angle between vectors RJ and XJ. be denoted by 1 *, it can be shown V that = x** or S * .l = ^ , . 1 * ( 12) */ Southwell, R# V., “Theory of Elasticity", pp* 265-266, Oxford, 1936* F ig u r e 13* S h £ A f t z , T fc* S. CONN p. F ig u re 14 Referring to Fig. 10* we may make use of eq u atio n (12) to establish the following relations. If the direction cosines of vector S are l f, m», n», and those of the normal OP to plane ABC are 1, m, n , th en H I * = p s l Rm» Z p9m (13) Rn» Z p^n A n d ,if 9 is the angle between R and its normal component lJ,then from (13), - D2 - 2 ,2 2 2 2 2 H - Ps 1 + P2 » -v Px a (14) H = RcosO = R (l* l -t mfm + n*n) or B = P3 l 2 t p2 m2 + px n2 (15) and, since T 2 - R2-N2, T2 = m2l 2(ps-p2)2 +■ l2n2(p3-Pl)2 + m2n2(p2-Pl)2 Expressed in terms of the direction angles c l and y , H becomes? N - §(Ps+ Pg) **4 (P2~P3)cos2 A t + sCp^^) (16) It is readily seen that the expression on the right of equation (16) is too complex for direct graphical construction* ^s is also the expression for T. However, it w ill be proved that all points R, whose coordinates on the Mohr Stress Diagram represent the stress components on planes whose normals make the same angle with one of the principal stress axes, describe a locus which is a concentric circle to the stress circle in the plane of that particular principal stress. For example, if the normals of the random planes in question make an angle oc with the OX or pg axis, then the locus of points R for those planes w ill be a eoncentric circle to the YZ circle (see Fig. 14). 26 In Fig* 14 the point represents the state of stars ss in the plane perpendicular to the XX plane* The normal of this random plane makes an angle* with the X-axis* Similarly, the point Q represents the state of stress in & plane perpendicular to the XZ plane and identically situated with respect to the X-axis as the first random plane. Both and Q are determined by the line XQ-jQ (laid off at an angle*, to the perpendicular erected at point X), whieh cuts the XX and XZ circles at and Q* If triangles YCQ and CQZ are solved for the value of CQ 2 and SQ^2, it will be found that both lines are equal in length and have the value S5X2 = CQ2 * i (p1-p2)2 + (P2-P3)(P1-Ps)oos2 Moreover, in Fig* 14, let R be a point representing the state of stress on any plane (see Fig* 8) whose normal makes an angle *. with the X-axis. Then CS2 2 p2 ) - s ) 2 = R2 + iCp^-hPg)2 -^ (P x ^ P g ) Substituting the values of R and H from equations (14) and (15) we have, upon re d u c tio n , CR2 Z ^ { p ^ p g ) 2 + (P2-P 3 )(P 1» P3)c o s 2* ( 1 8 ) Inasmuch as (18) is Identical with (17), it follows that CQj = CQ, * CR* Thus the stress on any plane In the octant shown in Fig* 10 whose normal makes an angle oL with the X-axis can be determined by a point on the arc of a circle with radius CQ or CQ^. By similar arguments it may be shown that concentric circles to the XX circle are loci of points representing the state of stress on planes whose normals make an angle of 7 w ith th e Z ax is* S HE. A R- c.o/*\p. F igu re 15* Mohr S tr e s s Diagram fo r G eneral S t a t e o f S t r e s s . (Method B) 27 Construction procedure# (Method B) From the foregoing considera tio n s we arrive at the following graphical construction for determining the state of stress at a point across a plane whose normal has direc tion angles c l , ./3 * and y with the axes of principal stresses, OX, OT, and OZ, if the values of the principal stresses are known: (1) Lay off along the horizontal axis the distances OX ~ p^j OT Z pgj OZ - p^* Describe the circles XZ, XT, and YZ* Erect perpendiculars XX^, YY-j_, ZZj to the horizontal at X, Y, and 2# (2) From XXX lay off XjXQ = oL . From Z2X lay off ZXZP * 7 (in a direction opposite to a ), (3) From the center, C, of the circle YZ, and with CQ as a radius, describe an arc. From the center, D, of circle, XY, and with DP as a radius, describe an arc to intersect the arc of radius CQ at point R* (4) The distance OR is the resultant stress, R, on the plane the direction angles of whose normal are ot, /j , and y « The projection of OR' on the vertical axis determines the magnitude of the tangential stress, T, on the plane, while the abscissa of point S is the magnitude of the normal stress on the plane* FwrthBT Discussion of Failure of Rooks* The Mohr Stress Diagrams furnish a valuable tool for the investi gation of experiments having to do with the failure of rock specimens under various conditions of loading* A cylindrical specimen tested to failure in simple tension w ill exhibit a fracture as shown in the sketch of Fig* 16a* A similar specimen failing under simple compression may e x h ib it a ty p ic a l ”c o n ica l f r a c tu r e *1 like the sketch of Fig* 16b* Other sp cimens tested to failure in a triaxial compression machine. a* b. Simple Tension* Simple Compression. Rupture at 1,000 p.s.i. Rupture at 17,000 p.s.i* e* Triaxial Compression. Triaxial Compression. PjL - 28,400 p.s.i* p-j_ = 5 2 ,4 0 0 p .s.i. p2 = p3 = 2 ,0 0 0 p .s.i. p3 * 1 0 ,0 0 0 p .s.i. F ig u re 1 6 . Specimens of Andesite-Tuff Breccia From Boulder Dam Tested to Failure Under Various Loads. (From U.S. Bureau of Reclamation Lab. Rept. N o. SP 6, July 10, 1945. Y. Jones & D. McHenry). 28 with the confining liquid prevented from entering the pores of the specimen by a sheath, or jacket, w ill exhibit quite similar conditions of failure as the simple compression test* The only major difference is the greater values of the principal stresses at failure* If we cut the ruptured specimens shown in Pig* 16 in half by a plane passed through the long axis of each of the specimens, the lines of intersection of the surfaces of rupture with the cutting planes will be as shown in Fig* 17* In all the tests mentioned here, the values of pg and p^are interchangeable in any two selected orthogonal direc tions in a horizontal plane* Therefore, for purpose of illustration, we may state that the stress pg acted across the cutting plane during all the tests* Then planes* passed through the rupture lines perpen dicular to the cutting plane w ill determine the direction along which failure of the specimen occurred (in the cutting plane, of course)* The Mohr Diagrams indicate at a glance the inclination of these planes with the plane of the maximum principal stress (Fig* 17)* Moreover, the normal and tangential components on the planes of failure may be determined readily from each of the stress circles* 1* Envelope of Rupture* On the compression side of the diagram the rupture line in simple compression and the failure lines of the specimens under several lateral loadings were taken* It follows that if many triaxial tests were made, the locus of the points of intersection of the failure lines with the stress circles will be an envelope to the stress circles* This envelope is commonly referred to ♦Note: Actually, the rupture lines appear quite irregular due to the heterogeneity of the structure of most rooks* The Mplanes ’1 through these rupture lines are those which most nearly approximate the trend of the rupture line* O- _o Q_ _o 6 _o M £ o o o o «0 «h o ' fO CQ L- I- Id ul J «i O u Q _ Mehr Mehr Stress Diagrams for Rock Specimens Shown in Figure X M® Z O «/>o ** iM ul z o ul - o a «» ' II Ui Li II J i-c- o_ »/) S c/) 29 as the envelope of rupture. The shape of the envelope approximates that of a pair of straight lines symmetrical about the horizontal axis* If the lateral confining pressure is high enough, the rock no longer fails by rupture but exhibits plastic flow* Therefore, the envelope exhibits the limiting values of elastic or nearly elastic action for the rock under any condition of confining pressure and a differentially greater pressure, p^* The tension side of the diagram in Fig* 17, on the other hand, shows the rupture line as being parallel to the plane of the principal stress, p^, in tension* If other stress circles were plotted with lateral tensions between zero and the simple tension value, p^, it is conceivable theoretically that all these stress circles w ill become progressively smaller and tangent to a vertical line drawn through point T* The circles could not be situated beyond the point T, for such a condition would give rise to a greater value of p^ than OT* It is inconceivable that, if a brittle substance such as rock is favorably disposed to break wider a simple tension p^, it would not be so disposed to break whenever this normal stress is encountered across any plane in the body under another condition of loading* Thus, the Maximum Stress Theory mentioned in Section A-l of this chapter would seem to be the most appropriate criterion for failure under loading conditions causing a tensile stress in the body equal to OT* Between the simple tension circle and the simple compression circle there is unfortunately not enough experimental evidence avail able to loo&te exactly the envelope of rupture* in line with the r discussion of the loadings mentioned in Sections B-3 and B-4 of this chapter, however, it is possible to predict with reasonable accuracy 30 the behavior of rooks between the lim its of simple tension and simple compression* We have seen that, on the tension side of the Mohr Diagram when all the principal stresses are tensile, the criterion of failure is the simple tensile strength of the material* On the compression side of the diagram, failure obviously occurs on a plane where the tangential or shear stress becomes large enough to overcome not only the shear resistance of the material but also the normal compressive stress component on the plane* Take, for example, the simple com pression circle* When - 45°, the shear stress on this plane is greater than on any other plane through a point in the body, but the value of the normal component is also large* As the value of / ranges from 45°-90°, the ratio of the shear component to the normal component increases as tan /* At a certain critical value of the shear stress trill s till be sufficiently large to overcome both the normal component on the plane and the shear resistance of the material, so that the specimen w ill fail along this plane* It is possible, under the loading described in Section B-*4, to so adjust the confining pressure, p, that the stresses across planes PX (see Fig* 9c) will always be in tension but less than the simple tensile strength of the material* For various values of p, the specimens w ill break across planes PA, where the normal and shear components of the rupture plane may be determined from the Mohr Diagram* At a certain value of the confining pressure, p, the point A on the corresponding stress circle w ill coincide with the vertical shear axis as the specimen breaks along plane PA* The ordinate of A for this case w ill determine the strength of the material for "pure shear,” and thus should be the shear resistance of the material* To SI my knowledge, such experiments have never been performed on actual rock specimens, but the results from such tests should prove quite valuable* It is probable that the locus of points A in this segment of the envelope of rupture would be a parabola-like curve connecting the points of the rupture envelope for simple tension and simple compression. 2. The form of the envelope of rupture. Having formulated some general ideas concerning the envelope of rupture in Pig# 17b, the question of comparing its shape with same simple mathematical curve such as a parabola or hyperbola arises. Such a question is mainly academic, for too many variables enter into the strength of rocks both in nature and in laboratory procedures to allow formulation of a precise curve for the envelope. Among the factors affecting the shape of the envelope are: ( 1 ) heterogeneity and.polyorystallinity of rocksj ( 2 ) porosity and moisture (or fluid) content of rocks ; and (3) compress ibility of rocks. The effect of each of these factors will be dis cussed in the following paragraphs# a* Heterogeneity and polyorystallinity of rocks. The non-uniform structure of most rocks and the presence ft of planes or directions of weakness in the crystals asking up the rocks and in the rock mass itself—due to foliation, jointing, rift and grain, etc.,—make anything like closely correlating results in a series of strength tests on rocks quite difficult to achieve* The disparity of test results can be attributed to variances of the rock specimens from one to the other as well as to errors introduced in testing technique. For this reason, it is just as illogical to combine the tests carried out on several specimens of granite from the same rock mass if the * * o o Ci Bft) V a© CO o o c*J «H o 43 >-o fefl « S-.ft) 43 CQ o 43 •r-5 CO eg r-f O u © o *•« pu, a T3 ft) ts oO £ 0 u + 3 to •H *d) mO o +5 O 2 lei I— test results differ appreciably as it is to combine the test results for a specimen of limestone and another of sandstone drawn from the same locality* The best results to be hoped for is an "average” value of strengths which may be used only as a guide for one particular rock mass* b* porosity and moisture (or fluid) content of rocks* It may be surmised from the discussion of the stress components along the planes of failure of rocks in various compression tests that, if moisture or fluids were present in the pore spaces of the rocks, the hydrostatic pressure of this fluid would serve to counter act or nullify the normal compressive stress components on the planes* In such a case, the impressed shear strength would not have to reach as high a value in order to overcome the cohesive effect of the remaining normal stress and the shear strength of the material. If the hydro static pressure of the pore fluids is sufficient to nullify completely the normal stress on the plane, then only the shear resistance of the rock would have to be overcome by the tangential component to produce failure* With the presence of moisture and its "lubricating” qualities, however, even the shear resistance of the rock my be redueed appre ciably, particularly in relatively unconsolidated shales# Thus, with the addition of moisture to the pore spaces, we find that the Mohr Stress Circles deviate from the original series for a dry specimen. Pig. 18 illustrates specimens of a porous, dry rock subjected to a series of .triaxial tests* The specimens are unjacketed, so that the confining fluid is free to enter the pores of the rock* As the confining pressure increases, more liquid enters the pores of the rock, thereby affecting the rock strength as explained above* We would not bo justified in drawing an envelope to this series of circles as we would for the dry speoimens because the circles shewn are actually equivalent to stress circles at different locations on the "dry *1 envelope. By transferring the stress circles for any of the moisture circles to the dry envelope we may determine from the diagram the extent to which the normal stress component has been counteracted by the hydrostatic pore pressure (see circles C and C* of Pig. 18). c. Compressibility of Rooks. The compressibility factor, K, for a material under a hydrostatic confining pressure according to the definition given in Chapter 1 is K * 3(l-2m) B Both m (poisson*s ratio) and E (Young*s Modulus) are assumed to be constants denoting a particular elastic property of a material* j / Terzaghi states that for dense soils and solid, granular materials Tersaghi, K., “Theoretical Soil Mechanics," pp. 369-370, J. ?/iley & Sons, 1947* such as sandstone the value of m ranges from about 0 .2 for low pressures to more than 0.5 for very high stresses* The value of m - 0.5 would be the theoretical limit for a state of incoispressibility to occur. Actually, rocks do approach such an incompressible state. It would seem, therefore, that this factor of incompressibility should give rise to a somewhat hydrostatic behavior of rocks under high confining pressures, thus reducing the value of the shearing resistance of the material* The resultant effect on the envelope of elastic failure 34 would be to curve it slightly away from a true straight line and toward the horizontal axis* Experimental data is yet insufficient to corroborate such a conjecture* 3* The Influence of the Intermediate Principal Stress* In postulating Mohr’s modification of the Stress-Difference Theory, it was tacitly assumed that the intermediate principal stress bore no influence on the failure of materials along planes parallel to this stress under the stress difference p^- p^. This assumption is justi fied because an examination of Fig* 15 w ill reveal that the greatest value of a shear stress associated with any given normal component will always be found on the XZ circle (i.e. on planes parallel to the direction of the intermediate principal stress)* In the case of simple tension, simple compression, and even in triaxial compression tests, Pg — Pg, and failure may occur on any plane properly inclined to p^* Hence we may observe the conical fracture shape typical of many rock specimens tested in this manner* 4* Summary o f Methods of F a ilu re of Rooks. The b r i t t l e n a tu re of rocks makssthem favorably inclined to break under a tensile stress. Rocks will fail whenever the strength of the rock in tension is exceeded, this strength being determined from a simple tension test. Rocks will also fail under other conditions of loading whenever the shearing stress on any plane in the rock is sufficient to overcome the shearing resistance of the material and the normal stress component on the plane. If the normal stress component is tensile, the required shearing stress on the plane is less than that required if the normal stress v/ere compressive. LIBRARY C6 M ADO SCHOOL OF MINES Two other types of failure may be produced .4 s during triaxial or other compression tests* The first type of fracture, called V an "extension” fracture, forms parallel to the direction of the Bridgman, P. W., "Reflections on Rupture”, Journal of Applied Physics, vol. 9, pp. 517-528, 1938. maximum principal stress while it is in compression. These fractures are said to be secondary, after the rock starts to fail along the stress-diffsrence planes. Small wedges formed by the intersection of these planes act to split the rock in tension in much the same manner as an axe splitting wood. The second type of failure, called "release” i f fractures, occur parallel to the maximum principal stress planes 17------:------Griggs, D. T., "Deformation of Rocks Under Eigh Confining Pressures”, Journal of Geology, vol. 44, pp. 541-577, 1936. after a specimen is removed from a triaxial compression test under high confining pressure. These fractures may also be regarded as induoed tension fractures brought about by the relatively rapid release of a compressive load. 36 Chapter 4 The Physical Behavior of Rock Masses Under Statio and Dynamic Loadings The m ilitary engineer and the tunneling or mining engineer are interested in selecting advantageous sites for the location of under ground openings. Mining or tunneling operations may be conducted in any type of ground, but mine development work and tunneling are confined wherever possible to ground which stands well with a minimum of support. The m ilitary engineer has a somewhat wider latitude in the selection of sites for new excavations, yet it may become necessary to utilize abandoned stopes, development d rifts, or tunnels to house underground installations. In addition, the military engineer is confronted with the problem of designing an underground opening which is able to with stand the effects of tremendous dynamic loads such as bomb blasts which may be imposed on h is rock f o r t r e s s . The th eo ry o f e l a s t i c i t y may be applied by the military engineer in seeking practical answers to his design problems, but such theory, based as it is on certain radically simplifying assumptions, is not always reliable. Elastic action is usually premised upon the assumption that the material under investiga tion is isotropic and homogeneous. Moreover, in order to solve most elasticity problems it is necessary to assume that all the elastic constants of the material actually are constants, and that there is a definite relation between the stresses acting on the material and the corresponding defoliations. It has already been pointed out that rocks are not homogeneous. However, if the body of rock under consideration is large in relation to its texture and structure, the matter of figure 19a. Radial Stress Contour Pattern for Circular Tunnel, Pi = 15 P3 : 5 Contour Interval : 1 Figure 19b. Tangential Stress Contour Pattern for Circular Tunnel. Px = 15 Pg = 5 Contour Interval i 1 (Contour lin es less than fives shown only at top) Figure 19c* Shear Stress Contour Pattern for Circular Rock Tunnel • p^ - 15 p 3 = 5 Contour In terval: 1 (Shear s tr e s s e s shown occur on r a d ia l and ta n g e n tia l planes) 37 inhomogeneity decreases in importance* The actual variations of the elastic "constants , n suoh as E and m, under different environments has also been mentioned previously* Lastly, the forces with which the m ilitary engineer is concerned frequently are of such magnitude that elastic action of a rock mass may not exist.* From the foregoing considerations it seems best to confine atten tion mainly to a qualitative discussion of the character of the loadings to be expected upon underground openings from the static forces exist ing in the rock mass and from dynamic loads which may be impressed upon the rock formation* At the conclusion of this chapter, a brief discussion of the principles of similitude as applied to the m ilitary design of underground openings w ill be made* A* Static Stresses Around a Tunnel Opening. Figure 19 illustrates a cross section taken of an unlined hori zontal drift of considerable length in comparison to its cross-sectional dimensions* For simplification of calculations, the shape of the open ing is taken as a perfect circle, and the irregularities due to the rook outline have been disregarded* In order to further simplify the consideration of the character of the stresses surrounding the opening, the f oil owing additional assumptions are also made: 1* The rock mass behaves in a truly elastic manner* 2* The vertical and horizontal earth pressures produced by the weight of the rock mass and the weight of the overburden are constant throughout the region of the tunnel site prior to the excavation of the tu n n el* 3* That plane transverse sections are plane and parallel before and after excavation* 38 4. That the cross-section under consideration is sufficiently far removed from the ends of the drift to be free of the influence of the end boundary conditions. If we consider a small unit block of rock material located in the vicinity of the tunnel site before excavation, the following pressures may be computed to be acting upon the block. First, a pressure, p^, (which is the maximum principal stress for a point in the block) equal to the weight of the overlying column of rock resting upon the block, acts in a vertical direotion. A lateral pressure, p , exists on the O sides of the block. A consideration of the elastic deformation of the unit block under the influence of p and p leads to a value of p equal 1 5 3 to pj,. For, if we consider the deformations due to the separate pressures we find that: The deformations due to p^ are: ©1 Z 1 p (in the vertical direction). ¥ 1 02 " e3 ~ **E two horizontal directions). k* The deform ations due to p (s p ) a re : ______~ 2 ~3 e 2 = 1 P , E V e3 = - £ P3 The deform ations due to a re : e3 = 1 E e = e = - » p E The net strain, € in the horizontal direction of p^ is equal to the sum of the individually produced deformations, ©g, above. Also, by logical argument, we may infer that, at the interior of a large body such as a section of the earth, the net strain €g is equal to zero, for adjacent blocks to the particular one under consideration cannot be displaced to accomodate any lateral expansion; hence, they resist this expansion by a pressure of pg. Thus; 6 3 1 “ E Pi + i Ps * Ps - 0 . E E E Solving for pg we obtain: P3 = » P i (1 ) how th e s p e c ific change in volume, AY z g. 0f the unit block under the influence of the principal stresses is equal to the sum of the specific strains in the directions of the principal stresses. The specific strains, €g and are both>zero, however, leaving the value of the specific volume change equal to Solving for ^ in the above expressions we obtain; € » d * 1 pi * 2m pg 1 E E or e = 1 ( 1 - &a2 )p ( 2 ) ¥ The factor modifying p^ in expression (2) may be called the co efficient of compressibility for the rock. Thus the zone of rock around the proposed excavation is under an initial compression, the amount of which may be computed from equation (2). If the circular excavation is made in the rock, the rock surrounding the opening is then free to expand or squeeze into the opening. In so doing, the original stresses 40 present in the unexcavated ground are readjusted until a state of equilibrium is again reached with regard to the stresses impressed upon any portion of the unexcavated rock* If the amount of "squeezing in,” together with the cohesive strength of the rock, is insufficient to restore such an equilibrium, then the opening will collapse* On the other hand, if the amount of initial compression is great enough to build up equilibrium stresses which approach or exceed the strength of the rock, then "spalling," scabbing, or even rock bursts are likely to occur* The problem of determining the value of the stresses around a tunnel opening necessary to restore equilibrium is a diffioult one to solve analytically* With irregularly shaped openings the complexity of the problem increases greatly* In order to illustrate the nature of the stress distribution, a simple, circular opening has been selected. A simplified solution, based on the assumptions we have already made, V has been proposed by F. W. Hanna* A small section of the tunnel Hanna, F. W., "Stresses in Circular Holes in Dams", Trans* Amer* Soc* of Civil Engrs*, vol* 103, pp* 165-170, 1938* was taken to behave as & plate with a hole in it. In addition, the diameter of the opening was small in comparison with the depth of the tunnel below the surface. Moreover, the radius, r , of the opening was considered as zero in comparison to appreciable distances, r, from \ the center of the drift. Hanna,s Solution* Designate radial stresses (i.e. normal stresses on radial planes) by sr * Denote tangential stresses, or normal stresses on planes perpendicular to the radii of the opening, 41 by s^. The shearing stresses on each of these planes will be sg. Before the excavation the stress condition on any such planes through a point in the rock body may be determined by means of Mohr*s Diagram (see F ig . 6 ) from the expressionss sr z iKPq +- Pg) - 4 sq s h($i +■ P3 ) + i(Pi “ P3 ) 008 (tangential stress)(3)b and ss s - 4(p*i “ P*) sin 2© (shear stress on radial (3)o 0 or tangential planes) The angle © is the angle (measured counter-clockwise) which the radius to the point under consideration makes with the'plane of the maximum principal stress. The expressions in (3) may be considered as consisting of two componentsj ( 1 ) the constant components, i(Px + P 3 )f ( 2 ) th e com ponents involving functions of ©• The effect of the constant components in restoring equilibrium around the opening may be considered and then the effect of the components involving functions of © my be determined. Finally, the separate effects may be added together to determine the new values of sr , s^, and sg (after excavation). For the constant components of (3) we have an anaiagous condition to the problem of a thick, hollow cylinder subject to an external pressure. The partial values for the stresses sr, 8q, and sg due to the constant components are given bys 2 8* = i(Px+ P 3 X 1 - I ? ) =5 = Mpx + P3 H 1+_ 2) (4)b r «* = 0 (4)0 42 The equilibrium conditions dependent upon the functions involving Q can be satisfied by'means of an Airy Stress Function, F * (Cxr2 + + £3 + ^ 4) eos2©« The solution for these partial r 2 values of sp, se, and sg gives: 4 2 33r r »j;jr-4 r ri 0 _ 0 a" = -ICPj-PjXl + _J* - _ ^ ° ) 00s 2© (5 )a T “TT r r „ 4 3 r 0 M *V<% 0 (5)b _ ° ) 4 r 4 3 r 2r 2 0 _ (5)o 2 r r Character of Stress Distribution# Expressions (4) and (5) may be used to determine the radial, tangential, and shear stresses for various values of r (distance from center of the tunnel) and d# Results may be plotted using contour lines for equal stress values as shown in Figs# 19a, b, and c# In these sketches an index value of p^was selected as 15, while m was taken as 0#25 (typical of rocks), giving a value of 5 for the stress p • 3 The in fo rm atio n su p p lie d by Figs# 19a, b, and c is not sufficient, however, to determine the significance of the stress distribution around the tunnel# In accordance with the theory of failure of rocks postulated in Section C-4 of Chapter 3, it is necessary to determine the attitudes of the shear or tensile planes of failure with respect to the principal stress planes in order to locate the directions of potential weakness around the tunnel# The determination of the direc tions of the principal planes of stress can be made most conveniently /ORIGINAL EAR.TH S T « .C S ^ E \S PoTCNfiAL TeW4‘ \ poycH X iA u s h e a r , o r . CAJENSiOHi fRACfUR.ES Figure 19d. Pattern of Shear Weakness Planes Surrounding a Circular Tunnel Opening. Px = 15 Original stresses ps = 5 43 lay employing Mohr’s Stress Diagram to a sufficient number of points around the tunnel# From the principal stress planes, the shear or tensile planes of failure may be located. Fig. 19d is a sketch showing the attitudes of the shear planes of weakness around the top half of the tunnel* The shear planes ware taken at an angle of approximately 68 ° from the maximum principal stress plane. Such an angle may be assumed as being fairly representative for most rocks. Several points of interest should be noted in connection with Fig. 19d. Point A is a point of hydrostatic pressure (normal stress only in all directions). For the particular value of p^ selected, point B is a point of no stress whatsoever. For a smaller value of p,,, point B would mark a small zone of tensile stress, while a larger value of p would give rise to a compressive stress at point B. The region 3 from point B to point A is one of small compressive and shear stresses, which indicates that this region could easily be converted to one of tensile stress under suitable loading* The general inclination of the shear planes emanating from point A probably suggests the principle of the “natural arch” long recognized in mining practice. This arch tends to form over an underground opening during successive stages of subsi dence. If the zone between B and A were under tension or under slight compression, then the arch-shaped area below point A could be extracted without affecting markedly the stress distribution for other points surrounding the tunnel. The small effect which does occur on the stress distribution at other points progressively moves Point A vertically upward, thus enlarging the natural arch. The natural arch, then, is a tendency of the rook mass to distribute the static stresses around the opening. i TABUt I I Values of the Principal Stresses and of the Shear and Normal Components on the Planes of Weakness for Various Points around a Circular Tunnel Opening. r 0° 30° 45° 60° 90° p „ 0 0 0 0 0 5 p? 40 30 20 10 0 s£ 6.0 4.3 2.7 1.3 0 s“0 14.3 10.4 7.0 3.4 0 p3 4 3.0 1.6 1.0 1.2 6 p* 28 23.6 17.2 12.5 4.5 s„ 7.7 6.0 3.7 2.8 1.8 s“ 8.7 7.2 5.4 4.2 1.2 3 p3 5.6 3.0 1.4 1.0 3.0 7 pf 24 20.9 17.8 14.0 6.0 SR 8.5 4.8 3.6 2.8 3.5 s“3 6.6 5.4 5.8 4.5 1.2 p3 6.4 3.5 2.0 2.0 5.5 8 p-, 20 19.5 18.0 14.7 6.6 * sR 8.5 5.8 4.5 3.6 5.7 s 3 4.8 5.5 5.7 4.1 0.3 p3 6.6 4.0 2.3 2.8 6.5 9 p, 19 18.3 17.3 14.8 7.2 SR 8.5 6.0 4.3 4.4 6.6 as 4.5 5.0 5.0 4.0 0*4 p3 6.6 4.0 3.0 3.2 6.4 10 Pi 19 17.6 16.7 14.6 8.3 s n 8.5 6.0 4.9 4.6 6.7 a 3 4.5 5.0 4.8 4.2 ,0.8 p3 6.0 4.0 4.0 4.6 5.7 15 Pt 16.4 16.4 15.7 14.5 11.5 S„ 7.5 5.8 5.5 6.1 6.7 s 3 3.7 4.3 4.0 2.5 2.0 * Hydrostatic Pressure * 6.6 when r * 8 . 6 , 9 = 90° Table I I shows th e v alu es o f th e maximum and minimum p rin c ip a l stresses and the values of the normal stresses (s ) and shear stresses n (ss) on the shear planes of weakness for various values of 9• I t should be noted that at point C in Fig# 19d, the value of the maximum principal stress is 40 in a vertical direction (as compared to the value of p-^ before excavation - 15) while the minimum principal stress is zero. This point is therefore on© of potential danger for shear fractures or for extension fractures resulting from compressive over loads on the rock* B# Dynamic Loadings 1* General considerations* Several conditions of so-called ”static loading” of rocks have been considered in the previous dis cussions, In each case it was tacitly or expressly assumed that the resultant of the forces acting upon any portion of the body, regardless of the size of the portion selected, would always be equal to zero* For example, in the discussion of Mohr’s Stress Diagram, the elemental volumes considered were stated to be in a state of static equilibrium, with the resultant of the inertial forces, gravity forces, and external ly applied forces equal to zero* Again, in the case of the stress distribution around a tunnel opening, it was stated that static equi librium existed both before and after the excavation was made* If the resultant of the body forces is not zero but has seme finite magnitude and direction, a state of static equilibrium nolonger exists* The significance of this lack of equilibrium as applied to an elastic body such as a rock mass is the existence of a force tending to 45 produce a displacement in space of either the rock mass as a whole or of some portion of the rock mass. Moreover, in accordance with the fundamental theorem of mechanics, the unbalanced force is equal to the product of the mass of the body considered and the acceleration of the mass. Inasmuch as the mass of the body is simply a measure of the inertia of the body, the greater the mass of the body the less the acceleration produced by a given force. For example, a very large por tion of the earth surrounding the foeus of an impulse may, by virtue of its great inertia, resist the impulse to an extent that no displacement of the whole body v/ill take place. On the other hand, the rock mass is an elastic body, therefore the force of this same impulse will cause a certain amount of elastic deformation, first of the rock in the immed iate vieinity of the impulsive force and later at points in the rock at considerable distances from the focus of the impulse. It will be shown that such elastic deformations are propagated through the medium, whereas the rock mass as a whole does not move* These imbalanced forces which may act on a rock mass or a portion thereof may be termed dynamic loads, for they tend to produce accelera tion of a mass. Since inertial and gravitational influences eventually counteract the effects of dynamic loadings, forces of a dynamic charac ter imply a time of application of the unbalanced force. The effect of dynamic loads when acting on a rock mass to produce elastic deformations v/ithin the rock will be considered initially in the following discussion* Some aspects of dynamic loadings on rock which produce rupture of the rock will then be discussed. 2# State of Stress on a Body. Fig. 20 illustrates a small, elementary free body taken from the interior of an infinite, homogeneous, 6z Y- PLANE Figure 20 b Figure 21 46 e l a s t i c , ro ck medium* The c o o rd in ate axes* X, Y, and Z, a re tak en fo r convenience parallel to the edges of the parallelopiped whose dimen sions are Ax, Ay, and Az* The components of the resultant stresses on the faces of the "body may he resolved in any desired direction* For convenience we sha 11^ resolve the stress components parallel to the directions of the coordinate axes* Further, it is convenient to desig'r nate the stress components by a capital letter to denote the axis to which they may he parallel, with a subscript to indicate the particular plane on which a stress component may he acting. The plane, or face of th e the parallelopiped, is in turn designated by/axis to which it is perpen dicular. Thus, Xy describes the stress parallel to the X-axis and acting on the Y-plane, this plane being perpendicular to the Y-axis« In a general case, there will be nine stress components acting on the three front faces of the parallelopiped as shown in Fig* 20#- From a consideration of Fig. 13 it is evident that *y = Yz 2y = Tz (6) " Zx On the three faces not shown in Fig. 20, the stresses w ill not be the same as those on the visible faces. Take for example the hidden t face parallel to the X-plane face. The stress components on this face w ill consist of the three components Xx, Yx , and Zx (reversed in sense) plus tine rate of change of each of these components with respect to the X-direction multiplied by the distance Ax over which the change occurs. That is, the stresses on the hidden X-face are Xx + A x 3 x 47 kx zx + t!i!L • 3 X The element of volume* A V, considered possesses a quantity of mass# I f f> is the density of the body and u is the elastic displace ment of the body in the X-diraction produced by the unbalanced force acting on the body in the X-direotion, then it can be shown that 33^ ^Xy ZXz AY l^ x 3y In the above expression, R^, denotes the component of the body force acting in the X-direction# If the volume element is taken very 2 s m a ll,th e body fo rces may b e n eglected# The ex p ressio n p-A Vd u is the" expression for the unbalanced force on the body acting in the X-direction. Similarly expressions may be derived for the elastic displacements, v and w, in the T and Z directions* We then have the following expressions to represent the state of stress on the body: ?>Xy 2 + _ 1 + s p i j g d x 3y 3>z S t a Tx 3Yy 2 *T~" + T~~ + *r— */°“ 2 (7) ox dy oz 3t d 2X d zv 3 Z 2 + ------+ V — 2 d x & y 3 z 3t 3# Relations Between Stresses and Strains# If the strains in an elastic body remain small enough to permit the body to behave elas tically, it can be shown that the specific change in volume, Q$ o f the body brought about -through the aotion of normal stresses is equal to the sum of the specific strains in the directions of these stresses* The action of the shearing stresses on the body is such as to produce a change in the shape of the body but not in the volume of the body* If the strains in the x, y, and z directions are denoted by d u , dvy g>x dy d w respectively, and if the shearing strains are: 3 s d u , 3 u , 3v, 0 V, dw, d w ~0 y “^2 0x 0z “dx “T y j / Then it can be showxr by use of the elastic constants of the rock, that V Southwell, R* V*, "Theory of Elasticity", Oxford, 1936* Z A • 0 4 -2 G . 3 u a x Y s A S + 2G • J3_v 0 y 2 = A-9 + 2G . 0w * T T and xz(= zx ) = G(5u a®_ ) V T T + V V - + ) The values of A and G are given in the definitions of Chapter 1* 4* Propagation of Elastic Deformations Through A Rock Mass* An examination of equations (7) w ill show that the right hand side of each expression contains the accelerations of the displacements—u, v, and w—- in the x, y, and z directions brought about by the unbalanced force 49 acting on the small body of rook shown in Fig# 20# By substitution of equations ( 8) and (9) in equation (7) we obtain the relation of the unbalanced foroe to the respective strains, thus Y Heiland, C# A#, “Seismic Prospecting", Colorado School of Mines, 1939. 2 2 p d ti Z (A + G ). d Q + G v u “FI 2 a x 0 3 2v . ( A + G). d0 + G V2v (10) 1 “T— 2 "T— d t 5 y ^ a 2w - (A t G). c>© 4- G v 2w " s i 2 (where y represents d , d i d ) T 2 a *2 For simplicity of illustration we may consider that the unbalanced force on the body produces initially a deformation only in the In direction# This deformation, u, produces the only volume change, since v and w are zero# Thus, Q a a us <3>9 s d2\Xj d Q s 0; d_Q s 0 , dx dx d x^ dy dz Further, y2u s 3 2u ; y ^ Z d ^v; y^w z 'S x2 0 j2 ^ j2 Substituting the above values in equations (10) we have 3 2a = ( A -I- 2G) a 2u ( l l ) a ^ d t 2 3 x 2 a2* = eliz_ (u)b at 2 dx2 2 2 0 w = G 3 w ^ 1 7 T I P ( u ) o Although it was stated above that the unbalanced fore© produces only a deformation u, it can be demonstrated that equations ( 1 1 ) can be made to apply in the case of deformation in all three directions. To fully investigate the significance of equations (11) we con sider Fig. 21a, which shows'the instantaneous state of deformation on the parallelepiped of Fig. 20. The deformations are exaggerated in the sketch to show the effect of the unbalanced force or stress when it is first applied to the small body inside the larger elastic rock medium. Point A. represents the state of rest of one corner of the 2 left face of the parallelepiped before the dynamic stress, S * p d u , “Ft2 has been impressed upon the element. Bhen the load is applied, the left face is compressed against neighboring planes to the right, giving rise to a reactive stress which w ill obviously, after an extremely small lapse of time, equal the load first applied to the left face. In the meantime, the body at the left face is storing an amount of potential elastic strain energy in the In direction. After a lapse of the same interval of time necessary to cause a reactive stress upon a neighboring plane to the right of the left face, this portion of the body will also store potential elastic strain energy, for these planes will also be displaced in the X- direetion an amount equal to u. Thus the action of the small rock body in resisting the impressed dynamic load upon it can be summarized by stating that the impressed load causes a local elastic strain, the amount of which is determined by the quantity of resistive elastic strain energy which the body is able to build up at that locality. The building up of the potential strain energy, however, gives rise to the same amount of reactive stress on neighboring portions of the body as the original impulse. The reactive stress is then capable of producing the same amount of elastic strain as the original deformation. The impressed load and impressed energy brought on by the original impulse can then be said to be propagated or reproduced through the elastic medium by means of the elastic strain and strain energy of the medium. Equation (ll)a indicates that the magnitude of the elastic stress acting on any part of the rock body is proportional to tie rate of change of the strain at that particular point. The elastic stress varies in magnitude and direction from S (the originally applied dynamic stress) to 0 and to -S, depending upon 1 the slope of the strain curve at the point and instant under consideration. It is seen by an examination of Pigs. 20a and b that the initial compression caused by the dynamic load itself moves along in the in direction to the right under the influence of the propagated elastic force and propagated strain until, after a small amount of time has elapsed, the compressed portion of the rock may be at point B. Equation (ll)a furnishes the expression for determining the amount of time required for this compression to cover the distance from A to B. n For, if we cancel the partials, c> u, from either side of the equation and transpose d 7? and p , we have ( 12) Equation (12) is then the velocity of propagation of the compress ive strain in the X-direction. It is constant as long as the elastic constants of the rock shown under the radical themselves remain constant. A further examination of Figs. 21a and b show that when the plane containing point A is moved to a position A*, that portion of the body has stored up potential strain energy. If, at this point, the initial ly unbalanced impressed stress, S, were released suddenly, then the potential strain energy is converted into kinetic energy of movement toward point A”. When the plane containing A reaches its former state of rest, A, all of the potential strain energy is converted into kinetic energy. Eventually, this kinetic energy will be re-converted into potential strain energy at point A” and the plane will begin to move in the opposite direction back to A*. This type of motion of the plane A may be recognized as simple harmonic motion. If the portion of the rock body near the plane containing point A. is viewed as a unit, it will be seen that it undergoes a maximum compression at point A*, with regard to the x-direction, and a maximum extension in the x-direction at point A”. On the other hand, with regard to the y and z directions, this same portion of the body undergoes a maximum enlargement at the dis placement corresponding to A* and a minimum dimensioning at point A”. It will be seen that the left face has suffered an enlargement in both the y and z directions in relation to the neighboring planes of the body. This enlargement of the left face under the action of the force S reaches a maximum value when the displacement, u, has occurred. The potential strain energy in the y and z directions is stored in the portion of the body as shear strain energy, inasmuch as in these two directions the left face has been influenced to slide past its neighboring planes. Expressions lib and c show the relation of the induced elastic forces in the y and z directions tending to produce rotation of the body around axes parallel to the y and z directions. This rotative force is eventually neutralized by the reactive shear strain energy, which energy in turn is capable of performing elastic displacement on the material of the body to the right of the left face. X 4 *W if£9 _CQ p = < § © © TO O £ P Q 2 CJ •H © P cn i > <3 TO TO p r-4 P OTO *HTO •rl P P TO tH ea 53 Thus, the shear deformation causing the "hump** shown in Figs. 21(a) and (b) is also "propagated” elastically through the body. Equations lib and c indicate the oonstant velocity of propagation of these shear deformations as d x (13) It should be remembered that the shear displacements, v and w take place in a direction perpendicular to the x-axis, but the actual propagation of these deformations is in the direction of the x-axis. A comparison of (12) and (13) w ill show that the shear (or transverse) deformations in the case of rocks are propagated at a lower velocity than is the compression deformation. At distant points, therefore, the compression or longitudinal deformations w ill arrive first* e 5# Repeated Deformations. Mention has been made of the fact that the plane containing point A on the left face is free to move back to a position A” from A* after release of the dynamic stress. At point A” this small portion of the body is capable of producing the elastic stress, S, on neighboring portions by virtue of the fact that it has stored the equivalent potential strain energy at point A”. Thus, the continual displacement of this small portion-of the body back and forth in harmonic motion is sufficient to propagate further elastic deforma tions, u, in both compression and dilatation, and shear displacement, v, and w, in the x-direction. At a distant point in the interior of the elastic rock body we may observe an instantaneous "snapshot" of a succession of deformations on a partially cut-away portion taken at the point as shown in Fig* 22. Points A in Fig. 22 show the compressive deformations, points B are the rarefied portions of the rock. Chi the other hand, points C represent the shear deformations in the y and z directions propagated as a result of the original compressive deformations, while points D represent the shear deformations caused by the original dilatation deformations. Bote that the propagated shear deformations have lagged behind the compression and rarefaction deformations due to the slower velocity of the former. 6 . Wave Motions. The elastic deformations whose character istics have been discussed in the preceding paragraphs belong to the type of energy propagation known as "wave motion.” A rock mass will exhibit the two types of wave motion—longitudinal and transverse— whenever it receives an impulse as described from some source within the rock mass sufficiently removed from boundaries such as the earth*s surface or from such planes of discontinuity as bedding planes or different rock formations. The longitudinal (compression and rare faction) deformations are known as Primary or P waves inasmuch as they will arrive at a given point ahead of the shear, or Secondary, S v/aves. The velocity of the P waves given in (12) converted to other units is Vp “ -x /Bg .(!-») (W) and the velocity of the shear waves is T* = V s r & v ’ V - J 2- (15i Assuming an average value for m for rocks of .25, the velocity of the P and S waves become Vp : -\/l.20M (I 6 )a Vs Z -y^O.40 (16)1)' a. Other types of waves. The P and S waves are waves propagated through the interior of an elastic medium. Different types of waves may be propagated along 54 a boundary of discontinuity. The two types of discontinuity waves (frequently known as "surface waves") of special interest are the Kayleigh waves and the Love waves• The Rayleigh, or R wave, is a combined longitudinal and transverse wave wherein the particles oscil late in an elliptical orbit with the plane of oscillation perpendicular to the discontinuity plane# The longer axis of the ellipse is normal to the boundary# The Love, or Q wave, is a type of transverse wave with the plane of oscillation parallel to the discontinuity plane# Along the discontinuity, the surface waves travel with a velocity of propagation slightly smaller than that of the S waves# The elastic deformations produced by these waves and the amount of these deforma tions decreases with the distance from the boundary surface# The general classification of the surface waves falls under the heading of Main (M) or Longae (L) waves in earthquake records# b# Other Properties of Yfove Motion# In addition to the characteristics of wave motion already discussed, some other fundamental properties useful in the study of this type of energy propagation w ill be summarised below# D e fin itio n s — The p e rio d 0 T, of vibration in wave motion is the time required for a particular particle to complete on© cycle of vibra tion about its state of rest# The frequency, f, of vibration is the reciprocal of the period, or the number of vibration oycles completed, per second# The amplitude, a, of vibration is the maximum displacement of a particle from its state of rest (i#e# the displacement, u, in Fig# 21)# The phase of the vibration of the particle is Its instantaneous disposition in its vibratory path with respect to a particular time in the period# If two separate, vibrating particles oontlnually pass through corresponding points in their paths at the same points of time 55 they are said to be in phase* Otherwise these two particles are out of phase* If they reach maximum displacements in opposite directions in their paths at the same point of time they are in phase opposition* The wave length, A , is the distance in the direction of propagation of the wave motion between two points which are in phase on adjacent waves. In Fig. 22, the distance from one point A to the next point A, measured from left to right, is a . o. Velocity, Wave Length, and Frequency. It was shown in the discussion of the propagation of elastic deformations (Fig. 22a), that in the period, T, required for the left face to complete one cycle from the extreme forward position, u, back to the same position, the deformation or wave front had advanced a distance equal to the wave length, A * Therefore, the velocity of propagation, V, of a wave is V - A or V - A * f # T But the velocity of propagation of waves was shown to be a constant, therefore the product Af must also be constant. We should expect that in the same medium if the wave length were increased, then the frequency w ill be decreased, and vice versa* d. Huygen,s Principle. This principle is a further extension of the idea of pro pagation of elastic deformations. Each point in a wave front advancing through a rock mass is capable of propagating elastic distrubances of its own in the form .of little wavelets, the collective effect of each wavelet producing a new wave front. Another principle of wave motion states that two waves moving simultaneously through the same region w ill advance independently. 56 each producing the same elastic disturbance as if it were acting alone* The collective effect of both waves at a particular instant may be found by combining algebraically, point by point, the elastic deforma tions produced by each wave* By making use of the two principles stated above, it is possible to demonstrate that when a wave front reaches a boundary or surface of discontinuity, part of the energy associated with the wave is reflected back from the surface, the angle of incidence of the wave front being equal to the angle of reflection* Also, it can be shown that, if the wave front strikes the boundary between two rock formations having different wave velocities and is refracted into the second formation, the sine of the angle of incidence is to the sine of the angle of refraction as the velocity of the wave in the first medium is to the v e lo c ity in th e second medium* The collective action of two waves of the same frequency, in phase with each other and advancing in the ssune direction, is to produce a reinforcement wave of double the amplitude of one of the waves* Several such waves w ill produce amplitudes which are a product of the individual wave amplitudes and the number of individual waves. Two waves of the same frequency moving in the same direction, in phase opposition, produce destructive interference, and, if the amplitudes of the two waves are the same, the result is a complete annulment of the waves* e. Energy of Waves and Attenuation of Waves* By analysis of the kinetic energy possessed by a small mass as it sweeps through its state of rest during its vibratory motion, it can be shown that for a unit area of the wave front the wave intensity, I, or the amount of energy passing through a unit of area at the wave front is I s 2 t t 2Y/3 f 2 a2 (17) The elastic vibrations associated with energy propagation do not go on free from dampening or diminishing influences* The elastic sirain- energy is dissipated and the deformations are eventually stilled due to a number of causes. Some of the factors affecting the dissipation or spreading of the strain-cnergy at a distance from the source may be listed briefly as follows: (1) Geometric Divergence* The waves travel out from a source on a spherical front, therefore the energy per unit area at two points at different distances from the source will be inversely pro portional to the squares of the radii from the source to the two points. Absorption or Damping. Rapid vibration of particles produces heat with consequent loss of energy. Expression (17) shows that the intensity of energy is proportional to the squares of the frequency and amplitude of the vibration, therefore the greater the intensity of energy, the greater is the likelihood of loss of that energy by heat dissipation. Thus, low amplitude, low frequency waves will survive at a distance from the source in preference to high fre quency, high amplitude ones. It is also recognized that certain vis cous and plastic resistance to vibration is present in all types of rock, which tends to dampen the vibrations as the wave moves through the rock. It should be surmised that the less truly elastic a rock behaves, the greater is the damping effect in the rock# f • Transfer of Energy Between Two Different Rock Formations. Experience has shown that the transfer of energy from one rock formation to another is very poor if the factor, V/>, in equation j / (17) is very different in value for the two formations. Heiland gives Holland, C. A., op. oit., p. 104. 58 the following index values for the factor V/* for various type rooks: Hook Formation Y/3 (index Value) Basement Rocks 176 B a sa lt 150 Cap Rock 131 Limestone 108 Rock S a lt 100 Sandstone 95-63 Glacial Strata, wet 37 Glacial Strata, dry 20 Top S o il 11-4 The transfer of elastic energy, as my he deduced from, the above table, from a mantis of top soil to a hard bed-rock beneath w ill be very small j most of the energy is reflected back through the soil, g. How Much Energy Can Be Propagated Elastically? The theoretical derivations for the propagation of elastic strain assumed that the rock material behaved in a truly elastic manner under a given stress and that there were no other influences on the elastic properties such as hysteresis of strain or plasticity of certain constituents of th9 rock. Elastic propagation of energy implies, therefore, that only elastic strain-energy may be propagated through the rock medium. Inasmuch as rocks are inherently weak in tension, consequently the modulus of resilenoe of the rock in tension is rela tively small, and the rocks cannot propagate strain deformations which exceed the tensile elastic lim it. 7. Dynamic Loadings Involving Rupture. In the preceding para graphs, dynamic loadings were investigated from the standpoint of 59 elastic deformations of a small portion of a rock mass. If dynamic loadings are sufficiently great, part of the rock mass may fail by rupture* Instead of producing an elastic strain of a portion of a rock mass, large dynamic loads are capable of overcoming the cohesive strength of portions of the rock mass and of imparting translatory accelerations to the separated pieces* An example of a dynamic load ing capable of producing rupture is a "crater blast." Some of the basic principles associated with crater blasting will be considered briefly at this point. Figure 23a shows an explosive charge to be set off in a hole in sound rock at a specified distance below a horizontal ground line or "free face." The explosive is capable of producing an enormous amount of pressure upon the surrounding rock within a very short period of time* The actual amount of pressure produced depends upon the type of explosive used and its quantity and quality. The degree of con finement of the charge also affects the amount of pressure generated. The rate at which the pressure is produced is dependent upon the velocity of detonation of the explosive. The explosive pressure exerted upon the confining rock ohamber creates an unbalanced dynamic force on the rock mass in the vicinity of the explosion. This force tends to produce an acceleration of the rook surrounding the hole. In order to accelerate any portion of the rock mass in a given direction, however, the foroe must be capable of overcoming the mass inertia of that portion as well as the cohesive force binding the rock to the neighboring mass. We may consider in Fig. 23a that the mass inertia of the rock around the hole in any direction from the hole is directly proportional to the distance from o a . Pattern of Shear Weakness Directions Around a Crater B la st. S/'A.ALL. AAtblO ^ CHATL4E. LAdQC CM*AC*E. b. Figure 23. Outlines of Typical Crater Cross-Sections in Bock. 60 the hole to the surface or free face* Theoretically, then, the radial distance, OA, from the hole to the free face is the direction of least inertia. The inertia of the rock increases for any other radial direc tion in proportion to the secant of the angle between OA and the direction under consideration until, at the horizontal and below the hole, the inertia is theoretically infinite. If a small charge is set off in the hole, the acceleration of the rock mass will tend to be the greatest in the direction of OA. More over, if we consider the force of the explosion as being exerted radially in all directions from the hole, the only cohesive resistance which the rock near the surface in the vicinity of point A is able to offer to prevent rupture is the tensile cohesive strength of the rock. To the left and right of A the rock is also resisting the applied force by tensile stress, but, inasmuch as the inertia is increasing as we move away from A and the radially directed force is acting at an ever decreasing angle to the surface, we may expect that at some point such as B, the rock will fail by an extension fracture or by a shear and tension fracture to the left of B but will be able to maintain its coherence to the right of this point, ^he only damage sustained by the rock as a result of this relatively small charge will be a compres sion and crushing of the wall rook around the original hole and a small crater near the surface around point A. If a larger charge had been used initially, a somewhat larger tension crater would be formed. With progressively larger charges the tension crater will be deepened and widened, but at a certain amount of charge, the dynamic force will be capable not only of over coming the tensile strength of the rock but will also be able to overcome the shear resistance of the rock nearer the center of the 61 hole. The manner in v/hioh the shear failure occurs is identical with the manner of failure of rock under compressive, axial and lateral loads as discussed in Chapter 3 and in Section A of Chapter 4* Along & circular arc at a given radius from the center of the hole, we may expect that the lateral pressure will he the least where the length of the tangent to the free surface is the least, for the amount of lateral pressure is dependent upon the inertia of the rock mass in the direction of the lateral pressure. In Fig. 23a the circle OA represents the locus of points of tangency for circles of different radii from 0 where the lateral pressure is the least. Shear planes of failure will be inclined at angles to the plane of maximum principal stress as deter mined from static rock strength tests. Figure 23a shows the planes of incipient shear failure. Figure 23b shows typical crater holes for various sized charges. C. The Concept of Similitude as an Aid to the Study of M ilitary Geology. Many of the problems encountered in the study of m ilitary geology are of the type which defy exact analytical solution. The character of stress distribution around an irregularly shaped tunnel opening, the size of crater produced by a given quantity of explosive, and the pene tration of a projectile in rock are some of the phenomena of interest to the military engineer which are difficult to evaluate reliably. Frequently, a series of field experiments may be run for a particular problem, and from the observed data empirical relations may be estab lished to predict the manner of behavior of a rock mass under similar circumstances. In many cases, however, the time, expense, and diffi culty of control of field studies of considerable soope and magnitude 82 w ill preclude the collection of as much experimental data as might be desired* In order to supplement the data obtained from field work, recourse may be had to experimentation with small-scal® laboratory models* If the original phenomenon under investigation is faithfully reproduced in the model in accordance with the principles of sim ilitude, then the model w ill prove to be a valuable asset in shedding light upon the behavior of the prototype. Even if no model were constructed, it is still possible to reduce the original in the mind*s eye to & size where its behavior can be more readily viaualized and understood* The purpose of this discussion is to explain enough of the basic principles of similitude between a model and an original to enable one to apply these principles to & problem of primary importance and eurrent interest—namely, that of the effect of an atomic bomb upon an underground rock fortress. Detailed discussions of the construction of scale models may be found in any of the references given at the end or this section* It is a matter of cossnon knowledge that two bodies of the same material and of the same shape, but of different size, w ill not behave in identically the same manner under the influence of their weights alone* For example, a round steel bar, in diameter and weighing about 0*282 pounds per cubic inch, may have the following strength properties: a* Elastic limit (tension) 30,000 pounds/square inch b. Rupture Stress 60,000 ” « » o* Young*s Modulus 30,000*000 ** " ” If we take a two-foot length of this steel bar and support it as a simple beam at its ends, it will be found that the maximum tensile stress, developed at the mid-point of the span by the bending moment 63 induced by the weight of the bar, is about 325 pounds per square inch* Suppose we enlarge this bar, or construct another of the same material, so that the new bar is one hundred times as great as the smaller one. That is, we will increase every linear dimension one hundred times to produce the large bar. We then have a second bar* fifty inches in diameter, resting as a beam between supports 200 feet apart. By increasing the linear dimensions of the small bar one hundredfold, however, we have increased the weight of the large bar over the small one by one million times, so that the maximum tensile stress in the second is 32*500 p .s.i. Since we used the same material in both the large and small bars, the actual strength Of the materials in the two will not vary. But the stress in the large bar has reached the yield point of the material and a permanent stretching of the bar may have occurred, whereas the deflection or deformation of the small bar would be negligible* It is evident, then, that if we wanted the small bar and the large bar to be alike in behavior, we would have to increase the strength of the large bar in the same proportion as we increased the linear dimen sions • In other words, the large bar would have to be 100 times as strong as steel if it is to behave like the small one. Conversely, if the small bar were to behave in a manner similar to the large, its strength would have to be the strength of steel. Manifestly, it is impossible to obtain sim ilarity of action between the two bars as long as they are made of the same material and are resting alongside each other as simple beams in the condition described. An analysis of the dimensional formula for "weight" w ill reveal that it is actually a force, expressed as F = MLT~2 , -2 where the expression LT denotes the acceleration by which the gravity of the earth is attracting the mass of either of the two bars. When we increased the "weight" of the small bar in reaching the dimensions of the large one, we actually increased the total mass of the bar while the mass concentration, or density, remained the same* Also the expression LT —2 remained the same for the two bars, for the acceleration of gravity is constant (32.2 feet per second) for both masses regard less of the quantity of mass involved* For a purely mechanical system, we are therefore dealing with three fundamental quantities—•mass, length, and t ime—whenever we seek to make one body behave in a manner sim ilar to the behavior of another body. Not only must geometric sim ilarity be established, but also kinematic sim ilarity (involving distance and time) and force or dynamic sim ilarity (involving mass, distance, and time) must be obtained* It is obvious also, that in addition to geometric sim ilarity, we must have a corresponding distribution of mass between the two bodies if force sim ilarity is to be accomplished* If we take the dimensions of the first, or small bar, and compare them by ratios to the corresponding dimensions of the second, or large b a r , we have 2j L s \ » ^1 * r j ® • = a * h ^ where A B r , and >u. are the length, time, and mass ratios between the two bars, ^e may use these fundamental ratios to obtain the force 65 (or weight) and strength ratios between the two bars# Henoe Force ratio - ml^l^« s jjl.K t'2' m2^ 2 ^ Inasmuch as strength is usually denoted by a certain value of the inter nal stress, or force per unit area, which the body is able to withstand, we have? -/ _ - i Strength ratio “ stress ratio - A a. i In a similar manner we may derive the secondary ratios of energy, density, acceleration, velocity, etc#, simply by substituting the ratios involved in the dimensional formula for the particular quantity# Thus, the density ratio, 6 , is determined from the dimensional formula for density ( ) as 6 s /«-A and the acceleration ratio becomes A . It has been indicated that the gravitational acceleration ratio between the two steel bars in our example is equal to unity, for the acceleration of gravity is the same in each bar# For kinematic and force sim ilarity, all the accelerations in the two bars and all the forces, which are capable of producing accelerations of the two masses, must conform to the gravitational acceleration ratio# This means that we have the following fixed relations: * AT2: i, or A = r * and T s This restriction applies in a general case in model work where both the model and original are at the earth’s surface# In most problems associated with m ilitary geology the general case will apply# There fore, it will be convenient at this point to list the mechanical quantities of interest expressed in terms of the fundamental ratios and also in terms of the ratios of only density (5 ) and length ( A ), Mechanical Quantity General Case Model Ratios. Length A « 3 Mass m 8 K Time / - A*Y1/2- -3 - D ensity m-^z o V e lo city k t ’z ^ -z Acceleration AT z I Force A1 A. T z S \ pressure, stress, strength, ( Young*s Modulus of E lasticity m-K t z d A. Strain • s \ Work and Energy ><- r £ A4^ With the aid of the ratios given in the above table, it is pos sible to explain the disparity of action between the two steel bars in our example. The force ratio from the small bar to the large bar is S S? Z 1x100^ or one million, which is satisfied. However, the strength r a t i o , hKz 1x100 s 100 is not satisfied, for the two bars are made of identical m aterial. Therefore, from a strength standpoint, if from none other, the two bars will not exhibit similar behavior. In order to understand the behavior of the large bar by the use of a model the same size as the small bar, we must make use of some material which has a strength of JL_. the amount of the steel used in the two bars. WJ Steel in small sizes or masses is very strong, but the same steel in large masses may be very weak in relation to its ability to with stand the forces induced in it as a result of the gravitational attrac tion for its mass. The same remarks apply also to rocks, for a small, B 7 individual rock specimen may be quit© strong, but a portion of the earth’s orust made up of this same rock will be very weak* In order to understand the action of a rock mass resulting from forces induced within or impressed upon a portion of the earth’s crust, it is necessary to conceive a model which will reflect in a true manner the weakness of the original and also to scale down the forces acting upon the original in their corresponding ratio* Let us consider the example of an atomic bomb dropped on an under ground fortress or tunnel* We may say that the rock mass in which the tunnel is located is sound granite, out of which a horizontal opening SO feet in diameter has been excavated. Moreover, the tunnel is situated 200 feet directly below the bomb, which has penetrated to a depth of 20 feet below ground level* We wish to know what the effect of the explosion of the bomb w ill be on the rock fortress* By test borings we may determine the strength of specimens taken from the rock mss* Assume that we found a strength for the granite ♦ of 30,000 p«s*i* in compression and 2,000 p«s*i* in tension* We also determined that the modulus of elasticity of the rock was about 6,000,000 p*s*i* We then have enough data to establish the specifica tions for our model material. If we are to construct a model the size of the original, then A. s ,006. The model material we find suitable to use weighs 3/4 as much as the granite, so S s 0.75* ♦Note: The us© of mass terns to express a force is so customary that it w ill be used in this discussion also* In converting strength or stress from original to model, the corresponding forces are related to the same gravitational field, so the customary terminology for these quantities is permissible. 68 The various ratios of interest become: Length s A s .005 Velocity * \ /2" - .071 Strength and Modulus of Elasticity Z SK Z 0*75x.005 s0*375xl0 Energy = dA4 = 0.75(*005)4 * 0.468xl0“9 For a model material we must then satisfy the strength require ment of 113 p.s.i* in compression and 7*5-8 p .s.i. in tension* The modulus of elasticity should also be reduced proportionately* A material of the constituency of dry, tamped clay, or clay and sand mixed with plaster may serve the purpose for the model material. All of the dimensions of the original must be reduced to scale in the model* Thus the "bomb" in the model has penetrated to a depth of 1.2" and is located one foot above a 1*8" diameter "tunnel” exca vated from the clay material. The entire model should be sufficiently large to eliminate any "boundary effect" from the sides* In regions of complex geologic structures, the faithfulness of the reproduction of such structures in the model will naturally have a great bearing on the recreation of the original in the model. For our example, however, we shall assume .that the rock mass under consideration is practically "structureless •** Having reduced the size of the rook fortress and the strength of the rock itself to convenient dimensions, we must also reduce the size of the atomic bomb to more understandable and more fam iliar terms* Considerable secrecy s till shrouds the exact properties of the bomb, therefore we shall consider, merely for the purpose of this discussion, that it is roughly equivalent to 20,000 tons of TNT—a disclosure made public at the time the first bomb v/as dropped on Hiroshima. We shall then reduce 20,000 tons of TNT to model size* The mechanical energy developed by TNT is about the same as that o f 60fo dynamite, or 900 foot-tons per pound on the average. If we multiply 20,000 tons of TNT by the energy ratio we obtain 0.1875 pounds or 8.6 grams of TNT. The velocity of detonation of TNT—21,000 feet per second—must also be reduced by the velocity scale ratio. We obtain a velocity of detonation of the model explosive of about 1,500 feet per second, or about the rate of burning of black powder* From Peele’s Mining Engineer’s Handbook, Volume 1, Section 5, Page 16, we find that the energy of black powder is about 500 foot-tons per pound. If we multiply the 8.6 grams of TNT by the ratio of the energies of TNT to black powder, we w ill have the amount of the charge of black powder to use in the model* Charge of black powder * 8.6 x JL s 15.5 grams 5 Thus, our model, constructed on a scale” from the original rock mass by using dry, tamped clay, can be made to simulate the effect of an atomic bomb set off upon the rock fortress if we use only 15.5 grams of black powder in the model explosive charge. Even if no model were ever constructed, still the reduction of the original blast to more understandable terms will enable us to visualize more clearly in our imaginations the damage wrought by such a blast. 70 REFERENCES Bucky, P. B. (1931), Us© of Models for the Study of Mining Problems, Amer. Inst, of- Mining Met. Engr. Teoh. Pub. 425, Class A, Mining Methods, No* 44. (1934), Application of principles of Similitude to Design of Mine Workings, Trans. Aau Inst. Mining Met. Engrs•, Vol. 109, pp. 25-42* Gibson, A. H. (1924), Sim ilarity and Model Experiments, Engineering, March 21, 1924, p. 327. Groat, B* F. (1932), Theory of Sim ilarity and Models, Trans. Amer. Soc. Civil Engrs., Vol. 96, pp. 273-386 (with discussions by Bateman, Roop, others). Hubbert, M* K. (1937), Theory of Scale Models as Applied to the Study of Geologic Structures, Bull. Geol. Soo* of Amer., Vol. 48, pp. 1459-1520, Oct. 1, 1937. Newton, I. (1687), Principia Mathematics, Book II, Proposition XXXII, Theorem XXVI. APPENDIX List of Symbols Used Frequently in the Text. Botes Other symbols used In the text and not given in this list are either defined specifically at the time of their use or are so obvious in their meaning as to require no further definition* a - Amplitude of wave vibration* A - Cross-sectional area <*-,($,y - Direction angles of a normal to a plane* A lso , /3 z the angle which a random plane makes with the plane of pure shear 8 - Density ratio, in model study e - Elongation, or deformation E — Tensile modulus of elasticity (Young*s Modulus) € - S tr a in cQ - Strain at elastic limit f - Frequency of wave motion g - Acceleration of gravity, 32*2 ft./sec^* G - Modulus of Rigidity, or Shear Modulus. I — Wave intensity k — Modulus of Inoompressibility. E - Modulus of Compressibility. l*m#a - Direction cosines of angles a , and y respectively. A — Wave length* Also, length ratio in model study. A * Folias© Modulus of Elasticity (Lame's constant) m - Poisson*8 ratio* jx. - Mass ratio in model study 72 K - Normal stress component p - Pressure, or confining pressure - Maximum principal stress Pg - Intermediate principal stress Pg - Minimum or l e a s t p rin c ip a l s tre s s r ~ Radius S - Resultant stress f> - Density, or mass per unit volume S - S tre s s sn - Normal stress component on a random plane ss «. Shear or tangential stress component on a random plane s0 - Stress at ths elastic limit / - An an g le Also, an angle which a random plane makes with the plane of maximum principal stress* © - An angle Also, the specific change in volume = ^ Y T - Time o r p erio d o f wave v ib r a tio n Also, tangential or shear stress component on a plane. r - Time ratio, in model study u,v,w - Deformations parallel to the X, Y, and Z axes respectively* II - Modulus of Resilience V - Volume Also, velocity (of waves)* Vp - V e lo c ity of Prim ary Waves Vs - Velocity of Secondary or Shear Haves w - Weight of a unit volume of material Xy - Stress parallel to the X-dlrection and acting across a plane perpendicular to the Y-direction* BIBLIOGRAPHY Baliaer , G. G., and McHenry, D. (1947), Determination of Boundary Porosity by Triaxial Compression Tests of Concrete, U. S. Dept. Of Interior, Bureau of Reclamation, Lab. Rept. No. SP-15. Billings, M. P. (1947), Structural Geology, New York, Prentice-Hall. Bridgman, P. W. (1931), Dimensional Analysis, New Haven, Yale Univ. Press* .— (1938), Reflections on Rupture, Journal of Applied Physics, vol. 9, pp. 517-528. Buoky, P. B. (1931), Us© of Models for the Study of Mining Problems, Amer. Inst. Mining Met. Bngr. Tech. Pub. 425, Class A, Mining Methods, No* 44. ------(1934), Application of Principles of Similitude tc Design of Mine Workings, Trans. Am. Inst. Mining Met. Engrs., Vol. 109, pp. 25-42. Daly, R. A. (1940), The Strength and Structure of the Earth, Hew York, Prentice-Hall. Gibson, A. H. (1924), Sim ilarity and Model Experiments, Engineering, March 21, 1924, p. 327. Gregory, II. E. (1918), M ilitary Geology and Topography, Hew Haven, Yale Univ. Press. Griggs, D. T. (1936), Deformation of Rocks Under High Confining Pressures, Journal of Geology, Vol. 44, pp. 541-577. — ------, (1939), Creep of Rocks, Journal of Geology, Vol. 47, pp. 225-251. Groat, B. P. (1932), Theory of Sim ilarity and Models, Trans. Araar. Soc. Civil Engrs., Vol. 96, pp. 273-386 (with discussions by Bateman, Roop, others). Hanna, F. W. (1938), Stresses in Circular Holes in Dams, Trans. Amer. Soc. Civil Engrs., Vol. 103, pp. 163-170. Beiland, C. A. (1939), Seismic Prospecting, Colorado School of Mines. ------(1940), Geophysical Exploration, New York, Prentice-Hall. Eouwink, R, and Burgers (1940), Elasticity, Plasticity, and Structure of Matter, Cambridge (England), Univ. Press. LIBRARY COtSMAM SCHOOL OF MIHXS GOLDIN, COLORADO Hubbert, M. K. (1937), Theory of Scale Models as Applied to the Study of Geologic Structures, Bull. Gaol. Soc. of America, Vol. 48, pp. 1459-1520, Oct. 1, 1937. Jones, V., and McHenry, D. (1945), Tensile and Triaxial Compression Tests of Rock Cores froia the Passageway to Penstock Tunnel 1-4 at Boulder Dam, U. S* Dept, of Interior, Bureau of Reclamation, Lab. R ept. h o . SP—6. Love, A. E. H* (1934), A Treatise on the Mathematical Theory of Elas ticity , 4th Ed., Cambridge (England), Univ. Press. Macelwane, J* B. (1936), Introduction to Theoretical Seismology, Part 1 Geodynamios, lew York, J. Wiley and Sons, Inc. Mindlin, R. B. (1939), Stress Distribution Around a Tunnel, Trans. Amer Soc. Civil Engrs., Vol. 104, pp. 1714-1718. Mohr, 0. (1928), Abhandlungen aus dem Gebeite der Technischen Mechanik, 3d Ed., revised by K. Bayer and H. Spangenberg, Berlin, W. Ernst u. Sohn. Hadai, A. (1931), Plasticity, lew York, McGraw-Hill. Seely, F. B. (1946), Resistance of M aterials, Hew York, J. ’Riley and Sons, In c . Southwell, R. V. (1936), An Introduction to the Theory of Elasticity, London, Oxford Univ. Press. Terzaghi, K. (1943), Theoretical Soil Mechanics, lew York, J. Wiley and Sons, Inc. Timoshenko, S. (1934), Theory of E lasticity, lew York, McGraw-Hill. Todhunter, and Pearson (1S86), History of the Theory of E lasticity, Cambridge (England), Univ. Press. Tyrell, G. W. (1929), The Principles of Petrology, Hew York, E. P. Dutton and Co., Inc.