THE BEHAVIOR OP ROCKS MD ROCK MASSES
IB RELATION TO MILITARY GEOLOGY
By
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A t h e s is submitted to th e Faculty and the Board o f Trustees of the Colorado School o f Hines in p a r tia l f u lf illm e n t of the requirements fo r the degree o f Master o f Mining Engineering*
Signed a** Wilmot H* MeCutohen Golden, Colorado Date # 18d8 4 V
r
I
. Approvedi C* W, L ivin gston Golden, Colorado Date /<£- , 1948A bstract
Following a b r ie f in trod u ction g iv in g a gen eral c l a s s i f i c a t i o n of rooks in the e a r th 's c ru st which are o f In te r e st to the m ilita r y g e o lo g is t , Table 1 i s presented to summarise the e f f e c t of various fa c to r s on th e p h y sica l p rop erties o f rooks. Accompanying d e fin itio n s f a c i l i t a t e the in te r p r e ta tio n o f the ta b le and provide a reference o f terms used in sub sequent d isc u s s io n s .
The p rop erties o f e l a s t i c i t y and p l a s t i c i t y in rocks are trea ted in Chapter 2m Some other c h a r a c te r is tic phenomena o f importance in the study o f rock s tr e n g th s, such as creep , f a t ig u e , and endurance, are a lso
mentioned.
F ailu re o f rock specimens under s t r e s s i s given a d e ta ile d stu d y. F i r s t , th e accepted c l a s s i c a l th eo ries of f a ilu r e are s ta te d b r ie f ly j then the Mohr S tr e ss Diagram i s developed fo r a number o f types o f load in g s , ending w ith a general s ta t e o f s t r e s s . Using the Mohr S tr e ss Dia gram, a c a r e fu lly co n tr o lle d laboratory experiment on rook specimens i s analysed to deduce th e manner o f f a ilu r e o f rooks and th e form of th e envelope of ru p ture.
Chapter 4 d isc u sse s sev era l examples o f s t a t i c and dynamio loadings on rock m asses, including s t r e s s d is tr ib u tio n around a tunnel opening, propagation o f e l a s t i c s t r a in , and crater b la s tin g . The l a s t part of the chapter i s devoted to a d isc u ssio n of th e p r in c ip le s o f s im ilitu d e as they may be used in th e study o f m ilita r y g eo lo g y . A s p e o lf ic example in volvin g the deton ation of an atomic bomb above an underground tunnel i s
*
presented as. an i l l u s t r a t i o n o f th e a p p lic a tio n o f th ese s im ila r ity p r in c ip le s .
Page I
X
48
8 10X 0
12
IS 13 14 14 17 18 19 21 27 28 31 34 34 36 37 40 42 A A ym 44 43 47 48 33 53 58 61 71 73 TABLE OPCOBTENTS IntroductionHooke in the E arth's Crust D e fin itio n s
E l a s t i c i t y and P la s t ic it y o f Rooks E l a s t i c i t y
P l a s t i c i t y
Stages o f Deformation F atigue and Endurance
F ailu re of Hook Specimens Under S tr e ss C la s s ic a l Theories o f Failure Mohr's S tr e ss Diagram.
Simple ten sio n T r ia x ia l loading Pure shear
Pure shear and compression General s ta t e o f s tr e s s
Further d isc u ssio n o f f a ilu r e of rocks Envelope o f rupture
The form o f the envelope of rupture
Influence o f the interm ediate p r in c ip a l s tr e s s Summary of methods o f f a ilu r e
P h ysical Behavior of Rook Masses Under S ta tio and Dynamic Loadings
S t a tic S tr e sse s Around a Tunnel Opening Hanna1s s o lu tio n
Character o f s t r e s s d is tr ib u tio n Dynamic Loadings
General con sid eration s S ta te of s tr e s s on a body
R elations between s tr e s s e s and s tr a in s Propagation o f e l a s t i c deformations Repeated deformations
Wave motions
Dynamic loadings in volvin g rupture
The Concept o f S im ilitu d e as an Aid t o th e Study o f M ilita r y Geology
Appendix Bibliography
ILLUSTRATIONS
O p p o site
F ig u re No* S u b je c t Page No*
X S t r e s s - S t r a i n Diagram 9
2 Deformat ion-Tim e Diagram 9
3 Sim ple t e n s i o n 15
4 Mohr S t r e s s Diagram f o r Sim ple t e n s i o n 15
5 T r i a x i a l lo a d in g 18 6 Mohr S t r e s s Diagram fo r , t r i a x i a l lo a d in g 18 7 P ure S h ear 19 8 Mohr S t r e s s Diagram f o r p u re sh e a r 19 9 P u re s h e a r and co m p ressio n 20 10 G en eral S ta te o f S t r e s s 21
11 E q u ilib riu m o o n d itio n s on a body i n
g e n e r a l s t a t e o f s t r e s s 22
12 Mohr S t r e s s Diagram f o r g e n e r a l s t a t e o f
s t r e s s (Method A) 23
13 R e s o lu tio n o f S t r e s s Components 25
14 D e riv a tio n o f Mohr S tr e s s Diagram f o r
g e n e r a l s t a t e o f s t r e s s (Method B) 25 15 Mohr S t r e s s Diagram f o r g e n e r a l s t a t e of
s t r e s s (Method B) 27
16 Specimens o f ro c k t e s t e d t o f a i l u r e un d er
v a r io u s lo a d in g s 28
17 Mohr S t r e s s Diagrams f o r specim ens o f
> F ig u re 16 show ing en v elo p e o f r u p tu r e 29 18 In f lu e n c e o f m o is tu re and p o r o s ity on Mohr S t r e s s diagram 32 19 S t r e s s d i s t r i b u t i o n around a tu n n e l a* R a d ia l S t r e s s p a t t e r n 37 b . T a n g e n tia l s t r e s s p a t t e r n 37 o* S h ear s t r e s s p a t t e r n 37 d* P a t t e r n of s h e a r p la n e s o f w eakness 43 20 S t a t e o f s t r e s s on a body 46
F ig u re No*
21
22
S u b je c t
B l a s t i o d e fo rm a tio n s from a dynamic lo a d E l a a t i o d e fo rm a tio n s a t a d i s t a n t p o i n t from th e im pulse O p p o site Page Ho» 50 53 23 P a t t e r n o f s h e a r w eakness aro u n d a o r a t e r c h a rg e and o u tl in e o f t y p i c a l o r a t e r s 60
TABLES T able I I I The In f lu e n c e o f V ario u s F a o to rs on th e P h y s ic a l P r o p e r t ie s o f Rooks V alues of th e P r i n c i p a l S t r e s s e s and o f - th e S h e a r and Normal Components on th e P la n e s o f Weakness f o r V a rio u s P o in ts Around a C i r c u l a r Tunnol Opening
O p p o site Page No»
4
AC KNCMLB DGXENTS
I w ish t o e x p re s s my a p p r e c i a tio n f o r th e c o u r te s y e x te n d e d me by th e S t r u c t u r a l R esearch L a b o ra to ry o f th e U n ite d S t a t e s Bureau o f R e c la m a tio n , D enver, C o lo ra d o , i n making a v a i l a b l e th e r e s u l t s o f v a lu a b le e x p e rim e n ts p e r t i n e n t t o th e s u b je c t o f t h i s t h e s i s *
Thanks a r e a ls o due t o P r o f e s s o r s C* W# L iv in g s to n , H enry Baboook, and W* H. J u rn e y o f th e F a c u lty o f th e C o lo rad o S chool o f Mines f o r t h e i r h e l p f u l c r i t i c i s m s and s u g g e s tio n s *
1
THE "BEHAVIOR OF ROCKS AND ROCK MASSES IN RELATION TO MILITARY GEOLOGY
Chapter I
I n tr o d u c ti o n
Rocks In th a E arth * s C ru s t
The e a r t h ’ s c r u s t i s g e n e r a l l y d e f in e d a s th e s i l i c e o u s zone forming
th e o u te rm o s t la y e r o f th e e a r th * The d e p th o f th e c r u s t i s more o r l e s s a r b i t r a r i l y ta k e n a t a b o u t t e n m ile s * A t p r e s e n t* th e m i l i t a r y g e o l o g i s t i s co n ce rn e d w ith o n ly th e up p er p o r t i o n o f th e © rust* down to a d e p th o f s e v e r a l hun dred f e e t , w hich f a l l s w ith in t h e rang© o f m o d erate d e p th s a s p r e s e n t- d a y m ining o p e r a tio n s go* T h is u p p e r l a y e r
o f th e e a r t h c o n s i s t s o f " ro c k s " —h a r d , c o h e s iv e m a t e r i a l composed o f m in e ra ls o r a g g re g a te s o f m in e r a ls bound to g e th e r i n t o a d e f i n i t e u n i t . Rocks a r e i n tu r n g e n e r a l l y c o v e re d w ith a m a n tle o f decomposed* uncon s o l i d a t e d m in e ra l and o rg a n ic m a t e r i a l known a s s o i l * The b o u n d ary betw een th e lo w e r l i m i t o f th e m a n tle and th e u p p e r l i m i t o f th e "bed ro c k " i s u s u a ll y n o t w e ll- d e f in e d inasm uch a s th e u p p e r zone o f th e b e d -ro c k h as u s u a ll y undergone a l t e r a t i o n and d ecay from i t s o r i g i n a l s t a t e th ro u g h th e a c t i o n o f th e a g e n ts o f w e a th e rin g and e ro s io n * This d is c u s s io n w i l l b e co n ce rn e d m a in ly w ith th e p h y s ic a l p r o p e r t i e s o f the b e d -ro c k as i t i s fo u n d i n i t s more o r l e s s u n a l t e r e d , c o h e s iv e , and c o n s o li d a te d s t a t e *
A p p ro x im a te ly tw en ty m in e r a l s , c a l l e d " ro o k -fo rm in g m in e r a ls , 0 make up th e g r e a t m a j o r ity o f th e m in e ra l a g g re g a te s o f w hich ro c k s a r e composed* N o tab le among th e ro o k -fo rm in g m in e ra ls a r e t h e
t
f e l d s p a r s , m ic a s , p y ro x e n e s , a m p h ib o lo s, q u a r tz and c a l e i t e .
Rooks a r e g e n e r a l l y c l a s s i f i e d a c c o rd in g to t h e i r modes o f o r i g i n a s ( 1 ) ig n eo u s r o c k s , o r th o s e d i r e c t l y d e r iv e d from a magma* ( 2 ) s e d im e n ta ry r o c k s , o r th o s e s e c o n d a r ily d e riv e d ro c k s form ed th ro u g h th e d i s i n t e g r a t i o n , t r a n s p o r t a t i o n , and s u b se q u e n t in d u r a tio n o f p r e v io u s ly e x i s t i n g r o c k s ; and (3 ) m etam orphic r o o k s , o r ro c k s w hich w ere o r i g i n a l l y ig n eo u s o r s e d im e n ta ry and w h ich have b een p a r t l y o r w h o lly r e c o n s t i t u t e d a s a r e s u l t o f t h e a c t i o n o f h e a t and p r e s s u r e a t d ep th *
Each o f th e Inroad c l a s s i f i c a t i o n s o f r o c k s — ig n e o u s , s e d im e n ta ry , and m etaraorphie—may i n tu r n be s u b d iv id e d i n t o f u r t h e r g e n e ti c c a t e g o r i e s . The ig n eo u s ro c k s a r e , f o r exam ple, c l a s s i f i e d a s e x t r u s i v e (form ed
th ro u g h th e c o o lin g o f th e m o lte n magma on th e e a r t h ’ s s u r f a c e ) and i n t r u s i v e (form ed th ro u g h t h e c o o lin g o f th e magma i n a r e l a t i v e l y d e e p - s e a t e d e n v iro n m en t)* Examples o f e x t r u s i v e ig n eo u s ro o k s a r e r h y o l i t e s , b a s a l t s , s c o r i a , o b s id ia n , and o th e r v o lc a n ic g la s s e s * The i n t r u s i v e ig n e o u s ro c k s o f n o te a re g r a n i t e s , s y e n i t e s , m o n z o n ite s , g a b b r o s , and p e r i d o t i t e s * The i n t r u s i v e s a r e , i n th e o r d e r g iv e n , p r o g r e s s i v e ly more b a s ic in c o m p o sitio n * G ra n ite s ta n d s a s t h e m ost a c i d i c o f th e i n t r u s i v e ig n e o u s r o c k s , w h ile g a b b ro and d i o r i t e c o n ta i n a much l a r g e r p r o p o r tio n o f b a s i c m in e ra ls *
S e d im e n ta ry ro c k s a l s o f a l l i n t o two m ain s u b d iv is io n s : n am ely , ( 1 ) th o s e form ed th ro u g h th e c o n s o li d a tio n o f c l a s t i c fra g m e n ts
(s a n d s to n e s and s h a l e s ) and ( 2 ) th o s e form ed th ro u g h th e p r e c i p i t a t i o n o f s o l i d m a t e r i a l from s o l u t i o n ( lim e s to n e s ) * M etam orphic r o c k s , on th e o th e r h a n d , a r e u s u a l l y c l a s s i f i e d a c c o rd in g t o th e d e g re e o f d i a s tro p h is m t o w hich th e y have b een s u b je c te d * Low d e g re e o r low ”ra n k n m etam orphies c o n s i s t of su ch ro c k s a s s l a t e s , w h ile th e h ig h e r
5
ra n k m etam orphics ar© su ch ro c k s a s g n e is s e s and s c h i s t s #
A f u r t h e r d e t a i l e d c l a s s i f i c a t i o n o f ro c k s a c c o rd in g to mode o f o r i g i n o r m in e ra l c o m p o sitio n i s o f no s i g n i f i c a n t i n t e r e s t t o t h i s d i s c u s s i o n . However, any p a r t i c u l a r ro o k may e x h i b i t c h a r a c t e r i s t i c f e a t u r e s o f t e x t u r e and s t r u c t u r e w hich may a f f e c t i t s p h y s ic a l p ro p e r t i e s t o a marked d e g re e . E x tr u s iv e ig n e o u s ro c k s a r e c h a r a c t e r i s t i c a l l y f i n e g r a in e d and q u i t e o f te n e i t h e r g l a s s y o r s c o r ia c e o u s . An im p o rta n t s t r u c t u r a l f e a t u r e o f i n t r u s i v e ig n e o u s ro c k s i s th e p re s e n c e o f i n t e r lo c k in g c r y s t a l l i n e m in e r a l s . C l a s t i c s e d im e n ta ry ro c k s a re u s u a ll y c h a r a c t e r i s e d by m in e ra l p a r t i c l e s h e ld to g e th e r by a c em en tin g agent* M etamorphic ro c k s may e x h i b i t th e s t r u c t u r a l c h a r a c t e r i s t i c s o f b o th s e d im e n ta ry and ig n eo u s ro c k s a n d , in a d d i t i o n , have su ch u n iq u e f e a t u r e s as b a n d in g , f o l i a t i o n , s c h i s t o s i t y , and gneissos© s t r u c t u r e *
Rocks a t o r d in a r y c o n d itio n s o f te m p e ra tu re and p r e s s u r e , and even u n d er th e en v iro n m en t o f m o d erate d e p th s , a r e i n h e r e n t l y b r i t t l e sub s t a n c e s , I n d iv i d u a l ro c k specim en s a r e a b le t o w ith s ta n d a r e l a t i v e l y la r g e co m p ressiv e lo a d b e fo re r u p t u r e , b u t a r e n o t ic e a b ly weak i n s h e a r and ev en w eak er i n te n s io n * The ro o k m asses w hich c o n s t i t u t e th e e a r t h 1 s c r u s t a r e w eaker s t i l l when c o n s id e re d i n r e l a t i o n t o th e t e c t o n i c
f o r c e s t o w hich th e y a r e s u b je c te d . As a co nsequence o f th e a c t i o n o f th e s e e a r t h f o r c e s , c e r t a i n w ell-know n f e a t u r e s a s f o l d s , f a u l t s , j o i n t i n g , c le a v a g e , and f o l i a t i o n a r e p r e s e n t t o p ro d u ce p la n e s o f d is c o n t i n u i t y o r zones o f s t r u c t u r a l w eakness w hich s e r v e t o f u r t h e r red u ce th e s t r e n g t h o f th e ro c k m a ss. O th er s t r u c t u r a l c h a r a c t e r i s t i c s o f ro c k m asses su ch a s b e d d in g , d e g re e o f c o n s o l i d a t i o n , d e g re e o f a l t e r a t i o n , g r a d a t io n from one ty p e o f ro c k t o a n o t h e r , p o r o s i t y , and m o is tu re c o n te n t a f f e c t th e p h y s i c a l and m ech an ical p r o p e r t i e s o f ro c k m asses* The accom panying t a b l e (T ab le I ) , t o g e th e r w ith th e d e f i n i t i o n s g iv e n
TA B LS I (C o n t’ d ) •I'T^TT «T° C^xpox©A ^ v f r a r sepjdoioo iC^TPT^TH *ofq!i8H gJUOSSTO^ /'VTOf -*swTd L^Susj^g sntnpoj? s,S uno^ 09 © *H HP ft © P« O ft fft S © C © H u M
3
6M O ( P o +■ + ■ 4 --tJ © c •H n3S
® bO-P © © (4 5 “ <M © *—-> 31 © KJ ft O £ o -p K •© © H > I 4 “ + ♦ t rG ft P © •H •P «w 53 X> • 1 O •P 6 WT5 © d •H <—1 © © • P * P ft oj O ft M © ft bO«H p fl © O § A 3 S O R G P o O O © ft O rP ft O o a •Hi © © ft ft © P ft <3 .ft £ O © O rP ft •H C ft >> 05 © O ft o *H © H a rP ft © •P © ft G rP >P •H ■£ p p A r-i 25 © 05 bO , rP P* © © P ■3T3 © ft -P 53, ft © G •P •s © © ft P« O ft •P §# ■H r-i 23 O <sj a bO 'd +3 t i « © © p p «j C "© O »P O rP • O © © © ,M tH ^ O G a o P o u © •H ♦ o d tJ © p 05 X? •H r-i O © £ © O X o o « o p ft 4 4 -~4te 4 " 4 4 -4 ~ © © ftbO ® J f « id •© a? *P 3 A a g. ( 4 O © a p 3 d © b 2S d X <w cjs o o p © © t—i bO S3 d pt—r bO • r l u p d A P bJ3 ft •P 'fts
p o p rP © r-t r-i © h (0 ft* • U) ft £ © •p P 05 © ft § O A ***» © © §© u o © T d & © © s bO © t3 d o p © © © ft o £ tp “ h In d ic a te s in c re a s e s , w h il e — in d ic a te s d e c re a s e sIn fl u e n c e of V a ri o u s F a c to rs on P h y si c a l P ro p e rt ie s of R o c k s © o fl ft ft *H f t © f t ca r-i Q ft O P P CO © >» e •H ft © M W O r-i > P* O 03 P © P © ft © P H O © O P© ft *—I S3 ^—- © O © •HO^« U © 3 Vi (0 H © ft © ft © *4 © © ft O P M 55 D O C a p a b le of m or e p la s ti c or " e la s ti c o -v is c o u s " d e fo rm a ti o n s , § If c o a rs e r g ra in e d
b e lo w , w i l l s e r v e t o i l l u s t r a t e th e e f f e c t o f v a r io u s f a c t o r s on t h e p h y s ic a l b e h a v io r o f ro c k m asses# I t sh o u ld be remembered t h a t any number o f su ch f a c t o r s may b e o p e r a tin g a t any one tim e to p ro d u ce a r e s u l t a n t e f f e c t w hich may b e d i f f e r e n t from th e b e h a v io r u n d e r any one o f th e component f a c t o r s .
D e f in i tio n s
(S ee A ppendix f o r l i s t o f sym bols u sed h e r e in )
E l a s t i c i t y . The a b i l i t y o f a}b o d y , when deform ed t o a c e r t a i n
!
e x t e n t by a sy stem o f a p p lie d f o r c e s , t o r e g a in i t s o r i g i n a l sh ap e upon rem oval of th e f o r c e s . The d e g re e o f e l a s t i c i t y o f m a t e r i a l s i s m easured by th e amount o f d e fo rm a tio n ( s t r a i n ) , e x p re s s e d a s a f r a c t i o n o f th e o r i g i n a l l i n e a r d im e n s io n , w hich a body w i l l u n d erg o u n d e r a g iv e n s t r e s s ( f o r c e p e r u n i t a r e a ) w ith in th e ra n g e o f e l a s t i c a c t i o n o f th e m a t e r i a l . S t e e l , f o r ex am p le, i s a much l e s s e l a s t i c s u b s ta n c e th a n r u b b e r , b e c a u se a g iv e n s t r e s s i n s t e e l w i l l pro d u ce much l e s s e l a s t i c d e fo rm a tio n th a n w i l l th e same s t r e s s in ru b b e r# Inasm uch as i n any s tu d y o f e l a s t i c i t y , Hooke’ s law i s assumed to h o ld below th e e l a s t i c l i m i t — t h a t i s , f o r any m a t e r i a l th e a p p lie d s t r e s s i s p r o p o r tio n a l t o th e r e s u l t i n g s t r a i n — Young’ s modulus h a s b e en u sed t o d e s c r ib e t h e r e l a t i v e e l a s t i c i t y o f tw o m a t e r i a l s # S in c e Young’ s m o d u lu s, E Z w i l l have a much l a r g e r
s t r a i n
v a lu e f o r s u b s ta n c e s such as s t e e l th a n i t w i l l f o r a m a t e r i a l l i k e r u b b e r , i t i s r e a d i l y s e e n t h a t , t h e low er th e v a lu e o f t h e e l a s t i c m odulus, th e more e l a s t i c a s u b s ta n c e m ust b e . C h a r a c t e r i s t i c s o f e l a s t i c i t y w i l l be d is c u s s e d f u r t h e r I n th e n e x t c h a p t e r .
P l a s t i c i t y . The p r o p e r ty o f a b o d y , w h ereb y , when i t i s s u b je c te d to a d e fo rm a tio n by th e a p p l i c a t i o n o f a sy stem o f f o r c e s , i t rem ain s
deform ed upon rem oval of th e fo r c e s * In o th e r w o rd s , an i d e a l l y p l a s t i c body sh o u ld e x h i b i t no e l a s t i c re c o v e ry * A ll s o l i d s e x h i b i t b o th e l a s t i c i t y and p l a s t i c i t y t o v a r y in g d e g re es* S te e l u n d e r o rd in a ry c o n d itio n s o f te m p e ra tu re and p r e s s u r e e x h i b i t s e l a s t i c p r o p e r t i e s u n d e r s t r e s s e s below th e s o - c a l l e d " e l a s t i c l i m i t " (beyond w hich Hooke’ s law no lo n g e r h o l d s ) , th e n w ith in c r e a s in g s t r e s s e s i t d is p la y s p l a s t i c b e h a v io r u n t i l ru p tu r e * Rocks u n d er th e same c o n d itio n s show v e r y l i t t l e i f any p l a s t i c d e fo rm a tio n beyond th e e l a s t i c l i m i t and b e fo re ru p tu r e * However, w ith i n c r e a s in g d e p th and c o n f in in g p r e s s u r e , ro c k s may e x h i b i t a c o n s id e r a b le amount o f p l a s t i c d e fo rm a tio n *
* V is c o s ity * The r e s i s t a n c e o f f e r e d by a f l u i d t o th e r e l a t i v e m o tio n o f i t s p a r t i c l e s . A v is c o u s d e fo rm a tio n i s c h a r a c te r iz e d by a c o n tin u o u s d is p la c e m e n t o f th e p a r t i c l e s o f th e f l u i d body a t a r a t e p r o p o r tio n a t e t o t h e r a t e o f a p p l i c a t i o n and m agnitude o f th e a p p lie d s h e a r in g lo a d . Such dependence on th e ra t© o f a p p l i c a t i o n o f th e s h e a rin g lo a d d i s t i n g u i s h e s v is c o u s flo w from p l a s t i c f lo w , f o r in th e l a t t e r ty p e o f d e fo rm a tio n th e s h e a r s t r e s s n e c e s s a r y t o p ro d u ce p l a s t i c s h e a r i s v e r y n e a r ly in d e p e n d e n t o f th e r a t e o f sh ear* E l a s t i o o - v i s o o s i t y * A s u b s ta n c e i s c a l l e d " e l a s t i c o - v i s c o u s " when i t e x h i b i t s e l a s t i e p r o p e r t i e s (and p e rh a p s p l a s t i c p r o p e r t i e s a s w e ll) u n d e r lo a d s o f s h o r t d u r a t i o n , y e t u n d e r r e l a t i v e l y s m a ll lo a d s o f lo n g d u r a tio n deform s v is c o u s ly * In such m a t e r i a l s th e c o e f f i c i e n t o f
v i s c o s i t y i s q u i t e h ig h * An example o f e l a s t i c - v i s c o u s m a t e r i a l i s s e a l in g wax* Rooks a r e s a i d by some a u t h o r i t i e s to b e e l a s t i c - v i s c o u s *
C re e p * A slow d e fo rm a tio n o f a m a te r ia l when s u b je c te d t o r e l a t i v e l y s m a ll s t r e s s e s a c ti n g o v e r a r e l a t i v e l y lo n g p e rio d o f tim e .
w hich a body f a i l s t o e x h i b i t b e f o r e f a i l u r e under a g iv e n c o n d itio n o f lo a d in g . Rooks a t o r d in a r y c o n d itio n s o f te m p e ra tu re and p r e s s u r e a r e s a id t o be b r i t t l e .
H a rd n e s s . As u se d h e r e i n , h a rd n e s s r e f e r s to th e a b i l i t y o f a body t o s ta n d a b r a s i o n . Such a d e s c r i p t i v e term can o n ly be a p p lie d w ith r e f e r e n c e t o a g e n e r a l l y a c c e p te d s t a n d a r d , o r s c a le o f h a r d n e s s ,, t o w h ich a g iv e n m a t e r i a l i s compared* I n th e t e s t i n g o f h a rd n e ss o f m in e ra l s p e c im e n s, Moh’ s s c a le i s th e u s u a l l y a c c e p te d c r i t e r i o n . I n t h i s s c a l e , t a l c i s a s s ig n e d a h a rd n e ss o f 1 , gypsum 2 , c a l o i t e 3 , f l u o r i t e 4 , a p a t i t e 5 , o r th o c la s e 6 , q u a r tz 7 , to p a z 8 , corundum , 9 and diamond 10.
T o ug h n ess. The p r o p e r t y o f a body by v i r t u e o f w hich v/ork may b e done on th e body when s t r e s s e d beyond i t s e l a s t i c l i m i t .
S t r e n g t h . The p a r t i c u l a r c o m b in a tio n o f s t r e s s e s w hich a m a t e r i a l c a n u n d erg o b e f o r e f a i l u r e e i t h e r b y r u p tu r e o r p l a s t i c flo w u n d e r a g iv e n s e t o f c o n d itio n s o f tim e and te m p e r a tu r e . Fundam ental s t r e n g t h i s th e s t r e n g t h o f a m a te r ia l r e g a r d l e s s o f th e tim e p e r io d o v er w hich th e s t r e s s e s m y b e a p p l i e d .
P o r o s i t y . The amount o f v o id s i n a g iv e n volume o f m a t e r i a l i n r e l a t i o n t o th e amount o f s o l i d s i n th e same volum e.
E la s t ic C onstants.
a . Modulus o f E l a s t i c i t y (Y oung's M odulus) u> — s t r e s s s ** s t r a i n V b . P o is s o n 's r a t i o * The r a t i o o f th e l a t e r a l u n i t s t r a i n t o th e l o n g i t u d i n a l s t r a i n su ch a s fo u n d i n a t e s t specim en s u b je c te d t o a s im p le a x i a l lo a d . P o i s s o n 's r a t i o - m - € l a t e r a l I l o n g i t u d i n a l
7
c * s kQar modulus o r modulus o f r i g i d i t y # The r a t i o o f th e a p p lie d s h e a rin g s t r e s s t o th e r e s u l t i n g s h e a rin g s t r a i n #
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d . Modulus o f F l e x i b i l i t y # The r e c i p r o c a l o f th e modulus o f r i g i d i t y , o r A #
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e . Modulus o f volume e l a s t i c i t y # (Lama’ s c o n s ta n t)
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- - ( y t n ) ' ( r - - s )
f• C o m p r e s s ib ility f a c t o r # A p r o p o r t i o n a l i t y f a c t o r w h ich may be u sed to d eterm in e th e c o m p r e s s ib ili t y o f a m a t e r i a l u n d er a g iv e n c o n fin in g pressui*©. F o r h y d r o s t a t i c p r e s s u r e 5
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g# I n c o m p r e s s ib il ity f a o t o r ( k ) # A lso c a l l e d th e wB ulk M odulus, 11 i s th e r e c i p r o c a l o f t h e c o m p r e s s i b i l i t y f a c to r #
k* Modulus o f R e s i l i e n c e # The e n e rg y a b so rb e d p e r u n i t volume o f a m a te r ia l when s t r e s s e d to th e e l a s t i c l i m i t . F or specim ens t e s t e d i n sim p le t e n s i o n o r co m p re ssio n th e modulus o f r e s i l i e n c e , U, i s :
2 TT 1 1 “ • t ' l c e t ^ L . V e l o c i t i e s o f Wave p r o p a g a tio n . a# C o m p re ss io n a l, l o n g i t u d i n a l , o r p rim a ry w aves: *» = b# S h e a r, o r tr a n s v e r s e w aves: v s = V - y - n S e— 5— : - J - S L V 2w (l + m) Y w
C h a p te r 2
E l a s t i c i t y and P l a s t i c i t y o f Rocks
E l a s t i c i t y
G e n e ra l. An e l a s t i c d e fo rm a tio n i s one w hich d is a p p e a rs c o m p le te ly upon r e l e a s e o f th e s t r e s d c a u s in g th e d e fo rm a tio n . As a lr e a d y in d ic a te d u n d e r th e d e f i n i t i o n o f e l a s t i c i t y i n C h ap ter 1 , th e r e i s a d e f i n i t e r e l a t i o n , e s t a b l i s h e d by e x p e rim e n t, betw een e l a s t i c s t r e s s and e l a s t i c s t r a i n — t h a t i s , th e y a re p r o p o r tio n a l to one a n o th e r w i t h i n a c e r t a i n r a n g e . T his e l a s t i c p r o p e r ty o f p r o p o r t i o n a l i t y i s known a s Hooke’ s Law and i s one of th e fu n d a m e n ta l p r e c e p ts on w hich i n v e s t i g a t i o n s o f e l a s t i c b e h a v io r a r e b a s e d . A h a r d , r i g i d s u b s ta n c e su ch as s t e e l w i l l behave i n a t r u l y e l a s t i c m anner u n d e r a sm a ll s t r e s s a c c o rd in g t o th e d e f i n i t i o n o f e l a s t i c i t y , b u t a h ig h ly e l a s t i c s u b sta n c e such as ru b b e r w i l l n o t behave i n a t r u l y e l a s t i c m anner. When s t r e t c h e d s lo w ly , ru b b e r does n o t im m e d ia te ly r e t u r n to i t s o r i g i n a l sh ap e a f t e r th e r e l e a s e o f s t r e s s . I t would seem , t h e r e f o r e , t h a t th e h a r d e r and more r i g i d a m a t e r i a l i s , th e more t r u l y e l a s t i c i s i t s b e h a v io r u n d e r a s m a ll s t r e s s . Rocks a s e n c o u n te re d n e a r th e e a r t h ’ s s u rfa c e a l l y th e m selv e s more n e a r ly t o th e h a r d e r s u b s ta n c e s and behave i n a more o r l e s s t r u l y e l a s t i c m anner u n d er sm a ll s t r e s s e s *
S t r e s s - S t r a l n D iagram . An i d e a l i z e d ” s t r e s s - s t r a i n ” diag ram f o r a ro c k specim en i s shown i n F i g . 1 . The specim en has b e e n ' s u b je c te d t o a s lo w ly a p p l i e d , s im p le , a x i a l l y c o m p re ssiv e lo a d u n d e r o r d in a r y c o n d itio n s o f te m p e ra tu re and a tm o sp h e ric p r e s s u r e . I f th e ro c k specim en behaved i n a t r u l y e l a s t i c m anner, th e l i n e QA
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Figure 1
S tr esa -S tr a in Diagram fo r Rook,
p L A S j l C O G j r o a ^ N A f i O N E L A S - p C A f J S f t * E f f E C j I L A 4 J I C R.E COV E t Q • L o A O l K C i C O A M £ U C £ D t, : Lo a o i m^ ft e l e a s e d : U L j I M A f E <2. £ C O V E f t - Y
Figure 2
i n t h e diagram would be s t r a i g h t * A c tu a l specim ens t e s t e d , how ever, i n d i c a t e t h a t th e s t r e s s - s t r a i n r e l a t i o n s h i p i s n o t e n t i r e l y i n a c c o rd a n c e w ith Hooke’ s Law, s o t h a t th e l i n e QA e s t a b l i s h e d by
e x p e rim e n ts on ro o k s i s s l i g h t l y curved* Inasm uch a s th e a r e a u n d e r th e c u rv e d l i n e OA r e p r e s e n ts th e a c t u a l work done on th e sy ste m , i t i s e v id e n t t h a t l e s s p o t e n t i a l e n e rg y i s s to r e d up in th e body th a n w ould be p o s s ib le i f th e ro c k had behaved i n a t r u l y e l a s t i c manner*
The h o r i z o n t a l l y sh ad ed p o r t i o n o f Fig* 1 i n d i c a t e s th e amount o f p o s s i b l e e l a s t i c p o t e n t i a l e n e rg y w h ic h , th ro u g h a phenomenon c a l l e d R e l a x a t i o n " , i s n o t s t o r e d i n th e body d u rin g d e fo rm a tio n * R elax a t i o n may be d e s c rib e d a s th e p ro c e s s w hereby c e r t a i n p o r ti o n s o f th e ro c k make u se of some o f th e p o t e n t i a l s t r a i n e n e rg y a s h e a t e n e rg y o f p a r t i c l e v ib r a ti o n * M oreover, upon g ra d u a l r e l e a s e o f th e lo a d , t h e r o c k sp ecim en does n o t r e t u r n t o i t s o r i g i n a l sh ap e a lo n g th e same l i n e a s i n lo a d in g th e specim en* f o r a c e r t a i n s t r e s s on th e lo a d in g cu rv e OA i s a s s o c i a t e d w ith a g r e a t e r s t r a i n on th e u n lo a d in g cu rv e* The a r e a b etw een th e lo a d in g cu rv e and th e u n lo a d in g cu rv e i s th e amount o f s to r e d p o t e n t i a l e n e rg y l o s t as h e a t , w hich i s d i s s i p a t e d i n p ro du cin g a change i n th e i n t e r n a l s t r u c t u r e o f th e ro c k in lo a d in g i t t o a c o n d itio n c o rre sp o n d in g t o p o i n t A* T his lo s s o f e n e rg y i s c a l l e d " h y s t e r e s i s o f s t r a i n * "
I f a t p o in t A th e sp ecim en i s s u b je c te d t o f u r t h e r l o a d , i t would f i n a l l y f a i l by r u p tu r e a t a p o i n t B, c o rre sp o n d in g t o a s t r e s s v a lu e Bf • The e n t i r e d e fo rm a tio n b e fo re r u p tu r e would be a p p ro x im a te ly e l a s t i c , as a lr e a d y in d ic a te d *
C re e p * The d e fo rm a tio n i l l u s t r a t e d i n Fig* 1 i s b ro u g h t a b o u t on a specim en d u rin g a r e l a t i v e l y s h o r t tim e ex p erim en t* I n a d d i t i o n t o t h i s ty p e o f d e fo rm a tio n , i t may b e o b serv ed t h a t , i f th e specim en
10
i s deform ed t o a p o in t su ch a s A by th e c o rre sp o n d in g s t r e s s A*, and th e n i f th e s t r e s s i s k e p t c o n s ta n t o v e r a p e rio d o f tim e , th e ro o k w i l l c o n tin u e to deform in e v e r d e c r e a s in g amounts o v e r e q u a l tim e
i n t e r v a l s u n t i l th e d e fo rm a tio n v i r t u a l l y c e a se s* T h is phenomenon, c a l l e d c r e e p , i s th o u g h t t o be a n o th e r m a n if e s t a t io n o f r e la x a ti o n * I f th e s t r e s s A* i s removed a f t e r th e c re e p deform a.tion o c c u r s , th e specim en w i l l in tim e u s u a l l y r e g a in i t s o r i g i n a l s h a p e , o r n e a r ly so* T hus, th e ro c k may s t i l l be c o n s id e re d a s b e h av in g more o r l e s s e l a s t i c a l l y u n d er " c re e p d e fo rm a tio n s* " P l a s t i c i t y I f th e phenomenon o f r e l a x a t i o n i s e x te n d e d to a c o n s id e r a ti o n o f th e b e h a v io r o f ro c k s u n d er much h ig h e r c o n f in in g p r e s s u r e s th a n o rd in a ry a tm o sp h e ric p r e s s u r e — t h a t i s , a t p r e s s u r e s t o be found in th e e a r t h a t d e p th — i t w i l l be fo u n d t h a t th e ro c k specim en w i l l behave i n an e l a s t i c m anner u n d e r much h ig h e r s t r e s s e s and i n a d d i tio n w i l l u n d e r go a ty p e o f flo w d e fo rm a tio n from w hich i t w i l l n o t re c o v e r i t s
o r i g i n a l shape* The ty p e o f flo w d e fo rm a tio n w hich p ro d u ces a p erm anent " s e t " i n th e ro ck specim en i s c a l l e d " p l a s t i c flow *" I f th e c o n f in in g p r e s s u r e i s h ig h e n o u g h , th e ro c k specim en may undergo a c o n s id e r a b le amount o f p l a s t i c d efo rm a tio n *
S ta g e s o f D efo rm ation
Rocks a re made up o f a g g re g a te s o f m in e ra l m a t t e r , bonded to g e th e r by v a rio u s m eans. I t i s e v id e n t t h a t each o f t h e m in e ra l c o n s t i t u e n t s may e x h i b i t i n d iv id u a l e l a s t i c and p l a s t i c p r o p e r t i e s w hich make i t c o n c e iv a b le t h a t u n d e r a g iv e n lo a d in g some m in e ra l c r y s t a l s may be u n d e rg o in g p l a s t i c d e fo rm a tio n w h ile o th e r c r y s t a l s a r e s t i l l i n t h e i r
e l a s t i c r a n g e . This h e te r o g e n e ity of a c t i o n u n d e r s t r e s s has been
y
advanced by B u rg ers to e x p la in th e d e v ia tio n o f ro c k s and o th e r
V
Houwink, " E l a s t i c i t y , P l a s t i c i t y , and S t r u c t u r e o f M a tte r " , Cambridge U n iv . P r e s s , 1940.p o l y c r y s t a l l i n e m a t e r i a l s from a t r u l y e l a s t i c b e h a v io r . The in n e r te n s io n s a r i s i n g betw een t h e e l a s t i c a l l y and p l a s t i c a l l y deform ed p o r ti o n s may, i n a d d i t i o n t o th e d e fo rm a tio n s a lr e a d y d e s c r ib e d ,g iv e r i s e to an " e l a s t i c a f t e r - e f f e c t , " o r slow re c o v e ry from d e fo rm a tio n a f t e r r e l e a s e o f th e l o a d .
We may th e n summarize ( s e e P ig . 2) 'th e d e fo rm a tio n s p roduced on a t y p i c a l ro c k specim en u n d e r s u i t a b l e c o n d itio n s o f te m p e ra tu re and c o n f in in g p r e s s u r e as f o llo w s :
1 . S m all lo a d s p ro d u c in g t r u l y e l a s t i c a c ti o n o f a l l c r y s t a l l i n e m a tte r and b o n din g m a t e r i a l .
2 . I n c r e a s in g lo a d s t o produce a p l a s t i c d e fo rm a tio n o f a few o f t h e c r y s t a l s ( o r bonding m a t e r i a l ) , so t h a t th e r o c k a s a whole b eh av es a s e s s e n t i a l l y e l a s t i c b u t w ith p o t e n t i a l e l a s t i c a f t e r e f f e c t and h y s t e r e s i s p ro d u c e d .
3 . S t a t i c lo a d in g o v e r a p e rio d o f tim e ( c r e e p ) , g iv in g r i s e t o a s u f f i c i e n t r e l a x a t i o n and i n t e r n a l a d ju s tm e n t o f th e ro c k s t r u c t u r e to produce a more f a v o r a b le e q u ilib r iu m w ith th e e x t e r n a l lo a d s*
4* I n c r e a s in g lo a d s resum ed. D efo rm atio n o f th e c r y s t a l l i n e m a tte r and bonding a g e n ts i s so p re p o n d e ra n t t h a t , i f th e lo a d w ere r e l e a s e d , th e e l a s t i c p o t e n t i a l e n erg y o f th o s e c r y s t a l s s t i l l u n d e r g o in g e l a s t i c d e fo rm a tio n i s i n s u f f i c i e n t t o r e s t o r e th e ro c k t o i t s o r i g i n a l sh a p e . A "p erm an en t s e t " i s p ro d u c e d .
12
5 . F u r th e r in c r e a s in g o f lo a d in g . P l a s t i c d e fo rm a tio n s o c cu r t o su ch a d e g re e t h a t th e e l a s t i c e le m e n ts no lo n g e r a r e a b le t o form any c o n c e rte d r e s i s t a n c e t o th e d e fo rm a tio n , hen ce a p p r e c ia b le y i e l d i n g i s e v id e n c e d w ith s m a ll in c r e a s e o f lo a d . P l a s t i c flo w i s d o m in an t.
I f , b e f o r e th e s t a t e o f d e fo rm a tio n in 5 above b e g i n s , th e lo a d w ere g r a d u a lly r e l e a s e d , th e fo llo w in g s ta g e s o f re c o v e ry o f th e ro c k w ould be o b s e rv e d :
1 . An e l a s t i c re c o v e ry (accom panied by h y s t e r e s i s ) from a c o n s id e r a b le p r o p o r tio n o f th e d e fo rm a tio n p ro d u c e d .
2 . A re c o v e ry from c r e e p , com bined w ith th e e l a s t i c a f t e r - e f f e c t m en tio n ed p r e v i o u s l y , to f u r t h e r re d u c e th e am ount o f d e fo rm a tio n p ro d u c e d .
A p o r ti o n o f th e d e fo rm a tio n w i l l s t i l l rem ain w hich was b ro u g h t a b o u t by th e p l a s t i c flo w . The ro c k w i l l n o t r e c o v e r from t h i s p l a s t i c d e fo rm a tio n .
F a tig u e and Endurance
Under r e p e a te d lo a d in g c y c le s o f th e n a tu r e ^ u s t d e s c rib e d i t i s p ro b a b le t h a t th e chan ges b ro u g h t a b o u t upon th e i n t e r n a l s t r u c t u r e of th e ro c k by h y s t e r e s i s , p l a s t i c d e fo rm a tio n s o f some o f th e
c r y s t a l s o r bond in g m a t e r i a l , and l o c a l i z e d s t r a i n s w i l l b r in g a b o u t th e e v e n tu a l f a i l u r e o f th e m a t e r i a l u n d e r lo a d in g s below th o s e n e c e s s a r y to r u p tu r e th e ro c k d u rin g a s in g le t e s t . Yfhether th e s e f a c t o r s c o m p le te ly e x p la in t h e r e d u c tio n i n s t r e n g t h o f a ro c k u n d er r e p e a te d lo a d in g s i s p ro b le m a tic a l* However, i t i s s u f f i c i e n t t o know t h a t such r e p e a te d s t r e s s e s a c t u a l l y do re d u c e th e s t r e n g t h s o f ro c k s *
Chapter 3
F a i l u r e o f Rook Specim ens Under S t r e s s
Many v a r i e t i e s o f ro c k s have b een s u b je c te d t o s ta n d a r d l a b o r a to r y t e s t s t o d e te rm in e t h e i r a b i l i t y t o w ith s ta n d m e c h a n ic a lly in d u ced s t r e s s e s . U n f o r t u n a te ly , ro c k s have n o t b een t e s t e d a s e x h a u s tiv e ly a s have some o f th e more common s t r u c t u r a l m a t e r i a l s su ch a s s t e e l o r c o n c r e te ; n e v e r t h e l e s s , th e d a ta a v a i l a b l e from ro c k t e s t s i s s u f f i c i e n t t o j u s t i f y a t h e o r e t i c a l d is c u s s io n o f th e m anner o f f a i l u r e o f ro c k s u n d e r s t r e s s .
A . C l a s s i c a l T h e o rie s o f F a i l u r e
S e v e ra l t h e o r i e s o f f a i l u r e have been advanced to e x p la in th e c a u se o f th e commencement o f i n e l a s t i c a c t i o n o r th e manner o f f a i l u r e o f a m a t e r i a l . F a i l u r e o f e l a s t i c a c t i o n i s ta n ta m o u n t t o r u p tu r e i n a b r i t t l e m a te r ia l such a s r o c k u n d e r o r d in a r y c o n d itio n s o f tem p era t u r e and a tm o sp h e ric p r e s s u r e . B r i e f l y sum m arized, th e s e c l a s s i c a l t h e o r i e s o f f a i l u r e a r e :
1 . Maximum S t r e s s T h e o ry . T his th e o ry s t a t e s t h a t i n e l a s t i c a c t i o n b e g in s a t a p o in t o n ly when th e maximum norm al s t r e s s ( t e n s i l e ) a c r o s s any p la n e th ro u g h th e p o i n t ex ceed s th e s t r e s s c o rre s p o n d in g t o th e p r o p o r tio n a l l i m i t o f th e m a te r ia l i n a sim p le t e n s i o n t e s t .
2 . Maximum S t r a i n T h e o ry , T h is th e o r y p o s t u l a t e s t h a t t h e s t r a i n a s s o c i a t e d w ith th e f a i l u r e o f a m a t e r i a l i n a sim p le t e n s i l e t e s t m ust n o t be ex ceed ed a t an y p o i n t i n a body i f th e m a t e r i a l i s n o t to f a i l e l a s t i c a l l y .
3 . Maximum S tr a in -B n e rg y T h e o ry . The "modulus o f r e s i l i e n c e " ( s e e d e f i n i t i o n s ) i s used a s th e c r i t e r i o n o f e l a s t i c f a i l u r e in t h i s
/
14
t h e o r y . The maximum s tr a in - o n e r g y s t o r e d p e r u n i t o f volume by th e body u n d e r any com plex sy stem o f s t r e s s e s m ust n o t exceed t h a t w hich c a n be s t o r e d p e r u n i t volume by th e body as d e term in e d by a sim p le t e n s i o n t e s t .
4 . Maximum S tr e s s - D i f f e r e n c e Theory o r Maximum S h ear Theory* (G u e st* s Law ). T his th e o r y s t a t e s t h a t th e f a c t o r p ro d u c in g f a i l u r e o f e l a s t i c i t y i s th e g r e a t e s t s h e a r s t r e s s in th e m a t e r i a l ( o r t h e g r e a t e s t d if f e r e n c e betw een th e p r i n c i p a l s t r e s s e s ) .
j /
The maximum s t r e s s th e o ry h a s been m o d ifie d by 0 . M ohr. I n s te a d
^ M ohr, 0 . , “ T ech n isch e M echanik” , p p . 1 9 2 -2 3 4 , W. E r n s t u , S .# 3 rd E d ., B e r l i n , 1928.
o f p la c i n g r e l i a n c e on th e maximum s h e a r s t r e s s alo n e as a cau se o f f a i l u r e , Mohr s t a t e s t h a t o f a l l p la n e s h av in g th e same norm al
com ponent, t h a t w hich s u s ta in s the g r e a t e s t s h e a r s t r e s s i s th e p la n e on w hich i n e l a s t i c s t r a i n w i l l o c c u r . As f a r a s ro c k s a r e c o n c e rn e d , i t
seems m ost p l a u s i b l e to a c c e p t M o h r's h y p o th e s i s , w ith s e v e r a l im p o rta n t am endm ents, i n o r d e r t o e x p la in th e f a i l u r e o f t h e i r i n e l a s t i c a c t i o n . B efo re d is c u s s in g f a i l u r e f u r t h e r , i t i s a d v is a b le to e x p la i n i n d e t a i l Mohr’ s S t r e s s D iagram , w hich sh o u ld be u n d e rs to o d i n o rd e r to f u l l y a p p r e c ia te th e manner o f ro o k f a i l u r e s . Inasmuch a s m ost r e f e r e n c e s do n o t in c lu d e a th o ro u g h d is c u s s io n o f t h i s v a lu a b le g r a p h ic a l means o f r e p r e s e n t in g a s t a t e o f s t r e s s , a com prehensive a n a ly s is i s in c lu d e d i n th e fo llo w in g p a ra g rap h s*
Mohr’ s S t r e s s D iagram .
1 . Sim ple T e n s io n . F ig u re 3a i l l u s t r a t e s a ro c k body s u b je c te d t o a n a x i a l t e n s i l e l o a d , F , d i s t r i b u t e d u n ifo rm ly o v e r th e c r o s s
V J S f i S I ON dx <*r a .
F ig u r e 3
- S H £ * f c .F ig u r e 4
Mohr S t r e s s Diagram f o r F i g . 3
^ C O A A f R . e s S I OVIs e c t i o n o f th e body ta k e n a t p o in t P p e r p e n d ic u la r to th e d i r e c t i o n o f F* The " f o r c e i n t e n s i t y * ” o r " s t r e s s , ” a c r o s s such a p la n e i s p^ * F <• A.# w here A i s th e c r o s s s e c t i o n a l a re a* T h is p a r t i c u l a r p la n e i s known a s a " p r i n c i p a l s t r e s s p la n e ” f o r th e p o in t P , su ch a p la n e b e in g d e f in e d a s one on w hich th e r e i s no s h e a rin g o r ta n g en t i a l s t r e s s # I t can be shown t h a t f o r th e s t a t e s o f s t r e s s on th e many p la n e s w hich may b e p a ss e d th ro u g h p o i n t P u n d e r any sy ste m o f e x t e r n a l f o r c e s , th e r e w i l l be t h r e e and o n ly th r e e su ch p r i n c i p a l s t r e s s p la n e s * F u rth erm o re ^ th e t h r e e p r i n c i p a l s t r e s s p la n e s a re m u tu a lly p e r p e n d ic u la r . T h u s, th e d i r e c t i o n s o f th e th r e e p r i n c i p a l
s t r e s s e s — a c ti n g alw ays norm al t o t h e p r i n c i p a l p la n e s o f s t r e s s — a r e a l s o m u tu a lly p e rp e n d ic u la r* I n & g e n e r a l c a s e , none o f th e p r i n c i p a l s t r e s s e s w i l l be e q u a l, so t h a t , i f we d e s ig n a te th e s e s t r e s s e s as p ^ , p £ , an d p ^ , and i f P i > P2 > * V th e n p^ i s th e maximum p r i n c i p a l s t r e s s * pg i s th e in te r m e d ia te p r i n c i p a l s t r e s s , and pg i s th e minimum o r l e a s t p r i n c i p a l s t r e s s * I n th e p a r t i c u l a r ty p e o f lo a d in g shown i n Fig* 3 a , pgS pg - 0*
Through th e p o i n t P , o f Fig* 3 a , a random p la n e such a s a - a may be p a s s e d , making an a n g le / w ith th e p la n e o f th e maximum p r i n c i p a l s t r e s s * Upon th e s m a ll b o d y , dV, o f u n i t th ic k n e s s shown i n F ig s * 3a and 3b th e fo llo w in g e q u ilib r iu m o f f o r c e s e x i s t s ;
Z F X * 0 X F y ^Q
S g C o s/ d l* sn s i n / d l s nc o s / dx + s s s i n / dx = p^dx c o s / c o s /
S o lv in g th e above e q u a tio n s f o r s s and S&, th e sh o a r and norm al com ponents o f a r e s u l t a n t s t r e s s , B , on th e p la n e a - a , we o b ta in t
c 16
s s 8 J p i s i n 2/ ( 1 )
8n * i P i + "k Px ©os2/ ( 2 )
The r e l a t i o n s h i p betw een s s and sn may be re p r e s e n te d geo
m e t r i c a l l y by th e c o n s t r u c t i o n shown i n Pig* 4* The h o r iz o n ta l a x is i n th e draw ing m easu res t e n s i l e s t r e s s from th e o r i g i n , 0 , to th e l e f t , and th e v e r t i c a l a x i s r e p r e s e n t s s h e a r s t r e s s ( t h e s ig n o f th e s h e a r s t r e s s i s a r b i t r a r y ) . A le n g th 0 2 , e q u a l to th e m agnitude o f p i , i s l a i d o u t a lo n g th e h o r i z o n t a l a x is and w ith C a s a c e n te r a c i r c l e o f r a d iu s -g p^ i s d e s c r ib e d th ro u g h p o in ts 0 and 2* An a n g le 2 / i s m easured o f f from th e h o r i z o n t a l a x i s , i t s r a d iu s CA i n t e r s e c t i n g th e c i r c l e a t A. Thus th e a b s c is s a and o r d in a te o f p o in t A r e p r e s e n t th e m agnitude o f th e norm al and s h e a r com ponents o f s t r e s s on p la n e a - a s in c e e q u a tio n s ( 1 ) and ( 2 ) a r e s a t i s f i e d by th e g e o m e tric a l con s t r u c t i o n . L ength OA i s t h e r e s u l t a n t s t r e s s , R. The c i r c l e ,
t h e r e f o r e , r e p r e s e n ts th e l o c i o f a l l p o i n t s , such a s A , whose
c o o r d in a te s d e te rm in e th e s h e a r and norm al s t r e s s com ponents on p la n e s p a sse d th ro u g h p o in t P .
The random o r i e n t a t i o n o f th e specim en i n F ig . 3a and hence th e random d i r e c t i o n o f w ith r e s p e c t to t h e ax es of t h e Eohr d iag ram i n F ig # 4 i l l u s t r a t e th e method o f u s in g a “p o l e ," P , t o d e te rm in e th e p o in t A by draw ing p a r a l l e l l i n e s to th o s e o f F i g . 3a on t h e s t r e s s c i r c l e i n s t e a d o f la y in g o f f th e a n g le 2$ i n th e p ro c e d u re j u s t s t a t e d . A l i n e i s drawn from p o i n t 2 p a r a l l e l t o th e d i r e c t i o n o f th e maximum p r i n c i p a l s t r e s s p la n e i n Fig# 3a# I t s p o in t o f i n t e r s e c t i o n w ith th e s t r e s s p la n e i s c a l l e d th e " p o le " f o r th e c i r c l e . Any l i n e , su ch a s PA, drawn from th e p o le p a r a l l e l to a random p la n e th ro u g h p o in t P
i n th e specim en w i l l g iv e th e r e q u ir e d i n t e r s e c t i o n ( e . g . p o in t A) w ith th e s t r e s s c i r c l e f o r t h a t p la n e . I t i s more c o n v e n ie n t,
g e n e r a l l y , t o o r i e n t th e h o r iz o n ta l a x is i t s e l f p a r a l l e l to th e maximum s t r e s s p la n e so t h a t th e p o le P w i l l c o in c id e w ith th e i n t e r s e c t i o n o f th e c i r c l e w ith th e h o r i z o n t a l a x i s . The p o le i n th e Kohr d iag ram i s o f p a r t i c u l a r u se i n l o c a t in g th e d i r e c t i o n s o f th e p r i n c i p a l s t r e s s e s and p r i n c i p a l s t r e s s p la n e s i n c ase th e p r i n c i p a l s t r e s s d i r e c t i o n s a r e th e unknown q u a n t i t i e s .
2 . T r i a x l a l L o ad in g . C o n sid e r now a n o th o r c o n d itio n o f lo a d - in g such a s i s o b ta in e d in a " t r i a x i a l co m p ressio n " t e s t i n g m ach in e. I f an im p e rv io u s j a c k e t su rro u n d s th e specim en as i t i s immersed i n a l i q u i d u n d e r a c o n f in in g p r e s s u r e , p , and th e n th e specim en I s sub je c t e d t o an a x i a l p r e s s u r e , A p , in a d d itio n t o th e p r e s s u r e p , we have f o r a u n i t volume o f th e specim en c u t o u t by p la n e s p a r a l l e l t o th e d i r e c t i o n s o f th e p r i n c i p a l s t r e s s e s th e lo a d in g shown in F i g . 5 a . In t h i s s k e tc h = p + A p , and pg s - p . The s t r e s s pg i s a c tin g i n a d i r e c t i o n p e r p e n d ic u la r to th e p la n e o f th e p a p e r .
F o llo w in g th e p ro c e d u re o f r e s o lv in g th e f o r c e s n e c e s s a ry f o r e q u ilib r iu m o f a sm all volume dV ( F i g . 5b) i n a h o r i z o n t a l and v e r t i c a l d i r e c t i o n and s o lv in g f o r s fi and s g i n term s o f p ^ , p ^ , and we
I f we p l o t th e Mohr s t r e s s c i r c l e t o r e p r e s e n t th e r e l a t i o n g iv e n i n e q u a tio n s (7 ) an d ( 8 ) , i t w i l l a p p e a r as i n P i g . 6 . Kota t h a t th e s ig n o f th e s h e a rin g s t r e s s i s m e re ly c o n v e n tio n a l, d ep en d in g upon o b ta in :
«n * I ( P i + P3) + i ( P i - Pj)oos 2 /
s s = i (pt - P3 ) sin 2/
( 3 )
a.
« - P. ib
Figure 5
IsK&tOK 2 — CO*sp«.£$$iOW F ig u r e 618 w h e th er th e a n g le / i s g r e a t e r or l e s s th a n 9 0 ° , C om pressive s t r e s s e s a r e ta k e n in th e p o s i t i v e d i r e c t i o n o f th e h o r i z o n t a l a x i s , w h ile t e n s i l e s t r e s s e s a r e c o n s id e re d n e g a t i v e . QZ r e p r e s e n ts a g a in th e m ag n itu d e o f p ^ , l a i d o f f in a d i r e c t i o n to th e r i g h t o f th e o r i g i n ( t o s i g n i f y c o m p re s s io n ). OX r e p r e s e n ts th e v a lu e o f p , a l s o a 3
c o m p ressiv e s t r e s s . An i d e n t i c a l s t r e s s c i r c l e would have been ob ta in e d i f th e e le m e n ta l volume dV and th e p la n e a - a had b een d is p o s e d i n th e same a t t i t u d e w ith r e s p e c t t o p as i t i s w ith r e s p e c t to p
2 3 i n F i g . 5 a , inasm uch as p - p i n t r i a x i a l lo a d in g . 2 3 I n F i g . 6 th e o r i e n t a t i o n o f th e h o r iz o n ta l a x i s i s p a r a l l e l to th e p la n e o f th e maximum p r i n c i p a l s t r e s s , so t h a t th e p o le P , d e s c r ib e d i n F i g . 4 , c o in c id e s w ith p o i n t X in F i g . 6 * The l i n e XA i s p a r a l l e l to th e p la n e a - a o f F i g . 5 . The li n e OA r e p r e s e n ts th e m ag n itu de o f th e r e s u l t a n t s t r e s s , R, on th e p la n e a - a , and 9 i s th e a n g le betw een th e r e s u l t a n t and t h e norm al to t h e p la n e a -a *
3* Pure S h e a r. A n o th er ty p e o f s p e c i a l lo a d in g w hich may be im posed upon a body i s t h a t o f ’’p u re s h e a r . ** I f th e sp ecim en i n F i g . 7 i s s u b je c te d t o a co u p le a s shown, th e r e s i s t a n c e t o d e fo rm a tio n i s s e t up a s a s h e a rin g s t r e s s a lo n g th e p la n e a - a a s i t i s in f lu e n c e d t o move i n r e l a t i o n t o a p a r a l l e l p la n e a * - a * . The s h e a rin g s t r e s s i s I * P J A, assum ing t h a t th e s h e a rin g s t r e s s i s d i s t r i b u t e d u n ifo rm ly o v e r th e c r o s s - s e c t i o n a l a r e a A c u t o u t by th e p la n e a-& .*
* T h is a ssu m p tio n i s p ro b a b ly n o t J u s t i f i e d f o r th e s t r e s s c o n d itio n a t any p o in t th ro u g h o u t th e sp e c im e n , f o r i t i s p ro b a b le t h a t a com plex s t r e s s d i s t r i b u t i o n i s e v id e n t n e a r th e b o u n d a rie s o f th e sp ec im e n . The
same com plex s t r e s s d i s t r i b u t i o n p ro b a b ly e x i s t s i n t h e sim p le t e n s i l e t e s t sp ecim en a lr e a d y d e s c r ib e d , b u t i n e ac h c a s e , i f th e specim en i s s u f f i c i e n t l y l a r g e , th e fu n d a m e n tal d e f i n i t i o n f o r s t r e s s ^ L e .
s - F o rce d iv id e d by a r e a , sh o u ld be s u f f i c i e n t l y v a l i d f o r p o i n t s w e ll i n th e i n t e r i o r o f th e b o d y .
--- CL d * t i l l
F ig u r e 7
- X - -F i g u r e 8An e le m e n ta l cube in th e body, shown i n Fig* 7b i s a c te d upon by th e t a n g e n t i a l or s h e a r in g s t r e s s e s , T, on i t s s u rfa c e s * A random p la n e b - b , p a sse d th ro u g h th e elem en t w i l l , from th e re q u ire m e n ts o f e q u ilib r iu m o f th e f r e e b o d y , have a norm al t e n s i l e s t r e s s e q u al i n m agnitude to T a c tin g a c r o s s th e p la n e (w ith no s h e a r com ponent) i f
& - 450 , and a norm al co m p ressiv e s t r e s s e q u a l to T i n v a lu e i f
& - 1 3 5 °. Then th e c o m p ressiv e s t r e a s (= T, & *135°) may be c o n s id e re d th e maximum p r i n c i p a l s t r e s s and the t e n s i l e s t r e s s
P g (s T» /3 = 4 5 ° ) th e minimum p r i n c i p a l s t r e s s * Under th e c o n d itio n s o f lo a d in g s p e c i f i e d , th e in te r m e d ia te p r i n c i p a l s t r e s s i s s Q, w hich a l g e b r a i c a l l y i s a c t u a l l y in te r m e d ia te in v a lu e betw een p^ and p^*
The e q u ilib r iu m c o n d itio n on a sm a ll volume dV i n Fig* 7o may be e x p re s s e d i n te rm s o f sn , s g , T, and f o r any a s s ig n e d v a lu e o f
Q • From su ch c o n d itio n s we o b ta in th e r e l a t i o n s ; s n • T s i n 2& (5 )
s s - T cos 2j3 ( 6 )
On th e o th e r h an d , i f s„ and s„ w ere d e term in e d i n term s o f t h e i rJX 5 r e l a t i o n t o th e p r i n c i p a l s t r e s s e s , p^ and p ^ , and th e a n g le
( s 135° - /3 ) , th e r e s u l t i n g e q u a tio n s would be i d e n t i c a l w ith e q u a tio n s (3 ) and ( 4 ) .
’The s t r e s s c i r c l e f o r !lp u re s h e a r" i s shown in F ig , 8 , Note th e p o s i t i o n o f th e p o le P f o r t h i s diagram * I f t h e t e n s i l e s t r e s s p^ i s c o n s id e r e d th e maximum p r i n c i p a l s t r e s s , th e n th e p o le P w ould be throw n i n t o p o s i t i o n P* as shown*
4# Pure S h ear and Com pression* I f a specim en such as th e one shown i n Fig* 7a were " f i r s t s u b je c te d t o a u n ifo rm h y d r o s t a t i c p r e s s u r e .
b
VC .
4
a
p , b e fo re th e im p o s itio n o f th e c o u p le F , th e q u e s tio n a r i s e s as t o th e e f f e c t o f t h i s c o n f in in g p r e s s u r e upon th e Mohr Diagram o f Fig* 8 * I n c o n fo rm ity w ith th e d is c u s s io n u n d er P a r . 3 , l e t th e c o u p le F a o t i n t h e p la n e o f th e p a p e r as i n F i g . 9a* T hus, th e in te r m e d ia te p r i n c i p a l s t r e s s pg i s e q u a l to th e c o n fin in g p r e s s u r e , p* A ls o , l e t th e m agnitude o f p be l e s s th a n t h a t o f T . Then th e m ag n itu d es o f th e o th e r p r i n c i p a l s t r e s s e s w i l l bet
p-^ » T + p (c o m p re ssiv e s t r e s s ) j v. Pg = T - p ( t e n s i l e s t r e s s )
As b e f o r e , th e p la n e o f th e minimum p r i n c i p a l s t r e s s w i l l c o r r e s pond t o 0 - 4 5 ° , and th e p la n e of th e maximum p r i n c i p a l s t r e s s w i l l o c c u r when 0 ~ 1 3 5 °. S o lv in g f o r s n and s s (F ig * 9a) i n t e r n s o f T,
0 , and p , we o b ta in t en = T s i n 20 - p (7 ) j s s - T cos 2 0 ( 8 ) F ig u re 9c i l l u s t r a t e s th e Mohr Diagram f o r p la n e s p a r a l l e l to th e d i r e c t i o n o f p g , w h ile F ig . 9d shows th e s t r e s s r e l a t i o n s a c r o s s p la n e s p a r a l l e l to th e d i r e c t i o n o f p g . As m ig h t be s u rm is e d , b o th th e s t r e s s c i r c l e s of F i g s . 9c and d co u ld r e s u l t i n a specim en from an a x i a l c o m p ressiv e s t r e s s • p 4- T, a l a t e r a l c o m p ressiv e s t r e s s o f p^ s p , and a l a t e r a l t e n s i l e s t r e s s o f pg = T - p a t r i g h t a n g le s t o th e d i r e c t i o n o f p ^ . P r a c t i c a l d i f f i c u l t y would p e rh a p s be e n c o u n te re d in p ro d u c in g such s t r e s s e s , how ever, so t h a t i t w ould seem , fro m an
e x p e rim e n ta l p o in t o f v ie w , t h a t th e " i n d i r e c t ap p ro ach " u t i l i z i n g t r i a x i a l co m p ressio n and a superim posed couple would f u r n i s h th e means o f s e c u rin g su ch a c o n d itio n o f lo ad in g *
z
M — ^ P_ -/> o ^ ys>. oF ig u r e 1 0 ,
I t sh o u ld be n o te d fro m th e d is c u s s io n s o f p u re s h e a r and pu re s h e a r combined w ith c o n f in in g p r e s s u r e , t h a t , w henever a p la n e i n a body u n d er a s t a t e o f s t r e s s has o n ly a s h e a r o r t a n g e n t i a l s t r e s s a c t i n g upon i t w ith no norm al com ponent, th e n one o f th e p r i n c i p a l s t r e s s e s has a d i f f e r e n t s e n s e i n r e l a t i o n t o th e o th e r two p r i n c i p a l s t r e s s e s *
5* G en eral S t a t e o f S t r e s s * Thus f a r , s e v e r a l s p e c i a l c o n d i t i o n s o f lo a d in g have been c o n s id e re d * In each c a s e , th e p la n e s p a sse d th ro u g h a c e r t a i n p o in t i n a body in a s t a t e o f s t r e s s have been
p a r a l l e l t o one o f th e th r e e p r i n c i p a l s t r e s s d ir e c t i o n s * I t i s p o s s ib le to examine th f s t a t e o f s t r e s s on o th e r p la n e s a t random d i r e c t i o n s t o t h e p r i n c i p a l s t r e s s ax es as w e l l . Two m ethods o f c o n s tr u c ti n g M o h r's S t r e s s Diagram w i l l be p r e s e n te d . The f i r s t m ethod i s b a se d upon th e same “e q u ilib r iu m o f f o r c e s ” ap p ro ach u sed h e r e t o f o r e , w h ile th e seco n d method u t i l i z e s th e p r i n c i p l e o f
r e s o l u t i o n o f s t r e s s com ponents* a* Method A*
F ig u re 10 r e p r e s e n t s a v e ry s m a ll body a t p o in t P w hich i s bounded by p la n e s norm al t o th e th r e e p r i n c i p a l s t r e s s e s and by a f o u r t h p la n e , ABC, whose norm al i n t u r n makes d i r e c t i o n a n g le s
and Y w ith th e p r i n c i p a l s t r e s s d ir e c t i o n s * D enote th e d i r e c t i o n o f th e s t r e s s p^ by ZO* p^ by YG, p^ by XQ* F o r c o n v en ie n c e o f i l l u s t r a t i o n , a l l th e p r i n c i p a l s t r e s s e s i n t h i s d e r i v a t i o n a r e assumed c o m p re s s iv e , a lth o u g h th e p ro o f i s a p p li c a b le t o any s ig n s o f th e p r i n c i p a l s t r e s s e s . F u r th e r l e t
EMS7 fLH$. 0X1 P ° Y = P*
