Plasticity in rock mechanics

Int. J. mech. Sei.

PergamonPress. 1971. Vol.13, lap.291-297. Printedin GreatBritain

PLASTICITY

IN ROCK MECHANICS

~-I. LIFPMANlV Lehrstuhl B f'dr Mechanik, Teehnische Universit~t, Pockelsstrasse 14, D-33 Braunschweig, West Germany (Received

24 J u l y 1970, a n d i n revised f o r m 27 November 1970)

Dedicated to K. KLOTTER,Darmstadt, on the occasion of his 70th birthday 28 December 1971 Summary--The Coulomb yield condition and its associated flow rule is used to solve the title problem, i.e. what internal supporting forces would be necessary to prevent granular rock from breaking into horizontal, rectangular section coal-mining tunnels. The analysis based upon an upper and lower bound approach, shows several peculiarities so that it could be of interest in itself. 1. INTRODUCTION Tins paper concerns a practical problem in coal mining,* namely, by what forces can rock material be prevented from breaking and falling vertically into rectangular, horizontal tunnels. The first simple plane strain approach, which is presented here, uses the upper and lower bound techniques of classical rigid-plastic theory being based on a plastic potential, however with the following instructive peculiarities :1 (a) The flow rule associated with the Coulomb yield criterion gives rise to a volume dilatation. (b) Volume forces m a y not be neglected. (e) The upper and lower bound theorems yield conversely lower and upper bounds to the force required in this problem. (d) Velocity discontinuities along internal surfaces are admissible, but involve also jumps of the normal component. I t must be mentioned t h a t the Coulomb yield criterion 2,3 truly seems to represent a quite reasonable limit condition at least for the dry granular state in which the rock is assumed to be, but t h a t the applicability of the potential flow law is problematic. ~,5 One of the main objections against it is experimental evidence of the existence of volume compressibility, while the theory predicts an expansion only. Brown e therefore generalized it just by splitting off the volume change which has to obey a different equation. On the other hand, it could be shown t h a t a nearly arbitrary volume behaviour corresponds to the potential law, if its Coulomb form is dropped and replaced by a more complicated one (concerning the statics, see, for example, refs. 7 and 8) which may, moreover, allow for strain hardening or softening, or for rate effects. Mandl and Luque, g instead, preferred to drop the coaxiality between * This paper is a par~ of work stimulated and sponsored by the Bergbau-Forschungsinstitut, Essen (West Germany) and was presented to the Plasticity Teaching Symposium, Manchester (U.K.), 30 June-1 July 1970. 291

292

H. LIPPMANN

t h e tensors of stress a n d s t r a i n - r a t e , which m a y be done w i t h o u t violating t h e a s s u m p t i o n o f i s o t r o p y for p l a n e s t r a i n only. W e shall r e t a i n C o u l o m b ' s condition t o get a sufficiently simple a p p r o a c h which can t h e n be e x t e n d e d b y n u m e r i c a l m e a n s to m o r e realistic a p p r o x i m a tions. B u t we believe t h a t t h e first step p r e s e n t e d here is in f a c t s a t i s f a c t o r y since t h e r o c k before b r e a k i n g into t h e t u n n e l has o b v i o u s l y a m a x i m a l l y condensed state, so t h a t it n e v e r can condense b u t o n l y e x p a n d , in a c c o r d a n c e w i t h t h e analysis. T h e r e is a second a r g u m e n t due t o Sacchi a n d S a v e 1° which will be p r e s e n t e d here o n l y reservedly. Since we are m e r e l y looking for a statical limiting state, t h e n a t u r e of t h e flow law does n o t p l a y a n y role. So we m a y chose it a t will, o n l y for t h e p u r p o s e of simplifying t h e m a t h e m a t i c a l calculations. Finally, t h e r e is t h e question as to w h e t h e r t h e r o c k i n v o l v e d really possesses t h e g r a n u l a r consistency which is a s s u m e d for t h e t h e o r y ; this can be a n s w e r e d p o s i t i v e l y b y t h e a u t h o r ' s o w n observations. A n y a d d i t i o n a l f r a c t u r e process, if it occurs, would lead c e r t a i n l y to higher n e c e s s a r y loads so t h a t o u r e s t i m a tions are safe. O f course, t h e r e m a y exist a t o t h e r places quite different situations. C h e a t h a m zl has r e v i e w e d the l i t e r a t u r e on r o c k mechanics a n d Shield 1~ discussed t h e t h r e e - d i m e n s i o n a l Coulomb criterion w i t h its associate flow rule; Cox et a l . 13 considered a x i s y m m e t r i c d e f o r m a t i o n a n d specific p l a n e strain solutions are g i v e n in refs. 14-16; for a n i s o t r o p y , see ref. 17. One of t h e first i n v e s t i g a t i o n s on plastic limit design in a g r a n u l a r m a t e r i a l is due to D r u c k e r a n d Prager. 1 2. B A S I C E Q U A T I O N S In the case of isotropic plane strain, we adopt Coulomb's criterion, ~, 3 thus f = ½ ] (p= --p~)Z + 4~"~ It -- ½(p= + p ~ ) s i n ¢ -- c cos ¢ ~<0

(l)

with the associated flow rule z~ derived from the potential law, ~j~ = ,~(a.f/aa~), K~ = ~ i ( p _ p ~ ) 2 + ~ 2 ij K, = ~

[

( p ~ - p , ) ~ + 4-r~ I'~

(2)

X = X](p _p,)~+4r~ it, = const~>0: angle of internal friction; c = tonsil>0: cohesion; Px = -axx, p~ = - a ~ , pressures; ~ = - - a ~ , shear stress; K= = - ~ = z , K~ = - ~ , X = - ~ x ~ , strain rates; ~, see equation (4) below. In equations (2), the singular state of stress p~ = p~ = - c cotan¢, r = 0 is excluded. To do so, is also reasonable from the physical point of view otherwise, an all-sided isotropic expansion of material might be superimposed by which the grains loose their mutual contact and no longer form a solid continuum. Requiring the rate of dissipation density, A = K=p=+K~,p~+2x'r

= ~[] ( p z - - p ~ ) ~ + 4 " r ~ it-- (p~+p~) sine] = ~c eos¢,

(3)

to be non-negative in the plastic domain f = 0, we get, observing (2), = I (K,t--tct,)('Jr 4X ~' lt~>O

(4)

P l a s t i c i t y in r o c k mechanics

293

as well as t h e r a t e of compression, =

K,.+K,,

=

-~sin¢<0.

(5)

Thus, v o l u m e does steadily increase which, as n o t e d above, is c e r t a i n l y correct for t h e a p p l i c a t i o n considered in this paper. T h e well-known statics of g r a n u l a r media* is based on f -- 0 in c o m b i n a t i o n w i t h t h e conditions o f equilibrium,

ap~ ~r P=,} ~+~=

(6)

~r+Op~ Pj denotes v o l u m e forces.

3. T H E

UPPER-

AND

LOWER-BOUND

THEOREMS

These t h e o r e m s hold for a c o n v e x yield surface in stress space---as it is defined b y t h e Coulomb eriterion12--and m u s t be f o r m u l a t e d i n d e p e n d e n t of b o u n d a r y conditions (refs. 18 a n d 19). D e n o t e b y p = (p~, p , , ~) pressures or stresses, creating surface t r a c t i o n s T = ( T~, T,) a n d b y m e a n s of t h e e q u i l i b r i u m e q u a t i o n (6), v o l u m e forces P = (P=, P,). u = (u, v) are m a t e r i a l velocities a n d K = (K=, K,, 2X) t h e associated strain rates. T h e n t h e correlated p o w e r W can be expressed in t e r m s of t h e e x t e r i o r forces T, P a n d t h e velocities u as well as b y t h e i n t e r n a l p o w e r d e n s i t y A which depends on p, K, i.e. on p, u : W = W~(T, P, u) = W~(p, u).

(7)

T, P, u, p, K are assumed f r o m n o w on to define t h e " t r u e " state. The e q u a l i t y of e x t e r n a l a n d i n t e r n a l r a t e of dissipation, as f o r m u l a t e d in e q u a t i o n (7), holds e v e n u n d e r m u c h m o r e general b u t statical conditions. T h e k i n e m a t i c a l q u a n t i t i e s u* or u* are said to be admissiblet if, o n l y b y m e a n s of t h e flow rule a n d t h e yield condition, t h e r e can be f o u n d a t least one associated stress s t a t e p* (governing T*, P*). T h e n t h e " u p p e r - b o u n d t h e o r e m " W,(T, P, u*) < W~(p*, u*)

(8)

holds. I t m e a n s t h a t t h e t r u e s t a t e of stress dissipates, m u t u a l l y w i t h a n y admissible v e l o c i t y field, no m o r e e x t e r n a l w o r k t h a n t h e admissible k i n e m a t i c a l s t a t e w o u l d do in t h e interior b y its own. On t h e o t h e r hand, a pressure field p0 (creating t h e e x t e r n a l loads p0, T 0) is called admissible if it does n o t v i o l a t e t h e yield criterion, so t h a t f~< 0 according to e q u a t i o n (1). T h e n t h e " l o w e r - b o u n d t h e o r e m " becomes W~(To, po, u) ~ We(T, P, u).

(9)

This m e a n s t h a t t h e admissible s t a t e of stress produces a t m o s t as m u c h e x t e r n a l w o r k m u t u a l l y w i t h t h e t r u e velocities as t h e t r u e s t a t e of stress dissipates in t h e interior. E x t r e m u m conditions of a s o m e w h a t different k i n d are set u p in ref. 20. I f we t r y to solve t h e flow rule (2) a n d t h e yield criterion (I) w i t h respect to t h e stresses, t h e n we find, o b s e r v i n g (4), t h a t this would be possible if only * 2 + 4 x * { { s i n ¢ = 0. K.* + K,* + I (K.* -- K,)

(10)

Therefore, e q u a t i o n (10) represents t h e {only) condition of admissibility of t h e k i n e m a t i c a l field u*, K*. I t yields, using e q u a t i o n s (5), (4), (3), A* = --c~* c o t a n ¢ ,

~* = K*-bK~*.

A* is t h e p o w e r d e n s i t y a n d ~* t h e r a t e of compression. t " C o m p a t i b l e " according to ref. 18.

ill)

294

H. LIPPMANIV

Consider finally a surface S across which the velocity u* jumps by Au* = (Au*, Av*), see Fig. I. F r o m IAu* ] = ](Au*)~+(Av*) ~ ]i,equation (I0) and

Au* K~*~---Ax-x,

K'Z0,

X*--

1 Av* 2 Ay

it follows that in the limit Ax -> 0, so that Au* s i n e = ] Au* l"

"N

FIG. 1. Layer, thickness Ax, of velocity discontinuity S - S . Au*, Av*.

J u m p vector Au*, components

Therefore, a n y admissible jump vector Au* is inclined at angle ¢ to the jump surface S, and for ¢ ¢ 0 no simple shear discontinuity is possible. The shear zone produces volume!t I f we define by A,* = A*Ax and passing to the limit Ax-> 0, the power density per unit area at the jump surface S then we get from (11) and Au* = [A v*] t a n ¢ that A*, = c A u * cotan 4 = c l a y *

I

(12)

holds as in the plane strain deformation of a metal. 4. A P P L I C A T I O N The horizontal tunnel A B C D of rectangular cross-section and side lengths a, b, see Fig. 2, is internally supported b y a vertical surface pressure T to avoid a vertical break in of granular rock. We assume that it lies deep in the earth, i.e. h ~>(b]2) cotan 4,

(13)

and recognize b y (10) and Fig. 1 that the vertical constant velocity field u * > 0 in the triangle A E B , which disappears outside that triangle, is admissible. Dissipation takes place only at the jump surfaces A E , E B ; external power is generated b y T along A B a n d b y the volume forces P = (y, 0), (14) is the specific weight of the rock in A E B . Thus, the upper-bound theorem of equation (8) yields per unit width, yu*(b*/4) cotan 4 - T u * b <<.bcu* cotan 4 and gives a lower bound for T according to T I> {(b/4) ), - c} cotan 4"

(15)

t Since this effect is disregarded, the kinematics assumed in refs. 14 and 15 is not admissible.

Plastieity in rock mechanics

Y

"///////'

77-77-.77-77 / / / /

l P~-r

295

x'

®

\

®

I/

F~(L 2. H o r i z o n t a l r e c t a n g u l a r t u n n e l A B C D , a t t h e d e p t h h u n d e r t h e e a r t h . A d m i s s i b l e v e l o c i t y field u* i n t h e t r i a n g l e A B E , a d m i s s i b l e p r e s s u r e fields i n t h e d o m a i n s 0, 1, 2 b o u n d e d b y circles.

A s a n e x a m p l e : i t b e c o m e s n e g a t i v e or does n o t r e q u i r e a n y s u p p o r t for 7 = 2"5 × l 0 s kp/m 8, i f b ---- 2 m, ¢ = 30 °, w i t h for solid r o c k s c = l 0 Tkp/m ~. F o r ideal g r a n u l a r m a t e r i a l c = 0, t h e (small) p r e s s u r e T~> 0.2165 Icp/cm 2 m u s t b e a p p l i e d .

296

H. LIPP~N A n admissible pressure state 1a° m a y be established b y

and

~+Y9

= 0,

~ + ~ - y = o,

which gives using (14) a n d (6) po = p .

(17)

Thus, t h e v o l u m e forces corresponding to the assumed pressures are e q u a l to t h e t r u e ones, a n d t h e a u x i l i a r y pressures ~ = (Px, ]5~, ~) are equilibrated w i t h o u t a n y v o l u m e forces. Therefore, in polar co-ordinates r, ~, see Fig. 2, ~ = (/3~,/3~, ~¢~) and t h e conditions of equilibrium become

~-+7(~-~0)

= o,

- & - + 7 ~ o = 0.

(18)

N o w t h e following stresses, different in t h e ring domains 0, 1, 2 of Fig. 2, are seen to be radially continuous a t t h e i r boundaries r = R0, r = R ; t h e y disappear at x = 0 and fulfil (18) ( k , R = const): domain0:

P~ = Pa = P z = P~ = - 2 k i n ( R / R o ) ,

d o m a i n 1:

~

d o m a i n 2:

~Sr = P ~ = ~ a = / 5 ~ = ~ = ~ =

= -2kin(R/r),

~

= 2k+i5~,

~

~r~ = ~ = 0;

= 0;

0.

I f R is arbitrarily fixed b e t w e e n (h + ½a) a n d R 0 = ½(a~+ b~)i, a n d k is defined b y k = ~(h+~a-R) sin¢+c cos¢ 1 + (2 In ( R / R e ) - l] sin ¢ '

(19)

t h e n observing e q u a t i o n (16), t h e condition (1) for admissibflity~f can be shown to hold everywhere. Therefore, t h e surface pressure created b y t h e a b o v e stresses of d o m a i n 0 a n d e q u a t i o n (16) inside the tunnel, T O = ~ h - 2 k in (R/Re),

(20)

m a y be considered to be an u p p e r b o u n d

T°>~T of t h e t r u e a v e r a g e pressure T. F o r t h e lower-bound t h e o r e m , (9) reduces to - T o u ~< - Tu, since t h e contributions of t h e v o l u m e forces p0, p are, because of (17) equal at b o t h sides. A s s u m i n g t h e t u n n e l to be at a great d e p t h in t h e earth, i.e. h i> a, b a n d f u r t h e r m o r e R = ½(a + h) a n d [ln (R/Ro)] sin ¢ >~ 1, so t h a t we get T < T O~ (Fh/2) - c c o t a n ¢. This gives for instance, h a l f t h e weight ~h of t h e r o c k a b o v e t h e t u n n e l for an ideal granular material, i.e. w i t h c = 0. I f we should, however, a d o p t again t h e d a t a already used to illustrate t h e i n e q u a l i t y (15), i.e. ~ = 2.5 × 10 a k p / m a, b = 2 m , a n d additionally a = 2 m, h - 1000 m a n d solid r o c k c = 107 k p / m ~, t h e n T O w o u l d again become n e g a t i v e which m e a n s t h a t no supporb of t h e ceiling of the t u n n e l w o u l d be necessary a t all. REFERENCES 1. D. C. DRUC~_ER a n d W. P~AGER, Q. a p p L M a t h . 19, 157 (1952). 2. L . M i ~ , D e r F e l s b a u , Vol. 1. E n k e Verlag, S t u t t g a r t (1963). 3. A. CAQUOT u n d J . K~RISET., G r u n d l a g e n d e r B o d e n m e c h a n i k , 3rd edr~. Verlag, Berlin (1967).

Springer-

Because of its invariance, it remains like (] 6), u n c h a n g e d if T ----Tx~ is replaced b y vra, a n d t h e subscripts x, y b y r, ~ respectively.

Plasticity in rock mechanics

297

4. G. GVD~HUS, Elastoplastische Stoffgleichungen f'tir trockenen Sand. HabflitationThesis, Karlsruhe, 1970. 5. J. KRAVTCHENKOand P. M. SmI~YS (eds.), Rheology and Soil Mechanics. Symposium, Grenoble, 1964. Springer-Verlag, Berlin (1966). 6. E. H. B~ow~, Applied Mechanics (Edited b y H. GSR~LER), pp. 183--191, Congress, Munich, 1964. Springer-Verlag, Berlin (1966). 7. H. SCHLECH~VEG, Z. angew. Math. Mech. 38, 139 (1958). 8. I-I. SCHLECHTWEG,Z. angew. Math. Mech. 39, 82 (1959). 9. G. ~ L and R. FE~t~hNDEZ LUQUE, Gdotechnique 20, 277 (1970). 10. G. SACCHI and M. SAve, Meccanica 3, 43 (1968). 11. J. B. CHEATH.¢~, JR., Appl. Mech. Rev. 20, 923 (1968). 12. R. T. SHIELD, J. Mech. Phys. Solids 4, 10 (1955). 13. A. D. COX, G. EASON and H. G. HOPKINS, Phil. Trans. Roy. Soc. London A 254, 1 (1961). 14. J. B. CHEATHAM,P. R. PASLAY and C. W. G. FULCHER, Trans. A S M E J. appl. Mech. 35, 87 (1968). 15. P. R. P A s t Y , J. B. CHEATHAMand C. W. G. FUZCHER, Trans. A S M E J. appl. Mech. 35, 95 (1968). 16. E. A. M ~ S H ~ , Acta mech. 3, 82 (1967). 17. J. BOSCHATand D. RADE~--KOWC,Z. angew. Math. Mech. 42, T89 (1962). 18. M. SAYIR and It. ZI~GLER, Z. ange$. Math. Mech. 29, 78 (1969). 19. I. F. COLLINS, J. Mech. Phys. Solids 17, 323 (1969). 20. D. RADENKOV/C, C. r. Acad. Sci., Paris 252, 4103 (1961).

22