Rock plasticity in conditions of variable deformation rates

ROCK MECHANICS AND ROCK PRESSURE

ROCK PLASTICITY IN CONDITIONS OF VARIABLE DEFORMATION RATES

A. N. Stavrogln and A. G. Protosenya

Many problems in the development of mineral deposits, and the construction of subway lines and tunnels in seismically active regions are associated with the solution of problems of elastoplastlc-wave propagation in massive rocks. The consideration of these problems is based on dynamic dependences between the stresses and strains.

The influence of the deformation rate ~ on the dependence between the stress u and strain has been of interest to investigators for a long time. The first approach was the theory

of elastoplastlc-wave propagation of [i], which is based on the relation

a - ~(~,), C1)

where ~ is a known function.

This theory relates to a finite class of processes, and can be used at very large and very smell deformation rates.

To approximate the theoretical calculations to the experimental data, a second approach was formulated [2-4]. This is based on the assumption that there exists a dependence

�9 (a, o, ~, ~ ) - O. (2)

E q u a t i o n s o f s t a t e w e r e p r o p o s e d t o d e s c r i b e t h e b e h a v i o r o f s o i l s [ 5 ] , on t h e b a s i s t h a t t h e d e n s i t y o f t h e medium p and t h e p r e s s u r e P a c t i n g on i t a r e r e l a t e d a s f o l l o w s

p = l , Cp), ( 3 )

where f~ is a known function.

One of the principal assumptions in these equations is the dependence of the second in- variant of the stress deviator on the pressure P.

In the present work, on the basis of experimental data, the basic assumptions of the determining equations of rock-plasticity theory are chosen, taking account of rock deformation.

Experimental Data

With the aim of establishing the influence of thedeformatlon rate on the behavior of rock experimental investigations of a large range of rocks in conditions of complex stress states of the form o~ ~ oa = us have been performed in the Laboratory of Dynamic Strength and High Pressures of VN~II, [6-8]. The deformation rate varied by 10-12 orders of magnitude from i0 -I~ to 10 2 sec-1. Samples in core form were deformed by a force applied in the axial direction under equal values of lateral pressure a2 = as, which remained constant during the course of a given experiment.

The dependence of the strength z s - (oI -- a~)/2 is shown in Fig. la, as well as the elastic limit z e (see Fig. Ib) for white, uniformly granular marble of the Koelg deposit on the rate of longltudlnal (axial) deformation ~, with various lateral pressures a2: 1) a2 = 0; 2) 20! 3) 50; 4) i00 MPa. It is evident from the figures that, with increase in deformation rate ~, the strength and elastic limit increase. The experimental curves for the re- sldual deformation AE~ on log ~ at the strength limit with different values of the lateral pressure o2 are shown in Fig. 2, and analogous curves for the residual bulk deformation of expansion 0 in Fig. 3; for the conventional notation, see Fig. i. It is evident from Figs.

G. V. Plekhanov Iiining Institute, Leningrad. Translated from Fizlko-Tekhnicheskle Prob- lemy Eazrabotkl PoleznykhIskopaemykh, No. 4, pp. 3-13, July-August, 1983. Original article submitted June i, 1982.

0038-581/83/1904-0245507.50 O 1984 Plenum Publlshlng C o r p o r a t i o n 245

~=, M P z

fSO

50 2

~ ' re , MPa ' �9 �9 b �9 �9 Q 0 ~

! r , ~ 1 7 6 4 _ --.- . 0 �9

_ - - - - _ - - - - . r - - - ' - . . . . . -

....... / _

-~0 ' ' -~ ' ' -,~ ' ' -,~ ' ' -~ ' ' 6~,

Fig. 1

2 and 3 that the residual deformation and plastic disintegration depend in a complex manner on the deformation rate and on the magnitude of the lateral pressure o2. The given results for marble are typical for a wide range of rocks, the structure of which may be characterized as uniformly granular.

Choice of Defining Equations of State

The most important features of the process of rock deformation with different rates and forms of stress state are the dependences of the strength, the elastic limit, and the plastic disintegration on the deformation rate.

The well-known Zhurkov kinetic equation [9] is used to describe the dependences of z s and T e on ~,

(4) e~ = 80 e - ( "~

Equation (4) reflects the kinetic nature of the deformation and destruction of solids. With specific model representations, the constants appearing in this equation take on physical meaning. The quantity ~o is a frequency characteristic of the microprocesses of deformation and destruction and may be numerically equated to the maximum physically possible rate of deformation and destruction of a Solid. The constant uo is interpreted as the activation energy of the processes of diffusion, sublimation, etc., in a crystal lattice that is unloaded by external forces. The quantities T and K signify, respectively, the absolute (Kelvin) temper- ature and the Boltzmann universal constant. The constant 7 is a structure-sensltlve coefficient.

The use of Eq. (4) in a given problem requires a series of reservations. The main objec- tion results from the indeterminacy in the concept of the activation energy uot since the concept of a gram-mole is not defined. Without considering these questions is detail, Eq. (4) can be used as an empirical dependence that describes the exper~,~=ntal results in Fig. 1 reasonably accurately and completely. The bundle of strength and elastic-l~m~t rays emanate from a single pole. Each ray corresponds to a series of experiments obtained with any single value of the lateral pressure on. As the pressure o= is increased, the rays become steeper. The ray curvature is determined by the quantity 7: As y is reduced, the ray curvature in- creases. Any of the rays is described by Eq. (4), with the corresponding value of the coefficient 7. Other constants appearing in Eq. (4) remain unchanged for all the rays.

The conditions of the l!m~tlng states for any single value of the deformation rate ~x " const are described by equations of the form [11, 12]

% = T~e so, (6)

246

/ L '-'-a ' " - ' - - ' - ' ' ' - o - ' - e -4 -2 d~e,

F i g . 2 0 2

where Ts, Te~ A, B are constants; c = Omin/Omax; r = I/2(Omax -- Omin).

Equation (5) describes the strength and Eq. (6) the elastic limit. At the strength limit, T is denoted by v s and at the elastic limit by T e-

Equations (5) and (6) are obtained by vertical cross section of Fig. 1 at any constant value of the deformation rate ix. Change in r in Eqs. (5) and (6) is accompanied by change

O o and Ze, which may be interpreted as the strength and elastic limit in uniaxial compres- in T s sion. The constants A and B are taken to be independent of the deformation rate, in the first approximation. As shown by experiment [7], this approximation holds with an accuracy suffi- cient for practical purposes over a broad range of variation in deformation rate.

o and o The dependence of the constants r s z e on the deformation rate ix can be written in the following form on the basis of Eq. (4)

+-, , . ( . ' - '11 ' <'> t ~o/J II '

,~: [,.. + xr,,f~11. '-. (o)

o 7" characterize the slope of the rays in Fig. i for o2 = 0 and are the The quantities 7s and e constants of the equations Ys " Y~ e-At and Ye = Y~ e-BC, where 7s and 7e are the quantities appearing in Eq. (4) for any of the rays.

Taking account of Eqs. (7) and (8), Eqs. (5) and (6) can be written in the form

[ �9 , , : k so/a ~Ts' (9)

[ �9 �9 o : " "~

\ "o /J ~ " (lO)

The values of the constants appearing in Eqs. (9) and (i0) for a series of rocks are given in [8].

In selecting the defining equations, the strength and elasticity conditions in Eqs. (9) and (i0) are written in terms of invariants of the deformation-rate deviator. The most

247

/ • #'tO~ltSO

/ \ 7

-8 -s -4 -2 o 2 ~, Fig. 3. Dependence of the residual increment in volt-,~ (dilatation) at the strength limit on the deformation rate with various lateral pressures a~. For the notation, see Fig. 1.

%,MPa~20o ~=f~

I | I I I 7~ - 8 - 6 - 4 - 2 0 2 ~ ,

Fig. 4

significant of these is the intensity H of the shear deformation rate or the rate # of maximum shear. For the experimental results and conditions considered here, these quantities may be written in the form

2 (;,_ ;,)= ~, (11)

where ~ --(,i - &~ (L * ,,- w

The correctness of using these invariants in Eqs. (9) and (i0) is confirmed by Fig. 4, where experimental values of the strength are given as a function of the deformation rate ~,, together with the dependence of the strength on the maximum shear rate (~, -- s The shear masnitudes are obtained on the basis of analysis of the results in Figs. 2 and 3.

As is evident from Fig. 4, the two dependences practically coincide. The only differ- ence is that the rate points # are somewhat shifted to the right. The dependence is approxi- mated, as in the case of the deformation rate, by a bundle of rays leaving the same pole. The rightward shift of the points in this case can be neglected. Taking account of the foregoing, the limiting-state conditions in Eqs. (9) and (I0) can be written in the form

re [ux+KT I. r

(12)

248

o1~'/ _

-B-

F i g . 5

�9 w and -w where uz~ Ys, Ye are constant.

The rate of rock dilatation under plastic deformation can be taken as the second defining equation. The experimental data shown in Fig. 3 indicate a significant dependence of the dilatation on the deformation rate.

The dilatation rate 8 is written in the form of a function of the maximum shear rate # and the invariants o and x

=/(-;, o, ~). (14)

In the simplest case, this condition can be written in the form

where A(o, T) is some coefficient,

~te that Eq. (15), taking account of the dilatation, was proposed in [13] as the second limiting condition for the deformation rate of soils. The experimental dependence of # on 8 obtained after analysis of the results in Table 1 is shown in Fig. 5: o2 - 0; x) o2 ~ 20 MPa; A) o2 " 50; 8) o2 = i00 MPa. The coefficient of residual transverse deformation ~ = AE2/AE,. As is evident from Fig. 5, a dependence of the form in Eq. (15) is a reasonable ap- proximatlon.

In establishing the relationship between the stress and strain, the general dependence between the deformation-rate deviator D~ and the stress deviator D o can be taken as the start- ing point

O. = ,p (O.). (16) 8

where ~ is some function.

In the first approximation, Eq. (16) can be limited to the linear relation obtained on series expansion in terms of the tensor D o

D. = SD,, (17) 8

where ,~ is a scalar depending on the invarlants D 6 and D o.

System of Equations

Note that c - ,~min/Omax = (o -- T)/(O + T), where

249

TABLE 1 i ii I

o,,Mh .=

0

~0

0 2O

0 2O 50

0 2O 50

0

50

0 2O 50

0 2O 5O

0

5O

0 50

0 5O

0 0

~,00

E~o ~,~

E4s ~',~o ~,oo 3,23 Z,~o 1,50 E50 ~',1o ~',86

~,48 T,44 ~',20

:,40

T.,O0 t,86 i',~2 0,40 1,07 1,29 1,87 t,47 t,86

t,98 ~,00 i ,~

~,04 2,90

2,88

i,72

2,22 t,08 i,80

t,40 t,62 0,93

2,50 2,50 1,t3

2,00 1,20 1,t2

i,50 1,70 0,86 1,30 l,t8 4,60 1,17

2,30 2,69

i i i i

--A

0,978 t,O00

0,973

i,i04 0,809 0,897

1,068 0,558 0,9~9

0,750 0,855 0,446

1,t43 1,t43 0,59t

t,000 0,636 0,585

0,800 0,889 0,387

0,696 0,624 1,464 0,621 1,091 t,187

~,781 K4~ ?,t49 ~,375

K~ K~ ET00 K~

KT~ Kms E9~ K~

~,~0

K~7 T,3~

T,268 2,29 "[,79o 0,761 1,404

2,038 2,200

1,988 2,430

lu d

T,m Ir,477 i ',9~

Ksoo

~,043 ;r,8~o ~,~o "~,786 ~,~o T,9to T,~53 "~,450 ~,790

~,o82 ~,o~ ~,aoo

~,887 T, t26 ~',49o

T,8oi 2,240

T,38o 0,604 i,080 2,204 2,000 2,026 2,500

t o = -~ (a,,,ax + ~mi.).

Introducing Eq. (18) into Eq. (12), it is found that

F(o, T, ~ ) = 0,

where, for simplicity, ~s has been replaced by T.

Plane Deformation State. form

/

where ~ is the angle between oz and the axls ox.

The dilatation condition is written in the form

(18)

(19)

The stress components in the plastic range are written in the

(20)

250

6+4=a. V(L- (21

The velocity components are written in the form

~u J - - " g = ~ ' u ~in 2 ~ ,

where ~ is the angle between r and the axis ox.

It is assumed that Eq. (19) can be solved with respect to

�9 - ~(a, ~). ( 2 3 )

The s t ress components in Eq. (20) and the deformat ion- ra te components i n Eq. (22) are identical w i t h the plasticity condition in Eq. (23).

Introducing the stress components in Eq. (20) into the equation of motion of the medium it is found that

g..: XsJn2~ = pox

(24)

�9 a~ " { , aa+2T~/~+,:

-67 = - - PoY + P' \ " IF + " ~ T OV .'/

where po is the density~ X, Y are the projections of the mass forces on the coordinate axes.

The system of equations of motion can be written analogously if Eq. (19) may be solved with respect to

o=-o (T , ~). In considering the relation between the stress and deformation rate, various approaches

may be discussed. In the first stage~ it will be ass-med that the coaxlality condition is satisfied for the stress deviators and the deformation-rate devlators] for plane deformation, neglecting the elastic components in the plastic region, it may be written in the form

2T~ a~ a; = ctg2~ = ctg2~. (25)

Oy ~ O.

The dilation condition in Eq. (21) may be written in the form

r ~. - - s lnv 0.~ ] " ( 2 6 )

It follows from Eqs. (25) and (27) that

~ --~-. + - ~ - ~ �9 (27)

251

The solution of the plane problem of dynamic-plasticity theory reduces to finding o, ~, ~, from Eqs. (24)-(26), in which

(28)

C a l c u l a t l n $ an Underground Exp los ion wi th D e s t r u c t i v e Shear S t r e s s . The d e f i n i n g equa- t i o n s proposed above f o r the dynamics o f i n e l a s t i c rocks can he used i n c o n s i d e r i n g the mechan- i c a l e f f e c t s o f underground e x p l o s i o n s . Unde rg round-exp los lon phenomena have been i n v e s t i - ga ted by many prominent S o v i e t and n o n - S o v i e t s c i e n t i s t s ! f o r a r ev i ew, see [14, 15] a n d e l s e - where .

R e t a i n i n g the f o r m u l a t i o n o f the problem, the s o l u t i o n g iven i n [14] r e g a r d i n g the c a l - c u l a t l o n o f an underground e x p l o s i o n wi th a d e s t r u c t i v e shea r s t r e s s i s now g e n e r a l i z e d to the case when the c o n d i t i o n o f rock s t r e n g t h depends on the d e f o r m a t i o n r a t e .

The e x p l o s i v e sou rce i s r e p r e s e n t e d i n the form o f an expanding s p h e r e , the gas p r e s s u r e in which v a r i e s a d i a b a t i c a l l y , wi th a c o n s t a n t a d i a b a t i c c o e f f i c i e n t 70

p c ~ f p a C a o / a ) "~ (29)

where Po is the initial gas pressure; no, a are the initial and current radii of the cavity, respectively.

The Coulomb strength condition is written as a function of the deformation rate. Taking account of Eq. (7), which can be interpreted as the strength under uniaxial compression, a linear approximation is written for the regions of maximum principal stress in the form

qr--~o = ~npO(~r "~O)-~ 2(1--sinp~ [ (~--TO)] T~ u, + K ? I . , (30)

~tere po is the angle of internal friction.

It is assumed that, at the shock wavefront, the stress satisfies the plasticity condition in Eq. (30). In this case, the shock wavefront coincides with the destruction front.

The dilatation condition in Eq. (15) takes the following form with symmetric motion

�9 . o) O~ + , ( o . " , (31) 2 7 = A ~'57"r 7

and hence

�9 ~. (t) ( n 2 - - A U ~- , r n ~_= ~ ) . (32)

I t i s assmued here t h a t the c o e f f i c i e n t A(T, u) c h a r a c t e r i z i n g the d i l a t i o n s i s c o n s t a n t : ^ ( ' r , ~) = ^ = c o n s t .

In the p l a s t i c r e g i o n between the e x p l o s i o n c a v i t y and the w a v e f r o n t , the e q u a t i o n o f mot ion is as follows

,,r r =P~ , -T / -~ T'r/" (33)

where p i s a c o n s t a n t

2 5 2

teb

6

4

2

. _

%%

' 11 ! ! �9 ! , ii = ! I I I I I

0 0,5 1,0 1,5 2,0 ],nO_. Oo

Fig. 6. Curves when the deformation rate is (continuous) and is not (dashed) taken into account in the strength condition in Eq. (30).

The density p is variable, but, within the framework of the given theory of underground explosion, the solution at constant p - Po is of interest.

The solution of Eq. (33), taking account of Eqs. (30) and (32), is

C, (t) u, n F X, (j) ~-, ,~s (t) ,-~ 1

~ + v ~ ~ 4 J +

4 1 I ~ [ 4 I ~- y~(l+~)(i+,) - t-F-n) . __-- • (34)

a~ o 8or"+x J

where B " sin pop Cz(t) is an arbitrary function of the time.

At the cavity wall~ the boundary condition in Eq. (29) holds, and at the shock wavefront it is assumed that denslficatlon of the material occurs by a certain constant value

s = t ~ (35) p"

Then the relations for the mass and momentum fluxes can be written in the form

~(R) m e~, O,n--pJ~. (36)

Expressing ACt) and A'(t) in terms of a and ~(a) as in [14]~ and satisfying the boundary conditions in Eqs. (28) and (36), the equation of motion of the cavity boundary under the condition tha t (~/a) '+ '4 : | , C~/R)'+'~ 1 is found

where

2v~ 6

8Z'vl i~ 0

4 1

m = !+----~" " vz ;o / (t + p ) ( l + . ) ;

(37)

253

415 . (. + i) s t+-'~ =.-I .=m

) + ' - , , , _

( '-.') t a - -1 _ s l - ~ ,e

I+D+ I-~

-- / 415 ~,;

=1 I {-~I t ) , -1 - ..4,T ~ ,r - -

'1' == 2 p. ,.. (,~-'~11 __ s 'v,) '|

ET I

,t,= = 2 ~ ( t +IS) ~,~ .'('~-+ t)

T~p +i--

The e q u a t i o n o f m o t i o n o f t h e c a v i t y b o u n d a r y i n Eq. (37) is n o n l i n e a r , and c a n be e v a l - u a t e d u s i n g a c o m p u t e r . I n t h e p a r t i c u l a r c a s e when t h e i n f l u e n c e o f t h e d e f o r m a t i o n r a t e on the strength of the medium is not taken into account, it is considerably simplified:

d~= -[-=~= ~' P ( . ) + ~ (38) ='~"

and differs from the corresponding equation of [14] by a factor on the rlght-hand side Y1. For this case, its solution takes the form

a" =, c,, + ~ -t. P' =="-~' P (=~ -- 3'1t,) p (a),. (39)

�9 *-here C= is an arbitrary constant.

Consider an Lmderground explosion in limestone with the following initial data= p = 2.5"I0 -s, 0o = 30"; ~o = 3-18"i0-=4; r = l'10-s; y~ = 37"I0 -a kcal'cm=/mole.kg! ul = 25.9 kcal/mole! KT = 0.588 kcal/mole. For an angle of internal friction p" - 30=, Y: " Y~- The initial cavity radius uo is taken to be unity. The energy of, the explosion is taken to be E = 7 - i 0 " kglcm.

The calculations have been performed for three cases of the distribution of the energy of explosion initially between the energy of the compressed gas E n and the kinetic energy W. In the first case, the energy of the explosion is distributed equally between the compressed- gas energy and the kinetic energy: E n - W - 0.SE! in the second, it goes mainly to the compressed-gas energy E n = 0.gE, W - 0.1E; and in the third, mainly to the kinetic energy E n - 0.1E, W = 0.9E.

For comparison, calculations are performed in cases where the influence of the deformation rate on the rock strength is and is not taken into account.

Analysis of the results of the calculations shows that increase in strength of the rock with increase in deformation rate in Eq. (29) leads to considerable decrease in the cavity size. For the given initial characteristics of the medium and the explosion, its size is de- creased severalfold (see Fig. 6, where: 1) E n - 0.1E; W = 0.9E; 2) 0.SE, 0.SE! 3) 0.9E, 0.1E. In qualitative terms, reduction in velocity of the cavity boundary is noted with increase in its dimension in the cases where the deformation rate both is and is not taken into account

254

in the strength condition. Analogous results are obtained in calculating underground explosions in dlabase, marble, and other rocks.

It is noted in the calculations that the cavity size depends on the distribution of the explosion energy between the energies E n and W at time t - 0. However, calculations for the same initial data but with friable media with cohesion show that the theoretical curves of the velocity of the cavity wall with increase in its size converge. This results agrees with the data of calculations performed in [14].

The given calculations for the deformation rate of the cavity boundary are qualitative in character. Comparison with the data of full-scale observations entails the considerations of all the stages by which an underground explosion occurs. However, the given generalized solution of the problem can be used to develop and deepen the model concepts and theory of underground explosions.

The given experimental data on the plasticity of rock in cases with various deformation rates, strength conditions, and defining equations provide the basis for the development of a dynamic theory of the plasticity and destruction of rocks and methods of calculating the dynamics of plastic waves and destruction waves.

LITERATURE CITED

1. Kh. A. Rakb--mtulln, Prikl. Mat. Mekh., 9, No. 1 (1945). 2. V. V. Sokolovsk/i, Prikl. Mat. Mekh., 12, No. 3 (1948). 3. L. E. Malvern, J. Appl. Mech., 18, No. 2 (1951). 4. V. K. Novatskll, Wave Problems of Plasticity Theory [Russian translation], D~r, Moscow

(1978). 5. S. S. Grlgoryan, "Basic concepts of soll dynamics," Prlkl. Mat. Mekh., 24, No. 6 (1960). 6. A. N. Stavrogln and E~ D. Pevzner, '~4ethods and results of investigating the properties

of rock with change in deformation rate and the forms of stress state," in: Proceedings of VNIMI [in Russian], No. 85, Leningrad (1972).

7. A. No Stavrogln and E. D. Pevzner, '~fechanical porpertles of rocks with bulk stress states and various deformation rates," Fiz.-Tekh. Probl. Razrab. Polezn. Iskop., No. 5 (1974).

8. Catalog of Mech-n~cal Properties of Rocks with Broad Variation of Forms of Stress State and Deformation Rate [in Russian], VND~7, Leningrad (1976).

9. V. R. Regel', A. I. Slutsker, and E. E. Tomashevskll, Kinetic Nature of the Strength of Sollds [in Russ lan], Nauka, Moscow (1974).

10. A. N. Stavrogln, "The strength and plasticity of rocks under high pressures subjected to a strain rates," High Temp.-4Ligh Press., 9, No. 6 (1977).

Ii. A. N. Stavrogin, "Limiting states and deformation of rocks," 7zv. Akad. Nauk SSSR, Fiz, Zem., No. 12 (1969).

12. A. N. Stavrogin and A. G. Protosanya, Plasticity of Rocks [in Russlan], Nedra, Moscow (1979).

13. V. M. Nikolaevskil, "Relation of bu lk and shear plastic deformations and shock waves in soft soils,"'bokl. Akad. Nauk SSSR, 177, No. 3 (1967).

14. V. N. Rodlonov, V. V. Adushkin, V. N. Kostyuchenko, et al., Mechan/cal Effect of Under- ground Exploslon [in Russian], Nedra, Moscow (1971).

15. V. M. Kusnetsov, Mathematical Models of Explosions [in Russian], Nauka, Moscow (1977).

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Rock plasticity in conditions of variable deformation rates