Mechanics of porous and fractured media

J

o

r

where ZB is the dissipation function of the body with a crack, / is a crack growth rate, 7* is an irrecoverable part of surface energy discharge for unit length incre­ ment, £ is the specific energy discharge (inside the Gibbs layers along the crack edges). We shall assume that y0, 7* and £ do not depend on the force P. One can get the following set of equations P .„ •„ •» •» «n = — 5 (ue + u p ) 8n T r 8$ =

1

8S * - 58T ShT ;

T r

9$

9$ . 99 «./ + — rje &u Suee + + hT bT 9/ bu 9u dT

\ dl 9/

dl 9/

bT 9r

)/

du duee

(dWe r> 9y0 \ . +( + + 2 \ — - dx )&T ; \\ dT Jo I) dT Jo dT dT

/ 1 9Zfi 2 9-y* . 2 \ 1 3Zfl .„ .„p 8* = / 1 9Z -fi + 2 9-y* /. + -21 )\ 8 / + 1 9Z„ rf" p 8u .

\\ rr

a./ 9/

rr

9/ 9/

rr //

9up rr 9u

Introduction of these expressions into the Onsager principle (1.80) will give us the following result 1 / — IP IP — T \

dWee\ \ e e 9W 1 / 9dW^e fr> dy 9700 \ . -e ) 8 +— u + — [S [5 + + -)hu + 2 \J — - d b c l S r o dT du ) T \ dT )

1 / 9 Z f l \ .„ 1/ dWe •. 9> dya0 p 5 u Pp + — |[ P + 8u + — (I + —r*—£ 22Tr — ^— T \ 9up / T\ 9/ 97 dl dT 9 Zfi 9(>„ + Y*) . \ — - 2£ - 2 — —£ )8/ = 0 9/ 9/ / +

(1.83) (1.83)

Basic Concepts of Continuum Mechanics 31

§ 1.3.2. Criterium of crack growth in rate form The equation (1.83) gives us two constitative relations bW aw e p = = a«ee e

bu

37

abZ z „R i_ a^ buP

Jo Jo bT ar

altogether with condition of crack growth, see209'212 dWe \ bW

3 / 3/ \

B

bt

)

2

2

b(y0 + 7*) d(y >*) _wo /*/ j . _ 3/

2

dy by0 _J± 37

T

.

(1.84)

Thus, the crack is growing if the sum of the changes of dissipation and elastic energy per unit of crack increment is equal to the sum of specific surface dissipa­ tion and rate of irreversible work which is necessary for the transformation of volume material particle into surface ones. The concentrated dissipation 2£ has to be independent of the crack growth /, but it can be a function of crack growth rate /. For viscous dissipation one has

■-■^(TMTO--?^') ■- J ?(T) , '(T')- i 21 *L1?'(7')

(1.85)

**>

because 6 = ft/E is the time of relaxation, and is a suitable estimation for d, y = y0 + 7#The crack growth is accompanied by the reconstruction of internal texture of material near the fracture surface. The energy y is needed for such a work and 7* is a specific viscous dissipation for this reconstruction. Further this band of the material with changed texture will dissipate mechanical work in a different way than before reconstruction. So, one can explain the difference of surface us and 203

volume u = /J„ viscosities. Here [ (l ] = us - Uv For the differential equation (1.84) with / as unknown variable the initial condi­ tion (/ = / 0 at t = 0) can be determined by the solution of problem for elastic body with a crack.

32 Mechanics of Porous and Fractured Media Let us consider some particular cases. If ZB = £ = 0 then the criterion (1.84) will become the well known criterion of fracture of an elastic body: bWe 3/

= 2-y 2y '

(1.86)

where the Griffith energy 2-y includes the dissipative (Orowan's) component 7*. If, vice versa, We = 0, then the criterion of crack growth in a viscous body203 is a consequence of the general criterion (1.84): bZR 3/ 3/

= 2? . = 2£

(1.87)

§ 1.3.3. Crack growth in viscoelastic plane Let us consider the crack growth in viscoelastic Maxwellian isothermic plane, which is subjected to tension traction P. Then specific elastic energy and specific dissipation can be estimated in the following manner: P2

P2

e

W = IE ,

Z„ B = 2u

.

(1.88)

Because the superfluous elastic energy and rate of dissipation in the body are proportional to the area of stress concentration that is to I2 then -B, 1 We = -B. 1

P2l2 IE IE

+ C, + C. , '' ''

ZBRB = B22 B

2

P2l2 2n 2n

C, + C 22

(1.89)

2

where Bx, B2 are numerical coefficients and C,, C2 are constants independent of the crack length /. Introduction of the expressions (1.89) into the crack growth criterion (1.84) gives the following differential equation P2 d/ B, — B. At E df

P2 B72 — U

2{ = 0 / + 2£

(1.90)

Basic Concepts of Continuum Mechanics 33

if y = const. Let us consider the case when i

= const. ,

that is, / does not influence upon surface dissipation: / = const, in the estimation (1.85). Then the solution of (1.90) has the exponential form:

I/ = / /«, * + (/„ ( / „ - / J e x p f ^ i - ft ) \BX u I fl2i>2

P2

52

(1.91)

U

It means that the viscoelastic crack is growing if its initial length l0 has become larger than the threshold value /*, at which the dissipation change for unit crack length reaches the level of the surface specific dissipation. If l0 < /*, then the crack length appears to be diminishing. Let the length of the critical elastic Griffith crack be denoted by lG that is P22 Bll—— lGG=2y = 2y

.

(1.92)

E

Then the crack growth is possible because of the elastic energy release if / 0 > /*, lG. On the other hand, if lG> 10 > /*, then the solution (1.91) will give the increase of crack length up to lG size. At / 0 = lG the equation (1.90) gives the starting velocity of the crack tip

O-93)

1(0) == —r H( — 1(0) — -- -— r 1 ) )— — 2 BtP \yE \7E Bl / u Bj For the second possible case:

I ==nJ % »* i one can get the following solution of the equation (1.90) P E {IB,Bt P22 E \ I1== L exp [ —; — t ) . 2 >o exp — — — — — 2 *\B2 P2 - l^E n t)I . \B2

P - 2\l*E

n

)

(1.94)

34 Mechanics of Porous and Fractured Media So, the constant crack growth velocity is possible under constant tension P if the surface dissipation itself depends on crack velocity. If the surface dissipation is constant, it is necessary to diminish the tension traction to get the uniform crack growth. At last, the equation (1.90) gives the condition of viscous model validity, that is the crack has to be much longer than its increment for relaxation time

Q»io). § 1.3.4. Thermodynamics of steady state vicinity of a crack tip Let us consider now the conditions for a moving crack in a plane problem. For contour rr

r

= = roo ++ r0l0l. ++ IJr( ++ rrftft ,

which binds the A{_0 zone (Fig. 1.2), one can write the balance of total energy202 3

I

+

(•

2

-)p
0

+ okjVk -q]]nJiT+

" IK'

\

+

■ 2 'H--";)

QdA

(1.95)

A

1

i-o

where «.• d r is the oriented element of the contour length with external normal n.-. The product ak: n= dTmeans the specific traction at the element dr. If there are no force, mass and heat action at the crack edges T ■ and C : akj rtj d r = 0 ,

(Vf - vf)ttj d r = 0 ,

qj n} df = 0

the balance is equivalent to the difference I}_0 of the contour integrals at 1} and r o :

LI

t>(e +

= — = dt

f

JAAAi-0 i-0

\

pie \

k

) ( V / - v y ) + o ^ --
+ ^!L) ^lL)

2 '

dA -

\

J

A,

i-O i-O

QdA .

Basic Concepts of Continuum Mechanics 35

Fig. 1.2. Path of integration in a stationary field around a crack tip (point 0). For stationary fields of variables e, v,. .. , in the moving zone At_Q without sources Q the contour integral

e+

v )+

lM v w ~ ' w = i

"^

Tlj&Y

{... }n ; -dr = const.

(1.96)

appears to be invariant. On the other hand, the area^40 bounded by the contour T0 is one-connected domain if the point 0 of fracturing is belonging to the body. If thefields,mentioned above are stationary in the vicinity A0 and Q is equal to zero, that is, there is no exit for energy from the body in the point 0, then the contour integral (1.96) along T0 has to be equal to zero. Therefore, for any contour I], including a sta­ tionary zone, one has

$ |p(6

+

"T^) 0 ^' ~ vf} + °k'Vk " ^

n dr =

/

°

(1.97)

for any rheological model of the body and for the singularity, connected with chosen rheology.

36 Mechanics of Porous and Fractured Media In the theory of fracture the specific integral energy e is usually identified with the internal energy ev of a volume particle. If however, one intends to account the difference of e and ev, then the balance (1.97) can be presented as follows

2(e)

-u*

+

2

) ( V / -" "/) + ff/t/ "A: '" Qf ■nfdr

= \ fi(ev - e ) ( t y - vf)nfdr T,

.

(1.98)

Consequently, the total energy influx to the tip of stationary growing crack is nonequal to zero only because of finite difference at the right-hand side of the equation (1.98) for those parts of the contour I], which are crossed by the particles, transfer from « volume » to « surface » states. Let us consider the contour T0 consisting of D ABCD (Fig. 1.2). Neglecting the fluxes through boundaries BC and AD one has the following equality

Q°(() = { *{** ~ «)(V ~ v / > / d r = (J r0

AB

"(** - evMM - ",)d*2

because particles with internal energy e = 6s are going in through the boundary CD and they are going out through the boundary AD (where nx = - 1) with another energy e = es. The velocity I = V% - v of the crack tip is determined relatively to the body material. If the finite limit Q°(e) =

Ita d^-0

=

lim

\

p(es - e„)(Vj - v,)dx 2

* -d

2[e\pU

= 2y0i

(1.99)

d •* 0

exists then it will correspond to the rate of work spent for creation of a new surface. The assumption

A(.)

=

2(e)

Basic Concepts of Continuum Mechanics

37

leads to resulting invariant form. (The nonequality Q°( . * Q, , could open the possibility for account of reloading of surface layers). The surface energy y0 is the function of state parameters of the Gibbs surface phase, and it means the possibility of account of imbrittlement due to moisture addition. For the proper treatment of this effect it is necessary to introduce the chemical potentials, corresponding to materials composition of the crack edges. The integral form of entropy balance has the following form

- 5L rij d r =

{P*<M---"/)

J

T

/-o

9

f

— \psdA bt J

-

J

T

~ */-o (1.100)

and \]>j_0 is a dissipation in the zone 1A- 0 ' */-. =

f

1

•„

dT

(L4 T

(1.101)

Hence, even for stationary fields in the zone/4;_0 and for absence of body thermal sources Q= 0, the IJ contour integrals for entropy fluxes are not invariants because \jjj_0 does not depend on the area size AiQ. Only in the case \J>,-_0 = 0 of adiabatic or isothermal state (qf- ~ dT/dx,- = 0) in the zone Aj_0 of elastic (eP- = 0) or quasielastic (e P. = 0 out of T0) states one has j

ps(Vf - Vj) --%-

rij&T =\

\p s(Vf - Vj) - ^-

rij d r = const. (1.102)

The constant at the right-hand side of this integral is determined by the condition

.5. r

s(V/- - "/> "

Hydr =

"*0

for the V0 is the contour bounding the stationary field in A0. Let us introduce the specific energy sv of the volume particle of the body dev = <% + Tdsv ,

pdifiv = akjdeekj

(1.103)

38 Mechanics of Porous and Fractured Media (1.104)

- 2 -dr it ++ °k>dev ■ ' " ' , i s = -~~d^ Then the integrals (1.102) and (1.103) can be rewritten as

i r («",(v - >,) - f« .dr = - * ;

i

+ i p( S v r °

0

S )(^.

-

vj)n/&r (1.105)

and the estimation for □ ABCD gives I />(sv - s)(U - vAndr

= 2 lim p[s] rf->0

id,

[s]

= st - sv .

For isothermal condition (T= T0) one has T0 j

{ p ^ t y - iy) - (ev - *„)ty - ij)n, dr (1.106) 22

71

= - T004>00 + (7o " Tr) = 7T)' ' ->

To 7o -~ 7TT

==

ll ii m m

d

P[s] T0d PM ^o

d -+0

and it is convenient to use the dissipation Z 0 in the point of fracturing:

Z0 = T0*0 = J OyiftdA ■

(1-107)

The finite result follows from the relations (1.106) and (1.107)

~ "/) +

$ \P(% + ~-)d^ = Z 0 + 2yT I ,

ff

T = T0

*/"*|"/dr (1.108)

§1.3.5. Path independent contour J-integrals Due to conditions of stationary fields in the coordinates y , y2 and to usual approximations:

v, » v, ,

M= / ,

v2 = o

Basic Concepts of Continuum Mechanics 39 one has

3 - d — = -l —

3'

ty

,

. bu; vt = / —L

dyt

where u{ is the displacement. Then the contour integral (1.108) has the wellknown form of t h e / - i n t e g r a l . 5 9 ' 1 2 1 ' 2 4 8 ' 2 6 5 9

■ y.(*+ 2

J=

r>-

- %

"/t

^ dr =

i

+ 2r0 . (1.109)

The first term on the right-hand side of (1.109) can be interpreted by the following estimation

Z0 = 2 f

ZAA

=

2(Z)0A0

(1.110)

where ( Z ) 0 is a mean value of specific dissipation Z in A0. The role of constitutative parameters for (Z> can be played by crack growth velocity / , linear scale V A0 and the rheological coefficients. If one assumes that the Griffith process of brittle fracture, which has linear scale yE is fully suppressed by the dissipation process, then \f~A^ » yE and the following estimation is valid

XP

I

er. ~

yfA

'

^/x - -d

If the plastic flow is concentrated at the crack tip (in the Irwin-Orowan particle), then (D ) is the first order homogeneous function



~ y/JPJP

~ -7=V

(1.111)

^1Q

and the integral (1.110) is reduced to the following expression j

0 fc/ eP. A A = 7B,s/~AQ I = 2y. /

(1.112)

40 Mechanics of Porous and Fractured Media The coefficient y^ is usually supposed to be proportional to yield limit Y of the material. If the viscous flow is concentrated at the crack tip vicinity then Z is the second order homogeneous function:

„0

^

e

',7

'/

e

~

',7

'1

i2 ~AI

and therefore the integral (1.110) is reduced to the following result: Z 00 = Z = (Z) (Z)00AA00

= =T}h ij/2

(1.114)

where rj is the coefficient of viscous resistivity, 12 ' proportional to dynamic viscosity IX of the material. In considered cases scale d ~ STAQ does not depend on / but does depend on the crack velocity / that is /—

d ~ ■s/Ao

** • I E E

and, generally speaking, the resistivity coefficient tj can be a function of / itself:

(Ed

Ed

v = u f i\ — 7 ; J ; - Ml

\j

Ml

Y \

/in /III

Y\

-E = " / ( \ — y ;— E ))

E )

\y

•

d-115)

E )

However, the expression (1.115) shows that i\ is a function of / only in the case of correlation of brittle fracture and viscous dissipation. If such a correlation is absent, that is, d » yE, then y is not a constitutative parameter and viscous resistivity tj plays the role of a material constant.

§1.3.6. Path-independent

rate contour integrals of fracturing

Let us differentiate the equation L = 0 of heat balance, that is AL d I d( — = — \p At At [ dt

dv, Off —~ ' dXj

+

da,—L dXj

= 0

(1.116)

Basic Concepts of Continuum Mechanics 41

For the stationary fields in the system of coordinates moving with the crack tip one can use the following representation

Vk = xk~

\

d _ A

.

—

~ bk

—

iWda 3

tot

b t — IK /i. — - vVj. . k

,

k

At df

dyk * d_yA

*

k

k

*

*

Therefore, the equation (1.116) can be reduced to the following one 3 ... 3c . . i { by bk p + bk a;,- V{ - bt qf ) = 0 3

. . .

3c

\ by bk p

if

ty daif —*dXj

.

. i

(1.117)

(1.118)

+ bk On V{ - bu q,- ) = 0

X

oyk

k

= o,

—-« 1 tyfc

k

"'

''

3«j.

It means that the further analysis is valid for finite displacement but for small strains. The result of differentiation of the entropy production equation (1.19) ds 93 q= d / a_\ — ip IP — + — J2. L _ n ji = o0 At \

At

bXj T

I

(1119)

gives the following equality in the moving system of coordinates: 32 ——dyfbyk

. . . q, bf bk ps - bk -LT

. 9n = - bj — . dyf

(1.120)

The difference of the equations (1.118) and (1.119) (after multiplication of the latter by T0) will give the equation for the specific free energy *p(e?., T), that is,

42 Mechanics of Porous and Fractured Media

|I ^ pp (( „„ + rTA.)s) + bkk oi/Vi^ -b - kb(i-fc ( l - -- ^^ .. j

^ -bk

. 3

T0 — %

Bvty —

9 T0\ + « — ( — )

= 0

(1.121)

Adhere the following expressions are valid e - 0sT0s = e -Ts e-T - Ts -(T- (T - T0)s = * (e*,T) (<£, T) + T^s TA=

T-T0

it dr dt

, fi(de p(de - T0ds) = ^otj d
At dr dt

3*,bXj dxi Ij

2 [bXj \\3JC;-ty

One can note that it is possible to use also the following transformations in the aquation (1.120)

-^—{**+£*)) i ^++n)) ^j _bfc^-{p(v,+ rA,)} =- ±- i -{„(„ d re,7

d^

df J J » % 0 o v

de„-

deR

, , - * .drJ L . „ d? J l - „" . drJ . ,

;

7

3

;/

bv

— b' * t y v , =_ bCy k 0 ; / — iL -— bk aif vf - \ Of- ~—

tyk

(1.123)

&k

/ dv, \ p L

W—

deR

I = % —*-

0.124)

= 2Z

Therefore the equation (1.121) can be presented in the following form:

a

(1.122)

0,22)

. /

d^

\ l

b, b, ff pp — — + + 2z 2z \\

+ +

a /.

9^ \

[[ bbk Of, — LL )) k Of, —

Basic Concepts of Continuum Mechanics 43

-wNwO-f>l -£KwO-f>l

(1.125)

-^m-)—m

= 0 .

It is useful to recall that d^

de,7 v

At

(1.126)

dt

Now one can integrate the equation (1.125) over the area (Fig. 1.2). The integrand will give us the equality of two contour integrals: (1.127)

\I fa nff d r = 1 fa nrij ddTr JJ T ,r, rr0 0 " where fa is the following flux:

»('-f)l

fa=-^\okl^-2z(l-^) d

\(,

bv

Li:

M

i

c

a

/M

Integration of the entropy equation (1.120) gives us the equality of two other independent contour integrals:

&">

dr =■I.

I, « dr

(1.128)

;

where _

ds

^'

At

d / a- \

Tf = - b, P— + —(—) 1 dt\T

I

.

+ b,n '

Because the steady state takes place inside the contour the contour integrals (1.127) and (1.128) have only one non-zero part determined by the dissipation in the small area A0. The latter .can be. accounted, in .the following way:

44 Mechanics of Porous and Fractured Media

d I y*0,TA)

\ I/«/dr = - l

b ■ -— tL4 = - 2 | b | —- |

TA

Here TA is the averaged temperature over the area A0. The next step is connected again with the discrimination of the volume and surface thermodynamic functions ev and es, sv and ss, ipv and tfis, Zv and Zs. Then the equaUties (1.127) and (1.128) will take the form of the invariant conditions for arbitrary 1} is the contour:

,. d r =

I,

fsr-z- dr = ;

where kj , 2.-

■

<» -

hj) rij

dr + 2 | b |

d / r0 d'

\TA

r*j (1.129)

^

'

"I/)'ij d r ■-

2|b|

d/T0

r*j

are fluxes of ev, sv, etc. Because

by"/ * b, v, = | b | ,

[T]

= T - Ts <* 0

one can get the following estimations for the right-hand sides of equations (1.129):

j (hf - h^rijdT = - 2b j

I — ( P M - TAp[i\) + 2ZV

+ / - ^ - - A [Z] .dy = 2b — rf^ = 0 ) + 2HTA)i -(1.130) Here again the following estimations were assumed 7o -

Pt*H .

7*b ~ (Z> 0 d 2 ,

f ~ [ Z ] d2 .

So, for the non-isothermal case it is necessary to use two independent contour integrals, one for the mechanical work

t J

,

+

LN2 - (£- )* *£(*)h

Basic Concepts of Continuum Mechanics 45 TT I1 —dd //( 1 - — o\ o\ ) <7> - tyo„ — &>i 11 «, d r dr IV1 - — 7 /] q *>f - a," —M by, f «.7 dr + i1r p 1 —

ff + I

=

)„ U I W 2 — J7 (^ = 0 ) + 0

2

(1.131)

a,, I -fy*{T ) ' + 2 | ( 7 i )

(I m )

A

= 2 — \70(y2 =0)+ ^ - 7 * ^ ) 1 + 2|(r x ) and another for heat balance:

f \p—v- -n„v dy++ I —(-M f 1I

ds„

J T .l

dr

„

f

d / q, \

dr

* l,^(l)"' " Jr. d f \ j /

H/ ; dr

d

yjl,

T.)

7 v = -2 -2 — — * ^

dr dr

T3A

.

(1.132)

§ 1.3.7. Particular forms of contour rate integrals In the absence of dissipation [Z] m

9bTr «?„ — * 0 , *y oynn

=Z V v = 0 ,, = Z

eeeeetje iji}. ==e<

the integration of (1.131) over the time will give us the generalization of (1.109) for non-isothermal case

)-(7'-r 0 ) S ,}d> 2 + j r j (| l( i - -y)«i Ii^ J {PV^.D-iT-T^s^dy, J

T, 1

but -°H ~ °ij P~\ T—

« ? dd rr = = 22yo(y = 00)) "- 22 £ 7(5) ^y(TA) 7o(j2 = «/ 2

(1.133) (1-133)

Here one can use the linear constitative laws of coupled thermoelasticity **>(?. (T, *ij) eif) = ~ - O (A" Bifeej klt / 8hW + G kl + ff "- Jj GG) ) ^e(j % G ecif^eit. ChT -3Kz e(T-T 0)e-- r . ) e - -f - 3** ^ e (r z 2

bT

(T - r T0) )22 I- ° .,

(T

i

7Q0

(1.134) (ii34)

46 Mechanics of Porous and Fractured Media sv = 3Kzee

T - T0 + ChT—

2 otj = (K - -G)ehij

+ 2Geif - 3Kze(T - TQ) bif

where ze is the thermodilatation coefficient, * is the thermal conductivity, CAr is the specific heat capacity at T= T0 . The expression (1.134) can be introduced212 into the invariants (1.132) and (1.133). The integral (1.132) which is necessary for the studying of the thermal field, has the form

f ( T ^ ^ - ^ -^ ~ P

rt/dr)

= --Qe == -2 ~ 2^y yy*(l,T ,(/,rxA) ..

7/

'

^

(1.135)

For treatment of the adiabatic thermoelastic cases it is convenient to use the Ericksen approximation,83 i.e. 1 *{eijt T0) - {T - T0)s « ^ , D + - C , l

r

(T - TQ)2 — 2 -

(1.136)

1Q

which gives the two following invariants

J=

« ) n f t * * T ^ -^—r» - * ^rn/dr f

1/

+ 1c

(T - T0)2\

= 2y(y2 = 0) + 2 A

ty

(1.137)

7,(7;)

Q = - ( {3/:ze r0 e + c„ r (r - r0)}«y dr = 2 A

r *(£)

■ (1.138)

Another interesting particular case is a problem of a moving crack in an iso­ thermal viscoelastic body when the integral (1.131) is reduced to the following form

{

\

de.-j

dv;

otf - 2 - dy2 - a , —*" by, r . I " dt

«,dT = 2/ — (y0 + 7*) + 2?(1.139) (1.139) ' &

Basic Concepts of Continuum Mechanics 47 which is equivalent to the invariancy of such a contour integral

'fc-U'^S51* »)•*«-*£*«"*

(1.140)

Nk*= lim (

I 2[Z] + p H [ ^ J a ) ] dK + 2

9?

by = 2 — + 2£ . dr Here [ y ] =
0- 141 )

"« = "« + <

and consequently the contour integral (1.140) can be rewritten as the sum of two terms dZRB 3Z

.

2 ee W bdlW

°lk

„

(1.142)

°'k

which are calculated separately 3Z„ az„ —2- = —?3/^ = 9 k . d2We

fr

dvP 3v£

I 2Z nk AT - amj —2- «,- d r - 2 - H/ dr JIr . 2z «fc dr - o m/ 3**. Jr.

= \

3**.

~ {^P(% T0)}nk dr -

The quantities Z B and dWe/dlk can be determined from the solution of a problem in stresses. For example, for the problem of a crack in viscoelastic plane, con­ sidered earlier, we have

48 Mechanics of Porous and Fractured Media bZBR —B= bl bl

bWe

K (1 + X)8M ' (1 + X)8M

3/ 9/

1 + y K2 = 1 + X 1 4£ 1 + X 4£

(1.143)

where K is the stress intensity coefficient, E is the Young's modulus, X = 3 - 4v is for plain strains, X= (3 - J<)/(1 + v) is for plane stress state, v is the Poisson's coefficient. Introduction of the expressions (1.143) reduces the invariant condition (1.142) into the different equation K2 . 2(1 + v) — ~ II ~ ~ H E U E

bK2 — — bl bl

by •. = 16(1 16(1 + + Xx)(« + — — d) d) = )(* + bd da

(1.144)

which solution determines the critical (fracturing) values K = Kc. If y = const., the crack is growing in a viscous way and Kc c~yJJ. K ->/% .

(1.145)

It means that Kc ~ / if £ ~ I2 and Kc ~ Vh if £ ~ I. For case of tension of viscoelastic plane with a crack we have K = P-Jir — 2 and the equation (1.144) takes the form 2%

W2

P2

2

2

1 + V

P2

.

/ = 0 n P 78r 11 + + Xv P 16 (1 +x)/x E . 2% / + / = 0 +x)lt 8 1and + X(1.146) E shows that f y = const. 16 The (1 comparison of (1.90) if y = const. The comparison of (1.90) and (1.146) shows that Bi

=

■n 16(1 + X )

/ +

IT

,

n 1 + v B-> — — 8 1 +x

2

(1.146) (1.146)

.

Usually, the material toughness is characterized by critical values Kc, which can be material constants (for the Griffith brittle-elastic fracture or the OrowanIrwin plastic dissipation) or functions of the crack velocity. The latter is the viscous fractured evidence.

Basic Concepts of Continuum Mechanics 49

In the case of negligible elastic part of strains when aki(deki/dt)=2D the 121 209 191 integral (1.139) is equivalent of the ^-integral ' (the C*-integral ). Very curious effect was mentioned in the experiments270 with limestone. It was shown that critical value K^ of a tensile crack was growing with pressure P, if it was higher than some threshold value: Kcl = 0.93MPaVAf

,

P < 20MPa ,

Kcl = 0.93MPaVM + 0.07^ ,

P > 20MPa .

This effect can be explained by the solid friction and the specific plasticity of rocks (see §2.1). Usually accepted data for tension cracks in rocks are the following ones: Geomaterials:

A^.MPaV^

Sandstone:

0 . 4 - 1.5

Dolomite:

0.71

Limestone:

0.36-1.24

Granite:

3-5

Critical values Kc11 for shear strains are to some extent higher. For example, for micrograined sandstone K* = 1.47 M Pa <J~m, but K™ = 4.75 M Pa sfln. Dynamical values of these parameters are also higher than static ones. For sand­ stone the following interval is known Kc\dyn) = (4.8 * 9.15)MPaVw" and it means the dependence of Kg on the rate effects. Generally speaking Kc(dyn) appears not to be a material function, but depends on the fracture process, see §3.3.andRef. [ 1 3 6 ] .

CHAPTER 2 DILATANCY OF GEOMATERIALS

§2.1. Inelasticity Mechanisms and their Modelling Presentation §2.1.1. Description of continuous fracture of the medium TWO METHODS can be used to describe an inelastic deformation of the porousfractured medium. Firstly, it is possible to study acts of generation and growth individual slip surfaces, pores, cracks by usual elastoplastic or viscoelastic math­ ematical models. It gives, for example, the opportunity to find out a criterion condition of individual defect appearance, to calculate its growth, and in principle even to account nonequilibrium of this elementary source of nonelasticity. The next step is the solution of collective interactions of defects (pores, cracks), which determine changes in the whole system. If the defect size is comparable with the length scale of the body, then the particular problem of fracture will be solved. If the body length scale is much larger than the defects size then the body can be represented as a sum of elementary volumes AV, and every one includes the re­ presentative ensemble of defects. The volume AV containing pores, cracks or slip surface, is supposed to be continuous, and for the finding of proper rheologi­ cal laws, it is necessary to evaluate mean parameters for the ensemble of inter­ acting defects. The second method for a study of inelasticity consists of the usage of phenomenological concepts of plasticity theory from the very beginning. However, it is necessary to introduce special corrections which are dictated by our ideas of elementary inelastic mechanisms and by corresponding rheological experiments with fracturing samples. In porous-fractured materials there is a mechanism, determined by internal nonelastic deformation, including internal fracture, corresponding to diminishing strength. The growth of a crack from a small initial defect was considered above. 50

Dilatancy of Geomaterials 51

The next mechanism is connected with a nonrecoverable flow of a solid material into a pore.

§2.1.2. Plastic flow into pores Let us suggest that the medium can be represented201 by a system of elastoplastic incompressible cubes of a solid matrix with spherical voids of a radius a at their centres. Approximately, it is possible to represent such a cube by a spherical cell with a radius equal to b. Let the pore be filled with a fluid with pressure P, and the cell compressed by external pressure - a. Such a pore can be surrounded by a zone of outer radius a , in which the plasticity condition is valid: I ar ~ °°eg I

=

Y y

o

where ar is the radial and ag is the tangential stress, Y0 is yield limit of the matrix. Inside the plastic zone the stress field is determined by the expression oar = = -P + 2Y 2r 0 sgn(or - cfc)ln a,)ln — . a o

r < < a

(2.1)

where sgn (ar - ot)= sgn(e r - e9) = HT. In outer elastic zone a < r <, b one has another stress distribution 4 a3 - a3 a,r = a + — G r—°— 2 - .. 3 r3

(2.2)

The position of elasto-plastic boundary is determined by the following condition G

,

4P = 2 — (a3 V

-

2

< " > " ,

■

(2.3)

o

If there is no plastic deformation, then a < a, and the expression will give us the needed connection between pressures P and - a: 4 a3 - a3 a+ P = - G a+ r—5j—°3 a3

(2.4)

5 2 Mechanics of Porous and Fractured Media together with the corresponding condition.■: a3 < a3 = 2G

a3 5 2G-Y0HT

.

If the plastic deformation takes place then a3 > a3 and the summation of the relations (2.1) and (2.2) (for r = a„) will give the proper result 2 /I 2G(a 2G(a33 - a3) \ - — K HrT (1 + In — K —3 v- ) 3 » \ Y0a HT )

(2.5)

The sign HT is also equivalent to Ha = sgn (a3 - a 3 ) = sgn [ - (a + P) ] , and hence the condition (-a)» P corresponds to a decrease of the pore diameter: sgn = 1. Under this condition the pore is increasing and sgn = - 1. The porosity can be expressed as a ration of cell radii: m=

a3 — Zr

.

(2.6)

The definition (2.6) corresponds to the number 3 n = - 34nb of pores for the unit volume of a medium. If the radius b is sufficiently large then the number of pores is small. If the radius a is also large, then pore number is small but each of them is roomy. From the incompressibility condition b3 - b3 = a3 - a3 one has a

l

m

o

l m

(2.7)

~

where m/(l - m) is so-called porosity coefficient. Let us identify a quantity - a = - a,-.- 5« with the mean pressure in the phase, and a pressure P with a pore fluid pressure. Then it is possible to introduce the effective pressure Pf = -(I

- m)(a + P)

Dilatancy of Geomaterials 53

Then the relations (2.4)—(2.5) give us the law of irreversible volume compression of the medium because of a solid pore influx

m 11 -- m

m 00 m — m

11 -- m m 00

Pf < < Pf P[ = m0

-

1 - m0 ,f

pf 4 4 G(l G(l -- m) m) 33

1=

/ mQ = ( 1 - m„ \ 1 - m0 m^ ffl, 5— = = 5— 1 - m^ 1 - m^

f,

(2.3)

y00(l (l -- m)sgn(a m)sgn(o + P) , m#

pf pf > > pf p/

,

m \

-

1 exp 1 - m/

2G 2G 2G + Y0 sgn (o + P) 2G + Y0 sgn (o + P)

Pf - pf 7— Pi mmnn — -- — 1 - m0 1 - m0

In the limit of small variations of the porosity Am = = m - m00 «. « m* ™* - m00 «

m0 ,

A„. m = m - m # «

mQ0 ,

m0

one can get: , 4 Am Pff = - - G0 , P 3 m0 iPf^ - pf P/ *

pxn

exp

* "

Am

f,

f

=

pi >

f,

Pi H m

2 f ff P Pff < < P( Pf = - y 0 (l - m)sgn(o + P) (2.9) 3

,

r

pi

^ r^ ^-

m ™0 - m m* 2m 0 °G - *

2G + y 0 (l - m 0 )sgn(o + P)

One can estimate the effective elastic volume modulus K of a porous medium, whose matrix is presented by the incompressible (Pl = const.) material:

5 4 Mechan ics of Porous and Fractured Media 1

dm

I>f (2.10)

4 K ~ -

m

o(1

-

m

o)G

If one accounts properly the change of the sign (o + P), it is possible to estimate also a residual porosity mp (after elastic reloading from a state with a value m). The simultaneous application of shear stresses will give the distortion of pore form and intensify the compressibility of the porous medium.ls 1

§2.1.3. Irreversible repacking of particles in contact For the study of repacking of grains and blocks of the geomaterials let us con­ sider the system of rigid particles in contact with solid friction action at relative slippage. The regular packing of uniform spherical particles (Fig. 2.1) can be mathematically modelled by a medium, every macropoint of which is characterized by vector directors f", tf, connecting the mass-centres of spheres in contact.217

Fig. 2.1. Repacking of particles in contact. The kinematic limitation for strain rates is connected with the rigidity of spheres and means that relative instant displacements of the sphere centres are orthogonal to the straight lines, connecting these centres. It also means that the distances between spheres in contact remain constant. Let us select coordinate axes 1, 2 in such a way that the line, inclined at 45° angle, plays simultaneously the role of bisectrix of the angle 25 between the vectordirectors. Then v = (IT/2) - 2(§0pyr;g/7tecf Material

Dilatancy of Geomaterials 55 ya

V y . l l

V

S, = cos — , 2

C, = sin — 2 2

tr = sm — , 2

t,

(2.11) = cos — . 2

The velocity of the relative displacement of two points, separated by a unit radius-vector, is given by

*f-IT1*" ' /'

Av/,=

6x

f

£-?-»/ i

(2 12)

-

6x

j

and the conditions of orthogonality of pairs Ai)" f" and Aij, f;- have the form

I r - r/sf = o, 2 - t r/{f = o.

(2.13)

Decomposition of the tensor of velocity gradient to an antisymmetrical rotation tensor J2;y and a strain rate tensor gives the following form of the conditions (2.13):

I M*Y = o,

2 e^rfr/ = o

<./

'./

(2.14)

Introduction of the expressions (2.11) into the equation (2.14) gives the kine­ matic limit condition e n + 2 e i 2 s i n r + e"M = 0 which reduces to the dilatancy condition for strain rate tensor e = Aer

,

A = sin v .

ey = 2 e

(2.15)

due to the symmetry e = e = e/2. Here A is the dilatancy rate or dilatancy coefficient. The phenomenon of dilatancy of rigid particles in contact was dis­ covered experimentally by O.Reynolds 246 in 1885. However, the mathematical description of this effect is under study one century later. One can account the change of dilatancy angle j>200

56 Mechanics of Porous and Fractured Media dv At

= 22ee_ cosy cos v

(2.16)

y

which plays the role of the parameter of hardening (softening). The condition for stresses at a limit state is formulated as follows. At the planes with vector-directors as normals, the stress components a^ f? and o« f.- have to be connected by the Coulomb solid friction law. For example, for {f we have a |I a? a « II ++ aa" tg tg v»v>„ == 00

< = % m. rf?= o °" = oif $tf ,

(2.17)

oa < 0

where ip is the true solid friction angle. Because of the problem symmetry we have a

anu == °22 aM == °a ',

a sin vj> ,, oa" = = aa ++ aann sin

a« == -o -onn cos cos »P> a«

and the limit condition has the form o„ = ffa.rr == -- aa oa ', °i2 12 =

aa

==

tg(*> + vH ) . **(*-#,) „ , 7*— sin sin(*>/f7)

•

(218>

The threshold value of angle 28 = 60° exists for considered microstructure. Really, the further deformation of packing, see Fig. 2.1, is prohibited and the microstruc­ ture becomes rigid. Only change of a sign of deformation can make the structure more loose. The experiments37'38 with very dense particulate media show re­ covery of irreversible repacking after change of shear sign. On the contrary, at maximum loose (critical) state (26 = JT/2) the rate of plastic volume deformation is equal to zero, the material becomes plastically incompressible, the dilatancy is absent. From this illustrative example one can conclude that a solid friction effective coefficient ip as well as a dilatancy rate A are functions of the state para­ meter 26 of the microstructure. §2.1.4. Relative microslip with solid friction for dilatant media Now, let us consider the third example of relative displacements along slip surface. According to the initial Coulomb concepts,62 such a surface is a boundary

Dilatancy of Geomaterials 57

for soil masses. Along this surface two conditions are presumed. One is thei existence of jump of tangential components of velocity displacements [vT } * 0 another is the limit condition 1 °T« 1 + °nn tg *„ =

chs

„

(2.19)

where an, ann are the tangential and normal stress components, chs^ is the cohesion at the slip surface. In the case of a medium with internal solid friction it is necessary to change the Mises condition (1.27) to the invariant of the Coulomb limit condition (2.19), and the plastic incompressibility to dilatancy condition of the (2.15) type. In this case the Coulomb friction angle has to be interpreted according to O. Mohr183 as the angle of internal solid friction at the limit equilibrium surface in every point (or in every element) inside the medium instead of its original understanding as the friction angle for the boundary separating two parts of masses slipping relatively to each other. Let us consider the plane problem. It is quite possible to think that the solid friction force R, acting at the limit equilibrium surface, is inclined to it at the angle 0 and is proportional to the force N in accordance with the Coulomb law, that is, R = (Ntgv>M + chs^sgnfl

(2.20)

the forces R and N being expressed by the components aTn and onn of a stress tensor. R = aTn cos 0 + ann sin 0 (2.21) N = arn sin 0 - ann cos 0 If the surface of limit equilibrium has the angle $ with the axis x2 of the maxi­ mum compression stress o2, then /IT

* = (--

0 + Hyyifiu \ 2

~ jsgn^.

(2.22)

58 Mechanics of Porous and Fractured Media

Let us assume the hypothesis of coaxiality of stress and strain rate tensor e,y. Then the dilatancy condition (2.15) and the condition, that in the coordinate system xl ,x2 the above mentioned tensors have their diagonal form, are the following ones *«

+

e

'22

=

H

y($u

~ *»)"» " •

H

7 = sSn(e'n -

^2)

(2.23) 6n

2 \3x 2

9x,/

where v is the dilatancy angle and the important point is a proportionality of volume strain rate to the modulus of shear rate. The latter is in accordance with rheological tests of geomaterials (§2.2). The equations (2.23) determine two characteristics of a velocity field: dx2

/ l\ + 4- H7y sin vp

(kx dX;

v 1- H Hyy sini»

(2.24)

which are inclined to the axis x2 at the angle = arctg 66 = arctg

/l — - sin r ITIT vv %m.v / = — - — .. V 1 + sin c 4 2

(2.25)

The velocity of a relative displacement of two particles of the medium, con­ nected by the unit radius vector r,, is determined by the rule AVJ = Ai>j = ij.Bjitjr- ,,

rr = = sin7 sin y ,, rr2 2 = =--C O cos S Ty

(2-26)

where y is the angle between r.- and the axis x2. The projections of the vector Av{ at the radius ri and E, such that ^ r- = 0 have the following forms 9*i . 9"2 2 r.-Av,- = — sin2 7 4cos -y y 1 ' ' 3*i dxl ' bx dx2 ' (2.27) 1 / 9K dv2 \ /•; Av,- = — ( — sin 2> . ' ' 2 \dx, dx2 I

Dilatancy of Geomaterials 5 9 Because at the characteristic line (2.42) and at its orthogonal direction y = ± 5, then for the relative velocity of particles /jAvy = 0, but £Av- = 0. In other words, the relative velocity of particles, which are laying at the characteristic line, is orthogonal to considered characteristics, see also Refs. [98, 229]. Let us assume that the relative velocity Ai>,- is collinear to the solid friction resistance force ( 0v. sgntf = s g n ( — l -

9v2 \ - ~ )

(2.28) (2.28)

.

Then, see Ref. [229], one has 8 = (n-/2) - /J - \f/ and due to the equation (2.22) it means (2.29)

0 = -7 + V„ + " • 4 4

Introduction of the expressions of aT„ and a„„ (2.21) and of the angle 0(2.28) into the Coulomb law, (2.20) gives the resulting condition of the limit state I "nr I

+

°nn *8 * = chs

(2.30)

chs^ cos v^ = chs cos (fi = Y ,

= =2 2^ ^ + +c v

It is equivalent to the following invariant form of yield condition aT = P sin

°11 +

ff

22

2

°r -

-j^r-

- "22f + 4a»a .

The dilatancy condition, which is an invariant analog of the equation (2.15), can be presented as e = |I eeTr | | sin sin v,v,

2 eeyy== VV((eenn -- ee2222))22 ++ 44 ee* 22

.

(2.32)

60 Mechanics of Porous and Fractured Media

§2.1.5. Coulomb slip surface as strong tangential velocity jump A. S. Kuznetsov and the author have considered a situation at a strong tangential discontinuity of displacement velocity field [vT] * 0. Such a discontinuity is the only proper mathematical model of the Coulomb slip surface, along which soil or rock masses can slip after loss of stability. In the simplest case of rigid plastic incompressibility the following system of equations controls the plane equilibrium limit state of such masses 9 c,, 9a„

9o,_ -(-I- — » ^ = 00 f,

—ii—ii9JCJ 9JCJ

9X dx22

9o„

9a,,

—«^. ++ — —22— » . == o0 9JCJ 9JCJ

(2.33)

99X X2 2

ffr = Psin v> + F

( 9v, 9v, — — ii

9JC. 9JCJ

dv,

(2.34)

9K,

\

/ 9v.

9v, \

^9v, - l^r)-<«»-"*>(-£: + 4)

9v, + + —2. —2- = = 00 ..

(2.35)

<"5> (2.36)

9dx X2,

Due to incompressibility condition (2.35) the continuity, see (1.46), of normal velocity component vn is valid at any possible discontinuity: r[ vv j l =< = v + _- vV-

==0o. .

(2.37)

The continuity conditions of contact stresses (1.46) at the jump [ «nt ] = 0 .

t »„ J

= 0

(2.38)

can be presented as cos(4/+ + 4- J/ J>~)) + sin v0 cos(\J>+ - \p \p~) ) = 0 (2.39) P+ - P~= (P'cos 2$~ - P+ cos 24/+) sin y + 7(cos 2«|T - cos 2<JO if one uses the well-known representation:

'

Dilatancy of Geomaterials 61 a„„ = - P - aT cos 2<J/ (2.40) oTT = - P + aT 008 24* ,

a

nT

= a

r

sin 2

^

where \J/ is the angle between o2 and the tangent to the strong discontinuity line. The stress field characteristics are determined by the equations (2.32)and(2.33), that is,

£L"

<■*

±


+

T}) 2

^2-41)

where tf is the angle between axes a2 and x2 and they cannot coincide with the strong discontinuity. However, the latter can coincide with one of the charac­ teristics of displacement velocity field

( ^ 2) / ni.rv

= tg(*±y)

(2.42)

which is a consequence of the equations (2.34) and (2.35). Here we are interested in the situation when the tangential velocity jump is accompanied by the discontinuity of the tangential normal stress: [oTT]

* 0 .

The above mentioned coincidence* means that the value in the relations (2.38) and (2.39) at the side « + » of the jump has to be equal to the angle TT/4 between the characteristic (2.41) and the axis a2. It means that 4i+ = ± — 4

(2.43)

and ,JT= ± ( ^ + - )

+ jar ,

k = 0,±1...

* The strong discontinuity can coincide also with the envelope of the family (2.41).

(2.44)

62 Mechanics of Porous and Fractured Media p-=

-H + (P + H) sec 2 vo,

(2.45)

corresponding to the relations (2.39). Here the quantities H and tpt are such that y = H sin v> ,

tg
(2-46)

In the Mohr plane the states « + )> are corresponding to the line MF, that is, to 2|i//+ | = TT/2 (Fig. 2.2), which is going through the tops of the Mohr circles.

Fig. 2.2. Definition of the friction angle w at the strong tangential discontinuity in the Mohr plane. Let us determine the solid friction angle at the tangential jump surface by the formulae I °T« I + °nn ^ fy

=

chs


( 2 - 47 )

according to the Coulomb concept.62 Then the second equality (2.46) gives the value iff such that * * < * . <

9 ■

(2.48)

The left equality means the concordance of limit states along the strong discon­ tinuity and out of it, inside the masses. The angle ifi^ is smaller than the internal angle
Dilatancy of Geomaterials 63

surrounding rock masses into the limit state (2.33), for example, due to the water decreasing of angle

§2.2. Elastoplastic Theory of Dilatancy § 2.2.1. Nonassociated plastic flow rule The increments of total strain de y of the medium under consideration are a sum of elastic def- and plastic deP strains de. = de° + deP. The increments of elastic strains are determined by stress increments da,y accord­ ing to the Hooke law (1.9). The plastic strain increments are connected with the stresses o,y themselves by the tensor linear relations defj = - (H - a) 8# d* + (aif - grfiy) dX

(2.49)

which are valid for the isotropic case.198 " 20 ° Here a = (o,y 5,y)/3 is the first stress invariant, and the kinematic coefficients X, | play the role of additional unknown variables. Therefore, it is necessary to formulate two additional equations198"200 and it is done in the form of algebraic equations for the stress tensor and the plastic strain rate tensor invariants: *„ = HraT

-f„

(a,a,Y)

% = e p - fe(eE,A)

= 0

= 0

where a = a(x), A=A(X) and X is the parameter of hardening. Recalling the results of §2.1, we can say that the medium is in the limit (Mohr) state, if f0 is proportional to a, and the stresses are corresponding to some point at the yield surface I Nikolaevskii V. N. (1987) Mechanics of geomaterials. Generalized models, "Advances in Science and Technology. Ser. Mechanics of deformable solids", Moscow: VINITI, v. 19, pp. 148-182 (in Russian).

64 Mechanics of Porous and Fractured Media *„ = ~jf

HT o r + OLO - Y = 0 .

(2.50)

Here Hf = sgn aT, a has the same form as in the equation (1.27), a is the internal solid friction coefficient, Y is the cohesion coefficient. In the stress space the yield surface (2.50) has the form of the Mises-Guber-Schleicher-Botkin cone, see Ref. [200]. It can be interpreted as the generalization of the solid friction law (2.31) for the three-dimensional case. The second limit condition is represented by the dilatancy relation 2

% = AeP - ~j=A HydeP

= 0

(2.51)

where H = sgn e£ ;

de

r = v T l ( d e f l " de")2 + '•• + 6(def2)2 + ••• T2

The conditions (2.50) and (2.51) are coinciding due to the relations (2.49), if 2 d* =

«AdX , Y = uH . 3 Then the constitutative law (2.49) reduces to the following connection for plastic strain increments: de

ii

=

{ °ii

+

J

YA b

H - (1 + J « A ) " *if } ^

(2.52)

where dX >0, if the yield condition (2.50) is fulfilled and there is a process of active loading with the hardening (3*/9X > 0): *„ = 0 ,

d'* 0 = — - da + —2- daT > 0 do aaT

(2.53)

Dilatancy of Geomaterials 65

or with softening of the medium (9
d'*

= — a - da + — da doT

dc

< 0 .

(2.54)

In the latter case the continuous deformation can be unstable. This possibility will be considered in §2.3. The neutral loading is characterized by the conditions *„ = 0 ,

d'* c = 0 ,

dX = 0

(2.55)

and the unloading process is corresponding to % < 0 ,

dX = 0 .

(2.56)

So, the difference between the medium softening and its unloading is deter­ mined by the position of stress state, at or under the instant yield surface. The constitutative laws (2.52) can also be presented by the flow rule form 3t*

deP. = i]

(2.57)

dX . a<jy

Here ^ is the plastic potential. * = aT - 2a A(H -of

= aT0(H - a0)(a - A)

(2.58)

noncoinciding with the function 4>a = 0. It means that the flow rule (2.57) is nonassociated with the surface $ 0 = 0. It is necessary to underline that the function \J/ is determined only in the crossing point o r o , a0 of the yield surface and the plastic equipotential (2.58). The latter is a fictitious one. Because of its noncoincidence with any yield surface, the constancy condition for the parameter X is not valid along the equipotential surface. The values of a, A, H correspond to the value X0 only in the cross point aT0, a0. The straight line tangential to the equipotential surface in the point aT0, a0, is orthogonal to the vector of the plastic increment dep, del* in the plane ep, e? If A < 0, then plastic equipotentials have the ellipsoidal form (Fig. 2.3) and plastic shear dep is accompanied by the plastic decrease of volume (dep < 0). If A > 0,

66 Mechanics of Porous and Fractured Media

Fig. 2.3. Idealized yield surface (solid lines), plastic equipotentials (broken lines) and vector increments of plastic strains (arrows) at the points oT0, o0. the plastic equipotentials have the hyperbolic form and the plastic shear de^ creates the plastic volume increase ( d e p > 0). The particular case a = A > 0 is corresponding to the degeneration of the hyper­ bola to straight lines and the plastic equipotential coincides with the yield surface (with the straight Mohr-Coulomb line). Then so-called associated flow takes place. Another particular case, A = 0 is corresponding to the critical state of the geomaterial, when the equipotential is the straight line parallel to the pressure axis in Fig. 2.3.

§2.2.2. Comparison with triaxial tests of sands Let us introduce now the plastic modulus h for a dilatant geomaterial. The condition (2.53) of active loading can be expressed as

2 / da dY \ d * = —7=HTT7 -da -da, + a da do = = [ - a — + — dX . T + \/l \ dX dX / \^3 V dX dX Further, the constitutive law (2.52) gives 1H.r ddo 2# a Tr //VT V T + ado ado dX=

2afc (H - o) 2ah (H - a)

Dilatancy of Geomaterials 67

where the instant plastic modulus h is introduced: /I da dY \ h = ( -a— + IA . \ dX dX /

(2.59)

With the help of (2.59) we can now find the plastic strain increments:231

ddeep P== — ( —j=^ HrTda d oT T ++ a ad d0o )

Th (\V3 vT"'

'

))

(2.60)

d ==

dff + a d 0 << f(vT^ )f(vT^d^'

Functions a(X) and A(X) are determined by the comparison of the theoretical construction with so-called standard triaxial tests of geomaterials when °U

den

=

°1 •

°22 =

= de1 ,

ff

33 =

a

3 •

°12

de22 = de33 = de3 ,

=

°13 =

ieu

ff

23 =

0

= de,, = de23 = 0 .

The elastic strain increments are determined by the expressions da, da, de,1e = —-*• - 2v —S- , E E

_ do, da, d <2 = (1 - v) —*- - v —lE E

.

The yield and dilatancy conditions (2.50) and (2.51) will be taken in the follow­ ing forms:

<M», " 03) + 7f(",

+ 2ff

F

3>=

(2.61) def + 2 def = — AH 1

3

-

2

/

(def - d e f ) . 1

The constitutive law (2.52) gives

■*

68 Mechanics of Porous and Fractured Media dd\\ ==

\/TdeP \ATdeP 2H A o, - oa22) 2ff y A(c,

It means that the volume plastic strain is playing the role of hardening parameter: dX= dep. Under the assumptions Y= const., a = a(ep), which are admissible for soils, the differentiation of the yield condition gives dep

,HyVl = (H V 3 - 2a) - 2

+ a- (H„yf3 - 2a) ( d o , / d a ) T _ —-» L. 3y/JH7(dtt/deP)0l

r

da,

(262)

where da / d o , is the incremental coefficient of the lateral pressure. Moreover, if cohesion is negligible ( 7 = 0 ) then the total strains are determined by the expressions de, 1 v da, sfl + Hy A —L = _ _ 2 ?- + i— da, E E da, 3# T A de2

P

~d~o2 ~ E

dep da,

(2 63)

1- v

da3

V^I - 2AH7

dep

E

~dal

6AHr

do,

During the triaxial tests the confining pressure o = o3 = const, is keeping constant but o, is decreasing, that is the vertical compression is growing. It means that do3 = 0. During the octahedral shear the condition a = const, is fulfilled, that is do 3 /do, = - 1/2. The data of measurements of a , a , dep and dep gives the possibility to find a and A directly, the expressions (2.61) being used. The dependence of a(A) is given in Fig. 2.4 for river sands according to the Bishop review paper.35 It is approximated by the expression (2.30), that is by
.

(2.64)

The experimental data308 for standard tests of loose sands which are consolidating under shear are given in Fig. 2.5 by curves 1 —4 and in Table 2.1. The data of the octahedral shear are given for the same sands by curves 5 and 6.

Dilatancy of Geomaterials 69

Fig. 2.4. Relationship between the internal solid friction coefficient a and the dilatancy rate A for river sands.

Fig. 2.5. Standard triaxial test data for loose and dense (broken line) sands.

7 0 Mechanics of Porous and Fractured Media Table 2.1. Data on failure of sand samples. Initial porosity factor

Test

Confining pressure fflj.MPa

Volume strain at failure e,%

Axial stress failure

Axial strain ex at failure

ffj at

Standard triaxial tests

1.05 1.02

0.10 0.10

4.0 3.3

0.316 0.332

22.6 21.0

Octahedrical tests

1.03

0.10

2.4

0.186

14.3

According to the standard triaxial tests the value

;

a, °1 + + 22 °a. 3

is a function of the ratio ep/eE., where e% is the critical value of volume strain defined here as the limit deformation before the sample failure. This function can be approximated by the simple linear expression

a = 0.27 0.27 ++ 0.47 0.47 ((ee pp//e£) eP)

(2.65)

which is illustrated by the additional nonuniform scale in Fig. 2.4. The illustrative calculations with.fi' = 300 M Pa, v = 0.25 were done 231 and they are presented here by curves 1 —6 in Fig. 2.6.

Fig. 2.6. Calculation of uniaxial compression (solid lines) and experimental data.

Dilatancy of Geomaterials 71

At the one-dimensional compression in the rigid cylinder the lateral deforma­ tion is equal to zero (de3 = 0). The incremental coefficient da / d a is given by the following form

da 3 _ E(y/3-2H7A)(\f3-2Hyu)(\/T+Hra) - lSA\f3(daldeP)o1v da, " E(y/J-2HrA)(y/3-2H7tt) - 18A>/3(da/deP)a1(l -v) (2.66) The numerical calculations of the differential equation (2.66) are given by curves 1, 2, and 3 in Fig. 2.6 in the form a

i

=

ff

i(Ve*)

for eg = 0.3, 1.0 and 3.3% correspondingly. The experimental data292 for the sand with m = 0.495 are marked by 4 open circles. They coincide with the calculations. According to the considered theory, the unloading process begins with elastic deformations but later, after sufficient decrease of ol, the reverse plastic flow will take place, and this effect is observed in experiments. However, here the reverse plastic flow is not calculated and possible character of curves 2 and 3 is given by broken lines 5. Curve 6 was calculated forir= 200 MPa a n d e ^ 1%. The dependence of lateral stress o on o is given by curve 7, calculated for E= 300 MPa, ef = 1% and marked by the symbol X.

§ 2.2.3. Variants of hardening laws for geomaterials In some papers 295 ' 147 the hardening parameter was chosen as the plastic shear X = eP .

Although in the above considered cases the usage of ep is mathematically equivalent to e?, the residual volume strains can be measured in a simpler way. The behavior of sands with initial dense states is more complex, see curves 7 and 8 for ( a - a ) in Fig. 2.5 because they have intervals of increase and decrease of the strength, that is, the domains of positive and negative values of the plastic modulus h are separated by peak strength state with a = am. The change of the sign of A takes place at some value of the volume plastic strain

7 2 Mechanics of Porous and Fractured Media

< = «*« which depends on pressure. If pressure is very high, the loosing of sands is im­ possible and in spite of dense state the further shear deformation is characterized by a negative dilatancy (contraction). Generally speaking, X = X(ep, ep), but the possible elastoplastic model for dense sands is determined by the following relations a = ul(eP),

A=Ai(
eP

< e£(
a = a2 (eP),

A = A2 («?) > 0 ,

eP

> e£ (a)

Usually e p ( a m ) > e ? and such a complication is probably connected with two modes of internal deformation processes. The first mode is the continuous dis­ tributions of strains, and the second one is characterized by the strain localization into shear bands (§2.4). In accordance with the preliminary discussion (§2.1), it is possible to assume that the yield surface (2.50) open to the positive direction of pressure axis, is corresponding to the internal slippage phenomena. The plastic fluxes connected with pore collapse will take place, when stress strain is corresponding to the "cap" yield surface (curve 3 in Fig. 2.7). Such yield surfaces can be described by a family of ellipses: <*„ (a, oT,m) ={~\

~ \z*-\

-1=0

(2.68)

Here a , aT0 are experimental functions of a porosity, and the traditional associated flow rule is used sometimes for (2.68), that is, def = — °aif

dX .

(2.69)

The surfaces for sands (2.68) are needing much higher stress levels than the slippage surfaces (2.50). On the contrary, in case of soft, very porous materials (and powders) the stress level, necessary for the plasticity of the (2.68) type is much lower than for the slippage. But in case of such geomaterials, such as clay or limestone, it is necessary to summarize two types of the yield surfaces. It is

Dilatancy of Geomaterials 73

the real physical reason of the Cam-clay model275 or of other "cap" models.200 In such models the generalized yield surface is composed partly of the surface (2.50), corresponding to the dilatant loosening from dense states and partly of the ellipsoidal surfaces, corresponding to the plastic collapse of pores. The latter part is presented by the convex "cap" surface of plastic states. Then the line of critical state | aT | = et^(H- o) plays the role of the boundary between these two types of deformation. However, the phenomenon of microslips and of pore plastic collapses may take place in a small scale at sufficiently low levels. It is a sublimit plasticity. The experimental study of sands295 shows that sublimit yield surfaces given in Fig. 2.7 by broken lines 3 corresponding to the plastic volume strains of the order of 0.4% and to the shear strains of 0.03%. At the same time the limit plasticity corresponds to the order of 0.8% of plastic shear. In Fig. 2.7 the solid lines 1 and 2 are pre­ senting values of oT and a, which correspond to the slippage in accordance with non-associated flow rule, and broken lines 3 — with associated flow rule. If the latter is true then the broken lines have to cross the a-axis at the angle of 90°. The sublimit increments of the volume and shear strains are connected with the stress increments deP =flado

+ flf doT (2.70)

deP = f2o da + f2T daT and because of the thermodynamical symmetry concepts (§1.1) one can expect that the empiric functions fig and fir are such that fv = f2aIn the boundary zone of the sublimit and the limit plasticity the corresponding strain increments have to be summed up, that is deP = fla da + fu

daT + V I de% I (2.71)

deP = f2a da + f2T dar +

aTd\.

However, the shear effects accompanying the pore collapse, are sufficiently small in comparison with corresponding effects of the limit plasticity. Therefore, the resultative constitutative laws of nonelastic deformation of porous media are the following ones196 deP

da + A | deP \

= do

7

74 Mechanics of Porous and Fractured Media

Fig. 2.7. General view of failure surfaces (or of peak strength) of geomaterial samples (1), of elastic limit surface (2), of pore collapse (3) and of stress-strain paths (4). dep = aT dX,

- — = flo

.

(2.72)

Here dX> 0 if the yield condition (2.50) is valid, and dX = 0 in opposite cases. The function F(a) is represented by the curve Oil (or OF), found in the experi­ ments with hydrostatic compression,289'308 see Fig. 2.7. These curves have different branches for loading OCll (or OAT) and unloading OCP. The initial parts of the unloading lines correspond to true elastic modulus which can be measured indepen­ dently during dynamic experiments. In general, the relations (2.70) — (2.72) are nonholonomic, that is, they cannot be interpreted independently of the equations of motion, because fia, fJT include the derivatives of the plastic potential (5.9) together with increments dX. If the loading path is prescribed, the functions ep = ep(o,T),

eP = eP(o,r)

(2.73)

can be measured. They can be used directly in particular cases when the loading path is the same one as during their measurements. It is the example of the de­ formation theory which assumes the one-parametrical loading.

Dilatancy of Geomaterials 75

§2.2.4. Elastoplastic dilatancy of rocks Intact rocks are deformed by stress action as soils in dense states although the intensity of stresses, which can create the irreversible strains has to be much higher. At the corresponding level the shear stress aT has to achieve the elasticity limit (curve 2 of Fig. 2.7 and 2.8), above which the dilatant cracks begin to grow. In Fig. 2.6 the characteristic curve of hydrostatic compression OT is given for intact rocks (granite), which is nontinearly elastic (reversible) if there are no initial pores and cracks. If the granite sample is compressed along OA line and afterwards the shear stress is applied along AB, then up to the point B' the compression will take place along hydrostatic curve OF. Only after crossing the surface of dilatancy beginning there will be the transition to the state B of loosening , even under the action of high mean pressure. The loading of a porous rock, for instance, of a sandstone, is characterized by curve Oil. The motion along this curve corresponds to the pressure increase. The application of a shear stress along the line CE will intensify its compaction in the beginning (from the point C up to the crossing-point D with the surface of the dilatancy appearance). Afterwards the dilatant loosening (from D to E) will take place. Under very high pressures only the shear intensification of compaction will take place (curve FG). The unloading process corresponds to the branch CP, which is going away from Oil-curve due to the pores collapse phenomenon. On the contrary, under very low hydrostatic pressure the sandstone is elastic, there are no pore collapses and the sandstone response to shear is a brittle one. The maximum strength of soil or rock samples corresponds to stresses at the failure surface (curve 1 of Fig. 2.7). Here the internal structure of the geomaterial is changing. In the case of rocks one can see the reorientation of microcracks and pores in some preferable direction or their uniform growth in the whole volume. In case of soils the system of slip bands appears or plastic nonlimited flow takes place in broad volume zones. These processes transform geomaterials into a new state, fractured or granulated which are characterized by a new yield surface with a lower position in the plane o r , a. The illustrative example is given in Fig. 2.8 for fracture of sandstones at the room temperature. The finite rock states were studied carefully and it gives the classification of nonelasticity and of fracture of geomaterials for different levels of pressure and temperature, which determine the type of failure see Fig. 2.9. For proper under­ standing of this scheme, which develops the earlier results,31'78 it is necessary to recall three types of failure, that is, of the growth of individual cracks (§1.3), of the creation of slip surfaces (§2.4) and of the volume yielding with solid friction (§2.4).

76 Mechanics of Porous and Fractured Media

Fig. 2.8. Surfaces of maximum strength (1), of elasticity limit (2) and of residual strength (3), according to R. N. Schock et al. Under very low confining pressure - a the microcracks, parallel to the com­ pression axis, that is, orthogonal to the tension axis, are growing. The edges of these microcracks do not come into contact with each other, cracks are open (dilatant), there is no solid friction forces between them, and therefore even very low addi­ tional shear forces can develop one of them into macrocrack dividing the sample into parts. This macrocrack is also parallel to the compression axis (the splitting effect115). With the growth of confining pressure the edges of the microcracks begin to come into contact with each other, the solid friction begins to act and the shear macrocrack will appear in the plane, where the shear stresses are reaching the limit solid friction condition (2.30). Therefore, the macrocrack is inclined and its orienta­ tion depends on the ratio of main stresses, angles of solid friction and dilatancy. However, qualitatively one can see that macrocrack tries to be close to the com­ pression axis. The creation of macrocracks is manifested by the stress drop sd inside the testing machine. The stress drop is a feature of the elasto-brittle fracture. In these two pressure-temperature intervals the solid friction between the edges is less than the strength of surrounding intact geomaterials. Therefore, if one creates the preliminary crack inside the sample, then the slip will take place under applied stresses. However, at higher pressures P = - a and temperatures T when the resistance forces of solid friction between crack edges are comparable with intact rock strength,

Dilatancy of Geomaterials 77

Fig. 2.9. Dependence of finite brittle fractured states of rocks on pressure and temperature level.

7 8 Mechanics of Porous and Fractured Media the small cracks are concentrating continuously into a band. This localization of irreversible cracks is finishing by the slip along this band. Another argument for the same order of intact rock strength and of the solid friction resistance is con­ nected with the stick-slip phenomena: the slippage along the crack is interacting with the fracture of intact edges of the same crack. The corresponding P, T is the interval for granite and gabbro46 is given in Fig. 2.10. The open symbols corre­ spond to the stick-slip conditions (II), the filled symbols to the smooth slips (I). The intermediate results are given also in Fig. 2.9. The standard continental geotherm (III) shows that the stick-slip phenomenon plays the important role for the Earth crust structure (§2.4).

Fig. 2.10. Boundary between smooth slip (I) and stick-slip (II) zones in comparison with standard continental geotherm (III). 1 - slip along main fracture, 2 - slip along preliminar cut, 3 - stick-slip.

The close rheological behavior of the crack edge contacts and of the surrounding geomaterial gives the possibility to use the continuum elasto-plastic dilatancy theory in this thermodynamical interval. The corresponding mathematical study of the localization phenomenon will be given in §2.4. Here the rheological behavior of rocks and particulated materials are very similar because the solid friction of material masses and at the slip surface (§2.1) has the same order. The fracture of samples is finishing by the rock state with non-zero residual strength,corresponding to the yield of a crushed material which exists inside the slip band.

Dilatancy of Geomaterials

79

Under higher pressures and temperatures the creation of macrocracks is totally impossible, because the solid friction becomes higher than the intact rock strength. It happens above the Orowan-Byerlee-Mogi boundary. The failure gives no stress-drop at all and has a character of the pseudoplastic flow. It is possible to assume that the deformation is similar to the superplastic one, because here the main mechanism is a slippage at contacts of grains, created by the net of small crack distributed over the sample volume. The finite states of rocks is known in geology as cataclastic ones. It is known that in the superplasticity the equiviscosity boundary exists in theP, T plane. This boundary divides the zone of smaller P and T where a material with more rough grains gives less resistance to shear than a more fine texture, and the zone of higher P, T condition where the fine texture of material is thermodynamically preferable. The geomaterial possess the same transition from cataclastic to milonitic states which play the important role in geology. At last, the experiments300 with granite show that transition from the crack net creation to the dislocation flow-type nonelasticity takes place at temperature of 600°C and it is nonsensible to the pressure changes. However, intergrain cracks depend on both pressure and temperature. At P, T the conditions above the boundary are determined by the following pressure-temperature values P

1.5 Gpa

1 GPa

0.7 GPa

T

500°C

600°C

800°C

there are no such cracks in the finite crashed granite state. Here the non-elasticity of granite is controlled by dislocations (and by isolated microcracks inside some in­ dividual mineral grains). It is a true crystalline plasticity. The change of elementary mechanisms of deformation is manifesting the transi­ tion from the brittle-dilatant finite states to the true-plastic ones and from the permeable to the impermeable states.' §2.2.5. Sophisticated mathematical models The data discussed above correspond to the usual laboratory strain rates (e ~ 10~4 c" 1 ). If the loading is slower at moderate temperatures the creep, connected with moisture transfer through the crack net inside the sample, takes place. The wetness decreases the Griffith surface energy of the sample, the crack ' Quartzites have no cataclastic states due to hard intermineral bounds according to J. M. Logan.

80 Mechanics of Porous and Fractured Media

growth initiates at the lower stress level and consequently the dilatancy will begin at lower shear stresses. It was found that the slower strain rates (~ 10"8 s'1) do not influence the strength limit of geomaterial and even on the localization. However, the temperature does influence the elasticity limit and relieves the dilatant deformation. High-temperature infracrystalline creep influences moderately on the nonrecoverable deformation of granites in the temperature interval from 200 to 600°C. The dislocation creep will take place at higher temperatures T> (2/3) TM, where TM is the melting point. At higher pressures (P> Pm) above the cross-point of the fracture limit and the solid friction line (Fig. 2.9) the peak strength appears to be independent of pressure and further dilatant increase of pore volume becomes impossible. The change of deformation and failure regimes has to be more drastic in porous rocks because in these cases the additional effect of pore collapse is essential. For instance, it was found that under high pressure action tuffs can get new states of pressed powder with higher internal solid friction than its original value. It is quite possible to develop quantitative elasto-plastic model for rock geomaterials. It was shown,125'126 that yield condition can be approximated by the form: ar = (Y(i) - a ( 0 a)SW + Af(o)X{i)

(2.1'4)

where i = \ corresponds to the plastic hardening regimes, i = 2 to plastic softening, i = 3 to residual state, and P < P™. For P > P™, the condition (2.74) has to be changed for ffr = 0™ = const.

(2.75)

Here X,- is the hardening parameter X,- ~ (2eP -eP)q(0

■

The dilatancy rates A;- are exponentially decreasing functions of stress invariant difference P - ar. It means that the dilatancy is increasing with the shear inten­ sity growth. Pressures suppress the dilatancy. It can be seen from the character of the dilatancy rate along the failure surface (Fig. 2.11). Of course, such complicated mathematical models can be used only for computer numerical calculations. All necessary data were published in papers' 125,126 1

Diltancy of Geomaterials 81

Fig. 2.11. Decrease of dilatancy rate A*- with pressure growth at failure surface for sandstones. Herep = 3/>/(2o°), a°-failure shear stress at o, = 0.

Fig. 2.12. Decrease of shear strength of sandstones with initial porosity growth.

82 Mechanics of Porous and Fractured Media for rocks. They show the essential dependence of parameters of the model on initial values of porosity. Figure 2.12 shows the shear strength decrease for sandstones under the hydrostatic pressure P = 150 MPa. Symbols 110,. . . , D% correspond to numbers of samples, tested by A. N. Stavrogin.288 Other data correspond to the Reports on testing of rock samples from the Nevada Test site 75 - 274,280 ' 289 where the mentioned American nuclear yields were exploded. So, the authors of [125 and 126 ] tried to use the yield surfaces (2.50) for description of all geomaterial plastic strains including sublim'it plasticity, mentioned above. More detailed description probably needs the yield surfaces with singular points that means also the proportionality of plastic strains, increments to stress increments.223 Of course, the type of yield surfaces depends on given levels of measurements and accurate calculations. The anisotropic elasto-plastic model was suggested by S. M. Kapustansky123'124 for initially given anisotropy. Moreover, due to localization the strain anisotropy effects play important roles in the hardening process.116'172 It is useful to note that the strain-softening process during rock deformation is definitely connected with crack appearance and simultaneous decrease of real intact area of stress acting inside the tested sample.48 The suggested model describes the fracturing body as a continuum with special softening rules and, as the result, the calculations will lead to prediction of discontinuous dis­ placement fields, see §2.4. Very sophisticated models will be developed for polycrystalline ices, which definitely have dilatant properties81'82 but with much more strong simultaneous temperature effect which leads to appearance of some rate-dependent (viscous) properties.320

§2.3. Limit Equilibrium and Flow of Dilatant Masses § 2.3.1. Plane plastic flows The equilibrium equations of plane plastic flows include three unknown stress components 3

*u

+

OXx

Bx2

=

F

i

(2.76) F

+ 9

*i

= ,

***

where body forces Ft are given.

Dilatancy of Geomaterials 83

The yield (2.50) and dilatancy (2.51) conditions have the following forms: i

(a 7 V^ de

fi

r—

T, n

+ d

sin^

- a 22 ) 2 + 4a*2 + —

<2 =

sin

( a n + a 22 ) =

tfsin*

" VCdeP - de£) 2 + 4(def2)2 .

(2.77) (2.78)

The angles of solid friction

/3(3 - A2) sin itf = a / V 3 - aA

A

,

A~3

(2.79)

sin v = A / V 3 - A2

The incremental constitutive law of plastic deformation has the following form: 1 o^ 6fc/ 5..(1 + sin^ sin v) (2.80) where d \ > 0 if the condition (2.77) is valid, i, j = 1,2. If the elastic strains are negligible then one can use expressions (1.5) for deP= de.- and to suggest that X = p. The well-known introduction189'285 of the angle 8 between the main compression axis a22 and the coordinate x axis gives au = -P + (P + # ) sin* cos 20 ff22 = -P - (P + H) sin * cos 20 - -- (P + / / ) sin * sin 26 ,

(2.81)

a, «,., . 2 ; ' The expressions (2.81) satisfy the yield condition (2.77) identically and reduce the equilibrium equations (2.76) to the following form ^

\

1

P =

bz

\ dP bP bz dP cos + — sin 26 cos 26 26 )) bP /I 3* bx,x bP dxn bP

bz bp + — sin 20 bp bx

2zsin20

b6 bx

bz bp — cos 26 — bP bx.

2z cos 26

b6

= - F,

(2.82)

84 Mechanics of Porous and Fractured Media dz — bP

sin 20

dz

+

dp

bP

I + 1 1 + V

dx,

cos 20

dp

dz dP

\ dP cos 20 1 I 3x

+ 2z cos 20

dx2

do dxx

-

9dp_ sin 20 — 9; dx.

dz dp

2z sin 20

99

= - F.

bx2

.

Here z = (P + H) sin

+ v,

dp dx.

+ v,

dp 9.x

dv, dv. + p — L + p —*• = 0 . 9.x:, 9.x-

(2.83)

The flow rule (2.80) gives the equation for velocity components themselves: dv, sin 20 — 9.x-.

dv, dv. dv, + cos 20 — - + cos 20 — - - sin 20 — 9.x-, dx. dx.

= 0 (2.84)

dv, dv sin 20 — - + sin v — dx. dx.

dv, dv. + sin v — l - + sin 20 — — = 0 dx. dx2

§ 2.3.2. Characteristical equations for plane plastic flow Let us express the system of plastic equilibrium (plastic flow) in the charac­ teristic form. It is possible 230 to show that characteristics of stress field are the following ones:

m

in 20 ± V l (dz/dP)1 sin = Ctg(0 ± £) - cos 20 + (dz/dP)

i, ii

where cos

2E

=

dz

dz

dP

dP

< 1

Velocity fields characteristics are different ones

(2.85)

Dilatancy of Geomaterials 85

m

- sin 20 ± V 1 - cos2 26

dx, \

cos 28 - cos 20

in.rv

= ctg(0 ± 25)

(2.86)

because cos 25 = sin v and

5= JL_JL 4

2

At last, the fifth characteristic (for density) coincides with the particle trajectory =

A

(2.87)

(:::).. In the absence of body forces the relations along the characteristic lines can be presented in the universal form {(cos 20 - cos 28) dx 2 - (cos 20 4- cos 28) dx2 + 2 sin 20 dXj d*2 }{(v2 d.^ - vx dx2) [ (- cos 20 - cos2e)dx 1 + sin2fidx 2 ] dp 4- (dz/dp) d/>(v1 dxx

(2.88)

+ v dJC,)dx, - z(v dx - v, dx. ) d 0 d x } - 2pcos28(dv, dx, 2.

A

i.

+ dv2 dx2)(djc

2

2

1

1

*

2 i

1

1

2

+ dx ) + (9z/3p) cos 28 d*2 = 0

In each particular case it is necessary to introduce into the equation (2.88) the corresponding expression of (dx Idx ). If the hardening effect is accounted, then characteristic relations (2.88) cannot be integrated separately and in general it is necessary to use numerical methods. If one introduces the characteristics (2.86) of the velocity fields into the relation (2.89), it will be reduced to the simple connection dVjdXj + dvadjc2 = 0

(2.89)

because of the equality of the first cofactor of (2.89) to zero. The equation (2.89) also has to be expressed by the formulae (2.86) and there­ fore it leads to the equations

86 Mechanics of Porous and Fractured Media 9v. 9v. -*- + ctg(0 + 8) — i . = 0 , 9/m 97 in

9v, 9v. — l - + ctg(0 - 8) —*- = 0 9?iv 9/ IV (2.90)

with the arc length la along characteristics as corresponding independent variables. Instead of the Cartesian components v , v of velocity here it is more con­ venient to use components V m and Vjy at the characteristic direction V m = v1 sin (5 - 6) + v2 cos (8 - 6) (2.91) V

iv

=

- vi

sin

(8 + ^) + v2 cos (8 + 6) .

Then the expressions (2.90) can be rewritten in the form 9V m 3(0 - 8) — - (Vry cosec 28 - V m ctg 28) = 0 9/ m 9/ m (2.92) 9V m b{6 + 8) —y±- + (V m cosec 28 - VIV tg 28) — = 0 . 9/jv

9/jy

If the angle of dilatancy v = const., then the equation (2.92) can be simplified d V m + (V r a tg v - VIV sec v) d.6 = 0 (2.93) d Vjy + (V m sec v - Vw tg v) d$ = 0 .

§2.3.3. Centered plastic flows Let us consider the plane plastic flow only along one of two families of velocity characteristics, that is Vm * 0 ,

Vjy = 0

(2.94)

In this case another family of velocity characteristics is a bunch of straight lines y,v=0,

dtf = 0 ,

0 = 8 + — = const.

(2.95)

Dilatancy of Geomateriah 87

Fig. 2.13. Velocity field of plastic flow at bunch family of characteristics OA,. . . , OB.

going from the centre O (Fig. 2.13). The first equation (2.93) gives us the following velocity projection: V m = i4exp(-»tgi»)

(2.96)

where A is a constant along a characteristic of the IVth family. The transition from Cartesian coordinates x , x to polar ones x

= r cos 0 ,

x

— r sin 0

with the pole at the point 0 transforms the equation (2.87) of the characteristic family III to the differential equation: dr — = ( - tg v) dO r the integral of which has the form of the logarithmic spiral r e x p ( - 0tg v) = const.

(2.97)

The velocity Va of considered flow has to be orthogonal to the characteristic line of the IVth family (its projection VIV = 0). Because the mentioned charac­ teristics is a bunch of straight lines, the velocity vector has to be tangential to the circles (broken lines in Fig. 2.12).229

88 Mechanics of Porous and Fractured Media §2.3.4. Density and stress characteristics The characteristic condition for a density along the particles trajectories has the following form (cos 26 + cos 25)

-^ + tg(*+ » ) l p - + tg(* - 5) v.

dp (2.98)

- pd(vf + v*)cos 28 = 0 Let us consider the stress field characteristics if dz/dp = 0. Then the relations (2.89) reduce to three individual equations, the first of which determines conditions along the stress characteristics: [ - (cos 26 + simfi) dx t + sin 26dx2 ] dP (2.99) - (P + H) sin y? d$ dx2 = 0 the second one (v dx = v dx2) determines the particles trajectories, and the third one as one can see, coincides with equation (2.99) of velocity characteristics. If the internal solid friction angle ifi is independent of density (bifi/dp = 0) then E = -(ifij2) + (ff/4), and the equation (2.85) can also be reduced to dx. / IT —L=ctgle±—+—) dx1 \ 4

ifi \

(2.100)

2 I

Introduction of (2.100) into relation (2.99) gives d(P + H) + 2(P + H)tg,fid0

= 0

(2.101)

the condition along the characteristic (2.86). Integration of the equation (2.101) will give P + H = Cexp(+0 tg?) .

(2.102)

If one accounts the body forces, then the equation (2.99) will have the more general form

Dilatancy of Geomateriah 89

d(P + H) + 2(P + H)tgifid8,

= — — (sin(ft +
COS Ifi

l

'

*

(2.103)

+ cos ( 0j + ^) d*2 J

where F1 = pg sin 8l, F2 = Pg cos flj, pg is the specific weight, 8 is the angle of of inclination of the axis x to the horizon. The system of equations (2.100)—(2.103) is the base for solutions of plane, so-called, statically determined problems,285 which can be solved independently of velocity distribution. However, in general cases it is impossible to avoid the studying of velocity field.

§2.3.5. Flow of particulated masses from plane hopper Let us consider the solution for radial flow of particulated solid material from the hopper.153 In this case the particle trajectories coincide with lines: 8 = const, in the polar coordinate system r, 8. The equations (2.82) have the following form 9 ,

,

1

9

\ P(l + sin ifi cos 2^) | + — — (P sinifi sin 2\J/) dr r b8 + 9

2P r

(2.104) sin ifi cos 2\J/ -I- pg cos

8=0

1 9 {P sin ifi sin 2\J/) + — ^(1 - sin^cos2>J/) r b8

dr +

IP r

sin^ sin 2y - pg sin 8 = 0

and they have to be added by equation (2.84) of dilatant velocity field:

dr

r

■w(*-#**&) 2

v/ \ 3f

r j

9v„ 1 dvr v, tg2*-^ - - ~ = — tg2* dr r 90 r

r2 \d$

) (2.105)

90 Mechanics of Porous and Fractured Media in the polar system and

sm

''

=

_

sin v cos2\/< .

(2.106)

Further, we shall find the solution of the following type _ 2 - s i n y0

P = pgrX{6) ,

v, = f(0)r

'TT^T

(2.107)

that is, the equations (2.104), (2.105) will reduce to three ordinary differential equations for $(8), X(0) and/(0). Numerical integration of the latter is fulfilled for the boundary condition of zero shear stresses along the symmetry axis:

*(0) = f and for the flow intensity /(0) = 1 . The third boundary condition is the coincidence of outer hopper boundary (a particle trajectory) 9 = 0* e = e,

with the characteristics of the equation (2.104). If the integration shows the existence of the zone, where the velocity field gives the negative dissipation of mechanical energy then it is assumed to be the stagnation zone and the position of its outer boundary

e — ** has to be determined.

Dilatancy of Geomaterials 91

Fig. 2.14. Stagnation zone boundary position for different values of internal friction and dilatancy angles.

In Fig. 2.14 the calculated values 0tt(vu) a r e given for the hopper with a horizontal bottom: $ = 90° altogether with experimental measurements of tfM for particulated materials with different values of initial effective angle ifi = 2ifi, + v. of solid friction. It can be observed that the dependence 0 (
**

0

v v

ft

for v = i> ~ 0 is a rather poor description of experimental data. However, this experimental scatter is only image and can be explained quite satisfactorily by the dependence 6tt on » if y> = const. So, the dilatancy effect is dominating in considered cases. The assumption on radial trajectories of particles is a good approximation for the vicinity of point sink from the hopper. The boundary of the stagnation zone appears to be the line of weak discontinuity of radial velocity, the dilatancy angle v being equal to zero at 0 = 0^ . In the axisymmetrical case the two equilibrium equations includes four in­ dependent stress components, and therefore statically determined problems were studied under assumption of full plastic equilibrium, when the stress state was cor­ responding to cross-lines of two sheets of the Coulomb-Tresca yield surface which had a pyramidal form.129 For the Mises-Botkin plastic yield cone it is necessary to complete the equation system by kinematic relation, that is, by the flow rule itself. Given the above analysis of radial particulated flows near hopper discharge hole can be extended to the axisymmetrical case. Radial flows are unstable for some

92

Mechanics of Porous and Fractured Media

combination of dilatancy and solid friction angles. The consideration of the alternative flows with introduced discontinuities has direct implications287 also in geological mechanics.189

§2.3.6. Noncoinciding of stress and velocity characteristics Traditionally it is assumed that the Coulomb friction law or limit equilibrium is valid at the stress characteristic surfaces which have to be simultaneously slip surfaces. Because the equations of velocity field (2.84) are linear, the slip surfaces coincide with velocity characteristics. But coincidence of stress and velocity characteristics (2.85) and (2.86) is possible only if sinv - (P + H)cosifi

d