mech. res. comm. vo1.4(5), 353-366, 1977. pergamon press ... · mech. res. comm. vo1.4(5), 353-366,...
TRANSCRIPT
MECH. RES. COMM. VOI.4(5), 353-366, 1977. Pergamon Press. Printed in USA.
STABILITY CONDITIONS FOR PROPAGATION OF A SYSTEM OF CRACKS IN A BRITTLE SOLID
Zden~k P. Ba~ant and Hideomi Ohtsubo* Department of Civil Engineering, Northwestern University, Evanston, Illinois 60201
(Received ii February 1977; accepted for print 26 August 1977.)
Introduction
In one proposed geothermal energy scheme [1,2], a large vertical main crack is produced in a hot dry rock mass by hydraulic fracture. To be able to remove heat from rock mass which is remote from the crack face, it is necessary to induce by cooling a secondary crack system normal to the wall of the main crack. Significant heat removal is possible only if the opening of secondary cracks is sufficient to allow rapid water circulation in them. The crack opening is wider, the larger is the spacing of cracks. The rate ~ of beat removal from secondary cracks by non-turbulent water circulation is roughly proportional to wS/h, where w = width of cracks and h = their spacing; w is, in turn, proportional to h, and so ~ ,- h ~. Likewise, crack spacing is of importance when dealing with shrinkage cracks in reinforced concrete, for the opening of such cracks has a decisive effect on the rate of corrosion of the embedded steel reinforcement and on the shear transfer capability of aggregate interlock on rough crack surfaces. Other problems in which crack spacing is of interest include the vertical cracking of lava beds extruded and solidified at ocean floor [3], as well as cracking of mud flats and perma- frost soils caused by drying [4,3].
Cooling of a homogeneous brittle elastic halfspace may be expected to produce a system of equally long parallel equidistant cracks normal to the surface. However, crack spacing is not unique according to the Grifflth criterion, and also other equilibrium solutions in which the length alter- nates fromone crack to another are possible. This suggests investigation of uniqueness and stability. It seems that stability questions have so far been considered only with regard to the propagation of a single crack and its direction of propagation [5-7] (Sih's criterion of maximum strain energy density, criterion of maximum energy release rate). This paper attempts to lay down foundations of stability analysis of a system of cracks for each of which the p~opagation direction is known. This problem is much less difficult than the problem of crack direction.
*On leave from Department of Naval Architecture, University of Tokyo, Bunkyo-Ku, Tokyo, Japan
Scientific Communication
353
MECH. RES. COMM. Vo1.4(5), 353-366, 1977. Pergamon Press. Printed in USA.
STABILITY CONDITIONS FOR PROPAGATION OF A SYSTEM OF CRACKS IN A BRITTLE SOLID
Zdenek P. Ba~ant and Hideomi Ohtsubo* Department of Civil Engineering, Northwestern University, Evanston, Illinois 60201
(Received 11 February 1977; accepted for print 26 August 1977.)
Introduction
In one proposed geothermal energy scheme [1,2J, a large vertical main crack is produced in a hot dry rock mass by hydraulic fracture. To be able to remove heat from rock mass which is remote from the crack face, it is necessary to induce by cooling a secondary crack system normal to the wall of the main crack. Significant heat removal is possible only if the opening of secondary cracks is sufficient to allow rapid water circulation in them. The crack opening is wider, the larger is the spacing of cracks. The rate H of heat removal from secondary cracks by non-turbulent water circulation is roughly proportional to w3/h, where w = width of cracks and h = their spacing; w is, in turn, proportional to h, and so H - h2 • Likewise, crack spacing is of importance when dealing with shrinkage cracks in reinforced concrete, for the opening of such cracks has a decisive effect on the rate of corrosion of the embedded steel reinforcement and on the shear transfer capability of aggregate interlock on rough crack surfaces. Other problems in which crack spacing is of interest include the vertical cracking of lava beds extruded and solidified at ocean floor [3J, as well as cracking of mud flats and permafrost soils caused by drying [4,3).
Cooling of a homogeneous brittle elastic halfspace may be expected to produce a system of equally long parallel equidistant cracks normal to the surface. However, crack spacing is not unique according to the Griffith criterion, and also other equilibrium solutions in which the length alternates from one crack to another are possible. This suggests investigation of uniqueness and stability. It seems that stability questions have so far been considered only with regard to the propagation of a single crack and its direction of propagation [5-7J (Sih's criterion of maximum strain energy density, criterion of maximum energy release rate). This paper attempts to lay down foundations of stability analYSis of a system of cracks for each of which the propagation direction is known. This problem is much less difficult than the problem of crack direction.
*On leave from Department of Naval Architecture, University of Tokyo, Bunkyo-Ku, Tokyo, Japan
Scientific Communication
353
354 ZDENEK P. BAZANT and HIDEOMI OHTSUBO
Conditions of Stability of a General Crack Configuration
Consider first the general case of a brittle elastic body which contains
a number of cracks of arbitrary shape (Fig. la). For the sake of simplicity,
assume that the body is in a state of either plane strain or plane stress, and
that propagation of the cracks is governed only by Mode I (opening mode)
stress intensity factors, K i [6], where subscript i refers to the i th crack
tip, i = I, 2, ...N. Also, assume that the cracks do not branch and that they
propagate in given directions along straight or curved trajectories. Let a i
denote the length of crack up to ~ts tip (Fig. la).
It is well known that the condition of stability of equilibrium of a
single critical crack of length a. is i
~Ki/~a i < 0 (i = I, 2 .... N) (I)
This holds for a general elastic body, and because a body with many cracks of
which only one extends is a special case of a general elastic body with one
extending crack, this condition also represents a necessary condition of
stability o~ a crack system. It is not at all clear, however, whether Eq. (I)
represents a sufficient condition, i.e., whether there are other conditions
that have to be satisfied to assure stability.
To investigate equilibrium and stability, it is necessary to consider the
work, W (more precisely, Helmholtz free energy), that would have to be sup-
plied to the body in order to extend the cracks; N a.
W = U(al,am,...aN;D) + ~ ~ i 2~ida~ (2) i=l 0
Here U = elastic strain energy of the body, 2~i = specific energy of extension
of the i th crack; and D = loading parameter. In particular, D will represent
here the penetration depth of cooling. If yielding and microcracklng near the
advancing crack tip were absent and the crack surfaces were not rough but per-
fectly plane, ~i would equal the surface energy of the material. But these
effects are always present and often they dissipate much energy; then ~i is a
constant which is much higher than the surface energy.
Consider now that the crack tips number i = l,...m extend (6a i > 0), the
cracks numbered m+l .... n close and shorten (6a. < 0), and the crack tips num- i
bered n+l,...N remain stationary (~a i = 0); 0 ~ m ~ n ~ N. This includes the
case m = n when no crack closes, and the case n = N when no crack remains
354 ZDENEK P. BAZANT and HIDEOMI OHTSUBO
Conditions of Stability of a General Crack Configuration
Consider first the general case of a brittle elastic body which contains
a number of cracks of arbitrary shape (Fig. la). For the sake of simplicity,
assume that the body is in a state of either plane strain or plane stress, and
that propagation of the cracks is governed only by Mode I (opening mode)
stress intensity factors, K. [6J, where subscript i refers to the ith crack ~
tip, i = 1, 2, •.. N. Also, assume that the cracks do not branch and that they
propagate in given directions along straight or curved trajectories. Let ai denote the length of crack up to its tip (Fig. la).
It is well known that the condition of stability of equilibrium of a
single critical crack of length a. is ~
~K. /~a. < 0 ~ ~
(i = 1, 2, ••. N) (1)
This holds for a general elastic body, and because a body with many cracks of
which only one extends is a special case of a general elastic body with one
extending crack, this condition also represents a necessary condition of
stability of a crack system. It is not at all clear, however, whether Eq. (1)
represents a sufficient condition, i.e., whether there are other conditions
that have to be satisfied to assure stability.
To investigate equilibrium and stability, it is necessary to consider the
work, W (more precisely, Helmholtz free energy), that would have to be sup-
plied to the body in order to extend the cracks; N a.
+ ~ S ~ i=l 0
2y. da! ~ ~
(2 )
Here U = elastic strain energy of the body, 2V. = specific energy of extension ~
of the ith crack; and D = loading parameter. In particular, D will represent
here the penetration depth of cooling. If yielding and microcracking near the
advancing crack tip were absent and the crack surfaces were not rough but per
fectly plane, 'Y. would equal the surface energy of the material. But these ~
effects are always present and often they dissipate much energy; then v. is a ~
constant which is much higher than the surface energy.
Consider now that the crack tips number i = l, ••• m extend (6a. > 0), the ~
cracks numbered m+l, ••• n close and shorten (6a.< 0), and the crack tips num~
bered n+l, •.. N remain stationary (6a. = 0); 0 ~ m ~ n ~ N. This includes the ~
case m = n when no crack closes, and the case n = N when no crack remains
STABILITY OF CRACK PROPAGATION 355
immobile. The work, AW, that would have to be supplied in order to change the
crack lengths by 6a i (at constant loading parameter D and for applied loads
doing no work) is a function of 6a.. This function must admit Taylor series l
expansion, i.e., m n ) AW = 6W + 62W + ...; 6W = i=l~ ~Ul + 2~i 6ai + j=m+l~ ~aj 6a.3 (3a)
n n ~2 U m ~Yi 6ai)2 n n 62W ffi ½ E E 6a 6a + ~ ~ ( = ½ E E 6a 6a. (3b)
i=l j=l ~aiSaj i J i 1 5ai i=l j=l Wij i 3
in which 6W and 62W are the first and second variations; and
5v i 52U + 2 -- H(6ai) (i,j = I, .n) (4) Wij = Wji = ~ai~aj 5a i 6ij ..
where 6ij = Kronecker delta and H = Heaviside step function, i.e., H(6ai) ffi I
where 6a i > 0 and H(6ai) = 0 when 6a. < 0. Usually the fracture properties of i
the body can be considered homogeneous, and then 5Ni/~a i = 0.
For the cracks to change their length in an equilibrium manner, 6W must
vanish for any 6a i. It is necessary to distinguish whether a crack extends
(6a.l > 0) or closes (6a i < 0). According to Eq. (2), 6W = 0 occurs if, and
only if
for 6a. > 0: - ~U ~U • 5a i = 2~i; for 6a i < 0: ~a i = 0 . (5)
Eq. (5) includes the well-known Griffith fracture criterion• Note that the
strain energy release rate is -~U/Sa i.
An equivalent form of Eq. (5) can be given in terms of the stress inten-
sity factor,. K. = lim (c~) for r ~ 0 where r = distance from the crack 1
tip and @ = transverse normal stress straight ahead of the crack. It is well
known [6] that for plane strain ~U/Sa i = -K~/E' with E' = E/(I-92) where E ffi
Young's modulus, 9 ffi Poisson ratio. Thus, Eq. (5) is equivalent to
for 6a i > 0: K i = Kc. ; for 6a i < 0: K i = 0 (6)
)½ ~ where Kcl = (2viE' = critical value of the stress intensity factor = frac-
ture toughness of the material Using ~U/Sa i -K~/E' • = , one may write
~K ~K,1 E' ~2u = ~ ffi -Kj (7) 2 ~ai~a j -Ki ~aj ~a i
Having stated the conditions of equilibrium, it is natural to ask whether
the equilibrium configuration is stable. The crack system is said to be
STABILITY OF CRACK PROPAGATION 355
immobile. The work, ~W, that would have to be supplied in order to change the
crack lengths by Oa. (at constant loading parameter D and for applied loads ~
doing no work) is a function of oa .• This function must admit Taylor series ~
expansion, i.e.,
~W m (bU ) n bU t ~ + 2V. oai + t ~ oa.
i=l ~jai ~ j=m+l aj J (3a)
n n ~2u m bYi 2 n n o2W ~ l:: l:: oa oa + 1:: (oai ) ~ t t w.joa oa. (3b)
j=l bai baj i j i=l ba. i=l j=l ~ i J i=l ~
in which oW and o2W are the first and second variations; and
(i,j = 1, ... n) (4)
where 0.. Kronecker delta and H = Heaviside step function, i.e., H(Oai
) = 1 ~J
where Oai
> 0 and H(oa.) = 0 when oa. < O. Usually the fracture properties of ~ ~
the body can be considered homogeneous, and then bv.lba. = O. ~ ~
For the cracks to change their length in an equilibrium manner, Ow must
vanish for any oai
• It is necessary to distinguish whether a crack extends
(Oa. > 0) or closes (oa. < 0). According to Eq. (2), oW = 0 occu~s if, and ~ ~
only if
for Oai > 0: - b~~ = 2Vi ; for oai < 0: ~~i = 0 . (5)
Eq. (5) includes the well-known Griffith fracture criterion. Note that the
strain energy release rate is -bu/ba .. ~
An equivalent form of Eq. (5) can be given in terms of the stress inten
sity factor, K. = lim (a/Znr) for r ~ 0 where r = distance from the crack ~
tip and cr = transverse normal stress straight ahead of the crack. It is well
known [6J that for plane strain bU/ba. = -K:/E' with E' = E/(1-v2
) where E = ~ ~
Young's modulus, v = Poisson ratio. Thus, Eq. (5) is equivalent to
for oa. > 0: K. = K ; for oai < 0: K. = 0 ~ ~ c. ~
(6) ~
where KCi = (2ViE')~ = critical value
ture toughness of the material. Using
of the stress intensity factor = frac
bU/bai = -Kf/E', one may write
E' T
bK bK = -K --1 = -K ~
i baj j bai (7)
Having stated the conditions of equilibrium, it is natural to ask whether
the equilibrium configuration is stable. The crack system is said to be
356 ZDENEK P. BAZANT and HIDEOMI OHTSUBO
a) b) c) d) e) 20
P1 ~ K 2 D/2h = 1.0
3 IS
a~ ~
a 4 " = ~ IO : ,~ , .~~ - ~ ,~ ~ ~ 1 ~ ~ ~, ~_ ~ . ~ ~ y'X2 ~
~ - . ~ k~.., ~ ~ ~ ~ ~. o ~1 o st.o~,~
i i ~ ~ ~ ~ i i i 0 . 0 0 .5 LO
a~/2h
1.0 ~ ~ ' ~ ) ' ' al= ag
., [ O ~ ~n ~ m "Vz 40
o.~ g ~ p~a K I
E~ i l l b~ ium czacks 30 ~ ~
0.0 , , , I ~ , I ~ O "J UnstabZe ~ 0.0 1.0 2,0 3,0 4.0
~ ~.,,, ~ o,~ '~ ,0 ~ ~ l > o ~
- ' 1 ~ f f 2h 2 o: 4h 2b 4h
~ I'1 1 I *1 ~ 10 ~ • ~ ~ ~ ~ ~ l ~ ~ a~/2~ = 1.0
~ " ~ ~ / 2 ~ ~ 2 . 0
0 I i i i ~ i , i ,
0.0 0.5 1.0
a2/2~
~
,a~ , ~ / / unstahle-v_~K~= 0 (c~ack a~ i ) I~Z . t z o n p ~ / s t a ~ " , K'LO j ) ~ ~ c l o s . )
f u ~ c / ~ ~ ,< ~ * " ~c ~ a , ~ ' " " , ~ - ~, < ~ = ~c
~ ~ ,~,~t~o. ~'
a2, al ,a2
FIG. i General and Special Crack Systems Investigated (a-e), Some Numerical Results (f,g) with Grid Used (h; 2h = Im), and Bifurcation of Equilibrium Path (i,j)
356 ZDENEK P. BAZANT and HIDEOMI OHTSUBO
a) 20r---------------------------~
H £) ~ 1.0
C. '" .... ~ 81
0 0.5
Equilibrium cracks
4.0 O.O'=-_ ......... _----c:":-_----L __ -=':-__ L-_-=": __ -'-_---,I
0.0 1.0 2.0 3.0
D/2h
~.-T.f('ID) h) TTTTT
I D- 4" 2. <.
=:I'"'- '" '·1 j j +1
I
J l
~ !'l. 1!!';::n1i;1
i)
15
~ 10 'e ~ ~
0.0
40
30
OJ
iOl 'e 20 z ~
~
10
0 0.0
g)
~ ~D-----J
al 2h
a2 2h
al
0.5
0.5
a2/2h = 1.0
D/2h = 1.0
a l l2h = 1.0
D/2h = 2.0
j) 0 unstab~l. K,=O (crack a,
closed) /
/
a, / K, < K, = Kc
\bi f ur::tion ..-•
FIG. 1 General and Special Crack Systems Investigated (a-e), Some Numerical Results (f,g) with Grid Used (h; 2h = 1m), and Bifurcation of Equilibrium Path (i,j)
1.0
1.0
STABILITY OF CRACK PROPAGATION 357
stable if and only if no a. can change without changing the loads (or D). i
Thus, stability is ensured if and only if the work AW that is done at any
admissible crack length increments 6a. is positive, for if this work is not i
done 6a. cannot occur. On the other hand, if AW < 0 for some 6a. energy 1 l'
is released, and when a release of energy is possible, changes 6a. will 1
occur spontaneously, AW being transformed into kinetic energy and ultimately
dissipated as heat (which follows from the second law of the thermo-
dynamics).
One well-known unstable situation arises when K i > Kci (or -~U/ba i > 2Vi )
for 6a.i > 0. Indeed, 6W < 0 for 6a.l > 0, and so K.I > Kci is impossible. Simi-
< 0 (or -bU/~a i < 0) for 8a. < 0 is also dnstable. There- larly, the case K i 1
fore,with regard to non-negativeness of 6W, it is necessary that 0 ~ -bU/ba i K
2V i or 0 ~ K i ~ Kci at all times. Combining the foregoing conditions and
Eq. (6), it follows that with regard to the first variation, 6W, only the
fol~owing variations 6a. are admissible: 1
for K i = Kci: 6a.~ ~ 0; for K.~ = 0: 6a.i ~ 0 (8a)
• < 6a. = 0 (8b) for 0 < K I Kci: i
If 6W = 0, stability will be ensured if
n n 26SW = ~ ~ W.. 6a. 6a > 0 for any admissible 6a i. (9)
i=l j=l 13 ~ J
Conversely, instability occurs if 62W < 0 for some admissible choice of 6a i.
The admissible increments 6a.i are given by Eq. (8). If matrix Wij is positive
definite, stability is assured. However, if W.. is not positive definite, the 13 crack configuration may or may not be unstable. It is unstable if 6SW < 0 at
K i = Kc. or K i = 0 for some admissible 6a i. Critical state occurs when 6SW = 0 i
at K i = Kc. or K i = 0 for some admissible 6a.. i 1
Array of P,arallel Coolin~ C~acks Penetratin~ a Halfspace
Consider now a homogeneous isotropic elastic halfspace which is initially
(at time t = 0) at constant temperature, T = To, and is then cooled at the
surface x = 0 to temperature T s. This produces an array of straight parallel
equidistant cracks normal to the surface (Fig. 1 b-d). The temperature field
is assumed to have the form T-T ° = f(x/D) (Ts-To) where D = D(t) = penetration
depth of cooling. If all heat is transferred by conduction through the solid,
STABILITY OF CRACK PROPAGATION
stable if and only if no a. can change without changing the loads (or D). 1.
Thus, stability is ensured if and only if the work 6W that is done at any
admissible crack length increments ca. is positive, for if this work is not 1.
done ca. cannot occur. On the other hand, if ~W < 0 for some ca., energy 1. 1.
is released, and when a release of energy is possible, changes ca. will 1.
357
occur spontaneously, AW being transformed into kinetic energy and ultimately
dissipated as heat (which follows from the second law of the thermo
dynamics).
One well-known unstable situation arises when K. > Kc. (or -bU/ba. > 2V.) 1. 1. 1. 1.
for ca. > O. Indeed, cW < 0 for ca. > 0, and so K. > Kc. is impossible. Simi-1. 1. 1. 1.
larly, the case K. < 0 (or -bU/ba. < 0) for ca. < 0 is also unstable. There-1. 1. 1.
fore,with regard to non-negativeness of oW, it is necessary that 0 s -bU/ba. s 1.
2V. or 0 S K. S Kc. at all times. 1. 1. 1.
Combining the foregoing conditions and
Eq. (6), it follows that with regard to the first variation, oW, only the
fol{owing variations ca. are admissible: 1.
for K. = Kc.: ca. ~ 0; for K. 1. 1. 1. 1.
for 0 <K. < Kc.: oa. = 0 1. 1. 1.
If cW 0, stability will be ensured if
n n
0: oa. S 0 1.
(Sa)
(8b)
E t W .. oa. oa > 0 for any admissible oai • 1.J 1. J' i=l j=l
(9)
Conversely, instability occurs if c2W < 0 for some admissible choice of oai .
The admissible increments Oa. are given by Eq. (8). 1.
If matrix W .. is positive 1.J
definite, stability is assured. However, if W .. is not positive definite, the 1.J
crack configuration mayor may not be unstable. It is unstable if c2 W < 0 at
Ki = Kc. or Ki = 0 for some admissible oai • Critical state occurs When c2 W = 0 1.
at K = K or K - 0 for some admissible ca .• i c i i - 1.
Array of Parallel Cooling Cracks Penetrating a Halfspace
Consider now a homogeneous isotropic elastic halfspace which is initially
(at time t = 0) at constant temperature, T = T , and is then cooled at the o surface x = 0 to temperature T. This produces an array of straight parallel
s equidistant cracks normal to the surface (Fig. 1 b-d).
is assumed to have the form T-T = f(x/D) (T -T ) where o s 0
The temperature field
D = D(t) = penetration
depth of cooling. If all heat is transferred by conduction through the solid,
358 ZDENEK P. BAZANT and HIDEOMI OHTSUBO
one has f(~) = erfc ~ = 2 ~ exp(- ) d~I/~, ~ = /'~ xlD, D = ~I'~, c = heat
diffusivity. Because T is constant along lines parallel to the surface, it
is logical to assume a periodic pattern of crack lengths. Accordingly, con-
sider that every other crack has one length, a2, and the cracks inbetween have
another length, al; a 2 • a I (Fig. Ic). Cracks of equal lengths (a 2 = al)
represent one possible equilibrium state. These states (not all necessarily
stable) are plotted in Fig.if on the basis of finite element calculations for
To-T s = 100°C (with error function T-profile),~ = 8 X 10 -6 per °C (linear
thermal expansion coefficient), E = 37600 MN/m 2, and ~ = 0.305 (2h=Im).
Various interesting properties of the parallel crack system can be ana-
lyzed even without numerical solutions of K i and Wij. Since K c is a con-
stant, a sufficient condition of stability is the positive definiteness of
= ~2U/~ai~aj matrix Wij , which requires that
WII W12 I W22 = WII > 0 and 0 (I0)
W21 W221 >
A critical state, corresponding to a bifurcation point on the basic equili-
brium path a I = a2, would arise if 62W = ½ Ei(~ j Wij6aj)6a i = 0 for some
admissible 6a.. This condition is satisfied if ~. W.. 6a. = 0 or • 3 ~3 3
WII 6a I + WI2 6a 2 = 0 ,
(11)
W21 6a I + W22 6a 2 = 0
in which WII = W22 and WI2 = W21. Setting the determinant zero, one has
WII ~22 " WI22 = 0, and using Eq. (7) this becomes (-K 2 ~K2/~a2)2 - (-K 2 ~/ 2
~al) = 0. Noting that WII = W22 and K 2 = K I for a 2 = al, one has W22 -
= 0, yielding (~K2/~a2)2 - (~K2/~al)2 = 0 as a condition of possible critical
state. Admissibility of the corresponding eigenvector (6al, 6a2) must be checked,
though. Since WII = W22, WI2 = W21, and at the same time W22 = ± WI2 ,
Eq. II suggests
6al/~a = + I (12) 2 --
as possible critical states.
The plus sign in Eq. (12) yields 6a I = 6a 2. In this case Eq. (II) re-
duces to (WII + WI2) 6a = 0 where 6a = 6a I = 6a 2. Noting that, by virtue
of the chain rule of differentiation, WII + WI2=(DWl/~al )~al/~a+(~wl/~a2)oa 2
358 ZOENEK P. BAZANT and HIDEOMI OHTSUBO
(Xl 2 one has f(~) = erfc S = 2 J ~ exp( -'11 ) d'l1//TI, ~ = FJ x/D, D = jITCt, c = heat
diffusivity. Because T is constant along lines parallel to the surface, it
is logical to assume a periodic pattern of crack lengths. Accordingly, con
sider that every other crack has one length, a2 , and the cracks inbetween have
another length, a l ; a2 ~ al (Fig. lc). Cracks of equal lengths (a2
= al
)
represent one possible equilibrium state. These states (not all necessarily
stable) are plotted in Fig.lf on the basis of finite element calculations for
To-Ts = 1000C (with error function T-profile),~ = 8 X 10- 6 per °c (linear
thermal expansion coefficient), E = 37600 MN/m2 , and v = 0.305 (2h=lm).
Various interesting properties of the parallel crack system can be ana
lyzed even without numerical solutions of Ki and W ..• Since K is a con-~J c
stant, a sufficient condition of stability is the positive definiteness of
matrix W.. ~2u/~a.ba., which requires that ~J ~ J
Wll
W12 W22 = W
ll > 0 and > 0 W2l W22
(10)
A critical state, corresponding to a bifurcation point on the basic equili-2
brium path a l = a2 , would arise if 5 W = ~ ~.(~. w .. oa.)5a. = 0 for some ~ J ~J J ~
admissible oa .• This condition is satisfied if t. W .. 5a. = 0 or ~ J ~J J
Wll 5al + W12 5a2 0
(11)
in which Wll
= W22 and W12
= W2l • Setting the determinant zero, one has
Wll W22 - w122 = 0, and using Eq. (7) this becomes (-K2 ~K2/eaz)2 - (-K2 ~/ ~al)2 = O. Noting that Wll = W22 and K2 = Kl for a2 = aI' one has W~2 - Wi2
= 0, yielding (bK2/ea2)2 - (~K2/bal)2 = 0 as a condition of possible critical
state. Admissibility of the corresponding eigenvector (oal , 5a2) must be checked~
though. Since Wn = W22 ' W12
= W2l ' and at the same time W22 = ± W12 '
Eq. 11 suggests
(12)
as possible critical states.
The plus sign in Eq. (12) yields 5al = 5a2 , In this case Eq, (11) re
duces to (Wll
+ W12
) oa = 0 where 6a = 6al = 6a2 , Noting that, by virtue
of the chain rule of differentiation, Wll + W12=(bWl/~al)bal/ba+(~Wl/ea2)oa2
STABILITY OF CRACK PROPAGATION 359
/~a=dWl/da with WI=~W/~aI=-KI2/E ' , one concludes that Eq. (ii) degene-
rates into the condition (dKl2/da) 6a = 0 or dKl/da = 0, which is a condi-
tion of instability of the basic equilibrium path, a I = a 2. The condition
of stability of this path is dKl/da < 0 or dWl/da > 0, which is analogous to
the, well-known stability condition for a single crack (Eq. I).
Bifurcation of the equilibrium path would be obtained if Eq. (12) admitted
the minus sign, i.e., 6a I = -6a 2. For Eq. (II) to allow this, WII and WI2 , and
thus also ~K2/Da 2 and ~K2/~al, would have to be of the same sign. Of these,
~K2/~a 2 must be negative, or else a critical state of another type, associated
with the first condition in Eq. (I0) would precede this bifurcation. As far
as ~K2/~a I is concerned, the finite element calculations described in the
sequel indicated that for the present crack system with a I = a 2 it is always
negat£ve, which also agrees with some intuitive considerations.
Since in the present case both ~K2/~a 2 and ~K2/~a I are negative, it ap-
pears that Eq. (Ii) would indeed admit the minus sign. However, this means
that either 6a I or 6a 2 must be negative, and this violates condition (8)
because K I = K 2 = K c. Hence, a bifurcation of the type given by Eqs. (II)
and (12) is ~een to be impossible.
The remaining possible critical state according to Eq. (I0) is given by
the condition W22 = WII = 0 or
[~K2/~a2]al = const. = 0 . (13)
The associated second variation is ~2W = ½ W22 (6a2)2 with 6a I = 0, and the
bifurcation ("instability") mode is
6a 2 > 0, 6a I = 0 . (14)
This mode also represents a bifurcation point on the basic equilibrium path
a I = a 2 (Fig. li). According to Eq. (Sb), K I does not have to remain equal to
K c but it may decrease, i.e., the tip of the crack a I may be unloading. In
fact K I ought to decrease after bifurcation since extension of crack a 2
should have a non zero effect on K I. It might be also of interest to note 2
that if ~K2/Oa 2 > 0 (or WII = W22 < 0), then WII W22 - WI2 or det (Wij) is
always negative.
Further light may be shed on the problem if the ,path of equilibrium
states is regarded as a function of parameter D (cooling penetration depth);
STABILITY OF CRACK PROPAGATION 359
z loa=dWl/da with wl=oW/oal=-Kl IE', one concludes that Eq. (11) degene-
rates into the condition (dKlZ/da) 6a = 0 or dKl/da = 0, which is a condi
tion of instability of the basic equilibrium path, al = aZ' The condition
of stability of this path is dKl/da < 0 or dWl/da > 0, which is analogous to
the. well-known stability condition for a single crack (Eq. 1).
Bifurcation of the equilibrium path would be obtained if Eq. (lZ) admitted
the minus sign, i.e., Oal = -8aZ' For Eq. (11) to allow this, Wll and WlZ ' and
thus also bKZ/baZ and OKZ/bal , would have to be of the same sign. Of these,
bKZ/baZ must be negative, or else a critical state of another type, associated
with the first condition in Eq. (10) would precede this bifurcation. As far
as bKZ/oal is concerned, the finite element calculations described in the
sequel indicated that for the present crack system with a l ~ aZ it is always
negative, which also agrees with some intuitive considerations.
Since in the present case both OKZ/baZ and ~Z/oal are negative, it ap
pears that Eq. (11) would indeed admit the minus sign. However, this means
that either 6al
or 6aZ must be negative, and this violates condition (8)
because KI = K2 = Kc' Hence, a bifurcation of the type given by Eqs. (11)
and (12) is seen to be impossible.
The remaining possible critical state according to Eq. (10) is given by
the condition WZZ = Wll
= 0 or
[bK/bazJal const. = 0 • (13)
The associated second variation is OZW ~ W2Z (6aZ) Z with 6al
O,and the
bifurcation ("instability") mode is
6aZ
> 0, Oa = 1 0 . (14)
This mode also represents a bifurcation point on the basic equilibrium path
al = a2 (Fig. Ii). According to Eq. (8b), Kl does not have to remain equal to
Kc but it may decrease, i.e., the tip of the crack al
may be unloading. In
fact Kl ought to decrease after bifurcation since extension of crack aZ
should have a non zero effect on Kl
• It might be also of interest to note
that if OK2/0aZ > 0 (or Wll = W22 < 0), then Wil W22 - wi2 or det (Wij
) is
always negative.
Further light may be shed on the problem if the ,path of equilibrium
states is regarded as a function of parameter D (cooling penetration depth);
360 ZDENEK P. BAZANT and HIDEOMI OHTSUBO
see (Fig. lj). Denoting W i = ~W/~a. where W is given by Eq. (~) the 1 '
equilibrium path is distinguished by the conditions W. = 0(i = 1,2). Deriva- i
tives W.l at equilibrium states are functions of a I and a 2. However, by con-
trast to W in Eq. (2), a. are generally not independent of D because along l
the equilibrium path the crack lengths a I and a 2 depend on D. Thus, W i
along the equilibrium path are implicit functions of D; i.eo,
[~W/~ai]D,const. = W.l [al(D), a 2 (D)] = 0 (i = 1,2) (15)
Assume that on the basic equilibrium path a 2 = a I there is a critical
point (bifurcation point) correspdnding to D = D (Fig. lj). Functions W. O ~
ought to admit Taylor series expansions at D = D • The cracks should also o
be in equilibrium at adjacent states with D sufficiently close to D o . If
both 6a I and 6a 2 are assumed to be positive then W. would have to be con- ' ~
stant for all such D-values. Consequently, dWi/dD = O, d2W./dD 2 = O, etc. i ~
must be true at D = D . This yields O
2 r.~._]
E L~a~. ~ a~ = 0 (i = 1,2) (16) j=l J
0 w
~'i] a'~ = 0 (i = 1,2) (17) E E aj~ 3 4+ E aj J
j=Ik=l 0 j=l ~ 0 where a~ = da./dD along the equilibrium path at D = D O . These equations re-
] ] present conditions of continuing equilibrium, analogous to those which follow
from the perturbation method of structural stability theory [8]. If they
admit solution for D ~ const., then a critical state is reached. Setting
a! ~ 6a., the condition in Eq. (16) is obviously identical to Eq. (II), i i
which is a bifurcation of a type that is inadmissible. However, there exists
no reason why a higher-order bifurcation governed by Eq. (17) could not take
place. Since Eq. (16) is not satisfied, such higher-order bifurcation would
' = ' " e., to the increments for the basic path have to conform to a I a2,L
(0a I = ~a2) and would have to take place at increasing D (i.e., at increas- ! !
ing cooling penetration depth). Therefore a I and a 2 in Eq. (17) must be
equal, and a higher-order bifurcation, with the secondary path being tangent
to the basic path (a I = a 2) at the bifurcation point, would occur if Eq. (17)
~! ~ ~! • admitted a solution with a I a 2
Eqs. (16) and (17) were written under the assumption that both 6a I and
~a 2 are positive (and K I = KZ = Kc). Consider now that ~a 2 > 0 and ~a I = O,
360 ZDENEK P. BAZANT and HIDEOMI OHTSUBO
see (Fig. Ij). Denoting W. = eW/ba. where W is given by Eq. (2), the I. I.
equilibrium path is distinguished by the conditions W. = O(i = 1,2). DerivaI.
tives Wi at equilibrium states are functions of al
and a2
. However, by con-
trast to W in Eq. (2), a. are generally not independent of D because along I.
the equilibrium path the crack lengths al
and a2 depend on D. Thus, Wi
along the equilibrium path are implicit functions of D;
[~W/ba·JD t = W. [al(D), a2 (D)J = 0 (i = 1,2) (15) I. ·cons. I.
Assume that on the basic equilibrium path a2
= al
there is a critical
pOint (bifurcation point) correspdnrling to D = D (Fig. Ij). Functions W. o I.
ought to admit Taylor series expansions at D D. The cracks should also o
be in equilibrium at adjacent states with D sufficiently close to D. If o
both cal and 6a2 are assumed to be positive, then Wi would have to be con-
stant for all such D-values. Consequently, dW./dD = 0, d2W./dD2 0, etc.,
must be true at D = D. This yields o
2 r- bW .-l ~' I.
j=l Lbaj~ 0
2 2 b2W.
a~ = 0 J
I; k~l [~aj ~~kl a~ 2 [bW. l '+ L: l.
'\ j=l ~aj~ j=l J 0
I. I.
(i 1,2) (16)
a'.' = 0 ( i = 1, 2 ) J
(17)
0 where a~ = da./dD along the equilibrium path at D = DO. These equations re-
J J present conditions of continuing equilibrium, analogous to those which follow
from the perturbation method of structural stability theory [8J. If they
admit solution for D ~ const., then a critical state is reached. Setting
a! ~ ca., the condition in Eq. (16) is obviously identical to Eq. (11), I. I.
which is a bifurcation of a type that is inadmissible. However, there exists
no reason why a higher-order bifurcation governed by Eq. (17) could not take
place. Since Eq. (16) is not satisfied, such higher-order bifurcation would
have to conform to ai = a2, i. e., to the increments for the basic path
(oa1
= Oa2
) and would have to take place at increasing D (i.e., at increas
ing cooling penetration depth). Therefore ai and ai in Eq. (17) must be
equal, and a higher-order bifurcation, with the secondary path being tangent
to the basic path (al
= a2
) at the bifurcation point, would occur if Eq. (17)
admitted a solution with a1 ~ aZ. Eqs. (16) and (17) were written under the assumption that both cal and
6a2
are positive (and Kl = KZ = Kc). Consider now that 6a2 > 0 and 6al
= 0,
STABILITY OF CRACK PROPAGATION 361
i.e., one crack stops growing. In this case Eqs. (16) - (17) still repre-
sent a possible condition of critical state, but in view of Eq. (Sb) it is
also possible that only K 2 remains at its critical value K c while K I decreases
below K . In fact, since bifurcation ~iven by Eq. (16) was shown Co be inad- C
missible (according to Eq. 8), it is not possible that K I and K 2 remain equal
K during bifurcation. Hence, it is necessary that C
~Kl/~a 2 < 0 (18)
~uring bifurcation. Thus ~WI/~D cannot be zero at D = DO, and only the con-
dition ~W2/~D = 0 applies as a condition of continuing equilibrium, yielding
~W 2 !
~a 2 a 2 = 0 (19)
where a s = da2/dD. Eq. (19) represents the condition of a criticial state,
provided that it holds true in the limit for D ~ const. This case is identi-
cal with E~. (13) derived from the condition 62W > 0.
It is seen that the present variational stability analysis yields the
condition in Eq. 13. This condition is identical to the elementary condition
in Eq. I, which is i,~,ediately obvious even without the variational analysis.
It remains to be seen whether, for the particular crack system at hand,
~2/~a 2 can indeed change its sign. Therefore, some finite element computations
have been carried out. The grid in Fig. If, composed of four-node quadri-
lateral elements, formed by condensing a block of four constant-strain
triangles, was used. The derivatives of potential energy (~W/~al, ~2W/~al~a2,
etc.) were calculated from their finite difference approximations, using
the potential energy (Eq. 2) in the whole grid for various crack length a I and
a 2. The temperature profile was approximated as parabolic, and the Young's
modulus E = 37,600 MN/m 2, the Poisson ratio ~ = 0.305 and the linear thermal
expansion coefficient ~ = 8 X 10 -6 per °C (all typical of granite) were used.
Some of the results are shown in Fig. Ig, in which the intersections of curves
K I and K 2 represent equilibrium states if K c = 22.8 MNm -3/2. In one of these
intersections the slope of the curve of K 2 versus a 2 is positive, which
violates Eq. 13 and indicates that the equilibrium is unstable. This proves
that instability due to the violation of Eq. 13 is indeed possible. However,
it is not at all clear from Fig. Ig that the instability governed by Eq. 13
STABILITY OF CRACK PROPAGATION 361
i.e., one crack stops growing. In this case Eqs. (16) - (17) still repre
sent a possible condition of critical state, but in view of Eq. (Sb) it is
also possible that only K2 remains at its critical value Kc while Kl decreases
below K. In fact, since bifurcation tilven by Eq. (16) was shown to be inad-c
missible (according to Eq. 8), it is not possible that Kl and K2 remain equal
K during bifurcation. Hence, it is necessary that c
(IS)
1uring bifurcation. Thus oWl/bD cannot be zero at D = DO' and only the con
dition oW2/bD = 0 applies as a condition of continuing equilibrium, yielding
(19 )
where a2 = da2
/dD. Eq. (19) represents the condition of a criticial state,
provided that it holds true in the limit for D -7 const. This case is identi
cal with Eq". (13) derived from the condition e?" > O.
It is seen that the present variational stability analysis yields the
condition in Eq. 13. This condition is identical to the elementary condition
in Eq. 1, which is immediately obvious even without the variational analysis.
It remains to be seen whether, for the particular crack system at hand,
~2/oa2 can indeed change its sign. Therefore, some finite element computations
have been carried out. The grid in Fig. If, composed of four-node quadri
lateral elements, formed by condensing a block of four constant-strain
triangles, was used. The derivatives of potential energy (ow/oal , o2w/oaloa2'
etc.) were calculated from their finite difference apprOXimations, using
the potential energy (Eq. 2) in the whole grid for various crack length al
and
a2 • The temperature profile was approximated as parabolic, and the Young's 2
modulus E = 37,600 MN/m , the Poisson ratio v = 0.305 and the linear thermal -6 0 expansion coefficient Q' = 8 X 10 per C (all typical of grani te) were used.
Some of the results are shown in Fig. Ig, in which the intersections of curves
Kl and K2 represent equilibrium states if Kc = 22.S MNm- 3/ 2 • In one of these
intersections the slope of the curve of K2 versus a2 is positive, which
violates Eq. 13 and indicates that the equilibrium is unstable. This proves
that instability due to the violation of Eq. 13 is indeed possible. However,
it is not at all clear from Fig. Ig that the instability governed by Eq. 13
362 ZDENEK P. BAZANT and HIDEOMI OHTSUBO
(or Eq. I) is the case which controls. Without a complete solution of the
crack problem, the higher-order bifurcation with common tangent (Eq. 17)
cannot be ruled out in a general case. In fact, on the basis of some crude
finite element calculations of the equilibrium path it was initially thought
that in the present problem this latter type of bifurcation (Eq. 17) occurred
before the bifurcation given by Eq. 13 and that it, therefore, controlled. On
the other hand, S. Nemat-Nasser intuitively expected the elementary condition
in Eq. I (or Eq. 13) to control. Following the present finite element calcula-
tions which proved that the bifurcation due to ~K2/~a 2 is possible, L. M. Keer I
et al. demonstrated by a complete analytical solution of the problem based
on singular integral equations that the bifurcation due to ~K2/~a 2 turning
zero is not merely a possibility but a phenomenon which does actually occur.
Simultaneously, refined finite element calculations were being performed by
the authors together with K. Aoh of University of Tokyo (to be reported
separately) and these calculations led to the same conclusion. These calcula-
tions and the work of Keer et al} also indicated that (for granite and for a
temperature drop of 100°C) the bifurcation is reached when a I and a 2 are
roughly equal 1.8 times crack spacing.
Since K I was shown to decrease after bifurcation, the equilibrium path
must have a straight horizontal segment of finite length after the bifurcation
point; see (Fig. li). Assuming that the trend remains unchanged, the segment
would end by a state in which K I = O, K 2 = Kc, and subsequently crack a I would
begin to close, 6a I < 0. The fact that the bifurcation for 5K2/~a 2 = 0 occurs
at constant a I means that in the plot of D versus a 2 the equilibrium path
must have a horizontal tangent at the bifurcation point; see (Fig. ~). If the
path continued as a straight horizontal line beyond the bifurcation point,
there would be infinitely many equilibrium crack lengths a 2 corresponding to
the same D and same al, and this would require the potential energy release
rate to be independent of D. Obviously, this is impossible. Hence, the path
of D versus a 2 after the birfurcation point (Fig. lj)must curve either upward
or downward. If it curved downward, it would mean that a longer crack a 2
corresponded to a smaller cooling penetration depth D (at constant al) , i.e.,
IManuscript "Growth and Stability of Thermally Induced Cracks in Brittle Solids", communicated to the authors by L. M. Keer, S. Nemat-Nasser and K. Parlhar of Northwestern University in September 1976.
362 ZDENEK P. BAZANT and HIDEOMI OHTSUBO
(or Eq. 1) is the case which controls. Without a complete solution of the
crack problem, the higher-order bifurcation with common tangent (Eq. 17)
cannot be ruled out in a general case. In fact, on the basis of some crude
finite element calculations of the equilibrium path it was initially thought
that in the present problem this latter type of bifurcation (Eq. 17) occurred
before the bifurcation given by Eq. 13 and that it, therefore, controlled. On
the other hand, S. Nemat-Nasser intuitively expected the elementary condition
in Eq. 1 (or Eq. 13) to control. Following the present finite element calcula
tions which proved that the bifurcation due to bK2/ba2 is possible, L. M. Keer 1 et al. demonstrated by a complete analytical solution of the problem based
on singular integral equations that the bifurcation due to bK2/ba2 turning
zero is not merely a possibility but a phenomenon which does actually occur.
Simultaneously, refined finite element calculations were being performed by
the authors together with K. Aoh of University of Tokyo (to be reported
separately) and these calculations led to the same conclusion. These calcula
tions and the work of Keer et al~ also indicated that (for granite and for a o
temperature drop of 100 C) the bifurcation is reached when a l and a2 are
roughly equal 1.8 times crack spacing.
Since Kl was shown to decrease after bifurcation, the equilibrium path
must have a straight horizontal segment of finite length after the bifurcation
pOint; see (Fig. Ii). Assuming that the trend remains unchanged, the segment
would end by a state in which Kl = 0, K2 = Kc' and subsequently crack al would
begin to close, 6al
< O. The fact that the bifurcation for bK2/ba2 = 0 occurs
at constant al
means that in the plot of D versus a2 the equilibrium path
must have a horizontal tangent at the bifurcation point; see (Fig. ~). If the
path continued as a straight horizontal line beyond the bifurcation point,
there would be infinitely many equilibrium crack lengths a2 corresponding to
the same D and same a , and this would require the potential energy release 1
rate to be independent of D. Obviously, this is impossible. Hence, the path
of D versus a2
after the birfurcation point (Fig. Ij) must curve either upward
or downward. If it curved downward, it would mean that a longer crack a2 corresponded to a smaller cooling penetration depth D (at constant a l ), i.e.,
lManuscript "Growth and Stability of Thermally Induced Cracks in Brittle Solids", communicated to the authors by L. M. Keer, S. Nemat-Nasser and K. Parihar of Northwestern University in September 1976.
STABILITY OF CRACK PROPAGATION 363
equilibrium extension of cracks a 2 would require withdrawal rather than supply
of energy. Therefore, if adjacent equilibrium states exist after the bifurca-
tion point, their path in Fig. lj must curve upward, i.e., with increasing
penetration depth of cooling the leading cracks must get longer, not shorter,
as may naturally be expected. It must be emphasized, however, that the shape
of the post-bifurcation paths in Figs. li and j has been deduced here only
qualitatively. Prior to formulating this qualitative deduction, the post-
bifurcation paths of the type shown in Figs. li and j were obtained quantita-
tively by Keer et al~ by means of a singular integral equation approach.
The possibility that every other crack (al) might close is suggested by
empirical observations of drying shrinkage cracks, e.g., in mud flats or in
concrete. This was also suggested by the behavior of cracks in an experiment
at Los Alamos Scientific Laboratory [9] in which a concrete slab was cooled
by liquid nitrogen and hexagonal crack patterns at the surface were made
easily observable by formation of nitrogen bubbles on evaporation from open
cracks. The possibility of crack closing was also evidenced at the outset of
the finite element work by the fact that for a sufficiently large value of
a2/2h the normal stress ~y along the line of symmetry between two adjacent
cracks (a I and a2) became compressive up to a certain depth from the surface.
This showed that on this line of symmetry it is possible to introduce a
closed crack up to a certain depth without causing any change of the stress
state in the entire elastic half-space. It follows that shorter closed cracks
may exist between opened leading cracks and this suggests that every other
crack (cracks al) may close after bifurcation. However, it does not follow
theoretically that every other crack must close. Keer et al.2 demonstrated by
analytical solution of the problem that cracks a I must indeed close after
bifurcation) So, it is certain that the spacing of the opened (leading) cracks
doubles whenever the ratio of the opened cracks to their depth reaches a certain
fixed 9alue (about 1.8). This type of behavior, which has been suggested
before on the basis of empirical observations [3, 4], is favorable for the
afore-mentioned scheme for extracting geothermal heat, because it would mean
that the width of the opened cracks is proportional to the penetration depth
21bid.
3'1%e fact of closing is distinguished from the fact that K I must decrease after 5ifu=cation, which is here established by Eq. (18).
STABILITY OF CRACK PROPAGATION ~3
equilibrium extension of cracks a2 would require withdrawal rather than supply
of energy. Therefore, if adjacent equilibrium states exist after the bifurca
tion point, their path in Fig. lj must curve upward, i.e., with increasing
penetration depth of cooling the leading cracks must get longer, not shorter,
as may naturally be expected. It mugt be emphasized, however, that the shape
of the post-bifurcation paths in Figs. Ii and j has been deduced here only
qualitatively. Prior to formulating this qualitative deduction, the post
bifurcation paths of the type shown in Figs. Ii and j were obtained quantita-2
tively by Keer et al. by means of a singular integral equation approach.
The possibility that every other crack (al
) might close is suggested by
empirical observations of drying shrinkage cracks, e.g., in mud flats or in
concrete. This was also suggested by the behavior of cracks in an experiment
at Los Alamos Scientific Laboratory [9J in which a concrete slab was cooled
by liquid nitrogen and hexagonal crack patterns at the surface were made
easily observable by formation of nitrogen bubbles on evaporation from open
cracks. The possibility of crack closing was also evidenced at the outset of
the finite element work by the fact that for a sufficiently large value of
a2/2h the normal stress cry along the line of symmetry between two adjacent
cracks (al and a2) became compressive up to a certain depth from the surface.
This showed that on this line of symmetry it is possible to introduce a
closed crack up to a certain depth without causing any change of the stress
state in the entire elastic half-space. It follows that shorter closed cracks
may exist between opened leading cracks and this suggests that every other
crack (cracks a l ) may close after bifurcation.
theoretically that every other crack must close.
However, it does not follow
Keer et al.2 demonstrated by
analytical solution of the problem that cracks al
must indeed close after
bifurcation? So, it is certain that the spacing of the opened (leading) cracks
doubles whenever the ratio of the opened cracks to their depth reaches a certain
fixed value (about 1.8). This type of behavior, which has been suggested
before on the basis of empirical observations [3, 4J, is favorable for the
afore-mentioned scheme for extracting geothermal heat, because it would mean
that the width of the opened cracks is proportional to the penetration depth
2Ibid . 3 'fhe fact of clOSing is distinguished from the fact that Kl must decrease after bifu~cation, which is here established by Eq. (18).
364 ZDENEK P. BAZANT and HIDEOMI OHTSUBO
of cooling and that the flux of water through the cracks is proportional to
the square of the crack depth. These crude projections may, however, be greatly
modified when the effect of water circulation on the temperature profile and
possible development of eddy currents in the cracks is taken into account.
Conclusions
I. A system of cracks is stable if and only if the work AW needed to produce
any admissible crack length increments is positive definite. This is
assured if the second variation 62W of W is positive definite for all
admissible crack length increments.
2. A system of identical parallel equidistant cooling cracks propagating
into a halfspace can exhibit instability. The critical state is indi-
cated by the vanishing of the derivative of the stress intensity factor
of cracks a 2 with regard to a 2 at constant al, which is the same as the
well-known stability condition for a single crack considered separately.
At the critical state every other crack, of length al, ceases to grow
(6a I = 0) while the intermediate cracks of length a 2 = a I continue to
advance (6a 2 > 0) at constant temperature. The path of the equilibrium
states plotted in the space (al, a 2) or in the space (ai, D) then
bifurcates (D = penetration depth of cooling). After bifurcation, cracks
a I gradually close. The plot of a I versus D has a horizontal tangent at
bifurcation point.
3. Without numerical results, it cannot be ruled out that a higher-order
bifurcation, in which the bifurcating path and the main path a I = a 2 have
a common tangent, might be also possible for a system of parallel cracks.
4. Vanishing of the determinant of the second derivatives of work W with
respect to a I and a 2 does not cause bifurcation in a system of parallel
cracks because associated eigenvector (~al, 6a2) indicates negative 6a I.
Remark. - Equilibrium path bifurcation is characteristic of a perfect
crack system. An imperfect crack system, e.g., a system of cracks which are
almost but not exactly equidistant, would probably not exhibit bifurcation of
equilibrium path, just like an imperfect column does not. However, such a
case would be much more difficult to solve.
ZDENEK P. BAZANT and HlDEOMI OHTSUBO
of cooling and that the flux of water through the cracks is proportional to
the square of the crack depth. These crude projections may, however, be greatly
modified when the effect of water circulation on the temperature profile and
possible development of eddy currents in the cracks is taken into account.
Conclusions
1. A system of cracks is stable if and only if the work ~ needed to produce
any admissible crack length increments is positive definite. This is
assured if the second variation 52W of W is positive definite for all
admissible crack length increments.
2. A system of identical parallel equidistant cooling cracks propagating
into a halfspace can exhibit instability. The critical state is indi
cated by the vanishing of the derivative of the stress intensity factor
of cracks a2 with regard to a2 at constant aI' which is the same as the
well-known stability condition for a single crack considered separately.
At the critical state every other crack, of length aI' ceases to grow
(5al
= 0) while the intermediate cracks of length a2 = a l continue to
advance (5a2 > 0) at constant temperature. The path of the equilibrium
states plotted in the space (aI' a2) or in the space (ai' D) then
bifurcates (D = penetration depth of cooling). After bifurcation, cracks
al
gradually close. The plot of a l versus D has a horizontal tangent at
bifurcation point.
3. Without numerical results, it cannot be ruled out that a higher-order
bifurcation, in which the bifurcating path and the main path a l = a2 have
a common tangent, might be also possible for a system of parallel cracks.
4. Vanishing of the determinant of the second derivatives of work W with
respect to al
and a2
does not cause bifurcation in a system of parallel
cracks because associated eigenvector (6al
, 5a2 ) indicates negative 5al •
Remark. - Equilibrium path bifurcation is characteristic of a perfect
crack system. An imperfect crack system, e.g., a system of cracks which are
almost but not exactly equidistant, would probably not exhibit bifurcation of
equilibrium path, just like an imperfect column does not. However, such a
case would be much more difficult to solve.
STABILITY OF CRACK PROPAGATION 365
Acknowledgment
Support of the U. S. National Science Foundation under Grants AER 75-00187 and ENG 75-14848 to Northwestern University is gratefully acknowledged. The authors are grateful to Drs. S. Nemat-Nasser, L. M. Keer and K. S. Parihar for their many helpful con~ents and suggestions during the long period of gestation of this paper. The first author first communicated to them the present general variational analysis of stability early in 1976, and benefited substantially from the subsequent discussion with them. This led to the initial draft of this paper in July 1976, which contained the present conditions of stability and numerical results. Discussion of this draft with these colleagues greatly assisted in the preparation of the final manuscript.
References
I. M. C. Smith, R. L. Aamodt, R. M. Potter, D. W. Brown, Proc., 2nd United Nations Geothermal Energy Symposium, San Francisco, California (1975).
2. F. Harlow and W. Pracht, J. of Geophys. Res., 77, 7038 (1972). 3. C. R. B. Lister, Geophysics J. R. Astr. Sot., 39, 465-509 (1974). 4. A. H. Lachenbruch, J. of Geophys. Res., 66, 4273 (1961). 5. B. Cotterell, Intern. J. of Fracture, ~, 526-533 (1966). 6. J. F. Knott, Fundamentals of Fracture Mechanics, J. Wiley, New York (1973). 7. G. C. Sih (ed.), Methods of Analysis and Solutions of Crack Problems,
Noordhoff International Publ., Leyden, The Netherlands (1973). 8. J. G. Croll and A. C. Walker, Elements of Structural Stability, John Wiley
and Sons, New York (1972). 9. H. Murphy, R. M. Potter, Private Communication to Z. P. Ba~ant, Los
Alamos Scientific Laboratory, Dec. 18, 1975.
Appendix
The fact that in a system of parallel cracks the bifurcation associated
with the determinant condition in Eq. I0 is impossible is contingent upon
(a) bK2/ba I being negative, and (b) both cracks being at the point of extension.
Prior to completing the finite element calculations which confirmed that
bK2/ba I is always negative, S. Nemat-Nasser intuitively suggested to the authors
that it should always be so. Later it was thought that "it is generally true
that an extension of a given crack accompanied by no increase in applied loads
would result in a decrease of the stress intensity factor at other active
cracks, because such an extension decreases the overall stiffness of the
elastic body. ''4 Subsequently, however, an example of a cracked structure
for which bK2/~a I is positive has been found; hence the sign of ~K2/~a I is
not certain in advance, for the general case.
To show it, consider a horizontal simply supported continuous beam of
constant cross section (of depth H) and two equal spans (of length L), loaded
~L.M. Keer et al., loc. tit.
STABILITY OF CRACK PROPAGATION 365
Acknowledgment
Support of the U. S. National Science Foundation under Grants AER 75-00187 and ENG 75-14848 to Northwestern University is gratefully acknowledged. The authors are grateful to Drs. S. Nemat-Nasser, L. M. Keer and K. S. Parihar for their many helpful comments and suggestions during the long period of gestation of this paper. The first author first communicated to them the present general variational analysis of stability early in 1976, and benefited substantially from the subsequent discussion with them. This led to the initial draft of this paper in July 1976, which contained the present conditions of stability and numerical results. Discussion of this draft with these colleagues greatly assisted in the preparation of the final manuscript.
References
1. M. C. Smith, R. L. Aamodt, R. M. Potter, D. W. Brown, Proc., 2nd United Nations Geothermal Energy Symposium, San Francisco, California (1975).
2. F. Harlow and W. Pracht, J. of Geophys. Res., 77, 7038 (1972). 3. c. R. B. Lister, Geophysics J. R. Astr. Soc., 39, 465-509 (1974). 4. A. H. Lachenbruch, J. of Geophys. Res., 66, 4273 (1961). 5. B. Cotterell, Intern. J. of Fracture, 2, 526-533 (1966). 6. J. F. Knott, Fundamentals of Fracture Mechanics, J. Wiley, New York (1973). 7. G. C. Sih (ed.), Methods of Analysis and Solutions of Crack Problems,
Noordhoff International Publ., Leyden, The Netherlands (1973). 8. J. G. Croll and A. C. Walker, Elements of Structural Stability, John Wiley
and Sons, New York (1972). 9. H. Murphy, R. M. Potter, Private Communication to Z. P. Bazant, Los
Alamos Scientific Laboratory, Dec. 18, 1975.
Appendix
The fact that in a system of parallel cracks the bifurcation associated
with the determinant condition in Eq. 10 is impossible is contingent upon
(a) ~2/ba1 being negative, and (b) both cracks being at the point of extension.
Prior to completing the finite element calculations which confirmed that
~2/oal is always negative, S. Nemat-Nasser intuitively suggested to the authors
that it should always be so. Later it was thought that "it is generally true
that an extension of a given crack accompanied by no increase in applied loads
would result in a decrease of the stress intensity factor at other active
cracks, because such an extension decreases the overall stiffness of the 4 elastic body." Subsequently, however, an example of a cracked structure
for which bK2/Oal is positive has been found; hence the sign of OK2/Oal is
not certain in advance, for the general case.
To show it, consider a horizontal simply supported continuous beam of
constant cross section (of depth H) and two equal spans (of length L), loaded
4 L.M. Keer et al., loco cit.
366 ZDENEK P. BAZANT and HIDEOMI OHTSUBO
in the middle of the left span by downward load PI and in the middle of the
right span by equal but upward load P2 = -PI" The bending moments in the
middles of the left and right spans are M I = PIL/4 and M 2 = -M I. Assume
further that there are two vertical cracks , one reaching upward from the
bottom of the cross section in the middle of the left span to depth a I from
the bottom, and the second reaching downward from the top of the cross
section in the middle of the right span to depth a 2 = a I from top. Assume also
that K I = K 2 = K c. Let now the crack length a I be increased by 6a I while
keeping a 2 and the loads constant. Increase of a I will cause the left span to
become less stiff, and it will cause the left span to deflect downward. In a
continuous beam, this must cause the right span to deflect upward, and because
a 2 is constant, K 2 must increase, i.e. ~K2/~a I > O. Alternatively, this may
be also deduced by noting that the decrease of left span stiffness must cause
the bending moments to redistribute so that M I would decrease andl~lwould
increase; an increase of IM21at constant a 2 must cause K 2 to increase.
Likewise, condition (b), namely that both cracks are on the verge of
extension, does not have to always occur. Consider the same beam but with
different loads PI and P2 and with both cracks of lengths ~ = a 2 emanating
from the bottom of the cross section. Assume now that the beam is slender, so
that K I and K 2 are proportional to M I and M2, and that a I << H, a 2 << H, so
that crack lengths have negligible effect on the stiffness of the spans.
First, let loads PI = P2 = 1.0 be applied. This causes equal moments, M I =
~ = L/4, and assume that this creates equally long cracks a I = a 2 which are
both critical, K I = K 2 = K c. Subsequently, load P2 is changed to P2 = 1.3
and load PI is changed to PI = 0.3. This causes M to become zero while M 2 I
remains unchanged (M 2 = L/4). So, P2 = 1.3 and PI = 0.3 gives a state where
= and K I = O, crack a I being on the verge of extension and crack a 2 K 2 K c
being on the verge of closing. For checking stability on this state, one must
obviously consider 6a 2 ~ 0 and 8a I ~ 0 as the admissible 8a..i
In cases where condition (a) or (b) is reversed, the stability condition
det (Wij) > 0 cannot be dismissed a priori and must be evaluated to see whether
or not it is satisfied for all admissible 6a i.
366 ZDENEK P. BAZANT and HIDEOMI OHTSUBO
in the middle of the left span by downward load PI and in the middle of the
right span by equal but upward load P2 = -Pl. The bending moments in the
middles of the left and right spans are Ml = Pl L/4 and M2 = -MI. Assume
further that there are two vertical cracks, one reaching upward from the
bottom of the cross section in the middle of the left span to depth al
from
the bottom, and the second read:ing downward from the top of the cross
section in the middle of the right span to depth a2 = at from top. Assume also
that Kl = K2 = Kc· Let now the crack length a l be increased by &al while
keeping a2 and the loads constant. Increase of a l will cause the left span to
become less stiff, and it will cause the left span to deflect downward. In a
continuous beam, this must cause the right span to deflect upward, and because
a2 is constant, K2 must increase, i.e. bK2/0al > O. Alternatively, this may
be also deduced by noting that the decrease of left span stiffness must cause
the bending moments to redistribute so that Ml would decrease andlMzlwould
increase; an increase of IM21 at constant a2 must cause K2 to increase.
Likewise, condition (b), namely that both cracks are on the verge of
extension, does not have to always occur. Consider the same beam but with
different loads PI and P2 and with both cracks of lengths ~ = a2 emanating
from the bottom of the cross section. Assume now that the beam is slender, so
that Kl and K2 are proportional to Ml and M2 , and that a l «H, a2 «H, so
that crack lengths have negligible effect on the stiffness of the spans.
First, let loads PI = P2 = 1.0 be applied. This causes equal moments, Ml
Mz = L/4, and assume that this creates equally long cracks al = a2 which are
both critical, Kl = K2 = Kc. Subsequently, load P2 is changed to P2 = 1.3
and load PI is changed to PI 0.3. This causes Ml to become zero while M2
remains unchanged (M2 = L/4). So, P2 = 1.3 and PI = 0.3 gives a state where
K2 = Kc and Kl = 0, crack al being on the verge of extension and crack a2 being on the verge of closing. For checking stability on this state, one must
obViously consider &a2
~ 0 and &al ~ 0 as the admissible bai "
In cases where condition (a) or (b) is reversed, the stability condition
det (Wij
) > 0 cannot be dismissed a priori and must be evaluated to see whether
or not it is satisfied for all admissible &ai ·