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J. Mech. Phys. Solids,1967, Vol. IS, pp. 151 to 162. PcrgamonPress Ltd. printedn Great ritain.

A COMPARISON OF THE FRACTURE CRITERIA

OF GRIFFITH AND BARENBLATT

By J . R. WILLIS

Department of Applied Mathematics and Theoretical Physics, University of Cambridge

(Received 23rd January 1967)

SUMMARY

THE TENDENCY of a crack to extend under applied loads is governed by the cohesive forces acting

near the crack tip. The crack-extension criteria of Griffith and Barenblatt take account of the

cohesive forces in rather different ways, but are both of the same form. Therefore, if they are to

agree, the surface energy 2’ appearing in the Griffith criterion must be related in a deilnite way to

Barenblatt’s modulus of cohesion K. It is shown in this paper, from a detailed consideration of

the cohesive forces, that the required relationship holds, in an asymptotic sense, if the forces act

only over a short range; this is true in practice. Extension of the analysis to uniformly moving

cracks then shows that K is a function of velocity, even for a perfectly elastic-brittle body. This

has not been noted previously. Finally, the relative advantages of the two formulations are

compared.

1. INTR~DUOTJ~N

WHEN two adjacent lattice planes of a crystalline body are separated from their

equilibrium position, they attract one another according to some law of cohesion.

The cohesive forces increase up to a critical separation, beyond which they decrease.

I f the separation is large enough, the cohesive forces are negligible and the lattice

planes become the surfaces of a crack. I f it is very small, the body still possesses

cohesion and the separation is best regarded as a strain. The exact separation at

which a crack is defined is arbitrary and the most appropriate definition depends

upon the model of the body that is employed. I f a lattice model is used, thedefinition is immaterial but, for more tractable continuum models, it is clearly

important.

In this paper we consider the simplest possible model : a linearly elastic-brittle

continuum. For this model the cohesive forces are assumed to give the usual

linear stress-strain relations up to fracture, after which the crack faces attract one

another, the attraction being a function of their separation. This function is

expected to be clearly defined for new crack surfaces but will be reduced if they are

contaminated.

Considering for illustration just a normal relative displacement y, there is only

a normal attractive stress pnn and

Pnn <f(Yh (1-l)

with equality when the crack surfaces are clean and in particular when they are

new. I f the separation y is measured from equilibrium, f(0) = 0. However, it is

151



152 J. . WILLIS

more convenient to measure y from the separation at which a crack is defined to

start, any lower separation being regarded as a strain rather than a displacement.

In this case, f(0) # 0 and the zero level has to be defined appropriately. This

difficulty arises from approximating a lattice by a continuum, for which the relative

displacement at the crack tip has to be zero. Since the strain field is defined by an

averaging process, it can be seen that there is no real inconsistency.

There have been two approaches to the fracture of an elastic-brittle continuum.

The first, developed by GRIFFITH(1921), takes account of the cohesive forces by

assigning to all crack surfaces a surface energy T per unit area, which is the work

done against the cohesive forces in creating new surfaces. This energy then appears

in an energy balance equation. As outlined in $2 the resulting fracture criterion

can be expressed in a local form, which was first recognized by IRWIN (1957).

The other approach is that of BARENBL ATT 1959a, b), which accounts for the

cohesive forces more explicitly. This too is reviewed in $2. I t leads to a localfracture criterion exactly similar to that of Griffith. Therefore, if they are to yield

the same results, there must be a definite relationship between the physical constants

appearing in either formulation. This relationship was given by BARENBL ATT1962)

but it was derived under the assumption that the two criteria agree. We justify

this assumption theoretically in $3 by proving, from a detailed examination of the

cohesive forces, that the relationship cited is true in an asymptotic sense.

It is next observed that the analysis applies also, with only trivial modifications,

to a uniformly moving crack. From this it emerges that Barenblatt’s ‘ modulus

of cohesion ’ depends upon the speed of the crack and is therefore not so fundamental

a quantity as the surface energy which appears in the Griffith formulation.Finally, the relative advantages of either formulation are discussed.

2. REVIEW OF THE FRACTURE CRITERIA OF GRIFFITH AND BARENBLATT

This section contains a brief summary of the fracture criteria under examination.

Considering a crack C in a linearly elastic body, let the stress, strain and dis-

placement fields be prf, efj, uf respectively. Now SUpQose that the crack extends

to C + SC, with the boundary conditions unchanged. Let the new fields be

per + S~t3, etj -t- SeU, us + SW. Then the energy SU released in the extensionSC equals the work required to close SC. Thus.

where no is a normal to SC and [ Sut] is the relative displacement of the crack faces.

I f the extension SC is possible, the energy released must at least equal the increase

in surface energy 2TSC. Therefore,

SU > 2TSC. (2.2)

I f the inequality (2.2) is not satisfied for any SC, the crack cannot extend. I f it is

satisfied for some SC, the crack can, and therefore probably will, extend. This is

really an additional assumption which has been discussed by CRAGGS (1963).

The expression (2.2) is essentially Griffith’s fracture criterion. I t was originally



A comparison of the fracture criteria of Griffith and Barcnblstt 153

expressed as a global criterion and its local character, demonstrated by (2.2),

was first, recognised by I RWIN (1957). Also, in its local form, it is independent

of the type of boundary conditions that are applied. Thus, its use eliminates the

need for the careful analysis of SPENCER1965), who showed up the possible pitfalls

in applying Griffith’s criterion in its original form.

For simplicity we consider henceforth an infinite body, in plane strain, with a

plane crack occupying the position y = 0, x > 0. I t can be shown that, near the

crack tip,

pyy(-O)=N/++O(l) (2.3)

BS s -+ 0. Also, if the loading is symmetrical, pzy (x, 0) = 0 and

v Sl, f 0) = f 2 ; v, +I + 0 (Sl)f, (2.4)

where v is the y-component of displacement and v is Poissons’ ratio. Therefore

if the crack extends to LE= - h, where h is small,

6v (- s, * 0) = & 2(l - v,2/(h - s) + 0 (h3’2).P

(2.5)

Hence h h

au = 4

sp,, 2BvdS = 2 (1 - v) N2 + ds = l--v p ,&. (2.6)

CL P0 0

Griffith’s criterion (2.3) then yields

r(l -4N2>2TI *

P(2.7)

Another, independent, criterion of the form of (2.7) has been derived by

BARENBLATT1959a, b). He considers the cohesive forces in more detail and divides

the crack into two regions, an edge region of width d, over which the cohesive

forces are strong, and the remainder, over which they are negligible. He then

makes the following two hypotheses

(9

(ii)

The width d of the edge region is small compared to the size of the whole

crack.When the crack is about to propagate, the form of the normal section of

the crack surface in the edge region (and consequently the distribution of

the cohesive forces) does not depend upon the acting loads and is always

the same under given conditions of temperature, etc.

These hypotheses are plausible if the cohesive forces are large and also decay

rapidly to zero as the separation increases. In practice both of these conditions

are satisfied.

Since the stress at the tip of the crack is evidently finite, the cohesive forces

must adjust themselves so that they reduce to zero the stress concentration factor

N which the applied loads alone would produce. I f N is too great, the cohesive

forces cannot. cancel it and the crack will extend. In conjunction with hypotheses

(i) and (ii), this leads to the criterion

N>K /n w 3)



I.54 .J . R. WILLXS

for crack extension, where the ’ modulus of cohesion ’ K is a material constant.

I t is defined in terms of the acting cohesive forces in $3, where more detailed

formulae are presented.

From (2.7) and (2.8), if the two criteria are to agree, the relationship

jy2= fLTT

l-vWQ

must hold. This is examined theoretically in the following section.

3. PKOOF OF THE EQUIVALENCEOF THE Two CRITERIA

Consider an infinite body, under conditions of plane strain, with a crack occupy-

ing the position y = 0, X > 0. Let it be subjected to the loads

pzy = 0 , y = 0 , ; F 2 0 ,

p, , = g( d y = 0, ; 1: , 0, (3.1.)

pij -f 0, X2 + yz --f Co.1

The load g (x) is the sum of the applied loads, - p ( x ) , and the cohesive forces,

G (x). G (x) is not known a pr i or i , but obeys the law

G (X) <f[2~ (X)1, (3.2)

with equality when the crack is about to extend. The function f has a large

maximum value, fin, of the order of one-tenth of Young’s modulus (COTTRELL1964)and

f(2v) = 0 for 2v > 11, (3.3)

where h is probably one or two lattice spacings. Thus the width d of the edge

region of the crack is determined from the equation

2v (d) = h. (3.4)

It is shown by MCSKIIELISHVILI1953a) (or see the Appendix) that

(3.5)

(3.6)

It can be shown that, as X -+ - 0,a,

pyv (X, 0) = - _?. 1rl/-x s

gL;F -t g( 0) + O( $/ - c - 8 ) .

0

(3.7)

Therefore, if the stress at X = 0 is to be finite,

m

s!ib)dT- = -2/T sG(T)& o

0

oy/‘=’ (3.3)



A comparison of the fracture criteria of Griffith and Barcnblatt

In view of (sz), (3.8) may be written as

155

where

m

s

P (4 d7 < K

2/7’ ’

KoE mf[2v (41 d7 .

s 1/T0

(3.9)

(3.10)

Barenblatt’s second hypothesis ensures that the modulus of cohesion K is a

material constant.

Let us define m h

T = 3 fkd dy = 4 f(y) dy.s s

(3.11)

0 0

Clearly T’ is a material constant, being independent of any particular deformation.

However, we may also writed

T’ =s

j-[2v (z)] ; dx, (3.12)

for any monotone increasing v (x) with v (0) 0, being obtained from (3.4).

Now let ZI x) be the solution of (3.6), when the crack is about to propagate, i.e.

for G (7) =f[2V (T)]. A um q ue solution, monotone increasing in (0, d), is assumed

to exist on physical grounds. Therefore,

dv 1-v 1__= __-

IS

m?’ (7) 2/(T) dT _ df[2v (T)] d(T)

dx I.cr 4x 7-Z s

d7

7-X>

0 0m

P (T)dT df[2v )-jdT

s >

(3.13)

(1/+-x)- (dT)(T-z) ’0 0

since (3.8) is satisfied. Now suppose that p (T) < P for all 7 and that*

00

sp(T) dT = 0 (PL).

0

Then by hypothesis, P/p < 1 and L is a characteristic length, large compared

with lattice dimensions. Restricting attention now to values of ;1:= 0 (d), it is

evident that

while

cc

(3.14)

0

(3.15)

0

*Here, and in what follows, the statement a = 0 (b) means that a and b are o t he fame order of magnitude,

which is not quite ts usual mathematical meaning.



156 J . R. ILLIS

The estimate (3.14) would be easy to establish rigorously. The rigorous proof of

(3.15) ould depend upon an analysis of the non-linear equation (3.13) and does

not appear to be simple. However, (3.15) is certainly plausible and we wil l assume

that it holds, appealing to ’ physical intuition.’ Now invoking Barenblatt’s first

hypothesis, that d/L < 1, we have, for x = 0 (d), the approximate relation

dv_=-dx

(3.16)

when the crack is about to propagate. Thus, ZJ x) is determined, to within small

quantities, by (3.16) and the ‘ edge region ’ is indeed autonomous, the applied

loads being only required to give equality in (3.9). The consistency of Barenblatt’s

first hypothesis is readily established by deducing from (3.16) the approximate

inequality

v (x, + 0) < 20 - v)&Ax,7r CL

(3.17)

where2h - log{@ + l)/(h - 1)) = 0, (3.18)

so that

h = 2v d, + 0) < g! - ‘)h d = 0 (d)._- (3.19)7T P

Thus, if d is small, then h is small. It would be valuable to prove the converse

result, thereby proving Barenblatt’s first hypothesis, but this is a less simple task,

which again would depend upon an analysis of (3.13) or (3.16).Inserting (3.16) into (3.12) yields

T' = - -//[2u (z)] +z) drc: : $;j :‘p;) (1 + 0 [c /(;)I }. (3.20)=CL

0 0

Therefore,

T’--l--v

mP+ 0 ;i(;# (3.21)

0 0

by interchanging x and 7. Hence, by addition,

= zKs{ + 0 [f&j]} (3.22)

Since K = 0 [fm (441 = 0 [P (2/d)] and also

K =

mp(7)drs ___[P (@)I,

we may also write 0

T’ = zK2{1 + o(;)}. (3.23)



A comparison of the fracture criteria of Griffith and Barenblatt 157

Thus, if we identify T ’ with T , the equivalence of the two criteria is proven.

However, 2T is the work done in separating two lattice planes from their unstrained

position, while 2T’ is the work done in separating them from their strained position,

just prior to the formation of a crack between them. The definition of the start

of a crack is arbitrary to some extent, but clearly some strain prior to cracking

must be admitted, since otherwise the ‘ edge region ’ would extend to minus

infinity. Logically, the least objectionable definition is that the crack starts when

the.tensile stress equals f m, he theoretical tensile strength of the body. I t is then

fairly obvious that the tensile stress away from the crack tip is less than this value

and it is consistent to assume linear elasticity to hold up to this level of strain.

Beyond this level, a crack forms and the faces attract one another according to the

cohesive law (l.l), where f y ) ecreases as y increases. Defining the crack to start

at any lower stress level presents the inconsistency of assuming that linear elasticity

applies for strains in the body larger than those which are given special non-lineartreatment across the crack surfaces.

Therefore, there is some rather ill-defined discrepancy between the two theories,

which is small if the total area under f (y) approximately equals 2T’. In practice,

the difference is probably about 30 per cent (see e.g. TYSON 1966), if the crack is

defined in the ‘ logical ’ way, so that the two theories can only be made to agree if

logic is sacrificed to some extent. Alternatively, they could probably be made to

agree, in a logically consistent fashion, by admitting non-linear elastic behaviour of

the material near the crack tip. However, the details of such a calculation would

depend upon the precise form off(y) and it would therefore lack the generality

of the simple linear theory outlined here.

4. EXTENSION TO UNIFORMLY MOVING CRACKS

The foregoing analysis applies also to a crack moving uniformly with speed V

and driven by loads which move with it as, for instance, in wedging. Assuming that

the loads are known, the relevant solution for purely normal loads is given in the

Appendix. From (AGO), it may be seen that p,, (X , 0) is independent of the speed

V . Therefore, Barenblatt’s fracture criterion gives no relationship between the

loads and V unless the modulus of cohesion K depends upon V . It is fairly obvious

that K does depend upon V since, from (A21), v (X, 0), and hence the distribution

of the cohesive forces, depend upon V , even though this distribution may be

independent of the magnitude of the applied forces. This point appears to have

been overlooked by BARENBL ATTand CHEREPANOV1960) in their analysis of a

wedging problem. They took K to be a constant and their results should therefore

be slightly modified. The required function of V may be obtained by replacing

(1 - V)by Vs (1 - Vs/crs)f/css D (V) , in accordance with (A23), in the formulae

of $3. This yields

(4.1)

GRAGGS 1960)onsidered the problem of a crack being driven by a step-function

normal load and found a load-velocity relation by applying a Griffith-type energy

balance. The Barenblatt criterion, with K given by (4.1), yields the same relation,

if T’ and T are identified, as discussed in 93.



5. COIWLUDIP~G EMAE~S

It has been shown, from a consideration of the cohesive force law, that the

Barenblatt and Grifrith criteria are the same, provided that 1” is equated to 1’.

This requires the Barenblatt theory to sacrifice its logical consistency (in the linear

approximation). I t is also possible that T in the Grifbth theory should be defined

from some non-zero level of strain. I t seems unlikely, however, that it should be

modified by any large amount for most materials, since the singularity in the energy

density given by the linear solution is integrable and the strains are large only in a

small region. Verification of this would require a full non-linear analysis for a

particular force law. Such an analysis would be interesting from many points of

view, but would involve either a numerical solution, or, perhaps, a cumbersome

perturbation procedure; no simple analytic course is available.

The extension of the (linear) ana.lysis to uniform motion has revealed that

Barenblatt’s modulus of cohesion K depends upon velocity and the dependence has

been found. This result appears to be new. I t also raises the question of whether K

would depend upon acceleration in a more general situation. This possibil ity cannot

be ruled out and the safest course in tackling any new problem would be to employ

the Griffi th energy balance, even though the calculations may be a little more

unwieldy. I t can, in any case, always be expressed as a local criterion, if required in

this form (RICE 1965).

In conclusion, it appears that the Barenbtatt approach has no great advantage

over that of Griffith, if used just to predict crack growth. I ts advantage lies rather

in offering a more realistic picture of the stresses near the tip of the crack, therebyimproving knowledge of the basic mechanism of fracture. Unfortunately, there is

one last di&ulty to be raised in this context. Analysis of (A25) reveals that

pxx (X, 0) > p,, (X, 0) for all P > 0. Thus, since at some point of the X-axis,

p,, = fm, the body is overstressed in the X-direction and the stress pattern pre-

dicted by the linear analysis of the Appendix cannot be adequate. The only

exception to this is that, when V = 0, p,, =z yy on the X-axis. Calculations

based on the cohesive force distribution assumed by CKIBB and TOMKINS (1967)

have shown that, at V = 0.5 2, he material is overstressed within a radius of

about 05 cl of the crack tip. Evidently the complete answer again lies in a non-

linear analysis of the stresses near the crack tip. In this case, since the body isstrained beyond the value for which p zz = fm , stress relaxation must occur and the

mathematical problem would be analogous to one of plastic deformation, though

with a different physical interpretation.

APPE~VDIX

This appendix gives the stress field due to a uniformly moving crack, in plane strain, whose

position at time t is 2/ = 0, z -1. F’t > 0. The crack is driven by purely normal loads g (ir -+ Vt),

so that the boundary conditions are



A comparison of the fracture criteria of Griffith and Barenblatt 159

The method of solution leans heavily on that of CRAOGS 1960), except that here we solve a Hilbert

problem while Craggs employed a conformal transformation. The derivation is included both for

completeness and because the solution is not available elsewhere in the general form required.

BARENBIATT and CHEREPANOV (1960) employed a very similar method in solving their problem

of steady wedging.Introducing displacement potentials 4, $ so that

34 NJ 34 24u=__+-, n=---,bY 3y bx

the equations of motion of the body may be replaced by

(-42)

a24- = cp V-J 4,

iJ2#

312- = c22 v2 *,a2

where

Cl2 = (X + 2/4/p, e22 = CL/P.

Now set X = z + V1 and suppose 4 = 4 (X, y), # = 4 (X, y).

Define

(1 - V2/,12) +xX + #I/v 0,

(1 - V2/c22) #xx + SYV = 0.

z1 = X + i (1 - V2/c12)* y, z2 = X + i (1 -

(A3)

(Ad)

Equations (A3) then become

>(A5)

v2/c22p y. W-7

From (A5), there exist analytic functions WI (zI), ~2 (22) uch that

WI (21) = 4 + ix, w2 (22) = $ + iic, (A?

where ,r, K are the conjugate harmonies to 4, I/J , regarded as functions of X, (1 - V~/CI~)* ,

and X, (1 - V2/c22)* y respectively.

Craggs now shows that

Define

@-lpzz = (2 + v2/c22 - 2V2/c12) Qxx - 2 (1 - V2/cI2)* KXX,

&h-lp,y = - 2 (1 - Vs/,s)* xxx - (2 - V2/c22) $xx,

p-lpyy = - (2 - Vs/czs) 4xX + 2 (1 - v2/c22)* KXX.

W)

d 2 wa

wa (za) = F , dl = 1,2.OT

Then IV, (zc) --f 0 as lzcl --f co. Therefore, by Cauchy’s theorem,

m

sOL(T) 7~ ; ci = 1, 2,4 (z) > 0,

7-z--m

where the symbol 4 denotes ‘ the imaginary part of.’ Also,

co

t-J=’ s+‘rx4 dr

2?ri-; a=1,2,4(z)>O,

7 - z*--cc

where z* denotes the complex conjugate of z, so that

m

0s’ sV,* (7) dr

27ri 7-Z; a=1,2,9(z)>O.

--m

W)

WO)

(All)

Hence, from (A8),



160 J. R. WILLIS

00

1

s

PZ, (7) d7 2 (1 - Ys/cls)6 IV1 (a) (2 - Y2/W) I t’d 4%)

!z-P=--

7-Z 2i 2-co

00

1 PYY(T)dT (2 - v2/elqWl fz) Z(1 - Y2/c# Wz(2)_ ~Z.....

2k s

-.___-_p+--

7-Z 2 2i--m

(A159

Setting p,, (7) = 0 and pyv (7) = f (7) in (A12) and solving, we obtain

where

D (V) _I 4 { - V/4$)6 (I - P/e&” - (2 - IQ/@)%

The equation D (V) = 0 is the secular equation for Rayleigh waves.

At this point we depart from Craggs’ method. I t is easily shown that

(AM)

au----=3X

- (1 - P/Cis)*4 (Wl) - .B (W2) f

where the symbol 9 denotes ‘ t.he real part of.’ Therefore, by employing the Ylcmelj formulae

(see e.g. MUSRHELISHVII~I lWSb), we obtain

- F’s(1 - Ye/c& 1&(X9 +o+-&-- mf

7) dr

r>(Y) Yip -’s

7 - Y(A=)

--m

where tbe Cauehy principal value of the integral is understood.

The boundary conditions (Al) are now equivalent to

i s(7) dT i- “S(T)T

7 - x- = 0, x < 0,7-X

--m

Deilne

WV

Thus F (z) +O as \zI i + to and

F (X -f- io) + F (X - i0) = - 1s?(T)dT

?ri-, x<o*T--X

0

(A18)

The I I ilbert problem (Ala), with F(m) =1 0 and F (z) = 0 (z-* ) as z + 0, has the solution

{MUSK~ELIS~I~ 1953b)

where 42 is analytic in the x-plane cut along 41 (z) = 0, &J (z) < 0. Evaluating the integral wit11

respect to X then yields



A comparison of the fracture criteria of Griffi th and Barenblatt

co al

F(z)=-& -sw+ 1 gT)(2/T)dT~7-z 27x-i+As .

7-Z

0 0

Hence, by the Plemelj formulae,

p,, (X, 0) = f(X) = F (X + i0) - F (X - i0)

m

1 1=---

n (4-X) s

g (T) (2/r) d+ ;

7-X

x > o.

0

Also, IV1 (z), W2 (z) are known ifm

I(z) = 2; -s7) d7

7-Z

- aiis known. From (A16) and (A17),

m m

sg7) dr 1~ + F (z) = -

g (7) (4~) dr

7-Z 2&(4Z) s7-Z

0 0

Therefore, from (A15) and (A22),

- v2 (1 - V/c,+;(x, +o)=--

1

Cl2 L'(V) wc&v s

g (T) (v’T) d7,

7-X

x > o

0

Letting V + 0 in (A23) yields the static result (MUSKHELISHVILI1953a)

co

sg7) (2/d d+.

7-X0

161

t-419)

b4W

(-421)

b422)

(A23)

(A24)

The formulae (A20) and (A24) are used in 5 4. Another simple result, used in 5 5 is obtained

from the first of (AS) and (A13). Expressing 4, K n terms of WI , WZ and letting y --f + 0, it follows,

using the Plemelj formulae, that

pm X, 0) =2 (2 - VJ /c22) (V/Q2 - V2/c22)

DO’)

- 1 Pm (X, 0).

>

I t may be verified that, asV --f 0, pzz/pzlu -f 1

and that, for 0 <V < VR,

1 <p~~/pys < ~0,

where VR is the Rayleigh velocity.

BARENBLATT, G. I. 1959a

1959b

1962

BARENBLATT, G. I.

and CHEREPANOV,G. P. 1960

COTTRELL, A. M. 1964

CRAGGS, .W. 1960

1963

J. Appl. Math. Mech. 23, 622.

I bid. 23, 1009.

Ado. Appl. Mech. 7, 55.

J. Appl. Math. Mech. 24, 667.

The Mechani cal Properties of Matter (Wiley, New York).

J. Meek Phys. Soli ds 8, 66.

In Fr acture of Solids (Edited by DRUCK ER, D. C. andGILMAN, J . J .) (Interscience, New York).

CRIBB, J . L .

and TOMK INS, B.

GRIFFI TH, A. A.

IRWIN, G. R.

1967 J. Mech. Phys. Solids 15, 35.

1921 Phi l. T rans. R. Sot. A221, 163.

1957 J. Appl. Mech. 24, 361.

REFERENCES



162 J . R. WILLIS

MUSKEELISHVILI, N. I. 1953a Some Basic Problems of the Mathematical Theory of El asticity

(Noordhoff, Holland).

1953b Singular Integral Equations (Noordhoff, Holland).

RICE, J . R. 1965 Brown University Report,

SPENCER, A. J . M. 1965 Int. J. Engng Sci. 3, 441.TYSON, W. R. 1966 Phil . M ag. 14, 925.

a comparison of the fracture criteria of griffith and barenblatt

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