Materials and S t ruc tures /Matbr iaux et Constructions, 1991, 24, 202-209

Dimensional analysis for concrete in fracture

N . A . H A R D E R

University of Aalborg, Sohngaardsholmsvej 57, DK-9000 Aalborg, Denmark

A structural element of high strength in compression and extremely low strength in tension, such as concrete, is considered. Based on linear-elastic fracture mechanics a simple model law for ultimate load was derived by the author in 1986. This model law is shown also to be consistent with the non-linear fracture mechanical theory called the fictitious crack model.

N O M E N C L A T U R E

E V

Cry

GF C t , C 2

w G~

7u B ao

a

2a

a c

lo O" o

frOm

O" e

O" u

/3

Modulus of elasticity Poisson's ratio Yield stress Fracture energy Material parameters Displacement discontinuity Potential elastic energy relief Load factor Load factor at ultimate load Brittleness modulus Critical brittleness modulus Scale ratio Brittleness index Crack length for boundary crack Crack length for internal crack Characteristic crack length Characteristic length Stress field if the crack were not there Maximum tensile stress in ao-field Critical value of aOm O'0m at ~ = y. for a = a e

Constant, /3= 1 plane stress, /3 = 1 - v 2 plane strain Constant for similar ao-fields

1. I N T R O D U C T I O N

When testing non-reinforced concrete beams for ultimate load it is well known that a size effect is observed. Small test specimens are comparatively stronger than larger ones. This could mean that the ultimate stress was in fact not a real material property. However, Hillerborg [1] showed that this was not necessarily so. The size effect could be explained using fracture mechanics.

Linear-elastic fracture mechanics (LEFM), however, was not regarded a suitable model, mainly because this theory deals with fracture caused by an initial crack, but the condition for a new crack to develop is not included. Consequently, Hillerborg [1] introduced a new fracture mechanical model, later called the fictitious crack model (FCM) [2]. Using this model it was possible to explain the size effect in a qualitative way. But it was not possible to derive general analytical results such as model laws. The FCM is mainly applicable to numerical analysis.

0025-5432/91 �9 RILEM

However, a model law for ultimate load on similar test specimens was presented by the author at a Nordic seminar on fracture mechanics for concrete in 1986 [3]. It was thought that the model law could be useful in planning laboratory tests for such material properties as the fracture energy (see the RILEM Recommendation [4]). The model law was derived using some basic relations from LEFM and therefore, at this time, it was not seriously believed that it was consistent with FCM. However, further investigations have shown that in fact the model law is compatible with both LEFM and FCM theories.

The model law for ultimate load on a beam or any other test specimen is briefly explained as follows. All load parameters are multiplied by a load factor 7, which can assume all values from zero to the maximum value 7u at ultimate load. The model law for y, is

TuP = g']uM g ~ 1 for

g~- I /~ / z for

g = 1/~ 1/2 for

= Ip / l~ > 1

7c = Bp/Br

~ c < l < ~

I <~r <c~

1 <~_<~r

(1)

Bp is the brittleness modulus of the prototype and Be is the critical brittleness modulus. The function g = g ( a ) is shown in Fig. 1. The subscript P signifies prototype (the larger beam) and subscript M signifies model (the smaller beam). The two beams are geometrically similar. ~ is the scale ratio.

, /%

g \ ._~ f % < 1 Prototype and model ductile

- ~ Prototype and model brittle

z--Prototype l_ . . . . . . . ~ /brittle,

r- ~ ~ model ductile

I I

2 3

Fig. 1 The model law for ultimate load.

a c 4

Materials and Structures 203

The two beams are of the same material, hence the elastic modulus E, Poisson's ratio v, the fracture energy GF and the yield stress ay are the same in the prototype as in the model, ar being the only governing parameter of the model law, is called the brittleness index. In addition to Equation 1 there is a model law for each of the load parameters. In the following it is shown that the model law can be applied to FCM as well as to LEFM.

2. T H E F I C T I T I O U S CRACK M O D E L

For the sake of simplicity the discussion is restricted to the cases of plane stress or plane strain which cover so many situations from engineering practice. The FCM is based on the following assumptions:

(i) Materialproperties. The material is linear elastic up to the point where the maximum principal tensile stress al reaches the yield stress ay. The maximum principal compression stress can assume any value, i.e. the material is able to sustain infinite stresses in compression. The material properties are the modulus of elasticity E, Poisson's ratio v, the yield stress cry and the fracture energy G v-

(ii) Fracture.

crl <cry No fracture. Linear-elastic stress-strain relation.

%<era Impossible. cr} = % A displacement-discontinuity surface starts

developing.

The displacement-discontinuity surface is perpendicular to crl and is called a fictitious crack (FC) as long as there is a stress transfer across the crack. The displacement- discontinuity is called the crack opening w, and w is per- pendicular to the crack. The displacement-discontinuity calculated by the theory of elasticity also has a component tangential to the crack. This, however, is assumed to be

%

o

"w~g

(a) %

(b)

w

Oy.

c 1

0

C l

i - - . ~ w i

C2 wu Wu

(c) (d)

" r - W

Fig. 2 (a) cr-w relation of the Dugdale model; (b) linear cr-w relation; (c) bilinear a-w relation; (d) third-order polynomial a-w relation.

small compared with w. Hence, in this paper it is assumed that only tensile stresses cr develop across the crack. The constitutive law for these stresses is a non-reversible a -w relation

cr=J'(w) for w < w , c r = 0 for w , < w

For w. < w the crack is called an open crack. At unloading w is constant for all w < w, and a varies according to the linear theory of elasticity. Figs 2a-d show four estimates of the function <7 =f(w).

Fig 2a shows the a -w relation of the well-known Dugdale model [5]. Figs 2b and c show two cr-w relations applied by Petersson [6] in connection with finite-element calculations. Fig. 2d shows as a new proposal a third- order polynomial curve whose function a =f(w) is

cr = Nocry + NlCl + NzC 2

N ~ 1 - 3 ~ k W , /

N~= -77 . + 2 w _ \w.l ~ w,

F( w V ( ,,' V-1 N 2 = __ _ __ w u Lt, w.) t.,.) /

In all cases the area below the a -w curve is the fracture energy GF. Therefore, the cr-w relations shown in Figs 2a and b do not imply definition of any new material parameters because w, is determined from % and GF. The a -w relations in Figs 2c and d imply the definition of two new material parameters, ca and c 2.

As the FCM is described above, the stress field at the FC is always in mode I of LEFM, i.e. without any shear stress. This is a reasonable approximation to reality for a material with extremely low strength in tension.

In order to determine the condition for quasi-static propagation of a crack, consider a body with an initial crack and a given stress field a 0 (plane strain or plane stress), a 0 is the stress field existing if the initial crack were not there. Because of the crack the field cro changes to another field cr, and two FCs develop at the tips of the initial crack. In both fields ao and a no tensile stress exceeds the yield stress %.

Assume that one of the crack tips is in the state of changing from stable to unstable. The other crack tip is considered stable, and if there are other initial cracks all other crack tips but the one considered are assumed to be stable. The word 'stable' signifies that the crack does not propagate at constant displacement in a displacement- controlled loading process.

The crack tip is shown in Fig. 3. The length of the FC is b:

c r= jb r )<cry for O < x < b

a = a, for x = b

In case the body is without any initial crack, then the FC develops from a point in the field a o where the maximum principal tensile stress has reached ay. The condition for

204 Harde r

~ ,Y FC

b

Fig. 3 The FC at the initial crack tip.

the FC being in the state of changing from stable to unstable is

(i) that the tensile stress at the initial crack tip, x = 0, is zero; and

(ii) when the FC propagates at constant displacement in a displacement-controlled loading process by the small distance ~ in the direction of the x-axis in Fig. 3, then the potential elastic energy relief Gc is equal to the fracture energy GF.

Gv is a parameter sometimes written

of energy dissipation, which is

G~ = 2~

where 7 is the work necessary to create one unit of new one-sided crack surface.

Griffith [7] considered the work Gv as stored surface energy. Dugdale [-5] considered GF as energy dissipation caused by plastic strains. Others have the opinion that it is a mixture of the two. In any case the energy can never be relieved, and consequently it does not matter in which way the energy disappears.

For energy dissipation at other locations than the FC, Gv < Gr Energy dissipation could for example even take place in the testing machine. However, in the following discussion energy dissipation beyond the location of the FC is disregarded. Hence the fracture criterion is the Griffith assumption that

G~ = GF = 27

3. T H E P O T E N T I A L E N E R G Y R E L I E F G c

In order to calculate G~ imagine that the FC is closed by superimposing a linear-elastic stress field & When the d- field grows from zero to its final value the a-field changes. When the crack is closed a = 0 everywhere at the FC. At the same time material is replaced by a perfect linear- elastic material which is able to sustain infinite stresses. This is clearly a fiction but nevertheless useful. The work done by a + 6 over the FC is

Io(Ii ' ) W= W~+ W~= adw+~6w dx

This process of closing the FC is reversible, i.e. during the process the a-w relation is also considered as being elastic. Over the FC both a and 6 are pure tension in the y direction.

The idea is that the process shall be reversed, returning to the original stress field a after the initial crack has propagated a small distance ~. Now, when the FC is closed we have the stress field considered in LEFM. This

~ rt (r'0) x

Fig. 4 Polar coordinates.

shows that analytical determination ofG c can be made by applying LEFM.

In Fig. 4 the E-axis is in the direction of the initial crack. At a point (x,y)=(r,O) with the polar coordinates (r,v), r = x, the stresses from LEFM [8] are

Mode k

I,;i / v 3v'~ ax 4(2nx)1/2 ~,5 - cos ~ - cos 2 - ) +""

KI (3 v 3v) COSz + c ~ + . . . ay 4(2nX)U2

_ Kl sin ~ + sin + . . - zxy 4(2nx)1/e

Mode II:

( ") K~x _ 5 s i n 2 + 3 s in~ - + . - . a~ - 4(2nx) ~/z

KIj / . v 3v) a y - 4 ~ - - 3 s,n ~ - 3 sin-~--/+---

Klj // v 3v'~ r~,-- 4 ~ , c o s ~ + 3 cos~ - ) + . . -

K~ and K n are the stress intensity factors of the given stress field a 0. Combining modes I and II gives

1 cos ~v 3v) ,+~znx) k \5

+ K n - 5 s i n ~ + 3 s i n

a y - 4 ( 2 )1/2 Ki 3 c o s 2 + c o s ~-

+ Kll -- 3 s in~-- 3 sin

1 V [ . v , 3v'~ r~, - 4(2~v)1/2 Lg,/s'n 2 +sms~ )

( l + KH cos ~ + 3 cos

v is determined by putting z~y = 0:

( v 32) ( v 32) K~ s in~+s in + K u c o s ~ + 3 c o s = 0

Ifay < ax for this value of'v, then v is replaced by v - n/2. If now a~ < ay < 0, this means that the crack is stable. If

Mate r i a l s and S t ruc tu res 205

y

. . . . ~ X

6 4 ' g

Fig. 5 Crack propagation.

ax < ay and ay > 0, the problem of determining G~ can be overcome without any knowledge of the a-w relation at the FC.

Consider the situation where the crack has propagated quasi-statistically a small distance 6 along the x-axis as shown in Fig. 5. In a slice o f unit thickness the energy relief is

C ' 01 G~ 6 = J o 2 6 W d x (3)

(2nX) il2

R , = ~ K, 3 c o s S + c o s

+ K. - 3 sin z - 3 sin (4)

Rj is the stress intensity factor o f the stress field ao referred to the x ,y-coordina te system in which the singularity is in mode I.

For plane strain the displacement u at a point with the polar coordinate (r, c 0 (see Fig. 5) is [8]

R,r Ii2 [- 20~ T30~] u = 4 ( 2 ~ G [ ( B y - 7) sin + sin _a + " "

For plane stress Poisson's ratio v is replaced by v/(1 + v), and G = E/2(1 + v) is kept unchanged. Taking c~ = n and r = 3 - x gives

2 R i ( f i - x ~ i l 2 ( v _ l ) u. = - 6 - \-~7-~ )

Taking ~ = - n and r = 3 - x gives

(a- x],%_ u _ , - a \ 2 J z ]

Hence, for plane strain

" 4R i r 'i2 w = - u . + u_ . = - - - d - \ - - ~ / ( ~ - l )

2 8 & / a - x'~ ~/2

and for plane stress

4R,(3-x'~ i/2 - - I 8R, (6 -x '~ xi2 w= G \ 2 . ) l+v=g\ - -~7-~)

This result can also be written

w = E \ -~U) (5)

/7 = 1 for plane stress, fl = 1 - v 2 for plane strain. Using Equat ions 4 and 5 in Equat ion 3 gives

Gr E \ 2n J dx

2flR~ r ~ fiR2 - n 6 E J o \ x / -E I

Hence, in L E F M as well as in the FCM the following relation is obtained:

G~ f l K 2 (6) = E !

fl = 1 for plane stress,/7 = 1 - v z for plane strain.

4. CHARACTERISTIC LENGTH 1r AND THE BRITTLENESS M O D U L U S B

Consider an infinite, thin slice in homogeneous tension a o and a crack o f the length 2a perpendicular to the stress a o. In this case R, = Kx

KI = %(ha) 1/2 (7)

When a o is equal to the critical stress ac the crack is in the state o f changing from stable to unstable

K~r (8) O" 0 = 6c ----- ( x a ) l / 2

Kj~ is called the critical stress intensity factor. In this state Gc = GF and Equat ion (6) gives

=(EGF~ 1/2 (9) ~176 \~xa )

/7 = 1 for plane stress, /7 = 1 - v 2 for plane strain. For a o < ar the crack is stable. When a increases a c decreases. For a = ac, ac = a u. In L E F M a, is the yield stress, i.e. a , = ay. In F C M a u is the value o f ao at ultimate load for

a = a c.

ao - 3 ~ Bn \ ~ . /

EGF l~ = 2 (10)

O'y

The parameter lr is called the characteristic length. For a < ar the crack is stable.

It is seen that lc is a material property whose dimension is length. The above result can also be explained as follows. The condi t ion

can be written

a = a c = fin \au}

a 1 (o '~) 2 (11)

b is called the brittleness modulus, and b c is the critical brittleness modulus. For b < b c the fracture is said to be

206 H a r d e r

ductile. In LEFM this means that the fracture is a (yield) fracture, not influenced by the crack. In FCM b < b~ means that the crack is stable at ultimate load, i.e. the ultimate load is reached before the crack propagates at constant displacement in a displacement-controlled loading.

For b > bo the fracture is said to be brittle. In LEFM as well as in FCM this means that the fracture is a crack- instability fracture taking place at ultimate load, even in displacement-controlled loading.

In the case of a more complicated stress field o- o in a plane body of finite size the stress intensity factor K, in Equation 4 can be written approximately

K.! ---- O(;rom al l2 (12)

on the condition that the crack length 2a is small compared with the linear size of the body. 0 is a constant for a series of similar stress fields, replacing the constant n 1/2 in Equation 7. For a three-point bend test 0 ~ 2 when the initial crack length is small compared with the depth of the beam [9]. OOm is the maximum tensile stress in the stress field ao existing if there were no crack.

Using Equations 6 and 12 it is seen that in the general cases Equations 9, 10 and 11 are to be replaced by Equations 13, 14 and 15:

I ( E G F ~ '/2 (13) <'~ = \ - - y 2 )

E c , ,o (14) = - -

EG7 lr 2 fly

a _ (sV b = ~ = bc - flO 2 \ a . J (15)

A more general definition of the brittleness modulus is

L=Lb B = I c a

l is an arbitrarily chosen finite dimension of the body (if the body is an infinite plate l must be replaced by a).

Using B instead of b the condition for brittle fracture, a > ac, is

t . 14 _ _ L _ t ( % ' f B = l~ = e o v > B: - aflOZ \ ~ : ] (16)

In LEFM a n = ay is a material property. In FCM a, is the value of O'0m at ultimate load for a = a~. a , only depends on the a0-field (not on the actual crack length a) and is a constant for a series of similar ao-fields. Hence, the critical brittleness moduls Be is also constant for a series of similar %-fields. However, Be does not have the same value in FCM as in LEFM.

Equations 13, 14 and 16 are those necessary for deriving the model law. It must be remembered that these formulas are valid only for cracks that are small compared with the linear size of the body.

5. DIMENSIONAL ANALYSIS

Consider a linear-elastic plane body, with or without initial cracks, with fracture energy Gv. The loads consist of constant loads p, q, r and displacements u multiplied by a load factor ?, ranging from zero to ?n at ultimate load. The a - w relations are assumed to be linear. Using

__ 2 l r EGv/a r, ?n can be written as a function

Yn = 7,(P, q, r, u, l, a, l c, E, v, oy)

Taking force, F, and length, L, as fundamental dimen- sions, the dimensions of the parameters being the arguments of the function 7u are as follows:

p A concentrated force: F q A line force: F ' L -1 r A surface force: F. L - 2 u A displacement: L l A typical length: L a Length of a crack at the surface of the body or half

the length of an internal crack: L lr Characteristic length of the material: L E Modulus of elasticity: F" L-2 v Poisson's ratio: dimensionless ay Yield stress: F" L-2

Using Buckingham's n-theorem [10] with p and l as fundamental parameters the following model laws for complete similarity are obtained:

~ = 1 (a) 7M

q~v= (l_P_P ~ - I P--L (b) qM V ~ / p~

up le - (d)

U M lM

av lv - (e)

a M IM

-- (f) l~M lM

EM \IM/I PM

v.z = 1 (h)

V M

Gyp

O'y M

( l v "~- 2 pp (i)

Y"__Z = 1 (j) 7uM

Equations a to i determine the dimensions, loads and material parameters in the model. And from ?nM the unknown Ynv is determinSd from Equation j. Subscript P signifies prototype (the larger body) and subscript M signifies model (the smaller body).

M a t e r i a l s a n d S t r u c t u r e s 207

In practice, however, it is not possible to satisfy all Equat ions a to i at the same time. I f the model is made of the same material as the p ro to type it follows f rom Equat ion g that

P--L=(lP~ 2 (17) p=

and all model laws can be satisfied except Equat ion f which reads

(18) lcM loP

i.e. the brittleness modulus shall be the same in the model as in the prototype. This is a new, interesting proper ty o f the brit t leness modulus . This, however , canno t be achieved, because when using the same material in the model and the p ro to type the following is obtained:

==:!= ,= 1,,=

instead of, f rom Equat ion 18,

Now, using Equat ion j, y.p can be written

]Sup = g]suM

where the correction factor g can be found applying the following two models:

Model 1 is identical with the physical model and for this model

which means

B(M 1) = B p ~

and all model laws except Equat ion f are satisfied.

Model 2 is a fictitious model being the same as model 1 except for t~r being determined f rom Equat ion f:

= lop tp

which means

//(2) = Bp

Since all model laws for model 2 are satisfied it follows f rom Equat ion j that

. ( 2 ) ~(2) ~uM ]SuP=~uM=-Ti~]SoM=~oM

r u M

~(2) ]SuP ~uM g ~ . . ~

~(1) ]SuM ~uM

Defining the scale ratio a by

(19) = lr,

the following relations are obtained:

Model 1 Model 2

I(M1) = 0~- l i p lh2'=~- ' lp ,1) = Q , lo = 1lo,, eM ~" %M =

G(U=GFp Gv ~(2)=a-IGF P a-iG~ FM = XJFM

B~I) = ~ - 1 Bp B~ 2) = Bp

When calculating ,,(i) and ~ d 2 ) with a linear a - w re- / 'uM ~'uM

lation the only difference between model 1 and model 2 t'7~ (2) ,... - 1/"7_ (1 ) is that W~u 2) = ~'- 1,,,(u,v u , which follows f rom "JFN = ~ "n FM"

In order to find the funct ion g = g(~) a new parameter , the brittleness index ar is defined as ar = Bp/Br Using Equat ions 14 and 16 gives

% (20)

Hence, au = ap/~ = a c for e = er and

Bp m2~ ~ / . . s M c~r ~r162 < 1 --* Bp < B~

i.e. model 2 is ductile for ar < 1 and brittle for I < ar Fur the rmore

Bp ~(~) - ~ - lBp < B~ IXc = - - < ~ ----~ /.,i M -- B~

i.e. model 1 is ductile for ~r < ~ and brittle for cc < ~ . Using Equat ion 13 the following is obtained:

,

O~t/2 = ~X1/20"ce = ~ O" u (21)

~e

I (EG(F2)M~ 1`2 1 (EG(F1)M~ li2

1 " u 1

N o w consider

(22)

the two cases 7r < 1 and 1 < c% separately:

~c < 1 (i.e. the p ro to type is ductile): model 2 is ductile, and since ~ > 1, model 1 is ductile. Hence, assuming no or only a weak variat ion o f y u = ]S,(~) when the fracture is ductile,

7(2) uM ]SuP = - - = - - ~" 1 g ~,(1)

/uM ]SuM

1 < c~r (i.e. the p ro to type is brittle): model 2 is brittle. For c~c < c~ model 1 is ductile and Equat ion 22 gives

IS 2) rr(2) 1 uM ~ VcM g-~-~ , (1 ) - - = i /2 < I

~uM O'u 0~c

For ~ < ~r model 1 is brittle and Equat ions 21 and 22 give

IS(2) ~(2) 1 uM __ t~cM

g -- , , ( I ) ,,.r(1) 0~1/2 / uM VcM

208 H a r d e r

This result is shown in Fig. 1, from which it is seen that the function g = g(ct) in the model law

]2up = g]2uM

is fully determined by the value of the brittleness index %

Ur162 l~ ~,~) =a-7~

In LEFM % = %. In FCM % = aom at y = y. for a = ar is a constant in a series of similar ao-fields.

Generally 0 and a . are not known. Therefore, in FCM % must be determined by computer simulation of the load-displacement curve on similar models. Only a few calculations are necessary because, if for one trial g = 1, then % _< 1 a n d g = 1 for all ~ > 1. I f two calculations give the same result g = gr < 1 then (see Fig. 1) =r is determined by

1 0r c = _ ~ - gr

6. CHECK FOR I N T E R N A L CONSISTENCY

Since the model law for ultimate load was derived using the energy principle Gv = Gr the theory FCM can be checked for internal consistency in the following way. The ultimate load is determined numerically by computer simulation of the load-displacement curve in displace- ment-controlled loading on similar models for different values of the scale ratio cc The result should be the function g = g(a), being independent o f the value of G v.

7. D E T E R M I N A T I O N OF GF

In [4] it was proposed to determine G F by a three-point bend test on a notched beam. However, experiments in many different laboratories showed a variation of GF for the same material and it was doubted if G v was a real material property [11"]. I f not, then FCM must be abandoned. However, the variation of GF could be caused by misinterpretation of the test results. It is not easy to determine the fracture area correctly. It is really not a plane area. However, since the ultimate load depends on Gv it is possible to determine Gv in the following alternative way. Gv can be adjusted for a best fit of ultimate load calculated by computer simulation to ultimate load obtained from experiments. It is important to observe that the experiment need not be performed in such a way that the descending branch of the load- displacement curve is determined as it is required according to the R I L E M draft recommendation [4]. Since Equation (12) rests on the assumption that the initial crack length is small compared to the linear size of the body, the initial crack in the test beam must be small compared to the beam depth. This is not required in [4].

8. I M P R O V E D a - w R E L A T I O N

I f the a -w relation is taken as a third-order polynomial (see Equation 2 and Fig. 2d) it depends on four material

parameters au, lr c~ and c2:

. . . . Now 7u is a function of the parameters already considered p, q, r, u, l, a, l~, E, v and ay, and in addition the two new parameters c~ and c 2 with the dimension F L - 3. This gives a new model law for cl and c2

cMC---~P=(l~) -3P~PM

and using Equation 17

c. - - ~ _.~ ~ - 1

CM

Since c~ and c2 are material properties this model law is not satisfied when the prototype and the model are of the same material, as is the case with the model law Equation f, as explained above. Considering the models 1 and 2 the following relations are obtained:

Model 1 Model 2 l(1) = % = %

1 " ) - l~e 1~2~ M = c~- 1lr c M - -

g7(2) _~. 0 r /'7(1)FM ~ G F p "- ' FM

BU) = a - ~Bp B~ z) = Be M

,.(2) = o~el P C(1M ) .~. C1P t~ 1M

C(1) ~(2) = 2M = C2p t. 2M r

It is seen that Equations 21 and 22 still apply and that the function g = g(~) is the same as before. The fracture energy Gv is now a four-parameter function:

GF = Gday, w,, cl, c2)

a = Noay + Nlcx + N2c 2

_ _

\Wu/ \ w u / d

§ w+2cwr L w. \w.l

( w ) ' ] + - - _ _ WuC 2

W u

1 1 2 Gv = ~ WuO'y + ~ w•(c2 -- q )

Using the model laws for model 1 and model 2 with this expression for Gv gives

__ 1 W(1)O.. .1_ 1 i,,,{1)~2tc(lJ

1 tWI1)]2g C =-w(1)a2 aM y + ] ' ~ uMJ t 2p--elp)

Materials and Structures 209

1 1 2 G(~ = GFp = ~ WupO'y + ~ Wop(C2p - - Clp )

G(2)_ ]_.,~,(2),-e ..1- 1 lW(2)~2[C(2) FM - - 2 rVuMVy " "i'2 I uM' [ 2M - - C(12M ))

--I W(2) 0 " 1., (W(u2M) ) 2 0~( C 2 p CIp) = 2 ~M y + lZ

t ~ 2 G ~ = ~- ~G~p = ~ ~- ~ Wopa, + ~ ~- w.#2p - Clp)

from which (i)

WuM ~ Wup

W(2) uM ~ 0~- 1Wu P

W(1)..(1) ,~,(2)~(2) uMt~M ~ ,ruM,.. M

showing that the a-w curve for model 2 is affine with that of model 1.

Taking c~ = c 2 =c , G F becomes a two-parameter function

G F = GF(ay, )%)

not depending on c. However, the a-w relation is a three- parameter relation, which may be better than a two- parameter relation when used in computer simulation for calculating the ultimate load.

9. C O N C L U S I O N

It is shown that analytical determination of the potential energy relief G~ in the Griffith criterion for crack propagation, Gr = GF, is the same in the fictitious crack model [1, 2], as in linear-elastic fracture mechanics. Also the role of the brittleness modulus B is the same in the two theories, B < Be ~ ductile fracture, B > Be ~ brittle frac- ture. Furthermore, a simple model law for ultimate load derived from LEFM [3] is shown to be valid in FCM as well. However, the only governing parameter of the model law, the brittleness index c~r is different in the two theories. In LEFM ~ must be determined analytically, and this is possible only for special simple stress fields. In FCM ~o must be determined by computer simulation of the load-displacement curve for displacement-controlled

loading. It is shown how FCM can be checked for internal consistency and how the material parameters of the a-w relation of FCM can be adjusted for best fit of the ultimate load calculated by computer simulation to that obtained from experiments. A new proposal for the ~r-w relation, a four-parameter third-order polynomial relation, is presented. With this a-w relation an improved theoretical va lue of the ultimate load calculated by computer simulation can probably be achieved.

R E F E R E N C E S

1. Hillerborg, A., 'A model of fracture analysis', Report TVBM-3005 (Division of Building Materials, Lund Institute of Technology, Sweden, 1978).

2. Mod6er, M., 'A fracture mechanics approach to failure analysis of concrete materials', Report TVBM-1001 (Division of Building Materials, Lund Institute of Technology, Sweden, 1979)..

3. Harder, N. A., "Fracture mechanics and dimensional analysis for concrete structures', in Report TVBM-3025, 'Fracture mechanics of concrete', edited by A. Hillerborg (Lund Institute of Technology, Sweden, 1986).

4. RILEM Draft Recommendation, 'Determination of the fracture energy of mortar and concrete by means of three-point bend tests on notched beams', Mater. Struct. 18 (106) (1985).

5. Dugdale, D. S., 'Yielding of steel sheets containing slits', J. Mech. Phys. Solids 8 (1960) 100-104.

6. Petersson, P. E.,'Crack growth and development of fracture zones in plain concrete and similar materials', Report TVBM-1006 (Division of Building Materials, Lund Institute of Technology, Sweden, t981).

7. Griffith, A. A., 'The phenomena of rupture and flow in solids', Phil. Trans. R. S. A221 (1921) 163-198.

8. Hellan, K., 'Brudmekanik', 2. rev. udgave (Tapir, NTH, 1980).

9. Tada, H., Paris, P. C. and Irwin, G. R., 'The stress analysis of cracks' (Handbook, St. Louis, Missouri, 1973).

10. Langhaar, H. L., 'Dimensional Analysis and Theory of Models' (Wiley, 1951).

11. Hillerborg, A., 'Results of three comparative test series for determining the fracture energy G F of concrete', Mater. Struet. 18 (107) (1985).

R E S U M E

Analyse dimensionnelle du b6ton h la rupture

On montre que la dktermination analytique de l'knergie potentielle Gr = GF dans le critkre de Griffith pour la propagation des fissures, est identique clans le modOle f ic t i f de fissuration ( F C M ) et dans la mkcanique de la rupture linkaire klastique ( L E F M ). Le r6le du module de fragilitk B est kgalement le m~me dans les deux thbor&s, B < Be rupture en dOformation ptastique, B > B c ~ rupture fragile. De plus, on montre qu'une simple loi modkle d&ivke de L E F M pour la charge de rupture est kgalement valable pour FCM. Cependant, le seul paramktre prkpondbrant de la loi modkle, l'indice de fragilitb ~r diffkre dans les deux thbories.

Dans le L E F M ~c doit Ytre dklermink de fa fon analytique, ce qui n'est possible que dans le cas de champs de contrainte simple. Dans le FCM, on obtient ~ par simulation par ordinateur de la courbe de dbplacement de la charge pour une charge contr61be en dbplacement. On montre comment on peut vOrifier le F C M pour une cohkrence interne, et comment on peut adapter les paramktres matbriau de la relation cr-w du F C M pour une meilleure adbquation de la charge ultime calculOe par simulation par ordinateur par rapport ~ celle obtenue de fa fon expkrimentale. On prbsente une nouvelle expression de la relation a-w, relation polynomale du 3kme ordre ~ 4 paramktres. Grfzce ft cette relation a-w, on peut sans doute obtenir une meilleure valeur thkorique de la charge ultime calculbe par simulation par ordinateur.