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Fracture criteria for concrete: Mathematical approximations and experimental validation
Engineering Fracture Mechanics Vol. 35, No. l/2/3, pp. 87-94, 1990
0013-7944/90 $3.00 + 0.00 Pergamon Press plc.
Printed in Great Britain.
FRACTURE CRITERIA FOR CONCRETE: MATHEMATICAL APPROXIMATIONS AND EXPERIMENTAL VALIDATION J. PLANAS and M. ELICES Department of Materials Science, Escuela de Ingenieros de Caminos, Universidad polit&nica de Madrid, 28040_Madrid, Spain Abstract-Three well known fracture models (Cohesive Crack, Bazant’s Size-Effect Law, and Two-Parameter Shah’s Model) were analysed, and their size-effect prediction compared in the light of general and asymptotic size-effect equations previously developed by the authors. For notched three point bend specimens of usual laboratory sizes, the three fracture criteria are experimentally indistinguishable when usual scatter values are taken into account. For larger sizes, the fracture loads predicted by the models diverge appreciably and in the asymptotic limit Shah’s and Bazant’s model are more conservative than the Cohesive Crack Model by roughly 30%.
1. INTRODUCTION FRACTURE criteria are those mathematical conditions that, when satisfied, imply structural collapse under constant load. Whatever the potential criterion is, and whatever its physical basis, it must fulfill at least the condition of adequately capturing the structural size-effect over its range of applicability. This means that when a set of geometrically similar bodies are considered, subject to similar loading, the criterion must describe adequately the influence of the size on the maximum load. We may take a single linear dimension, D, to define the size of the geometrically similar structures, and define a nominal stress oN to characterize the load level. The oldest collapse criterion is that deduced from the Classical Limit Analysis, which leads to a very simple size-effect law: namely, that for geometrically similar structures the maximum nominal stress is constant. Linear Elastic Fracture Mechanics provided the first consistent criteria for handling fracture of precracked structures, and led to another simple size-effect law, in which the maximum nominal stress decreases as the inverse of the square root of the size. Unfortunately, size-effect for concrete over the usual range of sizes is not well described by either of the above laws[l]. This fact forced the invention, throughout the last decade, of a number of non-linear models to describe concrete fracture. Some of them make use of the essential concepts of LEFM introducing complementary hypotheses to achieve non-linear behaviour. The R-curve approach, the effective crack extension concept and the two parameter model of Shah and coworkers are examples of such approaches that can be called, tentatively, Modified Linear Elastic Fracture Models (MLEFM). Some models abandon the classical approach and try to describe progressive fracture by means of constitutive equations displaying softening and complementary hypotheses about strain localization. These are, essentially, Cohesive Crack Models and Smeared Crack Band Models, which we call Progressive Softening Models (PSM) and which are reviewed and compared with other models in[2]. The essential point that justifies this work is that while the physical foundations of the various models may be very different, they lead to mathematically similar results for some particular situations. Indeed, all models converge to classical LEFM for large sizes, to order l/D. Moreover, it has been proved recently[3] that for cohesive crack models a better approach, accurate to order (1 /D2), leads to a mathematical expression which coincides with that derived from an effective crack extension model, and also with Bazant’s Size-Effect Law and with the corresponding asymptotic behaviour of Shah’s Two-Parameter Model. Therefore, for large sizes the parameters of the different models may be adjusted to predict the same fracture behaviour to a given order of accuracy. 87
J. PLANAS and M. ELICES
88
While this has been proved rigourously for the large size range, it is not evident for the small size range. Nevertheless, one has to face the practical fact that all models have their advocates and none of them has been obviously disproved by experiments. The question arises, then, about the differences between models when the practical limitations of size are taken into account. For it may happen that different models give good mathematical approximations of the observed behaviour over the limited range of sizes found in practical testing. This paper is devoted to showing that three of the most widely used models for concrete fracture may give a good description of the size effect over the practical experimental size range, to such an extent that experimental scatter precludes the possibility of a single model being validated against the others. However, the prediction of the models does indeed diverge when much larger sizes are considered. 2. GENERAL
METHODOLOGY
In this paper we consider three models that have been independently proposed to RILEM as theoretical bases for fracture characterization of concrete. They are the Cohesive Crack Model[4,5], Bazant’s Size-Effect Law[ 1,6], and Shah’s Two-Parameter Model[7]. To be precise, we consider three point bend tests on notched specimens as depicted in Fig. 1, because this type of specimen has been extensively used in experimental research on fracture of concrete. We consider that the practical experimental beam depth, D, ranges between 10 and 40 cm, and that the initial crack (notch) length, a,, ranges from 0.2 to 0.5 of the beam depth. A typical concrete with the following characteristics is considered: Tensile Strength: f, = 3.21 MPa Elastic Modulus: E = 30.0 GPa Fracture Energy (according cohesive crack model):
Grc = 103 N/m.
The analysis takes the cohesive crack model as the main data input. Size effect on the practical experimental range defined above is obtained for this model and a,/D= 0.2 to 0.5. The parameters of the remaining models are determined to achieve a good fit over the practical experimental size range with the determined size effect for the cohesive crack model and the reference geometry, which is taken to be that corresponding to a,,/D = 0.5.Maintaining the parameters at the previously computed value, predictions for other a/D values are calculated and compared. Particular attention is paid to the predictions of the model for the asymptotic range
(D + co). In order to systematize the presentation of results, the comparison is graphic dimensionless representations of the maximum nominal stress intensity below, versus beam depth. A previous analysis carried out by the authors[8], Cohesive Crack model the general size effect law might be expected to take of decreasing powers of the size:
+ cl+ c2(D/kJ’ (f; /%llaX)* = co(D/l,,) L = EGclf:
+4D
t Fig. 1. Specimen geometry.
+ ...
always made using factor, to be defined showed that for the the form of a series (1) (2)
Fracture criteria for concrete
89
where coefficients ci are dimensionless depending on geometry and on material properties, and co must be adjusted to give LEFM results as D grows to infinity. Equation (1) may be recast in terms of the nominal stress intensity factor, Ku,, as ~G~~/(~~~~)2 = I+ C~(z~~/D)+
C2(~~/~)2
+
e..
(3)
where the K,, is defined as the stress intensity factor computed for the actual load and the initial crack length. Thus it appears advantageous to present size effect results in a Y vs X representation as Y =f(x);
Y = EG~~/(K,~~~~)2; X = UD.
(4)
This representation has the advantage of leading to a linear plot when only two terms of expression (3) are retained, while maintaining the infinite size region in focus (since X +O for D + 00). For some of the models, the stress intensity factor for an effective crack of length a, larger than the initial crack length %, has to be computed for the actual load. In the whole work, we represent this stress intensity factor as &(a) and compute it from the expression given by Tada et u1.[9]: K,(a) = u,.,D”~S(~/D)
(5)
irN= 6PIBD
(6)
S(cr) = 11.99 - a(1 - x)(2.15 - 3.93~~+ 2.7a2)]ar”‘(l + 2a)-‘(1 - CZ)-~‘~
(7)
where P is the central load and B the beam thickness (Fig. 1). The nominal stress oN may be eliminated in favour of the nominal stress intensity factor, KIN, so that eq. (5) turns out to read $(a)
(8)
= ~~~~(~/D)/~~
where S,, stands for S(u,JD). 3. COHESIVE CRACK MODEL The assumed softening curve is the modified exponential defined as follows in terms of the stress transferred though the crack faces, 0, and the crack opening W: a/f;=(l+A)exp(-Bf;w/G,)-A
for
0
g/f;=0
for
5G,/S, B w,
A = 0.~82896,
B = 0.96020.
The size-effect for small specimen sizes was computed by an influence method with a numerical program similar to that used by Petersson[lO], with the same influence matrices, but slightly modified algorithm to allow non-polygonal softening, and to speed-up the resolution, Forty equal elements are situated on the central section and nodal displacements (crack openings) are related to the nodal forces through the influence matrix. The independent variable in the computation is the cohesive crack length (i.e. the number of cracked elements). The program runs automatically until the peak load is surpassed, a smoothing 4th degree polynomial fitted to the last 5 points, and the corresponding peak load computed and stored. Systematic scanning of geometries (initial crack depths) and sizes are also automatized. The size-effect results for a selected set of values of so/D are plotted in dimensionless form in Fig. 2. Notice that in this plot the inverse of the square of the maximum nominal stress intensity factor is plotted vs I)-‘, as discussed above (eqs 3 and 4). The asymptotic size-effect was analysed through the method recently proposed by the authors[3]. The result may be written as: EG~c(&~man
)-' =
1 + (2S#,)Au,,D-’
(9)
90
J. PLANAS and M. ELICES ASYMPTOTAS =FC
7 -
I-~o.lD=05 U./D
0 0 0 0
.03
il-
_ K;N mox
r_------_-------___--I I I
RACTICAL
EXPERIMENTAL
SIZE
50 L5 80 30
----I RANGE
I
1
0
L
3
2
‘chlD
Fig. 2. Size-effect curves for three-point bend tests on notched beams, according to Cohesive Crack Model.
where S(a/D) is the shape factor for the stress intensity factor, given by eq. (7) S’(a/D) its first derivative with respect to the argument, and subindex 0 refers to initial crack length. Aacc, is the critical effective crack extension for infinite size, for this particular model, which is obtained as described in[2] (second subindex C refers to Cohesive, to distinguish this value from others similar found in other models). For the assumed softening curve we have Aaccm = 2.4805 l,, ,
(10)
The resulting asymptotic behaviour has also been plotted in Fig. 2 for a set of a,/D values. 4. BAZANT’S SIZE-EFFECT LAW Bazant’s Size-Effect law[l] may be put in the form =FB
(&+nax
)-Z=
1 +&D-l
(11)
where GFB stands for the fracture energy computed for Bazant’s model, which differs from G,,, and &, is a parameter with dimensions of length and being dependent upon material properties and geometrical shape, which must be adapted to each particular geometry. Using the values of fracture energy of the cohesive crack model, eq. (11) may be recast in a form suitable for comparison with the results presented in Section 3 as EGFC
(&nax
)-*
=
(GFc/%B)[~
+
(&lL,)(L,lD)l.
(12)
Equation (12) takes the form of a straight line in the X-Y plot defined in eq. (4). G,, is obtained once and for all by least square fitting of the results obtained in Section 3 for the reference geometry (a,/D = 0.5) over the practical size range previously defined (10 cm < D < 40 cm). This value is then used as constant for the other two geometries, and parameter A,/lch is determined by least square fitting. A summary of numerical values is given in Table 1. Figure 3 shows the plots of the fittings achieved for the extreme curves in Fig. 2 (u,/D = 0.3 and 0.5). It will be noticed that the fitting over the practical size range is very good, giving maximum differences between the two models below the 2.5% level, which makes it practically impossible to distinguish experimentally which of the two models is the more correct. However, the extrapolations to the large size range do diverge appreciably, Bazant’s model being more conservative by about 30%. Unfortunately, in order to detect appreciably differences (of about lo%, say, of maximum load), the experimental size range ought to be increased by a factor of 2.5, so that beams with depths between 10 and 100 cm ought to be tested, and this with a carefully controlled concrete to reduce experimental scatter to well below 10%.
Fracture criteria for concrete
91
Table 1. Summary of results for Baxant’s size effect law Geometry a,,/D a,/D q/D a,lD
Size range (cm)
GFB /GFC
10-40 l&40 l&40 lo-40 cc
0.52t 0.52t 0.52t 0.52t 0.52t
= 0.50 = 0.40
=0.30 = 0.20
all
Max. differencej in fracture load (%)
WC,
0.696$ 0.615$ 0.5761 0.5941 -
1.4 2.2 2.5 2.2 -28.1
tBest fit for the reference geometry, a,,/D = 0.5. $Best fit for each particular geometry. §Relative to Cohesive Crack Model.
5. SHAH’S
TWO-PARAMETER
MODEL
In Shah’s Two-Parameter Model[7] it is assumed that in a precracked structure a slow crack growth takes place under increasing load up to a certain crack extension Au, at which the maximum load is attained. This critical situation is assumed to occur when, simultaneously, the stress intensity factor at the extended crack tip reaches its critical value K$ and the crack opening at the initial crack tip, or CTOD, its critical value CTOD,,, where index S refer to Shah’s Model. In order to compare the results obtained from other models it appears convenient to define a conventional fracture energy GFS for this model as K& = (EGFS)“‘.
(13)
Particularizing eq. (8) for the critical situation, the stress intensity factor equation reads, using definition (13), (EG&”
= &max S(aJD
+ AaJD)l&.
(14)
Equation (14) provides one equation with two unknowns, namely, the maximum nominal stress intensity factor, and the critical effective crack extension. The CTOD critical condition provides the second equation to solve the problem. A convenient form of this second equation may be cast in the following way: From well known results in LEFM (see, eg. Knott[l l]), the opening, w, of an elastic crack of arbitrary length a at a distance r from its tip may be written, in plane stress, as ~(a, r) = 8(2x)-“‘E-‘K,(a)
-BAZANT -COHESIVE
&L(a/D,
r/D)
(15)
SIZE EFFECT LAW CRACK MODEL
2
%N Inax
EG, 'ch=--i_
‘1
I
1
3
4
5
kh’B
Fig. 3. Comparison of size-effect curves predicted by Baxant’s Size-Effect law and Cohesive Crack Model.
J. PLANAS and M. ELICES
92
where L(a/D, r/D) is a regular function which may be determined from elastic analysis and which satisfies the condition L(a/D, 0) = 1.
(16)
By setting a = a0 + Aa, and r = Aa, we obtain from eq. (IS) the general expression for the CTOD as a function of the stress intensity factor and the effective crack extension. In the critical situation, CTOD = CTOD,,, Aa = Au,, and K, (a,, + Au,) = (EGFS)‘12, and eq. (17) follows: CTOD,, = 8(2~)-“2E-‘(EG,s)“2
,/Aa,L(aJD
f Aa,/D, AaJD).
(17)
The problem is now completely defined. Assuming that function L is known, as well as the material parameters, 84 is obtained from eq. (17) and, after substitution, K,INmax is determined from eq. (14). Unfortunately no closed form exists for function K, nor for related functions for the geometry under study, and it has to be determined numerically. But before facing this problem we may write equations (17) and (14) in forms suitable for direct comparison with the results obtained for the cohesive crack model. We first notice that owing to the structure of eq. (I 7) and the property of function L given by eq. (16), we may define a derived material property, namely, the critical effective crack extension for infinite size, Aacsm, as A%,
= r~!?(CTOD~s)~/(32Grs).
With this definition, and after a little algebra, eqs (17) and (14) may be transformed X = (L iA%,
)K (Aa@ ;
Y = (G&G,,)
Yi (A@%
a0ID 1
(18) to read (19)
%/D)
(20)
+ Aa,l& Aa,/D)l*
(21)
where X and Y are the variables defined in eq. (4) and x1 (AacID;
aolD) = (Aa,lD)[&-,lD
Yi (AaJD;
ao/D) = [S(a,/D
+ AaJD)/SoJ2.
(22)
For a given geometry (fixed a,/D), eqs (19) and (20) are no more than a parametric representation of the size-effect curves, with parameter AaJD. Moreover, it is obvious that the X-Y curves, which depend on material parameters, are affine to the X,-Y, curves, which are size-independent. While a closed form is available for function Y, , function X, had to be obtained numerically. The same influence method previously used for cohesive crack problems was employed in the computations, except that cohesive forces were set to zero. Crack openings were computed at the nodal points for a set of crack lengths varying at 0.025 D intervals for a unit stress intensity factor. Equation (10) provided the definition of function L on a square grid with 0.025 grid spacing. Simple transfo~ations correspon~ng to eq. (21) lead to the definition of function X, on a grid of the same spacing as before. In order to determine the material parameters of Shah’s Model achieving a good fit between this and the Cohesive Crack Model, the affinity between X-Y and X,-Y, curves was exploited. In a log-log plot affinity transforms into a translation, so that, for the reference geometry, bilogarithmic plots were drawn for the X,-Y, curve and the cohesive crack results and a translation was found manually giving a good fit over the experimental range. The material parameters were found easily from the translation vector. The results are summarized in Table 2, and the size effect curves derived from Shah’s model are plotted in Fig. 4 for a,,/D equal to 0.3 and 0.5. The differences in maximum load as predicted by the two models is again very low on the practical experimental size range, although larger than for the Bazant model. The models diverge again for large sizes, the difference being slightly larger than that previously found for the Bazant model. It may also be noticed what while Shah’s model predicts an asymptotic behaviour similar to that found in cohesive crack models-eq. (9)-the asymptotic critical effective crack extension is some 33 times less than that found for the Cohesive Crack Model, 22 mm for the Two-Parameter Model versus 744 mm for the Cohesive Crack Model, when the concrete characteristics are those assumed in Section 3.
93
Fracture criteria for concrete Table 2. Summary of results for Shah’s two-parameter model Geometry aJD =0.50 aJD =0.40 cq,lD = 0.30 Q/D = 0.20 all
Size range (cm)
GFS /GFC
IO-40 IO-40 lo-40 lo-40 a,
0.48f 0.48f 0.4st 0.48t 0.48t
ks,
IL
Max. differen@ in fracture load (%)
0.0746: 0.07463 0.0746% 0.0746% 0.0746%
2.1 2.2 2.5 2.9 -30.8
tEtest fit for the reference geometry, aJD =0.5. $Bestfit for each particular geometry. $Relative to Cohesive Crack Model.
6. SUMMARY AND CONCLUSIONS Three well known fracture criteria are analysed in the light of general and asymptotic size-effect equations previously developed by the authors[3,8]. The cohesive crack model was characterized by the experimental fracture energy Grc and an exponential softening curve. The two parameters of Bazant’s Size Effect Law were drawn from least square fitting of the cohesive crack model for the reference geometry over a practical size range. Finally, the two parameters of Shah’s model were obtained in a similar way from the affinity between X-Y and X,-Y, plots, as explained before. These three fracture criteria were compared in Figs 3 and 4. From these numerical results some conclusions may be drawn: (I) Inside the practical size range (depth of beams between 10 cm and 40 cm), predictions of fracture loads from the three criteria are indistinguishable due to experimental scatter. A reasonable figure for maximum load scatter in these tests is 10% and the maximum differences computed are less than 3%, as shown in Tables 1 and 2. (2) Differences in fracture loads can be ascertained when testing larger size beams (more than 1 m depth). (3) Asymptotic behaviour of fracture loads for larger sizes depends on the model. For notched beams Bazant’s model seems to be more conservative than the cohesive model by about 30% (see Table 1). Shah’s model predicts an asymptotic behaviour mathematically similar to the cohesive model, although the asymptotic critical effective crack extension is much less than that found for the cohesive model. Again, like Bazant’s model, Shah’s model is more conservative by about 30%, as shown in Table 2.
-WO PARAMETER -COHESIVE CRACK
2
=FC
MODEL
MODEL
l-
KINmx
EG.,
'ch=--T ‘1
I
0
I
I
1
2
G
3 IchID
Fig. 4. Comparison of size-effect curves predicted by Shah’s Two-Parameter Model and Cohesive Crack Model.
94
J. PLANAS and M. ELICES
Acknow~ledgement-The
authors gratefully acknowledge financial support for this research provided by a Cooperative Research Project US-Spain Joint Committee for Science and Technology 83/071.
REFERENCES
VI Z. P. Bazant, Size effect in blunt fracture: concrete, rock, metals. J. Engng Me&. AXE 110,518-538 (1984). Pl M. Elites and J. Planas, Material models. Chap. 3 in Srafe of Ihe Art Report on Fracture of Concrete (Edited by L. Elfgren). RILEM TC-90 (1988). [31 J. Planas and M. Elites, Asymptotic analysis of the development of a cohesive crack zone in mode I loading for arbitrary softening curves. Proc. Fracture of Concrete and Rock (Edited by S. P. Shah and S. E. Swarm), Houston (1987). I41 A. Hillerborg, The theoretical basis of a method to determine the fracture energy G, of concrete. Muter. Srrucf. 18, 291-296 (1985). PI RILEM TC, Determination of the fracture energy of mortar and concrete by means of three-point bend tests on notched beams. Mater. Struct. 18, 285-290 (1985). PI Z. P. Bazant and P. Pfeiffer, Fracture energy of concrete: its definition and determination from size effect tests. Concr. Durability ACZ SP 100-8,89-109 (1987). [71 Y. S. Jenq and S. P. Shah, A two parameter fracture model for concrete. J. Engng Me&. AXE 111,1227-1241 (1985). @I J. Planas and M. Elites, Towards a measure of G,: an analysis of experimental results, in Fracture Toughness and Fracture Energy of Concrete (Edited by F. H. Wit~ann), pp. 381-390 (1986). [91 H. Tada, P. C. Paris and Cl. R. Irwin (eds), Stress Analysis of Cracks ~undbook. Del Research Corp. Hellertown, PA (1973). WI P. E. Petterson, Crack growth and development of fracture zones in plain concrete and similar materials. Report TVBM 1006, Lund, Sweden (1981). IllI J. F. Knott, Fundamenlals of Fracture Mechanics. Buttenvorths, London (1973). (Received for publication 16 November 1988)
0013-7944/90 $3.00 + 0.00 Pergamon Press plc.
Printed in Great Britain.
FRACTURE CRITERIA FOR CONCRETE: MATHEMATICAL APPROXIMATIONS AND EXPERIMENTAL VALIDATION J. PLANAS and M. ELICES Department of Materials Science, Escuela de Ingenieros de Caminos, Universidad polit&nica de Madrid, 28040_Madrid, Spain Abstract-Three well known fracture models (Cohesive Crack, Bazant’s Size-Effect Law, and Two-Parameter Shah’s Model) were analysed, and their size-effect prediction compared in the light of general and asymptotic size-effect equations previously developed by the authors. For notched three point bend specimens of usual laboratory sizes, the three fracture criteria are experimentally indistinguishable when usual scatter values are taken into account. For larger sizes, the fracture loads predicted by the models diverge appreciably and in the asymptotic limit Shah’s and Bazant’s model are more conservative than the Cohesive Crack Model by roughly 30%.
1. INTRODUCTION FRACTURE criteria are those mathematical conditions that, when satisfied, imply structural collapse under constant load. Whatever the potential criterion is, and whatever its physical basis, it must fulfill at least the condition of adequately capturing the structural size-effect over its range of applicability. This means that when a set of geometrically similar bodies are considered, subject to similar loading, the criterion must describe adequately the influence of the size on the maximum load. We may take a single linear dimension, D, to define the size of the geometrically similar structures, and define a nominal stress oN to characterize the load level. The oldest collapse criterion is that deduced from the Classical Limit Analysis, which leads to a very simple size-effect law: namely, that for geometrically similar structures the maximum nominal stress is constant. Linear Elastic Fracture Mechanics provided the first consistent criteria for handling fracture of precracked structures, and led to another simple size-effect law, in which the maximum nominal stress decreases as the inverse of the square root of the size. Unfortunately, size-effect for concrete over the usual range of sizes is not well described by either of the above laws[l]. This fact forced the invention, throughout the last decade, of a number of non-linear models to describe concrete fracture. Some of them make use of the essential concepts of LEFM introducing complementary hypotheses to achieve non-linear behaviour. The R-curve approach, the effective crack extension concept and the two parameter model of Shah and coworkers are examples of such approaches that can be called, tentatively, Modified Linear Elastic Fracture Models (MLEFM). Some models abandon the classical approach and try to describe progressive fracture by means of constitutive equations displaying softening and complementary hypotheses about strain localization. These are, essentially, Cohesive Crack Models and Smeared Crack Band Models, which we call Progressive Softening Models (PSM) and which are reviewed and compared with other models in[2]. The essential point that justifies this work is that while the physical foundations of the various models may be very different, they lead to mathematically similar results for some particular situations. Indeed, all models converge to classical LEFM for large sizes, to order l/D. Moreover, it has been proved recently[3] that for cohesive crack models a better approach, accurate to order (1 /D2), leads to a mathematical expression which coincides with that derived from an effective crack extension model, and also with Bazant’s Size-Effect Law and with the corresponding asymptotic behaviour of Shah’s Two-Parameter Model. Therefore, for large sizes the parameters of the different models may be adjusted to predict the same fracture behaviour to a given order of accuracy. 87
J. PLANAS and M. ELICES
88
While this has been proved rigourously for the large size range, it is not evident for the small size range. Nevertheless, one has to face the practical fact that all models have their advocates and none of them has been obviously disproved by experiments. The question arises, then, about the differences between models when the practical limitations of size are taken into account. For it may happen that different models give good mathematical approximations of the observed behaviour over the limited range of sizes found in practical testing. This paper is devoted to showing that three of the most widely used models for concrete fracture may give a good description of the size effect over the practical experimental size range, to such an extent that experimental scatter precludes the possibility of a single model being validated against the others. However, the prediction of the models does indeed diverge when much larger sizes are considered. 2. GENERAL
METHODOLOGY
In this paper we consider three models that have been independently proposed to RILEM as theoretical bases for fracture characterization of concrete. They are the Cohesive Crack Model[4,5], Bazant’s Size-Effect Law[ 1,6], and Shah’s Two-Parameter Model[7]. To be precise, we consider three point bend tests on notched specimens as depicted in Fig. 1, because this type of specimen has been extensively used in experimental research on fracture of concrete. We consider that the practical experimental beam depth, D, ranges between 10 and 40 cm, and that the initial crack (notch) length, a,, ranges from 0.2 to 0.5 of the beam depth. A typical concrete with the following characteristics is considered: Tensile Strength: f, = 3.21 MPa Elastic Modulus: E = 30.0 GPa Fracture Energy (according cohesive crack model):
Grc = 103 N/m.
The analysis takes the cohesive crack model as the main data input. Size effect on the practical experimental range defined above is obtained for this model and a,/D= 0.2 to 0.5. The parameters of the remaining models are determined to achieve a good fit over the practical experimental size range with the determined size effect for the cohesive crack model and the reference geometry, which is taken to be that corresponding to a,,/D = 0.5.Maintaining the parameters at the previously computed value, predictions for other a/D values are calculated and compared. Particular attention is paid to the predictions of the model for the asymptotic range
(D + co). In order to systematize the presentation of results, the comparison is graphic dimensionless representations of the maximum nominal stress intensity below, versus beam depth. A previous analysis carried out by the authors[8], Cohesive Crack model the general size effect law might be expected to take of decreasing powers of the size:
+ cl+ c2(D/kJ’ (f; /%llaX)* = co(D/l,,) L = EGclf:
+4D
t Fig. 1. Specimen geometry.
+ ...
always made using factor, to be defined showed that for the the form of a series (1) (2)
Fracture criteria for concrete
89
where coefficients ci are dimensionless depending on geometry and on material properties, and co must be adjusted to give LEFM results as D grows to infinity. Equation (1) may be recast in terms of the nominal stress intensity factor, Ku,, as ~G~~/(~~~~)2 = I+ C~(z~~/D)+
C2(~~/~)2
+
e..
(3)
where the K,, is defined as the stress intensity factor computed for the actual load and the initial crack length. Thus it appears advantageous to present size effect results in a Y vs X representation as Y =f(x);
Y = EG~~/(K,~~~~)2; X = UD.
(4)
This representation has the advantage of leading to a linear plot when only two terms of expression (3) are retained, while maintaining the infinite size region in focus (since X +O for D + 00). For some of the models, the stress intensity factor for an effective crack of length a, larger than the initial crack length %, has to be computed for the actual load. In the whole work, we represent this stress intensity factor as &(a) and compute it from the expression given by Tada et u1.[9]: K,(a) = u,.,D”~S(~/D)
(5)
irN= 6PIBD
(6)
S(cr) = 11.99 - a(1 - x)(2.15 - 3.93~~+ 2.7a2)]ar”‘(l + 2a)-‘(1 - CZ)-~‘~
(7)
where P is the central load and B the beam thickness (Fig. 1). The nominal stress oN may be eliminated in favour of the nominal stress intensity factor, KIN, so that eq. (5) turns out to read $(a)
(8)
= ~~~~(~/D)/~~
where S,, stands for S(u,JD). 3. COHESIVE CRACK MODEL The assumed softening curve is the modified exponential defined as follows in terms of the stress transferred though the crack faces, 0, and the crack opening W: a/f;=(l+A)exp(-Bf;w/G,)-A
for
0
g/f;=0
for
5G,/S, B w,
A = 0.~82896,
B = 0.96020.
The size-effect for small specimen sizes was computed by an influence method with a numerical program similar to that used by Petersson[lO], with the same influence matrices, but slightly modified algorithm to allow non-polygonal softening, and to speed-up the resolution, Forty equal elements are situated on the central section and nodal displacements (crack openings) are related to the nodal forces through the influence matrix. The independent variable in the computation is the cohesive crack length (i.e. the number of cracked elements). The program runs automatically until the peak load is surpassed, a smoothing 4th degree polynomial fitted to the last 5 points, and the corresponding peak load computed and stored. Systematic scanning of geometries (initial crack depths) and sizes are also automatized. The size-effect results for a selected set of values of so/D are plotted in dimensionless form in Fig. 2. Notice that in this plot the inverse of the square of the maximum nominal stress intensity factor is plotted vs I)-‘, as discussed above (eqs 3 and 4). The asymptotic size-effect was analysed through the method recently proposed by the authors[3]. The result may be written as: EG~c(&~man
)-' =
1 + (2S#,)Au,,D-’
(9)
90
J. PLANAS and M. ELICES ASYMPTOTAS =FC
7 -
I-~o.lD=05 U./D
0 0 0 0
.03
il-
_ K;N mox
r_------_-------___--I I I
RACTICAL
EXPERIMENTAL
SIZE
50 L5 80 30
----I RANGE
I
1
0
L
3
2
‘chlD
Fig. 2. Size-effect curves for three-point bend tests on notched beams, according to Cohesive Crack Model.
where S(a/D) is the shape factor for the stress intensity factor, given by eq. (7) S’(a/D) its first derivative with respect to the argument, and subindex 0 refers to initial crack length. Aacc, is the critical effective crack extension for infinite size, for this particular model, which is obtained as described in[2] (second subindex C refers to Cohesive, to distinguish this value from others similar found in other models). For the assumed softening curve we have Aaccm = 2.4805 l,, ,
(10)
The resulting asymptotic behaviour has also been plotted in Fig. 2 for a set of a,/D values. 4. BAZANT’S SIZE-EFFECT LAW Bazant’s Size-Effect law[l] may be put in the form =FB
(&+nax
)-Z=
1 +&D-l
(11)
where GFB stands for the fracture energy computed for Bazant’s model, which differs from G,,, and &, is a parameter with dimensions of length and being dependent upon material properties and geometrical shape, which must be adapted to each particular geometry. Using the values of fracture energy of the cohesive crack model, eq. (11) may be recast in a form suitable for comparison with the results presented in Section 3 as EGFC
(&nax
)-*
=
(GFc/%B)[~
+
(&lL,)(L,lD)l.
(12)
Equation (12) takes the form of a straight line in the X-Y plot defined in eq. (4). G,, is obtained once and for all by least square fitting of the results obtained in Section 3 for the reference geometry (a,/D = 0.5) over the practical size range previously defined (10 cm < D < 40 cm). This value is then used as constant for the other two geometries, and parameter A,/lch is determined by least square fitting. A summary of numerical values is given in Table 1. Figure 3 shows the plots of the fittings achieved for the extreme curves in Fig. 2 (u,/D = 0.3 and 0.5). It will be noticed that the fitting over the practical size range is very good, giving maximum differences between the two models below the 2.5% level, which makes it practically impossible to distinguish experimentally which of the two models is the more correct. However, the extrapolations to the large size range do diverge appreciably, Bazant’s model being more conservative by about 30%. Unfortunately, in order to detect appreciably differences (of about lo%, say, of maximum load), the experimental size range ought to be increased by a factor of 2.5, so that beams with depths between 10 and 100 cm ought to be tested, and this with a carefully controlled concrete to reduce experimental scatter to well below 10%.
Fracture criteria for concrete
91
Table 1. Summary of results for Baxant’s size effect law Geometry a,,/D a,/D q/D a,lD
Size range (cm)
GFB /GFC
10-40 l&40 l&40 lo-40 cc
0.52t 0.52t 0.52t 0.52t 0.52t
= 0.50 = 0.40
=0.30 = 0.20
all
Max. differencej in fracture load (%)
WC,
0.696$ 0.615$ 0.5761 0.5941 -
1.4 2.2 2.5 2.2 -28.1
tBest fit for the reference geometry, a,,/D = 0.5. $Best fit for each particular geometry. §Relative to Cohesive Crack Model.
5. SHAH’S
TWO-PARAMETER
MODEL
In Shah’s Two-Parameter Model[7] it is assumed that in a precracked structure a slow crack growth takes place under increasing load up to a certain crack extension Au, at which the maximum load is attained. This critical situation is assumed to occur when, simultaneously, the stress intensity factor at the extended crack tip reaches its critical value K$ and the crack opening at the initial crack tip, or CTOD, its critical value CTOD,,, where index S refer to Shah’s Model. In order to compare the results obtained from other models it appears convenient to define a conventional fracture energy GFS for this model as K& = (EGFS)“‘.
(13)
Particularizing eq. (8) for the critical situation, the stress intensity factor equation reads, using definition (13), (EG&”
= &max S(aJD
+ AaJD)l&.
(14)
Equation (14) provides one equation with two unknowns, namely, the maximum nominal stress intensity factor, and the critical effective crack extension. The CTOD critical condition provides the second equation to solve the problem. A convenient form of this second equation may be cast in the following way: From well known results in LEFM (see, eg. Knott[l l]), the opening, w, of an elastic crack of arbitrary length a at a distance r from its tip may be written, in plane stress, as ~(a, r) = 8(2x)-“‘E-‘K,(a)
-BAZANT -COHESIVE
&L(a/D,
r/D)
(15)
SIZE EFFECT LAW CRACK MODEL
2
%N Inax
EG, 'ch=--i_
‘1
I
1
3
4
5
kh’B
Fig. 3. Comparison of size-effect curves predicted by Baxant’s Size-Effect law and Cohesive Crack Model.
J. PLANAS and M. ELICES
92
where L(a/D, r/D) is a regular function which may be determined from elastic analysis and which satisfies the condition L(a/D, 0) = 1.
(16)
By setting a = a0 + Aa, and r = Aa, we obtain from eq. (IS) the general expression for the CTOD as a function of the stress intensity factor and the effective crack extension. In the critical situation, CTOD = CTOD,,, Aa = Au,, and K, (a,, + Au,) = (EGFS)‘12, and eq. (17) follows: CTOD,, = 8(2~)-“2E-‘(EG,s)“2
,/Aa,L(aJD
f Aa,/D, AaJD).
(17)
The problem is now completely defined. Assuming that function L is known, as well as the material parameters, 84 is obtained from eq. (17) and, after substitution, K,INmax is determined from eq. (14). Unfortunately no closed form exists for function K, nor for related functions for the geometry under study, and it has to be determined numerically. But before facing this problem we may write equations (17) and (14) in forms suitable for direct comparison with the results obtained for the cohesive crack model. We first notice that owing to the structure of eq. (I 7) and the property of function L given by eq. (16), we may define a derived material property, namely, the critical effective crack extension for infinite size, Aacsm, as A%,
= r~!?(CTOD~s)~/(32Grs).
With this definition, and after a little algebra, eqs (17) and (14) may be transformed X = (L iA%,
)K (Aa@ ;
Y = (G&G,,)
Yi (A@%
a0ID 1
(18) to read (19)
%/D)
(20)
+ Aa,l& Aa,/D)l*
(21)
where X and Y are the variables defined in eq. (4) and x1 (AacID;
aolD) = (Aa,lD)[&-,lD
Yi (AaJD;
ao/D) = [S(a,/D
+ AaJD)/SoJ2.
(22)
For a given geometry (fixed a,/D), eqs (19) and (20) are no more than a parametric representation of the size-effect curves, with parameter AaJD. Moreover, it is obvious that the X-Y curves, which depend on material parameters, are affine to the X,-Y, curves, which are size-independent. While a closed form is available for function Y, , function X, had to be obtained numerically. The same influence method previously used for cohesive crack problems was employed in the computations, except that cohesive forces were set to zero. Crack openings were computed at the nodal points for a set of crack lengths varying at 0.025 D intervals for a unit stress intensity factor. Equation (10) provided the definition of function L on a square grid with 0.025 grid spacing. Simple transfo~ations correspon~ng to eq. (21) lead to the definition of function X, on a grid of the same spacing as before. In order to determine the material parameters of Shah’s Model achieving a good fit between this and the Cohesive Crack Model, the affinity between X-Y and X,-Y, curves was exploited. In a log-log plot affinity transforms into a translation, so that, for the reference geometry, bilogarithmic plots were drawn for the X,-Y, curve and the cohesive crack results and a translation was found manually giving a good fit over the experimental range. The material parameters were found easily from the translation vector. The results are summarized in Table 2, and the size effect curves derived from Shah’s model are plotted in Fig. 4 for a,,/D equal to 0.3 and 0.5. The differences in maximum load as predicted by the two models is again very low on the practical experimental size range, although larger than for the Bazant model. The models diverge again for large sizes, the difference being slightly larger than that previously found for the Bazant model. It may also be noticed what while Shah’s model predicts an asymptotic behaviour similar to that found in cohesive crack models-eq. (9)-the asymptotic critical effective crack extension is some 33 times less than that found for the Cohesive Crack Model, 22 mm for the Two-Parameter Model versus 744 mm for the Cohesive Crack Model, when the concrete characteristics are those assumed in Section 3.
93
Fracture criteria for concrete Table 2. Summary of results for Shah’s two-parameter model Geometry aJD =0.50 aJD =0.40 cq,lD = 0.30 Q/D = 0.20 all
Size range (cm)
GFS /GFC
IO-40 IO-40 lo-40 lo-40 a,
0.48f 0.48f 0.4st 0.48t 0.48t
ks,
IL
Max. differen@ in fracture load (%)
0.0746: 0.07463 0.0746% 0.0746% 0.0746%
2.1 2.2 2.5 2.9 -30.8
tEtest fit for the reference geometry, aJD =0.5. $Bestfit for each particular geometry. $Relative to Cohesive Crack Model.
6. SUMMARY AND CONCLUSIONS Three well known fracture criteria are analysed in the light of general and asymptotic size-effect equations previously developed by the authors[3,8]. The cohesive crack model was characterized by the experimental fracture energy Grc and an exponential softening curve. The two parameters of Bazant’s Size Effect Law were drawn from least square fitting of the cohesive crack model for the reference geometry over a practical size range. Finally, the two parameters of Shah’s model were obtained in a similar way from the affinity between X-Y and X,-Y, plots, as explained before. These three fracture criteria were compared in Figs 3 and 4. From these numerical results some conclusions may be drawn: (I) Inside the practical size range (depth of beams between 10 cm and 40 cm), predictions of fracture loads from the three criteria are indistinguishable due to experimental scatter. A reasonable figure for maximum load scatter in these tests is 10% and the maximum differences computed are less than 3%, as shown in Tables 1 and 2. (2) Differences in fracture loads can be ascertained when testing larger size beams (more than 1 m depth). (3) Asymptotic behaviour of fracture loads for larger sizes depends on the model. For notched beams Bazant’s model seems to be more conservative than the cohesive model by about 30% (see Table 1). Shah’s model predicts an asymptotic behaviour mathematically similar to the cohesive model, although the asymptotic critical effective crack extension is much less than that found for the cohesive model. Again, like Bazant’s model, Shah’s model is more conservative by about 30%, as shown in Table 2.
-WO PARAMETER -COHESIVE CRACK
2
=FC
MODEL
MODEL
l-
KINmx
EG.,
'ch=--T ‘1
I
0
I
I
1
2
G
3 IchID
Fig. 4. Comparison of size-effect curves predicted by Shah’s Two-Parameter Model and Cohesive Crack Model.
94
J. PLANAS and M. ELICES
Acknow~ledgement-The
authors gratefully acknowledge financial support for this research provided by a Cooperative Research Project US-Spain Joint Committee for Science and Technology 83/071.
REFERENCES
VI Z. P. Bazant, Size effect in blunt fracture: concrete, rock, metals. J. Engng Me&. AXE 110,518-538 (1984). Pl M. Elites and J. Planas, Material models. Chap. 3 in Srafe of Ihe Art Report on Fracture of Concrete (Edited by L. Elfgren). RILEM TC-90 (1988). [31 J. Planas and M. Elites, Asymptotic analysis of the development of a cohesive crack zone in mode I loading for arbitrary softening curves. Proc. Fracture of Concrete and Rock (Edited by S. P. Shah and S. E. Swarm), Houston (1987). I41 A. Hillerborg, The theoretical basis of a method to determine the fracture energy G, of concrete. Muter. Srrucf. 18, 291-296 (1985). PI RILEM TC, Determination of the fracture energy of mortar and concrete by means of three-point bend tests on notched beams. Mater. Struct. 18, 285-290 (1985). PI Z. P. Bazant and P. Pfeiffer, Fracture energy of concrete: its definition and determination from size effect tests. Concr. Durability ACZ SP 100-8,89-109 (1987). [71 Y. S. Jenq and S. P. Shah, A two parameter fracture model for concrete. J. Engng Me&. AXE 111,1227-1241 (1985). @I J. Planas and M. Elites, Towards a measure of G,: an analysis of experimental results, in Fracture Toughness and Fracture Energy of Concrete (Edited by F. H. Wit~ann), pp. 381-390 (1986). [91 H. Tada, P. C. Paris and Cl. R. Irwin (eds), Stress Analysis of Cracks ~undbook. Del Research Corp. Hellertown, PA (1973). WI P. E. Petterson, Crack growth and development of fracture zones in plain concrete and similar materials. Report TVBM 1006, Lund, Sweden (1981). IllI J. F. Knott, Fundamenlals of Fracture Mechanics. Buttenvorths, London (1973). (Received for publication 16 November 1988)