Engineering Fracture Mechanics - Civil & Environmental Engineering
Engineering Fracture Mechanics 110 (2013) 281–289
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Comprehensive concrete fracture tests: Size effects of Types 1 & 2, crack length effect and postpeak Christian G. Hoover a,1, Zdeneˇk P. Bazˇant b,⇑ a
Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA McCormick Institute Professor and W.P. Murphy Professor of Civil Engineering and Materials Science, Northwestern University, 2145 Sheridan Road, CEE/A135, Evanston, IL 60208, USA b
a r t i c l e
i n f o
Article history: Received 7 February 2013 Received in revised form 25 July 2013 Accepted 3 August 2013 Available online 23 August 2013 Keywords: Cohesive crack model Scaling of strength Fracture testing Size effect law Flexural strength
a b s t r a c t The comprehensive fracture tests of notched and unnotched beams presented in the preceding paper are evaluated to clarify the size effects and fracture energy dissipation. The test results for beams with notches of various lengths agree closely with Bazˇant’s Type 2 size effect law and yield the value of the initial fracture energy Gf. Since nearly complete post-peak softening load–displacement curves have been obtained, the total energy dissipation by fracture can be accurately evaluated and used to determine the RILEM total fracture energy GF. For the present concrete, GF/Gf 2. The transition between the Type 1 and 2 size effects are determined and is found to be very different from that assumed in Hu and Duan’s boundary effect model. Determination of a universal size effect law fitting the beam strengths for all the sizes and all notch depths is relegated to a subsequent paper. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction and review of size effect and crack length effect The preceding paper [1] presented an introduction to the problem and reported comprehensive test data for fracture of concrete. The experimental program, also described in [1,2], consisted of 128 three-point bend beams with 4 different depths of 1.58, 3.66, 8.47 and 19.69 in. (40, 93, 215 and 500 mm), corresponding to a size range of 1:12.5. Five different relative notch lengths, a/D = 0, 0.025, 0.075, 0.15, 0.30 were cut into the beams, except that the notch a/D = 0.025 was skipped for the two smallest sizes because it would have been much shorter than the coarse aggregate diameter da and thus essentially equivalent to the case a = a/D = 0. A total of 18 different geometries (family of beams) were tested. The present paper will use these data to analyze the effects of size, crack length and postpeak softening. All the notations from the preceding paper are retained. The size effect on structural strength is generally described in terms of the nominal strength rNu of structure, which is a parameter of maximum (or ultimate) load Pu having the dimension of stress. It is defined as
rNu ¼ cN
Pu bD
ð1Þ
where D = characteristic size (or dimension) of the specimen or structure, b = specimen thickness, and cN = constant dimensionless coefficient chosen for convenience. If there is no stress-singularity, cN can be chosen so as to make rNu represent the maximum stress in the structure. According to the classical theories of elasticity and plasticity, the nominal strength of ⇑ Corresponding author. Tel.: +1 847 491 4025; fax: +1 847 491 4011. 1
E-mail addresses: [email protected], [email protected] (C.G. Hoover), [email protected] (Z.P. Bazˇant). Present address: Massachusetts Institute of Technology, Cambridge, MA 02139, USA.
0013-7944/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engfracmech.2013.08.008
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structure is independent of structure size, which came to be known as the case of no size effect. This is what is still assumed in most design codes and standards for concrete. The size effect is defined as the dependence of rNu on D when geometrically similar structures are compared. For perfectly brittle geometrically similar structures with similar cracks, linear elastic fracture mechanics (LEFM) shows that rNu decreases with structure size as D1/2. A salient feature of quasibrittle materials, containing at maximum load geometrically similar large cracks, is a size effect that is transitional between plasticity and LEFM. The transitional behavior is caused by the fact that the FPZ size, equal to several inhomogeneity sizes, is not negligible compared to the cross section dimension. The size effect can best be shown in the plot of log rNu vs. log D. When geometrically similar beams contain a deep notch or a pre-existing stress-free crack of large depth a relative to structure size D, the size effect is of Type 2, which represents a smooth transition from a horizontal line for small sizes (corresponding to plasticity or strength theory) to an inclined asymptote of slope 1/2 for large-sizes (corresponding to LEFM). It is well described by the approximate size effect law (SEL) [4]:
Bf
0
t rNu ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð2Þ
1 þ D=D0
0
Here Bf t and the transitional structure size D0 are empirical parameters to be identified by optimum fitting of measured rN values covering a broad enough size range. Eq. (2) was derived [5] by simple energy release analysis and later by several other ways, especially by asymptotic matching based on the simple asymptotic power scaling laws for very large and very small D [6,4]. The coefficients of Eq. (2) have been shown [7] to be approximately related to the LEFM fracture characteristics, as follows:
rNu
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E0 Gf ¼ gða0 ÞD þ g 0 ða0 Þcf
ð3Þ
2 where a = a/D = relative crack length, a0 = a0/D = initial rate function of pffiffi value of a, g(a0) = k (a0) = dimensionless energy release linear elastic fracture mechanics (LEFM), kða0 Þ ¼ b ðDÞK I =P, where KI = stress intensity factor, P = load; g0 (a0) = dg(a0)/da0; Gf = initial fracture energy = area under the initial tangent of the cohesive softening stress-separation law; E0 = E = Young’s modulus for plane stress and E0 = E/(1 m2) for plane strain (where m = Poisson ratio), and cf = characteristic length which represents about a half of the FPZ length and is proportional to Irwin’s [10] characteristic length l0 = E Gf/f0 t2 [10]. Note that the LEFM function g(a0) or k(a0) embodies information on the crack length effect and on the effect of structure geometry (or shape). Thus Eq. (3) is actually a size-shape effect law for Type 2 failures. The Type 1 size effect is observed in structures failing as soon as the macro-crack initiates from a smooth surface. In this case, the asymptotic large-size slope of the log–log size effect curve is, according to the cohesive crack model or nonlocal damage model, a horizontal line, provided that the Weibull statistical size effect is unimportant, which is the case for the three-point bend beams and is one reason for their selection. Why is it unimportant?—Because the zone of high stress is here relatively concentrated, even in absence of a notch, and this greatly restricts the chance for a critical crack to form at different locations of random strength. However, when the zone of high stress is large, as in four-point bending and even more in direct tension specimens, the statistical size effect is not negligible, and becomes important for large-sizes. Then the large size asymptote for Type 1 size effect is, in the log–log plot, a downward inclined straight line of a slope n/m which is much milder than the slope 1/2 for LEFM [11,12]; here m = Weibull modulus (typically 24) and n = number of spatial dimensions of fracture growth (usually 2). According to the cohesive crack model, the small-size asymptote must be a horizontal line and, for medium sizes, the size effect is a transition between this and the large-size asymptote (power laws). This is the so-called ‘Type 1’ size effect. It has two forms depending on whether or not the asymptotic Weibull statistical size effect is significant: [6]:
rN ¼
fr1
"
ls ls þ D
rN ¼ fr1 1 þ
rn=m
rDb D þ lp
rDb þ D þ lp
1=r
#1=r ðwith WeibullÞ
ðwithout WeibullÞ
ð4Þ ð5Þ
Here fr,1, Db, lp, ls and r are empirical constants to be determined from tests; Db = depth of the boundary layer of cracking (roughly equal to the FPZ size), fr,1 = nominal strength for very large structures (assuming no Weibull statistical size effects), ls = statistical characteristic length, and lp = material characteristic length that is related to the maximum aggregate size. Introducing lp is necessary for mathematical reasons, as a means to satisfy the asymptotic requirement of the cohesive crack model to have a finite plastic limit for D ? 0 while ensuring the effect of lp to be negligible for D lp. Note that lp differs from Irwin’s characteristic length l0 ¼ EGf =ft2 [10], which characterizes the length of the fracture process in the direction of propagation. When the crack at failure is neither negligible nor large, the size effect trend is expected to be some sort of a transition between the Types 1 and 2. To clarify this transition and thus obtain a combined, or ‘universal’, size-shape effect law is one key objective of the comprehensive tests. In absence of test data, such a universal law was attempted purely theoretically in [13,14]. These formulations as well as a new ’universal’ size-shape effect law are discussed in [3].
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Better understanding of the effects of structure size on the load capacity will improve the design of concrete structures. The present analysis of the comprehensive fracture tests will advance this objective. 2. Determination of fracture parameters and fitting by size effect law One advantageous feature of the Type 2 size effect tests is that the measured nominal strength values make it possible determine the initial fracture energy Gf and the material characteristic length cf. Setting Eq. (2) equal to (3), one can express the fracture parameters in terms of the size effect parameters B, f0 t, D0 and the LEFM function g(a0); 0 2
Gf ¼
ðBf t Þ D0 gða0 Þ ; E0
cf ¼
D0 gða0 Þ g 0 ða0 Þ
ð6Þ
For notched three-point bend beams [15–19], 2
gða0 Þ ¼ k ða0 Þ;
kða0 Þ ¼
pffiffiffiffiffi pD=S ða0 Þ a0
ð7Þ
ð1 þ 2a0 Þð1 a0 Þ3=2
For function pS/D(a0), Pastor [20] developed the following interpolation formula applicable to an arbitrary ratio of beam depth D to span S (i.e., the distance between support resultants):
pS=D ða0 Þ ¼ p1 ða0 Þ þ ð4D=SÞ½p4 ða0 Þ p1 ða0 Þ
ð8Þ
p4 ða0 Þ ¼ 1:9 ða0 Þ½0:089 þ 0:603ð1 a0 Þ 0:441ð1 a0 Þ2 þ 1:223ð1 a0 Þ3
ð9Þ
where
p1 ða0 Þ ¼ 1:989 ða0 Þð1 a0 Þ½0:448 0:458ð1 a0 Þ þ 1:226ð1 a0 Þ2
ð10Þ
For the present tests S/D = 2.176, which gives g (a0) and g0 (a0); see Table 1. Eq. (2) was fitted to the mean of each family of beams, using the trust-region-reflective optimization algorithm [8,9], with a a0 = 0.3 and a0 = 0.15, Gf and cf were then calPN 2 culated using Eq. (6). This procedure was repeated by applying weights wi ¼ r2 i¼1 ri , representing the inverse variance i = of each beam family normalized with the sum of the inverse variances for all N families (N = 4), to the means of each beam family. In all cases, this weighting gave worse CoV of curve fit (defined as the root-mean-square of errors divided by data mean) than the unweighted means. The statistics of the unweighted fittings, Gf, cf and the CoV of curve fit, calculated from both relative notch depths are shown in Table (1). The fracture energy Gf from both relative notch depths are essentially identical and the cf-values are also very close, suggesting that this procedure is valid for determining Gf and cf for a0 P 0.15. The strengths determined from the tests of crack initiation specimens were fitted using Eq. (5), and the resulting parameters are: Db = 2.88 in. (73.2 mm), lp = 4.98 in. (126.6 mm), fr,1 = 765 psi (5.27 MPa) and r = 0.52. The CoV of fit is 1.91%. The crack initiation data was also fit by applying weights wi, but the CoV was higher than the fit without weights. Fig. 1 presents all the size effect plots of log rNu vs. log D, showing the reduction of structure strength with increasing size, for all initial relative notch depths a0. The circles are the individual data points and the X’s are the means of each family of data. The +’s label the beams that failed dynamically and also beam Ae01, which remained loaded in the machine for about an extra hour due to the oil pump overheating, and beam Da05 in which the notch terminated in a compact group of large aggregates. These data points were not included in the optimal fit of the Type 1 or Type 2 size effect law, because their peak loads are not deemed reliable. The families of beams corresponding to a0 = 0.3 and a0 = 0.15 also show the optimal fit for the Type 2 size effect law, including the horizontal plasticity asymptote for very small beam sizes and the inclined LEFM asymptote, approached for beams of very large size.
Table 1 Statistics of Type 2 size effect law fitting with no weights and the resulting Gf and cf for a0 = 0.3 and a0 = 0.15.
a0 = 0.3 0 Bf t
Mean Standard deviation CoV (%) Lower 95% conf. limit Upper 95% conf. limit g (a0) g0 (a0) Gf (N/m) cf (mm) CoV of curve fit (%)
(MPa)
3.98 0.09 2.31 3.6 4.36 0.94 4.83 51.9 28.0 1.33
a0 = 0.15 0
D0 (mm)
Bf t (MPa)
D0 (mm)
143.8 14.1 9.78 83.3 204.0
6.03 0.25 4.15 4.95 7.10 0.41 2.69 49.8 21.0 2.39
137.0 24.72 18.05 30.60 243.0
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284
0.7
0.8
0.65 0.55
log (σN)
0.75
Bft
0.6
0.45
0.6
0.4
0.55
0.35
0.5
= 0.3
1
1.5
2
2.5
0.4
3
1
1.5
2
2.5
3
1
0.9 0.85
2
0.8
2
0.9
1
1
0.75
log (σN)
= 0.15
0.45
0.25 0.2
1
0.65
1
0.3
2
0.7
2
0.5
Bft
0.8
0.7 0.65
0.7
0.6 0.6
0.55
= 0.075
0.5
= 0.025 0.5
0.45 0.4
1
1.5
2
2.5
0.4
3
1
1.5
2
log (D)
2.5
3
log (D)
1 0.95
log (σN)
0.9 0.85 0.8 0.75
=0
0.7 0.65
1
1.5
2
2.5
3
log (D) Fig. 1. Size effect curves for all notch depths.
Fig. 2 shows the reduction in nominal strength as a0 increases for each beam size. Each data set also contains a fitted polynomial, which is not meant to serve as a predictive equation, but merely just to show the trend of how the strength changes versus the relative initial notch length. 3. Determination of fracture energy and its comparisons Fracture properties determined from Load–Displacement Diagrams: The work, WF, required to break the specimen completely, can be calculated from the area W0 under the curve of load (P) versus the load-point displacement (d). Since no weight compensation technique was used during the tests, the correction mgd0 for the work of self-weight needs to be added to W0, where m the mass, g is the acceleration due to gravity and d0 is the displacement that corresponds to zero load. For the smaller specimens, this correction is minimal because the mass is small. However for the largest specimens, this correction is not insignificant. The total fracture energy GF, corresponding to the total area under the stress-separation curve of the cohesive crack model, can then be estimated as
C.G. Hoover, Z.P. Bazˇant / Engineering Fracture Mechanics 110 (2013) 281–289
7
285
7 6.5
6
6 5.5
log (σN)
5
5 4
4.5 4
3
3.5
2 1
3
Depth is 500 mm 0
0.05 0.1 0.15 0.2 0.25 0.3
2
8
9
7
8
0
0.05 0.1 0.15 0.2 0.25 0.3
7
6
log (σN)
Depth is 215 mm
2.5
6 5 5 4
4
3 2
0
Depth is 40 mm
3
Depth is 93 mm 0.05 0.1 0.15 0.2 0.25 0.3
2
0
0.05 0.1 0.15 0.2 0.25 0.3
Fig. 2. Crack length effect curves.
Fig. 3. Shifted load vs load point Disp with fitted extrapolation function.
Table 2 Comparison of Gf, GF and cf calculated from different methods.
a0
Type 2 SEL fit 0.3
Gf (N/m) GF (N/m) cf (mm) a
51.87 – 27.97
Area under Pd curve 0.15
49.78 – 20.99
0.3
Prediction Eq’s 0.15
Mean
CoV (%)
Mean
CoV (%)
46.07a 96.94 –
31.75a 16.88 –
50.68 111.09 –
24.1 20.72 –
37.45 [24] 93.63 [24] 2.83 [24]
If the two questionable values are removed from the analysis, then the mean Gf = 42.55 (N/m) and the CoV = 15.75%.
56.09 [26] 85.37 [25] 12.54 [26]
C.G. Hoover, Z.P. Bazˇant / Engineering Fracture Mechanics 110 (2013) 281–289
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Beam Aa01
10
0
0
0.2 Beam Aa05
10
5
0
0.1
Force (kN)
0.2 Beam Ba03
6
0
0.5 Beam Aa06
10
0
2
2 0
0.1 Beam Ca01
3
5
4
0
0.2 Beam Ba01
4 2 0
0.1
0.2 Beam Ba05
6
0
4
2
2 0.1
0
0.2 Beam Ca03
3 2
2
1
1
1
1
0.1
4
0.2 Beam Ca05
3
0
0
0.05
0.1 Beam Ca06
3
0
0
0
0.1
4
Beam Ca07
3 2
2
1
1
1
1
2
0.05
0.1
Beam Da01
1.5
0.5 0
0
0.05
Beam Da02
0.1
0
0
0.1
0.2
Beam Da03
1.5
0
1
0.5
0.5
0.5
0
0
0.1 Beam Da06
1.5
1
1
0.5
0.5 0
0.05
0.1
0
0.05 Beam Da07
2
1.5
0
0
0
0.1
0
0.2
0.1 Beam Ca04
0
0.1 Beam Ca08
0
0.05
0.1
Beam Da04
1.5
1
2
1 0
0.1
1
0
0.05 Beam Da05
2
0
1.5
1
0
0
0.1
Beam Ba06
3
2
0
0
4
2
0
Beam Ba02
3
2
0
0.2
4
2
0
0.1
6
4
0
0
6
2
0
0.1 Beam Ca02
3
0.1
4
0 Beam Ba04
0
0
6
0.2
6 4
0
0
8
4
0
0
Beam Aa04
10
5
5
0
Beam Aa03
10
5
5
0
Beam Aa02
10
0
0.05
0.1 Beam Da08
1.5 1 0.5
0
0.1
0
0
0.05
Vertical Displacement (mm) Fig. 4. Force vs. vertical displacement graphs for the entire collection of a0 = 0.3 beams.
0.1
C.G. Hoover, Z.P. Bazˇant / Engineering Fracture Mechanics 110 (2013) 281–289
GF ¼
WF bDð1 a0 Þ
287
ð11Þ
where WF = W0 + mgd0. This estimate was proposed for concrete (under the name of fictitious crack model) by Hillerborg et al. [21] (following similar proposals for ceramics [22,23]) and is enshrined in a RILEM international recommendation [27]. Eq. (11) is only approximate. The main source of error is that, near the notch tip (and also near the end of the ligament), the energy to create the crack is not the same as it is in stationary propagation (the stationarity is required for GF to be equal to the J-integral). Also, the measured coefficient of variation (CoV) of scatter of GF is only slightly larger than that of Gf. This observation apparently differs from the conclusion in [24], in which the CoV of scatter of GF was found to be more than twice as large, but that work dealt with the scatter of combined data from many different concretes while the present study deals with one concrete only. According to the RILEM recommendation [21], WF is the area under the complete diagram of load versus load-point displacement. The diagram must include the effects of self-weight. The vertical displacement of the beams relative to the testing bed were recorded by two LVDT’s, one on each side of every beam. The effect of self-weight was added and the compliances of the support fixtures were subtracted in calculating each of these load–displacement diagrams. Because the initial increase of slope of the diagram can be caused only by the closing of voids or gaps in the seating of supports, a straight elastic line was drawn through the linear portion of the rising load–displacement diagram and was then extended down to the x-axis. The entire diagram was then shifted to the left so that this elastic line would intersect the origin. All the diagrams for beams with a0 = 0.3 and a0 = 0.15 were generated in this manner. GF can be accurately determined only when the load softens down to zero. This is usually difficult to achieve because either the test would have to run for a very long time or the measured reaction force would become indiscernible from the inherent noise in the testing machine. Since the measured softening curve does not extend to zero stress, it must be extrapolated. The extrapolation is assumed to be an exponential decay function Y = AeBx, which is integrable up to 1; Y = force, x = vertical displacement of beam, and A, B = constants to be calibrated by fitting the lower portion of the softening load–displacement diagram. A snapback (i.e., postpeak response in which the slope reverts to positive) was observed in many of the diagrams of load versus load-point displacement; see the right curve in Fig. 3. The total area enclosed between such diagram and the displacement axis is the dissipated energy. The area does not change by conversion to a diagram in which the elastic displacement is subtracted (or filtered out) from the total displacement, which may be more convenient for calculation (Fig. 3). This area has been accurately evaluated using the trapezoidal rule. Eq. (11) has then been used to calculate GF for each beam; see Table 2. The initial fracture energy Gf has also been determined from the diagram of the load versus the load-point displacement. Once the diagram of the load versus the LVDT displacement has been re-drawn, the initial steeply descending nearly straight part of the softening diagram was approximated in the least-square sense by the thick dashed straight line emanating from the initial peak, as shown in Fig. 4. The shaded area under this line, augmented by the effect of self-weight and divided by the average cross sectional area of the beam, represents the initial fracture energy, Gf. The overall mean and CoV for Gf and GF have then been calculated for all the beams with a0 = 0.30 and a0 = 0.15. The results are shown in Table 2. The statistics are significantly altered by beams Aa05 and Da07 Fig. 4, which are outliers exhibiting a much milder softening slope. They yield a Gf that is almost the double of the Gf value calculated from other beams of the same geometry. These two beams significantly increase the CoV and to a lesser extent also the mean Gf. However, no reason is seen for excluding these two beams as defective outliers, and so they have been retained in the statistics. 4. Summary and conclusions 1. Stable post-peak load–deflection diagrams with softening down to 90% and in some cases by more than 95% of peak have been achieved. This allows a reliable estimation of the entire tail of the load–deflection diagram. 2. The nominal strengths rN of beams with relative notch depths a = 0.30 and 0.15 can each be fitted closely by Bazˇant’s Type 2 size effect law (Eqs. (2) and (3)). 3. The nonlinear regressions of the logarithm of the mean of measured rN for each size versus logD, conducted separately for 0.30 and 0.15, give relatively small CoV (root-mean-square of errors divided by data mean), as low as 2.39% as seen in Table 1. A nonlinear regression of these results, run jointly for a = 0.30 and 0.15 (with different g0 and g 00 for each), gives a CoV of 4.0%, Gf = 0.28 lb/in. (Gf = 49.6 N/m) and cf = 0.88 in. (22.34 mm). It has been checked that applying weights PN 2 wi ¼ r2 i¼1 ri , gives Gf = 0.32 lb/in. (56.25 N/m) and cf = 1.17 in. (29.79 mm) but a CoV of 5.45%. i = 4. The measured data for beams with no notch (a = 0) agree well with the Type 1 size effect law (Eqs. (4 and 5)). 5. The values of initial fracture energy Gf calculated from the area under the initial steep segment of the measured cohesive softening curve match closely the Gf values obtained from the size effect, both for a = 0.30 and 0.15. This validates the concept of a bilinear softening law. 6. For the present concrete, the measured total-to-initial fracture energy ratio, GF/Gf, is about 2. 7. The measured data on nominal strength rN versus loga document a smooth transition from deep notches with a = 0.30 to the case of zero notch. This observed transition deviates significantly from the curve assumed by Hu and Duan. [28–33], as shown in [34,35].
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8. A notch of relative depth 0.15 used to be considered too shallow for fracture tests, but here the results are almost as good as for depth 0.30. 9. For the present concrete, the empirical equations of Becq-Giraudon [24] and Bazˇant and Oh [26] give better predictions of GF and Gf than does the CEB equation [25].
Acknowledgments Financial support from the U.S. Department of Transportation, provided through Grant 20778 from the Infrastructure Technology Institute of Northwestern University, is gratefully appreciated. Additional support for analysis was provided by NSF grant CMMI-1129449 to Northwestern University. Appendix A. Empirical prediction of fracture parameters from concrete strength and composition There has been a great desire to predict the fracture energy and characteristic length from concrete strength and mix parameters, but only very crude empirical formulas have been identified, as follows:
Gf ¼ a0
0:46 0:22 0:3 fc0 da w 1þ ¼ 0:21 lb=in: ð37:45 N=mÞ ½24 c 0:051 11:27
GF ¼ 2:5a0
0:46 0:22 0:3 fc0 da w 1þ ¼ 0:53 lb=in: ð93:63 N=mÞ ½24 c 0:051 11:27
ð12Þ
"
cf ¼ exp c0
0:019 0:72 0:2 # fc0 da w ¼ 0:11 in: ð2:83 mmÞ ½24 1þ c 0:022 11:27
ð13Þ
ð14Þ
2 da ¼ 0:32 lb=in: ð56:09 N=mÞ ½26 Gf ¼ 2:72 þ 0:0214ft0 ft0 E
ð15Þ
cf ¼ 1:811 þ 0:0143ft0 ¼ 0:49 in: ð12:5 mmÞ ½26
ð16Þ
f 0 0:7 2 c GF ¼ 0:0469da 0:5da þ 26 ¼ 0:49 lb=in: ð85:37 N=mÞ ½25 10
ð17Þ
where w/c is the water–cement ratio by weight, fc0 and ft0 are the compressive strength and tensile strength of the concrete, da is the coarse aggregate diameter, c0 = a = 1 and E is Young’s modulus. Although a proper statistical evaluation of these formulas requires considering many different concretes (as done in [24]), it may be of interest to check the error of these formulas for the present concrete. The results, given in Table 2, show that the estimates are all quite poor, although the Gf from [26] and GF from [24] are relatively best. References [1] Hoover CG, Bazˇant ZP, Vorel J, Wendner R, Hubler MH. Comprehensive concrete fracture tests: description and results. Engng Fract Mech 2013. http:// dx.doi.org/10.1016/j.engfracmech.2013.08.007. [2] Hoover CG, Bazˇant ZP, Wendner R, Vorel J, Hubler MH, Kim K, et al. Experimental investigation of transitional size effect and crack length effect in concrete fracture. In: Life-Cycle and sustainability of civil infrastructure systems: Proceedings of the 3rd International Symposium on Life-Cycle Civil Engineering (IALCCE), October 3–6, Hofburg Palace: Vienna Austria; 2012 [in press]. [3] Hoover CG, Bazˇant ZP. Universal size-shape effect law based on comprehensive concrete fracture tests. J Engng Mech 2012, [in press]. [4] Bazˇant ZP, Planas J. Fracture and size effect in concrete and other quasibrittle materials. CRC Press; 1998. [5] Bazˇant ZP. Size effect in blunt fracture: concrete, rock, metal. ASCE J Engng Mech 1984;110(4):518–35. [6] Bazˇant ZP. Scaling of structural strength. Massachusetts: Elsevier Butterworth-Heinemann; 2005. [7] Bazˇant ZP, Kazemi MT. Size dependence of concrete fracture energy determined by RILEM work-of-fracture method. Int J Fracture 1991;51:121–38. [8] Coleman TF, Li Y. On the convergence of reflective Newton methods for large-scale nonlinear minimization subject to bounds. Math Program 1994;67(2):189–224. [9] Coleman TF, Li Y. An interior trust region approach for nonlinear minimization subject to bounds. SIAM J Optim 1996;6:418–45. [10] Irwin GR. Fracture. Handbuch Phys 1958;6:551–90 (S. FlÃgge, ed., Springer, Berlin). [11] Weibull W. The phenomenon of rupture in solids proc. Roy Swed Inst Engng Res 1939;153:1–55. Ingenioersvetenskaps Akad. Handl., Stockholm. [12] Weibull W. A Statistical Distribution Function of Wide Applicability. J. Appl. Mech 1954;18:293–7. [13] Bazˇant ZP, Yu Q. Size effect in concrete specimens and structures: new problems and progress. fracture mechanics of concrete structures. In: Li VC, Leung KY, Willam KJ, Billington SL, editors. Proc., FraMCoS-5, 5th int. conf. on fracture mech. of concrete and concr. structures, Vail, Colo., vol. 1, IAFraMCoS; 2004. p. 153–62. [14] Bazˇant ZP, Yu Q. Universal size effect law and effect of crack depth on quasi-brittle structure strength. J Engng Mech 2009;135(2):78–84. [15] Tada H, Paris PC, Irwin GR. The stress analysis of cracks handbook. 2nd ed. MO: Paris Productions Inc.; 1985. [16] Srawley JE. Wide range stress intensity factor expressions for ASTM e 399 standard fracture toughness specimens. Int J Fracture 1976;12:475–6. [17] Murakami Y. Stress intensity factors handbook. New York: Pergamon; 1987. [18] Broeck D. The practical use of fracture mechanics. Dordrecht, The Netherlands: Kluwer; 1988.
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[19] Kanninen MF, Popelar CH. Advanced fracture mechanics. New York: Oxford University Press; 1985. [20] Elices M, Pastor JY, Guinea G. Nueva expresión del factor de intensidad de tensiones para la probeta de flexión en tres puntos. Anal Mecán Fract 1995;12:85–90. [21] Hillerborg A. The theoretical basis of a method to determine the fracture GF of concrete. Mater Struct 1985;18:291–6. [22] Nakayama J. Direct measurement of fracture energies of brittle heterogeneous material. J Am Ceram Soc 1965;48(11):583. [23] Tattersall HG, Tappin G. The work of fracture and its measurement in metals ceramics and other materials. J Mater Sci 1996;1(3):296–301. [24] Bazˇant ZP, Becq-Giraudon E. Statistical prediction of fracture parameters of concrete and implications for choice of testing standard. Cem Concr Res 2002;32(4):529–56. [25] Comite Euro-International du Beton CEB-FIP Model Code; 1991. [26] Bazˇant ZP, Oh BH. Crack band theory for fracture of concrete. Mater Struct 1983;16:155–77. [27] RILEM. Determination of the fracture energy of mortar and concrete by means of three-point bend tests on notched beams. Mater Struct 1985;18:285– 90. [28] Duan K, Hu X, Wittmann F. Scaling of quasi-brittle fracture: boundary and size effect. Mech Mater 2006;38:128–41. [29] Hu X. An asymptotic approach to size effect on fracture toughness and fracture energy of composites. Engng Fract Mech 2002;69:555–64. [30] Hu X, Duan K. Size effect: influence of proximity of fracture process zone to specimen boundary. Engng Fract Mech 2007;74:1093–100. [31] Hu X, Duan K. Size effect and quasi-brittle fracture: the role of FPZ. Int J Fract 2008;154:3–14. [32] Hu X, Duan K. Mechanism behind the size effect phenomenon. J Engng Mech 2010;136(1):60–8. [33] Hu X, Wittmann F. Size effect on toughness induced by crack close to free surface. Engng Fract Mech 2000;65:209–21. [34] Yu Q, Le JL, Hoover CG, Bazˇant ZP. Problems with Hu-Duan boundary effect model and its comparison to size-shape effect law for quasibrittle fracture. J Engng Mech 2010;136(1):40–50. [35] Hoover CG, Bazˇant ZP. Comparison of Hu-Duan Boundary Effect Model to Size-Shape Effect Law for Quasibrittle Fracture Based on New Comprehensive Fracture Tests. J. Engng. Mech, [in press].
Contents lists available at ScienceDirect
Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Comprehensive concrete fracture tests: Size effects of Types 1 & 2, crack length effect and postpeak Christian G. Hoover a,1, Zdeneˇk P. Bazˇant b,⇑ a
Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA McCormick Institute Professor and W.P. Murphy Professor of Civil Engineering and Materials Science, Northwestern University, 2145 Sheridan Road, CEE/A135, Evanston, IL 60208, USA b
a r t i c l e
i n f o
Article history: Received 7 February 2013 Received in revised form 25 July 2013 Accepted 3 August 2013 Available online 23 August 2013 Keywords: Cohesive crack model Scaling of strength Fracture testing Size effect law Flexural strength
a b s t r a c t The comprehensive fracture tests of notched and unnotched beams presented in the preceding paper are evaluated to clarify the size effects and fracture energy dissipation. The test results for beams with notches of various lengths agree closely with Bazˇant’s Type 2 size effect law and yield the value of the initial fracture energy Gf. Since nearly complete post-peak softening load–displacement curves have been obtained, the total energy dissipation by fracture can be accurately evaluated and used to determine the RILEM total fracture energy GF. For the present concrete, GF/Gf 2. The transition between the Type 1 and 2 size effects are determined and is found to be very different from that assumed in Hu and Duan’s boundary effect model. Determination of a universal size effect law fitting the beam strengths for all the sizes and all notch depths is relegated to a subsequent paper. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction and review of size effect and crack length effect The preceding paper [1] presented an introduction to the problem and reported comprehensive test data for fracture of concrete. The experimental program, also described in [1,2], consisted of 128 three-point bend beams with 4 different depths of 1.58, 3.66, 8.47 and 19.69 in. (40, 93, 215 and 500 mm), corresponding to a size range of 1:12.5. Five different relative notch lengths, a/D = 0, 0.025, 0.075, 0.15, 0.30 were cut into the beams, except that the notch a/D = 0.025 was skipped for the two smallest sizes because it would have been much shorter than the coarse aggregate diameter da and thus essentially equivalent to the case a = a/D = 0. A total of 18 different geometries (family of beams) were tested. The present paper will use these data to analyze the effects of size, crack length and postpeak softening. All the notations from the preceding paper are retained. The size effect on structural strength is generally described in terms of the nominal strength rNu of structure, which is a parameter of maximum (or ultimate) load Pu having the dimension of stress. It is defined as
rNu ¼ cN
Pu bD
ð1Þ
where D = characteristic size (or dimension) of the specimen or structure, b = specimen thickness, and cN = constant dimensionless coefficient chosen for convenience. If there is no stress-singularity, cN can be chosen so as to make rNu represent the maximum stress in the structure. According to the classical theories of elasticity and plasticity, the nominal strength of ⇑ Corresponding author. Tel.: +1 847 491 4025; fax: +1 847 491 4011. 1
E-mail addresses: [email protected], [email protected] (C.G. Hoover), [email protected] (Z.P. Bazˇant). Present address: Massachusetts Institute of Technology, Cambridge, MA 02139, USA.
0013-7944/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engfracmech.2013.08.008
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282
structure is independent of structure size, which came to be known as the case of no size effect. This is what is still assumed in most design codes and standards for concrete. The size effect is defined as the dependence of rNu on D when geometrically similar structures are compared. For perfectly brittle geometrically similar structures with similar cracks, linear elastic fracture mechanics (LEFM) shows that rNu decreases with structure size as D1/2. A salient feature of quasibrittle materials, containing at maximum load geometrically similar large cracks, is a size effect that is transitional between plasticity and LEFM. The transitional behavior is caused by the fact that the FPZ size, equal to several inhomogeneity sizes, is not negligible compared to the cross section dimension. The size effect can best be shown in the plot of log rNu vs. log D. When geometrically similar beams contain a deep notch or a pre-existing stress-free crack of large depth a relative to structure size D, the size effect is of Type 2, which represents a smooth transition from a horizontal line for small sizes (corresponding to plasticity or strength theory) to an inclined asymptote of slope 1/2 for large-sizes (corresponding to LEFM). It is well described by the approximate size effect law (SEL) [4]:
Bf
0
t rNu ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð2Þ
1 þ D=D0
0
Here Bf t and the transitional structure size D0 are empirical parameters to be identified by optimum fitting of measured rN values covering a broad enough size range. Eq. (2) was derived [5] by simple energy release analysis and later by several other ways, especially by asymptotic matching based on the simple asymptotic power scaling laws for very large and very small D [6,4]. The coefficients of Eq. (2) have been shown [7] to be approximately related to the LEFM fracture characteristics, as follows:
rNu
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E0 Gf ¼ gða0 ÞD þ g 0 ða0 Þcf
ð3Þ
2 where a = a/D = relative crack length, a0 = a0/D = initial rate function of pffiffi value of a, g(a0) = k (a0) = dimensionless energy release linear elastic fracture mechanics (LEFM), kða0 Þ ¼ b ðDÞK I =P, where KI = stress intensity factor, P = load; g0 (a0) = dg(a0)/da0; Gf = initial fracture energy = area under the initial tangent of the cohesive softening stress-separation law; E0 = E = Young’s modulus for plane stress and E0 = E/(1 m2) for plane strain (where m = Poisson ratio), and cf = characteristic length which represents about a half of the FPZ length and is proportional to Irwin’s [10] characteristic length l0 = E Gf/f0 t2 [10]. Note that the LEFM function g(a0) or k(a0) embodies information on the crack length effect and on the effect of structure geometry (or shape). Thus Eq. (3) is actually a size-shape effect law for Type 2 failures. The Type 1 size effect is observed in structures failing as soon as the macro-crack initiates from a smooth surface. In this case, the asymptotic large-size slope of the log–log size effect curve is, according to the cohesive crack model or nonlocal damage model, a horizontal line, provided that the Weibull statistical size effect is unimportant, which is the case for the three-point bend beams and is one reason for their selection. Why is it unimportant?—Because the zone of high stress is here relatively concentrated, even in absence of a notch, and this greatly restricts the chance for a critical crack to form at different locations of random strength. However, when the zone of high stress is large, as in four-point bending and even more in direct tension specimens, the statistical size effect is not negligible, and becomes important for large-sizes. Then the large size asymptote for Type 1 size effect is, in the log–log plot, a downward inclined straight line of a slope n/m which is much milder than the slope 1/2 for LEFM [11,12]; here m = Weibull modulus (typically 24) and n = number of spatial dimensions of fracture growth (usually 2). According to the cohesive crack model, the small-size asymptote must be a horizontal line and, for medium sizes, the size effect is a transition between this and the large-size asymptote (power laws). This is the so-called ‘Type 1’ size effect. It has two forms depending on whether or not the asymptotic Weibull statistical size effect is significant: [6]:
rN ¼
fr1
"
ls ls þ D
rN ¼ fr1 1 þ
rn=m
rDb D þ lp
rDb þ D þ lp
1=r
#1=r ðwith WeibullÞ
ðwithout WeibullÞ
ð4Þ ð5Þ
Here fr,1, Db, lp, ls and r are empirical constants to be determined from tests; Db = depth of the boundary layer of cracking (roughly equal to the FPZ size), fr,1 = nominal strength for very large structures (assuming no Weibull statistical size effects), ls = statistical characteristic length, and lp = material characteristic length that is related to the maximum aggregate size. Introducing lp is necessary for mathematical reasons, as a means to satisfy the asymptotic requirement of the cohesive crack model to have a finite plastic limit for D ? 0 while ensuring the effect of lp to be negligible for D lp. Note that lp differs from Irwin’s characteristic length l0 ¼ EGf =ft2 [10], which characterizes the length of the fracture process in the direction of propagation. When the crack at failure is neither negligible nor large, the size effect trend is expected to be some sort of a transition between the Types 1 and 2. To clarify this transition and thus obtain a combined, or ‘universal’, size-shape effect law is one key objective of the comprehensive tests. In absence of test data, such a universal law was attempted purely theoretically in [13,14]. These formulations as well as a new ’universal’ size-shape effect law are discussed in [3].
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Better understanding of the effects of structure size on the load capacity will improve the design of concrete structures. The present analysis of the comprehensive fracture tests will advance this objective. 2. Determination of fracture parameters and fitting by size effect law One advantageous feature of the Type 2 size effect tests is that the measured nominal strength values make it possible determine the initial fracture energy Gf and the material characteristic length cf. Setting Eq. (2) equal to (3), one can express the fracture parameters in terms of the size effect parameters B, f0 t, D0 and the LEFM function g(a0); 0 2
Gf ¼
ðBf t Þ D0 gða0 Þ ; E0
cf ¼
D0 gða0 Þ g 0 ða0 Þ
ð6Þ
For notched three-point bend beams [15–19], 2
gða0 Þ ¼ k ða0 Þ;
kða0 Þ ¼
pffiffiffiffiffi pD=S ða0 Þ a0
ð7Þ
ð1 þ 2a0 Þð1 a0 Þ3=2
For function pS/D(a0), Pastor [20] developed the following interpolation formula applicable to an arbitrary ratio of beam depth D to span S (i.e., the distance between support resultants):
pS=D ða0 Þ ¼ p1 ða0 Þ þ ð4D=SÞ½p4 ða0 Þ p1 ða0 Þ
ð8Þ
p4 ða0 Þ ¼ 1:9 ða0 Þ½0:089 þ 0:603ð1 a0 Þ 0:441ð1 a0 Þ2 þ 1:223ð1 a0 Þ3
ð9Þ
where
p1 ða0 Þ ¼ 1:989 ða0 Þð1 a0 Þ½0:448 0:458ð1 a0 Þ þ 1:226ð1 a0 Þ2
ð10Þ
For the present tests S/D = 2.176, which gives g (a0) and g0 (a0); see Table 1. Eq. (2) was fitted to the mean of each family of beams, using the trust-region-reflective optimization algorithm [8,9], with a a0 = 0.3 and a0 = 0.15, Gf and cf were then calPN 2 culated using Eq. (6). This procedure was repeated by applying weights wi ¼ r2 i¼1 ri , representing the inverse variance i = of each beam family normalized with the sum of the inverse variances for all N families (N = 4), to the means of each beam family. In all cases, this weighting gave worse CoV of curve fit (defined as the root-mean-square of errors divided by data mean) than the unweighted means. The statistics of the unweighted fittings, Gf, cf and the CoV of curve fit, calculated from both relative notch depths are shown in Table (1). The fracture energy Gf from both relative notch depths are essentially identical and the cf-values are also very close, suggesting that this procedure is valid for determining Gf and cf for a0 P 0.15. The strengths determined from the tests of crack initiation specimens were fitted using Eq. (5), and the resulting parameters are: Db = 2.88 in. (73.2 mm), lp = 4.98 in. (126.6 mm), fr,1 = 765 psi (5.27 MPa) and r = 0.52. The CoV of fit is 1.91%. The crack initiation data was also fit by applying weights wi, but the CoV was higher than the fit without weights. Fig. 1 presents all the size effect plots of log rNu vs. log D, showing the reduction of structure strength with increasing size, for all initial relative notch depths a0. The circles are the individual data points and the X’s are the means of each family of data. The +’s label the beams that failed dynamically and also beam Ae01, which remained loaded in the machine for about an extra hour due to the oil pump overheating, and beam Da05 in which the notch terminated in a compact group of large aggregates. These data points were not included in the optimal fit of the Type 1 or Type 2 size effect law, because their peak loads are not deemed reliable. The families of beams corresponding to a0 = 0.3 and a0 = 0.15 also show the optimal fit for the Type 2 size effect law, including the horizontal plasticity asymptote for very small beam sizes and the inclined LEFM asymptote, approached for beams of very large size.
Table 1 Statistics of Type 2 size effect law fitting with no weights and the resulting Gf and cf for a0 = 0.3 and a0 = 0.15.
a0 = 0.3 0 Bf t
Mean Standard deviation CoV (%) Lower 95% conf. limit Upper 95% conf. limit g (a0) g0 (a0) Gf (N/m) cf (mm) CoV of curve fit (%)
(MPa)
3.98 0.09 2.31 3.6 4.36 0.94 4.83 51.9 28.0 1.33
a0 = 0.15 0
D0 (mm)
Bf t (MPa)
D0 (mm)
143.8 14.1 9.78 83.3 204.0
6.03 0.25 4.15 4.95 7.10 0.41 2.69 49.8 21.0 2.39
137.0 24.72 18.05 30.60 243.0
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284
0.7
0.8
0.65 0.55
log (σN)
0.75
Bft
0.6
0.45
0.6
0.4
0.55
0.35
0.5
= 0.3
1
1.5
2
2.5
0.4
3
1
1.5
2
2.5
3
1
0.9 0.85
2
0.8
2
0.9
1
1
0.75
log (σN)
= 0.15
0.45
0.25 0.2
1
0.65
1
0.3
2
0.7
2
0.5
Bft
0.8
0.7 0.65
0.7
0.6 0.6
0.55
= 0.075
0.5
= 0.025 0.5
0.45 0.4
1
1.5
2
2.5
0.4
3
1
1.5
2
log (D)
2.5
3
log (D)
1 0.95
log (σN)
0.9 0.85 0.8 0.75
=0
0.7 0.65
1
1.5
2
2.5
3
log (D) Fig. 1. Size effect curves for all notch depths.
Fig. 2 shows the reduction in nominal strength as a0 increases for each beam size. Each data set also contains a fitted polynomial, which is not meant to serve as a predictive equation, but merely just to show the trend of how the strength changes versus the relative initial notch length. 3. Determination of fracture energy and its comparisons Fracture properties determined from Load–Displacement Diagrams: The work, WF, required to break the specimen completely, can be calculated from the area W0 under the curve of load (P) versus the load-point displacement (d). Since no weight compensation technique was used during the tests, the correction mgd0 for the work of self-weight needs to be added to W0, where m the mass, g is the acceleration due to gravity and d0 is the displacement that corresponds to zero load. For the smaller specimens, this correction is minimal because the mass is small. However for the largest specimens, this correction is not insignificant. The total fracture energy GF, corresponding to the total area under the stress-separation curve of the cohesive crack model, can then be estimated as
C.G. Hoover, Z.P. Bazˇant / Engineering Fracture Mechanics 110 (2013) 281–289
7
285
7 6.5
6
6 5.5
log (σN)
5
5 4
4.5 4
3
3.5
2 1
3
Depth is 500 mm 0
0.05 0.1 0.15 0.2 0.25 0.3
2
8
9
7
8
0
0.05 0.1 0.15 0.2 0.25 0.3
7
6
log (σN)
Depth is 215 mm
2.5
6 5 5 4
4
3 2
0
Depth is 40 mm
3
Depth is 93 mm 0.05 0.1 0.15 0.2 0.25 0.3
2
0
0.05 0.1 0.15 0.2 0.25 0.3
Fig. 2. Crack length effect curves.
Fig. 3. Shifted load vs load point Disp with fitted extrapolation function.
Table 2 Comparison of Gf, GF and cf calculated from different methods.
a0
Type 2 SEL fit 0.3
Gf (N/m) GF (N/m) cf (mm) a
51.87 – 27.97
Area under Pd curve 0.15
49.78 – 20.99
0.3
Prediction Eq’s 0.15
Mean
CoV (%)
Mean
CoV (%)
46.07a 96.94 –
31.75a 16.88 –
50.68 111.09 –
24.1 20.72 –
37.45 [24] 93.63 [24] 2.83 [24]
If the two questionable values are removed from the analysis, then the mean Gf = 42.55 (N/m) and the CoV = 15.75%.
56.09 [26] 85.37 [25] 12.54 [26]
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286
Beam Aa01
10
0
0
0.2 Beam Aa05
10
5
0
0.1
Force (kN)
0.2 Beam Ba03
6
0
0.5 Beam Aa06
10
0
2
2 0
0.1 Beam Ca01
3
5
4
0
0.2 Beam Ba01
4 2 0
0.1
0.2 Beam Ba05
6
0
4
2
2 0.1
0
0.2 Beam Ca03
3 2
2
1
1
1
1
0.1
4
0.2 Beam Ca05
3
0
0
0.05
0.1 Beam Ca06
3
0
0
0
0.1
4
Beam Ca07
3 2
2
1
1
1
1
2
0.05
0.1
Beam Da01
1.5
0.5 0
0
0.05
Beam Da02
0.1
0
0
0.1
0.2
Beam Da03
1.5
0
1
0.5
0.5
0.5
0
0
0.1 Beam Da06
1.5
1
1
0.5
0.5 0
0.05
0.1
0
0.05 Beam Da07
2
1.5
0
0
0
0.1
0
0.2
0.1 Beam Ca04
0
0.1 Beam Ca08
0
0.05
0.1
Beam Da04
1.5
1
2
1 0
0.1
1
0
0.05 Beam Da05
2
0
1.5
1
0
0
0.1
Beam Ba06
3
2
0
0
4
2
0
Beam Ba02
3
2
0
0.2
4
2
0
0.1
6
4
0
0
6
2
0
0.1 Beam Ca02
3
0.1
4
0 Beam Ba04
0
0
6
0.2
6 4
0
0
8
4
0
0
Beam Aa04
10
5
5
0
Beam Aa03
10
5
5
0
Beam Aa02
10
0
0.05
0.1 Beam Da08
1.5 1 0.5
0
0.1
0
0
0.05
Vertical Displacement (mm) Fig. 4. Force vs. vertical displacement graphs for the entire collection of a0 = 0.3 beams.
0.1
C.G. Hoover, Z.P. Bazˇant / Engineering Fracture Mechanics 110 (2013) 281–289
GF ¼
WF bDð1 a0 Þ
287
ð11Þ
where WF = W0 + mgd0. This estimate was proposed for concrete (under the name of fictitious crack model) by Hillerborg et al. [21] (following similar proposals for ceramics [22,23]) and is enshrined in a RILEM international recommendation [27]. Eq. (11) is only approximate. The main source of error is that, near the notch tip (and also near the end of the ligament), the energy to create the crack is not the same as it is in stationary propagation (the stationarity is required for GF to be equal to the J-integral). Also, the measured coefficient of variation (CoV) of scatter of GF is only slightly larger than that of Gf. This observation apparently differs from the conclusion in [24], in which the CoV of scatter of GF was found to be more than twice as large, but that work dealt with the scatter of combined data from many different concretes while the present study deals with one concrete only. According to the RILEM recommendation [21], WF is the area under the complete diagram of load versus load-point displacement. The diagram must include the effects of self-weight. The vertical displacement of the beams relative to the testing bed were recorded by two LVDT’s, one on each side of every beam. The effect of self-weight was added and the compliances of the support fixtures were subtracted in calculating each of these load–displacement diagrams. Because the initial increase of slope of the diagram can be caused only by the closing of voids or gaps in the seating of supports, a straight elastic line was drawn through the linear portion of the rising load–displacement diagram and was then extended down to the x-axis. The entire diagram was then shifted to the left so that this elastic line would intersect the origin. All the diagrams for beams with a0 = 0.3 and a0 = 0.15 were generated in this manner. GF can be accurately determined only when the load softens down to zero. This is usually difficult to achieve because either the test would have to run for a very long time or the measured reaction force would become indiscernible from the inherent noise in the testing machine. Since the measured softening curve does not extend to zero stress, it must be extrapolated. The extrapolation is assumed to be an exponential decay function Y = AeBx, which is integrable up to 1; Y = force, x = vertical displacement of beam, and A, B = constants to be calibrated by fitting the lower portion of the softening load–displacement diagram. A snapback (i.e., postpeak response in which the slope reverts to positive) was observed in many of the diagrams of load versus load-point displacement; see the right curve in Fig. 3. The total area enclosed between such diagram and the displacement axis is the dissipated energy. The area does not change by conversion to a diagram in which the elastic displacement is subtracted (or filtered out) from the total displacement, which may be more convenient for calculation (Fig. 3). This area has been accurately evaluated using the trapezoidal rule. Eq. (11) has then been used to calculate GF for each beam; see Table 2. The initial fracture energy Gf has also been determined from the diagram of the load versus the load-point displacement. Once the diagram of the load versus the LVDT displacement has been re-drawn, the initial steeply descending nearly straight part of the softening diagram was approximated in the least-square sense by the thick dashed straight line emanating from the initial peak, as shown in Fig. 4. The shaded area under this line, augmented by the effect of self-weight and divided by the average cross sectional area of the beam, represents the initial fracture energy, Gf. The overall mean and CoV for Gf and GF have then been calculated for all the beams with a0 = 0.30 and a0 = 0.15. The results are shown in Table 2. The statistics are significantly altered by beams Aa05 and Da07 Fig. 4, which are outliers exhibiting a much milder softening slope. They yield a Gf that is almost the double of the Gf value calculated from other beams of the same geometry. These two beams significantly increase the CoV and to a lesser extent also the mean Gf. However, no reason is seen for excluding these two beams as defective outliers, and so they have been retained in the statistics. 4. Summary and conclusions 1. Stable post-peak load–deflection diagrams with softening down to 90% and in some cases by more than 95% of peak have been achieved. This allows a reliable estimation of the entire tail of the load–deflection diagram. 2. The nominal strengths rN of beams with relative notch depths a = 0.30 and 0.15 can each be fitted closely by Bazˇant’s Type 2 size effect law (Eqs. (2) and (3)). 3. The nonlinear regressions of the logarithm of the mean of measured rN for each size versus logD, conducted separately for 0.30 and 0.15, give relatively small CoV (root-mean-square of errors divided by data mean), as low as 2.39% as seen in Table 1. A nonlinear regression of these results, run jointly for a = 0.30 and 0.15 (with different g0 and g 00 for each), gives a CoV of 4.0%, Gf = 0.28 lb/in. (Gf = 49.6 N/m) and cf = 0.88 in. (22.34 mm). It has been checked that applying weights PN 2 wi ¼ r2 i¼1 ri , gives Gf = 0.32 lb/in. (56.25 N/m) and cf = 1.17 in. (29.79 mm) but a CoV of 5.45%. i = 4. The measured data for beams with no notch (a = 0) agree well with the Type 1 size effect law (Eqs. (4 and 5)). 5. The values of initial fracture energy Gf calculated from the area under the initial steep segment of the measured cohesive softening curve match closely the Gf values obtained from the size effect, both for a = 0.30 and 0.15. This validates the concept of a bilinear softening law. 6. For the present concrete, the measured total-to-initial fracture energy ratio, GF/Gf, is about 2. 7. The measured data on nominal strength rN versus loga document a smooth transition from deep notches with a = 0.30 to the case of zero notch. This observed transition deviates significantly from the curve assumed by Hu and Duan. [28–33], as shown in [34,35].
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8. A notch of relative depth 0.15 used to be considered too shallow for fracture tests, but here the results are almost as good as for depth 0.30. 9. For the present concrete, the empirical equations of Becq-Giraudon [24] and Bazˇant and Oh [26] give better predictions of GF and Gf than does the CEB equation [25].
Acknowledgments Financial support from the U.S. Department of Transportation, provided through Grant 20778 from the Infrastructure Technology Institute of Northwestern University, is gratefully appreciated. Additional support for analysis was provided by NSF grant CMMI-1129449 to Northwestern University. Appendix A. Empirical prediction of fracture parameters from concrete strength and composition There has been a great desire to predict the fracture energy and characteristic length from concrete strength and mix parameters, but only very crude empirical formulas have been identified, as follows:
Gf ¼ a0
0:46 0:22 0:3 fc0 da w 1þ ¼ 0:21 lb=in: ð37:45 N=mÞ ½24 c 0:051 11:27
GF ¼ 2:5a0
0:46 0:22 0:3 fc0 da w 1þ ¼ 0:53 lb=in: ð93:63 N=mÞ ½24 c 0:051 11:27
ð12Þ
"
cf ¼ exp c0
0:019 0:72 0:2 # fc0 da w ¼ 0:11 in: ð2:83 mmÞ ½24 1þ c 0:022 11:27
ð13Þ
ð14Þ
2 da ¼ 0:32 lb=in: ð56:09 N=mÞ ½26 Gf ¼ 2:72 þ 0:0214ft0 ft0 E
ð15Þ
cf ¼ 1:811 þ 0:0143ft0 ¼ 0:49 in: ð12:5 mmÞ ½26
ð16Þ
f 0 0:7 2 c GF ¼ 0:0469da 0:5da þ 26 ¼ 0:49 lb=in: ð85:37 N=mÞ ½25 10
ð17Þ
where w/c is the water–cement ratio by weight, fc0 and ft0 are the compressive strength and tensile strength of the concrete, da is the coarse aggregate diameter, c0 = a = 1 and E is Young’s modulus. Although a proper statistical evaluation of these formulas requires considering many different concretes (as done in [24]), it may be of interest to check the error of these formulas for the present concrete. The results, given in Table 2, show that the estimates are all quite poor, although the Gf from [26] and GF from [24] are relatively best. References [1] Hoover CG, Bazˇant ZP, Vorel J, Wendner R, Hubler MH. Comprehensive concrete fracture tests: description and results. Engng Fract Mech 2013. http:// dx.doi.org/10.1016/j.engfracmech.2013.08.007. [2] Hoover CG, Bazˇant ZP, Wendner R, Vorel J, Hubler MH, Kim K, et al. Experimental investigation of transitional size effect and crack length effect in concrete fracture. In: Life-Cycle and sustainability of civil infrastructure systems: Proceedings of the 3rd International Symposium on Life-Cycle Civil Engineering (IALCCE), October 3–6, Hofburg Palace: Vienna Austria; 2012 [in press]. [3] Hoover CG, Bazˇant ZP. Universal size-shape effect law based on comprehensive concrete fracture tests. J Engng Mech 2012, [in press]. [4] Bazˇant ZP, Planas J. Fracture and size effect in concrete and other quasibrittle materials. CRC Press; 1998. [5] Bazˇant ZP. Size effect in blunt fracture: concrete, rock, metal. ASCE J Engng Mech 1984;110(4):518–35. [6] Bazˇant ZP. Scaling of structural strength. Massachusetts: Elsevier Butterworth-Heinemann; 2005. [7] Bazˇant ZP, Kazemi MT. Size dependence of concrete fracture energy determined by RILEM work-of-fracture method. Int J Fracture 1991;51:121–38. [8] Coleman TF, Li Y. On the convergence of reflective Newton methods for large-scale nonlinear minimization subject to bounds. Math Program 1994;67(2):189–224. [9] Coleman TF, Li Y. An interior trust region approach for nonlinear minimization subject to bounds. SIAM J Optim 1996;6:418–45. [10] Irwin GR. Fracture. Handbuch Phys 1958;6:551–90 (S. FlÃgge, ed., Springer, Berlin). [11] Weibull W. The phenomenon of rupture in solids proc. Roy Swed Inst Engng Res 1939;153:1–55. Ingenioersvetenskaps Akad. Handl., Stockholm. [12] Weibull W. A Statistical Distribution Function of Wide Applicability. J. Appl. Mech 1954;18:293–7. [13] Bazˇant ZP, Yu Q. Size effect in concrete specimens and structures: new problems and progress. fracture mechanics of concrete structures. In: Li VC, Leung KY, Willam KJ, Billington SL, editors. Proc., FraMCoS-5, 5th int. conf. on fracture mech. of concrete and concr. structures, Vail, Colo., vol. 1, IAFraMCoS; 2004. p. 153–62. [14] Bazˇant ZP, Yu Q. Universal size effect law and effect of crack depth on quasi-brittle structure strength. J Engng Mech 2009;135(2):78–84. [15] Tada H, Paris PC, Irwin GR. The stress analysis of cracks handbook. 2nd ed. MO: Paris Productions Inc.; 1985. [16] Srawley JE. Wide range stress intensity factor expressions for ASTM e 399 standard fracture toughness specimens. Int J Fracture 1976;12:475–6. [17] Murakami Y. Stress intensity factors handbook. New York: Pergamon; 1987. [18] Broeck D. The practical use of fracture mechanics. Dordrecht, The Netherlands: Kluwer; 1988.
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