# Average Fracture Energy for Crack Propagation in Postfire Concrete

Authors: Kequan Yu; Jiangtao Yu; Zhoudao Lu
Journal: Advances in Materials Science and Engineering
 (2013)

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## Abstract

Wedge-splitting tests of postfire concrete specimens were carried out in the present research to obtain the load-displacement curves. Ten temperatures varying from room temperature to 600°C were employed. In order to calculate the accurate fracture energy, the tails of load-displacement curves were best fitted using exponential and power functions. Three fracture energy quantities (fracture energyGF, stable fracture energy GFS, and unstable fracture energy GFU) with their variation tendency and their mutual relationship were determined to predict energy consumption for the complete fracture propagation. Additionally, the stable fracture work WFS was also calculated. All these fracture parameters sustain an increase-decrease tendency which means that the fracture property of postfire concrete shares the same tendency.


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## Body

## 1. Introduction

Since the application of fracture mechanics to concrete, the energy consumption for crack propagation in concrete has been a popular topic. For concrete, the specific fracture energyGF has been proven to be a useful parameter in the structure design and fracture behavior modeling. The specific fracture energy of concrete was defined based on a tensile test as the energy absorbed per unit crack area in widening the crack from zero to or beyond the critical value above which no stress can transmit [1]. Based on the work-of-fracture principle, three-point bending test [2], compact tension [3], and wedge-splitting method [4] were proposed as alternative methods to determine the specific fracture energy GF. It is computed as the area under the entire imposed load P and load-line displacement curve divided by the projected area of uncracked ligament, so the fracture energy GF represents the average or nominal energy consumption of concrete for an entire crack propagation process.The existence of fracture process zone FPZ ahead of a crack is now well accepted. Since the 1970s, it has been known that the evolution of the FPZ undergoes two distinct periods—precritical stable crack growth and unstable fracture process [5]. There is no doubt that crack propagation is accompanied by energy dissipation, and the motive for crack propagation comes from either work provided by the imposed load or released strain energy. Fracture energy is one appropriate consideration to describe the amount of energy consumed during crack propagation process.It is worth noting that the fracture energy can only represent the amount of average energy dissipation for entire crack propagation from crack initiation to complete failure without characterizing crack stable propagation and unstable fracture periods. So even withGF, it is still not clear how much energy is dissipated during those two crack extension periods. Xu et al. [6] proposed two new concepts the stable fracture energy GFS and unstable fracture energy GFU to describe fracture responses for different crack propagation periods. It is found that GFS kept constant for different ligament lengths, whereas GF and GFU showed the apparent size effect. But the accurate calculation of fracture surface remains unsolved. It is known that the true path of crack extension is tortuous, not straight as expected. The projected area underestimates the true fracture area. Hence, these parameters are actually nominal values.The fracture energy of postfire concrete has been studied by several researchers [7–12]. It is found that the residual fracture energy sustained an increase-decrease tendency with the turning point at approximately 450°C. The increase tendency is due to the energy dissipation of microcracks distributing in the concrete, whereas the thermal damage induced by high temperatures reduces the residual fracture energy. However, in these researches, the influence of loading-displacement tail was unknown or not considered.In present paper, wedge-splitting experiments of under ten temperatures levels varying from 20°C to 600°C and the specimens size of 230 mm × 200 mm × 200 mm with initial-notch depth ratios 0.4 are implemented [12]. Based on the work-of-fracture idea, the residual fracture energy GF is calculated considering the influence of load-displacement tail. Furthermore, the fracture energy consumption for crack stable extension and unstable extension, that is, GFS, GFU, is investigated. However, the true fracture surface remains undetermined and extremely difficult for specimens subjected to high temperatures, so these three fracture energy parameters are still nominal values. Hence, corresponding to GFS and GFU, the stable and unstable fracture work which neglect the fracture surface and their variation about temperatures are thus determined. From these parameters the fracture properties of postfire concrete could be described.


## 2. Fracture Energies for an Entire Crack Propagation Period

Based on the global energy balance principle, the work performed by a generalized forceP on its displacement will be transformed into energy: one part of energy is stored in the body in the form of strain energy and the other part is used for crack propagation. It can be mathematically written as [6]:
(1)GFiAi=Wi-ΔUi=∫0δiPdδ-ΔUi,
where GFi = average energy needed for unit crack propagation during period from crack initiation to any instant of time i; Ai = newly formed fracture area for this period; the product of these two quantities represents the energy absorbed for crack propagation during this period; Wi = work performed by the external force P during the same period; Ui = increase of elastic strain energy of the body until time i; δi = crack opening displacement corresponding to P. If time i is approaching the failure terminal point i = 1, where the load drops to zero, then GFi in (1) will reduce to
(2)GF=WAlig=∫0δ1Pdδb(h-a0),
where GF = RILEM fracture energy; W = total energy provided by the external force P for crack propagation; Alig = projected fracture area perpendicular to the tensile stress direction; b = specimen thickness; h = specimen height; a0 = initial crack length.During the stable crack propagation period, the load will increase up to its peak valuePmax with its corresponding displacement reaching CODc; see Figure 1. So, based on (1), the average energy absorbed during this period can be evaluated by
(3)GFS=WFSAS=WP-ΔUSAS=WOAMOAS=WOIMO-WAIMAAS=∫0CODcPdCODc-ΔUSAS,
where GFS by naming the stable fracture energy = average energy needed for the crack to grow unit area during the crack stable propagation; AS = change in fracture area; WFS=WOAMO= energy provided for stable crack propagation as shown in Figure 1 by the shaded area; WP=WOIMO= work performed by the external force P for crack increases its area AS; ΔUS = WAIMA = increased elastic strain energy of the body; and CODc = critical crack opening displacement.Figure 1
Energy consumption for an entire crack propagation period.From the definition of the stable fracture energyGFS, it is implied that the work from the imposed load P is expended in two forms during the crack stable propagation: one part is stored in the body in the form of the elastic strain energy (ΔUS) and the other part is the energy absorbed by the fracture zone mainly to counteract the resistance caused by the cohesive forces along the FPZ. In the same way, the unstable fracture energy, denoted by GFU, is defined based on energy equilibrium as
(4)GFU=WUAU=W-WSAU,
where WU = energy needed for the crack unstable propagation period and AU = change in fracture area. During this period, the stored energy is completely released until the deformation reaches its maximum value COD1 when the load closes to zero. Similar to the RILEM fracture energy GF, the establishment of the stable fracture energy GFS and unstable fracture energy GFU rests upon the implicit assumption that no other energy consumption occurs outside the fracture zone.


## 3. Experimental Program and Experimental Phenomena

In this test, concrete specimens were prepared using an ordinary silicate cement PO. 42.5 produced conforming to the Chinese standard. Coarse aggregate was calcareous crushed stone with a maximum size of 16 mm, and river sand was used as the fine aggregates and its maximum diameter was 5 mm. Details of the mix proportioning (by weight) used for concrete and some mechanical properties are Cement : Sand : Limestone Coarse aggregate : Water : fly ash = 1.00 : 3.44 : 4.39 : 0.80 : 0.26.Fracture properties of concrete were determined by means of the wedge splitting test [4]. The test setup and geometry of the specimen are schematically represented in Figure 2. Compared to three-point bending notched beams, the wedge-splitting test has following advantages. For the three-point bending beams, inaccurate measurement of load-point displacement and the self-weight of the specimen could influence the real value of the fracture energy. During the test, beams should be carefully handled due to their heavy weight. However, using the WS specimens, the recorded COD in a horizontal plane is not affected by the crushing of the specimen at the supports or some other factors. Besides, the WS specimens are simple and easily prepared in laboratories or on site.Figure 2
Test setup and geometry of specimens.A total of 50 concrete specimens with the same dimensions 230 × 200 × 200 mm were prepared; the geometry of the specimens and the test setup are shown in Figure2 (b=200 mm, d=65 mm, h=200 mm, f=30 mm, a0=80 mm, and θ=15°). All the specimens had a precast notch of 80 mm height and 3 mm thickness, achieved by placing a piece of steel plate into the molds prior to casting. Each wedge splitting specimen was embedded with a thermal couple in the center of specimen for temperature control.Nine heating temperatures, ranging from 65°C to 600°C (Tm = 65°C, 120°C, 200°C, 300°C, 350°C, 400°C, 450°C, 500°C, and 600°C), were adopted with the ambient temperature as a reference. Because it was recognized that the fracture behavior measurements were generally associated with significant scatter, five repetitions were performed for each temperature.An electric furnace with net dimensions 300 × 300 × 900 mm was used for heating. When the designatedTm was reached, the furnace was shut down, and the specimens were naturally cooled for 7 days prior to the test. It averagely took 50, 95, 135, 182, 218, 254, 294, 342, and 453 minutes for the specimens to reach the final temperatures, respectively (from 65°C to 600°C). Figure 3 shows the typical temperature history for several cases with different maximum temperatures. After heating, microcracks disperse on the specimen surface, especially for temperatures higher than 200°C (see in Figure 4).Figure 3
Typical temperature history of specimens.Figure 4
Microcracks on specimen surface (600°C).The fracture surfaces at different temperature intervals (20°C, 200°C, 350°C, 450°C, and 600°C) are shown in Figure5, which became lighter but more tortuous with increasing temperatures.Fracture surfaces of postfire specimens.
(a)
Fracture surface of 20°C
(b)
Fracture surface of 600°C
(c)
Fracture surfaces at typical temperatures (20°C, 200°C, 350°C, 450°C, and 600°C)A universal machine with a maximum capacity of 1000 kN was used to conduct the wedge splitting tests. During the test, the vertically applied loadPv and the crack opening displacement COD along the horizontal load line were simultaneously recorded through a data acquisition system. Referring to Figure 2, the splitting force Ph is actually the horizontal component of the force acting on the bearing. Taking the wedge angle into consideration, its relation with the recorded applied load Pv can be developed based on force equilibrium ignoring the small influence from the friction for different roller bearings:
(5)Ph=Pv2tanθ.To obtain the completeP-COD curves (shown in Figure 6), the test rate was fixed at 0.4 mm/min, such that it took approximately 20 minutes to complete a single test of specimens subjected to less than 300°C and 30 minutes for beyond 300°C.Pversus CMOD curves of specimens with temperatures.
(a)
P
h versus COD curves of specimens at 120°C
(b)
P
h versus COD curves of specimens at 450°C
(c)
P
h versus COD curves of specimens at 600°C
(d)
TypicalPv versus COD curves of specimens at all temperaturesThe fracture of specimen is essentially due to the bending moment caused by the horizontal splitting forcePh, vertical component 1/2Pv, and self-weight of the specimen. Two symmetrical supports are placed below the center of gravity of each half of the specimen. In doing so, the influence of the dead weight of the specimen and part of the vertical component force on the calculation of the fracture energies could be counteracted. Each roll axis is fixed at the same horizon as the lower plane of the groove and is very close to the center of gravity of each half of the specimen in the vertical position (shown in Figure 2). Due to the carefulness in the choice of the specimen geometry, the roll axis location, and the placement of the supports, the horizontal force Ph contributes most to fracturing the specimen. Therefore, Ph-COD curves were directly used in the calculation of the RILEM fracture energy GF, stable fracture energy GFS, and unstable fracture energy GFU.For our test results, thePh-COD curves could be easily obtained from the monitored Pv-COD curves and (5). Figure 6 contains the plots of Ph-COD curves for several temperatures and typical Pv-COD curves for all temperatures.From Figure6(a) to 6(c), it is found that with the increasing of temperature (20°C–600°C), the divergence between the curves for the same temperature is more significant. In particular 600°C, the ultimate load Pu of specimen WS50 is one time higher than the one of specimen WS47. Additionally, the whole loading process is not stable for specimens WS49 (a sudden snap-back) due to the thermal damage induced by high temperature. Figure 6(d) shows the typical P-COD curves of all temperatures. The ultimate load Pu decreases significantly with increasing temperatures Tm, whereas the crack mouth opening displacement (COD) increases with Tm. The initial slope of ascending branches decreases with heating temperatures, and the curves become gradually shorter and more extended.It is found that the ultimate loadPhmax decreases with the increasing temperatures, whereas the CODc increases with Tm (Figure 8). The average value of Pu decreases from 9.17 kN at ambient temperature to 7.92 kN at 120°C, 4.29 kN at 300°C, 3.16 kN at 450°C, and finally 1.38 kN at 600°C, with a final drop of 85%. The value of CODc increases from 0.178 mm at ambient temperature to 0.352 mm at 200°C, 0.901 mm at 400°C, and 1.848 mm at 600°C, nearly 10 times as the ambient value.


## 4. Experimental Results and Analysis

### 4.1. Determination of Residual Fracture EnergyGF

In the calculation of RILEM fracture energyGF, two things should be carefully considered. One thing is that the work done by the self-weight of loading device should be taken into account. In present experiments, the loading device is attached to the testing machine and the Ph-COD curve includes the self-weight; hence the work should not be calculated again. The other concern pertains to the tail part of the Ph-COD curve: the recorded point (Ph1, COD1) is just one point when the experiment stops, not the actual point of when the load drops to zero. Therefore, cutting the load-deflection tail may lead to noticeable inaccuracy in the true value of fracture energy. Thus, to account for these two factors, the actual RILEM fracture energy GF based on the work-of-fracture method becomes (see Figure 9)
(6)GF=WAlig=W1+W2t(h-a0),
where W1 = measured work enclosed by the Ph-COD curve until COD = COD1; Alig = fresh fracture area; W2 is the work caused by tail effect part. For W2, curve fitting technology based on Excel software was used to get its approximate value. In previous research from the test results at ambient temperature [13], it was found out that the descending branch of the Ph-COD curve after one-third of the peak load could be well described by power function (the coefficient of determination R2 for each curve is close to 1). This study indicates that for specimens subject to no more than 120°C, the power function is more accurate; however, for specimens subject to higher temperatures, exponential function is more suitable (see Figure 9).For power functionPh=β×COD-λβ, λ>0, where β, λ = constants for fitting curves. And then the unrecorded work W owning to cutting load-deflection tail can be written as
(7)W2=∫COD1∞β×COD-λdCOD=β(λ-1)×COD1λ-1.For exponential functionPh=m×e-n×CODm, n>0, where m, n = constants for fitting curves. And then the unrecorded work W owning to cutting load-deflection tail can be written as
(8)W2=∫COD1∞m×e-n×CODdδ=mnen×COD1.The parameters ofβ, λ for 20°C~120°C and m, n for 200°C~600°C, and the value of w1, w2 is listed in Table 1.Table 1
Parameters calculated from wedge-splitting test.
Specimen
Temperature
P
h
max (kN)
E (GPa)
a
c (mm)
β/m
λ/n
R
2
W
1
W
2
CO
D
c (mm)
COD1 (mm)
K
0 (kN/mm)
K
c (kN/mm)
WS1
20°C
8.304
15.30
0.107
1.484
2.273
0.970
5.231
0.389
0.174
1.926
87.081
79.219
WS2
9.407
20.51
0.097
3.131
1.206
0.998
10.468
1.140
0.205
4.914
116.710
113.648
WS3
10.379
20.66
0.114
2.464
1.117
0.991
8.699
1.818
0.195
3.280
117.610
110.633
WS4
7.884
18.88
0.112
1.599
3.224
0.964
5.128
0.137
0.152
1.903
107.480
98.911
WS5
9.364
15.45
0.107
2.048
1.683
0.991
6.970
0.735
0.199
2.657
87.940
80.103
Average
9.068
18.16
0.107
—
—
0.983
7.299
0.844
0.185
2.936
103.364
96.503
WS6
65°C
11.282
21.73
0.113
2.781
1.506
0.996
9.238
0.984
0.195
3.205
123.690
116.172
WS7
8.151
24.79
0.132
2.739
0.987
0.999
10.320
1.263
0.215
6.072
141.100
136.109
WS8
10.379
19.43
0.115
6.114
1.992
0.954
11.004
0.702
0.212
4.549
110.560
105.418
WS9
10.681
23.25
0.119
2.658
1.030
0.993
9.283
2.250
0.164
3.535
132.330
124.685
WS10
11.610
16.60
0.107
3.185
1.252
0.983
11.466
1.071
0.190
5.324
94.490
90.418
Average
10.421
21.16
0.117
—
—
0.985
10.262
1.254
0.195
4.537
120.434
114.560
WS11
120°C
8.353
10.65
0.095
2.477
1.085
0.990
8.423
1.094
0.191
3.418
60.638
57.254
WS12
8.226
11.87
0.107
2.667
0.890
0.999
10.624
1.803
0.224
6.443
67.564
65.212
WS13
7.631
9.48
0.119
4.340
0.998
0.980
13.227
2.487
0.357
5.641
53.926
50.511
WS14
7.302
15.42
0.117
2.139
1.153
0.998
6.934
1.357
0.198
3.534
87.758
82.839
WS15
—
—
—
—
—
—
—
—
—
—
—
—
Average
7.878
11.85
0.109
—
—
0.992
9.802
1.685
0.243
4.759
67.472
63.954
WS16
200°C
—
—
—
—
—
—
—
—
—
—
—
—
WS17
6.466
11.58
0.127
2.266
1.526
0.952
6.260
0.520
0.284
2.558
65.913
57.759
WS18
5.884
6.98
0.115
2.268
2.982
0.995
5.035
0.129
0.335
1.968
39.732
32.967
WS19
5.071
9.17
0.127
2.770
2.087
0.969
5.397
0.265
0.306
2.786
52.178
46.456
WS20
5.228
7.00
0.130
2.530
0.366
0.903
7.605
1.250
0.458
4.447
39.866
35.765
Average
5.662
8.68
0.125
—
—
0.955
6.074
0.541
0.346
2.940
49.422
43.237
WS21
300°C
3.341
2.45
0.121
4.023
0.361
0.981
10.019
0.491
0.792
3.101
13.939
11.004
WS22
5.513
3.49
0.117
4.090
0.300
0.987
13.570
1.105
0.667
5.065
19.869
17.251
WS23
3.371
1.91
0.113
6.985
0.610
0.982
7.727
0.475
0.672
4.957
10.854
9.384
WS24
4.761
1.99
0.105
4.285
0.345
0.990
12.552
0.987
0.728
6.816
11.350
10.138
WS25
4.076
4.03
0.137
2.172
0.206
0.963
11.532
1.667
0.869
6.874
22.960
20.438
Average
4.213
2.78
0.119
—
—
0.981
11.080
0.945
0.746
5.363
15.794
13.643
WS26
350°C
5.701
6.05
0.131
5.702
0.447
0.995
12.290
1.475
0.599
4.549
34.409
29.878
WS27
3.840
2.03
0.125
6.245
0.335
0.958
14.150
1.490
1.003
6.768
11.538
9.829
WS28
4.718
3.60
0.131
5.553
0.454
0.997
11.312
1.107
0.815
4.053
13.490
16.360
WS29
4.554
3.38
0.130
9.025
0.608
0.995
11.635
0.355
0.821
5.940
19.240
16.582
WS30
3.931
3.21
0.134
7.135
0.532
0.990
10.251
0.654
0.832
5.335
18.279
15.427
Average
4.549
3.65
0.130
—
—
0.987
11.927
1.016
0.814
5.329
19.391
17.615
WS31
400°C
3.584
2.56
0.136
3.934
0.307
0.999
11.737
1.033
0.921
6.937
14.557
12.391
WS32
3.039
1.42
0.126
21.138
0.828
0.992
9.300
0.226
0.904
5.614
8.084
6.331
WS33
3.228
2.12
0.114
2.100
0.224
0.995
8.533
1.575
0.842
5.819
7.348
6.419
WS34
4.476
1.71
0.111
4.208
0.322
0.929
13.388
0.986
0.985
5.940
12.035
8.463
WS35
—
—
—
—
—
—
—
—
—
—
—
—
Average
3.582
1.95
0.122
—
—
0.979
10.740
0.955
0.913
6.078
10.506
8.401
WS36
450°C
3.336
1.41
0.125
5.280
0.299
0.977
13.124
1.553
1.224
5.625
8.037
6.289
WS37
—
—
—
—
—
—
—
—
—
—
—
—
WS38
3.118
1.46
0.123
10.235
0.543
0.981
10.601
0.978
1.057
5.672
8.291
6.216
WS39
3.056
1.34
0.127
2.518
0.137
0.995
11.955
3.960
1.281
6.784
7.611
6.099
WS40
2.935
1.58
0.137
4.328
0.226
0.998
13.896
2.395
1.394
6.000
8.988
6.900
Average
3.111
1.45
0.128
—
—
0.988
12.394
1.663
1.239
6.020
8.232
6.376
WS41
500°C
—
—
—
—
—
—
—
—
—
—
—
—
WS42
2.153
1.76
0.146
3.056
0.245
0.976
7.762
3.054
1.284
5.153
7.251
7.506
WS43
2.857
1.09
0.119
3.648
0.306
0.979
10.092
1.342
1.174
6.533
6.179
5.069
WS44
1.929
0.75
0.109
2.234
0.153
0.968
8.758
4.119
1.225
6.591
4.243
3.651
WS45
1.838
1.48
0.147
2.300
0.186
0.996
7.826
3.012
1.354
6.225
6.078
6.591
Average
2.194
1.27
0.130
—
—
0.977
8.002
2.494
1.259
6.125
5.938
5.704
WS46
600°C
1.129
0.47
0.130
2.102
0.278
0.931
4.197
1.280
1.482
7.000
2.656
2.094
WS47
1.474
0.48
0.128
3.018
0.238
0.954
7.245
2.236
1.784
10.000
2.718
2.233
WS48
1.649
1.14
0.152
1.992
0.132
0.979
8.390
4.551
1.908
7.100
3.909
2.858
WS49
1.138
0.38
0.131
1.183
0.120
0.971
5.018
2.950
1.865
10.000
1.165
1.761
WS50
1.243
0.38
0.124
2.507
0.285
0.943
5.273
1.280
1.644
8.000
1.174
1.727
Average
1.326
0.57
0.133
—
—
0.956
6.025
2.460
1.737
8.420
2.324
2.135Some necessary test results are tabulated in Table1, including the maximum value of the horizontal load Phmax and its corresponding crack opening displacement CODc, the endpoint where Ph1 approaches zero and the crack opening displacement arrives at COD1, the initial slope of Ph-COD, that is, the initial stiffness K0, and the modulus of elasticity E.Hence, the residual fracture energy described by (1) could be calculated. The values are listed in Table 2 and are shown in Figure 10.Table 2
Fracture energies for crack propagation.
Specimen
Temperature
W
0 (Nm)
A
lig (m2)
G
F (Nm−1)
W
OIMA (Nm)
W
AIMA (Nm)
W
FS (Nm)
A
FS (m2)
G
FS (Nm−1)
W
FU (Nm)
A
FU (m2)
G
FU (Nm−1)
WS1
20°C
5.620
0.024
234.147
1.217
0.407
0.810
0.005
152.502
4.809
0.019
257.361
WS2
11.608
483.664
1.575
0.388
1.187
0.003
350.795
10.421
0.021
504.972
WS3
10.517
438.217
2.124
0.470
1.654
0.007
242.557
8.863
0.017
515.877
WS4
5.265
219.394
0.994
0.314
0.680
0.006
107.537
4.586
0.018
259.384
WS5
7.705
321.055
1.693
0.525
1.169
0.005
212.824
6.536
0.019
353.175
Average
8.143
339.295
1.521
0.421
1.100
0.005
213.243
7.043
0.019
378.154
WS6
65°C
7.666
0.024
425.907
1.791
0.535
1.255
0.007
281.835
6.411
0.017
480.239
WS7
8.687
482.619
1.076
0.241
0.835
0.010
113.926
7.852
0.014
768.132
WS8
8.779
487.745
1.475
0.509
0.966
0.007
208.612
7.813
0.017
602.385
WS9
8.649
480.507
1.027
0.440
0.587
0.008
118.113
8.062
0.016
656.640
WS10
9.402
522.360
1.419
0.733
0.686
0.005
210.596
8.716
0.019
614.693
Average
8.637
479.828
1.357
0.492
0.866
0.007
186.616
7.771
0.017
624.418
WS11
120°C
7.137
0.024
396.516
1.335
0.588
0.746
0.003
251.423
6.391
0.021
416.985
WS12
9.321
517.822
1.475
0.511
0.964
0.005
178.684
8.357
0.019
616.201
WS13
11.785
654.732
2.079
0.560
1.520
0.008
193.937
10.266
0.016
878.097
WS14
6.218
345.463
1.044
0.312
0.732
0.007
98.944
5.486
0.017
455.357
WS15
—
—
—
—
—
—
—
—
—
—
Average
8.615
478.633
1.483
0.493
0.991
0.006
180.747
7.625
0.018
591.660
WS16
200°C
—
0.024
—
—
—
—
—
—
—
—
—
WS17
6.780
282.509
1.522
0.345
1.176
0.009
123.021
5.604
0.015
385.812
WS18
5.163
215.135
1.556
0.525
1.031
0.007
146.955
4.132
0.017
243.300
WS19
5.662
235.913
1.252
0.270
0.982
0.009
104.764
4.680
0.015
320.014
WS20
8.854
368.937
1.828
0.365
1.463
0.010
145.364
7.391
0.014
530.467
Average
6.615
275.624
1.540
0.376
1.163
0.009
130.026
5.452
0.015
369.898
WS21
300°C
10.510
0.024
437.920
2.005
0.439
1.566
0.008
188.640
8.944
0.016
568.895
WS22
14.675
611.467
3.546
0.820
2.726
0.007
364.621
11.949
0.017
723.160
WS23
8.202
341.770
2.048
0.579
1.469
0.007
220.394
6.733
0.017
388.453
WS24
13.539
564.120
3.065
1.081
1.984
0.005
404.514
11.555
0.019
605.107
WS25
13.200
549.990
2.552
0.389
2.163
0.011
190.295
11.037
0.013
870.055
Average
12.025
501.053
2.643
0.662
1.982
0.008
273.693
10.044
0.016
631.134
WS26
350°C
13.765
0.024
573.540
2.689
0.511
2.179
0.010
211.652
11.586
0.014
845.346
WS27
15.639
651.638
2.589
0.703
1.887
0.009
207.566
13.753
0.015
922.314
WS28
12.420
517.489
3.827
0.919
2.908
0.010
317.145
9.512
0.014
664.527
WS29
11.990
499.602
2.900
0.611
2.289
0.010
228.349
9.701
0.014
694.230
WS30
10.905
454.375
2.796
0.475
2.320
0.011
216.312
8.585
0.013
646.739
Average
12.944
539.329
2.960
0.644
2.317
0.010
236.205
10.627
0.014
754.631
WS31
400°C
12.770
0.024
532.090
2.240
0.484
1.756
0.011
156.074
11.014
0.013
860.665
WS32
9.526
396.932
2.678
0.656
2.022
0.009
207.133
7.504
0.015
516.314
WS33
10.109
421.188
2.411
0.760
1.651
0.007
242.302
8.457
0.017
491.692
WS34
14.374
598.926
3.638
0.918
2.720
0.006
401.879
11.654
0.018
668.062
WS35
—
—
—
—
—
—
—
—
—
—
Average
11.695
487.284
2.742
0.705
2.037
0.008
251.847
9.658
0.016
634.184
WS36
450°C
14.677
0.024
611.530
3.255
0.779
2.477
0.009
277.088
12.200
0.015
809.983
WS37
—
—
—
—
—
—
—
—
—
—
WS38
11.579
482.451
3.284
0.669
2.615
0.009
303.750
8.964
0.015
582.382
WS39
15.914
663.097
3.281
0.704
2.578
0.009
272.015
13.337
0.015
918.284
WS40
16.291
678.790
3.847
0.532
3.315
0.011
292.743
12.976
0.013
1023.630
Average
14.057
608.967
3.417
0.671
2.746
0.010
286.399
11.869
0.014
833.570
WS41
500°C
—
0.024
—
—
—
—
—
—
—
—
—
WS42
10.816
450.668
2.080
0.355
1.725
0.013
138.244
9.091
0.011
831.501
WS43
11.434
476.413
2.890
0.743
2.146
0.008
272.496
9.288
0.016
576.031
WS44
12.877
536.544
1.495
0.475
1.021
0.006
170.514
11.856
0.018
652.188
WS45
10.838
451.571
2.720
0.308
2.412
0.013
186.562
8.426
0.011
785.892
Average
10.496
478.799
2.296
0.470
1.826
0.010
191.954
9.665
0.014
711.403
WS46
600°C
5.477
0.024
228.226
1.404
0.287
1.117
0.010
110.974
4.361
0.014
312.890
WS47
9.481
395.057
2.526
0.460
2.066
0.010
213.423
7.415
0.014
517.849
WS48
12.941
539.225
2.621
0.389
2.232
0.014
155.808
10.709
0.010
1107.140
WS49
7.968
331.988
1.514
0.635
0.879
0.010
115.238
7.088
0.014
491.494
WS50
6.554
273.074
1.749
0.773
0.976
0.009
151.878
5.578
0.015
342.831
Average
8.484
353.514
1.963
0.509
1.454
0.011
149.464
7.030
0.013
632.922Though the residual fracture energy at each temperature has significant scatter, Figure10 shows that the average values sustain an increase-decrease tendency with Tm. From 20°C to 450°C, average GF increases from 339.3 Nm−1 to 609 Nm−1, while the temperature reaches 600°C and the fracture energy falls back to 307.8 Nm−1. The detailed explanation would be seen elsewhere [12].


### 4.2. Determination of Stable Fracture EnergyGFS and Unstable Fracture Energy GFU

As an extension of fracture energyGF, two other energy-based fracture characteristics, the stable fracture energy GFS and unstable fracture energy GFU, are proposed to describe fracture responses for different crack propagation periods. The analysis shows that fracture energy GF is actually the weighed average of GFS and GFU, and GFS and GFU could be regarded as two components of GF. This is very helpful in understanding the whole crack propagation process from the aspect of energy consumption.To obtain the values ofGFS and GFU, the critical effective crack length ac should first be determined for the calculation of the fracture areas AS and AU. Herein, the value ac was computed based on the double-K fracture model [14]:
(9)a=(h+h0){1-(13.18E·b·c+9.16)1/2}-h0,
where c=COD/P is the compliance of specimens, E is modulus of elasticity, b is specimen thickness, h is specimen height, and h0 is the thickness of the clip gauge holder. For calculation of critical value of equivalent elastic crack length ac, the value of crack mouth opening displacement (COD) and P are taken as CODc and Phmax, respectively. Equation (9) is valid for 0.2–0.8 within 2% accuracy. The value of ac is reported in Table 1.In a typical load-displacement curve shown in Figure1, K0 is the initial stiffness of the load-displacement curve before the start of the fracture process. At critical state when the load arrives at its peak value Pmax and its corresponding displacement reaches CODc, the stiffness Kc may not be the same as the initial K0. The degradation of stiffness is the result of crack propagation. To make it simple, here, the value of K at any value of displacement is assumed to be linear with increased deformation [15]. 								(10)K=K0(COD1-COD)(COD1-COD0),
where COD1 = final deformation when the load P approaches zero and COD0 = displacement before which the stiffness is still kept at K0. At the origin of P-COD curve, the initial tangent stiffness is K0. The stiffness becomes degraded beyond the origin point; hence COD0 is presumed to be zero. It is clear from (10) that K=K0 when COD = 0 and K=0 when = COD1, and K is assumed to be a linear function with respect to the displacement. With the crack opening displacement COD = CODc, the stiffness Kc at this point could approximately be obtained through (10). Note that P-COD is a general term for load-displacement; in our case, the load is the horizontal force Ph and its corresponding displacement is the COD. Therefore, with K0, COD1, and CODc, the approximate degraded stiffness Kc can be gained from (10) and the result is shown in Table 1. Figure 11 shows that the values of stiffness Kc decrease monotonously with temperature due to the thermal damage induced by high temperatures.The stable fracture energyGFS and unstable fracture energy GFU can be derived by following (11)~(13). The specific values of GFS and GFU are compiled in Table 2:
(11)GFS=WFSAS=WOAMOAS=WOIMO-WAIMAAS,GFU=WUAU=W-WFSAU,(12)AS=B(ac-a0),AU=B(h-ac),(13)WAIMA=0.5Phmax2Kc.Figure12 shows the variation of stable fracture energy GFS with temperatures. Similar to the residual fracture energy GF, it also keeps an increase-decrease tendency with temperatures. Temperatures less than 120°C appear not to induce much thermal damage to concrete, so the cracking resistance almost keeps constant. The values of stable fracture energy GFS at these temperatures are 213 Nm−1, 186 Nm−1, and 180 Nm−1, respectively. The fracture surfaces tend to be more tortuous between 200°C and 450°C than those observed at lower temperatures (see Figure 4) and there exist several cracks competing to form the final fracture, so more energy was dissipated in these specimens. Additionally, the opening of microcracks on the surface and inside the specimens also dissipates energy (see Figure 3). Finally, higher heating temperatures would cause more micro cracks, dehydration, and decomposition and would degrade the resistance. After 450°C, cracking resistance continuously decreases with Tm.Table2 shows that the unstable fracture GFU also sustains an increase-decrease tendency with Tm, and its value is much larger than stable fracture energy. Two reasons for larger values of GFU are provided [5]. First, energy consumption besides the main fracture zone takes places for the whole fracture process. During the unstable fracture process, the energy consumption for plasticity or other nonlinear deformation beyond the main FPZ would be much higher, especially for specimens subjected to high temperatures. The other possible reason lies in the calculation of true fracture area. It is known that the interface (transition zone) between cement and aggregate is the weak link in microstructure for normal concrete, and crack propagation would proceed in the path where the energy needs are least. So the true path of crack extension is tortuous and the higher the temperature is, the more tortuous the fracture surface is (see Figure 4), not straight as expected. The projected area is used in the calculation of fracturing surface AS or AU, which underestimates the true fracture area. Since the crack experiences much longer distance for the unstable extension, which leads to a greater underestimation of the calculation of the newly fractured area AU, the calculated GFU is overestimated.Moreover, in view of (11) and (12), another expression for the fracture energy GF with respect to GFS and GFU can be followed:
(14)GF=WA=GFSAS+GFUAUA=GFS·ASA+GFU·AUA.
Since AS+AU=A, the fracture energy GF is the weighed average of GFS and GFU. However, for engineering application, the stable crack propagation is considered to be more important, because when the load exceeds the maximum value the whole structure would be in an unstable situation. Hence from Figures 7 and 9, it is concluded that the fracture property of postfire concrete sustains an increase-decrease tendency to 600°C, with a turning point at 450°C.Figure 7
Variation tendency ofPhmaxwith Tm.Figure 8
Variation tendency ofCODc with Tm.P-CMOD tail curve fitting of specimens.
(a)
Curve fitting for specimen of ambient temperature
(b)
Curve fitting for specimen of 200°CFigure 10
Residual fracture energy withTm.Figure 11
StiffnessKc with Tm.Figure 12
Stable fracture energy withTm.Furthermore, as mentioned above, the fracture surfaces ofA, AS, AU are project areas, so GF, GFS, GFU are nominal fracture energies. To avoid the violence of fracture surface, the variation tendency of stable fracture work WFS is determined (see Figure 10). Similarly, WFS also keeps an increase-decrease tendency at the same turning temperature of 450°C.


## 4.1. Determination of Residual Fracture EnergyGF

In the calculation of RILEM fracture energyGF, two things should be carefully considered. One thing is that the work done by the self-weight of loading device should be taken into account. In present experiments, the loading device is attached to the testing machine and the Ph-COD curve includes the self-weight; hence the work should not be calculated again. The other concern pertains to the tail part of the Ph-COD curve: the recorded point (Ph1, COD1) is just one point when the experiment stops, not the actual point of when the load drops to zero. Therefore, cutting the load-deflection tail may lead to noticeable inaccuracy in the true value of fracture energy. Thus, to account for these two factors, the actual RILEM fracture energy GF based on the work-of-fracture method becomes (see Figure 9)
(6)GF=WAlig=W1+W2t(h-a0),
where W1 = measured work enclosed by the Ph-COD curve until COD = COD1; Alig = fresh fracture area; W2 is the work caused by tail effect part. For W2, curve fitting technology based on Excel software was used to get its approximate value. In previous research from the test results at ambient temperature [13], it was found out that the descending branch of the Ph-COD curve after one-third of the peak load could be well described by power function (the coefficient of determination R2 for each curve is close to 1). This study indicates that for specimens subject to no more than 120°C, the power function is more accurate; however, for specimens subject to higher temperatures, exponential function is more suitable (see Figure 9).For power functionPh=β×COD-λβ, λ>0, where β, λ = constants for fitting curves. And then the unrecorded work W owning to cutting load-deflection tail can be written as
(7)W2=∫COD1∞β×COD-λdCOD=β(λ-1)×COD1λ-1.For exponential functionPh=m×e-n×CODm, n>0, where m, n = constants for fitting curves. And then the unrecorded work W owning to cutting load-deflection tail can be written as
(8)W2=∫COD1∞m×e-n×CODdδ=mnen×COD1.The parameters ofβ, λ for 20°C~120°C and m, n for 200°C~600°C, and the value of w1, w2 is listed in Table 1.Table 1
Parameters calculated from wedge-splitting test.
Specimen
Temperature
P
h
max (kN)
E (GPa)
a
c (mm)
β/m
λ/n
R
2
W
1
W
2
CO
D
c (mm)
COD1 (mm)
K
0 (kN/mm)
K
c (kN/mm)
WS1
20°C
8.304
15.30
0.107
1.484
2.273
0.970
5.231
0.389
0.174
1.926
87.081
79.219
WS2
9.407
20.51
0.097
3.131
1.206
0.998
10.468
1.140
0.205
4.914
116.710
113.648
WS3
10.379
20.66
0.114
2.464
1.117
0.991
8.699
1.818
0.195
3.280
117.610
110.633
WS4
7.884
18.88
0.112
1.599
3.224
0.964
5.128
0.137
0.152
1.903
107.480
98.911
WS5
9.364
15.45
0.107
2.048
1.683
0.991
6.970
0.735
0.199
2.657
87.940
80.103
Average
9.068
18.16
0.107
—
—
0.983
7.299
0.844
0.185
2.936
103.364
96.503
WS6
65°C
11.282
21.73
0.113
2.781
1.506
0.996
9.238
0.984
0.195
3.205
123.690
116.172
WS7
8.151
24.79
0.132
2.739
0.987
0.999
10.320
1.263
0.215
6.072
141.100
136.109
WS8
10.379
19.43
0.115
6.114
1.992
0.954
11.004
0.702
0.212
4.549
110.560
105.418
WS9
10.681
23.25
0.119
2.658
1.030
0.993
9.283
2.250
0.164
3.535
132.330
124.685
WS10
11.610
16.60
0.107
3.185
1.252
0.983
11.466
1.071
0.190
5.324
94.490
90.418
Average
10.421
21.16
0.117
—
—
0.985
10.262
1.254
0.195
4.537
120.434
114.560
WS11
120°C
8.353
10.65
0.095
2.477
1.085
0.990
8.423
1.094
0.191
3.418
60.638
57.254
WS12
8.226
11.87
0.107
2.667
0.890
0.999
10.624
1.803
0.224
6.443
67.564
65.212
WS13
7.631
9.48
0.119
4.340
0.998
0.980
13.227
2.487
0.357
5.641
53.926
50.511
WS14
7.302
15.42
0.117
2.139
1.153
0.998
6.934
1.357
0.198
3.534
87.758
82.839
WS15
—
—
—
—
—
—
—
—
—
—
—
—
Average
7.878
11.85
0.109
—
—
0.992
9.802
1.685
0.243
4.759
67.472
63.954
WS16
200°C
—
—
—
—
—
—
—
—
—
—
—
—
WS17
6.466
11.58
0.127
2.266
1.526
0.952
6.260
0.520
0.284
2.558
65.913
57.759
WS18
5.884
6.98
0.115
2.268
2.982
0.995
5.035
0.129
0.335
1.968
39.732
32.967
WS19
5.071
9.17
0.127
2.770
2.087
0.969
5.397
0.265
0.306
2.786
52.178
46.456
WS20
5.228
7.00
0.130
2.530
0.366
0.903
7.605
1.250
0.458
4.447
39.866
35.765
Average
5.662
8.68
0.125
—
—
0.955
6.074
0.541
0.346
2.940
49.422
43.237
WS21
300°C
3.341
2.45
0.121
4.023
0.361
0.981
10.019
0.491
0.792
3.101
13.939
11.004
WS22
5.513
3.49
0.117
4.090
0.300
0.987
13.570
1.105
0.667
5.065
19.869
17.251
WS23
3.371
1.91
0.113
6.985
0.610
0.982
7.727
0.475
0.672
4.957
10.854
9.384
WS24
4.761
1.99
0.105
4.285
0.345
0.990
12.552
0.987
0.728
6.816
11.350
10.138
WS25
4.076
4.03
0.137
2.172
0.206
0.963
11.532
1.667
0.869
6.874
22.960
20.438
Average
4.213
2.78
0.119
—
—
0.981
11.080
0.945
0.746
5.363
15.794
13.643
WS26
350°C
5.701
6.05
0.131
5.702
0.447
0.995
12.290
1.475
0.599
4.549
34.409
29.878
WS27
3.840
2.03
0.125
6.245
0.335
0.958
14.150
1.490
1.003
6.768
11.538
9.829
WS28
4.718
3.60
0.131
5.553
0.454
0.997
11.312
1.107
0.815
4.053
13.490
16.360
WS29
4.554
3.38
0.130
9.025
0.608
0.995
11.635
0.355
0.821
5.940
19.240
16.582
WS30
3.931
3.21
0.134
7.135
0.532
0.990
10.251
0.654
0.832
5.335
18.279
15.427
Average
4.549
3.65
0.130
—
—
0.987
11.927
1.016
0.814
5.329
19.391
17.615
WS31
400°C
3.584
2.56
0.136
3.934
0.307
0.999
11.737
1.033
0.921
6.937
14.557
12.391
WS32
3.039
1.42
0.126
21.138
0.828
0.992
9.300
0.226
0.904
5.614
8.084
6.331
WS33
3.228
2.12
0.114
2.100
0.224
0.995
8.533
1.575
0.842
5.819
7.348
6.419
WS34
4.476
1.71
0.111
4.208
0.322
0.929
13.388
0.986
0.985
5.940
12.035
8.463
WS35
—
—
—
—
—
—
—
—
—
—
—
—
Average
3.582
1.95
0.122
—
—
0.979
10.740
0.955
0.913
6.078
10.506
8.401
WS36
450°C
3.336
1.41
0.125
5.280
0.299
0.977
13.124
1.553
1.224
5.625
8.037
6.289
WS37
—
—
—
—
—
—
—
—
—
—
—
—
WS38
3.118
1.46
0.123
10.235
0.543
0.981
10.601
0.978
1.057
5.672
8.291
6.216
WS39
3.056
1.34
0.127
2.518
0.137
0.995
11.955
3.960
1.281
6.784
7.611
6.099
WS40
2.935
1.58
0.137
4.328
0.226
0.998
13.896
2.395
1.394
6.000
8.988
6.900
Average
3.111
1.45
0.128
—
—
0.988
12.394
1.663
1.239
6.020
8.232
6.376
WS41
500°C
—
—
—
—
—
—
—
—
—
—
—
—
WS42
2.153
1.76
0.146
3.056
0.245
0.976
7.762
3.054
1.284
5.153
7.251
7.506
WS43
2.857
1.09
0.119
3.648
0.306
0.979
10.092
1.342
1.174
6.533
6.179
5.069
WS44
1.929
0.75
0.109
2.234
0.153
0.968
8.758
4.119
1.225
6.591
4.243
3.651
WS45
1.838
1.48
0.147
2.300
0.186
0.996
7.826
3.012
1.354
6.225
6.078
6.591
Average
2.194
1.27
0.130
—
—
0.977
8.002
2.494
1.259
6.125
5.938
5.704
WS46
600°C
1.129
0.47
0.130
2.102
0.278
0.931
4.197
1.280
1.482
7.000
2.656
2.094
WS47
1.474
0.48
0.128
3.018
0.238
0.954
7.245
2.236
1.784
10.000
2.718
2.233
WS48
1.649
1.14
0.152
1.992
0.132
0.979
8.390
4.551
1.908
7.100
3.909
2.858
WS49
1.138
0.38
0.131
1.183
0.120
0.971
5.018
2.950
1.865
10.000
1.165
1.761
WS50
1.243
0.38
0.124
2.507
0.285
0.943
5.273
1.280
1.644
8.000
1.174
1.727
Average
1.326
0.57
0.133
—
—
0.956
6.025
2.460
1.737
8.420
2.324
2.135Some necessary test results are tabulated in Table1, including the maximum value of the horizontal load Phmax and its corresponding crack opening displacement CODc, the endpoint where Ph1 approaches zero and the crack opening displacement arrives at COD1, the initial slope of Ph-COD, that is, the initial stiffness K0, and the modulus of elasticity E.Hence, the residual fracture energy described by (1) could be calculated. The values are listed in Table 2 and are shown in Figure 10.Table 2
Fracture energies for crack propagation.
Specimen
Temperature
W
0 (Nm)
A
lig (m2)
G
F (Nm−1)
W
OIMA (Nm)
W
AIMA (Nm)
W
FS (Nm)
A
FS (m2)
G
FS (Nm−1)
W
FU (Nm)
A
FU (m2)
G
FU (Nm−1)
WS1
20°C
5.620
0.024
234.147
1.217
0.407
0.810
0.005
152.502
4.809
0.019
257.361
WS2
11.608
483.664
1.575
0.388
1.187
0.003
350.795
10.421
0.021
504.972
WS3
10.517
438.217
2.124
0.470
1.654
0.007
242.557
8.863
0.017
515.877
WS4
5.265
219.394
0.994
0.314
0.680
0.006
107.537
4.586
0.018
259.384
WS5
7.705
321.055
1.693
0.525
1.169
0.005
212.824
6.536
0.019
353.175
Average
8.143
339.295
1.521
0.421
1.100
0.005
213.243
7.043
0.019
378.154
WS6
65°C
7.666
0.024
425.907
1.791
0.535
1.255
0.007
281.835
6.411
0.017
480.239
WS7
8.687
482.619
1.076
0.241
0.835
0.010
113.926
7.852
0.014
768.132
WS8
8.779
487.745
1.475
0.509
0.966
0.007
208.612
7.813
0.017
602.385
WS9
8.649
480.507
1.027
0.440
0.587
0.008
118.113
8.062
0.016
656.640
WS10
9.402
522.360
1.419
0.733
0.686
0.005
210.596
8.716
0.019
614.693
Average
8.637
479.828
1.357
0.492
0.866
0.007
186.616
7.771
0.017
624.418
WS11
120°C
7.137
0.024
396.516
1.335
0.588
0.746
0.003
251.423
6.391
0.021
416.985
WS12
9.321
517.822
1.475
0.511
0.964
0.005
178.684
8.357
0.019
616.201
WS13
11.785
654.732
2.079
0.560
1.520
0.008
193.937
10.266
0.016
878.097
WS14
6.218
345.463
1.044
0.312
0.732
0.007
98.944
5.486
0.017
455.357
WS15
—
—
—
—
—
—
—
—
—
—
Average
8.615
478.633
1.483
0.493
0.991
0.006
180.747
7.625
0.018
591.660
WS16
200°C
—
0.024
—
—
—
—
—
—
—
—
—
WS17
6.780
282.509
1.522
0.345
1.176
0.009
123.021
5.604
0.015
385.812
WS18
5.163
215.135
1.556
0.525
1.031
0.007
146.955
4.132
0.017
243.300
WS19
5.662
235.913
1.252
0.270
0.982
0.009
104.764
4.680
0.015
320.014
WS20
8.854
368.937
1.828
0.365
1.463
0.010
145.364
7.391
0.014
530.467
Average
6.615
275.624
1.540
0.376
1.163
0.009
130.026
5.452
0.015
369.898
WS21
300°C
10.510
0.024
437.920
2.005
0.439
1.566
0.008
188.640
8.944
0.016
568.895
WS22
14.675
611.467
3.546
0.820
2.726
0.007
364.621
11.949
0.017
723.160
WS23
8.202
341.770
2.048
0.579
1.469
0.007
220.394
6.733
0.017
388.453
WS24
13.539
564.120
3.065
1.081
1.984
0.005
404.514
11.555
0.019
605.107
WS25
13.200
549.990
2.552
0.389
2.163
0.011
190.295
11.037
0.013
870.055
Average
12.025
501.053
2.643
0.662
1.982
0.008
273.693
10.044
0.016
631.134
WS26
350°C
13.765
0.024
573.540
2.689
0.511
2.179
0.010
211.652
11.586
0.014
845.346
WS27
15.639
651.638
2.589
0.703
1.887
0.009
207.566
13.753
0.015
922.314
WS28
12.420
517.489
3.827
0.919
2.908
0.010
317.145
9.512
0.014
664.527
WS29
11.990
499.602
2.900
0.611
2.289
0.010
228.349
9.701
0.014
694.230
WS30
10.905
454.375
2.796
0.475
2.320
0.011
216.312
8.585
0.013
646.739
Average
12.944
539.329
2.960
0.644
2.317
0.010
236.205
10.627
0.014
754.631
WS31
400°C
12.770
0.024
532.090
2.240
0.484
1.756
0.011
156.074
11.014
0.013
860.665
WS32
9.526
396.932
2.678
0.656
2.022
0.009
207.133
7.504
0.015
516.314
WS33
10.109
421.188
2.411
0.760
1.651
0.007
242.302
8.457
0.017
491.692
WS34
14.374
598.926
3.638
0.918
2.720
0.006
401.879
11.654
0.018
668.062
WS35
—
—
—
—
—
—
—
—
—
—
Average
11.695
487.284
2.742
0.705
2.037
0.008
251.847
9.658
0.016
634.184
WS36
450°C
14.677
0.024
611.530
3.255
0.779
2.477
0.009
277.088
12.200
0.015
809.983
WS37
—
—
—
—
—
—
—
—
—
—
WS38
11.579
482.451
3.284
0.669
2.615
0.009
303.750
8.964
0.015
582.382
WS39
15.914
663.097
3.281
0.704
2.578
0.009
272.015
13.337
0.015
918.284
WS40
16.291
678.790
3.847
0.532
3.315
0.011
292.743
12.976
0.013
1023.630
Average
14.057
608.967
3.417
0.671
2.746
0.010
286.399
11.869
0.014
833.570
WS41
500°C
—
0.024
—
—
—
—
—
—
—
—
—
WS42
10.816
450.668
2.080
0.355
1.725
0.013
138.244
9.091
0.011
831.501
WS43
11.434
476.413
2.890
0.743
2.146
0.008
272.496
9.288
0.016
576.031
WS44
12.877
536.544
1.495
0.475
1.021
0.006
170.514
11.856
0.018
652.188
WS45
10.838
451.571
2.720
0.308
2.412
0.013
186.562
8.426
0.011
785.892
Average
10.496
478.799
2.296
0.470
1.826
0.010
191.954
9.665
0.014
711.403
WS46
600°C
5.477
0.024
228.226
1.404
0.287
1.117
0.010
110.974
4.361
0.014
312.890
WS47
9.481
395.057
2.526
0.460
2.066
0.010
213.423
7.415
0.014
517.849
WS48
12.941
539.225
2.621
0.389
2.232
0.014
155.808
10.709
0.010
1107.140
WS49
7.968
331.988
1.514
0.635
0.879
0.010
115.238
7.088
0.014
491.494
WS50
6.554
273.074
1.749
0.773
0.976
0.009
151.878
5.578
0.015
342.831
Average
8.484
353.514
1.963
0.509
1.454
0.011
149.464
7.030
0.013
632.922Though the residual fracture energy at each temperature has significant scatter, Figure10 shows that the average values sustain an increase-decrease tendency with Tm. From 20°C to 450°C, average GF increases from 339.3 Nm−1 to 609 Nm−1, while the temperature reaches 600°C and the fracture energy falls back to 307.8 Nm−1. The detailed explanation would be seen elsewhere [12].


## 4.2. Determination of Stable Fracture EnergyGFS and Unstable Fracture Energy GFU

As an extension of fracture energyGF, two other energy-based fracture characteristics, the stable fracture energy GFS and unstable fracture energy GFU, are proposed to describe fracture responses for different crack propagation periods. The analysis shows that fracture energy GF is actually the weighed average of GFS and GFU, and GFS and GFU could be regarded as two components of GF. This is very helpful in understanding the whole crack propagation process from the aspect of energy consumption.To obtain the values ofGFS and GFU, the critical effective crack length ac should first be determined for the calculation of the fracture areas AS and AU. Herein, the value ac was computed based on the double-K fracture model [14]:
(9)a=(h+h0){1-(13.18E·b·c+9.16)1/2}-h0,
where c=COD/P is the compliance of specimens, E is modulus of elasticity, b is specimen thickness, h is specimen height, and h0 is the thickness of the clip gauge holder. For calculation of critical value of equivalent elastic crack length ac, the value of crack mouth opening displacement (COD) and P are taken as CODc and Phmax, respectively. Equation (9) is valid for 0.2–0.8 within 2% accuracy. The value of ac is reported in Table 1.In a typical load-displacement curve shown in Figure1, K0 is the initial stiffness of the load-displacement curve before the start of the fracture process. At critical state when the load arrives at its peak value Pmax and its corresponding displacement reaches CODc, the stiffness Kc may not be the same as the initial K0. The degradation of stiffness is the result of crack propagation. To make it simple, here, the value of K at any value of displacement is assumed to be linear with increased deformation [15]. 								(10)K=K0(COD1-COD)(COD1-COD0),
where COD1 = final deformation when the load P approaches zero and COD0 = displacement before which the stiffness is still kept at K0. At the origin of P-COD curve, the initial tangent stiffness is K0. The stiffness becomes degraded beyond the origin point; hence COD0 is presumed to be zero. It is clear from (10) that K=K0 when COD = 0 and K=0 when = COD1, and K is assumed to be a linear function with respect to the displacement. With the crack opening displacement COD = CODc, the stiffness Kc at this point could approximately be obtained through (10). Note that P-COD is a general term for load-displacement; in our case, the load is the horizontal force Ph and its corresponding displacement is the COD. Therefore, with K0, COD1, and CODc, the approximate degraded stiffness Kc can be gained from (10) and the result is shown in Table 1. Figure 11 shows that the values of stiffness Kc decrease monotonously with temperature due to the thermal damage induced by high temperatures.The stable fracture energyGFS and unstable fracture energy GFU can be derived by following (11)~(13). The specific values of GFS and GFU are compiled in Table 2:
(11)GFS=WFSAS=WOAMOAS=WOIMO-WAIMAAS,GFU=WUAU=W-WFSAU,(12)AS=B(ac-a0),AU=B(h-ac),(13)WAIMA=0.5Phmax2Kc.Figure12 shows the variation of stable fracture energy GFS with temperatures. Similar to the residual fracture energy GF, it also keeps an increase-decrease tendency with temperatures. Temperatures less than 120°C appear not to induce much thermal damage to concrete, so the cracking resistance almost keeps constant. The values of stable fracture energy GFS at these temperatures are 213 Nm−1, 186 Nm−1, and 180 Nm−1, respectively. The fracture surfaces tend to be more tortuous between 200°C and 450°C than those observed at lower temperatures (see Figure 4) and there exist several cracks competing to form the final fracture, so more energy was dissipated in these specimens. Additionally, the opening of microcracks on the surface and inside the specimens also dissipates energy (see Figure 3). Finally, higher heating temperatures would cause more micro cracks, dehydration, and decomposition and would degrade the resistance. After 450°C, cracking resistance continuously decreases with Tm.Table2 shows that the unstable fracture GFU also sustains an increase-decrease tendency with Tm, and its value is much larger than stable fracture energy. Two reasons for larger values of GFU are provided [5]. First, energy consumption besides the main fracture zone takes places for the whole fracture process. During the unstable fracture process, the energy consumption for plasticity or other nonlinear deformation beyond the main FPZ would be much higher, especially for specimens subjected to high temperatures. The other possible reason lies in the calculation of true fracture area. It is known that the interface (transition zone) between cement and aggregate is the weak link in microstructure for normal concrete, and crack propagation would proceed in the path where the energy needs are least. So the true path of crack extension is tortuous and the higher the temperature is, the more tortuous the fracture surface is (see Figure 4), not straight as expected. The projected area is used in the calculation of fracturing surface AS or AU, which underestimates the true fracture area. Since the crack experiences much longer distance for the unstable extension, which leads to a greater underestimation of the calculation of the newly fractured area AU, the calculated GFU is overestimated.Moreover, in view of (11) and (12), another expression for the fracture energy GF with respect to GFS and GFU can be followed:
(14)GF=WA=GFSAS+GFUAUA=GFS·ASA+GFU·AUA.
Since AS+AU=A, the fracture energy GF is the weighed average of GFS and GFU. However, for engineering application, the stable crack propagation is considered to be more important, because when the load exceeds the maximum value the whole structure would be in an unstable situation. Hence from Figures 7 and 9, it is concluded that the fracture property of postfire concrete sustains an increase-decrease tendency to 600°C, with a turning point at 450°C.Figure 7
Variation tendency ofPhmaxwith Tm.Figure 8
Variation tendency ofCODc with Tm.P-CMOD tail curve fitting of specimens.
(a)
Curve fitting for specimen of ambient temperature
(b)
Curve fitting for specimen of 200°CFigure 10
Residual fracture energy withTm.Figure 11
StiffnessKc with Tm.Figure 12
Stable fracture energy withTm.Furthermore, as mentioned above, the fracture surfaces ofA, AS, AU are project areas, so GF, GFS, GFU are nominal fracture energies. To avoid the violence of fracture surface, the variation tendency of stable fracture work WFS is determined (see Figure 10). Similarly, WFS also keeps an increase-decrease tendency at the same turning temperature of 450°C.


## 5. Conclusions

Energy consumption during an entire crack propagation period has been investigated, including the fracture energyGF, stable fracture energy GFS, unstable fracture energy GFU, and stable fracture work WFS. The conclusions are as follows.(1)
Wedge-splitting tests of ten temperatures levels varying from room temperature to 600°C and the specimen size of 230 mm × 200 mm × 200 mm with initial-notch depth ratios 0.4 have been presented. CompleteP-COD curves and the curve tails are obtained using exponential and power functions. For specimens subject to no more than 120°C, the power function is more accurate; for higher temperatures, exponential function is more suitable.(2)
Three fracture energy quantities corresponding to different aspects of fracture are proposed. The fracture energyGF in a general case only represents the average energy dissipation for an entire crack propagation process. GFS, the stable fracture energy, denotes the average energy absorption during crack stable propagation, and GFU is used to characterize the average energy consumption for crack unstable propagation and a higher value of GFU than GFS is observed. GF is actually the weighed average of GFS and GFU.(3)
However, for engineering applications, the stable crack propagation is considered to be more important. From the wedge-splitting tests of different temperatures, it is concluded thatGF, GFS sustain an increase-decrease tendency to 600°C, with a turning point at 450°C. Furthermore, the variation of stable fracture work WFS is determined and shares the same tendency with GF and GFS (Figure 13). All these three parameters mean that the fracture property of postfire concrete sustains an increase-decrease tendency.Figure 13
Stable fracture work withTm.



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*Source: 143208-2013-11-30.xml*