Advances in Materials Science and Engineering

Volume 2013 (2013), Article ID 143208, 13 pages

http://dx.doi.org/10.1155/2013/143208

Research Article

Average Fracture Energy for Crack Propagation in Postfire Concrete

Kequan Yu, Jiangtao Yu, and Zhoudao Lu

College of Civil Engineering, Tongji University, Shanghai 200092, China

Received 20 September 2013; Accepted 20 October 2013

Academic Editor: Filippo Berto

Copyright © 2013 Kequan Yu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 energy , stable fracture energy , and unstable fracture energy ) with their variation tendency and their mutual relationship were determined to predict energy consumption for the complete fracture propagation. Additionally, the stable fracture work 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.

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 energy 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 . It is computed as the area under the entire imposed load and load-line displacement curve divided by the projected area of uncracked ligament, so the fracture energy 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 with , 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 and unstable fracture energy to describe fracture responses for different crack propagation periods. It is found that kept constant for different ligament lengths, whereas and 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 is calculated considering the influence of load-displacement tail. Furthermore, the fracture energy consumption for crack stable extension and unstable extension, that is, , , 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 and , 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 force 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]:

where = average energy needed for unit crack propagation during period from crack initiation to any instant of time ; = newly formed fracture area for this period; the product of these two quantities represents the energy absorbed for crack propagation during this period; = work performed by the external force during the same period; = increase of elastic strain energy of the body until time ; = crack opening displacement corresponding to . If time is approaching the failure terminal point = 1, where the load drops to zero, then in (1) will reduce to

where = RILEM fracture energy; = total energy provided by the external force for crack propagation; = projected fracture area perpendicular to the tensile stress direction; = specimen thickness; = specimen height; = initial crack length.

During the stable crack propagation period, the load will increase up to its peak value with its corresponding displacement reaching ; see Figure 1. So, based on (1), the average energy absorbed during this period can be evaluated by

where by naming the stable fracture energy = average energy needed for the crack to grow unit area during the crack stable propagation; = change in fracture area; energy provided for stable crack propagation as shown in Figure 1 by the shaded area; work performed by the external force for crack increases its area ; = = increased elastic strain energy of the body; and = critical crack opening displacement.

Figure 1: Energy consumption for an entire crack propagation period.

From the definition of the stable fracture energy , it is implied that the work from the imposed load 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 () 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 , is defined based on energy equilibrium as

where = energy needed for the crack unstable propagation period and = change in fracture area. During this period, the stored energy is completely released until the deformation reaches its maximum value when the load closes to zero. Similar to the RILEM fracture energy , the establishment of the stable fracture energy and unstable fracture energy 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 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 Figure 2 ( mm,  mm,  mm,  mm,  mm, and ). 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 ( = 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 designated 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 Figure 5, which became lighter but more tortuous with increasing temperatures.

(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)

Figure 5: Fracture surfaces of postfire specimens.

A universal machine with a maximum capacity of 1000 kN was used to conduct the wedge splitting tests. During the test, the vertically applied load and the crack opening displacement along the horizontal load line were simultaneously recorded through a data acquisition system. Referring to Figure 2, the splitting force 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 can be developed based on force equilibrium ignoring the small influence from the friction for different roller bearings:

To obtain the complete - 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.

(a) versus curves of specimens at 120°C

(b) versus curves of specimens at 450°C

(c) versus curves of specimens at 600°C

(d) Typical versus curves of specimens at all temperatures

Figure 6: versus CMOD curves of specimens with temperatures.

The fracture of specimen is essentially due to the bending moment caused by the horizontal splitting force , vertical component , 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 contributes most to fracturing the specimen. Therefore, - curves were directly used in the calculation of the RILEM fracture energy , stable fracture energy , and unstable fracture energy .

For our test results, the - curves could be easily obtained from the monitored - curves and (5). Figure 6 contains the plots of - curves for several temperatures and typical - curves for all temperatures.

From Figure 6(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 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 - curves of all temperatures. The ultimate load decreases significantly with increasing temperatures , whereas the crack mouth opening displacement () increases with . 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 load decreases with the increasing temperatures, whereas the increases with (Figure 8). The average value of 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 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 Energy

In the calculation of RILEM fracture energy , 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 - curve includes the self-weight; hence the work should not be calculated again. The other concern pertains to the tail part of the - curve: the recorded point (, ) 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 based on the work-of-fracture method becomes (see Figure 9)

where = measured work enclosed by the - curve until = ; = fresh fracture area; is the work caused by tail effect part. For , 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 - curve after one-third of the peak load could be well described by power function (the coefficient of determination 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 function , , where , = constants for fitting curves. And then the unrecorded work owning to cutting load-deflection tail can be written as

For exponential function , , where , = constants for fitting curves. And then the unrecorded work owning to cutting load-deflection tail can be written as

The parameters of , for 20°C~120°C and , for 200°C~600°C, and the value of , is listed in Table 1.

Table 1: Parameters calculated from wedge-splitting test.

Specimen
Temperature
(kN)
(GPa)
(mm)
/
/

(mm)
COD₁ (mm)
(kN/mm)
(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
2.135

Some necessary test results are tabulated in Table 1, including the maximum value of the horizontal load and its corresponding crack opening displacement , the endpoint where approaches zero and the crack opening displacement arrives at , the initial slope of -, that is, the initial stiffness , and the modulus of elasticity .

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
(Nm)
(m²)
(Nm⁻¹)
(Nm)
(Nm)
(Nm)
(m²)
(Nm⁻¹)
(Nm)
(m²)
(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
632.922

Though the residual fracture energy at each temperature has significant scatter, Figure 10 shows that the average values sustain an increase-decrease tendency with . From 20°C to 450°C, average 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 Energy and Unstable Fracture Energy

As an extension of fracture energy , two other energy-based fracture characteristics, the stable fracture energy and unstable fracture energy , are proposed to describe fracture responses for different crack propagation periods. The analysis shows that fracture energy is actually the weighed average of and , and and could be regarded as two components of . This is very helpful in understanding the whole crack propagation process from the aspect of energy consumption.

To obtain the values of and , the critical effective crack length should first be determined for the calculation of the fracture areas and . Herein, the value was computed based on the double- fracture model [14]:

where is the compliance of specimens, is modulus of elasticity, is specimen thickness, is specimen height, and is the thickness of the clip gauge holder. For calculation of critical value of equivalent elastic crack length , the value of crack mouth opening displacement () and are taken as and , respectively. Equation (9) is valid for 0.2–0.8 within 2% accuracy. The value of is reported in Table 1.

In a typical load-displacement curve shown in Figure 1, 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 and its corresponding displacement reaches , the stiffness may not be the same as the initial . The degradation of stiffness is the result of crack propagation. To make it simple, here, the value of at any value of displacement is assumed to be linear with increased deformation [15].

where = final deformation when the load approaches zero and = displacement before which the stiffness is still kept at . At the origin of - curve, the initial tangent stiffness is . The stiffness becomes degraded beyond the origin point; hence is presumed to be zero. It is clear from (10) that when = 0 and when = , and is assumed to be a linear function with respect to the displacement. With the crack opening displacement = , the stiffness at this point could approximately be obtained through (10). Note that - is a general term for load-displacement; in our case, the load is the horizontal force and its corresponding displacement is the . Therefore, with , , and , the approximate degraded stiffness can be gained from (10) and the result is shown in Table 1. Figure 11 shows that the values of stiffness decrease monotonously with temperature due to the thermal damage induced by high temperatures.

The stable fracture energy and unstable fracture energy can be derived by following (11)~(13). The specific values of and are compiled in Table 2:

Figure 12 shows the variation of stable fracture energy with temperatures. Similar to the residual fracture energy , 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 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 .

Table 2 shows that the unstable fracture also sustains an increase-decrease tendency with , and its value is much larger than stable fracture energy. Two reasons for larger values of 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 or , 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 , the calculated is overestimated.

Moreover, in view of (11) and (12), another expression for the fracture energy with respect to and can be followed:

Since , the fracture energy is the weighed average of and . 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 of with .

Figure 8: Variation tendency of with .

(a) Curve fitting for specimen of ambient temperature

(b) Curve fitting for specimen of 200°C

Figure 9: -CMOD tail curve fitting of specimens.

Figure 10: Residual fracture energy with .

Figure 11: Stiffness with .

Figure 12: Stable fracture energy with .

Furthermore, as mentioned above, the fracture surfaces of , , are project areas, so , , are nominal fracture energies. To avoid the violence of fracture surface, the variation tendency of stable fracture work is determined (see Figure 10). Similarly, 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 energy , stable fracture energy , unstable fracture energy , and stable fracture work . 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. Complete - 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 energy in a general case only represents the average energy dissipation for an entire crack propagation process. , the stable fracture energy, denotes the average energy absorption during crack stable propagation, and is used to characterize the average energy consumption for crack unstable propagation and a higher value of than is observed. is actually the weighed average of and .(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 that , sustain an increase-decrease tendency to 600°C, with a turning point at 450°C. Furthermore, the variation of stable fracture work is determined and shares the same tendency with and (Figure 13). All these three parameters mean that the fracture property of postfire concrete sustains an increase-decrease tendency.

Figure 13: Stable fracture work with .

Acknowledgments

The State Key Laboratory of Disaster Reduction in Civil Engineering (SLDRCE09-D-02) and Young Scientist Project of Natural Science Foundation of China (NSFC) have supported this research.

References

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