Advances in Materials Science and Engineering

Advances in Materials Science and Engineering / 2013 / Article
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Brittle or Quasi-Brittle Fracture of Engineering Materials: Recent Developments and New Challenges

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Research Article | Open Access

Volume 2013 |Article ID 143208 | https://doi.org/10.1155/2013/143208

Kequan Yu, Jiangtao Yu, Zhoudao Lu, "Average Fracture Energy for Crack Propagation in Postfire Concrete", Advances in Materials Science and Engineering, vol. 2013, Article ID 143208, 13 pages, 2013. https://doi.org/10.1155/2013/143208

Average Fracture Energy for Crack Propagation in Postfire Concrete

Academic Editor: Filippo Berto
Received20 Sep 2013
Accepted20 Oct 2013
Published30 Nov 2013

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 [712]. 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.

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.

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).

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 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.

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.


SpecimenTemperature (kN) (GPa) (mm) / / (mm)COD1 (mm) (kN/mm) (kN/mm)

WS120°C8.304 15.30 0.107 1.484 2.273 0.970 5.231 0.389 0.174 1.926 87.081 79.219
WS29.407 20.51 0.097 3.131 1.206 0.998 10.468 1.140 0.205 4.914 116.710 113.648
WS310.379 20.66 0.114 2.464 1.117 0.991 8.699 1.818 0.195 3.280 117.610 110.633
WS47.884 18.88 0.112 1.599 3.224 0.964 5.128 0.137 0.152 1.903 107.480 98.911
WS59.364 15.45 0.107 2.048 1.683 0.991 6.970 0.735 0.199 2.657 87.940 80.103

Average9.068 18.16 0.107 0.983 7.299 0.844 0.185 2.936 103.364 96.503

WS665°C11.282 21.73 0.113 2.781 1.506 0.996 9.238 0.984 0.195 3.205 123.690 116.172
WS78.151 24.79 0.132 2.739 0.987 0.999 10.320 1.263 0.215 6.072 141.100 136.109
WS810.379 19.43 0.115 6.114 1.992 0.954 11.004 0.702 0.212 4.549 110.560 105.418
WS910.681 23.25 0.119 2.658 1.030 0.993 9.283 2.250 0.164 3.535 132.330 124.685
WS1011.610 16.60 0.107 3.185 1.252 0.983 11.466 1.071 0.190 5.324 94.490 90.418

Average10.421 21.16 0.117 0.985 10.262 1.254 0.195 4.537 120.434 114.560

WS11120°C8.353 10.65 0.095 2.477 1.085 0.990 8.423 1.094 0.191 3.418 60.638 57.254
WS128.226 11.87 0.107 2.667 0.890 0.999 10.624 1.803 0.224 6.443 67.564 65.212
WS137.631 9.48 0.119 4.340 0.998 0.980 13.227 2.487 0.357 5.641 53.926 50.511
WS147.302 15.42 0.117 2.139 1.153 0.998 6.934 1.357 0.198 3.534 87.758 82.839
WS15

Average7.878 11.85 0.109 0.992 9.802 1.685 0.243 4.759 67.472 63.954

WS16200°C
WS176.466 11.58 0.127 2.266 1.526 0.952 6.260 0.520 0.284 2.558 65.913 57.759
WS185.884 6.98 0.115 2.268 2.982 0.995 5.035 0.129 0.335 1.968 39.732 32.967
WS195.071 9.17 0.127 2.770 2.087 0.969 5.397 0.265 0.306 2.786 52.178 46.456
WS205.228 7.00 0.130 2.530 0.366 0.903 7.605 1.250 0.458 4.447 39.866 35.765

Average5.662 8.680.125 0.955 6.074 0.541 0.346 2.940 49.422 43.237

WS21300°C3.341 2.45 0.121 4.023 0.361 0.981 10.019 0.491 0.792 3.101 13.939 11.004
WS225.513 3.49 0.117 4.090 0.300 0.987 13.570 1.105 0.667 5.065 19.869 17.251
WS233.371 1.91 0.113 6.985 0.610 0.982 7.727 0.475 0.672 4.957 10.854 9.384
WS244.761 1.99 0.105 4.285 0.345 0.990 12.552 0.987 0.728 6.816 11.350 10.138
WS254.076 4.03 0.137 2.172 0.206 0.963 11.532 1.667 0.869 6.874 22.960 20.438

Average4.213 2.78 0.119 0.981 11.080 0.945 0.746 5.363 15.794 13.643

WS26350°C5.701 6.05 0.131 5.702 0.447 0.995 12.290 1.475 0.599 4.549 34.409 29.878
WS273.840 2.03 0.125 6.245 0.335 0.958 14.150 1.490 1.003 6.768 11.538 9.829
WS284.718 3.60 0.131 5.553 0.454 0.997 11.312 1.107 0.815 4.053 13.490 16.360
WS294.554 3.38 0.130 9.025 0.608 0.995 11.635 0.355 0.821 5.940 19.240 16.582
WS303.931 3.21 0.134 7.135 0.532 0.990 10.251 0.654 0.832 5.335 18.279 15.427

Average4.549 3.65 0.130 0.987 11.927 1.016 0.814 5.329 19.391 17.615

WS31400°C3.584 2.56 0.136 3.934 0.307 0.999 11.737 1.033 0.921 6.937 14.557 12.391
WS323.039 1.42 0.126 21.138 0.828 0.992 9.300 0.226 0.904 5.614 8.084 6.331
WS333.228 2.12 0.114 2.100 0.224 0.995 8.533 1.575 0.842 5.819 7.348 6.419
WS344.476 1.71 0.111 4.208 0.322 0.929 13.388 0.986 0.985 5.940 12.035 8.463
WS35

Average3.582 1.950.122 0.979 10.740 0.955 0.913 6.078 10.506 8.401

WS36450°C3.336 1.41 0.125 5.280 0.299 0.977 13.124 1.553 1.224 5.625 8.037 6.289
WS37
WS383.118 1.46 0.123 10.235 0.543 0.981 10.601 0.978 1.057 5.672 8.291 6.216
WS393.056 1.34 0.127 2.518 0.137 0.995 11.955 3.960 1.281 6.784 7.611 6.099
WS402.935 1.58 0.137 4.328 0.226 0.998 13.896 2.395 1.394 6.000 8.988 6.900

Average3.111 1.45 0.128 0.988 12.394 1.663 1.239 6.020 8.232 6.376

WS41500°C
WS422.153 1.76 0.146 3.056 0.245 0.976 7.762 3.054 1.284 5.153 7.251 7.506
WS432.857 1.09 0.119 3.648 0.306 0.979 10.092 1.342 1.174 6.533 6.179 5.069
WS441.929 0.75 0.109 2.234 0.153 0.968 8.758 4.119 1.225 6.591 4.243 3.651
WS451.838 1.48 0.147 2.300 0.186 0.996 7.826 3.012 1.354 6.225 6.078 6.591

Average2.194 1.27 0.130 0.977 8.002 2.494 1.259 6.125 5.938 5.704

WS46600°C1.129 0.47 0.130 2.102 0.278 0.931 4.197 1.280 1.482 7.000 2.656 2.094
WS471.474 0.48 0.128 3.018 0.238 0.954 7.245 2.236 1.784 10.000 2.718 2.233
WS481.649 1.14 0.152 1.992 0.132 0.979 8.390 4.551 1.908 7.100 3.909 2.858
WS491.138 0.38 0.131 1.183 0.120 0.971 5.018 2.950 1.865 10.000 1.165 1.761
WS501.243 0.38 0.124 2.507 0.285 0.943 5.273 1.280 1.644 8.000 1.174 1.727

Average1.326 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.


SpecimenTemperature (Nm) (m2) (Nm−1) (Nm) (Nm) (Nm) (m2) (Nm−1) (Nm) (m2) (Nm−1)

WS120°C5.620 0.024 234.147 1.217 0.407 0.810 0.005 152.502 4.809 0.019 257.361
WS211.608 483.664 1.575 0.388 1.187 0.003 350.795 10.421 0.021 504.972
WS310.517 438.217 2.124 0.470 1.654 0.007 242.557 8.863 0.017 515.877
WS45.265 219.394 0.994 0.314 0.680 0.006 107.537 4.586 0.018 259.384
WS57.705 321.055 1.693 0.525 1.169 0.005 212.824 6.536 0.019 353.175

Average8.143 339.295 1.521 0.421 1.100 0.005 213.243 7.043 0.019 378.154

WS665°C7.666 0.024425.907 1.791 0.535 1.255 0.007 281.835 6.411 0.017 480.239
WS78.687 482.619 1.076 0.241 0.835 0.010 113.926 7.852 0.014 768.132
WS88.779 487.745 1.475 0.509 0.966 0.007 208.612 7.813 0.017 602.385
WS98.649 480.507 1.027 0.440 0.587 0.008 118.113 8.062 0.016 656.640
WS109.402 522.360 1.419 0.733 0.686 0.005 210.596 8.716 0.019 614.693

Average8.637 479.828 1.357 0.492 0.866 0.007 186.616 7.771 0.017 624.418

WS11120°C7.137 0.024 396.516 1.335 0.588 0.746 0.003 251.423 6.391 0.021 416.985
WS129.321 517.822 1.475 0.511 0.964 0.005 178.684 8.357 0.019 616.201
WS1311.785 654.732 2.079 0.560 1.520 0.008 193.937 10.266 0.016 878.097
WS146.218 345.463 1.044 0.312 0.732 0.007 98.944 5.486 0.017 455.357
WS15

Average8.615478.633 1.483 0.493 0.991 0.006 180.747 7.625 0.018 591.660

WS16200°C0.024
WS176.780 282.509 1.522 0.345 1.176 0.009 123.021 5.604 0.015 385.812
WS185.163 215.135 1.556 0.525 1.031 0.007 146.955 4.132 0.017 243.300
WS195.662 235.913 1.252 0.270 0.982 0.009 104.764 4.680 0.015 320.014
WS208.854 368.937 1.828 0.365 1.463 0.010 145.364 7.391 0.014 530.467

Average6.615275.624 1.540 0.376 1.163 0.009 130.026 5.452 0.015 369.898

WS21300°C10.510 0.024 437.920 2.005 0.439 1.566 0.008 188.640 8.944 0.016 568.895
WS2214.675 611.467 3.546 0.820 2.726 0.007 364.621 11.949 0.017 723.160
WS238.202 341.770 2.048 0.579 1.469 0.007 220.394 6.733 0.017 388.453
WS2413.539 564.120 3.065 1.081 1.984 0.005 404.514 11.555 0.019 605.107
WS2513.200 549.990 2.552 0.389 2.163 0.011 190.295 11.037 0.013 870.055

Average12.025 501.053 2.643 0.662 1.982 0.008 273.693 10.044 0.016 631.134

WS26350°C13.765 0.024 573.540 2.689 0.511 2.179 0.010 211.652 11.586 0.014 845.346
WS2715.639 651.638 2.589 0.703 1.887 0.009 207.566 13.753 0.015 922.314
WS2812.420 517.489 3.827 0.919 2.908 0.010 317.145 9.512 0.014 664.527
WS2911.990 499.602 2.900 0.611 2.289 0.010 228.349 9.701 0.014 694.230
WS3010.905 454.375 2.796 0.475 2.320 0.011 216.312 8.585 0.013 646.739

Average12.944 539.329 2.960 0.644 2.317 0.010 236.205 10.627 0.014 754.631

WS31400°C12.770 0.024 532.090 2.240 0.484 1.756 0.011 156.074 11.014 0.013 860.665
WS329.526 396.932 2.678 0.656 2.022 0.009 207.133 7.504 0.015 516.314
WS3310.109 421.188 2.411 0.760 1.651 0.007 242.302 8.457 0.017 491.692
WS3414.374 598.926 3.638 0.918 2.720 0.006 401.879 11.654 0.018 668.062
WS35

Average11.695 487.284 2.742 0.705 2.037 0.008 251.847 9.658 0.016 634.184

WS36450°C14.677 0.024 611.530 3.255 0.779 2.477 0.009 277.088 12.200 0.015 809.983
WS37
WS3811.579 482.451 3.284 0.669 2.615 0.009 303.750 8.964 0.015 582.382
WS3915.914 663.097 3.281 0.704 2.578 0.009 272.015 13.337 0.015 918.284
WS4016.291 678.790 3.847 0.532 3.315 0.011 292.743 12.976 0.013 1023.630

Average14.057 608.967 3.417 0.671 2.746 0.010 286.399 11.869 0.014 833.570

WS41500°C0.024
WS4210.816 450.668 2.080 0.355 1.725 0.013 138.244 9.091 0.011 831.501
WS4311.434 476.413 2.890 0.743 2.146 0.008 272.496 9.288 0.016 576.031
WS4412.877 536.544 1.495 0.475 1.021 0.006 170.514 11.856 0.018 652.188
WS4510.838 451.571 2.720 0.308 2.412 0.013 186.562 8.426 0.011 785.892

Average10.496 478.799 2.296 0.470 1.826 0.010 191.954 9.665 0.014 711.403

WS46600°C5.477 0.024 228.226 1.404 0.287 1.117 0.010 110.974 4.361 0.014 312.890
WS479.481 395.057 2.526 0.460 2.066 0.010 213.423 7.415 0.014 517.849
WS4812.941 539.225 2.621 0.389 2.232 0.014 155.808 10.709 0.010 1107.140
WS497.968 331.988 1.514 0.635 0.879 0.010 115.238 7.088 0.014 491.494
WS506.554 273.074 1.749 0.773 0.976 0.009 151.878 5.578 0.015 342.831

Average8.484 353.514 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.

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.

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.

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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.


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