This paper investigates the unstable fracture toughness of specimens of different heights using the double-K model for three-point bending tests on notched concrete beams. It is shown that unstable fracture toughness exhibits a significant size effect. The modified maximum tangential stress (MMTS) criterion is used to explain the size effect of unstable fracture toughness. The MMTS criterion considers the higher order terms of the Williams series expansion of the stress field. The results show that the MMTS criterion can reasonably estimate unstable fracture toughness. It is recommended that the minimum height of the specimen be 200 mm when three-point bending tests on notched beams are used to determine unstable fracture toughness.
Fracture toughness is an important property of fracture resistance, especially for quasibrittle materials such as rock and concrete. It is involved in building structure safety, energy mining, and other fields. Therefore, many researchers and organizations have measured fracture toughness with various specimen configurations and methods. For example, the American Society for Testing and Materials (ASTM) [
This is because a fracture process zone (FPZ) exists at the crack tip, and due to its influence, the linear elastic fracture mechanics (LEFM) are no longer suitable for quasibrittle materials [
However, the effect of size on the double-K parameters is not sufficient, especially for small size test pieces. In addition to FPZ, the high-order terms of the crack tip stress field (in Williams’s series expansions) are also important factors influencing the stress intensity factor. For example, Aliha et al. and Ayatollahi et al. [
It should be noted that the actual fracture process of the crack is very complicated. Not only will the I model fracture but also the II model or even the III model may occur. For example, Golewski [
The purpose of this paper is to study the size effect of the unstable fracture toughness
Since a span-to-depth ratio (
The stress intensity factor can be obtained directly in Abaqus finite element software using the contour integral method [
The finite element models of the notched beam were simulated with plane stress conditions, as shown in Figure
Model of three-point bending tests on notched beams in Abaqus (
Parameters for simulating three-point bending test of notched beam in Abaqus and results of simulation.
CMOD (m) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
2.4 | 2.17 | 0.04 | 1 | 0.2 | 0.2 | 1 | 0.25 | 4.58 | 4.19371 | 2.70 | 1.62 | 1.41 |
2.4 | 2.17 | 0.04 | 1 | 0.3 | 0.3 | 1 | 0.25 | 4.09 | 6.53230 | 3.14 | 1.72 | 1.63 |
2.4 | 2.17 | 0.04 | 1 | 0.4 | 0.4 | 1 | 0.25 | 3.35 | 8.82119 | 3.32 | 1.93 | 2.02 |
2.4 | 2.17 | 0.04 | 1 | 0.5 | 0.5 | 1 | 0.25 | 2.35 | 1.05873 | 3.16 | 2.34 | 2.77 |
2.4 | 2.17 | 0.04 | 1 | 0.6 | 0.6 | 1 | 0.25 | 1.59 | 1.28538 | 3.05 | 3.05 | 4.15 |
The stress intensity factor and CMOD expressions for notched beams are as follows:
According to equations (
Relative error | Relative error | Relative error | ||||||
---|---|---|---|---|---|---|---|---|
0.2 | 1.6216 | 1.4434 | 0.8392 | 1.3780 | 1.4434 | 1.24 | 2.10 | 4.75 |
0.3 | 1.7192 | 1.5932 | 1.1324 | 1.6008 | 1.5932 | 1.25 | 2.06 | 0.47 |
0.4 | 1.9287 | 2.0189 | 1.5204 | 2.0288 | 2.0189 | 1.94 | 0.27 | 0.49 |
0.5 | 2.3389 | 2.7754 | 2.0985 | 2.7891 | 2.7754 | 1.48 | 0.78 | 0.49 |
0.6 | 3.0536 | 4.1445 | 3.0576 | 4.2170 | 4.1445 | 1.54 | 1.65 | 1.72 |
Both
In order to verify the validity of the method, the
It can be seen that the relative errors of the
Since
According to equation (
It can be seen that a span-to-depth ratio of 4 corresponds to larger
For the stress field at the tip of the model I crack (Figure
Stress components at crack tip shown in the rectangular coordinate system.
Since the coefficients
According to the maximum tangential stress (MTS) criterion [
For model I crack,
According to equation (
Before fracture failure of quasibrittle materials, there are inelastic zones (fracture process zones) at the tip of the original crack, as shown in Figure
Stress distributions for fictitious crack model (Hillerborg et al.) and equivalent elastic stress field.
The double-K model proposes a convenient method to determine the critical effective crack length
If the stress intensity factor is calculated using the MMTS criterion, the equivalent plastic zone length
For plane strain, Δ
Critical stress
Ignoring the high-order terms, the apparent fracture toughness
Therefore, the ratio
The ratio
Ayatollahi and Nejati [
According to the results in Table
Results calculated from the double-K model.
Label | Δ | |||||||
---|---|---|---|---|---|---|---|---|
Aa | 1.09 | 3.88 | 5.00 | 3.29 | 3.7548 | 4.66 | 1.08 | 1.4615 |
Ba | 4.69 | 3.89 | 2.15 | 3.36 | 4.1117 | 7.54 | 6.40 | 1.4596 |
Ca | 2.02 | 4.13 | 9.30 | 3.45 | 4.2080 | 7.53 | 3.58 | 1.1967 |
Da | 8.65 | 4.04 | 4.00 | 3.65 | 4.4377 | 7.90 | 1.80 | 1.0001 |
Dimensionless coefficient ratio
Ratio of high-order terms | Value |
---|---|
–0.32644 | |
–0.22786 | |
0.228546 | |
–0.39727 | |
0.916394 | |
–1.86374 | |
3.640217 | |
–9.39833 | |
23.50735 |
It is worth noting that ratio
Hoover et al. [
Geometric parameters of specimens for three-point bending tests on notched beams [
Label | Depth | Width | Span | |
---|---|---|---|---|
Aa01 | 500 | 38.13333 | 1089 | 0.3 |
Aa02 | 500 | 37.63333 | 1088 | 0.3 |
Aa03 | 500 | 41.77778 | 1088 | 0.3 |
Aa04 | 500 | 37.08889 | 1088.5 | 0.3 |
Aa05 | 500 | 38.88889 | 1089 | 0.3 |
Aa06 | 500 | 39.54444 | 1088 | 0.3 |
Ba01 | 215 | 40.24444 | 468 | 0.3 |
Ba02 | 215 | 38.38889 | 468.5 | 0.3 |
Ba03 | 215 | 39.16667 | 470 | 0.3 |
Ba04 | 215 | 39.11111 | 468.5 | 0.3 |
Ba05 | 215 | 38.8 | 469 | 0.3 |
Ba06 | 215 | 37.91111 | 469 | 0.3 |
Ca01 | 93 | 41.808 | 201.5 | 0.3 |
Ca02 | 93 | 41.474 | 201.5 | 0.3 |
Ca03 | 93 | 41.176 | 200.83 | 0.3 |
Ca04 | 93 | 40.12 | 201.4 | 0.3 |
Ca05 | 93 | 42.028 | 202.95 | 0.3 |
Ca06 | 93 | 41.118 | 202.42 | 0.3 |
Ca07 | 93 | 41.548 | 201.295 | 0.3 |
Ca08 | 93 | 41.4 | 201.84 | 0.3 |
Da01 | 40 | 40.292 | 86.82 | 0.3 |
Da02 | 40 | 40.898 | 84.85 | 0.3 |
Da03 | 40 | 40.776 | 87.01 | 0.3 |
Da04 | 40 | 40 | 86.03 | 0.3 |
Da05 | 40 | 40.074 | 87.14 | 0.3 |
Da06 | 40 | 40.21 | 87.07 | 0.3 |
Da07 | 40 | 39.882 | 86.6 | 0.3 |
Da08 | 40 | 40.794 | 86.43 | 0.3 |
Wendner et al. [
For convenience, the test data of the database are used directly, and the nominal stress-strain curves are transformed into a load-crack opening curve (P-CMOD), as shown in Figure
Average response curves of load versus crack mouth opening-displacement (P-CMOD) curves by three-point bending tests on notched beams with
The authors of this paper are very grateful to Hoover et al. and Wendner et al. for publishing the detailed test results.
The maximum load
In order to obtain the law that the critical effective crack length
Critical effective crack length ac versus height
For comparison purposes, the unstable fracture toughness
Equation (
This is a linear function about 1/
Linear regression for calculating
Then, the values of
The above values are substituted into equation (
Variation in values of
According to the results of Wendner et al. [
Variation in ratio
As can be seen from Figure
Thus, the values of
Values of
0.04 | 0.330 | 0.676 | 0.552 | 0.610 | 0.567 | 0.607 | 0.575 | 0.599 | 0.576 | 0.597 |
0.093 | 0.167 | 0.836 | 0.805 | 0.812 | 0.809 | 0.811 | 0.810 | 0.810 | 0.810 | 0.810 |
0.15 | 0.109 | 0.893 | 0.880 | 0.882 | 0.881 | 0.881 | 0.881 | 0.881 | 0.881 | 0.881 |
0.215 | 0.078 | 0.924 | 0.917 | 0.917 | 0.917 | 0.917 | 0.917 | 0.917 | 0.917 | 0.917 |
0.3 | 0.057 | 0.944 | 0.941 | 0.941 | 0.941 | 0.941 | 0.941 | 0.941 | 0.941 | 0.941 |
0.35 | 0.049 | 0.952 | 0.949 | 0.949 | 0.949 | 0.949 | 0.949 | 0.949 | 0.949 | 0.949 |
0.4 | 0.043 | 0.958 | 0.956 | 0.956 | 0.956 | 0.956 | 0.956 | 0.956 | 0.956 | 0.956 |
0.45 | 0.039 | 0.962 | 0.961 | 0.961 | 0.961 | 0.961 | 0.961 | 0.961 | 0.961 | 0.961 |
0.5 | 0.035 | 0.966 | 0.965 | 0.965 | 0.965 | 0.965 | 0.965 | 0.965 | 0.965 | 0.965 |
0.6 | 0.029 | 0.971 | 0.971 | 0.971 | 0.971 | 0.971 | 0.971 | 0.971 | 0.971 | 0.971 |
0.7 | 0.025 | 0.975 | 0.975 | 0.975 | 0.975 | 0.975 | 0.975 | 0.975 | 0.975 | 0.975 |
0.8 | 0.022 | 0.978 | 0.978 | 0.978 | 0.978 | 0.978 | 0.978 | 0.978 | 0.978 | 0.978 |
0.9 | 0.020 | 0.981 | 0.980 | 0.980 | 0.980 | 0.980 | 0.980 | 0.980 | 0.980 | 0.980 |
1 | 0.018 | 0.983 | 0.982 | 0.982 | 0.982 | 0.982 | 0.982 | 0.982 | 0.982 | 0.982 |
As shown in Figure The size effect law obtained according to the modification maximum tangential stress (MMTS) criterion is basically consistent with Bazant’s size effect law, and both can reasonably explain the test results of the unstable fracture toughness. However, the values of According to expression ( For
In this paper, three-point bending tests of notched beams with span-depth ratio The normalized stress intensity factor As the height The size effect of unstable fracture toughness can be reasonably explained by the MMTS criterion. Three-point bending tests on notched beams are used to determine unstable fracture toughness. The minimum height of the test piece is recommended to be 200 mm. Of course, the minimum height should be related to the The higher order terms of the stress field at the crack tip (Williams’s series expansion) are important factors affecting unstable fracture toughness. In addition to the A3 term, the higher order term (A5, A7, …) also plays a role, especially for small specimens. This is because more high-order terms can more accurately describe the crack tip stress field, which is consistent with Wei et al. [
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
JL analyzed the data and wrote the manuscript. ZWD helped to perform the experiments and revised this manuscript. ZPG and DCA guided the data analysis method and supervised the experiments.
This work was supported by the 2011 Collaborative Innovation Center for Green Development of Coal of Guizhou Province (Project no. [2016]02), Department of Education of Guizhou Province “125” Major Project ([2012]017), and the Joint Fund of Department of Science and Technology of Guizhou Province (Project no. [2014]7457).