In recent years, terrorist attacks and regional conflicts have increased and research on the resistance of building structures has been more highlighted. Q235B is a commonly used steel type in building structures, and there is an urgent need for research on its dynamic material properties. In this paper, Paul constitutive model was calibrated based on the material properties of Q235B steel. Based on the obtained results, it was found that the original model was not able to take into account temperature-softening effect. Therefore, a temperature-softening parameter was added into the temperature term of the original Paul model, and the modified M-Paul constitutive model was developed. Then, numerical prediction results obtained from Q235B steel penetration experiments using J-C (Johnson–Cook) and M-Paul constitutive models were compared. It was observed that J-C constitutive model significantly overestimated ballistic limit while estimations obtained from M-Paul constitutive model were closer to experimental results. This observation verified the correctness and effectiveness of M-Paul model and established a solid foundation for relevant research works on the impact resistance of building structures.
In recent years, terrorist attacks and regional conflicts have increased and research on the resistance of building structures has been more highlighted. Under dynamic loadings, the stress-strain relationships of materials become more complicated. Therefore, the selection of a reasonable and effective material constitutive relationship plays an important role in obtaining accurate simulation results. The Johnson–Cook (J-C) constitutive model [
Yang [
It can be seen in the above paragraphs that significant research works have been conducted on the constitutive relationships of materials and a large number of effective constitutive relations have been widely used in different engineering fields. Previous research on the properties of metal materials has been mostly based on J-C and modified J-C constitutive models. But, it is not entirely suitable for expressing the dynamic mechanical behaviors of structural steel and some better constitutive models should be found necessarily. At present, research on the dynamic mechanical behaviors of materials mostly adopts a combination of experiments and numerical simulations. Numerical simulations are frequently used, and their accuracies are verified by Taylor impact or target penetration tests. They are also commonly used to verify the accuracies of constitutive models and fracture criterion.
In this paper, the Paul constitutive model was calibrated by several material tests. It was found that the model could not take into account temperature-softening effect for the Q235B steel test specimen. Therefore, temperature-softening parameter was added into the temperature term of the model, and the modified M-Paul constitutive model was proposed. Moreover, it was observed in the test that flat-body projectile-penetrated Q235B steel target plate and the damage form as well as the ballistic limit of the target plate are obtained, which are subjected to the projectile penetration. The numerical model was developed using Abaqus software, and M-Paul, modified J-C, and J-C constitutive models were used in the simulation. It was observed that the J-C constitutive model significantly overestimated ballistic limit, while estimations obtained from the M-Paul constitutive model were closer to experimental results, and the accuracy of the proposed material model was verified.
The original Paul [
In this paper, the mechanical behavior of Q235B steel was described by the Paul constitutive model; therefore, the calibration of unknown material parameters of the Paul constitutive model depends on all kinds of material test results in literature [
The corresponding quasistatic item of the Paul constitutive model is shown in Equation (
Because the conversion formula from engineering stress to true stress is no longer applicable after necking [
After that, the numerical model was generated by using Abaqus/Standard, and user-defined material mechanical behavior (UMAT) of the Paul model was developed. Then, the initial value of
Load-displacement curve of smooth cylindrical specimen under different conditions.
In order to obtain the more accurate parameter value, this paper refers to the numerical optimization process in literature [
At reference temperature (room temperature), the Paul constitutive model is expressed as
At the yield point (
The coefficient
Yield stress at different strain rates.
At reference temperature, the Paul constitutive model can be expressed as
Flow-process diagram of Isight optimization of
The optimization programs are as follows: First, coefficient
The Paul constitutive model at the reference strain rate is shown as follows:
At the yield point (
The coefficient of
Yield stress at different strain rates.
Constitutive model and material parameters at yield point at different temperatures.
Constitutive model | Expression | Coefficient and value | |
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J-C model |
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Paul model |
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MJ-C model |
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M-Paul model |
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Similar to the temperature item of parameter
At reference strain rates, the M-Paul constitutive model can be expressed as
Equation (
Material parameters of Q235B steel.
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200 | 0.3 | 7800 | 0.9 | 469 | 8.33 |
293 | 1795 | 244.8 | 0.0317 | 210.1 | 740.7 | 1.989 | −0.0409 | 11.066 | 2.925 | 1.575 | 1.409 |
At reference temperature, the modified J-C constitutive model is similar to the J-C constitutive model. Therefore, the J-C constitutive model parameters of literature [
Flow curve at different strain rates. (a) 10 mm/min. (b) 50 mm/min. (c) 100 mm/min. (d) 200 mm/min.
By comparing the flow curves of Q235B steel at different strain rates, it is found that compared with the J-C constitutive model, the Paul constitutive model predicts the test better and could better reflect the mechanical behavior of Q235B steel at different strain rates. Especially at a high strain rate, the Paul model is more accurate in predicting the test equivalent stress-equivalent plastic strain curve than at the low strain rate, which can better simulate the mechanical behavior of Q235B steel at a high strain rate.
At reference strain rates, the flow curves of Q235B steel at different temperatures can be obtained by introducing the parameters of the modified J-C constitutive model and the M-Paul constitutive model calibrated above into equations (
Flow curve at different temperatures.
Since the extensometer used in the test cannot monitor the deformation of the specimen at high temperature, the displacement used in processing the data of the tensile test at high temperature is the beam displacement of the test machine [
Q235B target plate material is a 70 mm diameter rod produced by Shanghai Baosteel, which penetrates into a 6 mm thick target plate by the wire-electrode cutting and milling machine, and 12 round holes whose diameters are 3 mm are uniformly processed on the circle 31 mm from the center. The projectile material is hardened 9CrSi, and the average diameter and length of the cylindrical blunt projectile are 5.95 mm and 29.82 mm, as shown in Figure
Q235B steel target plates and blunt projectiles. (a) Target plates. (b) Blunt projectiles.
The tests were performed at a one-state compressed gas gun installed at School of Civil Engineering in Nanyang Institute of Technology; the initial and residual velocities of projectile were obtained by FASTCAM SA-Z high-speed camera. The initial velocity of projectiles is controlled by pressured gas.
The total 14 ballistic tests of Q235B steel target plate were completed. Among them, there were 5 tests in which the trajectory of the projectile was not horizontal before penetrating the target plate, so the number of effective tests is 9, and the results of the effective test are shown in Table
Results of the ballistic test for Q235B steel targets.
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Test results |
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B-1 | 5.95 | 29.84 | 6.482 | 234.51 | 0 | Embedded without plug |
B-3 | 5.95 | 29.88 | 6.496 | 373.70 | 153.5 | Plugging |
B-4 | 5.95 | 29.89 | 6.478 | 422.91 | 209.6 | Plugging |
B-6 | 5.95 | 29.83 | 6.501 | 327.55 | 126.5 | Plugging |
B-8 | 5.94 | 29.82 | 6.488 | 272.48 | 0 | Embedded, plug was basically shaped |
B-9 | 5.95 | 29.82 | 6.497 | 323.26 | 120.7 | Plugging |
B-10 | 5.94 | 29.82 | 6.489 | 303.16 | 102.0 | Plugging |
B-12 | 5.95 | 29.79 | 6.467 | 284.17 | 52.8 | Plugging |
B-14 | 5.95 | 29.77 | 6.461 | 283.80 | 50.8 | Plugging |
As Table
In the expression,
Initial versus residual velocity data and fitting curves of test and numerical simulation for Q235B steel targets.
In the two tests that the projectile did not penetrate the target, the projectile was all embedded in the targets, but the results of the two tests were slightly different. In test B-1 of low initial velocity (
Failure mode of steel target plate under projectile impact at different velocities.
Velocity (m/s) | Front surface | Rear surface |
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234.51 |
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272.48 |
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284.17 |
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327.55 |
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422.91 |
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Figure
Penetration process of target plate under projectile impact at
In the 7 tests when the initial velocity of the projectile is higher than the ballistic limit, the projectile penetrates the target plate and the failure model is the shear plugging. Moreover, different degrees of bulge occurred near the perforation on the back of the target plate, as shown in Table
When the initial velocity of projectile is slightly higher than the ballistic limit, the back concave crater shape of the target plate is quite regular. With the improvement of the projectile initial velocity, the uneven arch height began to appear near the perforation on the back of the target plate.
When
Figure
Fracture situation in the skin of the plugs. (a) Q235-Blunt-12. (b) Q235-Blunt-6. (c) Q235-Blunt-3. (d) Q235-Blunt-4.
Figure
Penetration process of target plate under projectile impact at
A half of a full-scale 3D finite element model was developed by Abaqus/Explicit, and the modeling process is referred to literature [
Finite element model of ballistic test for Q235B steel targets.
M-Paul, modified J-C, and J-C constitutive models were selected for the material strength model, and the modified J-C fracture criteria were adopted for the failure model. Because M-Paul constitutive model, modified J-C constitutive law, and fracture criterion are not provided in Abaqus material base, corresponding VUMAT subroutines should be developed to complete the input of material model by calling it during the numerical simulation. And, the M-Paul constitutive model is expressed as Equation (
Lin li et al. [
Material parameters of modified J-C constitutive model and fracture criterion.
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244.8 | 899.7 | 0.940 | 0.0391 | 1.977 | 1.515 | −43.408 | 44.608 | −0.016 | −0.0145 | 0.046 | 7.776 |
The J-C constitutive model is expressed as Equation (
Refer to the thermal-softening item (1 −
The parameters of thermal-softening items
Fracture strain at different temperatures.
In this paper, the numerical simulation of Q235B steel target plate under blunt projectile impact was performed. The residual kinetic energy of the projectile after perforation is output and converted into the residual velocity of the projectile.
The numerical simulation of the ballistic test was carried out by the M-Paul constitutive model and modified J-C fracture criterion.
The numerical simulation of the ballistic test was carried out by the modified J-C constitutive model and fracture criterion.
The numerical simulation of the ballistic test was carried out by the J-C constitutive model and modified J-C fracture criterion.
In cases 1 and 2, a total of 7 numerical simulations were completed, respectively. The initial-residual velocity and the states of projectiles and targets after impact are shown in Table
In Case 3, a total of 8 numerical simulations were completed. The initial-residual velocity, deformation, and failure of projectiles and targets are shown in Table
A comparison of ballistic limit between ballistic test and numerical simulation is shown in Figure
By observing the numerical simulation process of target plate subjected to projectile impact at different velocities under different working conditions, the test phenomenon is shown in Table
As Table
In addition, it can be seen from Figure
The above phenomenon indicates that with the improvement of strain rates, M-Paul and modified J-C constitutive models began to show difference in the prediction of material mechanical behavior. The numerical simulation result of Case 1 is more consistent with the test result; this illustrates that the mechanical behavior of Q235B steel can be better predicted by the M-Paul constitutive model.
Numerical simulation results of Case 1 and Case 2.
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Result |
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Case 1 | |||
C1-1 | 272.5 | 0 | Crater and rebound |
C1-2 | 283.8 | 0 | Crater and rebound |
C1-3 | 290 | 36.4 | Plugging |
C1-4 | 303.2 | 85.2 | Plugging |
C1-5 | 323.3 | 116.8 | Plugging |
C1-6 | 373.7 | 166.7 | Plugging |
C1-7 | 422.9 | 204.3 | Plugging |
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Case 2 | |||
C2-1 | 272.5 | 0 | Crater and rebound |
C2-2 | 283.8 | 0 | Crater and rebound |
C2-3 | 290 | 0 | Crater and rebound |
C2-4 | 303.2 | 18.7 | Plugging |
C2-5 | 323.3 | 107.6 | Plugging |
C2-6 | 373.7 | 158.3 | Plugging |
C2-7 | 422.9 | 201.3 | Plugging |
Numerical simulation result of Case 3.
Number |
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Result |
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C3-1 | 272.5 | 0 | Crater, rebound |
C3-2 | 283.8 | 0 | Crater, rebound |
C3-3 | 303.2 | 0 | Crater, rebound |
C3-4 | 323.3 | 0 | Crater, rebound |
C3-5 | 340 | 73.2 | Perforation, plugging |
C3-6 | 360 | 102.5 | Perforation, plugging |
C3-7 | 373.7 | 121.9 | Perforation, plugging |
C3-8 | 422.9 | 162.1 | Perforation, plugging |
Ballistic limits and model constants of numerical simulation for Q235B steel targets.
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Case 1 | 0.55 | 2.95 | 285.9 |
Case 2 | 0.55 | 2.91 | 299.8 |
Comparison of numerical simulation results in different case.
Case 1 | Case 2 | ||
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Stress distribution |
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The stress is distributed mainly in an annular region where the edge of projectile contacts the target | |||
Failure model | Back bulge | Back bulge | |
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Stress distribution |
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The stress is concentrated in the annual region where the edge of projectile directly contacts target plate and extends to the back of target plate | |||
Failure model | Shear plugging or adiabatic shear failure | Shear plugging or adiabatic shear failure | |
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Deformation of target |
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Highly localised bulge on both sides of the rear cavity; the height of rear bulge is uneven in different orientations. | Highly localised bulge on both sides of the rear cavity; the height of rear bulge is even in different orientations. |
Macrographs of target plate under projectile impact at
In this paper, the dynamic constitutive model of Q235B steel was studied systematically through theoretical derivations, numerical optimization, tests, and numerical simulation. The M-Paul model for characterizing Q235B steel mechanical behavior under larger strain and high strain rate and temperature is proposed. The tests and numerical simulations of Q235B steel target plate under blunt projectile impact were performed. The mechanical properties of material under impact was investigated, and the accuracy of M-Paul model was verified. The results are as follows: When the parameters of the Paul constitutive model are calibrated by combing theory with numerical optimization, it was found that thermal-softening item could not reasonably describe thermal-softening behavior of Q235B steel, so the original model was modified to obtain the M-Paul constitutive model. By comparing the prediction of flow curves under different strain rates and temperatures between M-Paul and Johnson–Cook constitutive models, it is found that M-Paul constitutive models can more accurately represent the dynamic mechanical behavior of Q235B steel. The ballistic test of Q235B steel target plate under blunt projectile impact was performed; three failure models, i.e., back bulge, shear plugging failure, and adiabatic shear failure, were concluded. It was found that with the improvement of the initial velocity of projectile, the phenomenon that the bulge of concave crater is jagged becomes more serious and, also, the maximum bulge height is increased. In addition, the tensile strain produced in the forming process of the plug exceeds the fracture strain of the material, and thus, cracks appear on the surface of the plug. The numerical simulation of Q235B steel target plate under blunt projectile impact was performed. By comparing the ballistic limit and the numerical simulation results, it was found that the ballistic limit error obtained by using the J-C constitutive model was very large, and then, compared with the modified J-C constitutive model, M-Paul can more accurately describe the dynamic mechanical property of Q235B steel under complex conditions such as large strain and high strain rates and temperature.
The data used to support the findings of this study are included within the article.
The authors declare that they have no conflicts of interest.
The authors wish to acknowledge the support from the National Science Foundation of Heilongjiang Province of China (no. LH2019E060) and the National Nature Science Foundation of China (grant nos. 51578515 and 11502120).