Study on Dynamic Constitutive Model of Polypropylene Concrete under Real-Time High-Temperature Conditions
Abstract
:1. Introduction
2. The Modified Dynamic Constitutive Model of Polypropylene Concrete
2.1. The Modified Z-W-T Nonlinear Viscoelastic Model
- The initial stage of the stress–strain curve of concrete under impact loading was nearly linear elastic [29], meaning that part I of Equation (1) can be approximately converted into a linear polynomial, as shown in Equation (2).
- Part Ⅱ of Equation (1) consists of two Maxwell element relaxation functions and have large differences in the relaxation time (θ1 and θ2), where the Maxwell element with relaxation time θ1 describes the mechanical behavior of the material at a low strain rate, and the Maxwell element with relaxation time θ2 describes the viscoelastic behavior of the material at a high strain rate (the order of magnitude for θ1 and θ2 are 10~102 s and 10−4~10−6 s, respectively). Other studies have shown that the mechanical properties of concrete material were obviously affected by the strain rate especially and they were sensitive at high strain rates [26].The strain rate of polypropylene concrete under the impact load 102 s−1 [30] resulted in a short observation time; in this case, the low-frequency Maxwell element could not be relaxed, showing linear springs characteristics, whereas the Maxwell element with a relaxation time of θ2 described the viscoelastic mechanical behavior of the material under high strain rate conditions. Therefore, a simple spring can be used to replace the low-frequency Maxwell element in the Z-W-T non-linear viscoelastic model in this situation (Equation (3)), and under a high strain rate, the Z-W-T nonlinear viscoelastic model can be expressed by Figure 1b. Equation (4) shows the equivalent treatment of two parallel elastomers and the adjusted Z-W-T nonlinear viscoelastic model (Equation (5)) is represented in Figure 1c. Although the final expression form of the modified Z-W-T nonlinear viscoelasticity model was similar to that of the Kelvin–Voigt model [31], the derivation processes of the two models were not the same.
- As a heterogeneous material, polypropylene concrete contains a large number of random polypropylene fibers and pores [32]. Therefore, damage factors should be considered when studying dynamic damage constitutive models of polypropylene concrete under impact load and thermal conditions. From the perspective of continuous damage mechanics, polypropylene concrete is assumed to be a continuous medium [33]. The composite damage amount D was introduced to measure the degree of damage experienced by the polypropylene concrete. In this case, Equation (6) describes a relationship according to the principle of strain equivalence [34]. Additionally, the modified Z-W-T non-linear viscoelastic model with the damage variable D (Equation (7)) can be obtained by substituting Equation (5) into Equation (6).
2.2. Damage Variable
3. Experiment and Results
3.1. Experimental Material
3.2. Experimental Equipment
3.3. The Solution of Real-Time High-Temperature Experiments
3.4. Experiment Theory
3.5. Typical Waveform and Dynamic Stress Equalization
3.6. The Relationship between Impact Air Pressure Level and Impact Velocity
3.7. Results of Experiment
4. Discussion
4.1. Effect of Thermal Conditions on Dynamic Mechanical Properties of Polypropylene Fiber Concrete
4.2. Validation of Constitutive Model and Determination of Parameters
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Characteristics | Density (g/cm3) | Length (mm) | Equivalent Diameter | Melting Point (°C) | Elongation at Break (%) | Elastic Modulus (MPa) |
---|---|---|---|---|---|---|
Parameter | 1.18 | 3–9 | 9–30 um | 220 | ≥15–25 | ≥13,000 |
Water | Cement | Fly Ash | Silica Fume | Sand | Pebble | Polypropylene Fiber | Water Reducing Admixture |
---|---|---|---|---|---|---|---|
160 | 234 | 108 | 18 | 714 | 1166 | 1 | 7.2 |
Material | Density | Elastic Modulus | Poisson Ratio |
---|---|---|---|
45GrNiMoVA | 7794 g/cm3 | 211 GPa | 0.285 |
Target | Removal | Before Experiment | After Experiment | Drop Rate (%) |
---|---|---|---|---|
700 | 704 | 683 | 672 | 4.55 |
500 | 503 | 489 | 483 | 3.98 |
300 | 306 | 297 | 290 | 5.23 |
100 | 105 | 101 | 98 | 6.67 |
Impact Air Pressure (MPa) | Impact Velocity (m/s) | Average Impact Velocity (m/s) |
---|---|---|
0.4 | 5.6 | 5.5 |
5.3 | ||
5.5 | ||
0.6 | 7.3 | 7.5 |
7.8 | ||
7.4 | ||
0.8 | 9.2 | 9.5 |
9.7 | ||
9.6 |
Average Impact Air Pressure | Temperature Grade | Calculated Parameters | Fitting Parameters | R2 | |||
---|---|---|---|---|---|---|---|
m | F0 | Ea | E2 | ||||
0.4 | 25 | 1.36 | 43.23 | 1.27 | −1.39 | 1.79 | 0.925 |
200 | 1.69 | 41.61 | 5.25 | −4.51 | 1.56 | 0.948 | |
400 | 2.11 | 33.65 | 3.19 | −3.35 | 2.45 | 0.964 | |
600 | 4.01 | 20.51 | 2.21 | −1.87 | 3.60 | 0.915 | |
800 | 3.92 | 9.56 | 1.48 | −2.65 | 2.67 | 0.939 | |
0.6 | 25 | 1.41 | 52.39 | 1.54 | −3.32 | 2.41 | 0.959 |
200 | 1.97 | 51.16 | 4.87 | −1.65 | 2.86 | 0.926 | |
400 | 3.47 | 38.76 | 4.36 | −3.78 | 3.75 | 0.959 | |
600 | 4.48 | 25.35 | 4.23 | −5.23 | 3.55 | 0.927 | |
800 | 4.29 | 8.61 | 6.79 | −5.11 | 1.37 | 0.949 | |
25 | 1.94 | 64.42 | 1.76 | −2.45 | 1.27 | 0.913 | |
0.8 | 200 | 2.58 | 61.15 | 1.53 | −3.12 | 1.53 | 0.914 |
400 | 3.66 | 45.36 | 1.04 | −2.12 | 3.69 | 0.981 | |
600 | 4.12 | 37.61 | 1.86 | −3.13 | 5.49 | 0.971 | |
800 | 4.24 | 21.23 | 1.29 | −2.76 | 1.42 | 0.915 |
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Li, R.; Liu, L.; An, H.; Wang, Y. Study on Dynamic Constitutive Model of Polypropylene Concrete under Real-Time High-Temperature Conditions. Appl. Sci. 2022, 12, 1482. https://doi.org/10.3390/app12031482
Li R, Liu L, An H, Wang Y. Study on Dynamic Constitutive Model of Polypropylene Concrete under Real-Time High-Temperature Conditions. Applied Sciences. 2022; 12(3):1482. https://doi.org/10.3390/app12031482
Chicago/Turabian StyleLi, Rui, Lei Liu, Huaming An, and Ya Wang. 2022. "Study on Dynamic Constitutive Model of Polypropylene Concrete under Real-Time High-Temperature Conditions" Applied Sciences 12, no. 3: 1482. https://doi.org/10.3390/app12031482
APA StyleLi, R., Liu, L., An, H., & Wang, Y. (2022). Study on Dynamic Constitutive Model of Polypropylene Concrete under Real-Time High-Temperature Conditions. Applied Sciences, 12(3), 1482. https://doi.org/10.3390/app12031482