*4.3. Average Strain Rate of 23.5 s*−*<sup>1</sup>*

The macro morphology of the ultra-early-strength cement-based material specimens in each stage of the three repeated experiments, when the impact velocity is 15 m/s (the average strain rate is 23.5 s<sup>−</sup>1), can be seen in Table 5. At an impact velocity of 15 m/s, the specimen was impacted within a very short time after the bullet was launched, accompanied by a loud noise. Afterward, it can be observed that the concrete specimens were a comminuted failure under the impact compression stress wave, and the specimens were broken into a large amount of powder and small pieces. Figure 7 shows the stress–strain curve and the average stress–strain curve obtained from three repeated experiments at an impact velocity of 15 m/s. It can be seen from the figure that the curve rises sharply at the beginning of the impact, and subsequently, the compressive strength of the specimen quickly reached the peak point due to its own strain rate sensitivity, and the second half of the curve began to gradually decrease after the end of the loading.

**Table 5.** Test results at an impact velocity of 15 m/s.


**Figure 7.** Stress–strain curve of 15 m/s.

#### **5. Discussion**

#### *5.1. Strain Rate Effect and Analysis of Compressive Strength*

Figure 8 is a summary of the stress–strain curves of ultra-early-strength cement-based materials at different strain rates. The impact velocities are 5 m/s, 10 m/s, and 15 m/s, and the corresponding average strain rates are 7.5 s<sup>−</sup>1, 15.3 s−1, and 23.5 s−1, respectively. Experimental results show that ultra-early-strength cement-based materials are strain-rate sensitive materials, and the stress–strain curves of the materials at different strain rates are significantly different. When the strain rate is 7.5 s−<sup>1</sup> (impact velocity 5 m/s), the stress–strain curve shows a yield platform, and it shows that under this loading condition, the ultra-early-strength cement-based material specimen enters an obvious yielding stage from the elastic stage.

**Figure 8.** Dynamic stress–strain curve at different strain rates.

After reaching the peak point, the specimen was damaged. The stress–strain curve cannot be unloaded to the zero point, indicating that during the loading process, damage evolution occurred inside the concrete, and the specimen produced plastic deformation.

When loading at 10 m/s and above, the stress strain curve of the ultra-early-strength cement-based material does not show an obvious yield platform, and it transitions directly from the elastic stage to the yield stage. An obvious strain softening phenomenon appears after the peak stress (the stress decreases with increasing strain), indicating that the specimen still has the load-bearing capacity. The stress–strain curve at the strain software stage is no longer the mechanical response of the initial complete material, but the specimen still has a certain residual strength at this time. At this stage, the stress–strain curve still has engineering significance for analyzing the damage and destruction of concrete structures under the explosion and impact load, so the curve of the strain-softening stage is still retained in data processing.

The phenomenon that the dynamic compressive strength of concrete materials increases with the increase in strain rate has been confirmed by extensive experiments, but there is no unified conclusion on the mechanism of the strain rate effect of strength. The increase in the strain rate causes an increase in the strength of concrete materials, which is generally caused by the free water viscosity effect and the crack propagation inertia effect. The inertial effect of crack propagation is generally caused by the concrete matrix and aggregates. Considering that the ultra-early-strength cement-based materials do not contain aggregates, which is different from ordinary concrete, the main reason that the dynamic compressive strength of ultra-early-strength cement-based materials increases with the increase of the strain rate may be the effect of free water viscosity inside the material.

The dynamic increase factor, DIF, the ratio of the dynamic strength to the static strength of concrete, is used to characterize the dynamic characteristics of brittle materials frequently. Table 6 shows the dynamic compressive strength of ultra-early-strength cement-based materials obtained by SHPB impact compression loading experiments at three different strain rates.

**Table 6.** Dynamic compressive strength under different strain rates.


In this paper, the DIF model Formula (2) of the concrete under the one-dimensional stress state approved by the Euro-International Committee for Concrete (the CEB) is used to fit the experimental data in Table 6.

$$DIF = \frac{f\_c}{f\_{co}} = \begin{cases} \left(\frac{\dot{\varepsilon}}{\overline{\varepsilon}\_s}\right)^{\alpha\_c} & \dot{\varepsilon} < k\\ \beta\_c \left(\frac{\dot{\varepsilon}}{\overline{\varepsilon}\_s}\right)^{\gamma\_c} & \dot{\varepsilon} \ge k \end{cases} \tag{2}$$

In the formula, *fc* is the corresponding compressive strength (MPa) when the strain rate is . *<sup>ε</sup>*, *fc*<sup>0</sup> is the static compressive strength (MPa), . *<sup>ε</sup>* is the strain rate, . *ε*s( . *<sup>ε</sup>*<sup>s</sup> = 3 × <sup>10</sup>−<sup>5</sup> <sup>s</sup><sup>−</sup>1) is the quasi-static strain rate, *αc*, *β*<sup>c</sup> and *γ<sup>c</sup>* are fitting parameters, and *k* is the critical strain rate. The fitting results are as follows:

$$DIF = \frac{f\_c}{f\_{co}} = 0.0018 \left(\frac{\dot{\varepsilon}}{\dot{\varepsilon}\_8}\right)^{0.5211} 7.5 \text{ s}^{-1} \le \dot{\varepsilon} \tag{3}$$

Figure 9 shows the relationship between the dynamic enhancement factor of ultraearly-strength cement-based materials and the strain rate. It can be seen from the figure that the dynamic compressive strength of the material has a significant strain rate effect, and as the strain rate increases, the dynamic compressive strength increases significantly.

**Figure 9.** Relationship between DIF and strain rate.

#### *5.2. Energy Absorption Density Analysis*

Toughness is the ability of a material or structure to absorb energy under a load until it fails. It not only depends on the bearing capacity but also on the deformation capacity [30]. The methods to determine the toughness index include the energy method, intensity method, energy ratio method, characteristic point method, etc. In this paper, the energy method was used, and the area enclosed under the stress–strain curve is used to represent the characterization method of absorbed energy. The energy absorption density can be calculated by Formula (4).

$$
\omega = \int \sigma \ast (\varepsilon) d\varepsilon \tag{4}
$$

Figure 10 shows the energy absorption density curve at different loading speeds obtained by Formula (4). It can be seen from the figure that under different impact speeds (strain rates), the energy absorption density increases with the increase of strain, and as the impact velocity increases, the growth rate of the energy absorption density accelerates. Figure 10a shows the relationship between energy absorption density and strain when the impact velocity is 5 m/s. The average energy absorption density at the stress peak point is about 1.5 × 105 J/m3, and the corresponding peak strain is 3100 με. When the impact velocity is 10 m/s, as shown in Figure 10b, the corresponding curve of each test piece is relatively concentrated, and the corresponding energy absorption density at the peak stress point is about 3 × <sup>10</sup><sup>5</sup> J/m3.

Relative to the impact velocity of 5 m/s, the energy density value doubles but the peak strain hardly changes. When the impact velocity increases to 15 m/s, the corresponding curves of each test piece are scattered slightly. As shown in Figure 10c, the curve becomes steep, compared with the low-speed impact, the corresponding energy absorption density at the peak stress point is about 2.3 × <sup>10</sup><sup>5</sup> J/m3, and the corresponding peak strain is about 1500 με. According to the energy density absorption value, peak stress, and corresponding peak strain of the specimen under different impact speeds, it can be concluded that, as the impact velocity increases, the peak stress rises, the energy absorption density increases and its growth rate accelerates, as shown in Figure 10d. The peak strain at an impact velocity of 15 m/s is lower than that of low-speed impact (5 m/s and 10 m/s).

(**c**)

**Figure 10.** *Cont*.

**Figure 10.** Energy absorption density–strain curve. (**a**) Impact velocity of 5 m/s. (**b**) Impact velocity of 10 m/s. (**c**) Impact velocity of 15 m/s. (**d**) Average value.

#### *5.3. Damage Evolution Process Analysis*

Under the impact load, cracks will appear inside the specimen, and the cracks will gradually propagate from the inside to the outside. When the cracks propagate to the boundary of the specimen, they will cause penetration and breakage. In this section, from the perspective of continuum mechanics, the region containing many scattered micro-cracks is regarded as a local uniform field, the overall effect of the crack in the field is considered, and the damage state of the uniform field is described by defining an irreversible related field variable, which is the damage variable *D*. Under uniaxial compression, the constitutive relation of concrete material after damage can be expressed as [31]

$$
\sigma = \text{E}\varepsilon (1 - D) \tag{5}
$$

where *σ* is stress, *ε* is strain, *E* is elastic modulus, and *D* is the damage variable. The damage variable is obtained by the transformation of Equation (5):

$$D = 1 - \frac{\sigma}{E\varepsilon} \tag{6}$$

The strain *ε* can be expressed as

$$
\varepsilon = \varepsilon\_{\mathcal{E}} + \varepsilon\_p \tag{7}
$$

where *ε<sup>e</sup>* is the elastic strain and *ε <sup>p</sup>* is the plastic strain. The elastic strain *ε<sup>e</sup>* can be derived from stress σ.

$$
\varepsilon\_{\ell} = \frac{\sigma}{E} \tag{8}
$$

The plastic strain *ε <sup>p</sup>* is expressed as

$$
\varepsilon\_p = \varepsilon - \frac{\sigma}{E} \tag{9}
$$

The relationship between the damage variable and the plastic strain under different loading speeds (strain rates) is shown in Figure 11. The damage variable *D* is calculated by calculating the elastic modulus *E* according to the linear elastic segment of the stress–strain curve and then substituting the stress and strain into Equation (6).

**Figure 11.** Damage evolution variable. (**a**) Impact velocity of 5 m/s. (**b**) Impact velocity of 10 m/s. (**c**) Impact velocity of 15 m/s. (**d**) Average value.

With the increase of the impact velocity, the crack propagation inside the specimen is hindered, and the evolution speed of the damage variable decreases, and under the same plastic strain, the damage variable corresponding to the high strain rate is lowered. When the stress of the concrete material is 70% of its peak value, it will enter the plastic stage [31]. According to the test results, when the ultra-early-strength cement-based material loses its bearing capacity, the corresponding damage variable is between 0.8 and 0.9, approximately.

According to the result of the analysis, the damage variable is related to strain rate . *ε* and plastic strain *ε <sup>p</sup>*. On this basis, the average value of the three damage variables under different strain rates was fitted. The comparison result of the fitting is shown in Figure 12. It can be seen from the figure that both are in a high degree of agreement. The fitting formula is as follows:

$$\begin{aligned} D &= 0.363 \times \ln(1 + A\varepsilon\_{\mathrm{P}})\\ A &= 2437 \ast \sin(0.054 \ast \dot{\varepsilon} + 1.39) \ \ 7.5 \,\mathrm{s}^{-1} \le \dot{\varepsilon} \le 23.5 \,\mathrm{s}^{-1} \end{aligned} \tag{10}$$

From the average value of the stress–strain curve of three repeated tests at different strain rates, the elastic modulus *E* (GPa) of the ultra-early-strength material at different strain rates is obtained. The relationship between elastic modulus *E* and strain rate is shown in Figure 13, and R<sup>2</sup> is 0.98. By fitting the elastic modulus *E* at different strain rates, it is found that the relationship between the strain rate and the elastic modulus can be expressed as:

$$E = 3.297\dot{\varepsilon} + 33.53 \ 7.5 \ s^{-1} \le \dot{\varepsilon} \le 23.5 \ s^{-1} \tag{11}$$

**Figure 12.** Comparison of damage variable fitting results.

**Figure 13.** The relationship between elastic modulus and strain rate.

By substituting the evolution variable expressions (8) and (10) under different strain rates into Equation (9), the stress–strain expression of the ultra-early-strength cement-based material can be obtained. The comparison between the results calculated by Equation (9) and the measured stress–strain results under different strain rates is shown in Figure 14. It can be found from the figure that the stress–strain curve calculated from the damage variables is in good agreement with the test results. Therefore, the internal damage evolution process of ultra-early-strength cement-based materials can be characterized by the damage variables.

**Figure 14.** Comparison of fitting results with test results.

#### **6. Conclusions**

In order to explore the dynamic mechanical properties of UR50 ultra-early-strength cement materials, an experimental study on the dynamic mechanical properties of ultraearly-strength cement-based materials at high strain rates was carried out using a largediameter SHPB. The macroscopic failure morphology, dynamic stress–strain curve, and the relationship between dynamic compressive strength and strain rate of specimens under different strain rates (impact velocity) were obtained. According to the experimental results, the main conclusions are as follows:


In general, UR50 ultra-early-strength cement-based materials are more brittle in shock compression and will undergo an overall fracture at low strain rates. The dynamic compressive strength increases with the increase of the strain rate and has an obvious strain rate strengthening effect. However, further research on UR50 ultra-high early-strength concrete (UHESC) under SHPB-impact tests should be conducted. Additional research is also needed for the UR50 concrete base with high-speed impact to explore impact performance. The results of this research will further the development of dynamic material simulation methods and material models.

**Author Contributions:** Formal analysis, Q.H. and X.S.; Investigation, X.W. (Xing Wang); Methodology, X.S.; Resources, J.Y.; Supervision and project administration, J.W.; Validation, X.W. (Xiaofeng Wang); Writing—original draft, Q.H. and J.Y.; Writing—review & editing, W.W. and Z.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Natural Science Foundation of China (Grant Nos. 11302261 and 11972201). This paper is also funded by the Key Laboratory of Impact and Safety Engineering (Ningbo University) project, Ministry of Education. The project number is CJ202011.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.
