Study on Mechanical Response and Constitutive Model of Rubber Concrete under Impact Load and Water Content Coupling

Abstract

Impact tests were implemented on concrete with five different types of rubber content utilizing a separated Hopkinson bar with a 50 mm diameter for investigating the mechanical performance of and damage variations in rubber concrete under the coupling effect of water content and impact load. The findings indicate that regular concrete is commonly stronger than rubber concrete, but rubber concrete has better plasticity. At the same time, with an increase in water content, the peak stress of rubber concrete increases gradually, among which RC-6-3 exhibits better mechanical behavior. Water content and rubber content have a significant influence on the fractal dimension of rubber concrete, showing that the fractal dimension is proportional to the amount of glue and inversely proportional to the water content, and RC-12-2 can be used to judge the damage degree of a specimen. As the content of water and rubber grows, so does the rate of energy use. The ratio of transmitted to incident energy decreases gradually as the rubber content increases and is enhanced as the water content is elevated. Among the specimens, tests on RC-9 resulted in the opposite conclusion. In the same water content state, there is a negative association between the content of rubber and the ratio of reflected energy to incident energy. In the natural state, RC-6 rubberized concrete exhibits a significant downward inflection point in the ratio of reflected energy to incident energy. Finally, based on the unified strength theory, a new damage constitutive model of rubber concrete is established and verified.

1. Introduction

In recent years, China has attached great importance to promoting a green economy and sustainable development, which has greatly promoted the transformation and upgrading of the engineering field [1]. With ordinary concrete as the main carrier, the particles of waste rubber tires are utilized as fine aggregate in concrete, which provides a new approach to improving concrete’s brittle physical characteristics [2]. Simultaneously, waste rubber is treated to a certain extent, having a protective effect on the environment [3]. In the construction of airport roads, bridges, highways and houses, rubber concrete bears impact loads and groundwater coupling environments, and the mechanical performance of rubber concrete varies considerably under different water content and impact load conditions. Therefore, the study of concrete with various rubber content under water content and impact load coupling has important theoretical and engineering value for safety evaluation and disaster warning in basic application engineering [4,5].

In the early stages, studies on rubber concrete materials mainly focused on the mechanical properties under static load, including uniaxial compression, tensile, bending, shear, and other mechanical properties tests under quasi-static low pressure. Many scholars at home and abroad have conducted research on the mechanical properties of rubber concrete under quasi-static load [6,7,8]. However, the actual mechanical damage of rubber concrete materials is not limited to simple quasi-static damage. For example, the damage of rubber concrete under dynamic high pressure, such as high-speed impact, collision, and blasting, belongs to dynamic damage. If only the mechanical properties under static low pressure are studied, there will be a large deviation. Moreover, with the development of testing technology and the accumulation of dynamic test data, more and more studies show that rubber concrete materials can better reflect the physical properties of materials under dynamic conditions. For instance A.O. Atahan et al. [9], M.M. Al-Tayeb et al. [10], B. Yang et al. [11], M.M. Al-Tayeb et al. [12], I. Khan et al. [13] studied and compared the mechanical properties of rubber concrete materials under quasi-static and dynamic conditions. This has prompted scholars at home and abroad to shift their research focus from studying the quasi-static low pressure properties of rubber concrete to studying the mechanical properties of rubber concrete under dynamic high pressure.

For the study of the dynamic mechanical properties of rubber concrete, W. Feng et al. [14], Liu et al. [15] and Pan et al. [16] studied the influence of rubber concrete materials under different strain rates. They found that the compressive strength of rubber concrete materials has strain rate sensitivity and established a constitutive model of rubber concrete. Liu et al. [17], T.M. Pham et al. [18] and Chen et al. [19] verified the rate-dependent dynamic mechanical properties of rubber concrete with different rubber content through dynamic mechanical tests and concluded that rubber content affects the mechanical properties of rubber concrete. A. Jahami et al. [20] and A. Gregori et al. [21] evaluated the effect of adding rubber particles on the mechanical properties of concrete and concluded that the mechanical properties of concrete are optimal when the rubber content is 10%.

With the development of testing technology and further research, the split Hopkinson bar (SHPB) is generally used at present. Because concrete material is a non-uniform brittle material, the test premise is to meet two basic assumptions. At the same time, in order to obtain the authenticity of the dynamic mechanical property data of concrete material, the test premise is to meet two basic assumptions. Therefore, a large-diameter SHPB rod was adopted for testing. W. Feng et al. [22] made a comparative analysis of ordinary concrete and rubber concrete under the impact load of large-diameter SHPB and concluded that the dynamic strengthening coefficient of rubber concrete with different dosages increases with the increase in strain rate. W. Zhide et al. [23], Liu et al. [24], Yang et al. [25], and Feng et al. [26] obtained the energy dissipation law and crushing characteristics of rubber concrete by using SHPB. T. Gupta et al. [27] and A.A. Thakare et al. [28] carried out impact loads on rubber concrete and observed the crushing characteristics of rubber concrete by scanning electron microscopy. However, roads, bridges and water conservancy projects are often affected by different water content. Some scholars have conducted research on dynamic and static loads. Liu et al. [29] studied the influence of water content on the elastic modulus of concrete through static compression tests and concluded that the elastic modulus increases with the increase in water content. Chen et al. [30] and P.G. Ranjith et al. [31] studied the effects of different loading rates and water content coupling on concrete strength.

In summary, the split Hopkinson pressure bar (SHPB) has been widely used by domestic and foreign scholars to research concrete’s dynamic mechanical characteristics, and significant results have been achieved. However, there are few studies on the effects of rubberized concrete under the coupled effects of water content and impact loading. Thus, under the effect of impact load, the energy transfer characteristics, fractal dimension, and dynamic mechanical performance of rubber concrete with varying rubber content and diverse water content are studied using the SHPB system. In addition, a constitutive model is established to provide a theoretical basis for applications in foundation engineering.

2. Experimental Design

2.1. Materials

The materials used in this work are as follows: cement, water, coarse aggregate, fine aggregate, admixtures, mixtures. In

Table 1. Material mix ratio.

ID	Mixing Ratio (kg/m3)							Rubber Content (%)
	Cement	Stone and Sand	Silica Fume	Water–Cement Ratio	Mineral Powder	Water Reducing Agent	Steel Slag Powder
RC-0	0.500	0.500	0.002	0.300	0.004	0.010	0.003	0
RC-3	0.500	0.470	0.002	0.300	0.004	0.010	0.003	3
RC-6	0.500	0.440	0.002	0.300	0.004	0.010	0.003	6
RC-9	0.500	0.410	0.002	0.300	0.004	0.010	0.003	9
RC-12	0.500	0.380	0.002	0.300	0.004	0.010	0.003	12

Note: specimen numbers RC-X are as follows: RC—ordinary concrete; X—equal volume of rubber particles instead of coarse aggregate percentage content.

, the contents of RC-0,RC-3,RC-6,RC-9 and RC-12 coarse aggregate (stone) are 0.25, 0.228, 0.207, 0.187 and 0.167, respectively, and the contents of fine aggregate (sand) are 0.25, 0.235, 0.22, 0.205 and 0.19, respectively [32]. One kind of ordinary concrete and four types of rubber concrete were prepared, and the specimen number was RC-X (X is the equal volume of rubber particles instead of coarse aggregate percentage content). Table 1 provides a detailed mix of materials. Figure 1a depicts the specimen’s fabrication procedure. According to the diameter of the Hopkinson system pressure rod used, a 50 mm diameter by 25 mm length cylinder was created for the concrete test block, and different moisture content states were adopted. The pouring sequence of test blocks is as follows: Firstly, rubber and sand particles are mixed for sixty seconds in a dry environment, then water and a water-reducing agent (50% of the total amount) are gradually added, and the mixing starts again for 60 s. Then, cement and other mixing materials are added to continue stirring, and water and a water-reducing agent (50%) are added to stir for 60 s. After mixing evenly, layers on the shaking table are applied to condense and shake it until it cures after the mixture is put into the mold. The curing was divided into the following steps: (1) The experiment is carried out in a dry, natural and saturated water environment. However, at room temperature, it is difficult to completely remove the moisture in the specimen, which affects the accuracy of the test data, so the drying box is used to dry the specimen. First, we adjust the stability, generally set the temperature to 60~110 °C, then put the test pieces into the oven, maintaining distance between the test pieces so as not to affect the drying effect, and then start the drying box. Throughout the drying process, an observation needs to be continuously carried out regarding the condition of the test pieces in the oven to ensure that the quality of the test pieces does not change while drying. And finally, when the test pieces reach a constant weight, this means that the specimen is completely dry (during operation, we set a 24 h drying time, which can meet the test requirements) and the drying is finished. (2) The specimen produced in the natural state is cured under standard conditions for 28 days. (3) The specimen produced in the saturated water state is placed in tap water for 48 h. Figure 1b displays the finished specimen.

2.2. Test Scheme

The SHPB device used in this test is shown in Figure 2. The lengths of the transmission rods and incident rods are 2300 and 2500 mm, respectively, with diameters of 50 mm.

2.3. Strain Rate Measurement and Data Processing

By installing a resistance strain sensor on the SHPB pressure bar, the strain–time $\left({\epsilon }_i\left(t\right),{\epsilon }_r\left(t\right),{\epsilon }_t\left(t\right)\right)-\left(t\right)$ curves of the three pulses (incident, reflected and transmitted) can be identified [33]. In accordance with 1-D elastic stress wave theory, the strain rate $\dot{\epsilon }(t)$, strain $\epsilon \left(\mathrm{t}\right)$ and stress $\sigma \left(t\right)$ of the specimen were calculated by Equation (1).

$$\{\begin{array}{l}\dot{\epsilon }(t)={\displaystyle \frac{C_0}{L}}\left[{\epsilon }_i\left(t\right)-{\epsilon }_r\left(t\right)-{\epsilon }_t\left(t\right)\right]\\ \epsilon (t)={\displaystyle {\int }_0^t\frac{C_0}{L}\left[{\epsilon }_i\left(t\right)-{\epsilon }_r\left(t\right)-{\epsilon }_t\left(t\right)\right]}dt\\ {\sigma }_t(t)={\displaystyle \frac{AE}{2A_P}}\left[{\epsilon }_i\left(t\right)+{\epsilon }_r\left(t\right)+{\epsilon }_t\left(t\right)\right]\end{array}$$

(1)

in which $E$ represents the modulus of elasticity of the compression rod, $A_p$ and $A$ denote the cross-sectional area of the press rod and specimen, separately, and $C_0$ is the 1D stress wave velocity in the compression rod. $L$ stands for the specimen’s initial length.

2.4. Dynamic Stress Balance

When conducting impact load tests with SHPB, the sample will face problems caused by the inertia effect [34], which will lead to the dispersion of the test results of the specimen. To minimize the dispersion effect [35], the waveform requires adjustments to achieve dynamic stress equilibrium between the two ends of the sample.

$$\{\begin{array}{l}{\sigma }_i\left(t\right)+{\sigma }_r\left(t\right)={\sigma }_t\left(t\right)\\ {\epsilon }_i\left(t\right)+{\epsilon }_r\left(t\right)={\epsilon }_t\left(t\right)\end{array}$$

(2)

To achieve dynamic balance, Zencker et al. [36] believe that the stress deviation in the block length direction can be neglected for uniform deformation. They presented the concept of the relative stress difference ratio as illustrated in Equation (3). When $R\left(t\right)$ < 5%, the axial stress equilibrium inside the block is considered to be achieved.

$$R\left(t\right)=2\left|{\displaystyle \frac{{\epsilon }_i\left(t\right)-{\epsilon }_t\left(t\right)+{\epsilon }_r\left(t\right)}{{\epsilon }_i\left(t\right)+{\epsilon }_t\left(t\right)+{\epsilon }_r\left(t\right)}}\right|$$

(3)

Stress equilibrium in rubberized concrete is demonstrated by the propagation of transmitted, reflected, and incident waves [37]. Dynamic stress equilibrium is also considered to be reached in the specimen when the incident wave matches the transmitted wave, as illustrated in Figure 3.

3. Experimental Results and Analysis

3.1. Stress–Strain Curve

The dynamic stress–strain curve provides the changing trends in mechanical properties of materials under stress, including elastic deformation, plastic deformation, hardening, fracture, and other mechanical properties, which has significant implications for material selection and engineering design. The stress–strain data of the rubberized concrete under impact load were determined by dynamic compression tests, as presented in Figure 4a–c. Rubber concrete under different moisture content undergoes the stages of elastic deformation, yield, hardening, and necking when bearing impact load. Rubber, as a flexible material, has obvious deformation characteristics when resisting impact on concrete, resulting in differences in the stress–strain curves of rubber concrete [38].

When rubber concrete is loaded, due to the small load, it presents a relatively obvious linear elastic stage. With the continuous increase in load, the strain gradually increases because the internal voids of the rubber concrete are compacted, and nonlinear deformation begins to appear. At this stage, with the change in rubber content, the curve shows significant fluctuations. The internal cracks accumulate continuously, and the cracks develop from micro to macro until the specimen is completely destroyed.

The stress–strain relationship curves of rubber concrete with varying water contents are displayed in Figure 5a–e. The graph shows that the stress–strain curve can be broadly classified into four stages: elastic deformation, yield, hardening, and necking. However, different moisture contents have a certain penetration effect on rubber concrete, and the softening characteristics are obvious, making the stress–strain curve different.

During the initial loading phase, the rubber concrete undergoes linear elastic deformation, no new cracks appear, and the stress rises evenly with strain. The slope of the curve reduces as the moisture content grows, resulting in local fluctuations in the curve. This is due to the fact that elevated water content in the rubberized concrete increases the complexity and randomness of cracks, resulting in nonlinear deformation. Therefore, the axial strain increases with the axial stress until the peak strength is achieved. At this point, the peak stress is the highest in the saturated water state, followed by the natural state, and the lowest in the dry state. Finally, the test block enters the fracture stage, where the fracture surface accelerates, and the test block cracks until it is destroyed.

3.2. Peak Stress

Dynamic peak stress refers to the maximum stress value experienced by a material under dynamic impact. When subjected to impact loads, rubber concrete rapidly develops stress concentration and deformation due to instantaneous high energy input. The study of dynamic peak stress has a significant influence on the failure performance and durability of materials. It is necessary to carry out dynamic strength and toughness testing of the material to ensure that it can withstand the dynamic load without failure.

Peak stress of rubberized concrete is directly proportional to water content and inversely to rubber content. As illustrated in Figure 6a,b.

From Figure 6a and Figure 7, it can be concluded that under the same moisture content conditions, the higher the rubber content, the smaller the corresponding peak stress. For example, in the saturated water state, the peak stresses of RC-0, RC-3, RC-6, RC-9, and RC-12 rubber concrete are 49 MPa, 47 MPa, 46 MPa, 41 MPa, and 40 MPa, respectively, decreasing by 4.1%, 2.1%, 10.9%, and 2.4%, respectively. In rubber concrete, there exists an interface slip zone between the coarse aggregate and the cement matrix [14]. Due to the presence of rubber powder, rubber concrete is more susceptible to cracking compared to ordinary concrete. As a result, the strength of rubber concrete is lower than that of ordinary concrete; however, it exhibits superior deformation ability. It can be observed from

Table 2. Energy results of different rubber content.

ID	WI(t)/J	WR(t)/J	WT(t)/J	WS(t)/J	WT(t)/WI(t)	WR(t)/WI(t)	Energy Utilization Rate	η/ (J/cm3)
RC-0-1	356.140	99.710	105.060	151.350	0.295	0.280	0.425	1.540
RC-0-2	396.320	103.830	129.200	163.280	0.326	0.262	0.412	1.660
RC-0-3	347.450	91.720	124.030	131.68	0.357	0.264	0.379	1.340
RC-3-1	368.980	102.940	107.370	158.660	0.291	0.279	0.430	1.610
RC-3-2	379.150	100.470	120.190	158.480	0.317	0.265	0.418	1.610
RC-3-3	386.390	103.160	125.570	157.640	0.325	0.267	0.408	1.600
RC-6-1	371.620	103.310	106.650	161.640	0.287	0.278	0.435	1.640
RC-6-2	372.450	100.930	113.220	158.290	0.304	0.271	0.425	1.610
RC-6-3	359.780	95.700	112.250	151.820	0.312	0.266	0.422	1.550
RC-9-1	364.150	99.770	100.500	163.860	0.276	0.274	0.450	1.670
RC-9-2	357.320	91.830	106.110	159.360	0.297	0.257	0.446	1.620
RC-9-3	365.510	93.570	112.940	158.990	0.309	0.256	0.435	1.620
RC-12-1	355.150	90.910	97.310	166.920	0.274	0.256	0.470	1.700
RC-12-2	349.180	88.340	99.510	161.320	0.285	0.253	0.462	1.640
RC-12-3	371.230	94.290	110.990	165.930	0.299	0.254	0.447	1.690

that the energy of rubber concrete undergoes significant changes under different water content conditions. In Figure 6a, 1 represents dry, 2 represents natural, and 3 represents saturated water.

As shown in Figure 6b, in the condition of saturated water, concrete with different rubber contents (0%, 3%, 6%, 9%, and 12%) exhibits peak stresses of 49 MPa, 47 MPa, 46 MPa, 41 MPa, and 40 MPa, respectively. This represents an increase of 5.1%, 16.5%, 18.6%, 16.2%, and 26.3% compared to the initial value. The peak stresses of RC-3 and RC-6, as well as RC-9 and RC-12 rubber concrete, are very close to each other, indicating that the water content will affect concrete’s peak stress. It can be intuitively found from Figure 6c,d that ordinary concrete has the highest peak stress in the saturated water state, while rubberized concrete with a water content of 12% has the lowest peak stress in the dry state, and the change rule conforms to the two-dimensional scatter diagram. As shown in Figure 7, this may be due to the Stefan effect and Newton internal friction effect of rubber concrete, and it is concluded that concrete’s peak stress is directly proportional to the content of water. Additionally, higher water content facilitates the filling of crack spaces in rubber concrete with free water, which enhances the material’s resistance to dynamic loads.

3.3. Peak Strain

Peak strain is one of the most critical measurements of maximum deformation that the brittle material can withstand. For the brittle material rubber concrete, we need to study the maximum deformation that the material can withstand. Figure 8a exhibits the association between raised rubber content and water content and the maximum deformation of concrete. As shown in Figure 8a, as the rubber and water contents increase, the peak strain of RC-0, RC-3, and RC-9 firstly increases and then decreases. In the natural state, the peak strain of RC-0, RC-3, and RC-9 increases by 47.6%, 18.3%, and 14.8%, respectively, compared to the dry state and by 2.8%, 4.0%, and 19.6% compared to the saturated water state. Water and rubber contents can strengthen the impact resistance of regular concrete. The peak strain of RC-6 tends to remain steady as water content increases, while that of RC-12 dramatically decreases with higher water content.

Figure 8b illustrates that as rubber content increased, the peak strain of rubber concrete under dry conditions initially increased and then decreased, while under natural conditions, the peak strain of rubberized concrete reduced first, then increased, and then decreased again. In addition, the peak strain of rubber concrete continues to decline in the saturated water state, among which the peak strain in the three water content states of RC-6 is close to 0.05. In the natural state, the peak strain of RC-9 increases by 11.6% and 1.9% compared with RC-6 and RC-12, respectively, while in the dry state, the peak strain of RC-9 decreases by 3.8% and 19.2% compared with RC-6 and RC-12, respectively, suggesting that the water content can suppress concrete deformation, but RC-9 shows a better strain capacity under the natural state. Because of the incorporation of rubber powder as an elastic material, the volume of brittle material of cement mortar is reduced, and the deformation ability of concrete is enhanced.

It can be intuitively found from Figure 8c,d that under different water content and rubber content, the change in peak strain is close to an “M” shape, and there is a transition at the rubber content of 6% and 9%.

3.4. Fractal Dimension Features

The concrete failure theory is mainly based on the stress–strain relationship while disregarding the impact of microstructure and the nonlinear performance of concrete materials. The principle of fractal theory can be utilized to characterize the damage process of concrete materials under complex stress conditions more comprehensively [39]. The degree of impact breakage of rubber concrete is characterized by quantitative analysis. A sample after the impact compression test is collected, and then the particle size screening is carried out. The specific process is as follows: a sieve with different particle sizes is used to screen the rubber concrete after impact crushing, and the fragments after screening are weighed, and then the rubber concrete is calculated by fractal dimension technology via the following formula.

In this study, the G-G-S fractal dimension (Gate–Gaudin–Schuhmann) calculation model was used, which was suitable for rubber concrete with fine-grained particles in the sample fracture morphology. The G-G-S distribution function [40] is expressed below:

$$y={\left({\displaystyle \raisebox{1ex}{$r$}\!\left/ \!\raisebox{-1ex}{$R_1$}\right.}\right)}^b$$

(4)

Here, $r$ is the granularity of a given fragment; $R_1$ denotes the largest particle size of the fragment; and $b$ is the distribution parameter of rubber concrete.

When the particle size is smaller than $r$, $M_r$ is the cumulative mass of the fragment, then the ratio of the cumulative mass of the fragment, $M_r$, to the total mass, $M$, also satisfies the G-G-S function. Then, Formula (4) can be converted into

$${\displaystyle \frac{M_r}{M}}={\left({\displaystyle \frac{r}{R_1}}\right)}^b$$

(5)

By solving the derivative of (5), we obtain

$$d{M}_r\propto r^{b-1}dr$$

(6)

Due to the relationship between block and mass increment,

$$dm\propto r^{3}dN$$

(7)

According to Turcotte’s research, the number of fragments with fractal dimension $D$, equivalent particle size $N$ and fragment particle size $r$ under a particle size of the fragment meet the following requirements:

$$N\propto r^{-D}$$

(8)

According to the definition of fractal, the fractal dimension $D$ can be represented as

$$D={\displaystyle \frac{\ln N_r}{\ln \left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$r$}\right.\right)}}$$

(9)

In which $N_r$ represents the number of fragments when the particle size is $r$.

Combined lines (4)~(9) are available

$$y={\displaystyle \frac{M_r}{M}}={\left({\displaystyle \frac{r}{R_1}}\right)}^{3-D}$$

(10)

This can be given by taking the logarithm on the two sides of Equation (10):

$$\ln {\displaystyle \frac{M_r}{M}}=\left(3-D\right)\ln {\displaystyle \frac{r}{R_1}}$$

(11)

Based on Equation (11), the mass and particle size data of broken particles were obtained in combination with the test, and the log–log graph of $\ln \left({\displaystyle \raisebox{1ex}{$M_r$}\!\left/ \!\raisebox{-1ex}{$M$}\right.}\right)-\ln \left({\displaystyle \raisebox{1ex}{$r$}\!\left/ \!\raisebox{-1ex}{$R_1$}\right.}\right)$ was drawn and linear fitting was carried out. The slope of $\ln \left({\displaystyle \raisebox{1ex}{$M_r$}\!\left/ \!\raisebox{-1ex}{$M$}\right.}\right)-\ln \left({\displaystyle \raisebox{1ex}{$r$}\!\left/ \!\raisebox{-1ex}{$R_1$}\right.}\right)$ was $3-D$; thus, fractal dimension $D$ was determined [41].

From Figure 9a–c, it can be seen that under different moisture content conditions, the double-log goodness of fit is high, and the fitting coefficient R² ranges from 0.89206 to 0.96426, showing a good linear correlation on the whole, indicating that the fragment distribution of test blocks after impact load has good fractal characteristics.

Figure 10a–e are the double-logarithm fitting curves of concrete screening with varying rubber content of RC-0~RC-12, respectively. We can summarize that as the content of water increases, the slope of the fitting curve progressively rises; the fitting coefficient R² in the saturated water state ranges from 0.93642 to 0.96426, and it is proportional to the rubber content. In the natural state, the fitting coefficient R² ranged from 0.89206 to 0.96258. The growth rate of the fitting coefficient is high, and the relationship between the fitting coefficient and rubber blending is nonlinear, and the fitting coefficient R² ranged from 0.91112 to 0.95432 in the dry state, thus meaning that the fitting coefficient was higher.

According to Figure 11a,b, under different moisture content states, the fractal dimension of RC-0~RC-12 gradually enhances as the content of rubber elevates. At a constant rubber content, the reduction in fractal size of rubberized concrete is associated with an increment in water content, which means that the sample is completely broken in the dry state, and the breaking degree of rubber concrete is intensified as the content of rubber grows. Conversely, in the natural state, the degree of rubber concrete breakage is relatively low. The fractal dimension of RC-12-2 shows an inflection point, from which it can be concluded that there is a correlation between the degree of fragmentation of the specimen and the fractal dimension, which can be measured by quantitative indicators. In order to more intuitively describe the distribution law of fragments of test blocks after impact load, a three-dimensional diagram is drawn, as displayed in Figure 11c,d. The increase in rubber content and water content will make the crack development of concrete occur more fully under impact load. (Note: the specimen numbers RC-X-Y are as follows: RC—ordinary concrete; X— the equal volume of rubber particles instead of coarse aggregate percentage content); and Y—environment (1 for dry, 2 for natural, and 3 for saturated water).

3.5. Energy Transfer Characteristics

The energy expended by a specimen in the SHPB test from loading to destruction can be utilized to represent the resistance of the specimen to destruction, and the energy generated by the test block during the whole process includes reflected, incident, absorbed and transmitted energies. The calculation formula is as below:

$$\{\begin{array}{l}{W}_I(t)=AEC{\displaystyle \underset{0}{\overset{t}{\int }}{\epsilon }_I^2}dt\\ W_R(t)=AEC{\displaystyle \underset{0}{\overset{t}{\int }}{\epsilon }_R^2}dt\\ W_T(t)=AEC{\displaystyle \underset{0}{\overset{t}{\int }}{\epsilon }_T^2}dt\end{array}$$

(12)

where $A$ is the cross-sectional area, $m^2$; $E$ represents the elastic modulus of the compression rod, $\mathrm{GPa}$; $C$ denotes the P-wave velocity; and ${\displaystyle \raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.}$; ${\epsilon }_I\left(t\right)$, ${\epsilon }_R\left(t\right)$ and ${\epsilon }_T\left(t\right)$ are the incident, reflected and transmitted strains, respectively.

During the test, assuming that all of the kinetic energy of the impact rod is transformed into the energy of the incident wave without considering other energy losses, the formula for calculating the specimen’s absorbed energy is as below:

$$W_S(t)=W_I(t)-W_R(t)-W_T(t)$$

(13)

In the formula, $W_s\left(t\right)$ is the sample’s absorbed energy, and $W_I\left(t\right)$, $W_R\left(t\right)$ and $W_T\left(t\right)$ are the incident, reflection and transmission energies separately, which can be calculated by the relationship between stress and time in the pressure rod.

The absorption energy is generally approximated as the crushing energy [42]. The ratio of absorbed energy to incident energy is utilized to express the energy consumption of specimen crushing, that is, the energy absorption rate, $\omega$.

$$\omega ={\displaystyle \frac{W_S\left(t\right)}{W_I\left(t\right)}}$$

(14)

To provide a clearer picture of the energy evolution of a sample during the failure process, the sample’s energy consumption density was determined by dividing the absorbed energy by the volume [43]. The formula for the calculation is as below:

$$\eta ={\displaystyle \frac{W_S\left(t\right)}{V}}$$

(15)

where $\eta$ denotes the energy consumption density, ${\displaystyle \raisebox{1ex}{$\mathrm{J}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{cm}}^3$}\right.}$ and $V$ stands for the volume of specimen, ${\mathrm{cm}}^3$.

According to Formulas (12)~(15), Table 2 displays the computed impact load resistance energy of rubber concrete with varying contents of rubber.

According to Table 2, the relationship between different rubber and water contents and the energy utilization ratio, including reflected, transmitted, and absorbed energy as well as energy consumption density, is illustrated in Figure 12, Figure 13, Figure 14 and Figure 15.

Figure 12a,b show that the rate of energy usage increases as the rubber content rises and decreases as the water content increases. The energy utilization rates for RC-0-1 to RC-12-1 are 0.425, 0.430, 0.435, 0.450, and 0.470, respectively, with increases of 1.18%, 1.16%, 3.45%, and 4.44%. For RC-0-2 to RC-12-2, the energy utilization rates are 0.412, 0.418, 0.425, 0.446, and 0.462, reflecting increases of 1.46%, 1.67%, 4.94%, and 3.59%. The energy utilization rates for RC-0-3 to RC-12-3 are 0.379, 0.408, 0.422, 0.435, and 0.447, with increases of 7.65%, 3.43%, 3.08%, and 2.76%. As both water and rubber content increase, the degree of breakage also rises, and the energy utilization rate is reduced due to crack expansion. Higher water content causes some of the input energy to be offset by free water in the sample, while increased rubber content results in more energy being released when encountering rubber particles, leading to greater damage. Thus, as rubber content increases, the fractal dimension increases accordingly, and the internal crack expansion of the sample changes from micro to macro. From Figure 12c,d, it can be concluded that the energy utilization rate becomes more efficient as the rubber and water contents increase.

Figure 13a,b show that the ratio of transmitted energy to incident energy decreases as the rubber content increases but increases as the water content rises. For RC-0-1 to RC-12-1, the transmitted energy/incident energy ratios are 0.295, 0.291, 0.287, 0.276, and 0.274, respectively, with decreases of −1.36%, −1.37%, −3.83%, and −0.72%. For RC-0-2 to RC-12-2, the ratios are 0.326, 0.317, 0.304, 0.297, and 0.285, reflecting decreases of −2.76%, −4.1%, −2.3%, and −4.04%. The transmittance/incident energy of RC-0-3~RC-12-3 was 0.357, 0.325, 0.312, 0.309, and 0.299, respectively, with an increase of −8.96%, −4%, −0.96%, and −3.24%. The transmitted energy/incident energy ratio for RC-9 shows the greatest decrease in the dry state and a smaller decrease in the natural state, being minimal in the saturated water state. From Figure 13c,d, it can be concluded that the presence of water and rubber in the concrete impedes the transfer of transmitted energy.

Figure 14a,b show that the ratio of reflected energy to incident energy for RC-0-1 to RC-12-1 is 0.28, 0.279, 0.278, 0.274, and 0.256, respectively, showing decreases of −0.36%, −0.36%, −1.44%, and −6.57%. For RC-0-2 to RC-12-2, the ratios are 0.262, 0.265, 0.271, 0.257, and 0.253, with increases of 1.15% and 2.26% and decreases of −5.17% and −1.56%. For RC-0-3 to RC-12-3, the ratios are 0.264, 0.267, 0.266, 0.256, and 0.254, with increases of 1.14% and decreases of −0.37%, −3.76%, and −0.78%. Under the same water content, the reflected energy/incident energy ratio decreases as the rubber content increases. In the natural state, there is a noticeable turning point of decline in the reflected energy/incident energy ratio after RC-6. From Figure 14c,d, it can be intuitively concluded that this phenomenon is likely due to the increased rubber content, which leads to a more complex fracture network within the rubber concrete, making it less effective at reflecting energy.

Figure 15 shows that the energy consumption density increases with the rubber content in the dry state. This indicates that adding rubber significantly enhances the test block’s ability to absorb energy and that varying the rubber content substantially affects the energy dissipation density of the material. However, the energy consumption density does not show a significant change in the saturated and natural states, suggesting that water content has no substantial effect on energy consumption density.

3.6. Constitutive Model

3.6.1. Unified Strength Theory

The unified strength theory is put forward and developed by Yu Maohong based on the double shear strength theory [44], and the basis for judging the damage is as follows: when the major stress on the double shear element’s surface and the two significant shear stresses on it both have impact functions that are greater than a certain threshold. Through many scholars’ theories, the primary stress expression of unified strength theory is obtained:

$$F=\left\{\begin{array}{l}{\sigma }_1-{\displaystyle \frac{\alpha }{1+b}}\left(b{\sigma }_2+{\sigma }_3\right)={\sigma }_t,{\sigma }_2\le {\displaystyle \frac{{\sigma }_1+\alpha {\sigma }_3}{1+\alpha }}\\ {\displaystyle \frac{1}{1+b}}\left({\sigma }_1+b{\sigma }_2\right)-\alpha {\sigma }_3={\sigma }_t,{\sigma }_2>{\displaystyle \frac{{\sigma }_1+\alpha {\sigma }_3}{1+\alpha }}\end{array}\right\}$$

(16)

where $\alpha$ is the compressive and tensile strength ratio of the material, $\alpha ={\displaystyle \raisebox{1ex}{$\left(1-\sin \phi \right)$}\!\left/ \!\raisebox{-1ex}{$\left(1+\sin \phi \right)$}\right.}$; $\phi$ is the internal friction angle of rubber concrete, $°$; ${\sigma }_1,{\sigma }_2,{\sigma }_3$ represent the first, second and third principal stresses, separately, $\mathrm{MPa}$; ${\sigma }_t$ is the uniaxial compressive strength of rubber concrete, $\mathrm{MPa}$; and $b$ is the intermediate principal stress coefficient, $0\sim 1$, illustrating how rubber concrete failure is impacted by the intermediate primary stress.

When $b$ is equal to $0$, the Mohr–Coulomb strength theory can be obtained. When $b=1$, it is the double shear strength theory that can be obtained. Yu Maohong [45] holds that when $b={\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}$, a new strength theory between the double shear strength criterion and the $M-C$ strength criterion is produced, which has certain significance and is more reasonable than the $D-P$ criterion, as shown in Figure 16. Therefore, this study takes $b={\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}$ for analysis and obtains a new failure criterion:

$$F=\left\{\begin{array}{l}{\sigma }_1-{\displaystyle \frac{4\alpha }{5}}\left({\displaystyle \frac{1}{4}}{\sigma }_2+{\sigma }_3\right)={\sigma }_t,{\sigma }_2\le {\displaystyle \frac{{\sigma }_1+\alpha {\sigma }_3}{1+\alpha }}\\ {\displaystyle \frac{4}{5}}\left({\sigma }_1+{\displaystyle \frac{1}{4}}{\sigma }_2\right)-\alpha {\sigma }_3={\sigma }_t,{\sigma }_2>{\displaystyle \frac{{\sigma }_1+\alpha {\sigma }_3}{1+\alpha }}\end{array}\right\}$$

(17)

It is assumed that rubber concrete conforms to the Weibull distribution [46]:

$$P\left(F\right)={\displaystyle \frac{m}{F_0}}{\left({\displaystyle \frac{F}{F_0}}\right)}^{m-1}\exp \left[-{\left({\displaystyle \frac{F}{F_0}}\right)}^m\right]$$

(18)

in which $P\left(F\right)$ and $F$ denote the strength distribution function and random distribution variable of rubber concrete and $m$ and $F_0$ are Weibull statistical distribution parameters.

$D$ is the damage variable:

$$D={\displaystyle {\int }_0^{F}P\left(x\right)dx={\displaystyle {\int }_0^F\frac{m}{F_0}}}{\left({\displaystyle \frac{x}{F_0}}\right)}^{m-1}\exp \left[-{\left({\displaystyle \frac{x}{F_0}}\right)}^m\right]dx=1-\exp \left[-{\left({\displaystyle \frac{F}{F_0}}\right)}^m\right]$$

(19)

3.6.2. Establish a Constitutive Model of Damage

Based on equivalent strain theory [47],

$$\{\begin{array}{l}{\sigma }_1^{*}={\displaystyle \frac{{\sigma }_1}{1-D}}={\displaystyle \frac{E{\epsilon }_1}{1-D}}+\upsilon \left({\sigma }_2+{\sigma }_3\right)\\ {\sigma }_2^{*}={\displaystyle \frac{{\sigma }_2}{1-D}}={\displaystyle \frac{E{\epsilon }_2}{1-D}}+\upsilon \left({\sigma }_1+{\sigma }_3\right)\\ {\sigma }_3^{*}={\displaystyle \frac{{\sigma }_3}{1-D}}={\displaystyle \frac{E{\epsilon }_3}{1-D}}+\upsilon \left({\sigma }_1+{\sigma }_2\right)\end{array}$$

(20)

where ${\sigma }_1^{\ast },{\sigma }_2^{\ast },{\sigma }_3^{\ast }$ are the effective stresses, $\mathrm{MPa}$; ${\sigma }_1,{\sigma }_2,{\sigma }_3$ denote separately the first, second and third nominal principal stresses, $\mathrm{MPa}$; $E$ is the elastic modulus, $\mathrm{GPa}$; ${\epsilon }_1,{\epsilon }_2,{\epsilon }_3$ are the elastic strains of rubber concrete; $D$ is the damage variable; and $\upsilon$ is the Poisson ratio of rubber concrete material.

Combining (17), (19) and (20), the following results are obtained:

$$D=\{\begin{array}{l}1-\exp \left\{-{\left[{\displaystyle \frac{{\sigma }_1^{*}-\frac{\alpha }{5}\left({\sigma }_2^{*}+4{\sigma }_3^{*}\right)}{F_0}}\right]}^m\right\},{\sigma }_2\le {\displaystyle \frac{{\sigma }_1+\alpha {\sigma }_3}{1+\alpha }}\\ 1-\exp \left\{-{\left[{\displaystyle \frac{\frac{1}{5}\left(4{\sigma }_1^{*}+{\sigma }_2^{*}\right)-\alpha {\sigma }_3^{*}}{F_0}}\right]}^m\right\},{\sigma }_2>{\displaystyle \frac{{\sigma }_1+\alpha {\sigma }_3}{1+\alpha }}\end{array}$$

(21)

Because of the impact test of rubber concrete, there is no lateral pressure, so

$${\sigma }_1=E{\epsilon }_d\exp \left\{-{\left[{\displaystyle \frac{{\sigma }_1^{*}-\frac{\alpha }{5}\left({\sigma }_2^{*}+4{\sigma }_3^{*}\right)}{F_0}}\right]}^m\right\}$$

(22)

Upon combining (20) and (22), the following is obtained:

$$\sigma =E\epsilon \exp \left\{-{\left[{\displaystyle \frac{0.93E\epsilon }{F_0}}\right]}^m\right\}$$

(23)

The parameters $F_0$ and m can be identified through stress–strain curves. Take the derivative of (23):

$${\displaystyle \frac{d\sigma }{d\epsilon }}|\epsilon ={\epsilon }_d=E\left\{\left[1-m{\left(-{\displaystyle \frac{0.93E\epsilon }{F_0}}\right)}^m\right]\exp \left[-{\left({\displaystyle \frac{0.93E\epsilon }{F_0}}\right)}^m\right]\right\}=0$$

(24)

When the stress–strain is at its peak, (24) can be simplified as follows:

$$\{\begin{array}{l}m={\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\ln \left(\frac{E_d{\epsilon }_d}{{\sigma }_d}\right)$}\right.}\\ F_0=0.93E_d{\epsilon }_d{m}^{{\displaystyle \frac{1}{m}}}\end{array}$$

(25)

The parameters can be obtained through (25), and then the constitutive model is verified.

3.6.3. Constitutive Model Verification

The dynamic stress–strain curves of concrete with varying rubber content under diverse contents of water were estimated through a modified damage constitutive model (

Table 3. Comparison of peak stress, peak strain, experimental and simulation results.

ID	Experiment		Simulation
	σd/Mpa	εd	σd/Mpa	εd
RC-0-1	38.530	0.039	38.500	0.391
RC-0-2	46.602	0.058	47.100	0.056
RC-0-3	49.000	0.057	45.000	0.056
RC-3-1	35.420	0.046	33.500	0.046
RC-3-2	40.350	0.054	38.000	0.052
RC-3-3	47.000	0.052	43.000	0.052
RC-6-1	32.000	0.051	31.200	0.049
RC-6-2	38.780	0.050	36.600	0.050
RC-6-3	46.000	0.050	45.400	0.049
RC-9-1	27.699	0.049	25.000	0.046
RC-9-2	35.290	0.056	33.000	0.053
RC-9-3	41.000	0.045	38.500	0.045
RC-12-1	24.000	0.058	22.000	0.058
RC-12-2	31.670	0.055	29.900	0.055
RC-12-3	40.000	0.041	38.000	0.040

lists the peak stress and peak strain estimated by the new constitutive model and those measured by the experiment), and a comparison was made between the test curves and the calculated findings of the model, as shown in Figure 17. Figure 17 shows that the modified dynamic damage constitutive model is well in line with the test curve, and the model proposed in this work is validated. The novel constitutive model offers a theoretical foundation for the practical use of rubber concrete by simulating the dynamic stress–strain properties of the material under impact load.

4. Conclusions

Impact tests were carried out on rubber concrete with different rubber content using the separated Hopkinson bar to study the mechanical properties and damage changes in rubber concrete under the coupling action of impact load and water content and obtain the dynamic mechanical properties, fractal dimension and energy transfer characteristics of rubber concrete. The main conclusions are as follows:

(1) The dynamic stress–strain curve of rubber concrete resembles that of regular concrete, including the elastic stage, yield stage, hardening stage and necking stage. Due to the difference in water content and rubber content, the stress–strain curve is different. Under different moisture content states, rubber concrete is more flexible than regular concrete, yet regular concrete often has a higher strength. When the content of rubber is constant, the rubber concrete has the highest peak stress in the saturated water state, the second in the natural state, and the least in the dry state.

(2) The distribution of rubber concrete fragments under water content and impact load has good fractal characteristics. Moisture content and rubber content have evident influence on the fractal dimensions of the concrete specimens. Rubber concrete sample fractal dimensions are proportionate to rubber content and inversely correlated with water content. The fractal dimension of RC-12-2 shows an inflection point.

(3) Both the water and rubber contents cause a rise in the energy consumption rate; the incident energy/transmitted energy steadily declined as the content of water grew and rose as the rubber content did. The transmitted energy/incident energy of RC-9 decreased the most in the dry state, followed by the natural state, and the least in the saturated water state. In the same water content state, the reflected energy/incident energy of different rubber content decreases as rubber content elevates. In the natural state, the incident energy/reflected energy after RC-6 shows an obvious decreasing inflection point.

(4) Based on the new dynamic damage constitutive model, the dynamic stress–strain curves of rubberized concrete under coupled water content and impact loading were derived. The modified dynamic damage constitutive model is well in line with the test curve, suggesting that the model can express accurately the stress–strain constitutive relation of rubberized concrete under the coupled effects of water content and impact loading.

5. Outlook

In this study, rubber concrete with varying rubber content was utilized to conduct an experimental investigation under the combined influence of impact load and water content. The study was constrained by a limited number of samples, loading conditions, and environmental factors. To further explore the applicability of rubber concrete, future research will encompass an array of approaches including a consideration of the mechanical properties at different rubber contents in complex environments involving varying temperatures, strain rates, and coupled dynamic and static loads. Additionally, numerical simulations will be employed to validate the validity of the tests.