Rubberized Concrete: Effect of the Rubber Size and Content on Static and Dynamic Behavior

: Rubberized concrete (RC) has received widespread attention due to its energy absorption and crack resistance properties. However, due to its low compressive strength, it is not recommended for structural applications. The rubber size and content affect RC’s mechanical properties. This study investigated and formulated the behavior of RC with different particle sizes and contents under dynamic and static loading. Quasi-static compressive and dynamic tests were conducted on RC with varying content of rubber (0–30%) and rubber sizes (0.1–20 mm). It was found that the rubber particle size was 0.5mm and the rubber content was 2%. An equation was derived from the experimental data to forecast the impact of rubber size and content on compressive strength. Additionally, by combining the literature and this research’s data, a model was established based on neural networks to predict the strength of RC. SHPB tests were carried out to study the stress–strain curves under dynamic load. The peak stress, fragment analysis, and energy absorption of RC with varying content of rubber and rubber sizes at three different strain rates (100 s −1 , 160 s −1 , and 290 s −1 ) were investigated. Equations describing the relationship between dynamic increase factor (DIF), rubber material content, and strain rate on different particle sizes were obtained by fitting. The DIF increased as the content of the rubber increased. By analyzing energy absorption data, it was found that the optimal ratio for energy absorption was RC-0.5-30, RC-0.1-30, and RC-10-30 at strain rates of 100 s −1 , 160 s −1 , and 290 s −1 . This study could be a good guideline for other researchers to easily select the content and size of the rubber in RC for their applications. It also has a positive significance in promoting the development of green building materials.


Introduction
Concrete is most commonly used in civil engineering and finds extensive application in industrial and civil structures, transportation infrastructure, and coastal and harbor developments [1][2][3].The primary constituents of concrete are derived from abundant and inexpensive geological resources, including sand, gravel, water, and cement, which are obtained by calcinating clay and limestone [4].Nevertheless, concrete exhibits drawbacks, including brittleness [5] and limited tensile strength [6].Concrete building structures experience more than just static loads during their utilization.They must also endure impact loads in certain special occurrences, such as earthquakes [7], blasts [8,9], and vehicular collisions with guardrails [10].The drawbacks of concrete significantly compromise the safety of building structures [11].
To improve the impact resistance of concrete, researchers have conducted many dynamic load tests on concrete.Dashun Cui et al. [12] analyzed the fracture process and crack propagation behavior of concrete specimens mixed with steel fibers and palm fibers using Split-Hopkinson pressure bar (SHPB) and DIC techniques and obtained their dynamic tensile properties and energy dissipation.The experimental results show that adding fibers can enhance the impact toughness of concrete, reduce the failure of the stress end of the specimen due to stress concentration, delay the failure time of the specimen, and effectively suppress crack propagation.Feng Shi et al. [13] analyzed and compared the dynamic uniaxial compression tests of self-compacting concrete (SCC) and normal concrete under different load strain rates and obtained the compression failure modes and stress-strain curves of the two types of concrete.Shiwen Feng et al. [14] used SHPB to analyze foam concrete's damage behavior and energy absorption characteristics under dynamic load.Tiecheng Yang et al. [15] conducted dynamic pressure tests on geopolymer concrete before and after water saturation using SHPB.The effects of water saturation and strain rate on the impact toughness of geopolymer concrete were studied.Multiple studies have also demonstrated that adding rubber to concrete can enhance the impact resistance of normal concrete [16,17].
How to solve and reasonably utilize solid waste materials has always been a key issue.M Manjunatha et al. [18] used industrial polyvinyl chloride (PVC) waste to replace cement in some M60 grade self-compacting concrete.Research has shown that this method not only helps reduce the cost of SCC but also does not affect its strength and durability.Waste rubber is also a conventional solid waste primarily derived from recycled tires [19].According to information, the yearly manufacturing of new tires surpasses 1.6 billion, whereas 1 billion new waste tires are generated annually [20].The primary method of disposing of these discarded tires is depositing them in landfills [21].Rubber exhibits high resistance to degradation and is highly flammable, posing significant environmental contamination risks [22].The RC offers a commendable and environmentally friendly resolution to this ecological issue by pulverizing discarded rubber tires into rubber powder or fibers, letting the rubber waste substitute for a portion of the aggregate of the concrete [23][24][25].This method not only solves the problem of environmental pollution but also enhances the mechanical properties of the concrete.
Rubber is an organic substance known for its high toughness [26] and low density [27].Specifically, rubber exhibits impact resistance [23,28].Mixing rubber as aggregate into concrete can significantly enhance its mechanical properties and ameliorate its inadequate toughness and other inherent properties.Shengtian Zhai et al. [29] discovered that RC's abrasion resistance tended to increase and decrease as the rubber percentage increased.RC exhibited a 14.9% enhancement in abrasion resistance compared to the control concrete.This improvement can be attributed mainly to rubber's ability to absorb the abrasion-generated energy.Nouran Yasser et al. [30] examined the durability of RC by conducting experiments on water absorption, chloride erosion, and sulfate erosion.They observed that RC exhibits improved workability and reduced density.A.F. Angelin et al. [31] examined the impact of rubber on the acoustic characteristics of RC and demonstrated a significant enhancement in acoustic attenuation, ranging from 30% to 43%.Wanhui Feng et al. [32] investigated the dynamic fracture toughness of RC using the drop weight test, which showed that the fracture toughness of RC increased to 30% at the time of the increase in rubber content.In the rest of the studies, the RC also showed excellent thermal insulation [33] and energy absorption properties [20,34].
The corresponding performance index of dynamic compression is significant in some high-strain rate states, such as explosions, earthquakes, collisions, and many other dynamic impacts [35].The study of materials under relevant high-strain rate states can be well carried out using the SHPB.Lei Pan et al. [36] conducted a study on the strength of RC in varying rubber content using simulation and the SHPB test.They found that the dynamic strength of RC increases as the strain rate increases.Zhiheng Liu et al. [37] examined the dynamic compression performance of self-compacting RC under multiple impact loads using the SHPB.They considered the effects of rubber content, impact velocity, and heat treatment temperature.The results indicated that including rubber aggregate improves the ductility and toughness of self-compacting RC.Thong M. Pham et al. [38] used SHPB to study the dynamic compression behavior of RC with two rubber particle sizes of 1-3 mm and 3-5 mm at 15% content.The RC containing a smaller rubber aggregate has higher static compressive strength but lower dynamic compressive strength than RC with bigger rubber particles.Although the dynamic impact of RC has been partially studied, there is still a lack of comprehensive and comparative studies on different rubber particle sizes and content.
While using rubber enhances the performance of normal concrete, the strength of RC generally exhibits a decrease in strength when rubber is added [2,[38][39][40].This has been investigated mainly due to the weaker interfacial transition zone (ITZ) created by the weaker bonding process of the rubber to the cement cementitious material [41,42].A slight decrease in concrete strength is acceptable within the scope of standard requirements because RC brings better impact resistance.However, a significant drop in strength is unacceptable, especially in some load-bearing structures, and unique application scenarios are often fatal [43].One of the most effective improvement methods is chemical or physical methods to improve the adhesion performance between rubber and cement [17,44,45].Another more effective scheme is to conduct a more comprehensive and regular investigation of the dynamic impacts and static properties of different rubber contents and particle sizes.At present, there are still a few relevant studies, and further exploration is needed.
This study involves doing a series of combination tests using various rubber contents (0%, 2%, 5%, 15%, and 30%) and particle sizes (0.1 mm, 0.5 mm, 2 mm, 5 mm, 10 mm, and 20 mm).The tests will employ static mechanical combination experiments and the split-Hopkinson pressure bar (SHPB) for dynamic impact.The aim is to examine the influence of varying rubber concentrations and particle dimensions on the dynamic impact behavior and static mechanical characteristics of RC.
In a static loading study, this paper investigates the damage process of 25 groups of RC with different content and sizes.Furthermore, this paper explores the reasons for the loss of strength.The relationship equation between the compressive strength of RC, the content of rubber, and particle size was derived based on data fitting.Meanwhile, a neural network learning method [46][47][48] was used to collect 197 sets of data from 23 studies, and a neural network model of particle size, replacement content, control concrete strength, and rubberized concrete strength was established to predict the static strength of RC.In the study of dynamic impact, the stress-strain curves sets of RC with different rubber admixtures at three different strain rates were analyzed using the SHPB test.They were fitted based on DIF and strain rate, and the DIF equations of RC with different rubber contents and different rubber particle sizes were established by the fitting method.The energy absorption capacity of RC was also analyzed.The work in this paper can demonstrate the static and dynamic mechanical behavior of RC, which is instructive for applying RC as a future building material.

Methodology
This article explores the influence of rubber particle size and content on the dynamic and static mechanical properties of RC.In previous literature research, it was found that as the rubber content exceeded 30%, there was a significant loss in the strength of RC.Therefore, the replacement content of rubber in this design is 2%, 5%, 15%, and 30%.At the same time, particle size exceeding 20 mm also causes a significant loss in the strength of RC.In order to explore the pattern, rubber particle sizes of 0.1 mm, 0.5 mm, 2 mm, 5 mm, 10 mm, and 20 mm were selected.An experimental program was created to observe several RC mixtures' dynamic and static mechanical properties, and the results were analyzed.The experimental program diagram is shown in Figure 1.
several RC mixtures' dynamic and static mechanical properties, and the results were analyzed.The experimental program diagram is shown in Figure 1.

Material Preparation
The concrete mix ratio is selected based on Chinese Code for the Mix Design of Ordinary Concrete (JGJ55-2011) [49].The cement used in this research is Shenyang Jindong P.O.42.5 Ordinary Portland cement, with a density of 3100 kg•m −3 .The chemical compositions and physical properties of the cement are shown in Table 1.The coarse aggregate natural gravel has a particle size of 5-15 mm.The grain gradation is continuous, with an apparent density of 2680 kg•m −3 .The fine aggregate is river sand, medium sand, and fineness module 2.6, with an apparent density of 2600 kg•m −3 .Tap water is used in this research.In order to prevent other factors from affecting the mechanical properties of concrete, concrete admixture is not used in the experiment.Rubber is produced by the Sichuan Huayi Rubber Company, and six different sizes of rubber (0.1 mm, 0.5 mm, 2 mm, 5 mm, 10 mm, and 20 mm) and five different replacement rates (0%, 2%, 5%, 15%, and 30%) are used to make the RC.By using the sieving method, the particle size of the rubber is controlled to meet the experimental requirements.We use high-precision electronic scales to weigh rubber particles to ensure the accurate control of rubber content.The rubber particle is shown in Figure 2. The single-factor experiment is used for research.This method has high experimental efficiency, can examine the interaction, is more conducive to controlling interference variables, and is closer to reality.The rubber comes from discarded tires, with an apparent density of 1120 kg•m −3 .The equal volume method is used to calculate the rubber content of the replacement aggregate.Due to the aggregate particle grading, the rubber size of 0.1-2 mm is replaced with fine aggregate, and rubber with a particle size of 5-20 mm is replaced with coarse aggregate [50].The concrete-mix ratio is shown in Table 2.

Material Preparation
The concrete mix ratio is selected based on Chinese Code for the Mix Design of Ordinary Concrete (JGJ55-2011) [49].The cement used in this research is Shenyang Jindong P.O.42.5 Ordinary Portland cement, with a density of 3100 kg•m −3 .The chemical compositions and physical properties of the cement are shown in Table 1.The coarse aggregate natural gravel has a particle size of 5-15 mm.The grain gradation is continuous, with an apparent density of 2680 kg•m −3 .The fine aggregate is river sand, medium sand, and fineness module 2.6, with an apparent density of 2600 kg•m −3 .Tap water is used in this research.In order to prevent other factors from affecting the mechanical properties of concrete, concrete admixture is not used in the experiment.Rubber is produced by the Sichuan Huayi Rubber Company, and six different sizes of rubber (0.1 mm, 0.5 mm, 2 mm, 5 mm, 10 mm, and 20 mm) and five different replacement rates (0%, 2%, 5%, 15%, and 30%) are used to make the RC.By using the sieving method, the particle size of the rubber is controlled to meet the experimental requirements.We use high-precision electronic scales to weigh rubber particles to ensure the accurate control of rubber content.The rubber particle is shown in Figure 2. The single-factor experiment is used for research.This method has high experimental efficiency, can examine the interaction, is more conducive to controlling interference variables, and is closer to reality.The rubber comes from discarded tires, with an apparent density of 1120 kg•m −3 .The equal volume method is used to calculate the rubber content of the replacement aggregate.Due to the aggregate particle grading, the rubber size of 0.1-2 mm is replaced with fine aggregate, and rubber with a particle size of 5-20 mm is replaced with coarse aggregate [50].The concrete-mix ratio is shown in Table 2.    Note: regarding RC-S-C, RC represents the rubberized concrete, S is the rubber size, and C is the rubber substitution rate, and NC represents the normal concrete (see Figure 2).
Rubber is a kind of surface hydrophobic material, and rubber particles easily float in the process of stirring, affecting the mix's performance.The method of mixing the RC is different from that of normal concrete.In this experiment, rubber particles, according to their particle size, were first mixed with aggregate and water.We made sure the rubber  Note: regarding RC-S-C, RC represents the rubberized concrete, S is the rubber size, and C is the rubber substitution rate, and NC represents the normal concrete (see Figure 2).
Rubber is a kind of surface hydrophobic material, and rubber particles easily float in the process of stirring, affecting the mix's performance.The method of mixing the RC is different from that of normal concrete.In this experiment, rubber particles, according to their particle size, were first mixed with aggregate and water.We made sure the rubber surface was thoroughly wet.Secondly, we mixed the cement to ensure the rubber and cement were evenly mixed.The process flow chart is shown in Figure 3.
After the mix, we place the mixture into the mold.After that, it is placed on a shaking table to expel the air bubbles.Then, we cover it with plastic wrap to prevent water loss.After 24 h, demold the sample and place it in a curing box with a humidity of not less than 95% and a temperature of 20 • C ± 2 for 28 days.After 28 days, the concrete specimens are subjected to water grinding and cutting processing to ensure that the surface's flatness, smoothness, and verticality meet the test requirements.
surface was thoroughly wet.Secondly, we mixed the cement to ensure the rubber and cement were evenly mixed.The process flow chart is shown in Figure 3.After the mix, we place the mixture into the mold.After that, it is placed on a shaking table to expel the air bubbles.Then, we cover it with plastic wrap to prevent water loss.After 24 h, demold the sample and place it in a curing box with a humidity of not less than 95% and a temperature of 20 °C ± 2 for 28 days.After 28 days, the concrete specimens are subjected to water grinding and cutting processing to ensure that the surface's flatness, smoothness, and verticality meet the test requirements.

Quasi-Static Compressive Test
The electro-hydraulic servo compressive strength testing machine tests compressive strength.We use high-speed cameras to capture the process of crack formation during loading.The sample sizes are 100 mm × 100 mm× 100 mm.The loading rate of the experiment is 0.5-0.8MPa per second.It is acceptable to use Formula (1) based on the Chinese standard (GB/T50081-2002) [51] to calculate the compressive strength of the samples.
, is the compressive strength of the cube specimen. is the sample failure load, and  is the specimen-bearing area.As the sample size is 100 mm × 100 mm× 100 mm, the final figure also needs to be multiplied by the conversion factor 0.95.

Dynamic Compressive Test
The split-Hopkins pressure bar (SHPB) tested the dynamic mechanical properties.The machine is shown in Figure 4.The sample sizes are φ100 mm × 50 mm.The basic principle of this test is the propagation theory of elastic waves in elastic rods.Based on the assumption of stress uniformity, the stress wave repeats 2-3 times in the sample and is

Mechanical Testing 2.2.1. Quasi-Static Compressive Test
The electro-hydraulic servo compressive strength testing machine tests compressive strength.We use high-speed cameras to capture the process of crack formation during loading.The sample sizes are 100 mm × 100 mm× 100 mm.The loading rate of the experiment is 0.5-0.8MPa per second.It is acceptable to use Formula (1) based on the Chinese standard (GB/T50081-2002) [51] to calculate the compressive strength of the samples.
f c,s is the compressive strength of the cube specimen.F is the sample failure load, and A is the specimen-bearing area.As the sample size is 100 mm × 100 mm× 100 mm, the final figure also needs to be multiplied by the conversion factor 0.95.

Dynamic Compressive Test
The split-Hopkins pressure bar (SHPB) tested the dynamic mechanical properties.The machine is shown in Figure 4.The sample sizes are φ100 mm × 50 mm.The basic principle of this test is the propagation theory of elastic waves in elastic rods.Based on the assumption of stress uniformity, the stress wave repeats 2-3 times in the sample and is equal everywhere.Using the assumption of stress uniformity (ε i (t) + ε r (t) = ε t (t)), we use the two-wave method [52].The compression strain rate .ε s (t), the compressive strain ε s (t), and the compressive stress σ s (t) can be calculated using the following Formula (2).
E b is the elasticity modulus of the elastic rod, τ is the time variable, t is the time, c b is the stress wave velocity in the rod, and r b and r 0 are the radius of the bar and the sample.l 0 is the length of the sample.ε i , ε r , and ε t are respectively the incident, reflected, and projected strains in the bar.
We use the φ100 mm SHPB system by adjusting the impact speed of the φ100 mm × 600 mm bullet, and we obtain three strain rates for each sample (the three strain rates are 70-300 s −1 ) to study the dynamic behavior of RC under different strain rates.
In order to ensure the accuracy of SHPB test results and constant strain rate loading during the test process, a brass plate was used as a pulse shaper [53][54][55].It was pasted at the center of the impact surface of the incident rod as a pulse shaper.Comparative experiments were conducted on pulse shapers of brass and copper of different sizes based on different speeds.After comparisons, brass pieces with a diameter of 30 mm and a thickness of 1 mm and 2.5 mm were chosen as pulse shapers.This technology helps shape the incident wave.The plastic deformation of the pulse shaper can physically filter out the high-frequency components of the incident wave, reducing wave dispersion.Extending the rise time of the incident wave and presenting an approximately half-sine pulse is crucial for achieving stress balance in the specimen, making the SHPB test results in this paper reliable.
is the elasticity modulus of the elastic rod,  is the time variable,  is the time,  is the stress wave velocity in the rod, and  and  are the radius of the bar and the sample. is the length of the sample. ,  , and  are respectively the incident, reflected, and projected strains in the bar.We use the φ100 mm SHPB system by adjusting the impact speed of the φ100 mm × 600 mm bullet, and we obtain three strain rates for each sample (the three strain rates are 70-300 s −1 ) to study the dynamic behavior of RC under different strain rates.
In order to ensure the accuracy of SHPB test results and constant strain rate loading during the test process, a brass plate was used as a pulse shaper [53][54][55].It was pasted at the center of the impact surface of the incident rod as a pulse shaper.Comparative experiments were conducted on pulse shapers of brass and copper of different sizes based on different speeds.After comparisons, brass pieces with a diameter of 30 mm and a thickness of 1 mm and 2.5 mm were chosen as pulse shapers.This technology helps shape the incident wave.The plastic deformation of the pulse shaper can physically filter out the high-frequency components of the incident wave, reducing wave dispersion.Extending the rise time of the incident wave and presenting an approximately half-sine pulse is crucial for achieving stress balance in the specimen, making the SHPB test results in this paper reliable.

Static Compressive Strength
In order to better explore the relationship between RC and rubber particle size and content, Figure 5 is drawn.It can be observed that the addition of rubber weakened the original strength of concrete.When the rubber content was increased from 0% to 30%, the compressive strength of different particle sizes decreased by 40-61%.The largest decrease in strength, with a drop ratio of 61%, was observed in the specimen RC-10-30.The specimen RC-0.5-2 exhibited the lowest strength decrease ratio, which was 11%.

Static Compressive Strength
In order to better explore the relationship between RC and rubber particle size and content, Figure 5 is drawn.It can be observed that the addition of rubber weakened the original strength of concrete.When the rubber content was increased from 0% to 30%, the compressive strength of different particle sizes decreased by 40-61%.The largest decrease in strength, with a drop ratio of 61%, was observed in the specimen RC-10-30.The specimen RC-0.5-2 exhibited the lowest strength decrease ratio, which was 11%.
Buildings 2024, 14, 1541 8 of Figure 5 demonstrates a non-linear decline in the compressive strength of RC as th rubber sizes range from 0.1 to 20 mm.The alteration in content has a greater effect on RC compressive strength than does the variation in particle size.Through the static loading test process, the results obtained by high-speed camer show that the increase in rubber reduces the fissure of concrete during the process.Th integrity of damaged RC specimens is significantly better than that of normal concre Figure 5 demonstrates a non-linear decline in the compressive strength of RC as the rubber sizes range from 0.1 to 20 mm.The alteration in content has a greater effect on RC's compressive strength than does the variation in particle size.
Through the static loading test process, the results obtained by high-speed cameras show that the increase in rubber reduces the fissure of concrete during the process.The integrity of damaged RC specimens is significantly better than that of normal concrete specimens, and this performance becomes more pronounced as the rubber content continues to increase, as shown in Figure 6.Adding rubber to normal concrete can improve the brittle damage of ordinary concrete and increase the toughness.

Compressive Strength Fitting Analysis
The relationship with the compressive strength of RC and the rubber content and size can be established by analyzing the strength of normal concrete.Since the influence of rubber content on the change in compressive strength is dominant, formula fitting is first performed based on the relationship between rubber content and compressive strength.After many verifications, it was found that the linear fitting effect was better during the fitting process.At the same time, when the rubber content is 0%, the test block is normal concrete, so the intercept of the linear fitting formula is set to 58.3, which is the strength of the control concrete (normal concrete).The parameters of different particle sizes in Table 3 are fitted to the formula.The specific fitting process is shown in Figure 8.After multiple verifications, it was found that the exponential fitting effect is better.They are shown in formula (3).
In the formula,  is the rubber's content,  is the rubber's size,  , is the quasi- The decrease in concrete strength caused by the addition and replacement of rubber can be explained by the following theories: (1) Rubber and concrete have different moduli of elasticity [56].The elastic modulus of concrete is about 5000-24,000 times that of rubber.This difference makes RC prone to cracking around the rubber during the loading process, thereby accelerating the damage to the concrete during the loading process.(2) By observing the surface of the RC with a microscope, as shown in Figure 7, it can be found that there is a gap between rubber aggregate and cement paste.Because of the hydrophobic properties of rubber, the interface adhesion between rubber aggregate and cement slurry is low, resulting in a reduction in strength.(3) Aggregates mainly play a role similar to human skeleton in concrete, so good grading can help the accumulation of concrete to form the best state.The increase in rubber has disrupted the grading between the original sand and stone [57].

Compressive Strength Fitting Analysis
The relationship with the compressive strength of RC and the rubber content and size can be established by analyzing the strength of normal concrete.Since the influence of rubber content on the change in compressive strength is dominant, formula fitting is first performed based on the relationship between rubber content and compressive strength.After many verifications, it was found that the linear fitting effect was better during the fitting process.At the same time, when the rubber content is 0%, the test block is normal concrete, so the intercept of the linear fitting formula is set to 58.3, which is the strength of the control concrete (normal concrete).The parameters of different particle sizes in Table 3 are fitted to the formula.The specific fitting process is shown in Figure 8.After multiple verifications, it was found that the exponential fitting effect is better.They are shown in formula (3).
In the formula,  is the rubber's content,  is the rubber's size,  , is the quasistatic compressive strength, and  , , is the control concrete (normal concrete) compressive strength, and the result of this experiment is 58.3.

Compressive Strength Fitting Analysis
The relationship with the compressive strength of RC and the rubber content and size can be established by analyzing the strength of normal concrete.Since the influence of rubber content on the change in compressive strength is dominant, formula fitting is first performed based on the relationship between rubber content and compressive strength.After many verifications, it was found that the linear fitting effect was better during the fitting process.At the same time, when the rubber content is 0%, the test block is normal concrete, so the intercept of the linear fitting formula is set to 58.3, which is the strength of the control concrete (normal concrete).The parameters of different particle sizes in Table 3 are fitted to the formula.The specific fitting process is shown in Figure 8.After multiple verifications, it was found that the exponential fitting effect is better.They are shown in Formula (3).
In the formula, C is the rubber's content, S is the rubber's size, f c,s is the quasi-static compressive strength, and f c,s,normol is the control concrete (normal concrete) compressive strength, and the result of this experiment is 58.3.

Compressive Strength ANN Analysis
Compared to normal cement concrete, RC properties have varying changes, and the mechanism of action is relatively complex.Making a good theoretical model for strength prediction takes time and effort.Artificial neural network models can be well applied to predict the compressive strength of RC due to their strong nonlinear mapping ability and adaptive learning and memory characteristics [58].The artificial neural network is a black box structure that can be established and applied to predict the compressive strength of waste RC.

Collection and Classification of Samples
We obtained data from the literature to increase the accuracy of the neural networks.A total of 222 RC data were obtained, among which 25 were static test data in this paper and 197 were obtained from other literature, all of which are shown in Table 4.
The first 70% of the confirmed data is used as the training set, and the last 30% is used as the validation and testing set. Figure 8.The fitting process between rubber content and compressive strength of RC (R 2 = 0.9).

Compressive Strength ANN Analysis
Compared to normal cement concrete, RC properties have varying changes, and the mechanism of action is relatively complex.Making a good theoretical model for strength prediction takes time and effort.Artificial neural network models can be well applied to predict the compressive strength of RC due to their strong nonlinear mapping ability and adaptive learning and memory characteristics [58].The artificial neural network is a black box structure that can be established and applied to predict the compressive strength of waste RC.

Collection and Classification of Samples
We obtained data from the literature to increase the accuracy of the neural networks.A total of 222 RC data were obtained, among which 25 were static test data in this paper and 197 were obtained from other literature, all of which are shown in Table 4.The first 70% of the confirmed data is used as the training set, and the last 30% is used as the validation and testing set.

Normalization of Sample Data
Due to the large distribution range of data, in order to avoid saturation of the neural network neuron output due to a certain net input absolute value being too large, and to avoid some data with small values being ignored, resulting in output distortion, it is necessary to perform scale transformation on the original data.It is necessary to normalize the data [75].This article uses Formula (4) to normalize the data, transform input and output parameters at the [0,1] area.
In the formula, x is the original data true value, x min is the minimum value of the data in the original data, x max is the maximum value of the data in the original data, and x new is the value after normalization.

Model Building and Training
The MATLAB R2018b neural network toolbox is used to make the network model.The neural network structure is shown in Figure 10.We set the network model parameters as follows: (1) Input layer.The number of input nodes is three, including rubber size, content, and compressive strength of control samples.(2) Output layer.The number of output nodes is one, which is the compressive strength of RC. (3) Hidden layer.The hidden layer is one layer, and the number of nodes in the hidden layer is eight.( 4) Training function.We adopt the Levenberg-Marquardt algorithm as a compromise between the Gauss-Newton and gradient descent methods to make the fastest adjustment possible.
The neural network training performance diagram is shown in Figure 9.At 30 epochs, we achieve good performance.According to the fitting effect drawing in Figure 9, R 2 = 0.95, and so the fitting effect performs well.Therefore, the neural network model can be used to predict the compressive strength of the RC.

Model Testing
In order to better verify the effect of the fitted model and the neural network model, 12 groups of data were randomly selected from the experimental data for testing.The corresponding prediction results are shown in Figure 11.The data predicted by the neural network need anti-normalization.This article uses Formula (5) to anti-normalize the data:

Model Testing
In order to better verify the effect of the fitted model and the neural network model, 12 groups of data were randomly selected from the experimental data for testing.The corresponding prediction results are shown in Figure 11.The data predicted by the neural network need anti-normalization.This article uses Formula (5) to anti-normalize the data: In the formula,  is the original data true value,  is the minimum value of the data in the original data,  is the maximum value of the data in the original data, and  is the value after normalization.

Model Testing
In order to better verify the effect of the fitted model and the neural network model, 12 groups of data were randomly selected from the experimental data for testing.The corresponding prediction results are shown in Figure 11.The data predicted by the neural network need anti-normalization.This article uses Formula (5) to anti-normalize the data: In the formula, x is the original data true value, x min is the minimum value of the data in the original data, x max is the maximum value of the data in the original data, and x new is the value after normalization.We fit the predicted values through the neural network prediction values of each group of the RC specimens.Comparing the predicted values with the actual measured values, the absolute error and relative error are shown in Table 5 below.We fit the predicted values through the neural network prediction values of each group of the RC specimens.Comparing the predicted values with the actual measured values, the absolute error and relative error are shown in Table 5 below.
It can be seen from the prediction results of the compressive strength of the randomly sampled neural network and the formula fitting that both the neural network and the formula fitting show good prediction effects.Most of the sample errors are small.The absolute errors are all less than 5 MPa, among which the average absolute error of formula fitting is 3.3 MPa, the average absolute error of the neural network model is 2.1 MPa, the average relative error of formula fitting is 9%, and the average relative error of neural network simulation is 5%.Thus, both approaches can predict the compressive strength of RC.

Dynamic SHPB Test Result
Figure 12 is the complete stress-strain curve of RC with different content and sizes and different strain rates.It can be found that the dynamic strength of RC and normal concrete increases with a continuous increase in the strain rate.With the increasing rubber content, the dynamic compression strength of RC shows a decreasing trend.The size of rubber particles has no obvious effect on the dynamic compressive strength of RC.It is mainly based on the fact that the normal concrete material is relatively uniform, and the added rubber destroys the uniformity of the normal concrete.The elastic modulus of the RC increases with an increase in strain rate.It is observed that RC can achieve excellent performance in the face of a high strain rate.
Figure 13 illustrates the fragmentation of the specimen under different impact rates.It can be observed that as the loading rate increases, the percentage of larger fragments decreases significantly.Higher loading rates lead to a continuous increase in the number of microcracks.At a speed of 17 m/s, the coarse aggregate fractures.However, as the RC content increases, the integrity of the fragments improves.This suggests that increased rubber content enhances the concrete's toughness and energy absorption capacity [20].
In Figure 13, the structural study of the interface after impact failure of RC is observed, and it is suggested that there may be extensive microcracks on the rubber particles.At the location where the rubber aggregate is broken and extracted, dents in the cement matrix can usually be observed.These dents enhance the bonding strength between rubber and cement.These actual impact test results contradict the weak effect of rubber and cement-based direct ITZ described in the literature.In addition, the widespread micro-cracks observed on the surface of the test block particles support the hypothesis that rubber particles in RC generate tensile strain during the crushing process.This result is consistent with the fact that there is stress transfer between the two-phase composite materials, and the rubber particles undergo tensile strain before failure.In addition, the extraction of rubber particles and micro-cracks on the surface of rubber particles are two toughening mechanisms that consume energy in the rubber concrete matrix, which do not exist in traditional concrete.As mentioned above, these mechanisms have helped RC significantly improve impact energy and fracture toughness.Further research on crack propagation under load may help reveal the mechanism by which rubber particles improve the fracture toughness of concrete.
Figure 14 shows that the dynamic compressive strength of RC increases almost linearly with increasing strain rates for different particle sizes.As the rubber content increases from 0 to 30%, the strength of the RC decreases.This is primarily because the original aggregates in the concrete mix are being replaced by rubber, which disrupts the particle size distribution of concrete.Additionally, the difference in elastic modulus between rubber and cement slurry causes cracks to develop, forming weak areas that reduce the dynamic strength of RC.
Figure 15 shows the relationship between the strain rate and dynamic compressive strength at different rubber contents and sizes.Regardless of the size of rubber particles, the strength reduction rate can gradually increase with an increase in the rubber volume fraction.The test block with the largest reduction is RC-10-30%, and the dynamic compressive strength of RC is reduced by 48-52%.In the dynamic impact process of the RC with different rubber particle sizes, the strength of the RC with a rubber particle size greater than 5mm is significantly reduced.This is compared to the RC that has a smaller particle size.This is because the direct contact area between cement slurry and rubber with a larger particle size is larger, resulting in weaker areas and increasing the formation of vulnerable areas, decreasing the dynamic compressive strength.No suitable fitting relationship was found between the strain rate and rubber content through analysis.
Buildings 2024, 14, 1541 14 of 26 mainly based on the fact that the normal concrete material is relatively uniform, and the added rubber destroys the uniformity of the normal concrete.The elastic modulus of the RC increases with an increase in strain rate.It is observed that RC can achieve excellent performance in the face of a high strain rate.Figure 13 illustrates the fragmentation of the specimen under different impact rates.It can be observed that as the loading rate increases, the percentage of larger fragments decreases significantly.Higher loading rates lead to a continuous increase in the number of microcracks.At a speed of 17 m/s, the coarse aggregate fractures.However, as the RC content increases, the integrity of the fragments improves.This suggests that increased rubber content enhances the concrete's toughness and energy absorption capacity [20].In Figure 13, the structural study of the interface after impact failure of RC is observed, and it is suggested that there may be extensive microcracks on the rubber particles.At the location where the rubber aggregate is broken and extracted, dents in the cement matrix can usually be observed.These dents enhance the bonding strength between rubber and cement.These actual impact test results contradict the weak effect of rubber and cement-based direct ITZ described in the literature.In addition, the widespread microcracks observed on the surface of the test block particles support the hypothesis that rubber particles in RC generate tensile strain during the crushing process.This result is consistent with the fact that there is stress transfer between the two-phase composite materials, and the rubber particles undergo tensile strain before failure.In addition, the extraction of rubber particles and micro-cracks on the surface of rubber particles are two toughening mechanisms that consume energy in the rubber concrete matrix, which do not exist in traditional concrete.As mentioned above, these mechanisms have helped RC significantly improve impact energy and fracture toughness.Further research on crack propagation under load may help reveal the mechanism by which rubber particles improve the fracture toughness of concrete.Figure 14 shows that the dynamic compressive strength of RC increases almost linearly with increasing strain rates for different particle sizes.As the rubber content increases from 0 to 30%, the strength of the RC decreases.This is primarily because the original aggregates in the concrete mix are being replaced by rubber, which disrupts the particle size distribution of concrete.Additionally, the difference in elastic modulus between rubber and cement slurry causes cracks to develop, forming weak areas that reduce the dynamic strength of RC.  Figure 15 shows the relationship between the strain rate and dynamic compressive strength at different rubber contents and sizes.Regardless of the size of rubber particles, the strength reduction rate can gradually increase with an increase in the rubber volume fraction.The test block with the largest reduction is RC-10-30%, and the dynamic compressive strength of RC is reduced by 48-52%.In the dynamic impact process of the RC with different rubber particle sizes, the strength of the RC with a rubber particle size greater than 5mm is significantly reduced.This is compared to the RC that has a smaller particle size.This is because the direct contact area between cement slurry and rubber with a larger particle size is larger, resulting in weaker areas and increasing the formation of vulnerable areas, decreasing the dynamic compressive strength.No suitable fitting relationship was found between the strain rate and rubber content through analysis.

DIF of the RC
The dynamic increase factor (DIF) is the ratio of dynamic compressive strength ( , ) to static compressive strength ( , ), which reflects the increase in the compressive strength of materials under impact loads.The calculation is Formula (6): According to the calculation results of the DIF formula in Table 6, there is a linear

DIF of the RC
The dynamic increase factor (DIF) is the ratio of dynamic compressive strength ( f c,d ) to static compressive strength ( f c,s ), which reflects the increase in the compressive strength of materials under impact loads.The calculation is Formula (6): According to the calculation results of the DIF formula in Table 6, there is a linear relationship between the DIF of empirical concrete and the logarithm of the strain rate [36].Therefore, for RC with different particle sizes, the logarithm of the strain rate is linearly fitted with DIF, and Figure 16 shows the fitting results.According to the graphic, the DIF of RC grows as the strain rate rises and follows a similar trend to that of normal concrete.In addition, with the increase in rubber content, the DIF of concrete with different rubber particle sizes increases gradually.This is mainly because, with the increase in rubber content, the dynamic strength of RC is significantly improved in the impact test.The logarithmic relationship between the compressive strength of the DIF and the strain rate of the RC can be used to derive the fitting formula.Formula (7) describes the link between the RC's DIF and strain rate: In the formula, .ε represents the strain rate, while the remaining variables, a, b, c, and d, are the fitting parameters.Refer to Table 7 for specific information.The changes in Formula ( 7) parameters of RC with different particle sizes and rubber content are shown in Figure 17. =  ×  +  (11) Parameter  in Formulas ( 8)- (11) refers to the content of rubber particles.Through parameter fitting, the DIF and strain rate fitting formulas of RC with six rubber sizes, 0.1 mm, 0.5 mm, 2 mm, 5 mm, 10 mm, and 20 mm, with a content of 0-30% can be obtained.The formula can be used to predict the dynamic properties of RC.The specific fitting parameters are detailed in Table 8.Through the fitting process, the DIF formula for the RC with different particle sizes can be obtained: Parameter C in Formulas ( 8)-( 11) refers to the content of rubber particles.Through parameter fitting, the DIF and strain rate fitting formulas of RC with six rubber sizes, 0.1 mm, 0.5 mm, 2 mm, 5 mm, 10 mm, and 20 mm, with a content of 0-30% can be obtained.The formula can be used to predict the dynamic properties of RC.The specific fitting parameters are detailed in Table 8.  8)- (11).

Analysis of Dynamic Toughness Index of RC
In this paper, ASTM C1018, the toughness evaluation method for ideal elastoplastic materials formulated by the American Society for Materials and Testing, is used to evaluate the bending toughness of the RC studied.The specific evaluation method is shown in Figure 18 below.The σ represents the peak force, 0.85σ represents the first crack force, A represents the first crack point, which corresponds to the first crack deflection δ.The greater the toughness index, the closer the RC is to the ideal plastic material.Formula ( 12) calculates the values of I 5 .
Buildings 2024, 14, 1541 22 of 26 represents the first crack point, which corresponds to the first crack deflection δ.The greater the toughness index, the closer the RC is to the ideal plastic material.Formula ( 12) calculates the values of  .It can be seen from Figure 19 that with the increase in rubber content, the toughness index of RC gradually increases.In all cases, it is superior to that of normal concrete.This indicates that with the increase in the content of rubber, the toughness of RC is continuously enhanced, and it can absorb more energy.In all groups, the RC with a 30% rubber content and a diameter of 0.1 mm exhibits the highest increase in toughness, improving by 31.9% compared to the control group of normal concrete.On the other hand, there is no clear regular relationship between the toughness coefficient of RC and the particle size of rubber.This may imply that the energy absorption It can be seen from Figure 19 that with the increase in rubber content, the toughness index of RC gradually increases.In all cases, it is superior to that of normal concrete.This indicates that with the increase in the content of rubber, the toughness of RC is continuously enhanced, and it can absorb more energy.In all groups, the RC with a 30% rubber content and a diameter of 0.1 mm exhibits the highest increase in toughness, improving by 31.9% compared to the control group of normal concrete.It can be seen from Figure 19 that with the increase in rubber content, the toughness index of RC gradually increases.In all cases, it is superior to that of normal concrete.This indicates that with the increase in the content of rubber, the toughness of RC is continuously enhanced, and it can absorb more energy.In all groups, the RC with a 30% rubber content and a diameter of 0.1 mm exhibits the highest increase in toughness, improving by 31.9% compared to the control group of normal concrete.On the other hand, there is no clear regular relationship between the toughness coefficient of RC and the particle size of rubber.This may imply that the energy absorption capacity of RC mainly depends on the content of rubber added rather than the particle On the other hand, there is no clear regular relationship between the toughness coefficient of RC and the particle size of rubber.This may imply that the energy absorption capacity of RC mainly depends on the content of rubber added rather than the particle size.However, to draw more accurate conclusions, more research may be needed to validate this hypothesis further.

𝐼 = 𝑆 𝑆
Based on the experimental results of this study, it can be concluded that RC can be used as an ideal new type of green solid waste concrete material.Although, the addition of rubber weakened the strength of some of the normal concrete, the energy absorption effect of RC was significantly improved.For example, at a particle size of 0.5 mm and a content of 2%, the quasi-static strength of RC had a loss of only 6 MPa compared to normal concrete, with a strength of 52.1 MPa.The design specifications for concrete structures (GB 50010-2010) [76] can meet all design scenarios, and their energy absorption effect is improved by 5.6% under high strain rates.

Conclusions
This study experimentally studied the static and dynamic mechanical behavior of RC with different rubber particle sizes and rubber content.Using the compressive strength test and SHPB test methods, we comprehensively analyzed static strength, dynamic strength, the DIF, energy absorption capacity, and fragment characteristics.The results of the current work can be summarized as follows: (1) Rubber reduces the static strength of concrete, and the strength of RC decreases with an increase in rubber content.However, the effect of rubber particle size on the compressive strength of RC is irregular.(2) This article presents empirical equations for the compressive strength of concrete with varying rubber content and particle sizes.It uses BP neural networks combined with the literature data for neural network learning.The formula and neural network learning model can be used to predict the static compressive strength of concrete with different rubber particle sizes and contents.(3) In the dynamic test, the peak strain decreases with the increase in rubber content.
When the rubber content is 30% and the rubber size is 10mm, the maximum strength reduction is 52.3%.The dynamic compressive strength of RC with a rubber particle size greater than 5mm is generally small, but the effect is insignificant.As the particle size increases, the interface transition zone becomes larger.(4) Compared to normal concrete, RC exhibits better impact resistance and energy absorption ability at high load rates.The main reason is that a large content of rubber particles is added to RC, which can play a role in energy absorption, slowing down the speed of crack propagation and progressive failure.(5) The DIF formula for RC with different rubber content and different rubber particle sizes was established through fitting methods.The formula can be used as a reference to predict the dynamic properties of RC under different strain rates in the future.(6) The toughness index of RC increases with an increase in rubber content.There is no clear relationship between RC's toughness index and the rubber particles' size.This phenomenon may mean that the energy absorption capacity of RC mainly depends on the added rubber content rather than particle size.
The dynamic and static fitting formulas and mathematical models mentioned in this paper can provide guidance for other researchers in choosing the content and size of rubber in the practical application of rubber concrete.
Lastly, future research works in this regard are recommended as follows.The first field is to improve the bonding performance between rubber particles and cement-based materials through physical or chemical modification, thereby enhancing the strength of the ITZ.The second field is to explore the tensile performance of RC using dynamic and static Brazilian discs and three-point bending tests.The third area is to study the durability performance of RC.The fourth area is to analyze the economic benefit of RC.

Figure 2 .
Figure 2. The photos of rubber particles of different sizes.

Figure 2 .
Figure 2. The photos of rubber particles of different sizes.

Figure 3 .
Figure 3. Mixing process of the RC.

Figure 3 .
Figure 3. Mixing process of the RC.

Figure 5 .
Figure 5. Compressive strength of RC: (a) compressive strength with different rubber size, (b) com pressive strength with different rubber content.

Figure 5 .
Figure 5. Compressive strength of RC: (a) compressive strength with different rubber size, (b) compressive strength with different rubber content.

Figure 6 .
Figure 6.Loading failure process of concrete under the static loading.

Figure 6 .
Figure 6.Loading failure process of concrete under the static loading.

Figure 6 .
Figure 6.Loading failure process of concrete under the static loading.

Figure 8 .
Figure 8.The fitting process between rubber content and compressive strength of RC ( = 0.9).

Figure 9 .
Figure 9. BP neural network prediction model for RC.

Figure 10 .
Figure 10.Neural network training results: (a) neural network training performance, (b) neural network training regression.

Figure 9 .
Figure 9. Neural network training results: (a) neural network training performance, (b) neural network training regression.

Figure 9 .
Figure 9. BP neural network prediction model for RC.

Figure 10 .
Figure 10.Neural network training results: (a) neural network training performance, (b) neural network training regression.

Figure 10 .
Figure 10.BP neural network prediction model for RC.

26 Figure 11 .
Figure 11.Comparison between predicted and actual results.

Figure 11 .
Figure 11.Comparison between predicted and actual results.

Figure 12 .Figure 12 .
Figure 12.Stress-strain curves of RC with different rubber content and size.

Figure 18 .
Figure 18.Calculation method for dynamic toughness index.

Figure 18 .
Figure 18.Calculation method for dynamic toughness index.

Table 1 .
Chemical compositions and physical properties of the cement.

Table 1 .
Chemical compositions and physical properties of the cement.

Table 2 .
Concrete mix proportions of RC with different rubber sizes and replacement content (unit: kg•m −3 ).

Table 2 .
Concrete mix proportions of RC with different rubber sizes and replacement content (unit: kg•m −3 ).

Table 3 .
Parameters of fitting curves of Eq.

Table 5 .
Prediction value and error value of compressive strength of the RC fitted by neural network and formula.

Table 5 .
Prediction value and error value of compressive strength of the RC fitted by neural network and formula.

Table 6 .
Concrete mechanical parameters test after dynamic and static.

Table 8 .
Parameters of fitting curves of Formulas (