Modeling of Damping Characteristics of Rubber Geopolymer Concrete Based on Finite Element Simulation

Abstract

The stacking of waste rubber tires has led to serious environmental pollution. As an attempt to reduce pollution, rubber tires have recently been ground into rubber particles and incorporated into the geopolymer concrete to enhance the damping characteristics of concrete. Thus, we designed this study to quantify the effect of rubber particles on improving the damping performance of geopolymer concrete. The free vibration simulation of a rubber geopolymer concrete cantilever beam at four different rubber replacement volume fractions under five different damage displacements was performed on the ABAQUS platform. The damping loss factor, energy consumption, and modal shape of the cantilever beams under different damage displacements, as well as different rubber replacement volume fractions, were analyzed. The results showed that rubber particles significantly enhanced the damping characteristics of geopolymer concrete, and a certain amount of rubber particles could enhance the total energy consumption of concrete. The damping loss factor of geopolymer concrete was not closely related to its modal shape but mainly related to damage displacement and rubber particle replacement volume fraction. Altogether, these findings provide some technical references for the vibration resistance design of rubber geopolymer concrete.

1. Introduction

Blast furnace slag is usually subjected to stacking, leading to environmental pollution [1,2]. To maximize the rational utilization of blast furnace slag and reduce the emission of dust and CO₂ during cement production, one of the most advanced methods is to grind the blast furnace slag into powder and mix it with an alkali activator to develop a geopolymer. Intrinsically, geopolymer is a new type of calcium-free aluminum siliceous cementitious material. The main method of synthesizing geopolymer is to combine alkaline solution, such as NaOH solution or Na₂SiO₃ solution, with active aluminosilicate to form a new low-carbon cementitious material through polymerization, which has the potential to be a partial substitute in Portland cement for constructing buildings [3].

With the rapid growth of the number of vehicles, the disposal of waste rubber tires has become a serious challenge to the urban environment worldwide [4]. Currently, the most commonly used disposal methods include stacking and incineration, which have a high risk of incurring serious environmental issues, such as the encroachment of land resources and pollution of groundwater [1,2,5]. To dispose of waste tires more sustainably and efficiently, several efforts have been made to partially or completely replace the fine aggregates with rubber tire powder to make concrete [6]. This treatment method can not only promote the full utilization of waste tires but also improve conventional concrete’s working performance, owing to the advantages of the favorable characteristics of rubber particles, such as high elasticity [7,8,9].

It is well-recognized that vibration is a major factor influencing the stability of the structure of concrete. Usually, under the combined action of wind, vehicles, and other dynamic loads, the concrete structure responds to generate dynamic vibrations that might inversely lower its stability [10,11]. Thus, damping has been widely used to reduce the vibrations to improve the stability of the concrete structure. The most common damping practice is incorporating rubber particles with varying volume fractions into cement concrete. A large number of experimental studies showed that adding rubber particles to cement concrete can be conducive to absorbing energy during vibration [12], improving the damping and energy dissipation capacity of rubber concrete [13,14], ductility, and impact resistance [15,16]. In addition to concrete cement, some other studies have attempted to add rubber particles into geopolymer concrete. It was shown that rubber particles could improve the damping characteristics of concrete cement mainly because of their elasticity. When they are distributed in the geopolymer concrete, like tiny springs, the “micro-spring body” can reduce the elastic modulus of the concrete, thereby reducing its internal energy consumption. For instance, Thong et al. [17] used three different volume fractions of rubber chips to replace fine aggregate and examined the dynamic compression performance of rubber geopolymer concrete under different replacement rates. Their results indicated that rubber geopolymer concrete had better impact resistance and energy absorption performance than conventional geopolymer concrete. Sun et al. [18] used NaOH solution to modify rubber particles and replace river sand volume fraction, and performed a concrete mechanic performance test and a cantilever beam under low reversed cyclic free vibration test, with the increase in rubber particle replacement rate, rubber geopolymer concrete surface fracture reduction, and elimination of stress concentration of the cantilever beam section gap. They concluded that the rubber particles could enhance the toughness and damping performance of geopolymer concrete.

Previous studies also indicated that rubber particles could enhance the damping performance of concrete. However, most of these efforts were focused on laboratory testing, and research on evaluating the damping performance of rubber geopolymer concrete through numerical modeling is limited. To fill this research gap, we designed this study to simulate the dynamic response of rubber geopolymer concrete in the form of a cantilever beam under different free vibration conditions through finite element analysis. The damping properties of the rubber geopolymer concrete beam were quantified based on simulation results of plastic deformation, energy dissipation capacity and damping loss factor of the cantilever beam under varying test conditions, with different damage displacements and rubber substitution rates.

2. Materials and Methods

2.1. Test Materials

With reference to previous studies [18], the raw materials used in this study included a 400 mesh (38 $\mathsf{\mu }$m) slag powder, the coarse aggregate of crushed stone with a maximum particle size of 20 mm, the fine aggregate of natural river sand with a sand ratio of 0.33, and rubber particles. Of note, the rubber particles were made from crushed waste tires, with a particle size of 40 mesh numbers (380 μm). Based on the methodology of Sun et al. [18], the raw materials were used to fabricate the cantilever beams for the laboratory damping test, which was also used as a numerical reference model for validating the accuracy of the finite element simulation results.

The size of the cantilever beam for the damping test was 100 mm (height) × 150 mm (width) × 1200 mm (length). Four Hot-rolled Ribbed Bar-400 (HRB400) ribbed longitudinal bars with a diameter of 8 mm were symmetrically mounted at the four corners of the beam section [19], with a longitudinal reinforcement ratio of 1.34%. The stirrup adopted a 4 mm steel wire with densified and non-densified area spacing. The stirrup ratios were 100 mm, 166.7 mm, and 0.18%, respectively. The reinforcement arrangement of the cantilever beam and cantilever loading device is schematically illustrated in Figure 1 and Figure 2, respectively. The mechanical properties of the rubber geopolymer concrete are summarized in

Table 1. Mechanical properties of the rubber geopolymer concrete.

Test Sample	Compressive Strength/MPa	Splitting Tensile Strength/MPa	Flexural Strength/MPa
GC	71.88	5.06	5.58
RGC-5	57.74	4.69	6.08
RGC-10	54.74	4.59	5.51
RGC-15	52.84	4.01	5.41

Note: GC represents the geopolymer concrete, while RGC-5, RGC-10 and RGC-15 represent the volume fractions of rubber particles replacing natural river sand, which were 5%, 10% and 15%, respectively.

.

2.2. Test Plan

To accurately measure the damping performance of each group of cantilever beams under different damage degrees, the cross-test method of low-cycle repeated loading and free vibration of the cantilever beam was adopted. The length of the cantilever end of the beam was 1000 mm. During the test, the cantilever end of the cantilever beam was loaded using the displacement control method, whereby the amplitudes of damage control displacement were designed at six different levels, comprising 0 mm (no damage), 5 mm, 10 mm, 20 mm, 30 mm and 40 mm, which were recorded as y0~y5 respectively, and had an upper limit of 40 mm. The loading pattern of the free vibration test is depicted in Figure 3. The test loading procedures were: (1) slowly loading the cantilever beam upward with a jack until the displacement was $y_i$; (2) unloading and turning the beam over 180° after continuing the load for 3 min, then load to $y_i$ and unload after continuing the load for 3 min, with each displacement amplitude loaded three times, followed by observing the development of cracks on the concrete beam surface after completing the loading cycle; (3) performing the free vibration test on the cantilever beam after each displacement cycle.

In this test, the end of the cantilever beam was tapped with a force hammer to make the cantilever beam vibrate freely. Then, the acceleration sensor was employed to capture the acceleration time history of the cantilever beam end. In this way, the low cycle repeated loading and free vibration of the cantilever beam under the corresponding damage control displacement was performed until all the tests were completed. The free vibration wave test was completed using the UT3416FRS-DY dynamic signal analysis system, UT4102 charge amplifier, LC-04A exciting hammer and CA-YD-185 piezoelectric acceleration sensor. The free vibration test device is shown in Figure 4.

2.3. Test Results

Within the first displacement of 5 mm (L₁ = 5 mm), the rubber geopolymer concrete cantilever beam had no displacement damage and its damping ratio increased with the replacement rate of rubber particles. However, when the damage displacement reached 10 mm (L₂ = 10 mm), cracks began to appear at the bottom of the fixed end of the cantilever beam. Once the loading was stopped, the cracks were immediately closed and could not be easily identified by visual observation. Under this condition, the damping ratio of each group of concrete cantilever beams greatly increased, and the damping ratio of specimen RGC-15 reached its maximum. When the damage displacement was 20 mm (L₃ = 20 mm), the number and size of cracks started to increase. The cracks tended to expand to the middle, and despite being closed after unloading, the cracks could still be observed visually. At this stage, the damping ratio of the other groups of specimens reached the maximum. When the damage displacement was 30 mm (L₄ = 30 mm), upper and lower cracks were formed, and some concrete blocks of the specimen GC fell off. After unloading, the surface crack recovery force of the specimen was poor, and the damping ratio of each group of specimens decreased significantly. When the damage displacement was 40 mm (L₅ = 40 mm), the crack at the fixed end of the concrete cantilever beam was about 2 mm. After unloading, the sound of reinforcement rebound could be clearly heard, and the plastic deformation could be readily detected. Based on the test results, we found that the rubber particles could significantly reduce the number of micro-cracks in the geopolymer concrete cantilever beam and maintain good integrity condition of the beam sample after loading. Thus, the addition of rubber particles could ensure the integrity of the concrete cantilever beam, reduce the friction loss of the internal interface of concrete and effectively improve the damping performance of the concrete. The development of cracks at the fixed end of the cantilever beam after repeated loading is shown in Figure 5.

3. Finite Element Simulation

As an excellent finite element software, the main advantage of ABAQUS is its ability to perform numerical analysis of nonlinear materials, which can better simulate the experimental characteristics of materials. To further analyze the damping performance of the rubber geopolymer concrete cantilever beam structure, finite element simulation was performed in this section to model the damping characteristics of the four types of cantilever beams used in the laboratory test based on the ABAQUS platform.

3.1. Constitutive Model

Using the ABAQUS software, three constitutive models were used to characterize the mechanical behavior of concrete under low constraint loadings, which included the Brittle Cracking Model (BCM), the Smeared Cracking Model (SCM), and the Concrete Damaged Plastic Model (CDPM) [20,21]. In the BCM model, the tensile behavior of tensile materials was assumed to have a linear elasticity and crack initiation was detected based on the ranking criterion that cracks would occur when the maximum principal tensile stress exceeded the tensile strength of brittle materials. This model was applied to materials such as ceramic, rock, glass and plain concrete to allow the crack surface to be perpendicular to the direction of the maximum principal tensile stress [22]. In this regard, BCM was not needed to reinforce the concrete. In the SCM model, the cracks were assumed to be uniformly diffused in the actual structure of the concrete, and the behavior of the concrete after cracking was simulated by modifying the softening section of the concrete tensile stress-strain relationship curve. This model could not characterize reciprocating stress behaviors, such as the degradation of unloading stiffness and the recovery of reloading stiffness of concrete under reciprocating load [23]. Compared to the BCM and SCM model, the CDPM model could simulate the constitutive relationship of concrete materials under reciprocating loadings due to its ability to capture the damage, crack development, crack closure, and stiffness recovery of materials in the reciprocating loading mode. The hysteresis law of the CDPM model was believed to be more practical on the premise of reasonably setting relevant parameters. The plastic damage model could simulate the mechanical behavior of the concrete or composite structures under reciprocating loadings. In addition, since the ABAQUS standard method usually encounters convergence issues during the analysis process, the concrete damage plastic model used in this study was more feasible.

The concrete plastic damage model used explicit criteria to assign the properties of the materials and simulated the main failure characteristics of the concrete, including the tensile and compressive failure characteristics. This model was originally developed by Lubliner et al. and was initially applied to the monotonic loading mode [24]. It was then improved by Lee et al. [25] to make it applicable to cyclic and dynamic loading modes. The CDP model was used to characterize the elastic and plastic properties of concrete, with the former mainly represented by the elastic modulus E and Poisson’s ratio μ. As the main research objective, the plasticity of the concrete in this paper was defined by the plastic parameters, compressive response and tensile properties. In addition, compression and tensile damage factors were selected. To accurately characterize the concrete plastic behavior, the invariant stress ratio ${{\rm K}}_c$ was incorporated into the CDP model. Five variables were also added to the model, namely the invariant stress ratio (${{\rm K}}_c$), flow potential offset ($\epsilon$), shear expansion angle ($\psi$), biaxial to uniaxial compressive ultimate strength ratio (${\sigma }_{b0}/{\sigma }_{c0}$), and coefficient of viscosity (μ).

The stress-strain formula proposed by Mattock et al. and Hognestad et al. [26] is shown in Equations (1)–(3), which can be used to simulate the behavior of concrete in compression mode and also describe the degradation behavior beyond the loading limit.

$${\sigma }_c=f_c^{\prime }\left[2\right(\frac{{\epsilon }_c}{{\epsilon }_0})-{\left(\frac{{\epsilon }_c}{{\epsilon }_0}\right)}^2]$$

(1)

$${\epsilon }_0=\frac{2f_c^{\prime }}{E_c}$$

(2)

$${\epsilon }_c^{in}={\epsilon }_c-\frac{{\sigma }_c}{E_c}$$

(3)

where ${\sigma }_c$ represents the concrete stress; $f_c^{\prime }$ represents the uniaxial compressive strength of concrete measured in the test; ${\epsilon }_c \mathrm{represents}$ the assumed strain value; ${\epsilon }_0 \mathrm{represents}$ the corresponding strain value at the maximum stress of concrete; $E_c \mathrm{represents}$ the compressive elastic modulus of concrete and; ${\epsilon }_c^{in} \mathrm{represents}$ the inelastic strain of concrete.

As shown in Figure 6, the uniaxial compression stress-strain curve started from the initial stress of ${\sigma }_{c0}=0.4f_c^{\prime } .$The concrete stress reached a maximum value $f_c^{\prime }$, when the corresponding strain value was ${\epsilon }_0$. When the compressive stress of concrete decreased, the compressive strain continued to increase to a critical value of ${\epsilon }_u.$

In regard to the tensile response of concrete, this study used the crack displacement expansion method of CEB-FIP specification [27]. The relationship curve between concrete tensile stress and crack width is shown in Figure 7.

The calculation equations are shown in Equations (4)–(7):

$$f_t=0.3{\left(f_t^{\prime }\right)}^{\frac{2}{3}}$$

(4)

$$W_{c1}=0.75\frac{G_f}{f_t}$$

(5)

$$W_{c2}=5\frac{G_f}{f_t}$$

(6)

$$G_f=0.03{\left(\frac{f_c^{\prime }}{10}\right)}^{0.7}$$

(7)

where $f_t$ and $G_f$ represent the tensile strength and fracture energy of concrete; $W_{c1}$ represents the crack width corresponding to the stress of 0.25 $f_t$, and; $W_{c2} \mathrm{represents}$the ultimate crack width.

To better characterize the degradation of concrete after crushing and tensile cracking, the compressive and tensile damage factors were used to simulate the degradation of elastic stiffness caused by the concrete damage. These two different damage factors were expressed by Equations (8) and (9), respectively.

$$d_c=1-\frac{{\sigma }_c}{f_c^{\prime }}$$

(8)

$$d_t=1-\frac{{\sigma }_t}{f_t^{\prime }}$$

(9)

Based on the data measured from the test, the stress-strain relationship and the damage factor of geopolymer concrete were determined based on their corresponding compression and tension (Figure 8 and Figure 9).

In this study, the damping characteristics, energy dissipation and damage response of the geopolymer concrete were investigated based on the numerical simulation. The simulation model considered the mechanical properties of geopolymer concrete and calibrated the parameters for characterizing the concrete properties. The calibrated parameters are summarized in

Table 2. Plasticity parameters used in the concrete-damaged plasticity model.

${\mathit{{\rm K}}}_{\mathit{c}}$	$\mathit{\epsilon }$	$\mathit{\psi }$	${\mathit{\sigma }}_{\mathit{b}0}/{\mathit{\sigma }}_{\mathit{c}0}$	$\mathit{\mu }$
38	0.1	1.16	0.667	0.076

.

3.2. Material Properties

As mentioned in Section 2.1, the test materials used in this study included geopolymer concrete, stirrups with a diameter of 4 mm, longitudinal reinforcement of HRB400 with a diameter of 8 mm, and rubber particles. The reinforcement conformed to the Mises yield criterion and adopted the double broken line strengthening model for calculation because this model was shown to better capture the mechanical behavior of reinforcement [28,29]. The inputs of the double broken line model included the elastic modulus, yield stress, final stress, reinforcement density, and Poisson’s ratio. The hoop and longitudinal reinforcement material properties are shown in

Table 3. Material properties of steel wire mesh and steel rebar.

Material	Diameter/mm	Mass Density/kg/m3	Elasticity Modulus/GPa	Poisson’s Ratio	Yield Strength/MPa
hoop-steel	4	7800	194.1	0.27	270
longitudinal-steel	8	7800	210	0.3	360

. The tensile and compressive damage plastic model parameters of the geopolymer concrete were measured from laboratory testing. It was used to simulate the tensile and compressive behaviors of concrete materials with a good convergence performance. Considering that the Mooney Rivlin model can simulate the mechanical behavior of rubber materials, it was selected to characterize the mechanical properties of the rubber materials used in this study. In this model, C10 was 0.381, C01 was 0.102, and the bulk modulus D1 was 0.00137. Based on previous studies [30,31,32], it is worth mentioning that this model was suitable for small and medium deformation of rubber materials and for situations where the strain was approximately 100% in tension mode and 30% in compression mode.

3.3. Finite Element Modeling

The geopolymer concrete beam and reinforcement skeleton used in the test were modeled as shown in Figure 10a,b. The rubber particles were imported into the ABAQUS platform through Python programming. The randomly distributed rubber particles are shown in Figure 10c. The C3D8R hexahedral linear reduced integral solid element was used to characterize the properties of concrete and rubber [33]. The concrete grid adopted a hexahedral topological structure, and the spacing of the grid edge species is 20. The rubber particles in concrete were divided into 1/8 spheres, and the 42,864 rubber particle grids selected used a hexahedral topological structure grid. While the T3D2 linear truss element was used to those of the reinforcement, the spacing of the grids was five. During the assembling process, the rubber particle component and geopolymer concrete component were combined into the integrated rubber geopolymer concrete, in which each reinforcement was incorporated into a unit of reinforcement cage. Referring to the test results of Elchalakani et al. [34], the bonding performance of geopolymer concrete was similar to that of ordinary Portland cement concrete. Thus, the “embedded element” function in ABAQUS simulated the ideal bonding between the reinforcement cage and the normal geopolymer concrete. To this effect, the translational degree of freedom (TDF) of the reinforcement cage was constrained by the corresponding TDF of normal polymer concrete [35,36].

The 1200 mm concrete beam was divided into two parts which were 1000 mm and 200 mm in length. The 200 mm end simulated the fixed end of the concrete beam. The displacement constraint was set as $u_x=u_y=u_z=0$, and the rotation constraint was set as${\delta }_x={\delta }_z=0$. To reflect the reality of the testing condition, the coupling point was established at the beam end, and the amplitude function was added at the cantilever end coupling point to impose different damage displacements, as shown in Figure 11.

4. Results and Discussion

4.1. Plastic Damage and Energy Consumption Analysis

4.1.1. Plastic Damage

As shown in Figure 12, the simulation results explicitly showed that the plastic damage regions of the geopolymer concrete and rubber geopolymer concrete were mainly concentrated at the edge of the fixed end. The distortion of the concrete grid at the edge of the fixed end indicated that the concrete was fractured, consistent with the test results. To reduce the plastic damage and failure of the concrete, the fiber cloth and energy dissipation angle steel were added to the concrete fixed end in the following analysis of this study.

4.1.2. Energy Consumption Analysis of the Rubber Geopolymer Concrete

Figure 13 shows the hysteresis curve of the rubber geopolymer concrete under different rubber substitution rates simulated by the ABAQUS finite element software. Although the bearing capacity of the geopolymer concrete without rubber particles was significantly higher than that of the rubber geopolymer concrete, the hysteresis curve of the rubber geopolymer concrete cantilever beam was relatively fuller, indicating that the vibration resistance of the rubber geopolymer concrete was better.

Theoretically, the hysteresis loop area in the hysteresis curve represents the energy dissipation capacity of the structure. A larger hysteresis loop area indicated better energy dissipation capacity of the structure. In this study, the energy dissipation capacity of the specimen was defined by the energy dissipation coefficient E. The calculation method of the energy dissipation coefficient was specified according to the Chinese standard of JGJ-101–2015 [37], as shown in Equation (10). For convenience, the calculation diagram of the energy dissipation coefficient is shown in Figure 14.

$$E=\frac{E_D}{E_S}=\frac{S_{\left(ABC+CDA\right)}}{S_{\Delta \left(OBE+ODF\right)}}$$

(10)

where $E_D=S_{\left(ABC+CDA\right)}$indicates the area corresponding to the energy dissipation of a complete ideal hysteresis loop; $E_s=S_{\Delta \left(OBE+ODF\right)}$indicates the area of a triangle corresponding to the maximum lateral load and the maximum horizontal displacement point of the hysteresis loop on the upper and lower sides.

The total energy consumption of rubber geopolymer concrete cantilever under repeated loading and the energy dissipation coefficient at different rubber particle replacement rates are summarized in

Table 4. Simulation results of energy consumption coefficients and total consumption energy.

Sample Label		GC	RGC-5	RGC-10	RGC-15
Total Consumption Energy/J		15,731.49	16,759.11	18,415.31	16,352.81
Energy dissipation coefficient	L1 = 5 mm	0.50	1.07	1.14	1.45
	L2 = 10 mm	0.95	1.45	1.62	1.95
	L3 = 20 mm	1.39	1.63	1.82	2.15
	L4 = 30 mm	1.51	1.74	1.95	1.74
	L5 = 40 mm	1.58	1.75	2.02	1.64

.

By comparing the total consumption energy of rubber geopolymer concrete under different rubber particle substitution rates, we found that adding rubber particles to the concrete significantly enhanced the structural energy consumption capacity of the geopolymer concrete cantilever beam. The energy dissipation coefficients of geopolymer concrete cantilever specimens incorporated with rubber particles were greater than those of normal geopolymer concrete. The energy dissipation coefficient of adding rubber particles with a substitution rate of 15% at early loading stages (L₁ = 5 mm, L₂ = 10 mm, and L₃ = 20 mm) was the highest, indicating that rubber particles played a significant role in resisting vibrations. When the damage displacement reached 30 mm (L₄ = 40), the energy dissipation coefficient of the rubber geopolymer concrete cantilever beam structure decreased with a rubber particle replacement rate of 15%. By comparing the test results and the ABAQUS simulation results, we observed that excessive rubber particles reduced the energy dissipation resilience of the structure in the failure stage. Thus, selecting an appropriate amount of rubber particles to replace some fine concrete particles could effectively restore the energy consumption capacity of geopolymer concrete. To this end, we concluded that the optimum rubber content added to the geopolymer concrete was 10% (Table 4).

4.2. Analysis of Damping Characteristics of Rubber Geopolymer Concrete

ABAQUS finite element software was used to extract the first-order modes and frequencies of the rubber geopolymer concrete with different rubber substitution rates under different damage displacements. The frequencies under each damage displacement are shown in

Table 5. First-order resonance frequencies under various damage displacements.

	${\mathit{y}}_{\mathit{i}}$ (mm)	0	5	10	20	30	40
Specimen Label
GPC	$f_{\mathrm{a}}$(Hz)	79.5	73.5	70	65	61	59
	$f_{\mathrm{n}}$(Hz)	77.3	74.1	69.3	62.5	56.4	53.5
	Error value (%)	2.77	0.82	1	3.84	5.9	9.32
RGPC-5	$f_{\mathrm{a}}$(Hz)	77	74	70	63	58.5	54
	$f_{\mathrm{n}}$(Hz)	74.8	73.4	68.7	61.5	56.2	51.7
	Error value (%)	2.86	0.81	1.86	2.38	3.93	4.26
RGPC-10	$f_{\mathrm{a}}$(Hz)	76.8	71.5	69	61.2	53.6	47.5
	$f_{\mathrm{n}}$(Hz)	73.5	71.8	68.6	58.7	51.2	46.5
	Error value (%)	4.3	0.42	0.58	4.08	4.48	2.11
RGPC-15	$f_{\mathrm{a}}$(Hz)	75.5	68.5	55.2	51.4	48.5	45.2
	$f_{\mathrm{n}}$(Hz)	72.3	71.1	62.1	53.8	48.3	45.9
	Error value (%)	2.91	3.8	12.5	4.67	0.41	1.55

.

By comparing the error values of each group of tests and simulations, the frequency error values in the non-destructive stage were found to be more than 2.5%. With an increase in damage displacement, the frequency error value of the rubber geopolymer concrete cantilever beam increased gradually relative to each group of the measured frequency value f_a, and the numerically simulated frequency value f_n, indicated that a certain uncertainty in the control of displacement damage existed during the test process. The controllability of the degree of damage was not sufficient for numerical simulation. Increasing the replacement rate of rubber particles increased the discreteness of the relative error between the measured frequency value and the simulated frequency value, which confirmed that the uneven dispersion of rubber particles in geopolymer concrete resulted in large discreteness of experimental data. Thus, an adequate mixing process is required when adding the rubber particles to geopolymer concrete.

It is well known that the material loss factor is an important parameter in quantifying the damping characteristics of the system and determining its vibration energy dissipation capacity [38,39,40]. This parameter represented the state of materials within a certain range and denoted the damping loss factor η. The damping loss factor of the geopolymer concrete cantilever beam at each damage stage was measured from the damping performance test following the guidance of the Chinese standard of GB/T 18258-2000 [41] and was expressed as shown in Equation (11):

$$\eta =\frac{\mathsf{\Delta }{f}_i}{f_i}$$

(11)

where $\mathsf{\Delta }{f}_i$represents the half-power bandwidth frequency (Hz) of the i-th mode of the cantilever beam, and $f_i$represents the resonant frequency (Hz) of the i-th mode of the cantilever beam. As shown in Figure 15, the half-power bandwidth indicated the distance between the corresponding two points when the amplitude-frequency response function dropped to $\sqrt{2}/2$times that of the peak, and the frequency difference between them was $\mathsf{\Delta }{f}_i=f_2-f_1$.

The loss factors of rubber geopolymer concrete with different volume fractions under different damage conditions were calculated, as shown in Figure 16. The results showed that the damping loss factor of the rubber geopolymer concrete cantilever beam first increased, then decreased at each damage stage. When the replacement rate of the rubber particles was less than 10%, the maximum value of the damping loss factor was roughly distributed between the damage displacement of 20 mm, and a slight decline in the slope of the loss factor was observed. When the replacement rate of rubber particles was 15%, the maximum value of the damping loss factor was between the damage displacement of 10 mm, and the downward trend was more significant. By comparing and analyzing the curve relationship, we observed that the substitution rate of rubber particles had a great influence on the damping loss factor.

4.3. The State Change of the Cantilever Beam under Different Modes

The first five modal changes of rubber geopolymer concrete were extracted using the ABAQUS software. The changes in shape and situation of geopolymer concrete in each group were similar. The first five modal changes of rubber geopolymer concrete with a rubber substitution rate of 10% were taken as an example in this study, and the corresponding results are summarized in

Table 6. The first five modal mode shapes of the RGC-10 specimen at different damage stages.

		First Mode	Second Mode	Third Mode	Fourth Mode	Fifth Mode
${\mathit{y}}_{\mathit{i}}$/mm
0
		f = 73.5 Hz	f = 433.59 Hz	f = 453.22 Hz	f = 580.04 Hz	f = 1141.2 Hz
5
		f = 71.8 Hz	f = 428.08 Hz	f = 447.30 Hz	f = 573.34 Hz	f = 1128.4 Hz
10
		f = 68.6 Hz	f = 404.12 Hz	f = 412.4 Hz	f = 524.23 Hz	f = 1063.4 Hz
20
		f = 58.7 Hz	f = 330.82 Hz	f = 335.9 Hz	f = 348.86 Hz	f = 821.62 Hz
30
		f = 51.2 Hz	f = 220.04 Hz	f = 277.44 Hz	f = 305.34 Hz	f = 682.44 Hz
40
		f = 46.5 Hz	f = 186.18 Hz	f = 241.67 Hz	f = 277.50 Hz	f = 621.01 Hz

.

Table 6 shows that the changes in the shape of the first-order mode at each damage stage were roughly the same. When the damage displacement was 5 mm, compared with the non-destructive stage, the bending direction of the second-order mode was different, and the shape and resonance frequency of other modes changed slightly but were within the bending state. When the damage displacement reached 10 mm, the fifth-order modes of specimens RGC-10 and RGC-15 began to show a tensile trend, while the fifth-order modes of specimens GC and RGC-5 were still in the bending state, and the frequency of concrete cantilever beams in each group decreased more significantly, indicating that the cracks had a great decisive factor for the later destruction of concrete. When the damage displacement reached 20 mm, the fifth mode of GC and RGC-5 also began to change to the tensile state. The resonant frequency of each mode decreased greatly in the second mode, which changed from the previous antisymmetric bending state to the torsional state. The fifth mode was developed from the last tensile state to the tensile necking state. When the damage displacement reached more than 30 mm, the fifth mode changed from the previous state of tension to the compression state, and the changing amplitude of resonance frequency gradually slowed.

The damping loss factor of the first five modes of the rubber geopolymer concrete cantilever beam in each group were calculated using the half-power bandwidth method, and the change is shown in Figure 17. It can be seen from Figure 17 that the contour lines mapped out by each surface diagram were almost in straight lines. The damping loss factor between the first five modes of each group of cantilever beams was not different, indicating that the damping loss factor of concrete was related to the displacement damage of cantilever beams, and the replacement rate of rubber particles was very slightly related to their modal vibration modes. Figure 17 also indicates that the damping loss factor of the fifth mode of each group was lower than that of the remaining four modes, which could be explained by the fact that the damping material consumed vibration energy in the expansion and contraction process [42].

5. Conclusions

This paper established the rubber geopolymer concrete cantilever model based on the ABAQUS finite element software. The proposed model successfully simulated the damping performance of the rubber geopolymer concrete cantilever under different rubber particle substitution rates and different damage displacements. In addition, the first five modes of its free vibrations were analyzed, based on which the following conclusions were obtained:

(1)

By comparing the numerical simulation results with the experimental results, we found that the rubber particles enhanced the damping performance of the geopolymer concrete, although some differences were observed.

(2)

The hysteretic curve and energy dissipation coefficient of geopolymer concrete cantilever beam under repeated loading showed that rubber particles could significantly enhance the energy consumption of concrete beams. When the replacement rate of rubber particles was 10%, the effect of energy dissipation restoring force was optimal.

(3)

Analysis of the variation of the first five natural frequencies of the cantilever beam with the damage displacement showed that for certain modes, the natural frequency decreased with an increase in damage displacement, with the decreasing speed gradually slowing. This was mainly because the increase in damage displacement gradually decreased the stiffness of the overall structure, resulting in a decrease in the natural frequency.

(4)

Analysis of the damping loss factor of the geopolymer concrete under different modal shapes showed that it was mainly related to the damage displacement and the rubber particle replacement rate rather than its modal shapes.

Altogether, this study showed that rubber particles could significantly enhance the damping characteristics and reduce the risk of transient damage of geopolymer concrete, which was significantly different from the damping characteristics of rubber geopolymer concrete with different volume replacement fractions of rubber particles. Thus, the influence of rubber particle size and surface treatment on the damping characteristics and mechanical properties of geopolymer concrete should be continuously enhanced to promote the application of rubber geopolymer concrete.