#### Abstract

This paper aims to propose predictive equations for the small-strain shear modulus () and small-strain damping ratio () of a granulated mixture with plastic and nonplastic materials to reduce the dynamic energy of the ground. Polyurethane bead (PB) and glass bead (GB) were used as the plastic and nonplastic materials, respectively. 180 resonant-column tests were conducted with various conditions affecting the dynamic properties, such as nonplastic particle content (PC), void ratio (*e*), particle-size ratio (), and mean effective confining pressure (). The results showed that and , respectively, increased and decreased as *e* decreased with increasing of material mixtures. In addition, decreased with an increase in PC, whereas increased. It was also found that of materials affected the changes in and . With an increase in , increased while decreased because small particles do not hinder the behavior of large particles as the size of larger particles increases. Finally, based on the results, new equations for estimating and of a granulated mixture with PB and GB were proposed as functions of PC, *e*, median grain size (*D*_{50}), and .

#### 1. Introduction

Dynamic soil properties, such as shear modulus (*G*) and material damping ratio (*D*), have been used as key parameters for evaluating seismic ground response and design of foundation subjected to cyclic or dynamic loading. The behavior of these parameters varies with shear-strain amplitude beyond the specific threshold value of shear strain. In other words, both *G* and *D* exhibit linear behavior at a very small strain level, whereas nonlinear behavior with increasing shear strain. *G* and *D* at very small strain amplitudes (10^{−3}%) are referred to as the small-strain shear modulus () and the small-strain damping ratio ().

There are two ways to reduce the dynamic energy of the ground and to prevent damage from the earthquake. The first way is to increase *G*, and the other is to increase *D* [1–21]. The factors affecting dynamic soil behavior include shear strain (*γ*), mean effective confining stress (), void ratio (*e*), plasticity index (PI), soil grain distribution, degree of saturation, frequency of loading, and the number of cycles. Its relation is quite complicated because these factors vary depending on the soil type and the ground conditions. Wichtmann et al. [20] and Choo and Burns [21] studied the dynamic properties according to differences in fine content. They suggested that is reduced by an increase in the fine content, in a way that depends on the specific fine content and particle-size ratio. Santamarina et al. [15] performed a theoretical analysis of packing of granular mixtures of different-sized particles because the shear wave velocity () or in a medium of material is dependent on packing arrangement and packing density. They demonstrated that the maximum size of small particles that can be placed in the pore between large particles is different according to packing conditions such as loose and dense packing.

The dynamic properties of heterogeneous materials mixed with a plastic material, such as a rubber by-product, have been studied to investigate the absorption of earthquake vibrations and the reduction of seismic forces [22–26]. They reported that soil-rubber mixtures reduce as rubber content increases for all , whereas increases as increases. *D* was generally increased by an increase in rubber content but decreased with . Although many researchers have studied the various factors affecting the dynamic soil behavior, there are still few studies on the dynamic property of mixtures of different types of materials that use the plastic material to reduce the dynamic energy.

The objective of this study is to investigate the dynamic properties of mixtures composed of plastic and nonplastic materials that are used to reduce the dynamic energy of the ground. The resonant column (RC) test was carried out with various polyurethane content (PC), void ratio (*e*), particle ratio (), and mean effective confining stress (). New models based on various factors, such as PC, *e*, median grain size (*D*_{50}), and , are introduced to predict and of the mixture.

#### 2. Previous Empirical Equations for Estimating and

To calculate and , the measurement of shear wave velocity () in the field and laboratory has usually been conducted using the cyclic direct shear test and RC tests, respectively. However, since these methods are both costly and time-consuming, it is difficult to use them in the design of small projects with a small budget. Hence, an empirical equation for predicting the dynamic soil properties such as and might be useful for preliminary design, design calculation of small project, and confirmation of the observed values [18, 27].

An empirical equation to predict on cohesionless soils proposed by Hardin and Richart [28] has been widely used aswhere *e* = void ratio; *p* = mean pressure; = atmospheric pressure (100 kPa); *A*, *a*, and *n* are constants (*A* = 690, *a* = 2.17 and *n* = 0.5 for round grains, and *A* = 320, *a* = 2.97 and *n* = 0.5 for angular grains). Some researchers have also proposed formulas to predict based on a function of the void ratio similar to Hardin’s formula [29–31]. Wichtmann and Triantafyllidis [18] proposed a formula that correlates Hardin’s formula with uniformity coefficient () to examine the influence of the grain-size distribution on . Choo and Burns [21] introduced the critical fine content () and intergranular void ratio () to evaluate the property of shear wave velocity on packing of a granular mixture composed of large and small silica particle sizes. In addition, empirical formulas for estimating considering the effects of and rubber content on rubber-sand mixture were proposed [26, 32].

Most empirical formulas for estimating *D* are composed of a relation of based on assuming that *D* value is proportional to [33, 34]. In addition, other equations were expressed by a polynomial formula [10, 35, 36]. Zhang et al. [17] proposed a formula related to considering PI and , and the polynomial formula for estimating *D* based on and was expressed aswhere *a*_{1}, *a*_{2}, and *a*_{3} = constants (*a*_{1} = 9.4, *a*_{2} = 26.5 and *a*_{3} = 17.1 for RC test, and *a*_{1} = 10.6, *a*_{2} = 31.6, and *a*_{3} = 21.0 for torsional shear test).

According to literature, previous empirical equations to predict can be overestimated or underestimated depending on the characteristics of the materials and type of test. In addition, there are very few empirical equations to predict for heterogeneous mixtures using plastic and nonplastic materials such as rubber and sand to reduce the seismic forces and earthquake vibration. Moreover, in most cases of proposed equations has been used as a reference value to estimate of soils. Although some researchers have studied the equations of dynamic properties for mixtures according to different particle sizes [20, 21], there is a dearth of studies on the empirical equation to predict the dynamic properties considering the significant factors, such as , FC, *e*, and *D*_{50}, for the binary mixtures composed of both plastic and nonplastic materials.

#### 3. Experimental Program

##### 3.1. Materials and Sample Preparation

Spherical glass bead (GB) and polyurethane bead (PB) of single particle sizes were selected to remove the particle shape from among the variables that might affect the dynamic properties. These non-cohesive materials were used to effectively control the particle-size ratio (, where is the median grain size of large particles (GB) and is the median grain size of small particles (PB)). GB and PB were purchased from B&K MEDIA Co, Ltd., and Doosung Chemis Co, Ltd., respectively. GB was prepared as five types in terms of particle size in order to consider the effect of particle size on the values of , and PB was used at a fixed size. The physical properties of the materials and the composition ratios of the samples are listed in Tables 1 and 2, respectively. The PB contents (PC) can be defined as the ratio of the volume of PB particles to the volume of the total mixture, with GB and PB given bywhere and are the volumes of PB particles and GB particles, respectively.

The grain-size distributions for each material are shown in Figure 1, from which one can observe that materials used are generally of single particle size. For the preparation of the RC test samples, GB and PB particles were thoroughly mixed for 5 min. Predetermined weight of mixture for each relative density condition was used to prepare the samples. The mixtures were poured into the mold by the air pluviation method with extra caution to minimize the segregation of the mixture. The same compactive energy was lightly applied to each layer for obtaining the target density. The sample size is approximately 51 mm in diameter and 105 mm in height.

##### 3.2. Experimental Setup and Procedure

In this study, the RC test was carried out to obtain the dynamic characteristics, and . Stokoe-type device is a fixed-free system, and the top is freely rotatable. The RC device consists mainly of confining pressure control system, excitation system, and displacement measurement system. All tests were performed in a dry condition under different relative densities, confining pressure, and particle-size ratio, as summarized in Table 3. The sample is placed inside the membrane in a pressure cell, and then predetermined confining pressure is applied to the sample as the isotropic condition for 20 min. RC tests were conducted with various shear strain magnitudes. In this study, the change of the sample height was monitored by using a proximitor to measure the volume change of the sample caused by the confining pressure. Based on the elastic-wave theory, the shear modulus was calculated by the shear wave velocity obtained from the resonant frequency. The damping ratio was calculated by the free-vibration decay method.

#### 4. Results and Discussion

##### 4.1. Factors Affecting of the Mixture

Based on the previous studies, we could propose a multiplicative model to predict of a spherical material mixture aswhere of materials with no influence of void ratio when is 1 kPa; = function of effect of PC on in mixtures; = function of effect of *e* on in mixtures (packing function); *n* = sensitivity of the curve of G_{max} depending on .

###### 4.1.1. Effect of Packing on

Choo and Burns [21] reported that the behavior of of the mixture cannot be expressed by a global void ratio (*e*) that defines the ratio of the volume of voids to volumes of small and large particles because *e* does not capture the mechanical behavior of large particles with smaller particles that are fully filled in the void formed by large particles. Therefore, intergranular void ratio () was defined by the ratio of the volumes of voids and small particles to the volume of large particles. In this study, was used to consider the mechanical behavior of large particles with small particles that are completely filled in the void between large particles. Figure 2 shows the effect of the void ratios (i.e., *e* and ) in GB-PB spherical mixtures. The relation of and *e* showed no clear trend with a large dispersion. On the other hand, tended to decrease with an increase of and PC and to increase with an increase of because PB has a small stiffness relative to GB. It means that the small particle (PB) hinders the behavior of large particles (GB). As shown in Figure 2, agreed well with compared to the relationship between *e* and . Consequentially, we used the equation for the effect of the void ratio on by using the packing function for round particles suggested by Hardin and Richart [28] and aswhere *E* is the constant of packing for round particle and its value is 2.17 [28].

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###### 4.1.2. Effect of Confinement () on

The value of in GB normalized by the packing function with for the round particles, as proposed by Hardin and Richart [28], was used in order to examine of spherical mixtures when was 100, 200, and 300 kPa. Figure 3 shows the relationship between the normalized by and for pure GB (i.e., PC = 0%). All normalized values of GB increased with an increase of confining pressure independent of the size of the material. The relation between normalized and shows a power function similar to Hardin and Richart’s equation (1), depended on the particle-size ratio (). Therefore, it is necessary to confirm the influence of the median particle size of GB (*D*_{50GB}) on the value of exponent *n*. The relationship between and *D*_{50} for pure GB is plotted in Figure 4(a), which shows that increased with an increase of *D*_{50GB} as . Furthermore, for verifying the change of *n* in terms of the plastic particle contents, *n* is plotted with the changes for PC and contents, as shown in Figure 4(b). The *n* tended to increase linearly with a constant slope as PC increased, regardless of , and its equation was obtained as . The values of these parameters change with the difference in . Consequently, the exponent *n* can be determined by considering the effects of *D*_{50GB} and PC aswhere *a* and *b* are 0.6 and 0.06, respectively. *c* is the gradient for the relationship between *n* and PC, and its value is 0.01 in this study.

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###### 4.1.3. Effects of Size Ratio () and PB Content (PC) on

The coefficient *A* in equation (4) represents the of the main materials with no influences by the void ratio and the confining pressure. Figure 5 shows curve with the changes of when . An experiment was carried out by changing the diameter of the pure GB mixture (i.e., PC = 0%) to evaluate the of mixtures according to . The value of when the converged to approximately 7.2 MPa for all conditions in this study, as shown in Figure 5. It implies that affects only the slope of the change curve with increasing confining pressure without affecting other factors.

The relationship between the first term () and PC is plotted in Figure 6 to access the of mixtures with the change of PC at . It was observed that linearly decreased with an increase of PC, and its gradient is −58.85 in this study as follows:

##### 4.2. Factors Affecting of the Mixture

of a spherical material mixture is affected by the properties of the material, void ratio, plastic index, and confining pressure [25, 26]. In this study, estimating formula for of a spherical material mixture was established as follows based on the literature:where of the material without the influence of void ratio when is 1 kPa; = function of effect of PC on in mixtures; = function of effect of *e* on in mixtures (packing function); *β* = sensitivity of the curve of depending on .

was insufficient to investigate the effect of the packing function, especially for heterogeneous materials. Therefore, it is necessary to normalize the measured value with the packing function to accurately analyze the influence of each factor on the proposed model. Note that, in this study, the packing function, (), at which the void ratio affects , adopted the same form on the model proposed by Hardin and Richart [28], in equation (5). However, the constant corresponding to the shape factor in the packing function should be calculated to apply Hardin’s packing function for . The shape factor was determined using multivariable regression analysis for each test condition. The *e*, , and PC were used as the independent variables for each measured values in order to estimate the shape factor.

Consequently, the value of the shape factor was presumed approximately 0.04 based on the multivariable analysis study, and the packing function of model can be expressed as

Note that, in this study, each variable affecting in the proposed model was established by the fixed packing function. Additionally, the reliability of the proposed packing function was verified by inversely estimating the constant of the shape factor based on the calculated variable.

###### 4.2.1. Effect of Packing on

Representative value for and are plotted along with *e* and in Figure 7 to investigate the influence of the void ratio on . As shown in Figure 7, increased with an increase of the void ratio under the equal condition of PC because the stiffness decreases when the mixture is loose and the behavior of large particles is impeded by increase of the soft and elastic materials, such as PB; the larger the PC, the larger the . Consequently, could be increased by increasing the plasticity. It can be observed in Figure 7 that both *e* and work in the same manner on based on the result. However, for , it was reasonable to suggest a model with *e* rather than in term of because both the large and small particles participate in the of the whole mixture. Therefore, the packing function () composed of *e* and the shape factor for affecting the void ratio on was proposed in this study as equation (9).

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###### 4.2.2. Effect of Confinement () on

The value of in pure GB material normalized by the packing function proposed in this study with *e* was plotted in Figure 8 against the change of to 100, 200, and 300 kPa, to assess the effect of *β* on . of all mixtures decreased with an increase in , and *β*, which is the sensitivity of , decreased with an increase of . The *β* exponent according to *D*_{50GB} of pure GB material is plotted in Figure 9(a) in order to verify the variation of *D*_{50GB}, not . The *β* value tends to exponentially decrease with an increase of *D*_{50GB} as . Moreover, the relationship between the *β* and PC is shown in Figure 9(b) to investigate the change of *β* with PC. It was observed that *β* increases with a gradient equal to that of the PC increase independent of as . These values of parameters changed as changes. Based on observed relationships considering the effects of *D*_{50GB} and PC, *β* can be calculated aswhere the value of parameter −*γ* is −0.14 and the values of parameters *δ* and are 0.02 and 0.38, respectively, in this study.

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###### 4.2.3. Effect of Size Ratio () and PB Content (PC) on

Coefficient *α* of the first term in equation (8) is the of the pure GB when there is no effect of the packing under unity confinement (i.e., ). To evaluate the of mixtures according to , an experiment was performed by changing the *D*_{50} of pure GB (i.e., PC = 0%). Figure 10 shows the relationship between and *D*_{50GB}. The value of , i.e., *α* value, under decreased with an increase of *D*_{50GB} as .

The effect of PC on , i.e., (), which is eliminated with the influence of *α*, is indicated with the increase of PC under the , as shown in Figure 11. It was observed that linearly increased with an increase of PC as . Note that parameter *μ* is the gradient for the relationship between of the unit confining pressure and PC, and its value is 4.3, in this study. To validate the proposed equation, the experiment with PC = 5.69% was additionally conducted and the result agreed well with the suggested relation as shown in Figure 11. Consequently, we confirmed that tends to increase because when a relatively rigid soft material such as polyurethane is mixed with a rigid material, the contact of the large particles is inhibited and the plasticity is increased as the content thereof increases.

#### 5. Formulation of Practical Equations for Estimating and

##### 5.1. Formulation for with Factors

By investigating the various factors, such as confining pressure, void ratio, and particle size, affecting the of spherical mixtures, we proposed equations (5)–(7). By substituting these equations into equation (4), the formulation for estimating the on the spherical material mixture of noncohesive plastic can be expressed as

The first term of the proposed equation represents the of material per when there is no effect on the packing. Parameters *A* and *d* are fitting parameters that change according to the composition of the material. Since the stiffness of PB is much smaller than that of GB, the stiffness of the whole mixture decreases with increasing PB contents. The second term indicates the effect of the packing condition on . It was shown that is in good agreement with compared to *e*. *E* is the shape factor for round particle and is used as 2.17. Given that the increment rate of confining pressure, the *n* was associated with the change of *D*_{50GB} and PC. The *n* can be obtained as 0.06 · · 0.01 · PC. Figure 12 compares the measured and the predicted by equation (11). The predicted data agreed well with the measured data. The coefficient of determination was very high, *R*^{2} = 0.91. In addition, to validate the proposed equation, extra experiments for two conditions (PC = 4.28% in = 1.1 and PC = 8.07% in = 4.2) are additionally conducted and their results fitted well with the proposed equation as shown in Figure 12.

##### 5.2. Formulation for with Factors

To corroborate the constant *C* in the packing function obtained by multivariable regression analysis, *C* was estimated by back analysis using the values for all tested mixtures with the confining pressure () and *β* in equation (8). As shown in Figure 13(a), *C* was about 0.042 in all conditions of and PC. Therefore, the result obtained from multivariable regression analysis to calculate the packing function is appropriate.

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By substituting equations (9) and (10) considering the effects for packing condition and confining pressure with PB contents (PC) on of spherical mixtures into equation (8), equation (12) to predict the of a spherical material mixture of noncohesive plastic can be suggested as

The first term of proposed equation (12) represents the of mixtures without the effect of packing under the unit confining pressure. Parameters *λ* and *μ* depend on the composition materials, GB and PB. Since the of GB is smaller than that of PB, of the mixture increased with increasing PC. The second term in equation (12) indicates the packing state when is measured. In addition, because all particles, both large and fine particles, contribute to , it was reasonable to use *e* rather than in this model, unlike the model. Based on multivariable regression analysis and back analysis, the sharp factor (*C*) of the packing function was identified as *C* = 0.04. The *β*, indicating the slope of the change curve in with confining pressure, was correlated with the change of *D*_{50GB} and PC. Figure 13(b) compares the measured and predicted by equation (12). As the same as , the extra experiments for two conditions (PC = 4.28% in = 1.1 and PC = 8.07% in = 4.2) were additionally carried out to substantiate the proposed equation, and their data are included with all tested data, as shown in Figure 13(b). The predicted and measured data showed good agreement with the high coefficient of determination, *R*^{2} = 0.80. Therefore, the applicability of the proposed equations could be confirmed based on this result.

Note that more experiments considering various contents of *D*_{50}, PC, , *e*, and are needed to widely use the proposed equation.

#### 6. Conclusions

Our experimental investigation carried out the parametric study for affecting and through the RC test. Based on the results, we proposed empirical equations to estimate and on the spherical mixture composed of the plastic and nonplastic materials. The following conclusions are drawn from this study:(1)The most important parameters affecting the binary mixture composed of the plastic and nonplastic materials are polyurethane content (PC), mean confining effective stress (), packing effect of void ratio (*e* and ), particle-size ratio (), and median grain size (*D*_{50}).(2)The effects of a void ratio in spherical mixtures (i.e., relative density reduction and an increase of confining pressure and the PB content) are dominant in the changes of which showed a linear behavior with an increase of . Moreover, the change of particle behavior caused by an increase of to a larger particle size also affects .(3)For of the mixture, *e* was used in the proposed prediction model because both the small particle (PB) and large particle (GB) contribute to of the whole mixtures. Moreover, *n* and *β* that present the sensitivity of the confining pressure can be expressed as a function of *D*_{50} and PC.

#### Notations

: | Small-strain shear modulus |

: | Small-strain damping ratio |

PB: | Polyurethane beads |

GB: | Glass beads |

PC: | Nonplastic particle content |

e: | Global void ratio |

: | Intergranular void ratio |

: | Particle-size ratio |

: | Median grain size of large particles (GB) |

: | Median grain size of small particles (PB) |

: | Mean effective confining stress |

D_{50}: | Median grain size |

: | Function of e on and in the mixtures |

: | Function of PC on and in the mixtures |

A and α: | and of materials with no influence of void ratio when |

n and β: | Sensitivity of the curve as and that depend on |

E and C: | Constant of packing for round particles on and in the mixtures |

a, b, and : | Fitting parameters of n value and β value—D_{50} curve |

c and δ, ε, : | Fitting parameters of n value and β value—PC curve |

λ and η: | Fitting parameters of α value |

μ and d: | Fitting parameters of PC on and when . |

#### Data Availability

The data of small-strain dynamic properties used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This work was financially supported by the Ministry of the Interior and Safety as Earthquake Disaster Prevention Human Resource Development Project.