Elastic-Viscoplastic Constitutive Model of Soil under Cyclic Loading

)e creep problems are often involved in soil under cyclic loading, and its behaviors of soil under cyclic loading sparks many arguments in current research field. To propose one newmodel to demonstrate these creep behaviors of soil under cyclic loading, the cyclic loading was simplified equivalently, and the elastic-viscoplastic model (EVPM) for soil under cyclic loading was established based on the Bingham model.)e yield criterion for soil under cyclic loading with constant amplitude was proposed following the simplified load. A constitutive equation based on the EVPMwas constructed by using the flow criterion related to the yield criterion.Meanwhile, the parameters of EVPM were identified and discussed. In addition, the case analysis of the EVPM was also performed. )e results indicate that the stable and destructive creep behaviors of soil under cyclic loading could bewell described by the recommendedEVPM, and the obtained parameters in the model exhibited a clear regularity with the increase of dynamic stress amplitude. Besides, the established model could be selected to predict the stable and destructive creep behavior of soil under cyclic loading.


Introduction
In geotechnical engineering, the building foundations often involves creep behaviors of soil under cyclic loading, such as earth dams under the conditions of cyclical changes in water level of reservoir and subgrade soil under traffic load and tunnels under high-speed railway load. e strength and creep characteristics of soil under cyclic loading are critical to the safety of engineering [1]. Under high-cycle stress, with the increase of loading time, the creep rate of soil becomes bigger; thus, the excessive deformation is caused. Meanwhile, the creep behavior will also occur under low-cycle stress. erefore, the research concerning the creep characteristics of soil under cyclic loading has certain theoretical significance for the long-term stability evaluation as well as design of geotechnical engineering. e creep behaviors of soil under cyclic loading are divided into stable, critical, and destructive creep ( Figure 1) in accordance with the ε(N) − N curves of soil under cyclic loading [2][3][4][5][6][7][8][9]. When the peak deviatoric stress σ 1 max − σ 3 of cyclic loading is less than the critical dynamic stress σ cr of soil, the creep rate soil is faster at the initial stage of loading; as the loading time is prolonged, the strain does not increase but stabilizes; in this case, the creep behavior of soil belongs to stable creep. When the peak deviatoric stress σ 1 max − σ 3 of cyclic loading is near the critical stress value σ cr of soil, the strain of soil sometimes increases and sometimes decreases with the increase of cyclic times N. In this case, the creep behavior of soil belongs to critical creep. When (σ 1 max − σ 3 ) > σ cr , the creep rate of soil increases until the soil is destroyed with the increase of N; at this time, the creep behavior of soil belongs to destructive creep behavior. Sometimes, the destructive creep behaviors show three creep stages of obvious constant speed, deceleration, and acceleration (Figure 2).
At present, many scholars have carried out a lot of research on the models which describe the strain accumulation curves of soil shown in Figures 1 and 2. Many achievements have been made in these studies. ere are two main aspects in the study of soil plastic cumulative model under cyclic loading, i.e., theoretical models and mathematical empirical models. e theoretical models mainly include plastic accumulation energy model [10], rheology theory model [11][12][13][14], modified Cambridge model [15], and HCA model [16]. Mathematical empirical models include stress-strain hyperbolic model [17], exponential model [18], modified hyperbolic model on the basis of hyperbolic model and exponential model [19], grey theory model [20], and semiempirical and semitheoretical model [21].
Although many constitutive models for soil under cyclic loading have been developed, many existing models cannot well reflect the plastic creep characteristics of soil under cyclic loading. erefore, the relevant research still needs to be strengthened. In this paper, according to the existing theory, the cyclic loading was simplified equivalently. A dynamic yield criterion of soil was proposed based on the simplified load. A new constitutive equation based on the improved Bingham model was established by using the flow criterion related to the dynamic yield criterion. Meanwhile, the creep constitutive equation was obtained. e parameters of the model were identified and application analysis was also conducted on the proposed model.

Cyclic Loading and Simplified Load.
In engineering, the most frequently encountered types of cyclic loading are sine-wave, cosine-wave, and rectangle-and triangular-wave cyclic loading. However, the rectangular-and triangular-wave cyclic loading can be transformed into sine-wave or cosinewave cyclic loading by Fourier transformation. erefore, in this study, the sine-wave cyclic loading equation (1) is taken as an example to study the creep behavior of soil under cyclic loading: where σ av is the vibration center of cyclic loading, that is, the average value of cyclic loading, σ d is the stress double amplitude of cyclic loading, ω is the angular frequency of cyclic loading, ω � 2πf (f is loading frequency), and t is the time of loading. For the sake of simplicity of the study [14], the cyclic loading (see equation (1)) acting on the soil was simplified into equation (2). However, the confining pressure was not considered in the simplified load: where σ max is the upper limit of cyclic stress and σ cr is the critical dynamic stress of soil.
In accordance with equation (2), when σ max > σ cr , the simplified load σ eq is larger than the average value σ av of cyclic loading; in this case, the destructive creep of soil will appear. When σ max < σ cr and σ eq < σ av , in this case, the stable creep of soil will appear. For equation (2), the equivalent stress σ eq increases with increasing frequency f and σ max of cyclic loading acting on soil. Although equation (2) was treated as a static form load on the surface, the properties of cyclic loading in essence can be reflected by it.
When we consider the confining pressure, the problem is studied under the state of principal stress. It is assumed that the direction of cyclic loading is consistent with the direction of the maximal principal stress σ 1 ; then, the cyclic loading under confining pressure can be written as where p c is confining pressure and σ st is static deviatoric stress. Based on the idea of equation (2), when we take the influence of confining pressure on creep behavior into consideration, the deviatoric stress of soil subjected to the cyclic loading shown in equation (3) can be equivalently simplified as where σ av is the average value of the deviatoric stress of cyclic loading, namely, the vibration center value of the load σ 1 (t) − σ 3 , σ 1max is the upper limit of cyclic loading,  occur, σ 3 is the confining pressure acing on soil, and g(f) is a function of frequency.
In equation (4), 0 < [(σ 1 max − σ 3 ) − σ cr ]/σ cr < 1; if equation (4) can be used to reflect the characteristic of cyclic loading, then there must be 0 ≤ g(f) < 1. If the g(f) value is too large, the simplified stress will exceed σ 1max ; as a result, the simplified stress is not consistent with the actual situation. For this reason, the frequency function in equation (4) is determined by where n is an integer and its value is determined according to the following procedure.
With relation 0 ≤ g(f) < 1, if cyclic loading is with different frequencies in the experiment when testing, the parameter n should satisfy where f max is the maximum frequency value of cyclic loading used during testing, and it is determined by where f 1 , f 2 , . . . , f n are the frequency values of cyclic load used during the test. From equation (6), it is well known that n ≥ 0 and n ≥ lgf max . When lgf max < 0, n value is 0. When lgf max > 0, n rounds up to the nearest multiple of lgf max .

Establishment of Dynamic Yield Criterion.
Because of the structural behavior and porosity of soil, the plastic volume and shear strain will occur when soil is subjected to cyclic loading. e creep behavior of soil under cyclic loading is related not only to its stress state but also to the characteristics of cyclic loading. Consequently, the dynamic yield function of soil under cyclic loading can be established in accordance with the equivalent load as follows: where J 2eq is the second invariant of equivalent stress deviator, I 1eq is first invariant of equivalent stress tensor, α eq I 1eq reflects the volumetric strain under cyclic loading, that is, the expansibility of soil, α eq is the equivalent influence coefficient of volume strain of soil, and K eq is the equivalent strength eigenvalue of soil under cyclic loading. e influence of hydrostatic pressure on the plastic volumetric strain of geomaterials is considered by αI 1eq of this criterion, and the influence of stress deviator on the shear strain of soil is reflected by �� J 2 . e parameters α eq and K eq in equation (8) is taken as the equivalent form of α and K of Druker-Prager criterion [22], i.e., where ϕ eq is equivalent internal friction angle of soil under cyclic loading and c eq is the equivalent cohesion of soil under cyclic loading.

e Definition and
Meaning of φ eq and c eq . φ eq and c eq are two dynamic indicators of soil under cyclic loading, that is, two strength indexes for soil resistance to destruction, which can be calculated in accordance with the Mohr-Coulomb criterion. e calculation process is as follows. Calculate the equivalent stress σ eq 1 of soil in the critical state according to equation (4) and then equivalent the Mohr-Coulomb diagram plotted, as shown in Figure 3. Finally, the φ eq and c eq values can be obtained from Figure 3 and also be solved by equation (10) according to the test data under two different stress states:

Elastic-Viscoplastic Model Based on the Bingham Model.
e Bingham model is composed of elastic element and ideal viscoplastic body in series, which cannot reflect the nonlinear creep behavior characteristics of geotechnical materials [23]. In order to describe the nonlinear creep behavior of soil under cyclic loading, a modified Bingham constitutive model (see Figure 4) is proposed by replacing the constant elastic and viscous elements in the Bingham model with nonlinear elements. A three-dimensional elasto-viscoplastic fatigue model (EVPM) for soil was established on the basis of the improved Bingham model. In Figure 4, G(N) is the elastic shear coefficient and η(N) is viscoplastic shear coefficient.

Elastic Element.
e relationships between dynamic strength, dynamic shear modulus, and the damping ratio of geomaterials under cyclic loading and the number of cycles N (or loads time t) were studied by many researchers such as Lai Xilei [24] and Mu Kun [25]. ese results show that the Advances in Civil Engineering dynamic parameters of geotechnical materials decrease with the increase of N (or loading time t), and most of them decay in the form of exponential functions. For this reason, it is assumed that the elastic shear fatigue coefficient G(N) in the model, shown in Figure 4, is where G 0 is the initial value of shear coefficient of elastic element, and its physical dimension is stress, and λ is a dimensionless parameter, which is used to reflect the change rule of elastic shear coefficient and the strain rate of soil at the stable creep stage, and it can be determined by test data and λ > 0.

Viscous Element.
Based on fractional calculus theory, the fractional element is defined and the fractional Kelvin-Voigt rheological model is constructed by Guo Jiaqi et al. [26]. It is found that the strain compliance J(t) of the fractional element is proportional to the rheological time t β , in which β is a constant and t is rheology time. Schiessel et al. [27] gave the strain compliance of the fractional unit proportional to t β by considering the time accumulation of viscous elements in the Kelvin model and constructed a fractional rheological constitutive equation by defining fractional units according to fractional derivative. From the above, it can be assumed that the coefficient η(N) of the viscous element in the model, shown in Figure 4, is where n is the strain rate parameter for soil destructive creep behavior, n ≠ 1, η 0 is an initial value of viscoplastic element, and η 0 ≠ 0, its physical dimension is stress·time n.

Constitutive Equation for Stable Creep of Soil under Cyclic
Loading. When the stable creep behavior occurs, the creep behavior is described by the nonlinear elastic element in the model, as shown in Figure 4. At this time, the model is degenerated to the classical nonlinear elastic model. According to the basic theory of elastic mechanics, the state equation of elastic element can be expressed as where S eq ij is the equivalent stress deviator, S eq ij � σ eq ij − σ eq m δ ij (δ ij is Kronecker delta mark), e e ij (N) is the strain tensor of elastic element, σ eq m is equivalent average stress, ε m is the volumetric strain, and K(N) is the fatigue bulk modulus of soil, assuming that its variation law is the same as that of G(N); accordingly, we have where K 0 is the initial value of volume coefficient. When the cycle number of load is N, the strain of the elastic element is where σ eq kk is the first invariant of the stress tensor. As indicated in equation (15)

Constitutive Equation for Soil Destructive Creep Behavior.
When the destructive creep of soil occurs, in the model shown in Figure 4, the soil strain is composed of elastic and viscoplastic strain. When the cycle number of cyclic loading is N, soil strain is In accordance with the basic principle of rheological mechanics, the viscoplastic flow rate in equation (16) can be determined by [28].
where η M is viscoplastic flow coefficient and 〈φ(F)〉 is calculated by equation (18). e main forms of φ(F) are equations (19) and (20), respectively [29]: e plastic strain of soil under cyclic loading can be determined by equation (17). With t � N/f and dt � dN/f, the viscoplastic flow rate of viscoplastic bodies in the model shown in Figure 4 can be written as where _ ε vp ij (t) is the viscoplastic flow rate of viscoplastic bodies in the model at time t, _ e vp ij (N) is the viscoplastic strain rate of the viscoplastic body in the model shown in Figure 4 when the number of cycles is N, F is the dynamic yield function of soil, and Q is the dynamic plastic potential function of soil.
φ(F) is calculated by equation (19) in this article, and taking b as 1 in processes of calculating, then equation (17) is expressed as If the flow rule related to yield criterion is adopted, then F � Q. When the destructive creep behavior occurs, the strain of viscoplastic body in the model can be obtained by integrating equation (22) as follows: When the destructive creep behavior occurs, it is assumed that the main deformation is plastic deformation. Without considering the volume strain and instantaneous elastic strain, the constitutive equation of elastic-viscoplastic fatigue based on the model shown in Figure 4 can be obtained from equations (15) and (23) as follows: When N≠0 and the different parameters G 0 , λ, and η 0 are chosen, equation (24) can be used to describe the destructive creep characteristics of soil under the condition of (σ 1 max − σ 3 ) > σ er . In addition, equation (24) can also be used to describe the three creep stages of soil under cyclic loading.

Constitutive Equation of Soil under
Triaxial Compression e dynamic triaxial compression test is an indoor test method for studying the creep behavior of soil. e rationality of the proposed model can be verified by the dynamic triaxial test data. erefore, the constitutive equation of soil under triaxial compression is obtained in view of equations (15) and (24).
According to equation (4), under triaxial cyclic compression, the equivalent stress state of soil subjected to where σ 3 is the confining pressure acting on the specimen. According to equation (8), the dynamic yield function under triaxial condition is F I 1eq , J 2eq , ϕ cq , ϕ eq , σ av � α eq σ eq 1 + 2σ 3 +

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By plugging equations (25) and (26) into equations (15) and (24), the constitutive equations of soil under triaxial compression are obtained. So, the constitutive equations of soil under triaxial compression are shown in equations (27) and (28), respectively:

Determination of Parameters.
1stop is leading software for nonlinear curve fitting and comprehensive optimization analysis in the world. It has been widely recognized in the field of nonlinear regression, fitting nonlinear curve, and parameter estimation of complex engineering models. In this paper, the software was used to solve the parameters of the proposed model. In equation (27), there are 3 parameters: G 0 , K 0 , and λ. In equation (28), there are 4 parameters: G 0 , λ, η 0 , and n. In order to verify the adaptability of the model, the PO (Powell Optimization) method and SA (Simulated Annealing) method in 1stop software are used to calculate the parameters of the constitutive equation for destructive and stable creep of soil under cyclic loading, respectively. e process of solving model parameters and fitting test data is as follows. According to equation (4), the equivalent stress under different stress amplitudes was obtained. When the stable creep behavior occurs, equation (27) and the corresponding experimental data were input into the software. When the destruction creep behavior occurs, the constitutive equation (28) and the corresponding test data were input, and then, the corresponding optimization algorithm for parameter calculation was selected. Finally, the curves and parameters of the proposed model could be obtained by running the program.

Calculation of EVPM Parameters for Laterite Mixed with Cement.
Kong et al. [30] used the British GDS dynamic triaxial test system to carry out cyclic test on laterite mixed with cement under sine-wave periodic load with frequency f � 5 Hz; the stress level is D � σ d /2τ f (τ f is the static shear strength) when testing. According to the test, the critical dynamic stress amplitudes σ dc of laterite mixed with cement under confining pressure p c of 25, 50, and 100 kPa are 216, 274, and 339 kPa respectively; the static shear strength τ f is 720 kPa, 917 kPa, and 1130 kPa, respectively. e test conditions are shown in Table 1. e vibration center of the axial dynamic stress is σ � p c + σ d , and the axial dynamic stress is According to equation (29), it is well known that σ 1max � σ 3 + σ d + (σ d /2), and the average value of the deviatoric stress under cyclic loading is σ av � σ d .
Under 25, 50, and 100 kPa confining pressure, when the critical state occurs, the average value of cyclic loading is 216, 274, and 339 kPa; then, the critical strength σ cr � σ dc + (σ dc /2) are 314, 411, and 508.5 kPa, respectively. e corresponding σ 1max at critical state are 457, 598, and 778 kPa, respectively. n in equation (4)  According to equation (10), the equivalent internal friction angle ϕ eq � 21.10 ∘ and the equivalent cohesion c eq � 78.42 kPa of laterite mixed with cement can be obtained by selecting the equivalent stress state under 25 and 100 kPa confining pressure. α eq � 0.12 and K eq � 76.73 kPa can be obtained from equation (9).
According to the obtained parameters, the constitutive equation (28) of laterite mixed with cement can be further simplified to

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According to the data from Table 1 and equation (4), the equivalent deviator stress σ eq 1 − σ 3 under 25 and 100 kPa confining pressure can be calculated; the results of equivalent deviator stresses under different dynamic stress amplitudes σ d are shown in Table 2.
In this research, the obtained parameters of the model of soil under the confining pressure of 100 kPa are shown in Table 3, and the comparison between model curves and test curves is shown in Figure 5.
From Figure 5 and Table 3, it can be seen that the elasticviscoplastic constitutive model based on the Bingham model can describe the stable and destructive creep behavior of soil under cyclic loading. e fitting precision of various deformation curves is good. e fitting correlation coefficient is above 0.870 and up to 0.999 under high and low dynamic stress. erefore, the constitutive equation established in this paper can be used to reflect the various creep behavior of soil under cyclic loading.

Analysis of EVPM Parameters of Laterite Mixed with
Cement. Under the confining pressure of 100 kPa, the obtained model curve is shown in Figure 5, and the fitted parameters are shown in Table 3. By analyzing the model parameters in Table 3, it can be found that when stable creep behavior occurs, the model parameters λ, K 0 , and G 0 decayed with the increase of σ eq 1 − σ 3 (Figures 6-8) and when destructive creep behavior occurs, λ, η 0 and G 0 decreased with the increase of σ eq 1 − σ 3 (Figures 9-11). When the upper limit of the cyclic loading is near the critical dynamic stress, n was close to 0, and when the cyclic load increased, n was close to a fixed value. R in Table 3 is correlation coefficient.

Application Analysis.
From the variation of model parameters under different dynamic stress amplitudes [31][32][33], when the confining pressure is 100 kPa, it is known that the value of each parameter varies obviously with σ eq 1 − σ 3 . According to the above analysis, it can be assumed that the parameter variation laws under any dynamic pressure and different dynamic stress are consistent with 100 kPa, but the numerical values are inconsistent. Under different confining pressures, the values of each parameter can be determined by a set of tests under specific confining pressures. Consequently, it can be assumed that the model parameters under a certain confining pressure with different dynamic stress amplitudes are determined by equations (31)-(33) when a stable creep behavior occurs: In equations (31)-(33), β s 1 , β s 2 , and β s 3 values can be obtained by plugging parameters K 0 , G 0 , and λ determined by a group test data from stable creep into equations (31)- (33). Of course, in order to get more accurate values, it can be determined by combining more test data.
When destructive creep occurs, the parameters G 0 , λ, and η 0 are determined by the following equations (34)-(36): In equations (34)-(36), β f 1 , β f 2 , and β f 3 values can be obtained by plugging parameters G 0 , λ, and η 0 determined by a group test data from destructive creep into equations (31)- (33). Of course, in order to get more accurate values, it can be determined by combining a set of test data.
When the upper limit of stress is close to the critical dynamic stress, the parameter n value tends to 0. Under the other different conditions of σ d , it is temporarily determined to be 1.350 in combination with Table 3, but the specific change law of n should be further studied in combination with a large number of experiments.
In this paper, when stable creep occurs, β s 1 � 1/3.44, β s 2 � 1/1.92, and β s 3 � 1/0.946 are obtained by plugging the parameter K 0 , G 0 , and λ value under the dynamic stress amplitude of 25 kPa and 25 kPa confining pressure into equations (31)- (33). When calculating, because the part result of calculation is slightly smaller, s is taken as 1/4. Finally, the parameters K 0 , G 0 , and λ under dynamic stress amplitude of 45, 65, 95, 150, and 225 kPa are calculated in Table 4. e deformation under dynamic stress amplitudes 45, 65, 95, 150, and 225 kPa is predicted by using the calculated parameters. e predicted results are compared with the experimental results, as shown in Figure 12, when stable creep behavior occurs.
When the destructive creep behavior occurs, in this paper, the parameter value of β f 1 , β f 2 , and β f 3 can be obtained by combining the experimental parameter value under confining pressure 25 kPa and dynamic stress amplitude 245 kPa with equations (32)-(34). β f 1 � 0.126, β f 2 � 0.302, and β f 3 � 6.476 × 10 − 5 are obtained by plugging the parameter K 0 , η 0 ,   Table 3: EVPM parameters of laterite mixed with cement under 100 kPa confining pressure [30]. Cumulative strain ε (%) According to these obtained parameters, the creep of soil under cyclic loading with a stress amplitude of 260 kPa is calculated, and the predicted results are compared with the experimental results, as shown in Figure 13.       e calculation results of soil deformation under confining pressure 25 kPa show that the proposed EVPM can predict the deformation of soil under cyclic loading. ere is a certain error between the calculated value of the model and the experimental value. e main reason for the error is that the experimental data are few, which leads to the inaccuracy of the changing rules of the parameters and the calculation error. Further research is needed in combination with experiments. However, the calculated results of the model are roughly consistent with the experimental data, which further illustrates the feasibility of the model.

Conclusions
In this paper, the elastic-viscoplastic model of soil under cyclic loading is established based on the Bingham model and the yield criterion of soil under equal amplitude cyclic loading is proposed. e major findings are summarized as follows: (1) Cyclic loading is simplified equivalently; a dynamic yield criterion of soil under cyclic loading is proposed in view of the simplified load. Based on the improved Bingham model, a EVPFM of soil is established by using the flow criterion related to the dynamic yield criterion. Meanwhile, the parameters of the model are determined and the adaptability of the model is verified. (2) e stable and destructive creep behavior of soil under cyclic loading can be reflected by the established EVPM, and the relevant parameters in the model can also be easily obtained by numerical methods. (3) When the peak deviatoric stress σ 1 max − σ 3 of periodic load is smaller than the critical dynamic stress σ cr value of soil, the model is a nonlinear model describing the stable creep behavior of the soil. Instead, the model is an EVPM describing the soil destructive creep behavior. (4) When stable creep behavior occurs, the model has three parameters, namely, the initial value of the elastic shear fatigue coefficient G 0 , the initial value of the volume fatigue coefficient K 0 , and the elastic attenuation coefficient λ. All three parameters decay with increasing equivalent deviatoric stress σ eq 1 − σ 3 . (5) When destructive creep behavior occurs, the model has four parameters, G 0 , λ, η 0 , and n, respectively. All three parameters decay with increasing equivalent deviatoric stress σ eq 1 − σ 3 . e law of change of n needs to be further studied. Data Availability e Microsoft Excel Worksheet data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest.