# Study on Cyclic Cumulative Deformation Characteristics and the Equivalent Cyclic Creep Model of Soft Clay

Authors: Wenbo Zhu; Guoliang Dai; Weiming Gong
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Suction caisson foundations can be used to anchor tension leg platforms. The soil at the bottom of the caisson undergoes both unloading and cyclic loading under wind and wave loads. However, the problem of cyclic cumulative deformation of soft clay under unloading has rarely been addressed. So, the strain cumulative deformation and strain softening characteristics of soft clay are studied by cyclic triaxial tests. The test results show that under low static deviator stress ratios and dynamic deviator stress ratios, the soil has a low level of strain accumulation and softening. As the dynamic deviator stress ratios increase, the cumulative cyclic deformation gradually increases, which rapidly develops in the early stage and tends to stabilize in the later stage. Moreover, the softening index gradually increases and is linearly related to the logarithm of the number of cycles. The cyclic cumulative deformation of the soil increases with increases in unloading stress and dynamic deviator stress, showing a creep characteristic of attenuation and then stabilization. Based on the tests, an equivalent cyclic creep model is established to describe the strain accumulation and softening of soil and verified through comparison with the test results. Then, the model is extended to a three-dimension model, and a finite element subroutine is developed for studying the strain cumulative deformation and strain softening characteristics of soft clay.


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## Body

## 1. Introduction

Suction caisson foundations can be used as anchoring foundations for tension leg platforms (TLPs). Under an uplift load, the soil at the bottom of the caissons is subjected to both axial unloading and cyclic loading superimposed by wind and wave loads, with cyclic loading inevitably leading to strain softening and cumulative cyclic deformation of the soft clay. Most of studies [1–8] have focused on the case where the initial deviatoric stress is greater than or equal to zero but rarely investigated the cyclic cumulation deformation of soft clay during unloading. Additionally, empirical models are mostly used for cyclic accumulation modeling but have difficulty describing soil strain softening, which is a problem shared by element creep models.Many studies have been conducted with the aim to accurately understand the characteristics of strain softening and the cumulative cyclic deformation of soft clay. In terms of soil strain softening, Andersen [9], Lee [10], Mitchell and King [11], Hyodo et al. [12], and Yasuhara [13] studied the dynamic properties of different types of undisturbed and remolded soil samples under cyclic loading and analyzed the effects of different influencing factors, such as cyclic stress ratio, overconsolidation ratio, frequency, and number of loading cycles on cumulative deformation and soil softening. Zhou and Gong [14] found that the higher the frequency was, the lower the degree of soil strain softening and the faster the strain development. Matasovic and Vucetic [15] established a relationship between softening parameters and cyclic strength to reflect the softening of soil by the change in softening parameters. According to the softening index defined by Idriss et al. and considering the effect of the initial consolidation shear stress, Yao and Nie [16] and Cai et al. [17] redefined the softening index to describe the degree of soil softening. Wang and Cai [18] investigated the influences of the cyclic stress ratio, vibration frequency, overconsolidation ratio, and consolidation ratio on the strain softening characteristics of saturated soft clay with cumulative plastic strain. In terms of strain accumulation, Monismith et al. [19] established a relationship between the cyclic cumulative strain of soil elements and the number of cycles through triaxial tests. On this basis, Li and Selig [20] introduced the static strength parameter of soil and proposed an empirical relationship for the cyclic cumulative strain. Based on the relationship proposed by Li and Selig [20], Chai and Miura [21] further established an empirical exponential relationship considering the effect of the initial static deviatoric stress of the soil element on the cyclic cumulative strain. For the first time, Huang et al. [22] introduced the concept of relative deviatoric stress level D∗, considering the cyclic cumulative strain resulting from the coupling effect of initial static deviatoric stress and cyclic stress. Hyde and Brown [23] and Yasuhara et al. [24] found that the behavior of the residual strain of clay generated under cyclic loading is very similar to that under creep loading and established a method for calculating the cumulative strain by making the number of loading cycles equivalent to time. Based on the theory of static creep, Wang et al. [25] proposed an empirical model for cumulative cyclic plastic axial strain of saturated soft clay under long-term cyclic loading. Dong [26] studied the creep characteristics of saturated soft clay under long-term cyclic loading, and by applying the empirical model theory of static creep to the study of cyclic creep characteristics, they established an empirical model for cumulative plastic strain with the ability to comprehensively consider the effects of dynamic deviatoric stress and initial static deviatoric stress levels. According to the variation characteristics of cyclic creep with the number of cycles, Qi [27] suggested an empirical model for cyclic creep calculation by making the number of cycles equivalent to time and following the parameter determination method of the Singh–Mitchell creep model. Moreover, Fujiwara et al. [28], Zhu et al. [29], Li and Huang [30], and Guo et al. [31] conducted long-term undrained cyclic loading tests on soft clays in different regions and analyzed the cyclic creep behavior of soft clay in depth under long-term repeated loading.The above studies are all focused on the case in which the axial deviator stress is greater than the radial deviator stress, whereas the cumulative cyclic deformation of soft clay during unloading has rarely been investigated. So, the strain accumulation and deformation characteristics of soft clay under axial unloading and cyclic unloading were investigated by the GDS dynamic triaxial test. Based on the tests, an equivalent cyclic creep model is established to describe the strain accumulation and softening of soil and verified through comparison with the test results. Then, the model is extended to a three-dimension model, and a finite element subroutine is developed for studying the strain cumulative deformation and strain softening characteristics of soft clay.


## 2. Cyclic Triaxial Tests

The test soil samples were prepared by the vacuum preloading method and the surcharge preloading method. As shown in Figure1(a), the soil sample preparation chamber is a bucket made of acrylic plexiglass, which has dimensions of 300 mm × 600 mm × 10 mm, to facilitate the observation of slurry settlement. After the soil sample molding, the soil samples were tested on a GDS dynamic triaxial test apparatus (Figure 1(b)). The test procedure is described as follows: (1) the B test was first performed; since the samples were prepared using the slurry method, the B values were above 98%, and there was no need for backpressure saturation; (2) isotropic consolidation was carried out according to the differently set confining pressures; (3) after consolidation was completed, the unloading was carried out at a rate of 0.1 kPa/min until the preset deviatoric stress was reached; and (4) cyclic loading was performed at a frequency of 0.1 Hz and with a sine waveform in the presence of the deviatoric stress. The test parameters are shown in Table 1.Figure 1
Dynamic triaxial tests. (a) The preloading device. (b) The GDS dynamic triaxial test system.
(a)(b)Table 1
Test parameters.
TypeConfining pressure (kPa)Unloading rate (kPa/min)Frequency (Hz)Static deviator stressDynamic deviator stressUnloading50/100/1500.1—0/0.4/0.5/0.6/0.8—0.10.4/0.5/0.60.1/0.2/0.3Cyclic loading50/100/150—0.10.80.10.100.3/0.4/0.5/0.6To determine the soil strength, the triaxial compression and extension tests of soft clay were performed under confining pressures of 50 kPa, 100 kPa, and 150 kPa. The stress-strain curves of saturated soft clay subjected to consolidated undrained (CU) triaxial tests are shown in Figure2. The Mohr–Coulomb ultimate strength indices of saturated clay were calculated to be c = 18.3 kPa and φ = 19.6°.Figure 2
Strength characteristics of soil.


## 3. Test Results

### 3.1. Cyclic Cumulative Strain

Figures3–5 show the cyclic cumulative strain curves under different static and dynamic deviator stresses. Figure 3(a) presents the cyclic cumulative strain curves under different dynamic deviator stress ratios. Under the dynamic deviator stress qd = 0.2qf, there was no significant strain accumulation due to the small static deviator stress. Under the dynamic deviator stress qd = 0.3qf, the cyclic cumulative strain increased rapidly at the beginning of the loading, and then, the rate of strain accumulation started to rapidly decrease, and the development of cumulative strain entered an attenuation stage and eventually reached a stabilization stage. Under the same confining pressure and the same static deviator stress, the cyclic cumulative strain increased with the increase of unloading stress, and as the dynamic deviator stress increased, the cyclic cumulative strain increased.Figure 3
Strain-time relationships (σ3 = 100 kPa). (a) qs = 0.4qf, (b) qs = 0.5qf, (c) qs = 0.6qf, and (d) qs = 0.8qf.
(a)(b)(c)(d)Figure 4
Strain-time relationships (σ3 = 50 kPa). (a) qs = 0.4qf and (b) qs = 0.5qf.
(a)(b)Figure 5
Strain-time relationships (σ3 = 150 kPa). (a) qs = 0.4qf and (b) qs = 0.5qf.
(a)(b)Figure3(c) shows the cyclic cumulative strain curves under different dynamic stress ratios and a static stress ratio qs = 0.6qf. With a dynamic deviator stress qd = 0.1qf and 0.2qf, all the curves exhibit the cyclic creep characteristics. With qd = 0.3qf, at the later stage, the slope of the cyclic cumulative strain curve did not tend to stabilize. Figure 3(d) shows the cyclic cumulative strain curves obtained under σ3 = 100 kPa, qs = 0.8qf, and qd = 0.1qf. Under high static deviator stress and low dynamic deviator stress, the cyclic cumulative strain increased rapidly with the increase in the number of loading cycles; at the later stage, the development of cumulative strain entered an attenuation stage and eventually reached stabilization. So, there was a large cyclic cumulative strain under high static deviator stress and low dynamic deviator stress.From the above test results, it can be seen that under low static deviator stress and low dynamic deviator stress ratio, the cyclic cumulative strain was small and tended to eventually stabilize with an increasing number of cycles. Under low static deviator stress and a high dynamic deviator stress ratio, the cyclic cumulative strain was mainly concentrated in the early stage with relatively fast cumulative cyclic deformation, and then, it gradually stabilized in the later stage. Under high static deviator stress and low dynamic deviator stress ratio, the cyclic cumulative strain of soil developed quickly in the early stage and gradually stabilized in the later stage.


### 3.2. Strain Softening Index

As the number of cycles increased, the hysteresis loops were gradually inclined toward the strain direction; as a result, the cyclic cumulative strain gradually increased, and the strength of the soil softened (Figure6). The soil shear modulus of the Nth cycle was calculated using the following equation.(1)GN=qN,max−qN,minεN,max−εN,min,where qN,max and qN,min are the maximum and minimum deviator stresses, respectively, of the Nth cycle; and εN,max and εN,min are the maximum and minimum deviator strains, respectively, of the Nth cycle.Figure 6
Definition of the softening index.Idriss et al. [32] were the first to introduce the concept of the softening index δ, which is defined as the ratio of the shear modulus of the Nth cycle to the shear modulus of the first cycle, that is,(2)δ=GNG1=τN/γτ1/γ=τNτ1.Andersen [9] redefined the softening index based on equation (2) by considering the initial deviator strain and also established the relationship between the softening index and the number of cycles:(3)δ=GNG1=εN,max−εN,minε1,max−ε1,min.The test results show that the strain softening was low under a small static deviator stress ratio and a small dynamic deviator stress ratio. As the number of cycles increased, the strain softening index gradually approached a constant value. A large dynamic deviator stress ratio led to a high degree of strain softening. Under the same dynamic deviator stress ratio, the softening index was smaller when the static deviator stress was higher. Therefore, a strain softening index equation, as shown in equation (4), was established in the present study based on the test data. The theoretically calculated values and the test values are compared in Figure 7. It can be seen that the fitting result is good. Moreover, the fitting parameters being a = 0.16, n1 = 3.6, n2 = 7.64, and n3 = 0.02. So, the process of soil strain softening under cyclic loading can be described by the following equation.(4)δ=1−αqdqfn11+qsqfn2ppan3lgwt,where p is the confining pressure, qs is the static deviator stress, qd is the dynamic deviator stress, and pa is the reference pressure, which is taken as 100 kPa here.Figure 7
Curves of softening index-number of cycles. (a)σ3 = 50 kPa, (b) σ3 = 100 kPa, and (c) σ3 = 150 kPa.
(a)(b)(c)


## 3.1. Cyclic Cumulative Strain

Figures3–5 show the cyclic cumulative strain curves under different static and dynamic deviator stresses. Figure 3(a) presents the cyclic cumulative strain curves under different dynamic deviator stress ratios. Under the dynamic deviator stress qd = 0.2qf, there was no significant strain accumulation due to the small static deviator stress. Under the dynamic deviator stress qd = 0.3qf, the cyclic cumulative strain increased rapidly at the beginning of the loading, and then, the rate of strain accumulation started to rapidly decrease, and the development of cumulative strain entered an attenuation stage and eventually reached a stabilization stage. Under the same confining pressure and the same static deviator stress, the cyclic cumulative strain increased with the increase of unloading stress, and as the dynamic deviator stress increased, the cyclic cumulative strain increased.Figure 3
Strain-time relationships (σ3 = 100 kPa). (a) qs = 0.4qf, (b) qs = 0.5qf, (c) qs = 0.6qf, and (d) qs = 0.8qf.
(a)(b)(c)(d)Figure 4
Strain-time relationships (σ3 = 50 kPa). (a) qs = 0.4qf and (b) qs = 0.5qf.
(a)(b)Figure 5
Strain-time relationships (σ3 = 150 kPa). (a) qs = 0.4qf and (b) qs = 0.5qf.
(a)(b)Figure3(c) shows the cyclic cumulative strain curves under different dynamic stress ratios and a static stress ratio qs = 0.6qf. With a dynamic deviator stress qd = 0.1qf and 0.2qf, all the curves exhibit the cyclic creep characteristics. With qd = 0.3qf, at the later stage, the slope of the cyclic cumulative strain curve did not tend to stabilize. Figure 3(d) shows the cyclic cumulative strain curves obtained under σ3 = 100 kPa, qs = 0.8qf, and qd = 0.1qf. Under high static deviator stress and low dynamic deviator stress, the cyclic cumulative strain increased rapidly with the increase in the number of loading cycles; at the later stage, the development of cumulative strain entered an attenuation stage and eventually reached stabilization. So, there was a large cyclic cumulative strain under high static deviator stress and low dynamic deviator stress.From the above test results, it can be seen that under low static deviator stress and low dynamic deviator stress ratio, the cyclic cumulative strain was small and tended to eventually stabilize with an increasing number of cycles. Under low static deviator stress and a high dynamic deviator stress ratio, the cyclic cumulative strain was mainly concentrated in the early stage with relatively fast cumulative cyclic deformation, and then, it gradually stabilized in the later stage. Under high static deviator stress and low dynamic deviator stress ratio, the cyclic cumulative strain of soil developed quickly in the early stage and gradually stabilized in the later stage.


## 3.2. Strain Softening Index

As the number of cycles increased, the hysteresis loops were gradually inclined toward the strain direction; as a result, the cyclic cumulative strain gradually increased, and the strength of the soil softened (Figure6). The soil shear modulus of the Nth cycle was calculated using the following equation.(1)GN=qN,max−qN,minεN,max−εN,min,where qN,max and qN,min are the maximum and minimum deviator stresses, respectively, of the Nth cycle; and εN,max and εN,min are the maximum and minimum deviator strains, respectively, of the Nth cycle.Figure 6
Definition of the softening index.Idriss et al. [32] were the first to introduce the concept of the softening index δ, which is defined as the ratio of the shear modulus of the Nth cycle to the shear modulus of the first cycle, that is,(2)δ=GNG1=τN/γτ1/γ=τNτ1.Andersen [9] redefined the softening index based on equation (2) by considering the initial deviator strain and also established the relationship between the softening index and the number of cycles:(3)δ=GNG1=εN,max−εN,minε1,max−ε1,min.The test results show that the strain softening was low under a small static deviator stress ratio and a small dynamic deviator stress ratio. As the number of cycles increased, the strain softening index gradually approached a constant value. A large dynamic deviator stress ratio led to a high degree of strain softening. Under the same dynamic deviator stress ratio, the softening index was smaller when the static deviator stress was higher. Therefore, a strain softening index equation, as shown in equation (4), was established in the present study based on the test data. The theoretically calculated values and the test values are compared in Figure 7. It can be seen that the fitting result is good. Moreover, the fitting parameters being a = 0.16, n1 = 3.6, n2 = 7.64, and n3 = 0.02. So, the process of soil strain softening under cyclic loading can be described by the following equation.(4)δ=1−αqdqfn11+qsqfn2ppan3lgwt,where p is the confining pressure, qs is the static deviator stress, qd is the dynamic deviator stress, and pa is the reference pressure, which is taken as 100 kPa here.Figure 7
Curves of softening index-number of cycles. (a)σ3 = 50 kPa, (b) σ3 = 100 kPa, and (c) σ3 = 150 kPa.
(a)(b)(c)


## 4. Equivalent Cyclic Creep Model

Figure8 shows the stress-strain relationship. During the cyclic loading process, there is not only cumulative cyclic deformation but also reversible deformation at a peak strain point εp and a valley strain point εv. However, the stress in the Merchant creep model is the static deviatoric stress, and there is no peak or valley point of the creep curve. Therefore, the stress in the Merchant creep model under cyclic loading should be the sum of static and dynamic deviator stresses, as expressed in the following equation:(5)σ=σs+σcycsinωt.Figure 8
Stress time curve.Figure9(a) shows the strain softening process by the test. The test cyclic creep curve had already accounted for the influence of strain softening, which was reflected by the increase in the difference between the peak and valley strain values in each cycle as the number of cycles increased. Figure 9(b) shows the strain softening process obtained by the Merchant model. The difference between εp and εv remained the same in the cyclic accumulation process. Therefore, the soil strain softening index cannot be reflected in the Merchant model. Note that in the test, the ratio of the difference between εp and εv in the first cycle to that in the Nth cycle of the cyclic accumulation process is the softening index. The cumulative cyclic deformation and strain softening can be more accurately reflected by multiplying the parameters in the Merchant model by the softening index. So, the equivalent cyclic creep model can describe the cyclic accumulation of soil, and the softening index can describe strain weakening characteristics of soft clay. Therefore, the improved Merchant model for equivalent cyclic creep is established as follows:(6)εt=σs+σcycsinwtδE0+σs+σcycδE11−e−E1/η1t.Figure 9
Strain curves-number of cycles. (a) Actual curve. (b) Traditional Merchant model.
(a)(b)Figure10 shows the equivalent cyclic creep model. The equivalent cyclic creep model is different from the Merchant model. The load applied on the Merchant model is static load, while the load applied on the equivalent cyclic creep Merchant model is dynamic load. The cyclic cumulative deformation can be regarded as the creep process of soil. A stress control switch can be paralleled on the Kelvin body. The stress control switch can ensure the maximum cyclic stress. It can ensure that the stress applied on the Kelvin body is the maximum cyclic stress, and the stress applied on the hooker body is the sum of static deviator stress and dynamic deviator stress. So, the equivalent cyclic creep model can not only describe the strain cumulative deformation characteristics but also describe strain weakening characteristics of soft clay.Figure 10
Equivalent cyclic creep model.The 3D constitutive equations of the equivalent cyclic creep model extended from its 1D equations are given in the following equation:(7)εn=εne+εnve,σne=σn,σnve=σn−σcycsinwt+σcyc,σnve=E1A−1εnve+η1′A−1ε˙nve,ε˙nve=1η1Aσnve−E1η1εnve,A=1−μ−μ−μ1−μ−μ−μ121+μ21+μ21+μ,where εn is the total strain at step n, εne is the instantaneous elastic strain of the Hookean body at step n, εnve is the viscoelastic strain of the Kelvin body at step n, ε˙nve is the viscoelastic strain rate of the Kelvin body at step n, and σn, σne, and σnve are the total stress, the stress of the Hookean body, and the stress of the Kelvin body, respectively, at step n.To implement the 3D constitutive model of the equivalent cyclic Merchant model through an FE subroutine, it is necessary to derive the stress-strain relationship in an incremental form for the equivalent cyclic Merchant model, i.e.,(8)ε˙n+1ve=ε˙nve+∂ε˙nve∂tΔtn,Δεnve=Δtn1−θε˙nve+θε˙n+1ve,where Δtn is the calculation step size at step n, and θ is the difference coefficient, which is taken to be 0.5 for the central difference scheme.From equations (7) and (8), there is(9)Δεnve=θΔtnAΔσnveη1+ε˙nveΔtn1−θE1Δtnη1,Δσnve=Δσn−Δtwσcyccoswt,Δεnve=BΔtnε˙nve+CnΔσnve,where B=1−θΔtnE1/η1, and Cn=θΔtn/η2′A.For the current incremental step, there is(10)Δσn=E0A−1εne,Δεne=1E0AΔσn,Δεn=Δεne+BΔtnε˙nve+CnΔσn−Δtwσcyccoswt,Δσn=D^Δεn−BΔtnε˙nve+CnΔtwσcyccoswt,D^=Cn+1E0A−1,where D^ is the Jacobian matrix in ABAQUS. According to the Jacobian matrix D^, the equivalent cyclic Merchant model can be obtained to analyze the stress-strain relationship.In ABAQUS, the 3D creep FE subroutine was validated through two analysis steps, i.e., application of the confining pressure and cyclic loading and using the strain-time curves for model validation. The total strain E33 was extracted, as shown in Figures11–13. A comparison of FE calculation results and experimental results found generally a good agreement between the two results. Therefore, the equivalent cyclic creep model can be used to analyze the strain accumulation and softening process of soft clay during unloading.Figure 11
Cumulative cyclic-strain curves (σ3 = 50 kPa). (a) qs = 0.4 qf, (b) qs = 0.5 qf.
(a)(b)Figure 12
Cumulative cyclic-strain curves (σ3 = 100 kPa). (a) qs = 0.4 qf, (b) qs = 0.5 qf.
(a)(b)Figure 13
Cumulative cyclic-strain curves (σ3 = 150 kPa). (a) qs = 0.4 qf, (b) qs = 0.5 qf.
(a)(b)


## 5. Conclusions

In this study, the strain softening and strain cumulative deformation of soft clay were characterized through cyclic triaxial tests. An equivalent cyclic creep model was established to describe the cyclic cumulative strain of saturated soft clay and then used to analyze the strain cumulative deformation characteristics and strain softening characteristics of soft clay. The main conclusions are described as follows:(1)
The strain accumulation and strain softening of the soil were low under low static stress ratio and low cyclic stress ratio. With the increase in the number of cycles and the cyclic stress ratio, the cumulative cyclic deformation gradually increased, developing quickly in the early stage and tending to stabilize in the later stage. The cyclic cumulative deformation of the soil increased with the increase in unloading stress and dynamic deviatoric stress, which was the same as the creep characteristics of the curve from the static unloading creep test under undrained conditions.(2)
The degree of soil strain softening was low under low static deviatoric stress and the dynamic deviatoric stress ratio. With the increase in the number of cycles and the dynamic deviatoric stress ratio, the softening index gradually increased and was approximately linearly related to the logarithm of the number of cycles. Based on the experimental results, a softening index equation considering static and dynamic deviator stresses was proposed.(3)
By introducing the Merchant model and considering the effects of dynamic deviator stress and the softening index, an equivalent cyclic creep model was established to describe the cyclic cumulative strain of saturated soft clay and validated through a comparison with the test results. Then, the model was extended to the 3D case, and the corresponding FE subroutine was developed and validated by the test results. Therefore, the equivalent cyclic creep model can be used to analyze the strain accumulation and strain softening process of soft clay.



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