# Undrained Deformation Characteristics of Soft Marine Clay Subjected to One-Way Cyclic Loading with Variable Confining Pressure

Authors: Lei Sun
Journal: Advances in Materials Science and Engineering
 (2022)

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## Abstract

To gain a better understanding of the undrained deformation characteristic of saturated marine clay soil subjected to vehicle cyclic traffic load, a sophisticated dynamic triaxial was used to conduct a variety of undrained one-way compression cyclic experiments with variable confining pressure (VCP) as well as constant confining pressure (CCP). The results indicate that, compared to CCP test results, VCP is helpful to raise the axial resilient modulus (Mr) and restrain the permanent plastic strain (εap) development of the specimens. By normalization analysis of the measured data of Mr and εap, the virtually unique correlation between normalized average resilient modulus, normalized permanent axial strain after 1,000 loading cycles, and normalized mean normal stress is established, respectively, regardless of the values of CSR. Additionally, the VCP influence on εap is quantified and fitted by a power law function, which can be used for subsoil deformation prediction and provides new insights into the mechanics of strain accumulation under undrained cyclic loading conditions.


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## Body

## 1. Introduction

Excessive subgrade deformation as a consequence of vehicular traffic has been one of the main causes of damage to transport infrastructures (such as motorways, railways, and airport runways) and financial losses which are founded on the deep soft clay area of southeast (SE) China [1]. Understanding and characterization of the deformation behavior of soft subsoil under traffic loading plays a crucial role in the construction and subsequent maintenance of transport infrastructures.The generation of axial strain in subsoil during repeated loading can be classified into resilient strains and permanent strains. It is the combination of these two components that determines the design and lifetime serviceability of a pavement [2]. Through literature review, cyclic triaxial tests have played a dominant role in the study of soil dynamic deformation behavior for practical reasons, although the effects of principal stress rotation and cyclic intermediate principal stress cannot be modelled in cyclic triaxial tests [3–8]. Resilient modulus and strain accumulation models obtained from cyclic triaxial tests can be used for the prediction of soil stiffness and permanent deformation subjected to traffic loading, which should be addressed in pavement design [9–14].In most of the cyclic triaxial tests in the literature, the confining pressure was maintained constant, and the traffic repeated load was characterized as a separate one-sided cyclic axial compression stress. In fact, as can be noted from Figure1, the dynamic stress field in the subsoil under cyclic loads induced by various transportation facilities includes both transient axial (σ11) and lateral (σ22) stress [15]. This phenomenon indicates that purely cyclic deviatoric loading is inadequate for the real stress path portrayed in the subgrade soil during traffic loading, which is more suitable for employing variable confining pressure cyclic triaxial test (VCP) [16, 17].Figure 1
Variation of stress components induced by traffic loads.To assess the VCP effect on the drained cyclic triaxial behavior of unbound granular materials (UGMs), RondóN et al. [16] presented an experimental study of the drained axial deformation characteristic of UGMs where variable confining pressure was applied at a constant cyclic deviator stress level. A notable finding of the study is that only under some special stress path conditions can the effect of confining pressure be negligible. Otherwise variable confining pressure promotes the strain accumulation in unbound granular materials in drained conditions. In addition, three forms of VCP tests performed by Sun et al. [17] on Toyoura sand suggested that the stress ratio as well as stress path length should be used for permanent deformation prediction. Cai et al. [18] as well as Sun et al. [19] discussed the influences of VCP stress paths on the axial strain evolution in saturated clay soils and concluded that the partially drained permanent deformation of soft clay subsoil is significantly impacted by the variable confining pressure.While most of the VCP effect investigations focus on the deformation behavior of subgrade soils such as unbound granular materials, Toyoura sand, and soft clay in free-to-drain or partially drained conditions, the undrained deformation behavior of natural soil is largely overlooked [20]. In particular, little is known about the undrained strain response of intact marine clay subsoil under one-sided cyclic compression load considering the VCP stress path condition.Extensive studies have provided considerable insight into the VCP effect on the strain response of subsoil. Although the existing undrained strain accumulation model has been widely accepted and used for settlement analysis, the influence of variable confining pressure is not considered in most cases. The strain accumulation model of subsoil under VCP stress path that matches traffic loading conditions is still a topic of interest in the field of soil dynamics which need further studying.The main purpose of this study is to evaluate the cyclic triaxial stress path effect on the resilient and permanent deformation response in soft marine clay during undrained one-way cyclic load conditions. Both CCP and VCP cyclic triaxial tests were conducted to establish empirical correlations between the resilient modulus as well as permanent axial strain and cyclic triaxial stress path, respectively. The research findings are helpful to determine the deformation caused by traffic load.


## 2. Cyclic Triaxial Test Program

In this study, normally consolidated soft marine clay was adopted for the undrained cyclic test program by the GDS dynamic triaxial device (10 Hz/2 kN) imported from England. A high-quality block sample of Wenzhou soft marine clay taken from about 11 m below the original ground level was used in this experimental investigation. The tested soil’s basic index properties determined according to ASTM standards are listed in Table 1.Table 1
Physical index of Wenzhou soft clay.
Index propertiesValuesSpecific gravity,Gs (g/cm3)2.71Natural water content,wn (%)59.7Initial density,ρ0 (g/cm3)1.65Initial void ratio,e01.62Liquid limit,wL (%)60Plasticity index,Ip37Clay fraction, (%)41Silt fraction, (%)55The standard cylindrical test specimens (d = 50 mm; h = 100 mm) are prepared by the hand-trim method from a large clay block. First, all the test soft clay specimens were installed in the triaxial cell and saturated by a 300 kPa magnitude backpressure until the B value was up to 0.97 and above; thereafter, isotropic consolidation was performed carefully at the effective confining stress p0′ = 100 kPa for 24 h. Finally, cyclic triaxial tests under CCP as well as VCP stress path conditions with a frequency of 0.1 Hz and cycle numbers of 1,000 were carried out while keeping the drainage line close throughout the duration of cyclic loading.The stress parametersp=σ1′+2σ3′/3 as well as q=σ1′−σ3′ were used to represent the test results, in which σ3′ = effective lateral stress, and σ1′= effective vertical stress. The testing program covered a lot of different values of the cyclic stress ratio (CSR = qampl/p0′) in different inclinations (ƞampl = qampl/pampl) in p-q space, as shown in Figure 2, where qampl and p0′ = cyclic deviatoric stress amplitude and effective mean principal stresses, respectively. Cyclic triaxial test conditions are summarized in Table 2.Figure 2
Stress paths of the tests inp-q space.Table 2
Scheme of CCP and VCP tests.
Test numberσ3ampl (kPa)qampl (kPa)CSRƞamplCycle numbersCCP10200.231,000VCP1-113200.21.01,000VCP1-223200.20.671,000CCP20250.2531,000VCP2-142250.250.51,000CCP335300.331,000VCP3-10300.31.01,000VCP3-220300.30.671,000CCP40400.431,000VCP4-167400.40.51,000CCP50450.4531,000VCP5-130450.451.01,000VCP5-275450.450.51,000CCP60550.5531,000VCP6-164550.550.671,000


## 3. Test Results of Analysis

### 3.1. Typical Deformation Behavior of Soft Marine Clay in CCP and VCP Tests

Figure3 presents typical VCP and corresponding CCP axial strain-time (εa − t) responses during the application of 1000 cycles of semisinusoidal deviatoric stress loading. Overall, all the axial strains (εat) show obvious periodicity under cyclic loading and develop similar trends with loading time t which could be decomposed into resilient axial strain (εar) and permanent axial strain (εap). Moreover, εap increases quickly during the initial stage of the cyclic loading test; since then, the growth rate gradually decreases and stabilizes, while εar stays approximately constant during cycling. From Figures 3(a) and 3(b), it can also be indicated that the generation of axial strain is dependent on not only the cyclic stress ratio CSR but also the value of parameter ƞampl. The rate of axial strain development in the case of ƞampl = 3 (CCP) was larger than those VCP results with ƞampl = 1 and ƞampl = 0.5, regardless of the amplitude of cyclic deviatoric stress. Under otherwise identical conditions, a higher level of deviatoric stress resulted in a larger axial strain. These observations suggest that both the stress level and stress path imposed significant influence on undrained axial strain response of soft marine clay.Figure 3
Typical axial strain development curves against loading time: (a) CSR = 0.25 and (b) CSR = 0.45.
(a)(b)Figure4 compares the typical CCP and VCP deviatoric stress-axial strain hysteretic loops in the case of CSR = 0.25 and 0.45. It can be found that both the CCP and VCP stress-strain loops exhibit hysteresis and cyclic stability, which move to the right with increasing loading cycle numbers at these stress levels. To gain a better comparison of the hysteretic loops at different ηampl values, the last hysteretic loops at N = 1,000 are highlighted. Obviously, the area of hysteretic loops corresponds to the energy dissipated into the soil in the CCP test (ηampl = 3) is larger than that in VCP tests (ηampl = 1, 0.5). Moreover, the slope of the stress-strain hysteresis curves, a characterization of the Young’s modulus of the soil recognized as a significant parameter for subgrade soils’ stiffness characterization [2], is higher in VCP tests compared with the corresponding CCP Young’s modulus.Figure 4
Comparison of typical axial cyclic stress-strain loops between CCP and VCP conditions: (a) CSR = 0.25; (b) CSR = 0.45.
(a)(b)


### 3.2. Variation of CCP and VCP Resilient Modulus

For a better view of the CCP and VCP undrained resilient performance of the soft marine clay, the resilient modulus calculated by CCP and VCP test data are quantified and compared, where the resilient modulusMr shown in Figure 4 is defined as qampl/εar according to the studies of Tang et al. [7]. The typical variation of Mr with loading cycles is plotted in Figure 5. From the figures, it can be found that all the resilient modulus (Mr) exhibited a constant level after a certain loading cycle number. From Figure 5(a), comparison of Mr values among CCP tests reflects that CSR has a crucial impact in the buildup of Mr. At the identical N, the test specimen undergoes a higher deviatoric stress level and always has a smaller magnitude of resilient moduli, and this is consistent with that concluded from resilient tests of granular materials (ballast) [21]. Figure 5(b) depicts the evolution of resilient moduli (Mr) against cycle numbers, with identical CSR but different values of ηampl, which indicates increasing Mr with decreasing ƞampl values.Figure 5
Effect of CSR and stress path on resilient modulus: (a)ηampl = 3 and (b) CSR = 0.3.
(a)(b)Using the processing method on resilient modulus by Gräbe et al. [2], Figure 6 gives the variation trend of the average resilient modulus of the whole cyclic loading process (Mrav) against CSR values. As illustrated in Figure 6, the VCP test leads to higher values of Mrav than the corresponding CCP test. Meanwhile, for the samples with identical ηampl values, the resilient moduli values (Mrav) decrease with increasing CSR values, and a logarithmic function can be used to fit their relationship as follows: (1)Mrav=AlnCSR+B,where parameters A and B are related to ηampl values.Figure 6
Relationships between resilient modulus and CSR values.Figure7 shows the average resilient modulus Mrav as a function of ηampl values (Figure 7(a)) and the mean normal stress pampl (Figure 7(b)). It is obvious in Figure 7 that the specimens display increasing modulus with decreasing ηampl values, whereas increasing with increasing mean normal stress pampl at a given CSR value.Figure 7
Average resilient modulus as a function of (a)ηampl and (b) pampl.
(a)(b)To quantify the degree to whichηampl values and mean normal stress pampl affected the resilient behavior of the specimens, the normalized average resilient moduli values Mrav/Mr,CCPav versus the ηampl values and normalized mean normal stress pampl/pCCPampl are given in Figure 8. It can be seen that Mrav/Mr,CCPav is exponentially related to the value of ηampl or normalized mean normal stress pampl/pCCPampl, regardless of the CSR values. These observations suggest that how VCP affect Mrav is independent on the magnitude of the deviatoric stress.Figure 8
Normalized average resilient modulus as a function of (a)ηampl and (b)pampl/pCCPampl.
(a)(b)


### 3.3. Variation of the CCP and VCP Permanent Axial Strains

Figure9 shows the typical evolution of measured εap with increasing loading cycle numbers in CCP and VCP test conditions. Generally, all the shapes and patterns of the curves are identical to each other except for the magnitudes. It is evident from Figure 9 that εap accumulates quickly in the initial phase, and then the rate generally decreases with loading cycle numbers. Figure 9 also demonstrates that the accumulated values of εap rely on both the CSR and ηampl values. As Figure 9(a) plots, greater CSR signified a larger εap at an identical number of loading cycles. Figure 9(b) compares permanent axial strain in the CCP with those measured in the VCP under an identical CSR value of 0.2. It is clear that VCP tests induce a smaller εap than the corresponding CCP test, which implies that variable confining pressure limits the test specimens’ permanent axial strain development subjected to one-way cyclic compression loading in undrained condition.Figure 9
Variation of permanent axial strains with cycle numbers: (a)ηampl = 3; (b) CSR = 0.2.
(a)(b)Figure10 further reports the correlation between the permanent axial strain for N = 1,000 (εa,1000p) and the mean normal stress amplitude (pampl), indicating that the VCP affects the axial strain accumulation values evidently. Clearly, the permanent axial strain magnitudes decrease with increasing mean normal stress levels.Figure 10
Relationships betweenεa,1000p and pampl.The normalized result of Figure10 is further plotted in Figure 11. Obviously, the normalized axial strain accumulated values after 1,000 cycles (εa,1000p/εa,1000p,CCP) has a logarithmic relationship with the normalized mean normal stress pampl/pCCPampl, regardless of different cyclic deviatoric stress conditions. This observation suggests that the VCP effect on the undrained accumulated strain values after a certain cycle number such as 1,000 is also independent of the magnitude of cyclic deviatoric stress.Figure 11
Relationship betweenεa,1000p/εa,1000p,CCP and pampl/pCCPampl.To quantify the degree to which parameterηampl affected the accumulated axial strain values of the test soil specimens. As shown in Figure 12, when all strain values are identical in cycle numbers, the values of permanent axial strain at stress paths parameter ηampl = 1, 0.5, and 0.67 (εa,VCPp) have a linear relationship with the counterparts εa,CCPp in CCP test conditions, respectively. Moreover, Figure 13 further plots the fitting parameters λ against the values of the stress path parameter ηampl. Here, λ is represented as the ratio between εap and εa,CCPp. A power law function was selected to express the fitting curve in Figure 13.(2)λ=mηampl/3n,where m and n = fitting parameters equal to 1 and 0.31, respectively. By combining equation (2), ηampl values and CCP permanent axial strains together, the following expression, considering the cyclic triaxial stress paths effect, can be suggested for the accumulated axial strain prediction of soft marine clay under traffic loading.(3)εap=ηampl/30.31εa,CCPP.Figure 12
Relationship between VCP permanent axial strains and CCP permanent axial strains.Figure 13
Relationship betweenλ and ηampl.


### 3.4. Empirical Model for Permanent Axial Strain

Figure14 shows all the measured permanent axial strain values εa,CCPp under the CCP stress path for 1000 cycles in a double logarithmic coordinate. It can be noted from Figure 14 that the test data approximate a straight line after 10 cycles, which can be fitted using a power law function of the form as follows:(4)logεa,CCPp=logεa,10p+klogN10.Figure 14
Experimental and predicted permanent axial strain in CCP tests.Equation (4) can be rewritten as follows:(5)εa,CCPp=εa,10pN10k,where εa,10p = permanent axial strain accumulated values at cycle no. 10 and k = the slope of the dashed line in Figure 14. The effects of CSR on the parameters εa,10p and k are listed in Table 3. It can be noted that parameter k is little influenced by the deviatoric stress magnitude at these stress levels, which is consistent with previous conclusions [10, 14]. Therefore, the k value can be considered as a constant of the mean value of 0.189 and applied in equations (5).Table 3
The fitting parametersεa,10p and k in equations (4).
CSRεa,10p (%)k0.20.1710.2120.250.2780.1700.30.3640.1600.40.6090.1870.450.6900.2150.551.3420.190Mathematically, the correlation between parametersεa,10p and CSR in Table 3 can be fitted by an exponential law function as follows:(6)εa,10p=0.065e5.495CSR.By combining equations (5) and (6) and parameter k (0.189), an empirical model for strain prediction of test soil specimens during undrained CCP cyclic loading conditions can be established as follows:(7)εa,CCPp=0.065e5.495CSRN100.189.Afterwards, combining equations (7) and (3), a unified model that incorporates the influences of cyclic triaxial stress paths can be expressed as follows:(8)εap=0.065e5.495CSRN100.189ηampl30.31.Figure15 presents the comparison of prediction results from equation (8) and the measured data. As can be noted, the empirical model shows good predictive capacity for all the measured strain results under different stress path conditions. The proposed model in equations (8) can be used for analytical modeling of axial strain accumulation and benefits the understanding of existing prediction methods about settlement evaluation of subsoil triggered by traffic load. Meanwhile, it should also be pointed out that the reason for the good predictions is that the correlation parameters of the empirical model were fitted by the measured data against whom the comparisons were made. Further verification and calibrateion should be carried out to improve the applicability of this model.Figure 15
Comparison of the measured and predicted permanent axial strain responses.
(a)(b)(c)(d)(e)(f)


## 3.1. Typical Deformation Behavior of Soft Marine Clay in CCP and VCP Tests

Figure3 presents typical VCP and corresponding CCP axial strain-time (εa − t) responses during the application of 1000 cycles of semisinusoidal deviatoric stress loading. Overall, all the axial strains (εat) show obvious periodicity under cyclic loading and develop similar trends with loading time t which could be decomposed into resilient axial strain (εar) and permanent axial strain (εap). Moreover, εap increases quickly during the initial stage of the cyclic loading test; since then, the growth rate gradually decreases and stabilizes, while εar stays approximately constant during cycling. From Figures 3(a) and 3(b), it can also be indicated that the generation of axial strain is dependent on not only the cyclic stress ratio CSR but also the value of parameter ƞampl. The rate of axial strain development in the case of ƞampl = 3 (CCP) was larger than those VCP results with ƞampl = 1 and ƞampl = 0.5, regardless of the amplitude of cyclic deviatoric stress. Under otherwise identical conditions, a higher level of deviatoric stress resulted in a larger axial strain. These observations suggest that both the stress level and stress path imposed significant influence on undrained axial strain response of soft marine clay.Figure 3
Typical axial strain development curves against loading time: (a) CSR = 0.25 and (b) CSR = 0.45.
(a)(b)Figure4 compares the typical CCP and VCP deviatoric stress-axial strain hysteretic loops in the case of CSR = 0.25 and 0.45. It can be found that both the CCP and VCP stress-strain loops exhibit hysteresis and cyclic stability, which move to the right with increasing loading cycle numbers at these stress levels. To gain a better comparison of the hysteretic loops at different ηampl values, the last hysteretic loops at N = 1,000 are highlighted. Obviously, the area of hysteretic loops corresponds to the energy dissipated into the soil in the CCP test (ηampl = 3) is larger than that in VCP tests (ηampl = 1, 0.5). Moreover, the slope of the stress-strain hysteresis curves, a characterization of the Young’s modulus of the soil recognized as a significant parameter for subgrade soils’ stiffness characterization [2], is higher in VCP tests compared with the corresponding CCP Young’s modulus.Figure 4
Comparison of typical axial cyclic stress-strain loops between CCP and VCP conditions: (a) CSR = 0.25; (b) CSR = 0.45.
(a)(b)


## 3.2. Variation of CCP and VCP Resilient Modulus

For a better view of the CCP and VCP undrained resilient performance of the soft marine clay, the resilient modulus calculated by CCP and VCP test data are quantified and compared, where the resilient modulusMr shown in Figure 4 is defined as qampl/εar according to the studies of Tang et al. [7]. The typical variation of Mr with loading cycles is plotted in Figure 5. From the figures, it can be found that all the resilient modulus (Mr) exhibited a constant level after a certain loading cycle number. From Figure 5(a), comparison of Mr values among CCP tests reflects that CSR has a crucial impact in the buildup of Mr. At the identical N, the test specimen undergoes a higher deviatoric stress level and always has a smaller magnitude of resilient moduli, and this is consistent with that concluded from resilient tests of granular materials (ballast) [21]. Figure 5(b) depicts the evolution of resilient moduli (Mr) against cycle numbers, with identical CSR but different values of ηampl, which indicates increasing Mr with decreasing ƞampl values.Figure 5
Effect of CSR and stress path on resilient modulus: (a)ηampl = 3 and (b) CSR = 0.3.
(a)(b)Using the processing method on resilient modulus by Gräbe et al. [2], Figure 6 gives the variation trend of the average resilient modulus of the whole cyclic loading process (Mrav) against CSR values. As illustrated in Figure 6, the VCP test leads to higher values of Mrav than the corresponding CCP test. Meanwhile, for the samples with identical ηampl values, the resilient moduli values (Mrav) decrease with increasing CSR values, and a logarithmic function can be used to fit their relationship as follows: (1)Mrav=AlnCSR+B,where parameters A and B are related to ηampl values.Figure 6
Relationships between resilient modulus and CSR values.Figure7 shows the average resilient modulus Mrav as a function of ηampl values (Figure 7(a)) and the mean normal stress pampl (Figure 7(b)). It is obvious in Figure 7 that the specimens display increasing modulus with decreasing ηampl values, whereas increasing with increasing mean normal stress pampl at a given CSR value.Figure 7
Average resilient modulus as a function of (a)ηampl and (b) pampl.
(a)(b)To quantify the degree to whichηampl values and mean normal stress pampl affected the resilient behavior of the specimens, the normalized average resilient moduli values Mrav/Mr,CCPav versus the ηampl values and normalized mean normal stress pampl/pCCPampl are given in Figure 8. It can be seen that Mrav/Mr,CCPav is exponentially related to the value of ηampl or normalized mean normal stress pampl/pCCPampl, regardless of the CSR values. These observations suggest that how VCP affect Mrav is independent on the magnitude of the deviatoric stress.Figure 8
Normalized average resilient modulus as a function of (a)ηampl and (b)pampl/pCCPampl.
(a)(b)


## 3.3. Variation of the CCP and VCP Permanent Axial Strains

Figure9 shows the typical evolution of measured εap with increasing loading cycle numbers in CCP and VCP test conditions. Generally, all the shapes and patterns of the curves are identical to each other except for the magnitudes. It is evident from Figure 9 that εap accumulates quickly in the initial phase, and then the rate generally decreases with loading cycle numbers. Figure 9 also demonstrates that the accumulated values of εap rely on both the CSR and ηampl values. As Figure 9(a) plots, greater CSR signified a larger εap at an identical number of loading cycles. Figure 9(b) compares permanent axial strain in the CCP with those measured in the VCP under an identical CSR value of 0.2. It is clear that VCP tests induce a smaller εap than the corresponding CCP test, which implies that variable confining pressure limits the test specimens’ permanent axial strain development subjected to one-way cyclic compression loading in undrained condition.Figure 9
Variation of permanent axial strains with cycle numbers: (a)ηampl = 3; (b) CSR = 0.2.
(a)(b)Figure10 further reports the correlation between the permanent axial strain for N = 1,000 (εa,1000p) and the mean normal stress amplitude (pampl), indicating that the VCP affects the axial strain accumulation values evidently. Clearly, the permanent axial strain magnitudes decrease with increasing mean normal stress levels.Figure 10
Relationships betweenεa,1000p and pampl.The normalized result of Figure10 is further plotted in Figure 11. Obviously, the normalized axial strain accumulated values after 1,000 cycles (εa,1000p/εa,1000p,CCP) has a logarithmic relationship with the normalized mean normal stress pampl/pCCPampl, regardless of different cyclic deviatoric stress conditions. This observation suggests that the VCP effect on the undrained accumulated strain values after a certain cycle number such as 1,000 is also independent of the magnitude of cyclic deviatoric stress.Figure 11
Relationship betweenεa,1000p/εa,1000p,CCP and pampl/pCCPampl.To quantify the degree to which parameterηampl affected the accumulated axial strain values of the test soil specimens. As shown in Figure 12, when all strain values are identical in cycle numbers, the values of permanent axial strain at stress paths parameter ηampl = 1, 0.5, and 0.67 (εa,VCPp) have a linear relationship with the counterparts εa,CCPp in CCP test conditions, respectively. Moreover, Figure 13 further plots the fitting parameters λ against the values of the stress path parameter ηampl. Here, λ is represented as the ratio between εap and εa,CCPp. A power law function was selected to express the fitting curve in Figure 13.(2)λ=mηampl/3n,where m and n = fitting parameters equal to 1 and 0.31, respectively. By combining equation (2), ηampl values and CCP permanent axial strains together, the following expression, considering the cyclic triaxial stress paths effect, can be suggested for the accumulated axial strain prediction of soft marine clay under traffic loading.(3)εap=ηampl/30.31εa,CCPP.Figure 12
Relationship between VCP permanent axial strains and CCP permanent axial strains.Figure 13
Relationship betweenλ and ηampl.


## 3.4. Empirical Model for Permanent Axial Strain

Figure14 shows all the measured permanent axial strain values εa,CCPp under the CCP stress path for 1000 cycles in a double logarithmic coordinate. It can be noted from Figure 14 that the test data approximate a straight line after 10 cycles, which can be fitted using a power law function of the form as follows:(4)logεa,CCPp=logεa,10p+klogN10.Figure 14
Experimental and predicted permanent axial strain in CCP tests.Equation (4) can be rewritten as follows:(5)εa,CCPp=εa,10pN10k,where εa,10p = permanent axial strain accumulated values at cycle no. 10 and k = the slope of the dashed line in Figure 14. The effects of CSR on the parameters εa,10p and k are listed in Table 3. It can be noted that parameter k is little influenced by the deviatoric stress magnitude at these stress levels, which is consistent with previous conclusions [10, 14]. Therefore, the k value can be considered as a constant of the mean value of 0.189 and applied in equations (5).Table 3
The fitting parametersεa,10p and k in equations (4).
CSRεa,10p (%)k0.20.1710.2120.250.2780.1700.30.3640.1600.40.6090.1870.450.6900.2150.551.3420.190Mathematically, the correlation between parametersεa,10p and CSR in Table 3 can be fitted by an exponential law function as follows:(6)εa,10p=0.065e5.495CSR.By combining equations (5) and (6) and parameter k (0.189), an empirical model for strain prediction of test soil specimens during undrained CCP cyclic loading conditions can be established as follows:(7)εa,CCPp=0.065e5.495CSRN100.189.Afterwards, combining equations (7) and (3), a unified model that incorporates the influences of cyclic triaxial stress paths can be expressed as follows:(8)εap=0.065e5.495CSRN100.189ηampl30.31.Figure15 presents the comparison of prediction results from equation (8) and the measured data. As can be noted, the empirical model shows good predictive capacity for all the measured strain results under different stress path conditions. The proposed model in equations (8) can be used for analytical modeling of axial strain accumulation and benefits the understanding of existing prediction methods about settlement evaluation of subsoil triggered by traffic load. Meanwhile, it should also be pointed out that the reason for the good predictions is that the correlation parameters of the empirical model were fitted by the measured data against whom the comparisons were made. Further verification and calibrateion should be carried out to improve the applicability of this model.Figure 15
Comparison of the measured and predicted permanent axial strain responses.
(a)(b)(c)(d)(e)(f)


## 4. Conclusions

This study illustrates experimental research with the aim to explore the undrained resilient and permanent deformation characteristics of normally consolidated soft marine clay due to different cyclic triaxial stress paths (CCP and VCP). The main conclusions are as follows:(1)
Under an undrained one-way cyclic triaxial load condition, the axial strain response is greatly affected by the cyclic deviatoric stress as well as variable confining pressure, and the coupling effect of the two can be efficiently characterized by the cyclic triaxial stress path parameterηampl.(2)
The average resilient moduliMrav of the soft marine clay specimens can be obviously increased due to variable confining pressure, while, under identical cyclic stress path conditions, Mrav reduced with increasing CSR. Furthermore, by normalization, a uniquely relationship between Mrav and pampl can be established, which is independent of the CSR.(3)
Under otherwise identical conditions, the amount ofεap increased with increasing CSR values but decreased with increasing ηampl value. Similarly, a virtually unique relationship can also be established between normalized permanent axial strains and normalized mean normal stress, which is independent of the CSR values.(4)
A comparison betweenMrav and εap reveals that the specimen with the higher resilient modulus resulted in a smaller permanent axial strain.(5)
An empirical model of the power function considering not only the cyclic triaxial stress path but also the effect of loading level and cycles is proposed. The model presents fairly good applicability for the accumulated strain prediction of soft marine clay under different values ofηampl, CSR, and N.



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