A plasticity-based constitutive model for the behavior of soil-structure interfaces under cyclic loading

Accepted Manuscript A Plasticity-Based Constitutive Model for the Behavior of Soil-Structure Interfaces under Cyclic Loading Massoud Hosseinali, Vahab Toufigh PII: DOI: Reference:

S2214-3912(17)30106-X https://doi.org/10.1016/j.trgeo.2017.10.001 TRGEO 145

To appear in:

Transportation Geotechnics

Received Date: Revised Date: Accepted Date:

14 June 2017 22 September 2017 1 October 2017

Please cite this article as: M. Hosseinali, V. Toufigh, A Plasticity-Based Constitutive Model for the Behavior of Soil-Structure Interfaces under Cyclic Loading, Transportation Geotechnics (2017), doi: https://doi.org/10.1016/ j.trgeo.2017.10.001

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A Plasticity-Based Constitutive Model for the Behavior of SoilStructure Interfaces under Cyclic Loading

Massoud Hosseinali1 and Vahab Toufigh2 1

Ph.D. candidate, Department of Civil Engineering and Environmental Engineering, The University of Utah, Salt Lake City, UT USA;

2

Ph.D., PE, Department of Civil Engineering, Sharif University of Technology, Tehran, Iran. Assistant Professor P.O. Box 11155-1639, Tehran, Iran ([email protected], Corresponding author).

1

Abstract In this study, a new plasticity-based constitutive model is proposed for the behavior of soil-structure interfaces under monotonic and cyclic loadings. The features of the interface including the strainsoftening behavior, phase transformation, steady-state, and dilatancy behavior were described by the proposed model with satisfactory accuracy. The proposed model does not require additional concepts such as damage or disturbance, and the model parameters can be obtained easily using straight-forward analyses of the results obtained from constant normal stress tests. Moreover, the results of the proposed model showed its capability in predicting the experimental results obtained from various interface test devices (direct shear, simple shear, and cyclic multi-degree-of-freedom) at different test conditions (constant normal stress and constant normal stiffness) and loading conditions (monotonic and cyclic).

Keywords: soil-structure interaction; soil-structure interface; strain-softening; plasticity; cyclic loading; constitutive model.

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1. Introduction The soil-structure interaction is involved in a wide range of geomechanical systems including piles, foundations, retaining walls, tunnels, etc. Therefore, to have a rational design and cost-effective construction, it is of great significance to conduct experimental investigations on the behavior of soilstructure interfaces and formulate appropriate constitutive models accordingly. For example, FEM analysis results of several high concrete-faced rockfill dams showed that the stress and deformation of the face slab are significantly affected by the behavior of the interface between it and the cushion layer (Zhang & Zhang 2009). The soil-structure interface consists of the surface of the structure which is in contact with soil in addition to a thin layer of adjacent soil. The behavior of this layer is significantly different from that of the rest of soil. The thickness of soil-structure interface (t) is approximately five times the soil’s average grain size, D50 (Uesugi et al. 1988). A schematic illustration of the soil-structure interface is depicted in Figure 1. In general, the interface strains along the tangential direction (u, in Figure 1). However, its behavior along the normal direction (v, in Figure 1) is also important and often complex so that many of the constitutive models neglect it.

Figure 1. Schematic illustration of soil-structure interface

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In practice, modelers often assume a perfect bond between soil and structure and neglect the relative displacement which is equivalent to assigning the soil behavior to the materials adjacent to the structure (t=0, in Figure 1). Although the interface involves a few general properties of soil, such as dilatancy, it exhibits a significantly different response to loading application from that of the neighboring soil by the constraint effect of nearby structure. Namely, the significant deviation may be induced if a constitutive model of soil is simply extended to a soil-structure interface (Zhang & Zhang 2009). Thus, the behavior of the interface and its accurate description should be investigated using the test and theory approaches. The earliest model proposed for interfaces is a non-linear stress-displacement relationship of the hyperbolic type in both the normal and tangential directions (Clough & Duncan, 1971). In some cases, constitutive models of soils were specialized for the interfaces (Desai & Fishman, 1991). These types of models are commonly used because of their simplicity. However, as mentioned earlier, this approach can cause significant deviation. Some other researchers proposed elastoplastic constitutive models for soil-structure interfaces (Shahrour & Rezaie, 1997; Fakharian & Evgin, 2000; Ghionna & Mortara, 2002; and De Gennaro & Frank, 2002). To be able to use the constitutive models, thin-layer isoparametric elements (Desai et al. 1984) must be used. Beside the constitutive models, special contact algorithms and interface elements were also proposed (Goodman et al., 1968 and Villard, 1996). Despite the fact that the research on soil-structure interfaces started from the early 1970s, this is still an interesting and ongoing research topic from various aspects. For example, Jafari and Toufigh (2017) recently investigated the interface between tire and pavement; Stutz et al. (2016) attempted to enhance a hypoplastic model for granular soil-structure interfaces; A two-surface plasticity model is proposed for partially saturated soils by Lashkari and Torkanlou (2016); Toufigh et al. (2016) investigated the interface between polymer concrete and sand; Xiao et al. (2017) and Yavari et al. (2016) investigated soil-concrete interface subjected to cyclic loading and temperature change. Zhao et al. (2014) conducted research on 4

frozen soil-structure interfaces, and Tiwari et al. (2010) investigated the strain-softening behavior in interfaces. Each of the models proposed earlier has certain advantages and disadvantages. Some of their drawbacks which led to the proposed model are: 1) Some of the models proposed for soil-structure interfaces need additional concepts such as damage or disturbance to be able to capture the softening behavior. 2) Some others are only verified with a certain type of tests such as constant normal stress condition, and therefore their applicability is limited. And 3) The parameter determination procedures are difficult in some cases and require complex tests. In this study, a new approach to plasticity-based constitutive modeling is proposed for a soil-structure interface which is capable of modeling the strain-softening behavior along the tangential direction as well as the dilatancy behavior along the normal direction. The proposed model can predict the results of various interface test devices such as the direct shear device, simple shear device, and cyclic multidegree-of-freedom device, or CYMDOF, proposed by Desai (1980) solely for testing interfaces. In addition, the parameter determination procedures are straight-forward, and it does not require additional concepts.

2. Modeling 2.1. Yield Function For a proper constitutive model for soil-structure interfaces, it is required to formulate an appropriate yield function and to use plastic flow rule. The basis of the proposed constitutive model is the commonly used Cam-Clay plasticity model of soils; this model assumes that the soil is isotropic, elasto-plastic, deforms as a continuum, and is not affected by creep. The yield function (F) of the Cam-Clay model is 5

 J F   p 'M 

2

j

  p0      1  0   p' 

(1)

Where p’ is the mean effective stress; J is the deviatoric stress; p0 is the hardening parameter; and Mj is the material parameter. In the case of two-dimensional interfaces between two bodies, the only stresses acting on an interface element are normal stress (  n ) and shear stress (  ) (See Figure 1). Thus, Eq. (1) can be rewritten as

F   2  M j2 p0 n  M j2 n2  0

(2)

In Eq. (2) as the hardening parameter (p0) approaches to zero, the yield function approaches to the ultimate envelope line (Critical State Line, CSL) with a slope of Mj. Investigating the results from several researchers showed that the peak envelope is generally linear (Figure 2).

Figure 2. Peak envelope using data from various researchers. 6

Although, in a few cases a curved line gives a better approximation. To take into account this curved CSL, a more generalized form of the constitutive model for joints including the curved ultimate line and various rates of approaching to the CSL is given by Desai & Fishman (1991). In this model, the yield function is

F   2   nn   nq  0 Where n, q, and and





a b

(3)

 are phase change and ultimate parameters; q=2 yields the straight ultimate envelope;

is the hardening function

(4)

Where “a” and “b” are the hardening parameters; and  is the accumulative irreversible displacement:

   (du i du i  dv i dv i )1/2

(5)

Where dui and dvi are increments of irreversible tangential and normal displacements of ith increment, respectively. Despite the advantages, the abovementioned models and several other plasticity based models for soilstructure interfaces have, they are incapable of modeling the strain-softening behavior as well as the dilatancy behavior. This is due to the assumption that soil-structure interface cannot experience a stress state above the steady-state envelope. The envelopes of peak and steady-state are illustrated in Figure 3. In the literature of soil-structure interface, there are several proposed criteria and approaches to define limits between rough and smooth interfaces. They are all unanimous in one feature that rough interface will exhibit strain-softening and dilatancy while smooth interfaces do not. In the case of rough soilstructure interfaces where, by definition they exhibit strain-softening, and dilatancy behaviors, the slope 7

of peak envelope is higher than that of the steady-state envelope. In the case of smooth soil-structure interfaces; however, there is no strain-softening or dilatation; the peak and the steady-state envelopes coincide.

Figure 3. Schematic illustration of ultimate envelopes

In this study, the proposed yield function is

F   2  f ( ) nn   nq  0

(6)

Where f ( ) is a new hardening function.

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2.2. Hardening function As given in Eq. (6), the hardening function is a function of accumulative irreversible deformations in the interface. Thus, to obtain this hardening function, the changes in stress state of interface element versus accumulative irreversible deformation was investigated. A schematic illustration of the hardening function is shown in Figure 4. It was observed that initially, the stress state in the interface element is below the steady-state envelope i.e., the hardening function has a positive value. Note, by setting hardening function equal to zero in Eq. (6), the yield function transforms into the equation of the steady-state line. This also means that positive values of hardening function are equivalent to points below the steady state line and negative values are equivalent to points above the steady state line. By increasing shearing deformations in the interface, the interface will behave differently based on its roughness. In the case of rough soil-structure interfaces, the stress state above the steady-state envelope is possible, i.e., with rough interfaces, a negative value is possible for hardening function mostly near peak shear stress. On the other hand, in the case of smooth soil-structure interfaces, the stress state always remains below the steady-state envelope meaning that the hardening function always has a positive value as depicted in Fig. 4. In both cases, the final state of stresses coincides with the steady-state envelope where an increase in shear deformation does not change stress state and hardening function further.

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Figure 4. Schematic illustration of the hardening function versus accumulative irreversible displacement.

A hardening function is required which is capable of capturing the behavior of both smooth interfaces and rough interfaces. The proposed hardening function is

A  3  B  2  C   D f ( )   0  Where

  ss   ss

(7)

ss is the accumulative irreversible displacement corresponding to the steady-state; A, B, C, and

D are interface parameters. This function consists of two parts where the first expression captures the behavior before

ss , and the latter will cover steady state where the hardening function coincide with

the steady state line. In other words, the hardening function is somehow proportional to the difference between the state of stresses and the steady state. 10

The use of associated plasticity results in excessive and unreasonable dilatancy. Therefore, the nonassociated plasticity is employed in this model where the plastic potential function (Q) is

Q ( )  C1  f (C2  )  nn

(8)

Where C1 and C2 are the coefficients of dilatancy reduction. After obtaining the yield and plastic potential functions, the constitutive matrix can be obtained using the plastic rule T

 Q  F  e C  C       e C  T  Q  e  F   F    Q    C                   e

C ep

 Q   

1/2

   

(9)

where C ep is the elastoplastic constitutive matrix; and C e is the elastic constitutive matrix

K Ce   n  0

0  K s 

(10)

where Kn and Ks are the normal and tangential stiffnesses, respectively. In this study, it is assumed that elastic shear and normal responses are uncoupled. Thus, the off-diagonal elements of the elastic constitutive matrix are zero. For two-dimensional soil-structure interfaces, elastoplastic incremental equation is

d  n  ep  d   C  

du  dv   

(11)

For a detailed derivation of Cep in Eq. (11), see Appendix A.

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2.3. Modification for cyclic loading The behavior of soil-structure interfaces under cyclic loading is subject to change due to the potential soil particles crushing and rearrangements after each cycle. Based on the analysis of the experimental data from Shahrour and Rezaie (1997), the effect of cyclic loading on the behavior of soil-structure interfaces depends upon loading conditions; generally categorized as 1) Constant normal stress and 2) Constant normal stiffness. For the case of constant normal stress, by increasing number of cycles, the amount of normal displacement tends to decrease exponentially (Figure 5(a)) while the ultimate shear strength is relatively constant.

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Figure 5. Variation of the behavior under cyclic loading based on the experimental results from Shahrour & Rezaie (1997) at a) constant normal stress and b) constant normal stiffness.

To take into account the effect of cyclic loading in the proposed model, following modification is required for the case of constant normal stiffness:

 F   F     exp[   (N 1)]        cyclic   monotonic where N is the number of cycles;



(12)

and  are the cyclic loading parameters.

On the other hand, in the case of constant normal stiffness, by increasing number of cycles, the ultimate shear strength tends to decrease exponentially (Figure 5(b)) while the normal displacement does not change. For the case of constant normal stress, the required modification is:

Q cyclic    exp[  (N 1)]  Q monotonic where N is the number of cycles;



(13)

and  are the cyclic loading parameters.

3. Determination of model parameters 3.1. Kn and Ks (elasticity parameters) The values of shear stiffness, Ks, and normal stiffness, Kn, are equal to the average unloading slopes of shear stress versus tangential displacement and normal stress versus normal displacement curves, respectively. However, in case the data of cyclic testing are not available, the initial slopes can also be used.

13

3.2. n, q, and  (plasticity parameters) The value of

 is equal to the slope of the ultimate envelope shown in Figure 3. For the linear ultimate

envelope q=2, while for the curved ultimate envelope q can be determined using

ultimate   nq /2

(14)

The phase parameter, n, is based on the stress state at a transition point where the normal displacement is zero. By substitution of f ( ) from Eq. (6) in the expression F

/  n  0 , “n” can be

obtained using

n

q 2 1 q  n

where  and

(15)



are stresses at the transition point.

3.3. A, B, C, and D (hardening parameters) The results of three direct shear tests from Al-Younis (2013) were analyzed to observe the variation of  with displacement. These variations are shown in Figure 6. Labels in this figure (SD5, SD8, SD11, ND1, ND2, and ND3) are the original sample names in Al-Younis (2013) on which the analysis is done.

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Figure 6. Variations of  with displacement Based on this observation a typical variation of accumulative irreversible displacement with tangential displacement is shown in Figure 7.

Figure 7. Typical variation of accumulative irreversible displacement with tangential displacement As can be seen, there is an initial accumulative irreversible displacement (  0 ) due to the application of initial normal stress. At first, the slope of this curve is close to zero, and it increases up to a certain value corresponding to  peak where after that total tangential displacement will be irreversible displacement. 15

C is a representative of the state in which the irreversible displacements begin. ss

is the accumulative

irreversible displacement corresponding to the steady-state. Its approximate value is equal to 1 and 1.5 times of

 peak

for the rough and smooth interfaces, respectively.

The parameters A, B, C, and D can be obtained by solving the following linear equations:

f (  0 )  A 03  B 02  C 0  D  F0

(16-a)

f (  C )  A C3  B C2  C C  D  Fb

(16-b)

f (  ss )  A ss3  B ss2  C ss  D  0

(16-c)

f (  ss )  3A ss2  2B ss  C  0 

(16-d)

Where the value of Fb for smooth interfaces is negative while its value for rough interfaces is positive. In case that q=n=2, Eq. (6) yields:

    f ( ) |  n |

(17-a)

which indicates that f ( )  

(17-b)

therefore

F0  max(f ( ))  

(17-c)

In other cases, F0 can be determined using Eq. (6) by substituting the values of  , corresponding to

0 .

Moreover, Fb is proportional to the value of F0 as:

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n ,

and



Fb 

|  (  C ) |  |  (  ss ) |  F0 |  (  ss ) |

(16)

3.4. A, B, C, and D (different approach) The four interface parameters (parameters of the hardening function) are coefficients of a third order polynomial and thus can be obtained from different approaches. One of them is based on values and slopes of the function at two given points which is described in Section 3.3. Another one is to use values of the function at four points which will be discussed herein. The value of  can be computed at every point on the experimental response curves (shear stress versus tangential displacement and normal stress versus normal displacement) by subtracting the reversible displacements from total displacements at every increment. Since F=0, solving

A 3  B  2  C   D 

 nq   2  nn

(18)

Simultaneously for four arbitrarily chosen points yields directly the value of parameters A, B, C, and D. Because the selection of points is arbitrary in this approach, care must be given to obtain parameters such that the resultant hardening function fits the experimental data to a best possible extent. The former approach has a better physical definition of parameters and is less likely to diverge from the experimental data. Thus, generally, the former approach is recommended.

3.5. C1 and C2 (coefficients of dilatancy reduction) To reduce the excessive dilatancy, these coefficients are required. The coefficient C1 compensates for the error in the locus of dilatancy occurrence on normal displacement versus tangential displacement curve and can be obtained through: 17

u  0)  v C1  v (   peak ) v(

(19)

On the other hand, the coefficient C2 compensates for the error in the values of normal displacement and can be obtained using:

C2 

3.6.

u (   peak )

(20)

v (   peak )



and  (cyclic loading parameters)

The cyclic loading parameters can be determined using the analysis of the ultimate shear stress versus number of cycles (Figure 5(b)) or the normal displacement versus number of cycles (Figure 5(a)) curves. For this purpose, based on the loading condition (constant normal stress or constant normal stiffness) solving one of the following equations for two arbitrary cycles yields these parameters:

   i   exp[  (N1)]

(21-1)

v  v i    exp[  (N 1)]

(21-2)

where  i and

v i are

the initial values of the ultimate shear stress and the normal displacement,

respectively.

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4. Results Using the proposed constitutive model, results of soil-structure interface tests from several researchers were modeled. To assess the generality of the proposed model, these experimental results were chosen so that they are from different testing devices including direct shear, simple shear, and cyclic multidegree of freedom (CYMDOF) and different types of materials are involved. The test conditions also covered constant normal stress and constant normal stiffness. The parameters used in subsequent sections are given in Table 1. Table 1. Model parameters used in verifications Section(s) Elasticity parameters

Plasticity parameters

4.1.1 & 4.1.4

4.1.2

4.1.3 & 4.1.5

4.2.1

4.2.2

4.1.6

Ks

0.15 n  118

2 n  150

250

250

230

150

Kn

500

100

1000

1000

850

500

n

3.45

3.95

3.65

3.65

3.70

2.97

q

2

2

2

2

2

2



0.319

0.119

0.218

0.201

0.257

0.30

b

0.00089

0.0009

0.00015

0.00002

0.00002

0.00089

b

Fb

-0.00006

0.0014

0.00013

0.000045

0.000028

-0.0003

C

4

2

2

2

2

2

ss

10.0

4.5

5.0

5.0

5.0

4.0

C1

0.0211

0.071

0.069

0.070

0.073

0.181

C2

1.0

1.5

1.0

0.7

1.0

1.0



1.0

-

10.76

-

-

1.0

F0

a

hardening function’s parameters

Dilatancy reduction parameters Cyclic loading

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parameters a

Using

b

these



0.0 parameters

the

values

0.1 of

A,

B,

C,

and

D

could

also

0.015 be

obtained.

The values of F0 and Fb were a function of normal stress to be able to capture the softening-behavior at

higher normal stresses. However, for brevity in the table, only the initial values were reported.

Verification of model using results from constant normal stress condition is done under section 4.1. The constant normal stiffness condition is covered under section 4.2. Sections 4.1.1, 4.1.2, 4.1.3, 4.2.1, and 4.2.2 are based on monotonic loading while sections 4.1.4, 4.1.5, and 4.1.6 verify cyclic loading tests.

4.1. Constant normal stress condition 4.1.1. Gravelly sand & FRP - Monotonic CYMDOF The soil-structure interface of this section is obtained from Toufigh et al. (2013) which is between Tanque Verde gravelly sand and fiber-reinforced polymer. The tests were conducted using a cyclic multidegree of freedom device at constant normal stress condition. Nine initial normal stresses are given in Toufigh et al. (2013). However, only two of them (210 and 875 kPa) are modeled in this study because the normal behavior was not available for tests with other initial normal stresses. The results are shown in Figure 8. In this case, the model could successfully capture the behavior along both tangential and normal directions.

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Figure 8. Comparison of model predictions and experimental results from Toufigh et al. (2013) for interface between sand and FRP

4.1.2. Sand & Steel - Monotonic direct shear The soil-structure interface of this section is between Fontainebleau sand and rough metal plate. The tests were performed using a direct shear device at constant normal stress condition by De Gennaro (1999). Three tests with different initial normal stresses (25, 50, and 100 kPa) are modeled in this case, the results of which are shown in Figure 9. Along tangential direction, model yielded a more accurate result for higher normal stress; there are differences in low normal stresses especially in strainsoftening. Along the normal direction, the model is showing dilatancy at lower tangential displacements compared to the experimental data, but it is accurate on the ultimate values.

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Figure 9. Comparison of model predictions and experimental results from De Gennaro (1999) for interface between sand and rough steel

4.1.3. Sand & Steel (glued) - Monotonic direct shear The soil-structure interfaces of this section obtained from Shahrour and Rezaie (1997) are between Houston dense and lose sands glued to steel block. The tests were performed using a modified direct shear device at constant normal stress condition. Two sets of three tests with different initial normal stresses (100, 200, and 300 kPa) were modeled in this study, the results of which are shown in Figures 10 and 11. Along the tangential direction results are accurate to a satisfactory level; yet along the normal direction, results are not very satisfactory for higher normal stresses.

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Figure 10. Comparison of model predictions and experimental results from Shahrour and Rezaie (1997) for dense sand glued to steel block

Figure 11. Comparison of model predictions and experimental results from Shahrour and Rezaie (1997) for loose sand glued to polished steel block

4.1.4. Gravelly sand & FRP - Cyclic CYMDOF Details of this soil-structure interface are given in section 4.1.1. The tangential behavior of the interface with the initial normal stress of 350 kPa is modeled in this section, the results of which are shown in

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Figure 12. Normal behavior was not considered since the experimental data was not available. Performance of model is very promising in that it captures loading and reloading path very accurately. Nevertheless, there are discrepancies in reloading path.

Figure 12. Comparison of model predictions and experimental results from Toufigh et al. (2013) for interface between sand and FRP under cyclic loading

4.1.5. Sand & Sand - Cyclic direct shear The behavior of interface between the smooth surface and loose sand under cyclic loading at constant normal stress was predicted using the proposed model. Experimental results of this section are obtained from Shahrour and Rezaie (1997). As can be seen from the results in Fig. 13, model was successful in capturing the values of shear stress and normal displacement for cyclic loading.

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Figure 13. The results of (a) the model, and (b) the experiments from Shahrour and Rezaie (1997)

4.1.6. Steel & Gravel - Cyclic Large-scale test apparatus For a better verification, another sample of cyclic loading is obtained from Zhang and Zhang (2008). The soil-structure interfaces of this section were between steel and gravel. The tests were conducted at constant normal stress condition using a large-scale test apparatus developed by authors. The tangential and normal behavior of cyclic test with the initial normal stress of 700 kPa is modeled in this section;

25

results of which are shown in Figure 14. The proposed model was also successful in capturing the values of shear stress and normal displacement in this case.

Figure 14. The results of (a) the model, and (b) the experiments from Zhang and Zhang (2008)

26

4.2. Constant normal stiffness condition 4.2.1 Sand & Steel - Monotonic simple shear The soil-structure interfaces of this section were sand-steel interfaces. The tests were conducted by Evgin and Fakharian (1997) using a modified simple shear device named C3DSSI at constant normal stiffness condition. Three tests with different initial normal stresses (100, 200, and 300 kPa) are modeled in this section; results of which are shown in Figure 15. The model captured variations of normal stress very accurately.

Figure 15. Comparison of model predictions and experimental results from Evgin and Fakharian (1997) for the steel-sand interface.

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4.2.2. Sand & Structure - Monotonic direct shear Another set of constant normal stress tests were obtained from Ghionna and Morata (2002). The tests of this investigation were conducted using the direct shear device at constant normal stiffness condition. Three tests with different initial normal stresses (100, 200, and 300 kPa) are modeled in this study; results of which are shown in Figure 16. The model predicted variations of normal stress accurately.

Figure 16. Comparison of model predictions and experimental results from Ghionna and Morata (2002).

5. Summary and Conclusion In this study, a new approach of the plasticity-based constitutive model was proposed for the behavior of soil-structure interfaces under monotonic and cyclic loading. The proposed model does not require 28

additional concepts such as damage or disturbance, and the model parameters can be obtained easily using constant normal stress tests. Moreover, results of the proposed model showed its capability in predicting the experimental results of various interface test devices (direct shear, simple shear, and CYMDOF) at different test conditions (constant normal stress and constant normal stiffness) and loading conditions (monotonic and cyclic). The results showed that 1. The model captures variations of normal stress for tests at constant normal stiffness very accurately, 2. The model’s agreement with experimental data for cyclic loading is excellent in terms of the shape and the area under shear stress versus shear displacement diagram as well as the amount of decay in normal displacement after each cycle; however, it has discrepancies on the reloading path, 3. And in case of monotonic loading, the proposed model is capable of capturing the strainsoftening and dilatancy behaviors to a satisfactory level.

Funding This research did not receive any specific grant from funding agencies in the public, commercial, or notfor-profit sectors.

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Appendix A Using the equations presented in the manuscript, the final form of Eq. (11) is derived here in this section with details. For the case of non-associated plasticity where F  Q :

dF  n nn 1 (A  3  B  2  C   D )   q  nq 1    d  2 

(A1-a)

dQ C 1n nn 1 (AC32  3  BC22  2  CC2   D )    d  0 

(A1-b)

dF   nn 1 (3A  2  2B   C ) d

(A1-c)

and consequently the final form of Cep matrix is

 C 1K n2 n  q nn 1 nq 1   nn 1n (f ( ))(f (C2  )) K   n C ep   Fdenom  0

2C 1K n K s n nn 1 f (C2  )   Fdenom   Ks

(A2)

where

Fdenom   nn 1 (3A  2  2B   C ) C12 n 2 n2( n 1) f (C2  )0.5 (A3)

 K n  nn 1C 1n ( q  nq 1  n nn 1f ( )f (C2  )) Thus,

 C K 2 n  q nn 1 nq 1   nn 1n (f ( ))(f (C2  )) d  n   K n  1 n Fdenom  d       0 

2C 1K n K s n nn 1 f (C2  )   du  Fdenom  dv     Ks

On the other hand, for the case of associated plasticity where F  Q :

dF dQ  n nn 1 (A  3  B  2  C   D )   q  nq 1     d d  2 

32

(A4)

while dF/ d  is the same as Eq. (A1-c), the final form of Cep matrix is

C ep

 C 12 K n2 n 2 q n2( n 1) f (C 2 )1/2 K   n Fdenom  0 

 0   K s 

(A5)

where

Fdenom   nn 1 (3A  2  2B   C ) C12 n 2 n2( n 1) f (C2  )0.5 (A6)

 K n n2( n 1)C 12 n 2 f (C2  )0.5 Thus,

 C 2 K 2 n 2 q n2( n 1) f (C 2 )1/2 d  n   K n  1 n Fdenom  d       0 

33

 0  du   dv    K s 