A constitutive model for unsaturated cohesionless soil

Unsaturated Soils: Theory and Practice 2011 Jotisankasa, Sawangsuriya, Soralump and Mairaing (Editors) Kasetsart University, Thailand, ISBN 978-616-7522-77-7

A constitutive model for unsaturated cohesionless soil-structure interfaces A constitutive model for unsaturated cohesionless soil-structure interfaces A. Lashkari

Department of Civil & Environmental Engineering, Shiraz University of Technology, Shiraz, Iran, [email protected]

A. Lashkari

Department of Civil & Environmental Engineering, Shiraz University of Technology, Shiraz, Iran, [email protected]

ABSTRACT: An elasto-plastic constitutive model for cohesionless soil-structure interfaces is introduced here. To consider the coupled influence of density, suction, and net normal stress on the behavior of unsaturated interfaces, the model is formulated within the critical state compatible SANISAND platform. In general, the model formulation is inspired by the model of Chiu & Ng (2003) for unsaturated granular soils. For saturated case where suction is zero, the presented formulation reduces to the state dependent interface model of Lashkari (2010, 2011). It is shown that the model predictions are in good agreement with experimental data. KEYWORDS: suction; constitutive equation; critical state; interface; sand; silt; SANISAND 1 INTRODUCTION The mechanical behavior of soil-structure interfaces is an influential factor in the load-deformation and the bearing capacity of geostructures. Investigations have revealed that the mechanical behavior of interfaces is affected by a number of factors including density, normal stress, surface roughness of structure and normal stiffness as well as mineralogy and soil gradation (e.g., Uesugi & Kishida, 1986; Evgin & Fakharian, 1996; Shahrour & Rezaie, 1997; Ghionna & Mortara, 2002; Hu & Pu, 2004). Along with the massive expansion of the knowledge on the behavior of interfaces, remarkable advances have been achieved in constitutive modeling of interfaces. Among the recent works, Shahrour & Rezaie (1997) proposed a Bounding Surface plasticity model for interfaces subjected to cyclic shear. Ghionna & Mortara (2002) suggested an elasto-plastic interface model with two Cam Clay type constitutive surfaces. Liu et al. (2006) and Lashkari (2010, 2011) considered state dependency of interfaces in Generalized Plasticity and bounding surface frameworks. In arid or semi-arid regions, unsaturated interfaces are common in engineering problems such as retaining structures and reinforced soils with unsaturated backfill and piles embedded in partially saturated soils. Studies have indicated that matric suction due to partial saturation may drastically affect the engineering behavior of soils (e.g., Fredlund & Rahardjo, 1993). As a result, experimental investigation of the mechanical behavior of unsaturated interfaces and subsequent development of constitutive models for unsaturated interfaces appears inevitable. Miller & Hamid (2007) studied the behavior of unsaturated in-

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terfaces using modified direct shear apparatus. Recently, Hamid & Miller (2008) introduced modifications to the platform of Navayogarajah et al. (1992) and suggested a constitutive model for unsaturated interfaces. Manzari & Dafalias (1997) introduced a plasticity model, so-called SANISAND model, originally for the state dependent behavior of cohesionless soils. Since then, this platform has been developed and applied to various problems. Among them, Li (2002) and Taiebat et al. (2007) considered state dependent cyclic shearing and particle crushing. Chiu & Ng (2003) modified the SANISAND platform to include unsaturated soils. Recently, Lashkari (2010, 2011) proposed a SANISAND interface model. Herein, considering the work of Chiu & Ng (2003), formulation of a SANISAND interface model for unsaturated cohesionless interfaces is presented. The predictive capacity of the interface model is verified by direct comparison of the model predictions with experimental data on dry and unsaturated interfaces. 2 DEFINITION OF BASIC TERMS In this study, n, , ua, uw, and e are respectively introduced as normal stress, tangential stress, pore air pressure, pore water pressure and void ratio. The Two Independent Stress Variables theory is adopted here (e.g., Fredlund & Morgenstern, 1977; Fredlund & Rahardjo, 1993). Four state variables are used in formulation of the proposed interface model:



net

 u n a



(1)

s u u a w v  1 e

where, net, s, and v are net normal stress, matric suction, and specific volume respectively. The inclusion of specific volume in conjunction with the critical state soil mechanics concepts enables the interface model to describe the volume change behavior of interfaces over wide ranges of density, suction, and net normal stress. H and V are respectively introduced as tangential and normal relative displacements measured with respect to the interface plane. Assuming that the strains are uniformly distributed within the interface zone, the average normal (n) and tangential (t) strains are calculated by:

where K ne and K te are the normal and tangential elastic stiffness of the interface under consideration: v(  net  sH ( S r ))  e e Kt   Kn K ne 

(5)

where  is a scalar parameter and  is the slope of the unloading-reloading line in the -lnn plane. It is assumed that  is not affected by variation of suction. The function H(Sr) considers the influence of degree of saturation on the elastic response: H ( Sr ) 

Sr  Sr 0 1  Sr 0

(6)

are Macualey brackets. For a typical scalar where x, x  x when x  0 , and 0 otherwise. Sr0 is the degree of saturation corresponding to residual suction in Soil-Water Characteristic Curve.

(2)

n  V / t ; t  H / t

where t is the interface thickness. Experimental studies have indicated that the interface thickness is 5 to 10 times of mean grains diameter (e.g., Boulon, 1989; Hu & Pu, 2004).

3.3 Definition of yield functions Similar to Li (2002) and Chiu & Ng (2003), a wedge shaped yield function for the shearing mechanism is selected:  ( s )  0 f s      net  M ( s )  

3 THE MODEL FORMULATION 3.1 Decomposition of strain rates In the elasto-plasticity theory, each strain rate is decomposed into the elastic and plastic portions. It is assumed that the plastic strains are due to shearing and compression mechanisms (e.g., Alonso et al., 1990; Wheeler, 1996; Chiu & Ng, 2003; Thu et al., 2007). Herein, strain rates are decomposed as: n  ne  np  ne  nsp  ncp

(3)

t  te  tp  te  tsp  tcp

(7)

where M(s) and (s) are respectively the gradient and the intercept of the yield function in -net plane (see Fig. 1). In Equation (7), , so-called stress ratio, is a hardening parameter which indicates the shearing yield function opening. The second yield function, a vertical cut-off, is defined as the plastic mechanism due to compression which may be activated as a result of the increase in net normal stress, reduction of suction or both: f c   net ( s )   0 ( s )  0

(8)

where superscripts “e” and “p” stand respectively for the elastic and plastic parts. In addition, subscripts “s” and “c” indicate the plastic strain rate due to shearing and compression yielding mechanisms.

where 0(s) is the yielding net stress at suction s (see Fig. 1). Moreover, the second yield function must also be introduced in v-net plane. According to Alonso et al. (1990), a family of isotropic normal compression lines is defined by:

3.2 Elasticity theory for unsaturated interfaces

 ( s )  v  N ( s )  ( s ) ln net  patm 

Different approaches exist in the literature for simulation of the elastic response of unsaturated soils (e.g., Alonso et al., 1990; Chiu & Ng, 2003; Sun et al., 2008). Analogous to Chiu & Ng (2003), the following constitutive laws are used for the elastic response of unsaturated soil-structure interfaces: ne  te 

d (  net  s H ( Sr ))  K ne  K te

H ( S r )  Sr ) S r  s H ( Sr ))

 (  net  H ( S r )s  s v(  net

(4)

(9)

where (s) and N(s) are the gradient and intercept of normal compression line in v-net(s) plane. patm is the atmosphere pressure (i.e. 101 kPa) which plays the role of a reference pressure. The application of Equations (4), (6) and (9) with some ordinary mathematical operations leads to the following relationship for load-collapse curve which is the projection of the compression mechanism on the net-s plane (see Chiu & Ng, 2003):

314



 p  atm 

( 0 )    ln  0 ( 0 )     ( s )  s H ( Sr  ( s )     0 N ( 0 )  N ( s )   ( s ) ln  0 patm   patm 

)  

(10)

 Critical State line M(s)  1 (s)

fc yield function

fs yield function

- (s)/M(s)

0(s)

net

3.4 Calculation of plastic strain rates According to the elasto-plasticity theory, plastic strains are calculated by: qps   s vps  qps d s

In above equations, s and ds are respectively the loading index and dilatancy function associated with the shear plastic mechanism. In the same manner, c and dc are related to the compression plastic mechanism. The extended Li (2002) dilatancy rules suggested by Chiu & Ng (2003) are adopted here:

M( s )  p d c  vcp  ( s )   d 2 ( s )    qc

(12)

1 K ps

 f s f f   net  s   s  s   net

c 

1 K pc

 f c f   net  c s   net

 s  

 s  

(15)

 M ( s )e  n ( s )  K ps  h0 K te   1    M( s ) K pc  v 0 ( s )d 2 ( s ) 

(16)

where h0 is a model parameter.

For calculation of the variations of the water content and the degree of saturation, specific water volume or air void ratio must be participated in constitutive modeling of unsaturated soils (e.g., Wheeler, 1996). In this study, definition of Wheeler (1996) for air void ratio is adopted:  (0 )  ea  v  v w  A( s )   ( s ) ln 0  patm 

(17)

where ea is the air void ratio and vw(=1+Sr e) is the specific water volume. A(s) and (s) are parameters and 0(0) is the value of 0(s) at zero suction. 3.7 Suction dependent parameters

In above equations, m, d1(s) and d2(s) are the model parameters. (s) is the state parameter of Been & Jefferies (1985) which is extended by Chiu & Ng (2003) to include unsaturated granular soils:    ( s )     ( s )  v  vc ( s )  v  ( s )  ( s ) ln 0   patm  

s 

3.6 Inclusion of specific water volume (11)

vpc  qpc d c

  p    d s  vsp  d1( s ) e m ( s )   M ( s )   qs 

3.5 Calculation of loading indices In the elasto-plasticity theory, each loading index is calculated by imposing the consistency condition to its corresponding yield function as follows:

where Kps and Kpc are respectively the plastic hardening moduli associated with the shear and compression yielding mechanisms:

Figure 1. The model yield functions (after Chiu & Ng, 2003).

qpc   c

  patm   patm    A1  A0 e n ( s )  (14)   0( s )   0 ( s )  M ( s ) 

where n, A0 and A1 are model parameters.

where N(0) and (0) are the values of N(s) and (s) at zero suction.

1

d1( s )  A0

(13)

According to Lashkari (2010), the following relationship for d1(s) is selected to enables equation (12) for proper simulation of the strong contraction observed in the volume change behavior of soilstructure interfaces immediately after shearing:

315

A number of the model parameters such as M(s), (s), (s), N(s), (s), (s), A(s), (s), d1(s) and d2(s) are functions of suction. It is suitable to introduce some interpolate functions to calculate the values of these parameters for any given suction value in given domains of applicability. Among different suggestions in the literature, the following relationship is selected (e.g., Alonso et al., 1990):





Q( s )  Q( 0 ) 1  rQ e   s  rQ



(18)

where Q(s) is an arbitrary suction dependent parameter and Q(0),  and rQ are parameters. Q(0) is the value of Q(s) at zero suction. For very large values of suction, Q(s) tends to rQQ(0).

Tangential Stress, t (kPa)

4 THE MODEL VERIFICATION The interface model has 18 parameters which can be determined using the straightforward methods introduced in Lashkari (2010, 2011) and Chiu & Ng (2003). It is interesting to note that in some cases depending on the saturation (dry, partially saturated, or saturated) condition and imposed stress path, the model may be applied to simulation of the mechanical behavior of interfaces without the need to complete calibration of the parameters. In the following sub-sections, the model is first evaluated against tests preformed on dry interfaces. Consequently, the model is verified with direct comparisons versus unsaturated interface tests. 4.1 The model verification using dry interface tests Hu & Pu (2004) studied the behavior of dry Yongdinghe sand-steel interfaces using a modified direct shear apparatus. Samples were air-pluviated with the initial relative density Dr≈90% (v≈1.691), and a rough (Rmax=0.5 mm) thick low-carbon steel plate was use as the structure. The steel plate and sand interface was 60 mm in length and 53 mm in width. Due to the larger dimensions of the steel plate, the area of interface remained constant during shearing. The sand sample was placed in a transparent Plexiglas shear box with inside graduated grids to facilitate visual observation of particles displacements near the interface. The interface model is calibrated against this set of experiments and the model parameters are given in Table 1. Considering that the interfaces are dry, the model can be operated with the less number of parameters. Table 1. Values of the model parameters in simulations shown through Figures 2-6. ______________________________________________ Parameter Hu & Pu (2004) Hamid & Miller (2007) ______________________________________________  0.032 0.10  1.0 0.25 Sr0 0.04 M(s) 0.61 0.609+0.0013s (s) 0 (0) 2.033 0.87 r 2.0  0.039 (s) 0.033 0.165 A0 1.5 0.55 A1 0.40 0.40 h0 0.20 0.20 m(s) 0.50 0.50 n(s) 2.10 2.0 _____________________________________________

(a)

400 300

n = 400 kPa 200

n = 200 kPa

100

n = 100 kPa

0 0

3

6

9

12

Normal Displacement, V (mm)

Horizontal Displacement, H (mm) 1

n = 100 kPa

0.8

n = 400 kPa

0.6 0.4

(b)

0.2 0 -0.2

0

3

6

9

12

Horizontal Displacement, H (mm)

Figure 2. Comparisons of the model prediction with three constant normal stress tests on dry Yongdinghe sand-steel interfaces (data reported by Hu & Pu, 2004).

For three constant normal stress tests, comparisons of the model predictions with the experimental data of Hu & Pu (2004) are presented in Fig. 2. In all simulation, the interface thickness is assumed 5 mm. It is observed that the model simulated the peak and the post-peak response of all interfaces well. In addition, the state dependent ingredients of the model enable it to provide reasonable predictions using a unit set of parameters. 4.2 The model verification using unsaturated interface tests Miller & Hamid (2007) developed a modified direct shear device to investigate the mechanical behavior of unsaturated interfaces. They conducted a series of tests on unsaturated Minco silt-steel interfaces under suction values of 20, 50 and 100 kPa and 105, 140, and 210 kPa of net normal stresses. In their study, Miller & Hamid (2007) used a commercially available direct shear device to perform tests. It was an automated device that uses feedback from vertical and horizontal load cells and displacement transducers to provide real-time control of vertical and horizontal loading. The shear box had a circular cross section and accommodated a soil sample that was 30 mm thick and 63 mm in diameter. Pore water pressure was controlled using a computer controlled stepper motor pump capable of maintaining pressure with ±1 kPa and detecting volume change ±1 mm3. Air pressure in the air chamber was controlled using a regular and burdon tube pressure gage with a resolution of

316

0.7 kPa. Identical samples were prepared with initial specific volume and degree of saturation v≈1.66 and Sr≈0.83 respectively. Then, they were subjected to normal stress until the values of net normal stresses reach the target values described above. Prior to shearing, each sample passed a so-called equalization phase in which pore air and pore water pressures were reached the desired values. During the equalization phase, the specific volume

200

(a)



160

of samples changed due to water movement. Using data of Miller & Hamid (2007), specific volume and degree of saturation after equalization phase are estimated v≈1.466 and Sr≈0.86. Finally, the samples were subjected to shear stress. During the shearing phase, the values of net normal stress and suction were kept unchanged. The proposed interface model is calibrated for simulation of this set of tests. Values of the model parameters are given in Table 1 and

120

80

40

0

(a)

150

snet = 210 kPa 100

snet = 105 kPa

50

0 0

2

4

6

8

0

Tangential Displacement, H (mm)



0.2

(b)

0.1 0.05 0 2

4

6

8

-0.05 -0.1

snet = 105 kPa

8

(b)

snet = 210 kPa

0.1 0.05 0 0

2

4

6

8

-0.05 -0.1



Figure 3. The model prediction versus an unsaturated Minco silt-steel interface test with net=140 kPa and s=50 kPa (data reported by Miller & Hamid, 2007). 160

Figure 5. The model prediction versus two unsaturated Minco silt-steel interface tests with s=100 kPa (data reported by Miller & Hamid, 2007). 160

(a)



6

0.15

-0.15

120

80

40

(a)

120

80

40

0

0 0

2

4

6

0

8

0.2


0.15 0.1 0.05 0 2

4

4

6

0.15

(b)

0

2

8




4


0.15

0

2

6

8

-0.05

(b)

0.1 0.05 0 0

2

4

6

8

-0.05 -0.1 -0.15

-0.1



Figure 4. The model prediction versus an unsaturated Minco silt-steel interface test with net=140 kPa and s=100 kPa (data reported by Miller & Hamid, 2007).

317

Figure 6. The model prediction versus an unsaturated loose Minco silt-steel interface test with net=105 kPa and s=20 kPa (data reported by Miller & Hamid, 2007).

the interface thickness is taken 1 mm. For a test with net=140kPa and s=50 kPa, simulations are compared with the experimental data in Fig. 3. For another test with the same value of normal stress and s=100 kPa, similar comparisons are presented in Fig. 4. Considering Figures (3) and (4), it is observed that the increase in matric suction leads to the increase in the peak tangential strength and dilatancy. In Fig. 5, comparisons of the predictions and experimental data of two samples with s= 100 kPa, but different values of net normal stress (net=105kPa and 210 kPa) are illustrated. As expected, the increase in net normal stress leads to the decrease in dilatancy. All the interfaces shown in Figs. 3-5 are denser than critical. Hence, we observe dilation in their volume change behavior. For a sample looser than critical, predictions are drawn against experiments in Fig. 6. In general, the volume change response of this interface is contractive and peak does not take place in strength. It is observed that the model predictive capacity for samples in loose state is acceptable. It is worth mentioning that all simulations illustrated through Figs. 3-6 are obtained by the use of a unique set of parameters. 5 CONCLUSIONS A constitutive model for unsaturated cohesionless soil-structure interfaces was introduced in this paper. In order to consider the combined influence of density, matric suction, and applied net normal stress, the model was formulated within the SANISAND platform which is an elasto-plastic critical state compatible constitutive model. The complete formulation of the model was presented. The interface model predictions were directly compared with the experimental data on dry and unsaturated interfaces. Comparing with the experiments, reasonable predictions were obtained using a unique set of parameters for each interface type. REFERENCES Alonso, E.E., Gens, A., and Josa, A. (1990). A constitutive model for partially unsaturated soils. Géotechnique, Vol. 40, No. 3: 405-430. Been, K., and Jefferies, M. G. (1985). A state parameter for sands. Géotechnique, Vol. 35, No. 2: 99-112. Boulon, M. (1989). Basic features of soil structure interface behavior. Computers and Geotechnics, Vol. 7: 115-131. Chiu, C.F., and NG, C.W.W. (2003). A state-dependent elasto-plastic model for saturated and unsaturated soils. Géotechnique, Vol. 53, No. 9: 809-829. Evgin, E., and Fakharian, K. (1996). Effect of stress path on the behavior of sand-steel interface. Canadian Geotechnical Journal, Vol. 33: 853-865. Fredlund , D.G., and Morgenstren, N.R. (1977). Stress state variables for unsaturated soils. ASCE Journal of Geotechnical Engineering, Vol. 103, No. GT5, 447-466. Fredlund, D.G., and Rahardjo, H. (1993). Soil mechanics for unsaturated soils. John Wiley, USA.

Ghionna, V. N., and Mortara, G. (2002). An elastoplastic model for sand-structure interface behavior. Géotechnique, Vol. 52, No. 1: 41-50. Hamid, M.T., and Miller, G.A. (2008). A constitutive model for unsaturated interfaces. International Journal for Numerical and Analytical Methods in Geomechanics. Vol. 32, No. 13: 1693-1714. Hu, L., and Pu, J. (2004). Testing and modeling of soilstructure interface. ASCE Journal of Geotechnical and Geoenvironmental, Vol. 130, No. 8: 851-860. Lashkari, A. (2010). A state dependent model for sandstructure interfaces. Proc. 7th European Conference on Numerical Methods in Geotechnical Engineering. Benz & Nordal (eds.), Trondheim, Norway: 2-4 June, 9-14. Lashkari, A. (2011). A SANISAND-Structure interface model. Iranian Journal of Science & Technology (IJST), Transaction of Civil & Environmental Engineering, 35(C1), 15-34. Li, X.S. (2002). A sand model with state-dependent dilatancy. Géotechnique, Vol. 52, No. 3: 173-186. Liu, H., Song, E., and Ling, H.I. (2006). Constitutive modeling of soil-structure interface through the concept of critical state soil mechanics. Mechanics Research Communications, Vol. 33: 515-531. Manzari, M.T., and Dafalias, Y.F. (1997). A critical state two surface plasticity model for sands. Géotechnique, Vol. 47, No. 2: 255-272. Miller, G.A., and Hamid, T.B. (2007). Interface direct shear testing of unsaturated soil. Geotechnical Testing Journal, Vol. 30, No. 3: 182-191. Navayogarajah, N., Desai, C.S., Kiousis, P.D. (1992). Hierarchical single surface model for static and cyclic behavior of interfaces. ASCE Journal of Engineering Mechanics, Vol. 118, No. 5: 990-1011. Shahrour, I., and Rezaie, F. (1997). An elastoplastic constitutive relation for the soil-structure interface under cyclic loading. Computers and Geotechnics, Vol. 21: 21-39. Sun, D.A., Sheng, D., Xiang, L., and Sloan, S.W. (2008). Elastoplastic prediction of hydro-mechanical behavior of unsaturated soils under undrained conditions. Computers and Geotechnics, Vol. 35: 845-852. Taiebat, M., and Dafalias, Y. F. (2008). SANISAND: Simple anisotropic sand plasticity model. International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 32, No. 8: 915-948. Thu, T.M., Rahardjo, H., and Leong, E-C. (2007). Elastoplastic model for unsaturated soil with incorporation of the soil-water characteristic curve. Canadian Geotechnical journal, Vol. 44: 67-77. Uesugi, M. and Kishida, H. (1986). Influential factors of friction between steel and dry sands. Soils and Foundation, Vol. 26, No. 2: 29-42. Wheeler, S.J. (1996). Inclusion of specific water volume within an elasto-plastic model for unsaturated soil. Canadian Geotechnical Journal, Vol. 33: 42-57.

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