Experimental investigations and constitutive modeling of cyclic interface shearing between HDPE geomembrane and sandy gravel

Geotextiles and Geomembranes 47 (2019) 269–279

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Experimental investigations and constitutive modeling of cyclic interface shearing between HDPE geomembrane and sandy gravel

T

W.J. Cena,∗, E. Bauerb, L.S. Wena, H. Wanga, Y.J. Suna a b

College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing, 210098, China Institute of Applied Mechanics, Graz University of Technology, Graz, 8010, Austria

ARTICLE INFO

ABSTRACT

Keywords: Geosynthetics Geomembrane Interface Cyclic interface shear test Constitutive model Damping ratio

This paper presents the results of experimental investigations and constitutive modeling of cyclic interface shearing between HDPE geomembrane and cohesionless sandy gravel. A series of cyclic interface shear tests was performed using a large-scale cyclic shear apparatus with servo controlled system. Particular attention was paid to the influences of the amount of shear-displacement amplitude, number of cycles, shear rate and the normal pressure on the mechanical response. The experimental results show that the path of the shear stress against the cyclic shear displacement is strongly non-linear and forms a closed hysteresis loop, which is pressure dependent, but almost independent of the shear rate. For small shear-displacement amplitudes, the obtained damping ratio is significantly greater than zero, which is different to the behavior usually observed for cyclic soil to soil shearing. In order to describe the pressure dependency of the hysteresis loop using a single set of constitutive parameters, new approximation functions are put forward and embedded into the concept of the Masing rule. Further, a new empirical function is proposed for the damping ratios to capture the experimental data for both small and large cyclic shear-displacement amplitudes. The included model parameters are easy to calibrate and the new functions may also be useful in developing enhanced constitutive models for the simulation of the cyclic interface shear behavior between other geosynthetics and soils.

1. Introduction As an important branch of geosynthetics, HDPE (high-density polyethylene) geomembranes are used as barriers against liquid and gas flow in many geotechnical applications like for instance water reservoirs, water, oil and gas storage tanks, sealing of tunnel linings, liners and covers for landfills (e.g. Müller, 2010; Koerner, 2012; Pavanello et al., 2018). Further to the function of geomembranes as cutoffs against liquid and gas flow, in many applications the mechanical interface behavior between geomembrane and soil is also an important issue, in particular when relative motions along the interface lead to a possible instability of the whole system (e.g. Stamatopoulos and Kotzias, 1996; Palmeira, 2009; Saravanan et al., 2008, 2011). Experiments show that the interface friction angle between geomembrane and sand/gravel materials mainly depends on the normal stress, the surface roughness and deformability of the geomembrane, the angularity, surface roughness, moisture content and packing density of the grains of the adjacent soil. The size of the grains, i.e. the number of grain contacts on the geomembrane per unit area, may also have an influence on

the interface friction angle (e.g. Fuggle and Frost, 2010; Frost et al., 2012; Vangla and Latha, 2015). To investigate the interface behavior, various test devices are available such as the direct interface shear box test, inclined plane test and ring shear test. With the latter arbitrary large shear displacements can be applied, which also allows the investigation of the residual friction angle. As different test types can be characterized by differences in the boundary conditions, the evolution of the shear stress and shear deformation in the soil can be influenced by the individual mechanism of the test type. For instance, the displacement field of the granular soil close to the interface is usually nonlinear and different for the ring shear device and direct shear box device (Garga and Infante Sedano, 2002; Ebrahimian and Bauer, 2012, 2015). In this context the size of the specimen may also show a significant scale effect as reported for instance by Vieira et al. (2013a). Such effects can also be caused by inhomogeneous deformations in the soil body when shear strain localization takes place in the soil body close to the interface (e.g. Boulon, 1989; Palmeira and Milligan, 1989; Frost et al., 1999). However, for low normal stress and a very smooth and undamaged surface of the geomembrane, the grains only slide against the

Corresponding author. E-mail addresses: [email protected] (W.J. Cen), [email protected] (E. Bauer), [email protected] (L.S. Wen), [email protected] (H. Wang), [email protected] (Y.J. Sun). ∗

https://doi.org/10.1016/j.geotexmem.2018.12.013 Received 26 February 2018; Received in revised form 19 December 2018; Accepted 22 December 2018 0266-1144/ © 2019 Elsevier Ltd. All rights reserved.

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W.J. Cen, et al.

geomembrane and no shear band develops in the adjacent soil body (Prashanth et al., 2016). Under higher normal stresses a change of the shearing mechanism from sliding to scouring or plowing in the deformed geomembrane can be detected, which is more pronounced for a smaller mean grain size (e.g. Punetha et al., 2017). To gain further insight into the complex mechanism, numerical studies using DEM or enhanced constitutive models are helpful (e.g. Ebrahimian and Bauer, 2012, 2015; Tran et al., 2013; Liu et al., 2014; Hegde and Roy, 2018). In the last few decades, considerable experimental research has been conducted to investigate the mechanical behavior of the interface between different types of geosynthetics (e.g., geomembranes, geotextiles, geogrids, geosynthetic clay liners, and other geocomposites) and soils (cohesionless sand, gravel, broken rock, clay and other cohesive soils) under monotonic shearing (e.g. Liu et al., 2009; Bacas et al., 2015a, 2015b; Aldeeky et al., 2016; Chai and Saito, 2016; Choudhary and Krishna, 2016; Useche Infante et al., 2016; Vangla and Gali, 2016), and large databases have been created (e.g. Zornberg et al., 2005; Dixon et al., 2006; Cen et al., 2018a). But limited experiments have been carried out to investigate the interface behavior under cyclic shearing. Athanassopoulos et al. (2010) presented the results of a cyclic shear test of a secondary containment liner system composed of sand/ GCL/sand. The experimental data indicated that the hysteretic stressdisplacement behavior of the GCL interface is similar to natural soils and shows strength and stiffness degradation as well as a reduction in the damping ratio with an increasing number of cycles. Fox et al. (2011) conducted several shear tests on multi-interface geomembrane liner specimens to assess effects of damage from cyclic loading. The findings indicated that severe geomembrane damage can result from shear displacement against a compacted subgrade soil with gravel, and shear stiffness was found to be essentially constant. The damping ratio was again found to decrease with an increasing number of cycles. Liu et al. (2016) conducted a series of cyclic and post-cyclic shear tests on soilgeogrid interfaces considering different normal stresses and cyclic shear amplitudes. They observed that the interface shear strength increases with cyclic shearing. Wang et al. (2016) examined the particle size effect on the cyclic shear behavior of a coarse soil-geogrid interface in a series of cyclic shear tests. The results indicated that the interface exhibits cyclic hardening and the interface damping ratio decreased with an increasing number of cycles and with the relative density of the granular material. The characteristic properties of geosynthetic interfaces are the evolution of the hysteresis behavior and the corresponding damping ratio under different normal stresses and cyclic shear-displacement amplitudes. While for very small shear-displacement amplitudes in soil to soil cyclic shear experiments, the damping ratio usually almost vanishes, i.e. the mechanical response is elastic, there is experimental evidence from cyclic interface shear tests that for small shear-displacement amplitudes the damping ratio is also significantly different from zero (e.g. Nye and Fox, 2007; Vieira et al., 2013b). Although the mechanism under small shear amplitude is still not well understood, the experimental results are important for appropriate constitutive modeling of the interface behavior as also shown in the present paper. Based on the laboratory tests, several constitutive models based on elasto-plasticity, generalized plasticity, and hypoplasticity were developed to model the non-linear interface behavior under monotonic or cyclic shearing. These models can also be applied to simulate the properties of geosynthetics-soil interfaces (e.g. Shallenberger and Filz, 1996; Gomez and Filz, 1999; Esterhuizen et al., 2001; Boulon et al., 2003; Anubhav and Basudhar, 2010; Stutz et al., 2017). In practical empirical models utilizing the Masing rule are also frequently used to describe the hysteresis loop under cyclic interface shearing. Masing rule is based on the assumptions that the shear stiffness on each loading reversal is equal to the initial shear stiffness of the virgin loading curve, or so-called backbone curve, and that the hysteresis loop is similar to the backbone curve. For modeling the backbone curve either empirical functions of the hyperbolic type (Kondner, 1963; Hardin and Drnevich,

1972b) or of the Ramberg-Osgood type (Ramberg and Osgood, 1943) are frequently used. In the present paper the hyperbolic function by Kondner is used to model the backbone curve and Masing rule is applied to approximate the hysteresis loop based on the data from cyclic interface shear tests. In particular, the present paper is organized as follows. Section 2 presents the results of the experimental investigations of cyclic interface shearing between geomembrane and sandy gravel using a large-scale cyclic shear apparatus. Particular attention is paid to the influences of the amount of shear-displacement amplitude, number of cycles, shear rate and the normal pressure on the evolution of the shear stress within the prescribed cyclic shear displacements. In Section 3 the experimental results are adapted to a constitutive model based on the Masing rule for describing the one dimensional non-linear cyclic interface behavior. Different approximation functions for the pressure dependency of the constitutive parameters involved are proposed and their performance is evaluated by comparing the numerical results with the experimental data. For modeling hysteretic damping the function by Hardin-Drenvich (1972a, 1972b) is extended to cases with non-vanishing hysteresis for small cyclic shear amplitudes. 2. Experimental investigations 2.1. Test apparatus A servo controlled shear box device is used for cyclic interface shear tests. The overall view of the large shear box apparatus and the loading frame is shown in Fig. 1. The equipment consists of the cyclic shear device, the vertical loading device and the data acquisition system. The cyclic shear device consists in particular, of the horizontally moveable, displacement-controlled lower box and the upper box with the square cross-section of 360 mm × 360 mm and a height of 100 mm. In order to avoid relative displacements between the shear box frame and the geomembrane specimen during cyclic shearing, the geomembrane is glued on to the top surface of the lower box with an epoxy adhesive and also anchored on the lateral edges with bolts and steel clamping blocks as shown in Fig. 1(c). The upper box is fixed in space in a horizontal direction, but it can move freely in the vertical direction. The cylindrical opening for the soil specimen has a diameter of 300 mm. For the present investigation the upper box is filled with sandy gravel as shown in Fig. 1(d). Smooth bearings are placed between the shear boxes to minimize friction and to keep a small gap between the surface of the geomembrane specimen and the upper shear box so that the upper box will not ride on the geomembrane during cyclic shearing. The vertical load device consists of the actuator, which is connected with the force sensor. The force sensor captures a load range of 0–100 kN and the maximum error is less than 0.2%. The applied horizontal shear force and vertical load are controlled by the servo control system with an error less than 1%. The horizontal shear rate can be triggered within the range of 0.01–5.00 mm/min. The motion of the lower box is recorded with a displacement sensor with an error less than 0.1%. A data acquisition instrument is used for automatic data collection of cyclic displacement, shear force and vertical pressure. Data can be recorded with a frequency of 2 s. 2.2. Test materials and experimental program The smooth HDPE geomembrane used has a nominal thickness of 0.75 mm, a density of 0.94 g/cm3, a yield strength of 12.78 MPa and a yield elongation of 14.7%. The size of geomembrane specimen is greater than the contact area of sandy gravel to maintain constant area of failure during the cyclic shear process (Hsieh and Hsieh, 2003). In this study, air-dry sandy gravel was used with a moisture content of 3.3%. The maximum dry density of the sandy gravel is 2.25 g/cm3 and the minimum dry density is 1.68 g/cm3. The sandy gravel was filled into the upper shear box and compacted in four layers by using a hammer to hit the steel plate until the dry density reached 1.96 g/cm3. 270

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Fig. 1. Cyclic shear apparatus: (a) installation of the cyclic shear apparatus in the laboratory; (b) schematic view of cyclic shear apparatus; (c) geomembrane fixed on the lower shear box and (d) upper shear box filled with sandy gravel. Table 1 Test program. Scheme

I-A

I-B

I-C

II

III

Shear amplitude (mm) Number of cycles Shear rate (mm/min) Vertical pressure (kPa)

0.1–5.0 9 2.0 50

0.1–5.0 9 2.0 100

0.1–5.0 9 2.0 200

2.0 10 2.0 100

2.0 4 0.5–4.0 100

different vertical pressures, shear-displacement amplitudes from 0.1 to 5.0 mm were considered. To investigate different numbers of shear cycles and shear rates, a shear-displacement amplitude of 2.0 mm was chosen. Five test schemes were designed, as summarized in Table 1. 2.3. Test results Figs. 3–5 show the cyclic shear stress versus the shear displacement obtained for different displacement amplitudes and different vertical pressures (Schemes I–A, I-B and I-C). In order to visualize more clearly the experimental data, the curves are drawn for small displacement amplitudes in Figs. 3(a)–5(a) and for larger displacement amplitudes in Figs. 3(b)–5(b). The path of the shear stresses at the state of reversal shearing versus the corresponding displacement amplitude is the socalled “backbone curve” of cyclic shear tests and drawn in Figs. 3(c)–5(c). It is obvious that the cyclic shear stress-displacement relations of geomembrane-sandy gravel interface is strongly nonlinear. For a full

Fig. 2. Grain size distribution curve of fine sand (used by Cen et al., 2018b) and sandy gravel (investigated in this paper).

Fig. 2 shows the grain size distribution of the sandy gravel as well as the grain size distribution of the fine sand in previous research (Cen et al., 2018b). The size of sandy gravel particles ranges from 0.07 to 20 mm and the mean grain diameter is approximately 5 mm. A series of cyclic interface shear tests was carried out under various conditions to investigate the mechanical response of a geomembranesandy gravel interface (ASTM D5321). For the cyclic shear tests under 271

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W.J. Cen, et al.

(a)

(a)

(b)

(b)

(c)

(c)

Fig. 4. Cyclic shear behavior under vertical pressure of 100 kPa (Scheme I–B): (a) shear stress versus shear displacement u for small shear amplitudes (0.1–1.0 mm); (b) shear stress versus shear displacement u for large shear amplitudes (2.0–5.0 mm); (c) backbone curve.

Fig. 3. Cyclic shear behavior under vertical pressure of 50 kPa (Scheme I–A): versus shear displacement u for small shear amplitudes (a) shear stress (0.1–1.0 mm); (b) shear stress versus shear displacement u for large shear amplitudes (2.0–5.0 mm); (c) backbone curve.

approximation of experimental data made in the Masing rule as discussed in Section 3.2. The results obtained for the normal stress of 200 kPa indicate a certain tendency to strain softening. The size of the hysteresis loop depends on both displacement amplitude and pressure level. While for smaller displacement amplitudes the shear resistance at

displacement cycle, the curve forms a closed hysteresis loop, where the branches for loading and reversal loading are relatively centrally symmetric to the origin of the coordinate system. This observation supports the eligibility of the assumption made for the analytical 272

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Fig. 6. Evolution of mobilized interface friction angle pressures.

(a)

(b)

Fig. 7. Cyclic shear strength

f

under different vertical

versus vertical pressure

n.

Table 2 Peak shear stress and cyclic interface friction angle for different geomembranesoil interfaces and for different vertical stresses. Interface

Vertical Stress (kPa) 50

100

200

Interface friction angleb

Peak shear stress (kPa) Geomembrane-sandy gravel Geomembrane-fine sanda a b

40.6

82.04

144.5

36.7

40.5

73.9

134.7

34.7

Cen et al., (2018b). Average value corresponding to the linear approximation used in Fig. 7.

mobilized interface friction angle tends towards the same residual value for all experiments. The maximum shear stress depending on the normal stress can be approximated by a straight line as shown in Fig. 7. With respect to the inclination of the line a cyclic interface friction angle of c = 36.7 is obtained. Table 2 shows a comparison between the cyclic interface shear parameters of the geomembrane-fine sand interface (Cen et al., 2018b) and the geomembrane-sandy gravel interface. It becomes evident that the cyclic interface shear parameters also depend on the different grain size distributions (Fig. 2) and the grain properties. The peak shear stress and consequently the interface friction angle are higher for the geomembrane-sandy gravel interface. While under a vertical pressure of 50 kPa the differences between the peak shear stresses of these two

(c) Fig. 5. Cyclic shear behavior under vertical pressure of 200 kPa (Scheme I–C): versus shear displacement u for small shear amplitudes (a) shear stress (0.1–1.0 mm); (b) shear stress versus shear displacement u for large shear amplitudes (2.0–5.0 mm); (c) backbone curve.

reversal states depends on the amount of the shear displacement, the maximum shear stress is almost independent for larger displacement amplitudes. This means that for larger shear displacements an almost steady stress state can be observed. The comparison of the normalized backbone curves (Fig. 6) also shows that for larger displacements the 273

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Fig. 8. Shear stress

versus shear displacement u (Scheme II).

Fig. 10. Illustration of the backbone curve I and the cyclic hysteresis loop according to the Masing rule for the unloading branch II and the reloading branch III.

origin of the coordinate system. This assumption is in good agreement with the own experimental observations. According to the Masing rule the reloading branch can be constructed by scaling the backbone curve (curve I in Fig. 10) by a factor 2 and the unloading branch is obtained by rotating the reloading curve by 180°. The backbone curve, also termed skeleton curve, represents the states of loading reversals, which are coincidental with the initial stress-displacement curve obtained under monotonic shearing. Masing rule also assumes that for a constant shear amplitude the shape of the corresponding cyclic loop is independent of the number of shear cycles. Consequently, the shear stiffness on each reversal state is equal to the initial value Kmax of the backbone curve. The following hyperbolic equation is used for modeling the initial loading curve, i.e. the backbone curve (Kondner, 1963): Fig. 9. Shear stress (Scheme III).

u a + bu

=

versus shear displacement u for different shear rates

(1)

The cyclic secant shear stiffness K of the backbone curve is defined as the ratio of the shear stress to the corresponding shear-displacement amplitude u, i.e.:

interfaces are small, under high vertical pressures, e.g. 100 kPa and 200 kPa, particles of sandy gravel may locally deform the geomembrane, which results in an evident increase of the maximum shear resistance. The results obtained for different numbers of shear cycles and different shear rates are shown in Fig. 8 and Fig. 9 (Schemes II and III), respectively. It can be seen that for the investigated materials the number of shear cycles and the amount of the shear rates have no clear influence on the cyclic interface behavior which is in accordance with the observations for instance by Ferreira et al. (2016). It should be noted that the present investigation focuses on the cyclic behavior of geomembrane-sandy gravel interfaces for a small range of low shear rates, which does not allow conclusion about the behavior under much higher shear rates.

K=

u

=

1 a + bu

(2)

where the model parameter a and b are related to the following limits: for u = 0 : Kmax = 1/ a for u : u = 1/b (using the rule by Bernoulli-L’Hospital) In particular, Kmax denotes the initial shear stiffness and u is the asymptotical shear stress. As shown in Fig. 11 parameters a and b can also be calibrated based on the representation of the experimental data according to the function:

1 u = = a + bu K

3. Constitutive modeling and calibration

(3)

Then parameter a is the intercept, and b is the slope of the fitting line. The obtained values of the parameters are strongly influenced by the vertical stress n as shown in Table 3. Figs. 12 and 13 show the comparison of the experimental data with the approximated backbone curve and the decrease of the secant shear stiffness with an increase of the shear-displacement amplitude, respectively. It is obvious that the decrease of the shear stiffness is dominant at the beginning of shearing up to shear displacement of 2 mm. Also the influence of the pressure level on the shear stiffness is

3.1. Backbone curve and cyclic shear stiffness In the following the interface behavior under symmetric cyclic shearing is modeled based on the Masing rule. An essential assumption of the Masing rule is the similarity of the backbone curve, i.e. the curve I, with the shape of the unloading and reloading branches of the cyclic hysteresis loop as illustrated in Fig. 10. The unloading branch, i.e. curve II, and reloading branch, i.e. curve III, are centrally symmetric to the 274

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W.J. Cen, et al.

Fig. 13. Decrease of the secant shear stiffness K with an increase of the sheardisplacement amplitude u .

Fig. 11. Plot of 1/K versus shear-displacement amplitude u under different vertical pressures. Table 3 Parameters related to Eq. (1) and Eq. (2). n

(kPa)

50 100 200

a (10−3 mm/kPa)

b (10−3 kPa)

R2

Kmax (kPa/mm)

5.60 4.35 3.02

23.65 11.42 6.43

0.997 0.998 0.989

178.6 229.9 331.1

Fig. 14. Approximation of Kmax and tion.

Kmax =

1 = K1 a

n

relationship in logarithmic representa-

n w

n

pa

(4)

where K1 and n are the dimensionless parameters, w is the unit weight of water, and pa denotes the atmospheric pressure. Fig. 14 shows that Eq. (4) represented in a logarithmic diagram can fit the experimental data by a straight line, in which K1 = 24513 and n = 0.45. The analysis of the experimental data shows that the pressure dependency of parameter a can alternatively be modeled by the following exponential function:

Fig. 12. Approximation of the backbone curves of geomembrane-sandy gravel interface under different vertical pressures.

more pronounced for smaller shear displacements, i.e. the shear stiffness is much higher for higher vertical pressures. With an increase of the shear-displacement amplitude the influence of the pressure level on the shear stiffness is reduced.

a(

n)

= a0 e

( n / a1)

(5)

where a0 and a1 are the material parameters. In Eq. (5) the values a0 = 6.81 × 10−3 kPa and a1 = 239 kPa are obtained by using a standard optimization procedure. It is obvious that the approximation with only 3 experimental points in Fig. 15 does not clearly show significant differences in the predictions obtained from Eq. (4) and Eq. (5). As shown in Fig. 16, however, the predictions using the combination of Eqs. (5) and (11) yields slightly better results for larger shear displacements than the combination of Eqs. (4) and (10), which are frequently used (see below). For the pressure dependency of parameter b two different approaches are also discussed. A common relation is (Duncan and Chang, 1970):

3.2. Modeling pressure dependency of parameters a and b For modeling the pressure dependency of parameters a and b in a unified manner, different concepts are investigated in the following. To begin with parameter a , Eq. (2) for u = 0 is considered. A function frequently used for approximation of the influence of the pressure on the initial stiffness Kmax is the one proposed by Clough and Duncan (1969, 1971), which is expressed as (Reddy et al., 1996; Seo et al., 2004): 275

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W.J. Cen, et al.

dependent parameter b according to Eq. (6) is:

1.261

b ( n) =

(10)

n

For parameter b, which depends on the vertical pressure n (Table 3), the following relation can be obtained using a standard optimization procedure:

1.18

b ( n) =

(11)

n

In Fig. 16 the pressure dependency of parameters a and b in Eq. (1) are compared with the experimental results for three different normal stresses. The combination of Eq. (4) with Eq. (10) leads to almost the same results as the combination of Eq. (5) with Eq. (10). On the other hand the combination of Eq. (5) with Eq. (11) shows a slightly better agreement with the experiments than the combination of Eq. (4) with Eq. (10). From the evaluation of the different procedures investigated it can be concluded that Eq. (5) leads to a more accurate result for the pressure dependency of parameter a, and the optimization procedure for parameter b using Eq. (11) is much simpler than the one for Eq. (6).

Fig. 15. Comparison of approximation Eq. (4) with Eq. (5) for parameter a depending on vertical pressure n .

3.3. Performance of the interface model under cyclic shearing Based on the hyperbolic equation (1) and the pressure dependency of parameters a and b the backbone curve reads for unloading states m

=

a(

n)

um + b(

(12)

n ) um

and for reloading states m

n)

R¯f

= n

tan

(6)

c

with the average failure ratio R¯ f , i.e.

1 R¯ f = n

n i=1

=

(i ) f (i ) u

=

(i ) n

tan

(i ) f ,

=

um b(

n)

(13)

um

a(

n)

u + um + b( n) (u + um)/2

m

(14)

i.e. (8)

c

and the ultimate shear stress (i ) u

n)

(7)

the peak shear stress (i ) f

a(

respectively. In Eqs. (12) and (13) the shear displacement um and the corresponding shear stress m denote the state quantities where reversal loading takes place. With respect to the pressure dependent parameters a and b the comparison of Eqs. (12) and (13) with the experimental data is shown in Fig. 17. The modeling of the shape of the hysteresis loop is based on the Masing rule as outlined in Section 3.1. Thus, the unloading branch II and the reloading branch III are assumed to be centrally symmetric and similar to the backbone curve I. In particular, the reloading branch III and also the unloading branch II are twice as large as that of the backbone curve I. Hence, the hysteresis branches can be expressed by

Fig. 16. Comparison of experimental data with different approximation functions for the pressure dependent parameters a and b.

b(

=

(i ) u ,

i.e. (9)

1/ b(i)

Table 4 lists the peak shear stress

(i ) f

and the ultimate shear stress

under different vertical pressures of 50 kPa, 100 kPa and 200 kPa. The average value of R¯ f = 0.94 can be calculated. Thus, the pressure (i ) u

Table 4 Calculation of the average failure ratio R¯ f . (i) 1 2 3

(i ) n

50 100 200

(kPa)

(i ) f

(kPa)

40.5 82.04 144.5

(i ) u

(kPa)

42.28 87.54 155.51

R¯ f

0.94

Fig. 17. Backbone curves for unloading and reloading states under different vertical pressures. 276

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W.J. Cen, et al.

interface shearing between geomembrane and sandy-gravel. 3.4. Formulation of the damping ratio In this paper the dissipated energy during cyclic shearing is reflected through the following dimensionless damping ratio (Hardin and Drnevich, 1972a):

=

=

=

Fig. 18. Simulation of cyclic shear behavior of the interface under different vertical pressures: (a) 50 kPa, (b) 100 kPa and (c) 200 kPa.

for the reloading curve III, and by

u um + b( n) (u um)/2

m

1

K = Kmax

ult

1

a a + bu

(17)

0

1 + ku

+

ult

0

1 + ku

1

a a + bu

( 1+ 2 n )

(18)

In Eq. (18), 0 is the damping ratio for u = 0, ult denotes the ultimate value of the damping ratio for u , n is the vertical stress, k , 1 and 2 are constitutive parameters. The values of the parameters are summarized in Table 5 and the numerical simulations are compared with the experimental data in Figs. 19 and 20. It is shown that the proposed model can well simulate the damping behavior of a geomembrane-sandy gravel interface. In order to show the superiority of the proposed model, Fig. 20 shows a comparison between the measured values of / max and K / Kmax and the numerical simulations of the Hardin-Drenvich model. It can be observed that the conventional Hardin-Drenvich model cannot reproduce the relation between damping ratios and shear stiffness, while the modified model can well capture the essential damping behavior of a geomembrane-sandy gravel interface.

(c)

n)

ult

where the ultimate damping ratio ult is related to the asymptotical value under an theoretically infinitely large shear-displacement amplitude u. The quantities a and b are pressure dependent as discussed in 0 , the damping ratio Section 3.2. For small shear amplitudes, i.e. u obtained from Eq. (17) falls to zero. An almost vanishing damping ratio is related to an elastic behavior and can usually be observed for very small cyclic shear amplitudes of granular soil materials. The present cyclic interface shear tests, however, show that for small shear-displacement amplitudes within the range of 0.1 < u < 1.0 mm the damping ratio is approximately 0.2–0.3 and thus, significantly different from zero as clearly visible in Fig. 19. It can therefore be concluded that for small cyclic shear amplitudes the mechanisms of soil-to-soil shearing and soil-to-geomembrane interface shearing are basically different. The damping ratio tends to decrease for small shear-displacement amplitudes of less than 1 mm and increases for larger shear-displacement amplitudes. The amount of the damping ratios obtained for small shear-displacement amplitudes lies within the range also reported by other researchers (e.g. Yegian et al., 1998; Vieira et al., 2013b). In order to reasonably take into account the damping behavior of a geomembrane-sandy gravel interface also under small cyclic shear displacements, the following modified model for the pressure dependent damping ratio is proposed:

(b)

a(

(16)

In Eq. (16) AL denotes the total area enclosed by the hysteresis loop of a full shear circle A-B-A, and AT is the area of triangle O-A-um-O as illustrated in Fig. 10. In particular AL is related to the energy dissipation of a full shear circle and AT represents the maximum energy that can be stored in an idealized linear system with same shear amplitude um and the corresponding secant shear stiffness K. To model the hysteretic damping of soil, the following model is widely used in soil dynamics (Hardin-Drenvich 1972b):

(a)

=

1 AL 4 AT

(15)

for the unloading curve II. Fig. 18 shows that the model can simulate for different vertical pressures and a single set of constants the cyclic 277

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small shear-displacement amplitudes, i.e. 0.1–1 mm, the damping ratio is significantly greater than zero and shows a decreasing trend. For greater shear-displacement amplitudes, the damping ratio increases with an increase of the shear-displacement amplitude. This property observed in the interface shear box tests is different to the behavior usually observed for granular materials under small cyclic shear amplitudes. The backbone curves related to loading and unloading paths are almost centrally symmetric, which allows a relatively simple and sufficiently accurate approximation of the experimental data using the Masing rule. In particular, the backbone curve is modeled in a simplified manner with the hyperbolic equation by Kondner. For the calibration of the constitutive parameters, different pressure dependent approximation functions are investigated. The comparison with the experimental data shows that the proposed functions allow an easier and sufficient accurate calibration of the pressure dependent behavior using only a single set of parameters. As for small cyclic shear displacements, the damping ratio observed in experiments cannot be adapted to the function by Hardin-Drenvich, an extended empirical function is proposed. It is demonstrated that the new approximation function can well reflect the damping behavior of the interface experiments carried out between HDPE geomembrane and sandy gravel under both small and large cyclic shear-displacement amplitudes. The proposed approximation functions may also be useful in developing and calibration of enhanced constitutive models for simulation of the cyclic interface shear behavior between other geosynthetics and soils.

Fig. 19. Damping ratio depending on the shear-displacement amplitude u and the vertical pressure n . Table 5 Fitting parameters for Eq. (18). k 1.25

1

0

ult

0.32

0.72

1.84

2

0.0124

Acknowledgements The authors wish to thank the National Natural Science Foundation of China (Grant No. 51679073), the Natural Science Foundation of Jiangsu Province (Grant No. BK20141418), and the Priority Academic Program Development of Jiangsu Higher Education Institutions. References

Fig. 20. Relationship between /

max

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and K / Kmax .

4. Conclusions To investigate the interface behavior between HDPE geomembrane and cohesionless sandy gravel, a series of cyclic interface shear tests was performed using a large-scale cyclic shear apparatus with servo controlled system. Particular attention was paid to the influences of the amount of shear-displacement amplitude, number of cycles, shear rate and the normal pressure on the mechanical response. The experimental results show that the path of the shear stress against the cyclic shear displacement is strongly non-linear and forms a closed hysteresis loop. After repeated shear cycles conducted under different normal stresses, i.e. 50 kPa, 100 kPa and 200 kPa, and shear-displacement amplitudes in the range of 0.1–2.0 mm, no obvious damages were visible on the geomembrane. Within 10 shear cycles no clear degradation of the mobilized shear resistance was observed. The hysteresis loops are pressure dependent, but almost independent of the shear rate. For large shear displacements the mobilized interface friction angle tends towards a stationary value which is almost independent on the pressure level. The damping ratio is little lower for a larger normal stress. For 278

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