Nonlinear elastic model for compacted clay concrete interface

Abstract

In this paper, a nonlinear elastic model was developed to simulate the behavior of compacted clay concrete interface (CCCI) based on the principle of transition mechanism failure (TMF). A number of simple shear tests were conducted on CCCI to demonstrate different failure mechanisms; i.e., sliding failure and deformation failure. The clay soil used in the test was collected from the “Shuang Jang Kou” earth rockfill dam project. It was found that the behavior of the interface depends on the critical water contents by which two failure mechanisms can be recognized. Mathematical relations were proposed between the shear at failure and water content in addition to the transition mechanism indicator. The mathematical relations were then incorporated into the interface model. The performance of the model is verified with the experimental results. The verification shows that the proposed model is capable of predicting the interface shear stress versus the total shear displacement very well.

Keywords

interface modeling / friction / soil structure interface / soil structure interaction / simple shear test

Introduction

Interfaced compacted clay and concrete are often encountered in infrastructure works such as earth rockfill dams and retaining walls. One of the factors that affect the behavior of compacted clay concrete interface (CCCI) is the moisture/ water content of clay, which reduces the shear strength of clay. Since clay is cohesive, the friction in contact with the engaged concrete is to be influenced by the adhesion effect and, thus, shear failure may occur in the body of clays. This concept is not always correct and is valid for the interface between low density, high water content clay and the concrete interface. The interface failure manners should depend on the density and water content of clay, plus the roughness of the concrete surface. The unsaturated soils used in dams, retaining walls, and other works mostly present high densities and are compacted. Thus, it is paramount to study the stress-displacement relationship of CCCI consisting of clays of wide-ranging densities and water contents, and concretes of varied roughness.

In the case of the interface between high-dry-density clay and concrete, if substantial displacement (i.e., sliding displacement) occurs in the contact area, the water content of clay may increase the sliding strength and reduce the deformation strength in the limited range of water contents and normal stresses for the rough concrete interface. The reason for that may be attributed to the increasing compressibility of clay due to its increasing water content, which offers full contact between the clay and concrete asperities. At the same time, the increasing water content of clay makes the clay concrete interface lose part of its interface shear strength.

Background

A simple shear apparatus was used by researchers in characterizing the CCCI behavior since it has the ability to isolate the deformation from the sliding displacement measurement. The earliest use of the simple shear apparatus was recorded six decades ago when an NGI-type was used. Kjellman [1] tested the shear strength of clay in Sweden using a cylindrical sample confined within a reinforced rubber membrane. Roscoe [2] used a simple shear type utilizing stiff hinged edges instead of geomembranes. The NGI simple shear apparatus was developed in the “Norwegian Geotechnical Institute” and used in a series of researches by Bjerrum et al. [3], who used it widely to investigate the behaviors of the highly sensitive Norwegian quick clay. The cylindrical sample of 8 cm diameter and 1 cm height was confined with a spiral winding of wire having a diameter of 0.15 mm and being wound at 25 turns per cm. It was developed by Goh et al. [4] to study the soil concrete interface. Badhu [5] developed it and used it with a geomembrane. It was used also by Paikowski et al. [6] in a dual shear interface test and by Evgin et al. [7] in a two-directional shearing apparatus. It was developed and used with rectangular or circular thin metal plates by Uesugi et al. [8-11] to investigate the behavior of the interface between a steel plate and different construction materials.

The dynamic simple shear test (DSST) was developed in GHI, formerly known as the Geohohai Institute, by Pu [12]. It was used in a series of researches. Zhang et al. [13] studied the soil structure interface and developed the relation between the interface shear stress and shear strain using a hyperbolic model. Wang et al. [14] used it to investigate the effect of degree of saturation on the clay concrete interface using one type of concrete roughness. This apparatus was modified and utilized to test the compacted clay concrete interface in this study. Tsubakihara et al. [15] studied the friction between normally consolidated clay and a steel plate using a simple shear and shear box direct shear test apparatus. In their apparatus, the pore water pressure can be calculated during the test, and the roughness of steel was studied as a factor influencing the relation of shear stress-strain. They postulated that the interface shear sliding in the case of roughness less than the critical value dominates the behavior, while the interface shear agrees with the shear stress within soil in the case of surface roughness greater than the critical value. In addition, they stated that the shear speed affects the maximum interface resistance.

Other researchers have utilized a simple shear and shear box device to test the interface between sand-mild steel (such as Uesugi et al. [8]), the influential factor on friction between sand and steel (such as Uesugi et al. [9]), the particle behavior near the interface layer (such as Uesugi et al. [10]), and the friction between dry sand and concrete (such as Uesugi et al. [11]). The literature review about this apparatus suggests that it is used for sand or more sensitive clay. In this study, it was used for high-density clay with different water contents to investigate the effect of water content on the mechanism of failure.

With respect to modeling, the hyperbolic model has been used extensively since it gives good results, and is simple and suitable for computation in finite element analyses [16]. A hyperbolic function was suggested to represent the stress-strain curves by Konder [17] and developed by Duncan [18] to depict the relationship between stress and strain. This model was applied to different soils such as silt soil by Stark et al. [19], the earth pressure of compacted soil by Duncan et al. [16] and Seed et al. [20], reinforced soil by Ebeling et al. [21], and the vertical shear load on nonmoving walls by Filz et al. [22,23]. Zhou et al. [24] developed a hyperbolic model to take into account the degree of roughness for the sand steel interface under a relatively high load.

Laboratory test

A simple shear apparatus was used to study CCCI since it has the ability to measure the deformation displacement and the sliding displacement independently. Two boxes were manufactured and filled with concrete for this study. Figure 1 shows the schematic view of the simple shear test apparatus used in this study. Table 1 shows the properties of clay such as liquid limit (L.L), plastic limit (P.L), and dry density. The clay samples were prepared with controlled values of high dry density and water content. It was prepared directly in the frame of rings; then the constant normal load was applied directly after preparing the sample. The experiments were performed using three normal stress magnitudes: 50 kPa, 100 kPa, and 150 kPa; and three water content ratios: 10%, 16%, and 21%.

The results of the applied shear stress versus total shear displacement show that the strength increases with the normal stress in a linear form within the range of the normal stress used in the test. Figure 2 shows the relationship between the shear stress and the normal stress. Water content has an effect, as depicted in Fig. 3, which indicated that the water content may increase the interfacial strength. It is seen that the water content 16% may be considered as a critical value in the range of tested values, at which the interfacial strength increases more than elsewhere.

A high water content may increase the deformation, while it may decrease the sliding displacement. Figure 4 shows the relationship between the deformation and the sliding displacement. The relationship was not uniform and the points were scattering, but it can be recognized that the sliding was the substantial displacement except in the case of W_c=21%, where the deformation was the substantial. The kind of displacement that controls failure was very important to recognize the type of mechanism failure. Figure 5 shows the relation between the total displacement and the applied interface shear stress. It was seen that there was no apparent peak shear stress value. A shear stress corresponding to 15% shear displacement was identified as the peak shear stress value.

Referring again to Fig. 3, the value of a water content of 16% that may represent a critical value may be considered as a base for the transition mechanism failure. It was found that at 21%, the mechanism of failure was in the body of the sample, and so there are two types of failure: sliding and deformation.

Interfacial shear strength

The traditional criterion of the interfacial strength of Coulomb was compatible with the results of the shear stress-displacement obtained from the simple shear test for relatively rough concrete surface (see Fig. 2). The two components of strength were adhesion and the slope of the linear relation “angle of contact friction”. Each line has parameters different from the other and depends on the water content of clay. This relation is presented as follows:

(1)\tau =c_a+{\sigma }_n\tan \delta ,

where τ is the applied shear stress, is the normal stress, δ_n is the normal stress, δ is the angle of contact friction, and c_a is the adhesion of clay in contact with concrete.

Two types of failures were identified; i.e., sliding shear failure and deformation shear failure. It is valuable to propose an equation relating the shear stress at failure, normal stress, and water content. The following equation was proposed according to the results from Figs. 2 and 3:

(2){\displaystyle \frac{\tau }{\sigma }}=(D-S)w_T+wS,w>w_T,

where S is the slope of the first line in the shear water content relation (see Fig. 3), D is the slope of the second line in the shear water content relation (see Fig. 3), and w_T is the critical water content at which the failure mechanism transforms into another one.

It was noticed that both D and S depend on the critical water content of 16% that was selected according to experimental inspection to be used as a recognition criterion between the sliding and deformation failure. In the case of a sliding failure or of no recognition of failure by water content, Eq. (2) is derived as the following:

(3)

In the case of D=S, the relation becomes

(4){\displaystyle \frac{\tau }{\sigma }}=wS.

Transition mechanism failure (TMF)

To establish the idea of TMF in mathematical equation form and then incorporate it into a constitutive model, the following question needs to be answered. How can the types of mechanism failure be distinguished as a deformation or a sliding mechanism failure? There are two methods to answer this question.

The first method states that a deformation failure occurs when the deformation displacement continues until the end of the test while sliding displacement stops at a limited point. A sliding failure occurs when the sliding displacement continues until the end of the test while deformation displacement stops at a limited point. The second method gives another definition that depends on characteristics gathered from laboratory experiments. The deformation and sliding continue together through the experiments until a point near failure but at different rates. By this definition, the mechanism failure can be determined by the ratio of one type of displacement to another displacement type.

The best method to express the mechanism of failure is to represent it as a percentage of the deformation to the sliding displacement that was measured during the test. With this criterion, the percentage of the deformation to the sliding is between zero and 100%. The extreme values of zero and 100% are excluded.

Determining the type of mechanism failure depends on defining the ratio of deformation to sliding that is desired to be an indicator for the transition mechanism failure. In the following steps, depending on the relation between the ratio of deformation to sliding, the normal stress, interface shear stress at failure, and water content are adopted (see Figs. 2-4).

(5)\xi =\alpha +\beta \zeta ,

where \xi ={\displaystyle \frac{d_d}{d_s}},\xi ={\displaystyle \frac{\eta }{\varpi }},\eta ={\displaystyle \frac{\tau }{\sigma }},\varpi ={\displaystyle \frac{w}{w_T}}.

The substitution of \xi, \xi, and then \eta and \overline{w} into Eq. (5) yields

(6)\begin{array}{l}{\displaystyle \frac{\eta }{\varpi }}=\alpha +\beta {\displaystyle \frac{d_d}{d_s}},{\displaystyle \frac{\tau }{\sigma }}{\displaystyle \frac{w_T}{w}}=\alpha +\beta {\displaystyle \frac{d_d}{d_s}},\\ t_f=\overline{w}\left(\alpha +\beta {\displaystyle \frac{d_d}{d_s}}\right){\sigma }_n,\end{array}

where d_d is the value of the deformation displacement at the end of the test; d_s is the value of the sliding displacement at the end of the test; σ_n is the normal stress; \overline{w} is the water content divided by the critical water content; and \alpha and \beta are the fitting constants that can be found using the least square from Fig. 6, which shows the relation between \xi and \xi. The relation is assumed to be approximately linear. In spite of the approximation, the results obtained were accepted. The non-accuracy that may be noticed next in the curve of the 100 kPa normal stress belongs to this approximation.

This relation is a new relation and considered as an introduction to important researches in order to take the deformation and sliding displacements, mechanism of failure, and water content into account when the simple shear test is used.

Constitutive model derivation

The hyperbolic function was found to be more appropriate for the results of CCCI, which are in agreement with the hyperbolic model of Clough et al. [25]. The developed model is based on the hyperbolic function, which is used widely in stress-strain curve simulation.

The total shear displacement can be decomposed into two components, sliding displacement and deformation displacement, {\mathrm{\Delta }}_r={\mathrm{\Delta }}_s+{\mathrm{\Delta }}_d. The hyperbolic function that describes the interface shear stress and total displacement is expressed as follows:

(7)\tau ={\displaystyle \frac{{\mathrm{\Delta }}_T}{a+b{\mathrm{\Delta }}_T}},

where {\mathrm{\Delta }}_s is the sliding displacement, {\mathrm{\Delta }}_d is the deformation displacement, a is the reciprocal of the initial interface shear displacement tangent, and b is the reciprocal of the ultimate shear stress:

(8)a={\displaystyle \frac{\mathrm{1}}{G_i}},b={\displaystyle \frac{\mathrm{1}}{{\tau }_u}}\mathrm{.}

Figure 7 shows the linear transformation of the hyperbolic function for three cases of normal stress, 50 kPa, 100 kPa, and 150 kPa; and three cases of moisture water content, 10%, 16%, and 21%. The equation for each line was found by curve fitting where the slope (b) and intercept (a) can be identified.

The variation of the initial shear stiffness with the normal stress can be expressed by a general relation suggested by

(9)G_i=K_I{\gamma }_w{\left({\displaystyle \frac{{\sigma }_n}{p_a}}\right)}^n,

where K_I is a dimensionless stiffness number, n is the stiffness exponent, γ_w is the unit weight of water, and p_a is atmospheric pressure. Since there is no apparent peak stress in the results of CCCI in these tests, the interface shear failure τ_f will be taken as the shear stress opposite to 15% of the total shear displacement:

(10)R_f={\displaystyle \frac{{\tau }_f}{{\tau }_{\mathrm{ult}}}},

where R_f is the shear failure factor.

The substitution of Eqs. (8), (9), and (10) into Eq. (7) gives

(11)\tau ={\displaystyle \frac{{\mathrm{\Delta }}_T}{{\displaystyle \frac{1}{K_I{\gamma }_w{\left({\displaystyle \frac{{\sigma }_n}{p_a}}\right)}^n}}+{\displaystyle \frac{R_f}{{\tau }_f}}{\mathrm{\Delta }}_r}}.

The substitution of Eq. (6) into Eq. (11) yields

(12)\tau ={\displaystyle \frac{{\mathit{\Delta }}_T}{{\displaystyle \frac{\mathrm{1}}{K_I{\gamma }_w{\left({\displaystyle \frac{{\sigma }_n}{p_a}}\right)}^n}}+{\displaystyle \frac{R_f}{\overline{w}\left(\alpha +\beta {\displaystyle \frac{d_d}{d_s}}\right){\sigma }_n}}{\mathrm{\Delta }}_r}}.

After some simplification steps, it becomes

(13)\tau ={\displaystyle \frac{d_s{\mathrm{\Delta }}_T}{{\displaystyle \frac{d_s}{K_I{\gamma }_w{\left({\displaystyle \frac{{\sigma }_n}{p_a}}\right)}^n}}+{\displaystyle \frac{R_f{\mathrm{\Delta }}_T}{\overline{w}(\alpha d_s+\beta d_d){\sigma }_n}}}}.

Differentiating Eq. (13) and eliminating {\mathrm{\Delta }}_T lead to the tangent stiffness equation:

(14)K_{\mathrm{st}}=K_I{\gamma }_w{\left({\displaystyle \frac{{\sigma }_n}{p_a}}\right)}^n{\left(\mathrm{1}-M_I{\displaystyle \frac{\tau }{{\sigma }_n}}\right)}^{\mathrm{2}},d_s\ne \mathrm{0},\overline{w}\ne \mathrm{0},

where K_st is the tangential stiffness, M₁ is called the mechanism failure indicator (MFI)

(15)M_I={\displaystyle \frac{R_f{D}_{\mathrm{ts}}}{\overline{w}}},

and D_ts is the deformation to sliding factor, which is equal to

(16)D_{\mathrm{ts}}={\displaystyle \frac{d_s}{\alpha d_s+\beta d_d}}.

Identification of model parameters

The proposed model requires the determination of five parameters. All parameters can be simply determined through experimental tests. These five parameters are K, n, R_f, α, and β. The stiffness number K and power n can be calculated depending on the relation between the initial stiffness and the normal stress. The failure ratio R_f is computed as the ratio of the failure shear stress to the ultimate shear stress. α and β are the parameters of TMF, which can be obtained from fitting the relation presented in Fig.6. The last two parameters are influenced by the normal stress and water content. The parameters of adhesion and angle of contact were imbedded inside the parameters of TMF mathematically.

Table 2 lists the model parameters calculated with a computer program developed for this purpose. The sign of β belongs to the slope of the curves in Fig. 6, where the x-axis d_d/d_s is calculated depending on Fig. 4. Figure 4(a) may give a misleading result in the case of a normal stress of 50 kPa. The figure shows a high value of deformation compared to the two cases of 100 kPa and 150 kPa normal stresses. Table 3 shows the other factors that were referred to in this paper, such as M_I and D_ts.

Figure 8 shows the relation between the MFI parameter (M_I) and water content for the three values of normal stress. Generally, the figure shows a linear response with a negative slope. When the value of the water content is 21%, M_I get close one where the mechanism of failure was deformation mechanism failure. Figure 9 shows the indirect relation between the MFI (M_I) and the normal stress for the three values of water content, 10%, 16%, and 21%. It is obvious that the figure shows a horizontal line. The lowest line is in the case of a water content of 21%, where the value of the parameter (M_I) is approximately equal to 1.

Verification of the proposed model

The nonlinear interface parameters listed in Table 2 represent the behavior of the interface during shear loading. To check the validity of the model, laboratory test results can be backward predicted using the obtained parameters. Backward prediction includes the response of the shear stress versus the total shear displacement. It was presented in Fig. 10 for three values of normal stress (50 kPa, 100 kPa, and 150 kPa) and three values of water content (10%, 16%, and 21%). It was verified that the proposed model is capable of predicting the interface shear stress versus the total shear displacement very well. In Fig. 10(c), though the parameters used in the prediction are computed from the same test presented in this figure, the prediction for the case of (σ_n=100 kPa) does not exactly match the observation. This is due to the fact that the relation of TMF in Fig. 6 was taken to be linear while it is not really linear.

Conclusions

In the present study, the behavior of the interface between compacted clay and concrete was investigated. A nonlinear elastic model was proposed, taking into account the effect of the water content and type of failure. The model has five parameters that are simply determined through an interface shear test. It was verified that the proposed model is capable of predicting the interface shear stress versus the total shear displacement very well.

The following points were concluded during the program of study:1) The traditional Coulomb’s equation between shear failure and normal stress is valid. 2) In the range of the obtained results, there are two types of failure: sliding and deformation failure. The critical water content that can be depended on to recognize the type of failure was found to be 16%. 3) The relation between interface shear failure and water content was proposed depending on the critical water content. 4) The ratio between the deformation and the sliding displacement at failure was considered as the direct mechanism failure indicator and was incorporated in the proposed model.

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