1 Introduction

Number of river valley projects are being planned and implemented in India and abroad to store the natural water flowing in the rivers and use it later for different purposes viz. power generation, irrigation and flood control etc. Concrete, Masonry and Earth and rockfill dams are being constructed to store the river water. However, now a days, rockfill dams are widely being constructed all over the world because of their inherent flexibility, capacity to absorb large seismic energy, adaptability to various foundation conditions and economical as well. The behaviour of the rockfill materials used in the earth and rockfill dam is of considerable importance for the analysis, safe and economical design of rockfill dams. The prototype rockfill material used in the dam construction is large in size and it is not feasible to test it directly in the laboratory. Therefore, some kind of modeling technique is often used to reduce the size of particles so that the specimens prepared with smaller size particles can be prepared and tested in the laboratory. Among all available modeling techniques, the parallel gradation technique (Lowe 1964) is most commonly used and the same has been adopted in the present study. The behaviour of the rockfill materials has been reported by number of researchers. Marsal (1967), Marachi et al. (1969), Venkatachalam (1993), Gupta (2000), and Abbas (2003), Honkanadavar (2010), Varadarajan et al. (1997, 1999, 2002, 2003, 2006), Honkanadavar (2011, 2012a, Honkanadavar et al. 2012) have performed laboratory tests on various rockfill materials with different confining pressures. They concluded that the strength parameter, ϕ varies with the particle size. They had also observed that the trend of behaviour of alluvial rockfill material is opposite to that of blasted quarried rockfill material. For blasted quarried rockfill materials the angle of internal friction, ϕ decreases with increase in dmax and for alluvial rockfill material the angle of internal friction, ϕ increases with increase in dmax. HISS models have been used successfully to model the behavior of various materials (Desai 2001; Desai et al. 1986, 1995a, 1995b, Desai and Ugai 1996; Desai and Varadarajan 1987; Varadarajan and Desai 1987, 1993; Najjar et al. 1994; Varadarajan et al. 1994, 1997, 1999, 2002a, 2002b, 2003, 2006; Gupta 2000; Abbas 2003; Honkanadavar 2010; Xiao et al. Xiao et al. 2011a, 2011b, 2012; Honkanadavar 2012). They predicted the stress–strain-volume change behaviour of materials and compared with the observed results. From the predicted and observed results they found that both observed and predicted results match closely.

In the present research work, two types rockfill materials (alluvial and blasted quarried) have been considered. Renuka dam is a 148 m high earth and rockfill dam constructed using alluvial rockfill material across river Giri in Himachal Pradesh, India and Salma dam is a 107.5 m high earth and rockfill dam constructed using quarried blasted rockfill material across river Hari Rud, Afghanistan. The rock type of alluvial rockfill is limestone and quarried blasted rockfill is metamorphic gneiss. The details of project, location and types of rock are presented in Table 1. This paper deals with the laboratory study of stress–strain-volume change behaviour, determination of shear strength parameters, prediction of the behaviour for rockfil material using non-associative HISS constitutive model and comparison of laboratory and predicted behaviour of both projects rockfill materials.

Table 1 Rockfill Materials Used
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2 Experimental Investigations

2.1 Rockfill Materials Used

In the present research work modeled alluvial and blasted modeled quarried rockfill materials from Renuka dam site, India and Salma dam site, Afghanistan are obtained and used for research work. The details of the materials, location and rock types are given in Table 1. The alluvial rockfill material consist of rounded to sub-rounded particles in shape and quarried blasted rockfill material consist of angular to sub-angular in shape. The particle shapes of modeled alluvial (Renuka dam site) and blasted modeled quarried rockfill (Salma dam site) materials are shown in Figs. 1 and 2 respectively. Laboratory tests were conducted to determine the index properties of both types of rockfill materials and are given in Table 2. Prototype gradation rockfill material consists of maximum particle sizes of 1,000 and 600 mm for Renuka dam and Salma dam respectively. Rockfill material with such a large particle size is not feasible to test in the laboratory. Among all the available modeling techniques, the parallel gradation technique (Lowe 1964) is most commonly used and the same has been adopted in the present study to model the prototype rockfill materials. Modeled rockfill materials with dmax of 80, 50 and 25 mm for Renuka and Salma dam projects were obtained using parallel gradation technique. The prototype and modeled gradation curves for Renuka and Salma dam rockfill materials are shown in Figs. 3 and 4 respectively.

Fig. 1
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Modeled Alluvial Rockfill Materials from Renuka Dam project, Himachal Pradesh, India

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Fig. 2
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Modeled Quarried Blasted Rockfill Materials from Salma Dam project, Afghanistan

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Table 2 Index Properties of both Alluvial and Quarried Blasted Rockfill Materials
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Fig. 3
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Prototype and Modeled Grain Size Distribution Curves for Renuka Dam Project

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Fig. 4
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Prototype and Modeled Grain Size Distribution Curves for Salma Dam Project

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The values of index properties of alluvial rockfill material shows lessor values as compared to blasted quarried rockfill materials. This is due to the reason that the alluvial rockfill material formed as a result of disintegration of the hill rock in the upper himalaya, carried away the same by water, impact, rolling, sliding action in the riverbed and depositing the only hard core portion of the rock particles. However, the blasted rock is fresh, angular, having minor cracks and loose surface corners gives the index values higher as copmpared to alluvial rockfill material. As these values of both projects material are within the permissible limit, both materials can be used as rockfill material in the construction of rockfill dam.

2.2 Experimental Programme

2.2.1 Drained Triaxial Shear Tests

Relative density tests were conducted and determined minimum and maximum dry densities for both projects modeled rockfill material. Using minimum and maximum dry densities, the packing density of the specimen for triaxial shear test was determined corresponding to 87 % relative density. A specimen size of 381 mm diameter and 813 mm height is prepared and consolidated drained triaxial shear tests have been conducted on all the modeled rockfill materials collected from both the projects. Details of the triaxial equipment used in the present research work are given in Fig. 5. For testing, a dry density corresponding to 87 % relative density is adopted. Three confining pressures i.e. 0.4, 0.8 and 1.2 MPa were used for testing each dmax of modeled rockfill material. The quantities of various fractions of modeled rockfill materials required to achieve the gradation for preparing the specimen at the specified density was determined by weight. The specimen was prepared using a split mould and two rubber membranes of 1 mm inner and 3 mm outer. The specimen was compacted in six equal layers using mechanical vibratory compactor. The specimen was saturated by allowing water to pass through the base of the triaxial specimen and using a top drainage system for removing air voids. Conventional triaxial compression (CTC) tests were conducted in two phases; consolidation and shearing. In the first phase, required consolidation pressure is applied to the specimen and recorded the change in the volume of the specimen by allowing the water to drain out. In the second phase, keeping the confining pressure constant, the specimen was sheared till the failure under strain controlled test with a rate of loading 1 mm/min. During the test, for every strain level, deviator stress and volume change is recorded. The specimen is considered as failed when the specimen stops taking the load or when axial strain reaches 15 % whichever is earlier. From the test results, stress–strain-volume change behaviour was studied and presented.

Fig. 5
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Triaxial Compression Test Setup (381 mm dia. and 813 mm high)

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2.2.1.1 Stress–Strain-Volume Change Behaviour

The details of projects, maximum particle size (dmax), confining pressure, σ3, axial strain, ε1 and volumetric strain εv at failure for peak deviator stress are presented in Table 3. Stress–strain-volume change behaviour of six modeled rockfill materials tested for three confining pressures viz. 0.4, 0.8 and 1.2 MPa for Renuka dam and Salma dam projects are shown in Figs. 6, 7, 8, 9 and 10 and 11. From the deviator stress v/s axial strain relationship, it is observed that the behaviour of both materials is non-linear, inelastic and stress level dependent. For a dmax, the deviator stress and axial strain at failure increases with increase in confining pressure for both the materials. For a confining pressure, the deviator stress increases with increase in dmax for alluvial rockfill materials and it decreases with increase in dmax for quarried rockfill material. From the observed stress–strain curves of all the dmax tested, it is observed that the kind of mode of failure of specimen is brittle shear failure. The volume change behaviour shows compression during the initial stage of shearing and dilation effect is observed with further shearing for all the dmax. No strength reduction phenomenon or strain-softening behavior is observed when the stress state passes the phase changing line. The dilation effect decreases with increase in confining pressure and dmax for both the materials. Similar behavior has been observed and reported by Varadarajan et al. (1997, 1999, 2002, 2003, Varadarajan and Abbas 2006); Honkanadavar (2010); Honkanadavar and Sharma (2011, 2012a); Honkanadavar et al. (2012b).

Table 3 Axial and Volumetric Strains at Failure (Peak Deviator Stress) for Renuka Dam and Salma Dam Projects
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Fig. 6
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Stress–Strain-Volume Change Behaviour of 25 mm dmax for Renuka Dam Project Material. a Stress–strain behaviour b Volume change behaviour

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Fig. 7
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Stress–Strain-Volume Change Behaviour of 50 mm dmax for Renuka Dam Project Material. a Stress–strain behaviour b Volume change behaviour

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Fig. 8
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Stress–Strain-Volume Change Behaviour of 80 mm dmax for Renuka Dam Project Material. a Stress–strain behaviour b Volume change behaviour

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Fig. 9
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Stress–Strain-Volume Change Behaviour of 25 mm dmax for Salma Dam Project Material. a Stress–strain behaviour b Volume change behaviour

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Fig. 10
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Stress–Strain-Volume Change Behaviour of 50 mm dmax for Salma Dam Project Material. a Stress–strain behaviour b Volume change behaviour

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Fig. 11
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Stress–Strain-Volume Change Behaviour of 80 mm dmax for Salma Dam Project Material. a Stress–strain behaviour b Volume change behaviour

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2.2.1.2 Evaluation of Shear Strength Parameters

Mohr–Coulomb failure criterion has been adopted for determining the angle of internal friction at failure. Failure envelopes in mean stress (σm) v/s deviator stress (σ1 − σ3) space are plotted for all the dmax tested with different confining pressures for Renuka and Salma dam materials. The angle of internal friction, ϕ is determined from the best fit straight lines of these plots for both the modeled rockfill materials. Figure 12 shows a typical plot for Renuka dam site alluvial modeled rockfill material for maximum particle size (dmax) of 25 mm. It is observed that the cohesion, c for all the drained tests with a dmax of both project rockfill materials is observed as zero. The ϕ-values are tabulated in Table 4.

Fig. 12
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Relation between Mean stress (σm) v/s deviator stress (σ1-σ3) space for the dmax of 25 mm of Renuka dam materials

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Table 4 ϕ-values of Alluvial and Quarried Rockfill Materials
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2.2.2 Effect of Particle Size on Strength Parameter, ϕ

The strength of rockfill material used in the construction of rockfill dams is affected by number of factors such as mineral composition, particle size, shape, gradation, relative density and surface texture of the particles (Venkatachalam 1993; Gupta 2000; Abbas 2003; Honkanadavar 2010, 2011, 2012a, 2012b). The effect of maximum particle size, dmax on strength parameter, ϕ for both project materials are shown in the Fig. 13. It is observed that the angle of internal friction, ϕ increases with increase in dmax for Renuka alluvial and ϕ decreases with increase in dmax for Salma dam quarried rockfill material. The values of angle of internal friction for Shah Nehar project (alluvial), Purulia dam (quarried) and Parbati dam project (quarried) rockfill materials reported by Abbas (2003) are also superimposed in Fig. 13. They show similar trend as that of Renuka riverbed and Salma quarried rockfill material. Due to higher interlocking of angular particles, the strength of quarried rockfill material is more than the rounded rockfill material for the same dmax.

Fig. 13
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Variation of Angle of Internal Friction with Maximum Particle Size (dmax)

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3 Constitutive Modeling

Hierarchical single surface (HISS) model developed by Desai and co-workers (Desai and Siriwardhane 1980; Desai and Siriwardhane 1980; Desai et al. 1991) based on elasto-plastic theory has been used to characterize the behavior of rockfill materials. The non-associative δ1 model (Desai and Wathugala 1987; Desai 2001; Varadarajan et al. 1999, 2003, 2006; Gupta 2000; Abbas 2003; Honkanadavar 2010; Honkanadavar and Sharma 2013) gives the plastic potential function as

$$Q = \left[ {\frac{{J_{2D} }}{{P_{a}^{2} }}} \right] - \left[ { - \alpha_{Q} \left[ {\frac{{J_{1} }}{{P_{a} }}} \right]^{n} + \gamma \left[ {\frac{{J_{1} }}{{P_{a} }}} \right]^{2} } \right]\left( {1 - \beta S_{r} } \right)^{m}$$
(1)

where,

$$\alpha_{Q} = \, \alpha \, + \, \kappa \, \left( {\alpha_{0} {-} \, \alpha } \right) \, \left( {1 - r_{v} } \right)$$
(2)

in which κ is non-associative parameter, α0 is α (hardening parameter) at the beginning of shear loading and

$$r_{v} = \frac{{\xi_{v} }}{\xi }$$
(3)

where, ξ is plastic strain trajectory, and

$$\xi_{v} = \int {\frac{{\left| {d\varepsilon_{v}^{p} } \right|}}{\sqrt 3 }}$$
(4)

where, dɛ p v is the incremental plastic volumetric strain and ξv is volumetric part of ξ.

3.1 Material Parameters

The procedure for the determination of material parameters required in the HISS model has been described in detail in various references (Varadarajan and Desai 1993; Gupta 2000; Desai 2001; Abbas 2003; Varadarajan et al. 2003, 2006; Honkanadavar 2010). The procedure is briefly presented herein.

3.1.1 Elastic Parameters

The elastic parameter, modulus of elasticity (initial modulus) of rockfill material, E is determined from the observed stress–strain response using the hyperbolic relationship (Kondner 1963),

$$\left( {\sigma_{1} - \sigma_{3} } \right) = \frac{{\varepsilon_{1} }}{{a + b\varepsilon_{1} }}$$
(5)

where, (σ1 − σ3) is deviator stress, ε1 is axial strain, a is constant (= inverse of modulus of elasticity, E) and b is constant (= inverse of ultimate strength, (σ1 − σ3)ult). Equation (5) can be rewritten as

$$\frac{{\varepsilon_{1} }}{{\sigma_{1} - \sigma_{3} }} = a + b\varepsilon_{1}$$
(6)

The values of ε1/(σ1 − σ3) are calculated from the initial part of the stress–strain curve and are plotted against ε1. The intercept of the best fit line in the transformed plot is obtained as the value of the constant a. The reciprocal of the constant a gives the modulus of elasticity of rockfill material, E.

The modulus of elasticity, E is expressed as a function of the confining pressure (Janbu 1963) as

$$E = kP_{a} \left[ {\frac{{\sigma_{3} }}{{P_{a} }}} \right]^{{n^{\prime}}}$$
(7)

where, k and n ′ are the modulus number and modulus exponent of the material respectively, σ3 is the confining pressure and Pa is the atmospheric pressure. To determine the modulus number, k and modulus exponent, n ′, the plots between, E/Pa and σ3/Pa on log scale are made for all sizes of alluvial and quarried rockfill materials. The values of k and n ′ are determined as antilog of intercept on y-axis and slope of a best fit line respectively. The lateral strain is determined from the initial part of axial strain v/s volumetric strain curve and then Poisson’s ratio, ν is determined as the ratio of lateral strain to the axial strain.

3.1.2 Ultimate Parameters

For most of the geological materials m is found to be −0.5 (Desai et al. 1986). Therefore, in the present study, the value of m has been taken as −0.5. The ultimate parameters γ and β can be related with angle of internal friction, ϕ (Desai 2001) as

$$\beta = \frac{{1 - p^{2/m} }}{{1 + p^{2/m} }}$$
(8)

where,

$$p = \frac{{\tan \theta_{C} }}{{\tan \theta_{E} }}$$
(9)
$$\tan \theta_{C} = \left[ {\sqrt {\gamma \left( {1 - \beta } \right)^{m} } } \right]_{C} = \frac{2}{\sqrt 3 }\left[ {\frac{{\sin \varphi_{C} }}{{3 - \sin \varphi_{C} }}} \right]$$
(10)
$$\tan \theta_{E} = \left[ {\sqrt {\gamma \left( {1 + \beta } \right)}^{m} } \right]_{E} = \frac{2}{\sqrt 3 }\left[ {\frac{{\sin \varphi_{E} }}{{3 + \sin \varphi_{E} }}} \right]$$
(11)

In the present analysis, triaxial compression tests are conducted and friction angle in compression (φ C ) has been determined for all the dmax of alluvial and quarried modeled rockfill materials. The friction angle in extension (φ E ) is assumed equal to the friction angle in compression i.e. φ C  = φ E (Gupta 2000; Abbas 2003; Honkanadavar 2010). First, β is determined from Eq. (8) and then γ is computed using Eq. (10) or (11) for the alluvial and quarried modeled rockfill materials tested.

3.1.3 Phase Change Parameter

The phase change parameter, n is related to the state of stress at which the material starts dilating. Therefore, the point at which dilation starts is corresponding to dɛ p v  = 0 and therefore, the phase change parameter, n can be determined as

$$n = \frac{2}{{\left[ {1 - \left[ {\frac{{J_{2D} }}{{J_{1}^{2} }}} \right]} \right]\frac{1}{{F_{S} \gamma }}}}$$
(12)

where, J 1 is first invariant of stress tensor, J 2D is the second invariant of deviator stress tensor, F s is shape function and γ is the ultimate parameter. The point at which dɛ p v  = 0 is determined and n is calculated using Eq. (12). An average of all n values for different tests conducted for a dmax is taken as overall value of n for that size of rockfill material. Similarly, n value is calculated for all the dmax of alluvial and quarried modeled rockfill materials.

3.1.4 Hardening Parameters

In the present study, α is assumed as the function of ξ as

$$\alpha \, = \, \frac{{a_{1} }}{{\xi^{\eta }_{1} }}$$
(13)

Taking natural log on both sides of Eq. (13) gives,

$$\ln \, \left( \alpha \right) \, = \, \ln \, \left( {a_{1} } \right) \, {-} \, \eta_{1} \ln \, \left( \xi \right)$$
(14)

A plot α v/s ξ is drawn on logarithmic scale to get a1 and η1 as antilog of intercept and slope respectively of the best fit line. The hardening parameters for all the dmax of alluvial and quarried modeled rockfill materials were computed.

3.1.5 Non-associative Parameter

The non-associative parameter, κ in Eq. (2) is determined based on the following equation:

$$\left( {\frac{{d\varepsilon_{v}^{p} }}{{d\varepsilon_{11}^{p} }}} \right) = \frac{{\left( {3\frac{\partial Q}{{\partial J_{1} }}} \right)}}{{\frac{\partial Q}{{\partial \sigma_{11} }}}}$$
(16)

where, dɛ p11  = axial plastic strain increment; σ 11 = axial stress, and dɛ p v  = volumetric plastic strain increment. The ratio of dɛ p v /dɛ p11 can be obtained from the slope of the observed dɛ p11 v/sdɛ p v response by choosing a point in the ultimate state. The value of αQ [from Eq. (2)] which is represented on the right-hand side of Eq. (16) can then be found as the left-hand side is now known. Using this value along with α and rv at ultimate condition, the average value of κ is determined.

Following the above mentioned procedure, material parameters for both Renuka dam alluvial and Salma dam quarried rockfill materials have been determined and are presented in the Table 5. Using the determined material parameters, stress–strain–volume change behaviour for all the dmax tested with different σ3 is predicted using HISS model. The predicted behaviour is compared with the observed behaviour from the laboratory triaxial shear tests. From the comparison, it is observed that both results match closely. The observed and predicted stress–strain–volume change behaviour for both projects rockfill material is shown in Figs. 14, 15, 16, 17, 18 and 19.

Table 5 Material Parameters for HISS Model
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Fig. 14
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Observed and Predicted Stress-Strain-Volume Change Behaviour of 25 mm dmax for Renuka Dam Project Material

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Fig. 15
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Observed and Predicted Stress-Strain-Volume Change Behaviour of 50 mm dmax for Renuka Dam Project Material

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Fig. 16
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Observed and Predicted Stress-Strain-Volume Change Behaviour of 80 mm dmax for Renuka Dam Project Material

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Fig. 17
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Observed and Predicted Stress-Strain-Volume Change Behaviour of 25 mm dmax for Salma Dam Project Material

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Fig. 18
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Observed and Predicted Stress-Strain-Volume Change Behaviour of 50 mm dmax for Salma Dam Project Material

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Fig. 19
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Observed and Predicted Stress-Strain-Volume Change Behaviour of 80 mm dmax for Salma Dam Project Material

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4 Conclusions

Two types of modeled rockfill materials viz. alluvial from Renuka dam, India and blasted quarried from Salma dam, Afghanistan were collected and consolidated drained triaxial tests were conducted. Stress–strain–volume change behaviour for all the dmax were studied. It is observed that the stress–strain behaviour is non-linear, inelastic and stress level dependent for all the dmax of both project materials. The deviator stress and the axial strain at failure increases with increase in confining pressure for all the tested materials. It is also observed that the effect of σ3 is same on both types of material. From the volume change behaviour, it is observed that the volume decreases during the initial shearing and with further shearing material experiences dilation effect. This dilation effect reduces with increase in σ3 and dmax for both projects material. The shear strength parameter, angle of internal friction (ϕ) increases with increase in dmax for alluvial rockfill material and reverse trend is observed for blasted quarried rockfill material.

Rockfill materials have been characterised by using heirarchical single surface (HISS) model based on elasto-plasticity. Stress–strain–volume change behaviour has ben predicted for all the dmax of both alluvial and quarried rockfill materials. The predicted and observed (laboratory test results) stress–strain–volume change results were compared. From the comparison, it is observed that both results match closely. Hence, it is suggested that the HISS model can be used successfully to characterise both alluvial and blasted quarried rockfill materials.