Computers and Geotechnics · 2020. 10. 11. · τ σ II I I const 9 9 SMP SMP 12 3 3 (13) where τ SMP and σ SMP are the shear and normal stresses on the SMP, and I 1, I 2 and I

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Computers and Geotechnics

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Research Paper

Practical nonlinear constitutive model for rockfill materials with applicationto rockfill dam

Si-hong Liua, Yi Suna,⁎, Chao-min Shena, Zhen-Yu Yinb

a College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing 210098, ChinabDepartment of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

A R T I C L E I N F O

Keywords:Rockfill materialsDilatancyIntermediate principal stressNonlinear constitutive model

A B S T R A C T

In this paper, a nonlinear constitutive model for rockfill materials is proposed to account for the coupling in-fluence of the mean effective stress p and the deviatoric stress q on the deformation of rockfill materials. In themodel, the stress-dilatancy relationship derived from the microstructural changes of granular materials isadopted, and the strength nonlinearity of rockfill materials is considered by using a logarithmic relationshipbetween the peak friction angle and the mean effective stress. The SMP criterion is incorporated into the modelto consider the influence of the intermediate principal stress. The good performance of the proposed model isdemonstrated through modelling triaxial tests on rockfill materials from a rockfill dam. In addition, the FEMsimulated deformation of a real CFRD using the proposed model agrees well with the monitored data.

1. Introduction

Rockfill dams have been widely adopted due to the inherent flex-ibility and adaptability to different foundation conditions. In addition,due to the increasing construction technology, the rockfill dams havebecome the most economical dam type. As the main component of thedams, rockfill materials is of profound importance to the stability andthe safe operation of the dams. In general, the strength and deformationproperties of rockfill materials are very complicated [1–4]. For ex-ample, peak shear strength decreases with the confining stress in-creasing, exhibiting a non-linear function of confining stress; shear in-duced volume deformation (contraction or dilatation) is not negligible.Apparently, an ideal constitutive modelling of rockfill materials needreasonably reflect these complex behaviors.

So far, many constitutive models for rockfill materials have beendeveloped. These models can be classified as (1) nonlinear hypoelasticmodels [5,6], (2) incrementally nonlinear models [7,8], (3) elasto-plastic models [9–17], (4) hypoplastic models [18–21], and (5) mi-cromechanics-based models [22–25]. In general, the first four cate-gories are phenomenological models, which are commonly adopted inengineering practice due to their efficiency in finite-element analyses.

Nonlinear hypoelastic models, especially Duncan-Chang Model [5]and K-G models [6,26–28], are mostly adopted in the finite-elementanalyses of rockfill dams owing to the simplicity and easily-under-standable concept of the models. However, the Duncan-Chang Model

cannot take the influence of the intermediate principal stress intoconsideration as the Mohr-Coulomb failure criterion of soils is used.Moreover, the dilatancy of rockfill materials cannot be reflected in themodel as the shear-induced volume change is not considered. As a re-sult, the Duncan-Chang Model sometimes overestimates the settlementsof rockfill dams when it is adopted in the finite element analysis forrockfill dams, especially for medium and low dams. Domaschuk andVilliappan firstly proposed a K-G model in 1975 [6], and later manyimprovements [26–28] have been made to describe the coupling in-fluences of mean effective stress p and deviatoric stress q on the de-formation of rockfill material.

This paper presents a simple nonlinear constitutive model forrockfill materials that can consider the coupling influence of the meaneffective stress p and the deviatoric stress q and the influence of theintermediate principal stress on the deformation of rockfill materials.The determination method for the model parameters is suggested.Afterwards, the proposed constitutive model is adopted in finite ele-ment method to simulate the deformation of a real concrete-facedrockfill dam (CFRD) with adequate monitoring data.

2. Nonlinear constitutive model

2.1. Framework of the model

In this study, a nonlinear hypoelastic constitutive model for rockfill

https://doi.org/10.1016/j.compgeo.2019.103383Received 10 July 2019; Received in revised form 20 August 2019; Accepted 2 December 2019

⁎ Corresponding author.E-mail address: [email protected] (Y. Sun).

Computers and Geotechnics 119 (2020) 103383

0266-352X/ © 2019 Elsevier Ltd. All rights reserved.

T

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materials attempts to be developed. In the constitutive modeling, thegeneral relationship between incremental strains and incrementalstresses is given by

=dε C σ dσ( )kl ijkl mn kl (1)

where Cijkl are complementary constitutive tensors (or moduli) that arestress level dependent. The stress-strain behavior is more convenientlydescribed using the parameters p, q, εv and εs defined as

=

= − + − + −= + +

= − + − + −

⎫

⎬

⎪⎪

⎭

⎪⎪

+ +p

q σ σ σ σ σ σε ε ε ε

ε ε ε ε ε ε ε

( ) ( ) ( )

( ) ( ) ( )

σ σ σ

v

s

( )3

12 1 2

22 3

23 1

2

1 2 32

3 1 22

2 32

3 12

1 2 3

(2)

In p-q stress space, Eq. (1) can be derived as

= +dεdpK

dqJv 1 (3a)

= +dεdpJ

dqGs 2 (3b)

where K is a bulk modulus, representing the volumetric stiffness withrespect to dp; J1 is a shear dilatancy modulus (coupling modulus), ac-counting for the volumetric strain produced by an increment dq; G is ashear modulus that controls shear strain with respect to dq; and J2 isanother coupling modulus, accounting for the shear strain produced byan increment dp. Generally, the coupling moduli J1 and J2 are different,and it’s not easy to determine them separately from experimentallyobserved stress-strain data. For the sake of simplicity, either = ∞J2 or

= =J J J1 2 was assumed by some scholars [26,28]. For the assumptionof = ∞J2 , the model cannot incorporate shear strains generated bychanges in mean effective stress p. Also, the assumption of = ∞J2 leadsto the non-symmetry of the general matrix of the model. As a result, it isdifficult to use some existing efficient linear equation solvers in FEM.These two shortcomings can be avoided in the assumption of

= =J J J1 2 . Therefore, this paper adopted the assumption of = =J J J1 2 ,that is to say, the −dp dεs coupling and the −dq dεv coupling arecontrolled by the same J modulus. Positive dilatancy, that is, expansionduring shearing, is associated with J < 0. Compression duringshearing produces J > 0.

2.2. Derivation of hypoelastic K, G and J modulus functions

2.2.1. Bulk modulus KThe bulk modulus K relates the volumetric strain εv to the mean

effective stress p, which is commonly determined though isotropiccompression tests. The results of isotropic compression tests on rockfillmaterials indicate that the relationship between the volumetric strain εvand the mean effective stress p is more reasonably expressed with anexponential function [29]

⎜ ⎟ ⎜ ⎟= ⎡

⎣⎢

⎛⎝

⎞⎠

− ⎛⎝

⎞⎠

⎤

⎦⎥ε C

pp

ppv t a

n

a

n0

(4)

where C n,t are the parameters fitting experimentally observed stress-strain data; p0 is an isotropically initial stress and pa is the atmosphericpressure.

Differentiating Eq. (4) yields

= −dε C npp

dpp

( )v ta

n

a

1

(5)

Then, the bulk modulus K can be obtained from its definition

= = −Kdpdε

pC n

pp

( )v

a

t a

n1

(6)

Assuming that =K C n1/( )b t and = −n n11 , Eq. (6) is rewritten as

⎜ ⎟= ⎛⎝

⎞⎠

K K Pppb a a

n1

(7)

2.2.2. Shear modulus G and coupling modulus JIt is noted that the shear modulus G in Eq. (3b) controls the shear

strain with respect to dq under dp = 0. Ideally, it should be determinedfrom the shear tests under the constant mean effective stress p, whichare seldom carried out in practice. Usually, conventional triaxial testsare carried out to determine the model parameters. Therefore, beforegiving the expression for G, we define first the shear modulus de-termined from conventional triaxial tests as GTC.

Rockfill materials exhibit a nonlinear frictional behavior with anasymptotic relationship between the deviatoric stress q and the shearstrain εs. Here, the −q εs relation measured in conventional triaxialtests is supposed to be fitted with a hyperbolic function, expressed as

=+

q εa bε

s

s (8)

where a and b are the constants whose values can be determined ex-perimentally. To be more specific, they can be related to the initialtangential shear modulus GTCi ( =a G1/ TCi) and the asymptotic value qultof the deviatoric stress q where the curve −q εs approaches at infinitestrain ( =b q1/ ult), respectively.

It is commonly found that the asymptotic value of the deviatoricstress q is larger than the shear failure strength qf by a small amount.This would be expected, because the hyperbola remains below theasymptote at all finite values of strain. The asymptotic value qult may berelated to the shear failure strength qf, however, by means of a factor Rfas shown by

=q R qf f ult (9)

Under the triaxial condition, the shear modulus =G dq dε/TC s by thedefinition, can be obtained by differentiating Eq. (8) and combining Eq.(9):

= = − = −Gdqdε a

bq G Rqq

1 [1 ] (1 )TCs

TCi ff

2 2

(10)

Experimental studies have shown that the initial tangential shearmodulus GTCi varies with the mean effective stress p. Referring toJanbu’s study [30], it may be expressed as

⎜ ⎟= ⎛⎝

⎞⎠

G K PpPTCi G a a

n2

(11)

where KG is a material modulus, and n2 is the exponent determining therate of variation of GTCi with p. Both KG and n2 are dimensionless andmay be determined readily from the results of a series of triaxial tests byplotting the values of GTCi against p on log-log scales and fitting astraight line to the data.

In Eq. (10), the shear failure strength qf is usually related to themean effective stress p, which depends on the failure criterion. Thecriterion of the Extended Mises type =q M pf f was adopted for the shearyield and shear failure of soils in the Cam-clay model, and many othermodels, where =M φ φ6 sin /(3 - sin )f under the triaxial condition (φ isthe peak friction angle). If the criterion of the Extended Mises type isadopted, substituting Eq. (11) into Eq. (10) yields

⎜ ⎟⎜ ⎟= ⎛⎝

⎞⎠

⎛⎝

− ⎞⎠

G K PpP

Rq

M p1TC G a

a

n

ff

22

(12)

However, as exprimental evidence shows, the Extended Mises cri-terion grossly overestimates strength in triaxial extension, and alsoresults in incorrect intermediate stress ratios in π plane. It is known thatthe failure of soil can be reasonably explained by the SMP criterion[31], which is written as

S.-h. Liu, et al. Computers and Geotechnics 119 (2020) 103383

2

= − =τσ

I I II

const99

SMP

SMP

1 2 3

3 (13)

where τSMP and σSMP are the shear and normal stresses on the SMP, andI1, I2 and I3 are the first, second and third stress invariants.

In this study, we adopt the SMP criterion instead of the ExtendedMises criterion. To this end, the transformed stress tensor σ~ij proposedby Yao et al. [11,12] is used, which can transform the SMP criterioninto an Extended Mises type criterion in the new stress transformedstress space. The transformed stress tensor σ~ij is expressed as

= + −

=

⎫

⎬⎭− − −

σ pδ σ pδ

q

~ ( )ij ijqq ij ij

cI

I I I I I I2

3 ( ) / ( 9 ) 1

c

1

1 2 3 1 2 3 (14)

By using the transformed stress tensor σ~ij, the SMP criterion can beexpressed as

=q M p~ ~f f (15)

Matsuoka et al. [32] introduced the SMP criterion into the Cam-claymodel by replacing the stress tensor σij with the transformed stresstensor σ~ij. Similarly, the SMP criterion (Eq. (15)) is incorporated into theshear modulus of Eq. (12) through σ~ij, leading to

⎜ ⎟⎜ ⎟= ⎛⎝

⎞⎠

⎛⎝

− ⎞⎠

G K PpP

Rq

M p

~1

~~TC G a

a

n

ff

22

(16)

By considering =G dq dε/TC s, Eq. (3b) can be rewritten as

= +G J

dpdq G

1 1 1TC (17)

Combining Eq. (3a) and Eq. (3b), one can obtain

⎜ ⎟⎛⎝

+ ⎞⎠

= +dεdε J G

dqdp K J

dqdp

1 1 1 1vs (18)

Assuming =ξ dq dp/ and =D dε dε/v s, we can derive the shearmodulus G and the coupling modulus J from Eqs. (17) and (18).

=− +

GKG ξ

Kξ DKξ GTC

TC

2

2 (19)

=−

JKξG

DKξ GTC

TC (20)

Fig. 1 gives the typical stress-strain relation of granular materialsduring shearing in a triaxial test. It demonstrates that the sample isusually compressive at the beginning of shearing and gradually turns tobe dilative. The stress ratio η(=q/p) corresponding to the phasetransformation point from contraction to dilatation is denoted as M,which can be related to the phase transformation friction angle ψ with

= −M ψ ψ6 sin /(3 sin ). Considering this typical stress-strain

characteristic and the microstructure change of granular materials, Liuet al. [33] proposed a stress-dilatancy equation, expressed as

= =−

+

+ +D dε

dεmM mη

m η( 1)v

s

m m

m

1 1

(21)

where =η q p/ , and m is an experimentally fitting constant. Eq. (21) canreasonably describe the volumetric change of granular materials fromthe initial compression ( >dε 0v when <

Fig. 2. Comparison between experimental and predicted results of triaxial tests on Tankeng CFRD materials: (a) Cushion; (b) Transition; (c) Rockfill I; (d) Gravel; (e)Rockfill II; (f) Alluvium (Sand gravel).

S.-h. Liu, et al. Computers and Geotechnics 119 (2020) 103383

4

by fitting the curve −q εs of the drained tests under different confiningstresses, and the stress-dilatancy parameter (m) can be measured basedon the experimental −D η curve. The peak friction angle and the phasetransformation friction angle φ and ψ can be determined from the stressratios at the peak stress state and phase transformation state using

=M φ φ6 sin /(3 - sin )f and = −M ψ ψ6 sin /(3 sin ), respectively. Thefriction angle parameters (φ0, φΔ , ψ0, ψΔ ) can be obtained by fitting thecurves −φ p and −ψ p.

2.5. Experimental verification

A series of drained triaxial tests have been conducted on the con-struction materials and the site alluvium of the Tankeng CFRD that aredescribed in detail in Section 3.1. The triaxial test results in Fig. 2 de-monstrate that the shearing dilatation is more significant under lowconfining pressures compared to under high confining stresses. Theyare simulated using the proposed model with the set of model para-meters listed in Table 1. As shown in Fig. 2, the model responses arebroadly in good agreement with the experimental data, indicating theperformance of the proposed model with the simplification ( = =J J J1 2 )and the stress-dilatancy equation proposed by Liu et al. [33].

3. Application

3.1. The Tankeng project

As an example, the proposed nonlinear constitutive model was ap-plied in the FEM analysis of the Tankeng concrete-faced rockfill dam(CFRD). The Tankeng CFRD of 162 m in height, is located on the middlereaches of the Oujiang River, Zhejiang Province, China. The layout ofthe dam is shown in Fig. 3. The dam crest is 505 m long and 12 m wide.The upstream concrete face is composed of into 42 slabs. Each slab is12 m wide and the adjacent slabs are waterproofed with vertical joints.The slabs are connected to the toe slabs located on the abutmentsthrough peripheral joints.

Fig. 4 shows the typical cross-section of the dam corresponding tosection I-I in the layout. The upstream and downstream slopes of thedam are 1:1.4 and 1:1.55 (average), respectively. The thickness d of theconcrete face slab is 0.3 m at the top elevation of 167 m and varieslinearly with a function of = +d H0.3 0.0035 downwards the slope,where H is the vertical distance to the top elevation of 167 m. Behindthe concrete face slab, a cushion zone with a horizontal width of 3.0 mis placed, which is composed of sands and gravels with a maximumgrain size of 6 cm. Between the cushion zone and the main rockfill zone,a 5.0 m horizontally wide transition zone with a maximum grain size of30 cm is provided to prevent fine particles of the cushion zone fromentering the pores of the rockfill. The main dam body consists of rockfillI, gravel and rockfill II zones. The rockfill I zone provides a support for

Table 1The nonlinear model parameters for different materials of Tankeng CFRD.

Material Dry density (kN/m3) m φ0 φΔ ψ0 ψΔ Kb n1 KG n2 Rf

Cushion 22 0.75 57.0° 11.8° 43.7° 1.4° 129 0.33 1515 0.42 0.79Transition 21.5 0.72 50.7° 11.7° 43.9° 1.2° 513 0.16 1452 0.38 0.63Rockfill I 21.2 0.85 51.3° 12.2° 44.7° 1.2° 380 0.15 1288 0.46 0.65Gravel 21.5 0.67 51.7° 10.2° 44.8° 1.7° 717 0.19 1099 0.48 0.71Rockfill II 20.7 0.71 53.2° 12.5° 47.1° 1.7° 176 0.58 1369 0.31 0.63Alluvium 20.2 0.86 52.8° 6.3° 46.2° 3.5° 126 0.19 821 0.44 0.89

Fig. 3. Layout of the Tankeng CFRD.

S.-h. Liu, et al. Computers and Geotechnics 119 (2020) 103383

5

hydrostatic loads transmitted from the cushion zone and transitionzone. The gravel zone is surrounded by rockfill I and II zones since ithas relatively lower shear strength. The dam foundation has 20–30 mthick alluvium of sand gravels. The gradations of the dam constructionmaterials and the foundation alluvium are given in Fig. 5.

3.2. FE analysis model

Fig. 6 presents the FE model for the Tankeng dam considering theconstruction and the first impounding sequence. It is noted that the FEmodel contains the foundation alluvium and the bottom of the alluviumlayer is vertically restrained in the calculation. The construction stagewas simulated by 13 loading steps following exactly the constructionprocedure of the project (1–11, 13–14 steps). The hydrostatic load wasapplied on the upstream surface in 2 steps (12, 15 steps), in accordance

with the reservoir level records, to reflect the impact of the first im-pounding and water level fluctuation during 2008–2012. In summary,the three-dimensional FE model contains 7162 eight-node elements, 15loading steps and six rockfill materials.

The rockfill materials of the Tanleng CFRD were described using theproposed nonlinear model, and the model parameters are listed inTable 1. The concrete face slab was described using the linear elasticmodel with an elastic module of 30 GPa, Poisson’s ratio of 0.167 anddensity of 25 kN/m3. The interface between the concrete face slab andthe cushion layer was described using Goodman elements and thetangential stress-displacement relationship at the interface was char-acterized by the Clough-Duncan model, the details of which were seenin [39–41]. The parameters of the Clough-Duncan model K0, n, Rf and φadopted in the calculation are 3500, 0.56, 0.74 and 36°, respectively.The vertical and peripheral joints were simulated using a pair of nodesthat can be combined into a single node due to compressive stress ap-plication and separated into two independent nodes due to tensile stressapplication.

In 3D FE analysis, the constitutive relation is usually expressed inVoigt stress space, which has been given in Appendix A for the proposedconstitutive model. A incremental algorithm is used to solve non-linearfinite element equations and a symmetric successive over relaxation(SSOR) method is used to solve finite element control equation in thisFE analysis.

3.3. Results and discussion

Fig. 7 shows the contours of the calculated dam body deformation atthe maximum cross section after the first impounding. It can be seen

Fig. 4. Typical cross section of Tankeng CFRD.

Fig. 5. Gradation curves of alluvium and dam construction materials.

Fig. 6. Construction stages and three-dimensional mesh of Tankeng CFRD.

S.-h. Liu, et al. Computers and Geotechnics 119 (2020) 103383

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that the maximum settlement occurred nearly in half of the dam height(Fig. 7a). The maximum settlement of 116 cm accounts for 0.72% of thedam height, which is within the range as observed in most of CFRDs[37]. Under the action of the water pressure on the upstream concreteface slabs, the overall trend of the horizontal displacement is towardsthe downstream with the maximum magnitude of 22 cm at the down-stream (Fig. 7b). Fig. 8 presents the contours of the calculated concreteface slab deflection after the first impounding. The maximum deflectionis 33 cm, occurring in the middle of the upstream face slabs. Reference[37] presents the face slab deflection measured in the 87 case historieswith respect to dam heights. Statistical results show that the face slabdeflection in most cases is less than 0.40% of the dam height, and morethan half cases are less than 0.2% of the dam height. Very few cases that

were constructed using low-strength rockfills exhibit the face slab de-flection values up to 0.6% of the dam height. The calculated face slabdeflection 33 cm of the Tangken CFRD accounts for 0.2% of the damheight, within the range of the statistical results for most of CFRDs [37].Fig. 9 compares the calculated dam settlement and the slab deflectionwith the monitored data at the maximum cross-section after the firstimpounding. The settlement at the monitoring point V3-2 was notmeasured because the monitoring gauge had been damaged before thecompletion of the dam. Anyway, it can be observed that both the cal-culated settlements of the dam and the calculated deflection of theconcrete face slab agree basically with the monitored ones.

Fig. 10 compares the simulated settlement evolution at the mon-itoring point V2-3 in the middle of the maximum cross section with themonitored data. It demonstrates that the calculated settlement evolu-tion at the point V2-3 agrees basically with the monitored one with asignificant increase during the construction and an insignificant in-crease under the action of the water filling. As the creep deformation ofthe rockfills was not taken into account in the calculation, the calcu-lated settlement-time curve was slightly lower than the monitored onein Fig. 10.

Fig. 11 shows the comparison of the calculated dam settlement withthe monitored data along the longitudinal section V-V after the firstimpounding. The settlement values are presented in the form of frac-tional numbers beside the monitoring points, in which the numeratorand the denominator denote the monitored values and the calculatedvalues, respectively. It can be seen that the calculated settlement ateach monitoring point (VC1 to VC8) is close to the monitored one. Thecalculated maximum settlement occurs nearly in the middle of the damheight at the maximum section. As we know, for the dam built directlyon rock foundations, the maximum settlement of the dam would occurat nearly 2/3 dam height. In the calculated project, the dam was builton an alluvium foundation as shown in Fig. 4. The weight of the dambody and the water pressure will induce the settlement deformation ofthe alluvium foundation, leading to the downward movement of thelocation for the maximum settlement of the dam body. So, the max-imum settlement of the dam body shown in Figs. 7(a) and 11 occurnearly in the middle of the dam height, as reported in [42–44].

In this rockfill dam, the horizontal displacements on the down-stream slope have been monitored with a relatively high accuracy. Themonitored horizontal displacements after the first impounding agreeroughly with the calculated ones, as shown in Fig. 12. The maximumhorizontal displacement occurred near the 2/3-height of the dam.Along the dam height, the distribution of the horizontal displacementson the downstream slope is similar to that of the deflections of the faceslabs.

In summary, both the calculated deformation of the dam body andthe deflection of the concrete face slabs agree roughly with the

Fig. 7. Contours of the calculated deformation of the dam body at the max-imum section after the first impounding (unit: cm).

2823

1813 8

3

33

Fig. 8. Contours of the calculated concrete face slab deflection after the firstimpounding (unit: cm).

Fig. 9. Comparison of the numerically calculated dam settlement and the slab deflection with the measurement at the maximum cross section after the firstimpounding.

S.-h. Liu, et al. Computers and Geotechnics 119 (2020) 103383

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monitored ones, indicating the rationality of the proposed nonlinearconstitutive model for rockfill materials.

4. Conclusions

(1) A nonlinear constitutive model for rockfill materials that can reflectthe coupling influence of the mean effective stress p and the de-viatoric stress q on the deformation of rockfill materials was pro-posed. The model is of simple form but can account for the dila-tancy behavior, the strength nonlinearity of rockfill materials aswell as the influence of the intermediate principal stress. There are10 parameters involved in the model, which can be determined byconventional tests.

(2) The validity of this nonlinear constitutive model was verified bymodelling the triaxial tests on 6 kinds of rockfill materials used in

the Tankeng CFRD. This model was confirmed to be effective inreproducing basic features of rockfill materials, such as volumetricchange due to dilatancy and a nonlinear frictional behavior.

(3) This model was applied in the 3D FE calculation for the TankengCFRD. The calculated deformation of the dam body and the de-flection of the concrete face slabs are in good agreement with thein-situ measurements, indicating that the proposed model couldeasily implemented in a 3D FE simulation and is able to capture themain mechanical responses of rockfill materials in a rockfill dam.

Acknowledgements

This work was supported by the “National Key R&D Program ofChina” (Grant No. 2017YFC0404800), and the “National NaturalScience Foundation of China” (Grant Nos. U1765205 and 51979091).

Fig. 10. Settlement–time curves at the monitoring point V2-3.

Fig. 11. Comparison of the calculated dam body settlement with the measurement along the longitudinal section V-V after the first impounding.

Fig. 12. Comparison of the calculated horizontal displacements with the monitored ones on the downstream slope after the first impounding.

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Appendix A

In FE calculation, the constitutive model established in p-q stress space should be expressed in Voigt stress space.Eq. (1) is rewritten as

=dσ D σ dε( )ij ijkl mn kl (A.1)

where Dijkl is the inverse of Cijkl that are stress level dependent. Under the isotropic condition, Dijkl can be expressed as

= + + + + + + + ++ + + + + +

+ + + +

D σ A δ δ A δ δ δ δ A σ δ A δ σ A δ σ δ σ δ σδ σ A δ σ σ A δ σ σ A δ σ σ δ σ σ δ σ σδ σ σ A σ σ A σ σ σ A σ σ σ A σ σ σ σ

( ) ( ) () (

)

ijkl mn ij kl ik jl jk il ij kl ij kl ik jl il jk jk il

jl ik ij km ml kl im mj ik jm ml il jm mk jk im ml

jl im mk ij kl ij km mi im mj kl im mj kn nl

1 2 3 4 5

6 7 8

9 10 11 12 (A.2)

where ⋯A A A, , ,1 2 12 are coefficients related to stress invariants, and δ is the Kronecker delta (when i = j, =δ 1ij ; when ≠i j, =δ 0ij ). It is assumedthat the coefficients A5 to A12 are related to higher-order stress invariants and their values equal zero. Then Eq. (A.2) can be simplified as

= + + + +D σ A δ δ A δ δ δ δ A σ δ A δ σ( ) ( )ijkl mn ij kl ik jl jk il ij kl ij kl1 2 3 4 (A.3)

Substituting Eq. (A.3) into Eq. (A.1) yields

= + + +dσ A δ dε A dε A σ dε A δ σ dε2ij ij kk ij ij kk ij kl kl1 2 3 4 (A.4)

Under the triaxial stress state, Eq. (A.4) can be expressed as

= + + + += + + + +

=

⎫

⎬⎭

dσ A dε A dε A σ dε A σ dε σ dεdσ A dε A dε A σ dε A σ dε σ dε

dσ dσ

2 ( 2 )2 ( 2 )

kk kk

kk kk

11 1 2 11 3 11 4 11 11 22 22

22 1 2 22 3 22 4 11 11 22 22

33 22 (A.5)

and the increments of p, q, εv and εs can be written as

== −= +

= −

⎫

⎬

⎪⎪

⎭⎪⎪

+dpdq dσ dσdε dε dε

dε dε dε2

( )

dσ dσ

v

s

( 2 )3

11 33

11 3323 11 33

11 33

(A.6)

Combining Eq. (A.5) and Eq. (A.6) yields

= + + + +

= +

⎫⎬⎭

( )dp A A pA pA dε A qdεdq A qdε A dε3

v s

v s

123 2 3 4 4

3 2 (A.7)

The inverse expression of Eq. (3) can be written as

= −= − +

⎫⎬⎭

dp Kdε Jdεdq Jdε Gdε

¯ ¯¯ ¯

v s

v s (A.8)

where moduli K̄ , J̄ , Ḡ can be represented by K , J , G as follows

=

=

=

⎫

⎬

⎪⎪

⎭⎪⎪

−

−

−

K K

G G

J

¯

¯

¯

JJ KG

JJ KGKGJ

J KG

22

22

2 (A.9)

From Eqs. (A.7) and Eq. (A.8), coefficients A1–A4 can be obtained

= − +

=

= = −

⎫

⎬

⎪

⎭⎪

A K G J

A

A A

¯ ¯ ¯pq

G

Jq

129

2

2¯3

3 4¯

(A.10)

Substituting Eq. (A.10) into Eq. (A.4) yields

⎜ ⎟= ⎛⎝

− + ⎞⎠

+ − −dσ K Gp

qJ δ dε Gdε J

qσ dε J

qδ σ dε¯ 2

9¯ 2 ¯ 2

3¯ ¯ ¯ij ij kk ij ij kk ij kl kl

(A.11)

Eq. (A.11) is the expression of stress-strain relationship in Voigt stress space, which can also be written in a matrix form

=dσ D dε{ } [ ]{ } (A.12)

which can be expanded as

S.-h. Liu, et al. Computers and Geotechnics 119 (2020) 103383

9

⎧

⎨

⎪⎪

⎩

⎪⎪

⎫

⎬

⎪⎪

⎭

⎪⎪

=

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

⎫

⎬

⎪⎪⎪

⎭

⎪⎪⎪

dσdσdσdσdσdσ

D D D D D DD D D D D DD D D D D DD D D DD D D DD D D D

dεdεdεdγdγdγ

0 00 00 0

11

22

33

12

23

31

11 12 13 14 15 16

21 22 23 24 25 26

31 32 33 34 35 36

41 42 43 44

51 52 53 55

61 62 63 66

11

22

33

12

23

31 (A.13)

where [D] is a symmetric stiffness matrix, and items in the matrix can be expressed as

= + = = + += + = = + += + = = + +

= = =

= = = = = =

= = = = = =

= = = = = =

⎫

⎬

⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪

−

−

−

D α α D D α α αD α α D D α α αD α α D D α α α

D D D

D D D D D D

D D D D D D

D D D D D D

2 ;2 ;2 ;

G

Jσq

Jσq

Jσq

11 1 3 12 21 2 3 4

22 1 4 23 32 2 4 5

33 1 5 31 13 2 3 5

44 55 66¯3

41 42 43 14 24 34¯

51 52 53 15 25 35¯

61 62 63 16 26 36¯

12

23

31(A.14)

in which

= +

= −

= + −

= + −

= + −

⎫

⎬

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

α K G

α K G

α σ σ σ

α σ σ σ

α σ σ σ

¯ ¯

¯ ¯

( 2 )

( 2 )

( 2 )

Jq

Jq

Jq

149

229

3¯

3 22 33 11

4¯

3 11 33 22

5¯

3 22 11 33 (A.15)

References

[1] El Dine BS, Dupla JC, Frank R, Canou J, Kazan Y. Mechanical characterization ofmatrix coarse-grained soils with a large-sized triaxial device. Can Geotech J2010;47(4):425–38.

[2] Xiao Y, Liu H, Chen Y, Jiang J, Zhang W. Testing and modeling of the state-de-pendent behaviors of rockfill material. Comput Geotech 2014;61:153–65.

[3] Xiao Y, Liu H, Chen Y, Jiang J. Strength and deformation of rockfill material basedon large-scale triaxial compression tests. I: Influences of density and pressure. JGeotech Geoenviron Eng 2014;140(12):04014070.

[4] Xiao Y, Liu H, Liu H, Chen Y, Zhang W. Strength and dilatancy behaviors of densemodeled rockfill material in general stress space. Int J Geomech2016;16(5):04016015.

[5] Duncan JM, Chang CY. Nonlinear analysis of stress and strain in soils. J Soil MechFound Div, ASCE 1970;96(SM5):1629–53.

[6] Domaschuk L, Villiappan P. Nonlinear settlement analysis by finite element. JGeotech Geoenviron Eng 1975;101(7):601–14.

[7] Darve F, Labanieh S. Incremental constitutive law for sands and clays: simulationsof monotonic and cyclic tests. Int J Numer Anal Methods Geomech1982;6(2):243–75.

[8] Darve F, Flavigny E, Meghachou M. Yield surfaces and principle of superposition:revisit through incrementally non-linear constitutive relations. Int J Plasticity1995;11(8):927–48.

[9] Roscoe KH, Schofield AN, Wroth CP. On the yielding of soils. Géotechnique1958;8(1):22–53.

[10] Gajo A, Muir Wood D. A kinematic hardening constitutive model for sands: themultiaxial formulation. Int J Numer Anal Methods Geomech 1999;23(9):925–65.

[11] Yao YP, Sun DA, Matsuoka H. A unified constitutive model for both clay and sandwith hardening parameter independent on stress path. Comput Geotech2008;35(2):210–22.

[12] Yao YP, Hou W, Zhou AN. UH model: three-dimensional unified hardening modelfor overconsolidated clays. Geotechnique 2009;59(5):451–69.

[13] Kong Y, Xu M, Song E. An elastic-viscoplastic double-yield-surface model for coarse-grained soils considering particle breakage. Comput Geotech 2017;85:59–70.

[14] Yao YP, Liu L, Luo T. A constitutive model for granular soils. Sci China Tech Sci2018;61(10):1546–55.

[15] Yang ZX, Xu TT, Li XS. J2-deformation type model coupled with state dependentdilatancy. Comput Geotech 2019;105:129–41.

[16] Liu SH, Shen CM, Mao HY, Sun Y. State-dependent elastoplastic constitutive modelfor rockfill materials. Rock Soil Mech 2019;40(8):2891–8. [in Chinese].

[17] Yao YP, Liu L, Luo T, Tian Y, Zhang JM. Unified hardening (UH) model for clays andsands. Comput Geotech 2019;110:326–43.

[18] Niemunis A, Herle I. Hypoplastic model for cohesionless soils with elastic strainrange. Mech Cohes-Frict Mater 1997;2(4):279–99.

[19] Maier T. Nonlocal modeling of softening in hypoplasticity. Comput Geotech

2003;30(7):599–610.[20] Weifner T, Kolymbas D. A hypoplastic model for clay and sand. Acta Geotech

2007;2(2):103–12.[21] Mašín D. Hypoplastic Cam-clay model. Géotechnique 2012;62(6):549–53.[22] Chang CS, Hicher PY. An elasto-plastic model for granular materials with micro-

structural consideration. Int J Solids Struct 2005;42(14):4258–77.[23] Yin ZY, Chang CS, Hicher PY. Micromechanical modelling for effect of inherent

anisotropy on cyclic behaviour of sand. Int J Solids Struct2010;47(14–15):1933–51.

[24] Yin ZY, Zhao JD, Hicher PY. A micromechanics-based model for sand-silt mixtures.Int J Solids Struct 2014;51(6):1350–63.

[25] Shen CM, Liu SH, Wang LJ, Wang YS. Micromechanical modeling of particlebreakage of granular materials in the framework of thermomechanics. Acta Geotech2019;14(4):939–54.

[26] Yin JH, Saadat F, Graham J. Constitutive modelling of a compacted sand–bentonitemixture using three-modulus hypoelasticity. Can Geotech J 1990;27(3):365–72.

[27] Sun T, Gao XZ. Containing dilatancy and strain softening of earth’s K-G model. RockSoil Mech 2005;26(9):1369–73. [in Chinese].

[28] Cheng ZL, Jiang JS, Ding HS, Zuo YZ. Nonlinear dilatancy model for coarse-grainedsoils. Chin J Geotech Eng 2010;32(3):460–7. [in Chinese].

[29] Sun DA, Huang WX, Sheng DC, Yamamoto H. An elastoplastic model for granularmaterials exhibiting particle crushing. Key Eng Mater 2007;340–341:1273–8.

[30] Janbu N. Soil compressibility as determined by oedometer and triaxial tests.Proceedings of the European conference on soil mechanics and foundation en-gineering, Wiesbaden. 1963. p. 19–25.

[31] Matsuoka H. On the significance of the spatial mobilized plane. Soils Found1976;16(1):91–100.

[32] Matsuoka H, Yao YP, Sun DA. The Cam-clay models revised by the SMP criterion.Soils Found 1999;39(1):81–95.

[33] Liu SH, Shao DC, Shen CM, Wan ZJ. A microstructure-based elastoplastic con-stitutive model for coarse-grained materials. Chin J Geotech Eng2017;39(5):777–83. [in Chinese].

[34] Pastor M, Zienkiewicz OC, Chan AHC. Generalized plasticity and the modelling ofsoil behaviour. Int J Numer Anal Methods Geomech 1990;14(3):151–90.

[35] Wang ZJ, Liu SH, Vallejo L, Wang LJ. Numerical analysis of the causes of face slabcracks in Gongboxia rockfill dam. Eng Geol 2014;181:224–32.

[36] Wen LF, Chai JR, Xu ZG, Qin Y, Li YL. Monitoring and numerical analysis of be-haviour of miaojiaba concrete-face rockfill dam built on river gravel foundation inchina. Comput Geotech 2017;85:230–48.

[37] Wen LF, Chai JR, Xu ZG, Qin Y, Li YL. A statistical review of the behaviour ofconcrete face rockfill dams based on case histories. Géotechnique 2018;68(9):1–61.

[38] Sukkarak R, Pramthawee P, Jongpradist P, Kongkitkul W, Jamsawang P.Deformation analysis of high CFRD considering the scaling effects. Comput Geotech2018;14(3):211–24.

[39] Goodman RE, Taylor RL, Brekke TL. A model for the mechanics of jointed rock. JSoil Mech Found Div, ASCE 1968;94(SM 3):637–59.

S.-h. Liu, et al. Computers and Geotechnics 119 (2020) 103383

10

http://refhub.elsevier.com/S0266-352X(19)30447-1/h0005http://refhub.elsevier.com/S0266-352X(19)30447-1/h0005http://refhub.elsevier.com/S0266-352X(19)30447-1/h0005http://refhub.elsevier.com/S0266-352X(19)30447-1/h0010http://refhub.elsevier.com/S0266-352X(19)30447-1/h0010http://refhub.elsevier.com/S0266-352X(19)30447-1/h0015http://refhub.elsevier.com/S0266-352X(19)30447-1/h0015http://refhub.elsevier.com/S0266-352X(19)30447-1/h0015http://refhub.elsevier.com/S0266-352X(19)30447-1/h0020http://refhub.elsevier.com/S0266-352X(19)30447-1/h0020http://refhub.elsevier.com/S0266-352X(19)30447-1/h0020http://refhub.elsevier.com/S0266-352X(19)30447-1/h0025http://refhub.elsevier.com/S0266-352X(19)30447-1/h0025http://refhub.elsevier.com/S0266-352X(19)30447-1/h0030http://refhub.elsevier.com/S0266-352X(19)30447-1/h0030http://refhub.elsevier.com/S0266-352X(19)30447-1/h0035http://refhub.elsevier.com/S0266-352X(19)30447-1/h0035http://refhub.elsevier.com/S0266-352X(19)30447-1/h0035http://refhub.elsevier.com/S0266-352X(19)30447-1/h0040http://refhub.elsevier.com/S0266-352X(19)30447-1/h0040http://refhub.elsevier.com/S0266-352X(19)30447-1/h0040http://refhub.elsevier.com/S0266-352X(19)30447-1/h0045http://refhub.elsevier.com/S0266-352X(19)30447-1/h0045http://refhub.elsevier.com/S0266-352X(19)30447-1/h0050http://refhub.elsevier.com/S0266-352X(19)30447-1/h0050http://refhub.elsevier.com/S0266-352X(19)30447-1/h0055http://refhub.elsevier.com/S0266-352X(19)30447-1/h0055http://refhub.elsevier.com/S0266-352X(19)30447-1/h0055http://refhub.elsevier.com/S0266-352X(19)30447-1/h0060http://refhub.elsevier.com/S0266-352X(19)30447-1/h0060http://refhub.elsevier.com/S0266-352X(19)30447-1/h0065http://refhub.elsevier.com/S0266-352X(19)30447-1/h0065http://refhub.elsevier.com/S0266-352X(19)30447-1/h0070http://refhub.elsevier.com/S0266-352X(19)30447-1/h0070http://refhub.elsevier.com/S0266-352X(19)30447-1/h0075http://refhub.elsevier.com/S0266-352X(19)30447-1/h0075http://refhub.elsevier.com/S0266-352X(19)30447-1/h0080http://refhub.elsevier.com/S0266-352X(19)30447-1/h0080http://refhub.elsevier.com/S0266-352X(19)30447-1/h0085http://refhub.elsevier.com/S0266-352X(19)30447-1/h0085http://refhub.elsevier.com/S0266-352X(19)30447-1/h0090http://refhub.elsevier.com/S0266-352X(19)30447-1/h0090http://refhub.elsevier.com/S0266-352X(19)30447-1/h0095http://refhub.elsevier.com/S0266-352X(19)30447-1/h0095http://refhub.elsevier.com/S0266-352X(19)30447-1/h0100http://refhub.elsevier.com/S0266-352X(19)30447-1/h0100http://refhub.elsevier.com/S0266-352X(19)30447-1/h0105http://refhub.elsevier.com/S0266-352X(19)30447-1/h0110http://refhub.elsevier.com/S0266-352X(19)30447-1/h0110http://refhub.elsevier.com/S0266-352X(19)30447-1/h0115http://refhub.elsevier.com/S0266-352X(19)30447-1/h0115http://refhub.elsevier.com/S0266-352X(19)30447-1/h0115http://refhub.elsevier.com/S0266-352X(19)30447-1/h0120http://refhub.elsevier.com/S0266-352X(19)30447-1/h0120http://refhub.elsevier.com/S0266-352X(19)30447-1/h0125http://refhub.elsevier.com/S0266-352X(19)30447-1/h0125http://refhub.elsevier.com/S0266-352X(19)30447-1/h0125http://refhub.elsevier.com/S0266-352X(19)30447-1/h0130http://refhub.elsevier.com/S0266-352X(19)30447-1/h0130http://refhub.elsevier.com/S0266-352X(19)30447-1/h0135http://refhub.elsevier.com/S0266-352X(19)30447-1/h0135http://refhub.elsevier.com/S0266-352X(19)30447-1/h0140http://refhub.elsevier.com/S0266-352X(19)30447-1/h0140http://refhub.elsevier.com/S0266-352X(19)30447-1/h0145http://refhub.elsevier.com/S0266-352X(19)30447-1/h0145http://refhub.elsevier.com/S0266-352X(19)30447-1/h0150http://refhub.elsevier.com/S0266-352X(19)30447-1/h0150http://refhub.elsevier.com/S0266-352X(19)30447-1/h0150http://refhub.elsevier.com/S0266-352X(19)30447-1/h0155http://refhub.elsevier.com/S0266-352X(19)30447-1/h0155http://refhub.elsevier.com/S0266-352X(19)30447-1/h0160http://refhub.elsevier.com/S0266-352X(19)30447-1/h0160http://refhub.elsevier.com/S0266-352X(19)30447-1/h0165http://refhub.elsevier.com/S0266-352X(19)30447-1/h0165http://refhub.elsevier.com/S0266-352X(19)30447-1/h0165http://refhub.elsevier.com/S0266-352X(19)30447-1/h0170http://refhub.elsevier.com/S0266-352X(19)30447-1/h0170http://refhub.elsevier.com/S0266-352X(19)30447-1/h0175http://refhub.elsevier.com/S0266-352X(19)30447-1/h0175http://refhub.elsevier.com/S0266-352X(19)30447-1/h0180http://refhub.elsevier.com/S0266-352X(19)30447-1/h0180http://refhub.elsevier.com/S0266-352X(19)30447-1/h0180http://refhub.elsevier.com/S0266-352X(19)30447-1/h0185http://refhub.elsevier.com/S0266-352X(19)30447-1/h0185http://refhub.elsevier.com/S0266-352X(19)30447-1/h0190http://refhub.elsevier.com/S0266-352X(19)30447-1/h0190http://refhub.elsevier.com/S0266-352X(19)30447-1/h0190http://refhub.elsevier.com/S0266-352X(19)30447-1/h0195http://refhub.elsevier.com/S0266-352X(19)30447-1/h0195

[40] Clough GW, Duncan JM. Finite element analyses of retaining wall behavior. J SoilMech Found Div, ASCE 1971;97(SM 12):1657–72.

[41] Qu Y, Zou D, Kong X, Xu B. A novel interface element with asymmetric nodes and itsapplication on concrete-faced rockfill dam. Comput Geotech 2017;85:103–16.

[42] Özkuzukiran S, Özkan MY, Özyazicioğlu M, Yildiz GS. Settlement behaviour of aconcrete faced rock-fill dam. Geotech Geol Eng 2006;24(6):1665–78.

[43] Zhou W, Hua J, Chang X, Zhou C. Settlement analysis of the Shuibuya concrete-facerockfill dam. Comput Geotech 2011;38(2):269–80.

[44] Xu B, Zou D, Liu H. Three-dimensional simulation of the construction process of theZipingpu concrete face rockfill dam based on a generalized plasticity model.Comput Geotech 2012;43:143–54.

S.-h. Liu, et al. Computers and Geotechnics 119 (2020) 103383

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Practical nonlinear constitutive model for rockfill materials with application to rockfill damIntroductionNonlinear constitutive modelFramework of the modelDerivation of hypoelastic K, G and J modulus functionsBulk modulus KShear modulus G and coupling modulus J

Nonlinearity of shear strengthDetermination of model parametersExperimental verification

ApplicationThe Tankeng projectFE analysis modelResults and discussion

ConclusionsAcknowledgementsmk:H1_16References