Modeling Cyclic Behavior of Rockfill Materials in a Framework of Generalized Plasticity

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Modeling the cyclic behaviour of rockfill materials within the

framework of generalized plasticity

Zhongzhi Fu1, Shengshui Chen2 & Cheng Peng3

1 Geotechnical Engineer, Geotechnincal Engineering Department, Nanjing Hydraulic Research Institute. Nanjing 210024, Jiangsu Province, China (corresponding author). E-mail: [email protected] 2 Professor, Key Laboratory of Earth-Rock Dam Failure Mechanism and Safety Control Techniques, Ministry of Water Resources. Nanjing 210029, Jiangsu Province, China. E-mail: [email protected] 3 Ph.D. student, Geotechnincal Engineering Department, Nanjing Hydraulic Research Institute. Nanjing 210024, Jiangsu Province, China. E-mail: [email protected]

Abstract: Typical triaxial compression experiments were revisited to investigate the essential mechanical behaviour of rockfill

materials to be reflected in constitutive modeling, such as the nonlinear dependences of the strength and the dilation on the

confining pressure and the accumulation of permanent strains during cyclic loading. The mathematical descriptions of the axial

stress-strain behaviour during initial loading, unloading and reloading were formulated respectively, which enables us to rep-

resent the hysteresis loops directly without recourse to complex concepts and parameters. The axial stress-strain model was

then incorporated into the constitutive framework of generalized plasticity for the modeling of cyclic behaviour of rockfill ma-

terials. This task was fulfilled by defining the elastic modulus, the plastic flow direction tensor, the loading direction tensor and

the plastic modulus for different loading conditions. In particular, the plastic flow direction tensor was derived based on a

stress-dilatancy equation considering the influence of loading direction, and the representation of the plastic modulus was es-

tablished in terms of the tangential modulus and the elastic modulus by using the special constitutive equations under axisym-

metric stress states. The cyclic model proposed in this study has three distinct features: First, the hysteresis behaviour and the

accumulation of permanent strains were unifiedly described under the framework of generalized plasticity. Second, all the

loading phases were treated as elastoplastic processes so that no purely elastic regions exist in the principal stress space. Third,

the introduction of two ageing functions for the consideration of hardening effect facilitates the controlling of the magnitudes

of permanent strains. There are totally 13 parameters in the model, all of which can be determined easily from (cyclic) triaxial

compression experiments. To check the capabilities of the model in reproducing the monotonic and cyclic behaviour, typical

triaxial compression experiments were simulated with the constitutive equation. Satisfactory agreement between the experi-

mental results and the corresponding model predictions lent sufficient creditability to the effectiveness of the proposed model,

which further stimulates us to extend the model for more complex stress paths and apply the model in practical engineering in

the future.

Keywords: rockfill materials, cyclic behaviour, generalized plasticity, constitutive model

International Journal of Geomechanics. Submitted May 25, 2012; accepted March 7, 2013; posted ahead of print March 9, 2013. doi:10.1061/(ASCE)GM.1943-5622.0000302

Copyright 2013 by the American Society of Civil Engineers

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Introduction

Rockfill materials are widely used in earth-rock dams as the shells, which support and stabilize the impermeable systems like

concrete slabs and clay core walls (US Army Corps of Engineers 2004). The tight interaction between the dam shells and the

impermeable systems indicates that the behaviour and safety of earth-rock dams largely depends on the properties of the

rockfill materials used. For instance, the wetting induced deformation of rockfill materials during the first impounding is re-

sponsible for the formation and propagation of the longitudinal cracks at the dam crest (Fu et al. 2012) and the creep settle-

ments of the shells during operation is the cause for the long-term compressive failure of concrete slabs (Zhou et al. 2010).

Site investigations on Zipingpu concrete face rockfill dam (CFRD) after the great Wenchuan earthquake (2008) also accentu-

ates the great influence of the behaviour of rockfill materials on the overall seismic response of rockfill dams (Chen and Han

2009). The contraction of the dam shell and the residual settlement of the dam crest were interpreted as the results of the

volumetric contraction of rockfill materials during repeated loading as observed in cyclic experiments on granular materials

(Zambelli et al. 2012; Indraratna et al. 2011). Although good performance of the Zipingpu CFRD in the Wenchuan earthquake

re-proved the high resistance of this type of dam to earthquakes, the seismic safety of earth-rock dams has drawn substantial

attentions afterward because of the frequent occurrence of unexpected strong earthquakes around the world. Moreover, large

amounts of earth-rock dams in China, built, under construction and under designing, locate in regions of high earthquake in-

tensity (International Commission on Large Dams 2010). The predictions of the amplification of the ground acceleration and

the permanent deformation are of paramount significance in evaluating the seismic safety of such structures, and a proper

constitutive model describing the cyclic behaviour of rockfill materials plays a central role in a credible prediction.

After the pioneering study conducted in 1970s (Seed and Idriss 1970; Seed 1979; Hardin and Drnevich 1972a, 1972b), the

equivalent visco-elastic model has been widely accepted in analyzing the seismic response of earth-rock dams

(Feizi-khankandi et al. 2009; Uddin 1999; Gazetas and Dakoulas 1992). The hysteresis loops in the shear stress – shear strain

plane are approximated by a series of ellipses. The inclination and the slenderness of these ellipses are controlled by the

equivalent shear modulus and the damping ratio, respectively, both of which depend nonlinearly on the magnitude of the cy-

clic shear strain. Due to the introduction of the viscous damping, the mechanical response of the considered material is

rate-dependent and the employment of elasticity makes the direct consideration of residual strains impossible (Kramer 1996).

Generally, empirical relations should be added to take the accumulation of permanent strains into account (Martin et al. 1975;

Byrne 1991).

Another approach widely employed in seismic analysis is the so-called cyclic nonlinear model (Kramer 1996), in which the

actual stress – strain path during cyclic loading is directly followed by defining a backbone curve for initial loading and by

introducing mapping rules for unloading and reloading processes (Masing 1926; Finn et al. 1977). A great advantage of such

cyclic nonlinear models over equivalent visco-elastic models is the ability to represent the development of permanent shear

strains (Wu 2001). Nevertheless, most of these nonlinear models do not allow for the determination of shear induced volu-

metric strains that can lead to plastic hardening under drained condition or to development of pore pressure under undrained

condition. Additional relations should also be introduced to account for the accumulation of volumetric strain and buildup of

pore pressure as in equivalent visco-elastic models.

Great efforts have also been devoted to modeling the cyclic behaviour of granular materials within advanced constitutive

frameworks, such as bounding surface plasticity (Dafalias 1986; Bardet 1986; Khalili et al. 2005; Yao et al. 2011), hypoplas-



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ticity (Niemunis and Herle 1997), generalized plasticity (Zienkiewicz et al. 1999; Ling and Yang 2006; Ling and Liu 2003),

and subloading surface plasticity (Hashiguchi 1989; Hashiguchi and Chen 1998). In most of these advanced models, the me-

chanical behaviour of the soils are assumed rate-independent and the energy dissipation during a stress cycle does not depend

on the rate of loading. These advanced constitutive models allow considerable flexibility and generality in modeling the re-

sponse of soils to cyclic loading. However, their description usually requires many more parameters than equivalent vis-

co-elastic models or cyclic nonlinear models. Evaluation of these parameters can be difficult and the parameters obtained

from one type of test may not applicable to other stress paths (Kramer 1996). Although the use of advanced constitutive mod-

els will undoubtedly increase, these practical problems have, to date, limited their use in geotechnical earthquake engineering

practice. On the other hand, establishing a constitutive model, with acceptable number of physically meaningful parameters

that can be determined economically, still deserves extensive attentions and great efforts.

The main motivation of this paper is to develop a practical cyclic constitutive model for rockfill materials in the context of

generalized plasticity. Particular attentions were focused on the unified representation of the hysteresis behaviour and the

volumetric contraction during cyclic loading. The response of rockfill materials to cyclic loading was analyzed and the axial

stress – strain behaviour during loading, unloading and reloading were described with simple parameters well known in dam

engineering. The plastic flow rule and the loading direction tensor were derived and the representation of the plastic modulus

was established using the special forms of constitutive equation for axisymmetric stress states. The cyclic model was then

incorporated into a finite element procedure to reproduce the typical experimental observations introduced previously.

Throughout the paper, the sign convention used in soil mechanics was adopted, i.e. compressive stress or strain is positive.

The basic assumption employed in this study is the rate-independence of the mechanical behaviour of rockfill materials as

was similarly used in the abovementioned advanced models.

Observations made from monotonic and cyclic triaxial experiments

To provide an abundant database for the constitutive modeling of the mechanical behaviour of rockfill materials, triaxial ex-

periments carried out on different materials (Li 1988; Chen et al. 2010; Pradhan et al. 1989) were collected. In this section,

typical experimental observations were revisited to lay a physical foundation for the elastoplastic constitutive modeling.

Strength and dilatancy behaviour observed from monotonic loading

The material under consideration is constituted by rufous sandstones, which are slightly weathered according to geological

investigations. Due to the limitation of the triaxial apparatus, the prototype material was scaled down and the grains with a

diameter greater than 10 mm were sifted out. Fig. 1 shows the grain size distribution and the basic physical properties of the

prepared material. Fig. 2 plots the typical triaxial experimental results conducted under three different confining pressure, i.e.

100 kPa, 500 kPa and 1000 kPa. All the specimens contracted and showed a hardening behaviour at the beginning of shearing.

However, further loading resulted in a volumetric expansion when the mobilized friction angle (see Fig. 2 for definition) ex-

ceeded a certain value (known as the constant volume friction angle (Braja 2008)). Strain softening behaviour was also ob-

served after the peak state, especially when the confining pressure was low. Both the peak friction angle and the constant

volume friction angle decrease when the confining pressure is increased. This trend is more clearly shown in Fig. 3, and the

background underlying this nonlinear behaviour is essentially the particle breakage during loading (Chen et al. 2010). Same



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properties were also observed in experiments on sands and the influence of particle breakage was similarly emphasized (Lade

2006).

As can be seen in Fig. 3, the relationship between the peak friction angle, φf, and the confining pressure, σ3, can be well

represented by the following equation:

3logf oap

(1)

in which pa denote the atmospheric pressure; φo and ∆φ are two parameters that can be determined directly in Fig 3.

The constant volume friction angle, ψc, shows analogous dependence on the confining pressure and it can be evaluated

similarly, i.e.

3logc oap

(2)

Herein ψc and ∆ψ are another two parameters. With the increase of the confining pressure, the difference between φf and ψc

diminishes (Fig. 3) and the dilatancy behaviour during shearing is evidently suppressed due to severe particle breakage.

Elastoplastic behaviour observed from cyclic loading

Fig. 4 shows the typical results obtained from cyclic triaxial experiments on the same sandstone mentioned previously. Three

distinct features can be observed:

(1) Volume contraction always occurred when the loading direction was reversed regardless of the deviatoric stress level

where the loading was reversed. Since the mean pressure decreases when the axial stress is reduced and the elastic volumetric

strain is expansive, the volume contraction can only be interpreted as elastoplastic behaviour. Therefore, the familiar postulate

that the unloading and reloading processes are elastic will be abandoned in this study, and we assume that elastoplastic de-

formation occurs throughout the loading, unloading and reloading processes.

(2) The average unloading lines (AUL, the line connecting the reverse point and the point where the axial stress is reduced

to the radial stress in triaxial experiments) are almost parallel to each other under the same confining pressure, irrespective of

the deviatoric stress where the axial loading is reversed. The inclination of the AULs was often interpreted as the elastic

modulus. However, the term average unloading modulus will be used instead and it will be denoted by Eau in this paper.

(3) Although the average unloading modulus was not influenced evidently by the magnitude of the deviatoric stress, it

shows clear dependence on the confining pressure, i.e. the higher the confining pressure is, the steeper the average unloading

lines are and the higher the average unloading modulus is. This feature is similar as the dependence of the initial loading

modulus, Eis, (ε1→0) on the confining pressure as illustrated in Fig 5.

Stress - dilatancy equations for cyclic loading

A stress-dilatancy equation is one of the kernels of elastoplastic models and plenty of formulations have already been sug-

gested in the past (e.g. Rowe 1962; Li and Dafalias 2000). Moreover, complicated factors like fabric anisotropy (Wan and

Guo 2004) and rotation of principal axes (Gutierrez and Wang 2009) could also been taken into consideration in appropriate



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ways. However, most of the previous studies focused on the behaviour of soils during monotonic loading and the

stress-dilatancy behaviour during unloading and reloading attracted fewer attentions.

Pradhan (1989) was among the few scientists who have investigated the relation between the dilatancy ratio dg (the ratio of

the plastic volumetric strain increment to the plastic deviatoric strain increment) and the mobilized stress ratio η (the ratio of

the deviatoric stress to the mean effective stress) for sands during cyclic triaxial experiments. It was found that the

stress-dilatancy relation for loading and that for unloading could be approximated satisfactorily by two parallel lines and the

dilatancy ratio underwent a discontinuous change when the axial stress was reversed as shown in Fig 6. This clearly indicates

that the dilatancy ratio depends not only on the current stress state but also on the incremental stress or the loading direction.

To take this point into consideration, the following loading angle, θ, will be used in this study:

arccos : arccos ij ij

ij ij ij ij

s dsdd s s ds ds

s ss s

(3)

in which sij and dsij are deviatoric parts of the current stress tensor σ and its increment dσ, respectively. The loading angle

may be geometrically interpreted as the angle formed by the deviatoric stress vector and its increment on the π plane as

shown in Fig. 7. In triaxial compression stress states, the loading angle equals to zero during axial loading and it equals to π

during axial unloading. Radial loading and unloading gives a value of π and zero, respectively. Analogously, in triaxial exten-

sion stress states, the loading angle equals to zero during radial loading and it equals to π during radial unloading. Axial load-

ing and unloading gives a value of π and zero, respectively.

By virtue of the loading angle, the stress dilatancy equations applicable to loading and unloading processes may be unified

as follows:

cos 1 cosg goc

d dM

(4)

in which dgo is a parameter and the constant volume stress ratio, Mc, can be evaluated using the constant volume friction angle,

i.e.

6sin

3 sinc

cc

M (5)

It can be verified, with the constant dgo = Mc, that during axial loading Eq. (4) degrades to the familiar form, i.e. dg = (Mc – η),

which was initially proposed for Granta-gravel (Schofield and Wroth 1968) and it can be simplified as dg = -(Mc + η) for axial

unloading.

It deserves to emphasize that in classical elastoplastic theory the plastic flow direction is determined by the potential func-

tion and the dependence of the dilatancy behaviour on the loading direction can not be reflected directly. On the contrary, the

plastic flow direction in generalized plasticity is obtained by defining a stress-dilatancy equation explicitly, which allows

considerable flexibility in constitutive modeling as will be demonstrated later.



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Hardening behaviour observed from cyclic loading

Fig. 8 shows the grain size distribution and the basic physical properties of another kind of prepared rockfill material exca-

vated from the site of a pumped storage power station. The main content of the material is slightly weathered quartz monzo-

nite. Cyclic triaxial compression tests (constant confining pressure and sinusoidal axial pressure with a frequency of 0.1 Hz)

were carried out on saturated specimens (300mm×700mm) (Fu and Ling 2009; Chen et al. 2010). Plotted in Fig. 9 are the

accumulation of the axial strain and the volumetric strain during cyclic loading. Both the axial strain and the volumetric strain

accumulate much faster under a higher dynamical stress, σd, and this can be explained by the wider disturbance and more

pronounced reposition of the granular particles within the specimens under a higher deviatoric stress. The amplitude of the

cyclic axial strain tends to stabilize after a few loading cycles regardless of the dynamical stress applied to the specimen.

However, the amplitude of the cyclic volumetric strain shows some fluctuation as can be seen in Fig. 9(b), which, to some

extent, implies the difficulty in accurately measuring the dynamical volumetric behaviour in experiments.

Another important finding from Fig. 9 is that both the axial strain and the volumetric strain increase rapidly during the first

few cycles. The accumulation rate, however, decreases with the increase of the loading cycles, indicating a hardening behav-

iour during the cyclic loading. Similar phenomenon was also observed in cyclic loading tests on ballast (Indraratna et al.

2012). This characteristic is the result of densification of the specimen and hints that the volumetric strain or the void ratio

must be taken as a state variable in addition to the stress variables in modelling the cyclic behaviour of rockfill materials. A

common strategy is to introduce a function that takes the plastic volumetric strain as the variable into the representation of the

plastic modulus (Zienkiwicz et al. 1999; Ling and Yang 2006; Ling and Liu 2003). This concept was also employed in this

study; however, the hardening effect was modeled by incorporating two ageing functions, which take the total volumetric

strain as variables, into the representation of the tangential modulus and the stress-dilatancy equation, respectively. It will be

shown in the next sections that the introduction of ageing functions largely facilitates the controlling of the permanent strains

during cyclic loading.

Mathematical description of the axial stress - strain behaviour

One of the most important behaviour of rockfill materials during cyclic loading is the hysteresis effect, i.e. the stress – strain

loops B-C-E-F-G as depicted in Fig. 10. In many advanced models, this behaviour is captured by defining different plastic

moduli for loading, unloading and reloading processes. In this section, the representations of the tangential modulus at dif-

ferent stages will be defined instead.

Initial loading: A-B-H path

It was shown in Fig. 5 that the tangential modulus at the very beginning of initial loading, Eis, increases with the confining

pressure and it can be evaluated via the following equation according to the suggestion by Janbu (1963):

3

n

is aa

E k pp

(6)

Herein, k and n are two parameters. With the increase of the mobilized stress ratio, the tangential modulus decreases nonline-

arly and it vanishes gradually when the peak stress ratio is approached. This property can be modeled by the following equa-



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tion:

31n

t af a

E k pM p

(7)

where α is a dimensionless constant and Mf denotes the peak stress ratio, i.e.

6sin

3 sinf

ff

M (8)

Eq. (7) is similar to the one suggested by Duncan et al. (1970) except the inclusion of the intermediate principal stress. At the

peak states, i.e. η = Mf, the tangential modulus vanishes and the axial strain develops infinitely without additional axial stress

increment. Therefore, the failure surface given by Eq. (7) can be mathematically defined by η = Mf, and it is a cone in the

principal stress space. However, using the generalized stress transformation technique by Yao et al. (2004) allows the incor-

poration of advanced failure criterion such as the one proposed by Matsuoka et al. (1999).

Unloading: B-C-D path

As pointed out previously that the tangent of the average unloading line BD does not clearly depend on the stress ratio where

the axial loading is reversed. However, an evident dependence on the confining stress, σ3, is observed from Figs. 4 and 5, and

therefore we assume that the average unloading modulus, Eau, can be evaluated using a similar power function as Eq. (6), i.e.

3

n

au au aa

E k pp

(9)

herein, kau is another parameter that can be determined from Fig. 5. Although the inclinations of the two approximation lines

in Fig. 5 are slightly different, the same index n is used in Eqs. (6) and (9) for the sake of simplicity and convenience in pa-

rameter calibration.

Reinspection on the path BCD indicates that unloading the specimen from point B will bring it to a special stress state, C,

where the tangent of the unloading path is parallel to the AULs and the tangential modulus can be calculated using Eq. (9).

Herein, we further assume that the stress ratio at this point can be determined as follows:

u m (10)

where βu is a constant (0 < βu < 1) and ηm memorizes the maximum stress ratio during the past loading history.

Further unloading from Point C will bring the specimen to an isotropic compression stress state (Point D on path BCD)

again. Although the stress state at Point D is the same as that at Point A, the internal fabric of the specimen has been changed

due to disturbance and reposition of particles during the stress cycle, which consequently results in a plastic hardening effect

as pointed out previously. To take this effect into account, we assume that the unloading modulus at point D can be estimated

by Eq. (6) along with an ageing function, i.e.

31

n

ris v a

a

E G k pp

(11)



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in which the ageing function 1rvG reads:

1 expr

r vv

v

Gc

(12)

herein, cv is a parameter controlling the rate of hardening and rv denotes the volumetric strain at the nearest reversal point. In

the following part, we always regard k as a discrete parameter that should be updated from its initial value with the ageing

function once a stress reversal occurs.

Since the tangential modulus at the lower Bound D and the controlling point C are known, the tangential modulus during

the whole unloading process can be interpolated (Fig. 11) by a linear equation, i.e.

t u au u isE A E B E (13)

with the following coefficients:

; u mu u

u m u m

A B (14)

Reloading: E-F-G-H path

The representation of the tangential modulus during reloading can be similarly established. First, we assume that the

specimen is reloaded from Point E, where the stress ratio is denoted by ηr as shown in Fig. 10. Continuous loading will

bring the specimen to another special stress state, F, where the tangent of the reloading path is parallel to the AULs and

the representation of the tangential modulus is also given by Eq. (9). Herein, another parameter, βr, is introduced to con-

trol the stress ratio at this state, i.e.

rr

m r

(15)

Eq. (15) gives the stress ratio at the controlling point F, i.e. r r m r .

Further reloading will lead the specimen to a stress ratio approaching the maximal historical one memorized by ηm as

shown in Fig. 10, exceeding which the stress stain relation, according to the experimental observations (Li 1988), approxi-

mately follows the curve for initial loading. For a continuous transition of the tangential modulus from a reloading stage to an

initial loading stage, Eq. (7) will be used to calculate the tangential modulus at the transition (or memory) point G, i.e.

31n

mtm a

af

E k ppM

(16)

in which fM denotes the peak stress ratio at the memory point B where the maximal stress ratio is recorded. It should be bore

in mind that the parameter k in Eq. (16) has already been updated with the ageing function (12).

Obtaining the tangential modulus at the controlling point F and the memory point G, the representation of the tangen-



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tial modulus during the reloading process can be established using a linear interpolation (Fig. 11), i.e.

t r au r tmE A E B E (17)

where the two coefficients read:

;1 1

r r m rmr r

m r r m r r

A B (18)

Triaxial extension: D-I path

If the specimen is further unloaded from point D along the path DI in Fig. 10, a triaxial extension stress state will be achieved

and the stress path DI is absolutely strange to the specimen compared with the unloading path BCD and the reloading path

EFG. Rockfill is a kind of material that has a certain memory of the stress history during unloading and reloading pro-

cesses, and this fact has been reflected in Eqs. (13) and (17), where the maximal historical stress ratio, ηm, plays an im-

portant role in the combination coefficients. However, when the specimen is brought to the strange stress path DI, the

memory of the point B will be gradually swept out and we will treat this path as an initial loading one in this study. Once

the stress ratio along path DI surpasses the maximal historical value ηm, the memory of point B will be absolutely eliminated

and a new memory point will be recorded. Based on the above assumptions, the tangential modulus along the path DI can be

evaluated according to Eq. (7). Since the tangential modulus at Point D evaluated via Eq. (13) is the same as that given by Eq.

(7), the axial response is continuous when the stress state is shifted from triaxial compression, passing point D, to triaxial

extension.

For the representation of the axial stress-strain behaviour under cyclic loading, four modulus parameters in addition to the

strength parameters, i.e. k, kau, n and α has been introduced in this section, among which k controls the modulus during initial

loading and kau controls the inclination of the hysteresis loops. The shape or fatness of the hysteresis loops were controlled by

the parameters βu and βr. All these parameters can be determined based on (cyclic) triaxial compression experiments. We will

explain the routine for parameter identification further in next sections.

The representations of the tangential modulus established for different loading conditions will be of vital significance in

constructing the representation of the plastic modulus in the next section. Furthermore, the criteria distinguishing loading,

unloading and reloading will also be discussed.

Constitutive modeling the cyclic behaviour of rockfill materials

The framework of generalized plasticity will be employed for the constitutive modelling of the cyclic behaviour of rockfill

materials. This framework was firstly proposed by Pastor et al. (1990) and it has been successfully used in modeling the un-

drained behaviour of sands exposed to cyclic shearing (Zienkiewicz et al. 1999; Ling and Yang 2006; Ling and Liu 2003).

Discarding the yield function, the plastic potential function and the hardening rule, which are three keystones in classical

elastoplasticity, the generalized plasticity gains large flexibility and applicability in modeling complex behaviour of geo-

materials by defining the loading direction tensor, the plastic flow direction tensor and the plastic modulus (a scalar) instead.

The generalized constitutive equation reads (Zienkiewicz et al. 1999):



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: :

:: :

e eg fe

ef g

d dH

D n n Dσ D ε

n D n (19)

which is the strain-driven form and the inverse stress-driven form reads:

1:e

g fd dH

ε C n n σ (20)

In Eqs. (19) and (20) Ce and De are fourth-order tensors describing the elastic behaviour of materials (Ce = De -1); ng is the

normalized plastic flow direction tensor and nf the normalized loading direction tensor, both of which have a magnitude of 1;

H stands for the so-called plastic modulus which has a dimension of stress. In this section, the above quantities will be de-

fined to substantiate the constitutive equation.

The elastic behaviour

The elastic behaviour of rockfill materials can be determined by different combinations of parameters, such as the shear

modulus, Ge, and the bulk modulus, Ke, or the Young’s modulus, Ee, and the Poisson ratio, v. For the latter combination, the

indicial elastic stiffness tensor reads (Ottosen and Ristinmaa 2005):

1 1 2 2 1

e e eijkl ij kl ik jl il jk

v E ED

v v v (21)

in which δij is the Kronecker sign. For rockfill materials the Poisson ratio can be assumed a constant for simplicity, i.e. v ≈

0.2~0.4. Therefore, only the representation of the Young’s modulus needs to be specified. In this study, we assume that the

Young’s modulus equals to the average unloading modulus during initial loading while during unloading and reloading it is

proportional to the tangential modulus and the ratio equals to (kau/k) for unloading and (kau/k)·(1-η/Mf)-α for reloading, i.e.

3 initial loading

unloading

1 reloading

n

au aa

aue t

aut

f

k pp

kE E

k

kE

k M

(22)

Eq. (22) indicates that axial plastic strain develops at the very beginning of initial loading, and the portions of the elastic

part and the plastic part change with the stress state. Plastic strains also accumulates during unloading and reloading process-

es, however, the portion of which in the total axial strain increment keeps constant during unloading and for reloading condi-

tion it equals to the portion under the initial loading condition with the same stress state. Eq. (22) is somewhat complex;

however, it does not introduce additional parameters for the elastic behaviour.

The plastic flow direction tensor

Since the normalized plastic flow direction tensor specifies the relative values of the incremental plastic strain components

and it has a norm of one, the following expression can be used as a start point:



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pij

g pij

d

d

εn

ε (23)

With the assumption of coaxiality (the principal axes of the plastic strain increment coincide with the principal axes of the

current stress tensor), the incremental plastic strain tensor can be expressed as follows:

31

3 2ijp p p

ij v ij s

sd d d

qε (24)

in which q is the generalized shear stress, i.e. 3 2ij ijq s s ; pvd and p

sd denote the volumetric and the deviatoric strain

increments and the superscript ‘p’ refers to plastic strain, i.e.

2 1 1:

3 3 3

p pv ij ij

p p p p ps ij v ij ij v ij

d d

d d d d d (25)

Substituting Eq. (24) into Eq. (23) yields the explicit expression of the plastic flow direction:

2

313 2

1 33 2

ijg ij

g

g

sd

q

dn (27)

where the dilatancy ratio dg could be evaluated with Eq. (4) along with another ageing function, i.e.

2 cos 1 cosrg v go

c

d G dM

(28)

herein, the ageing function 2rvG reads:

2 expr

r vv

d

Gc

(29)

in which cd is a significant parameter that controls the accumulation rate of the volumetric strain during cyclic loading,

and rv has the same meaning as that in Eq. (12).

The loading direction tensor

In classical elastoplasticity theory, the loading direction is defined using the yield function and the flow rule is associative if

the plastic potential function is the same as the yield function or the plastic direction tensor is identical to the loading direc-

tion tensor. Nevertheless, many experiments invalidate an associative flow rule and indicate that a non-associative flow rule is

more reasonable for granular materials like sand and rockfill (Kin and Lade 1988a, 1988b). Herein, the loading direction ten-

sor is similar as the plastic direction tensor, i.e.



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2

313 2

1 33 2

ijf ij

f

f

sd

q

dn (30)

in which the factor df is calculated according to the following equation:

f fd M (31)

By virtue of Eqs. (30) and (31), Eq. (20) can be rewritten as follows:

: ffp

g gf

M dp dqdd

H H

n σε n n

n (32)

It is clear that the loading direction tensor is, in essence, introduced to control the contribution of compression and shearing to

the development of plastic strains during complex loading.

The plastic modulus

To establish the representation of the plastic modulus, the axial stress-strain behaviour during cyclic triaxial experiments will

be revisited. To this end, Eqs. (27) and (30) are inserted into Eq. (20) and the following matrix equation is obtained for the

stress state σ1 > σ2 = σ3:

1 1 1

2 2 2

3 3 3

1 3 31 1 1 1

1 3 2 3 23 3

1 3 2 3 2

T

g f

g fe g f

g f

d v v d d d dd v v d d d d

E Hd v v d d d d

n n (33)

In cyclic triaxial experiments dσ2 = dσ3 = 0 (herein, σ2 and σ3 should be interpreted as horizontal stresses rather than the prin-

cipal stresses) and the first line in Eq. (33) can be rewritten as follows:

1 1 1

3 31 1

3 3

g f

e g f

d dd d d

E H n n (34)

By virtue of the physical meaning of the tangential modulus, i.e. 11 1tE d d , the plastic modulus can be expressed in

terms of the tangential modulus and the elastic modulus, i.e.

13 3 1 1

3 3

g f

t eg f

d dH

E En n (35)

in which the norm of the plastic flow direction tensor and that of the loading direction tensor are given as follows:



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2

2

1 33 2

1 33 2

g g

f f

d

d

n

n (36)

Similarly, for the stress state σ1 < σ2 = σ3, we can obtain the following representation for the plastic modulus by simple

mathematical operations, i.e.

13 3 1 1

3 3

g f

t eg f

d dH

E En n (37)

in which the norms of the plastic flow direction tensor and the loading direction tensor can also be calculated via Eq. (36).

It deserves to point out that the plastic modulus given by Eqs. (35) and (37) depends not only on the current stress state but

also on the stress increment because the flow direction tensor determined by Eqs. (27) and (28) shows a clear dependence on

the direction of the incremental stress. Moreover, Eqs. (35) and (37) can be unified by incorporating the Lode angle ϑ , i.e.

13cos3 3cos3 1 1

3 3

g f

t eg f

d dH

E En n (38)

where the Lode angle ϑ can be determined by the following equation (Bauer et al. 2010):

3

6trcos3

s s s

s (39)

in which tr(∙) denotes the trace operation on a second-order tensor. It is simple to verify that Eq. (38) degrades to Eq. (35)

under the triaxial compression stress state (ϑ = 60º) and it reduces to Eq. (37) under the triaxial extension stress state (ϑ =

0º).

The loading-unloading-reloading criteria

As conveyed in the previous sections, the determination of the tangential modulus, the elastic modulus and consequently the

plastic modulus requires the distinguishing of loading, unloading and reloading. This is fulfilled with the aid of the yield

function, f, in classical elastoplasticity, i.e. the process is defined as loading if nf : dσ > 0 (nf = ∂f/∂σ) and it is treated as un-

loading when nf : dσ < 0. The identity nf : dσ = 0 gives the so-called neutral loading condition.

Despite of the simplicity of the above criterion, it is not appropriate for cyclic loading unless the memory effect is taken

into consideration. In the current model, a process is treated as initial loading if the stress ratio equals to the maximal histori-

cal value, ηm, and the stress ratio is still increasing, i.e.

& 0 initial loadingm d (40)

Unloading happens if the following condition is satisfied:

& 0 unloadingm d (41)



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In the case η < ηm & dη > 0, two possible loading conditions exist. For example, a specimen is first sheared from isotropic

stress state A to the stress state B and subsequently unloaded to the state C as shown in Fig. 12. Both the path CD and the path

CE fulfill the condition η < ηm & dη > 0. However, the behaviour along both paths are substantially different. In this study, it

is assumed that reloading happens when the stress point on the deviatoric plane moves towards the memory point where the

maximal stress ratio is recorded. On the contrary, initial loading happens if the stress point moves far away from the memory

point on the deviatoric plane. This assumption can be mathematically expressed as follows:

0 reloading

& 00 initial loading

mij ij ij

m mij ij ij

s s dsd

s s ds (42)

in which mijs denotes the deviatoric stress tensor at the memory stress state where the maximal stress ratio is registered.

In fact, the criterion (42) distinguishing reloading and initial loading has already been employed in describing the axial

behaviour when the isotropic compression stress state was crossed. This is more clearly shown in Fig. 10. If the axial stress is

increased after the specimen is unloaded to the isotropic stress state D, then a reloading behaviour will present. On the other

hand, initial loading happens if the axial stress is further decreased along the stress path DI.

Calibration of the model parameters

Excluding the elastic Poisson ratio, there are totally 13 parameters in the proposed dynamical constitutive model and they can

be divided into 5 groups as summarized in Table 1. All these parameters can be determined step by step based on convention-

al triaxial experiments as listed in Table 1.

The peak friction angle parameters, φo and Δφ, can be calibrated directly from Fig. 3, in which the peak friction angle un-

der different confining stress are plotted. In particular, φo is the peak friction angle when the confining stress equals to the

atmospheric pressure and Δφ is the inclination of the approximation line in the semi-logarithmic diagram. The constant vol-

ume friction angle parameters, ψo and Δψ, can be determined in a similar way.

The modulus parameters, k and n, can be determined by plotting the tangential modulus at the beginning of triaxial shear-

ing and the confining pressure in a double-logarithmic diagram as illustrated in Fig. 5, where k · pa is the initial modulus

when the confining stress equals to the atmospheric pressure and n is the inclination of the approximation line. The parameter

kau can also be determined by plotting the average unloading modulus and the corresponding confining stress in the same

double-logarithmic diagram. The parameter α controls the degradation of the tangential modulus with the increase of the mo-

bilized stress ratio, and it can be determined by plotting the normalized tangential modulus, i.e.3

nt a aE k p p , with the

mobilized stress ratio in a diagram and then fitting the experimental data.

The physical meaning of the parameter dgo is the dilatancy ratio at the instant of η = 0, therefore it can be calibrated direct-

ly from Fig. 6. In addition, the inclination of the strain paths drawn in Fig. 2 can also be used to estimate the value of dgo,

however, one should keep in mind that the dilatancy ratio dgo only relates to the plastic strains.

The hysteresis parameters, βu and βr, are introduced to manipulate the shape or fatness of the hysteresis loops in cyclic tri-



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axial compression tests, and the values of which essentially influence the energy dissipation behaviour of rockfill materials

under investigation. Both of them have a common range of [0.0, 1.0] and should be identified simultaneously by best fitting

the hysteresis loops presented in cyclic loading experiments.

The ageing parameters, cv and cd, are introduced to control the accumulation of the permanent strains. The former controls

the axial strain by modifying the modulus parameter k when reversal loading happens (see Eq. (11)) and the latter controls the

volumetric strain by influencing the stress dilatancy equation during cyclic shearing. Both of them can be calibrated using the

information about permanent strains as illustrated in Fig. 9.

As discussed in the above paragraphs, most of the parameters introduced in the proposed model have simple and clear

physical meanings and they can be calibrated by reorganizing the experimental data and simple calculations. This satisfactory

feature makes it possible to realize a convenient application of the model in practical engineering.

Verification of the proposed model

In this section, typical results obtained from triaxial experiments on two kinds of rockfill materials (marked as material I and

material II hereafter), including static monotonic loading, quasi-static cyclic loading and dynamical cyclic loading, were ex-

tracted to identify the constitutive parameters. Then predictive calculations were carried out with the proposed constitutive

model. The experimental results and the model predictions are compared in the following subsections.

Static monotonic loading

Figs. 13 and 14 show the experimental results obtained from static monotonic loading tests on the aforementioned two rock-

fill materials, based on which the parameters related to monotonic loading behaviour were calibrated and listed in Table 2.

Numerical simulations of these tests were also performed and the results were plotted in Figs. 13 and 14 for comparison.

All the model predictions agree satisfactorily with the experimental data, which seems to indicate the capability of the model

in reproducing the basic strength and dilatancy behaviour of rockfill materials. For example, the higher deviatoric stress level

and the severer volumetric contraction during shearing under a higher confining stress, and the more pronounced shear dila-

tancy under a lower confining stress.

Quasi-static cyclic loading

The bright circles plotted in Fig. 15 are the experimental data obtained from cyclic triaxial compression tests (σ3 = 0.5MPa

and 1.0MPa) on material I. The results under the confining stress of 0.5MPa were used to calibrate the parameters related to

cyclic loading behaviour, and the obtained values are given as follows:

kau=770.0; βu=0.5; βr=0.8; cv=1.00; cd=0.02.

Based on the parameters given above and those listed in Table 2, the cyclic triaxial compression tests with variable stress

amplitudes were numerically simulated and the corresponding model predictions were also plotted in Fig. 15. Both the hyste-

resis loops and the accumulation of permanent strains were successfully captured by the model. In particular, the volume

contraction behaviour when reversal loading happens was reproduced quantitatively. It can also be seen from Fig. 15 that

during cyclic loading the axial stress-strain curves of unloaded and reloaded materials approximately follow the original



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curves under monotonic loading condition, once the current stress ratio exceeds the maximal historical one. This salient fea-

ture was also reproduced successfully with the proposed model.

Dynamical cyclic loading

The curves plotted in Figs. 16(a) and 16(c) are the experimental data obtained from cyclic triaxial compression tests on mate-

rial II with a confining stress σ3 = 3.0MPa and stress amplitudes of 0.33σ3 and 0.66σ3, respectively. The results with stress

amplitude of 0.33σ3 were used to calibrate the extra parameters related to cyclic loading behaviour of the rockfill material

under investigation. The calibrated values are given as follows:

kau=2280.0; βu=0.50; βr=0.75; cv=0.007; cd=0.007.

It is important to note that the values of the parameters cv and cd here are much lower than that of the material I calibrated

previously. This difference may indicate that the behaviour of rockfill materials are essentially influenced by the rate of load-

ing (Indraratna et al. 2012; Porcino et al. 2011), since the cyclic triaxial compression tests carried out on material I are qua-

si-static and those conducted with material II are dynamical ones with a frequency of 0.1 Hz. This feature hints that different

values of the parameters cv and cd may need to be calibrated for static and dynamical problems separately, although the con-

stitutive equations for both conditions are the same.

Based on the parameters given above and those listed in Table 2, the dynamical triaxial experiments with constant stress

amplitudes were simulated and the corresponding predictions were also plotted in Figs. 16(b) and 16(d). Once again, the hys-

teresis effects and the accumulation of permanent strains were satisfactorily reproduced by the model. Fig. 17 shows the evo-

lution of strains with the increase of loading cycles under the two dynamical stress amplitudes. The predictions made by the

model were also plotted in this figure for comparison. It can be seen that the development of the axial strain and the volumet-

ric strain as well as the dependence of their magnitudes on the dynamical stress amplitude can also be reproduced quantita-

tively with the proposed cyclic model, which seems to validate the effectiveness of the model in predicting the permanent

deformation of rockfill dams during an earthquake.

Summaries and conclusions

In this paper, experimental observations made from typical triaxial compression tests were collected and summarized. Partic-

ular attentions were focused on modeling the strength, dilatancy and hardening properties of rockfill materials during static

and cyclic loading. The mathematical model describing the axial stress-strain behaviour during initial loading, unloading and

reloading were established with the aid of some simple and physically meaningful parameters. Different formulations of the

tangential modulus for initial loading, unloading and reloading processes made the hysteresis effect straightforward to be re-

produced.

Employing the concept of generalized plasticity, an elastoplastic constitutive model, unifiedly describing the behaviour of

rockfill materials under monotonic and repeated loading, were established by defining the elastic stiffness tensor, the plastic

flow direction tensor, the loading direction tensor and the plastic modulus (a scalar). The criteria distinguishing loading, un-

loading and reloading were also suggested with the maximal historical stress ratio and the increase or decrease of the current



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one. In the model, the volumetric behaviour was captured by defining stress-dilatancy equations for different loading condi-

tions and constructing the flow direction tensor accordingly. The plastic hardening behaviour of rockfill materials presented

in cyclic loading were modeled by introducing two ageing functions in the representation of the tangential modulus and that

of the stress-dilatancy equation. The two ageing parameters embraced in the ageing functions could effectively control the

accumulations of the axial and volumetric strains. There are totally 13 parameters in the proposed model, all of which have

clear physical meanings and can be determined based on routine (cyclic) triaxial compression tests.

To check the performance of the model in reproducing the experimental observations, typical triaxial compression experi-

ments were simulated. It was found that most of the experimental results, obtained from static monotonic loading tests, qua-

si-static cyclic loading tests and dynamic loading tests, could be reproduced satisfactorily with the proposed model. In partic-

ular, the hysteresis behaviour and the accumulation of permanent strains could be captured simultaneously by the model

without recourse to any additional empirical relations as was often introduced in equivalent viscoelastic models. Good

agreement between the model predictions and the experimental data also inspired us to apply the model to practical rockfill

dams so as to reproduce or predict their performance during construction, impounding and earthquake shaking.

It deserves to point out finally that a real earthquake often includes dynamical compression (vertical) effect and cyclic

shearing (horizontal) effect. The former can be represented by dynamical triaxial compression tests and the latter by cyclic

simple shearing experiments. However, like most of the previous models, the model proposed in this paper was also based on

the experimental observations under axisymmetric stress states. Verification of the model was carried out purely using the

triaxial experimental results. Applicability of the model in the cyclic simple shearing condition was not verified at this stage

due to the lack of available experimental results on rockfill materials. We believe that a model suitable for axisymmetric

stress state is a reasonable start point for the extension to complex stress states, and a simple shearing test serves as quite a

valuable experiment for the consideration of complex stress states. For this reason, we proposed a basic model applicable to

triaxial stress states in this paper and are preparing to conduct cyclic simple shearing experiments on rockfill materials for the

verification and improvement of the basic model.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 51209141 & 91215301). The

provision of experimental results by Professor Li G. X. from Tsinghua University and by Senior Engineer Fu H. from Nanjing

Hydraulic Research Institute are also greatly acknowledged.

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Matsuoka H., Yao Y. P., and Sun D. A. (1999). “The Cam-Clay models modified by the SMP criterion.” Soils Founds., 39(1), 81-95.



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Captions of figures.

Fig. 1. The grain size distribution and the physical properties of the material prepared by Li (1988)

Fig. 2. Typical results rearranged from monotonic triaxial compression experiments by Li (1988).

Fig. 3. φf vs. σ3 and ψc vs. σ3.

Fig. 4. Typical results rearranged from cyclic triaxial experiments by Li (1988).

Fig. 5. Eis vs. σ3 and Eau vs. σ3.

Fig. 6. Dilatancy ratio vs. mobilized stress ratio.(Bright and black dots were extracted from Pradhan et al. (1989))

Fig. 7. The geometrical interpretation of the loading angle θ.

Fig. 8. The grain size distribution and the physical properties of the material prepared by Chen et al. (2010)

Fig. 9. Accumulations of the axial strain and the volumetric strain in cyclic triaxial compression tests (ρd=2.26g/cm3; σ3=3.0 MPa; σ1=7.5MPa).

Fig. 10. Axial strain vs. mobilized stress ratio.

Fig. 11. The tangential modulus during unloading and reloading.

Fig. 12. Distinguishing of reloading and initial loading.

Fig. 13. Monotonic triaxial compression tests on material I (shapes are experimental results (Li 1988) and curves are model predictions).

Fig. 14. Monotonic triaxial compression tests on material II (shapes are experimental results (Fu et al. 2010) and curves are model predictions)

Fig. 15. Quasi-static cyclic triaxial compression tests on material I (bright circles are experimental results (Li 1988) and curves are model predictions)

Fig. 16. Dynamical triaxial compression tests on material II (ρd=2.26g/cm3; σ3 = 3.0MPa; f=0.1Hz)

((a) and (c) are experimental results (Fu et al. 2009); (b) and (d) are model predictions)

Fig. 17. The accumulation of permanent strains during cyclic triaxial compression tests on material II (ρd=2.26g/cm3; σ3 = 3.0MPa; f=0.1Hz)

(thin curves are experimental results (Fu et al. 2009) and thick curves are model predictions)



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Table 1 Parameters involved in the dynamical constitutive model

Description Involved parameters Experiments

Friction angles , , ,o o

Static triaxial experiments including unloading Modulus parameters , , ,auk k n

Dilatancy parameters dgo

Hysteresis parameters ,u r Cyclic triaxial compression tests

Aging parameters ,v dc c

Accepted Manuscript Not Copyedited



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Table 2 Parameters related to monotonic loading behaviour

φo (º)

Δφ (º)

ψo (º)

Δψ (º) k n α dgo

Material I 47.1 6.1 43.4 3.4 320.0 0.53 0.8 1.8 Material II 52.8 8.0 45.5 3.9 900.0 0.50 0.8 1.7




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90 100

80 70 60 50 40 30 20 10

0

Diameter (mm)

Perc

enta

ge fi

ner (

%)

10.0 2.0 0.5 0.1 5.0 1.0 0.2

D 50 = 2.6mm C u = 63 G = 2.55

e max = 0.48 e min = 0.20 D r = 0.75

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0

10

20

30

40

50

0 2 4 6 8 10 12 14 16 18 201 (%)

0.5

1.0

1.5

2.0

Mob

ilize

d fr

ictio

n an

gle

(°)

m(%

)v

Volu

met

ric st

rain

3 = 3.0 MPa3 = 2.0 MPa3 = 1.2 MPa3 = 0.8 MPaConstant volume

friction angle

m = atan 1 3-1 3+( )

Note:

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35

40

45

50

300.8 1.2 1.6

(°)

fc

1.0 1.43

pa( )log

c

3f ~3~

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0

0.5

1.0

1.5

2.0

2.5

3.0

0 2 4 6 8 10 12 14 16 18 20

1-

3(M

Pa)

1 (%)

average unloadinglines (AULs)

1.0

2.0

3.0

4.0

(%)

v

unloading contraction

3 = 0.1 MPa

3 = 0.5 MPa

Acc

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3.5

4.0

4.5

5.0

3.00.8 1.2 1.61.0 1.4

3pa( )log

isE p a( )

log

auE p a( )

log

&

3~3~isE

auE

Acc

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+1.5 -1.5

-1.2

+2.0

0

d 1 > 0 d 1 < 0

p = const.

A

BC

D

EF

d =d

gvp

d sp =

d 1p d 3

p2+d 1

p d 3p-( )2

3

=1-

3

1+

23) (3

pq=

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1

2 3

s ds

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90 100

80 70 60 50 40 30 20 10

0

Diameter (cm)

Perc

enta

ge fi

ner (

%)

10.0 2.0 0.5 0.1 5.0 1.0 0.2

D 50 = 18mmC u = 11G = 2.71

e max = 0.30e min = 0.15D r = 0.84

Acc

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d



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0

0.2

0.4

0.6

0.8

1.0

0 3 6 9 12 15 18 21 24 27 30Loading cycles N

a(%

)

(a)

0

0.1

0.2

0.3

0.4

0.5

0 3 6 9 12 15 18 21 24 27 30Loading cycles N

v(%

)

(b)

d = 0.66 3

d = 0.33 3

d = 0.66 3

d = 0.33 3

Amplitude ofcyclic strain

Acc

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1A

B

C

DE

F

1

H

ITriaxial extension

Triaxialcompression

Gm

mu.

r

mr. r-( )r+

Acc

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Et

B

C

D

mmu.0

Eau

Eis

Et

E

F

mr0

Eau

Etm

(a) Unloading (b) Reloading

G

mr. r-( )r+

Acc

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2Triaxial extension

1

3

A

B

C

D

E

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0 0.6 1.2 1.8

2.0 1.0

2.4 3.0 3.6 4.2

0 2 4 6 8 1 0 1 2 1 4 1 6 (%)

1 3

) a P M

(

3.0 4.0 5.0 6.0

) %

(

3 = 1.0 MPa

0.3 MPa

3 = 1.0 MPa

0.5 MPa

0.3 MPa

0.5 MPa

0 = 0.290

Acc

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0 2 4 6

1.0 0.5

8 1 0 1 2 1 4

0 2 4 6 8 1 0 1 2 1 4 1 6 (%)

1 3

) a P M

(

1.5 2.0 2.5 3.0

) %

(

3 = 3.0 MPa

2.0 MPa

3 = 3.0 MPa

0.8 MPa

2.0 MPa

0.8 MPa

0 = 0.199

Acc

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0

0.3

0.6

0.9

1.2

1.5

1.8

0 2 4 6 8 10 12 14 16 18 20

1-

3(M

Pa)

1 (%)1.0

2.0

3.0

4.0

(%)

v

3 = 0.5 MPa

2.1

2.4

0

0.5

1.0

1.5

2.0

2.5

3.0

0 2 4 6 8 10 12 14 16 18 20

1-

3(M

Pa)

1 (%)1.0

2.0

3.0

4.0

(%)

v

3 = 1.0 MPa

3.5

4.0

(a) (b) Acc

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2.0

2.5

3.0

3.5

4.0

4.5

5.0

00.1

0.2 0.3 0.4 0.5

1-

3)aP

M(

1 (%)0.05

0.10

0.15

0.20

)%(

v

d = 0.33

5.5

6.0

6.5

3

(a) (b)

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0 0.1 0.2 0.3 0.4 0.5

1-

3)aP

M(

1 (%)0.05

0.10

0.15

0.20

)%(

v

5.5

6.0

6.5

Experiment

d = 0.33 3

Simulation

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0 0.1 0.2 0.3 0.4 0.5

1-

3)aP

M(

1 (%)0.05

0.10

0.15

0.20

)%(

v

5.5

6.0

6.5

(d)

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0 0.1 0.2 0.3 0.4 0.5

1-

3)aP

M(

1 (%)0.05

0.10

0.15

0.20

)%(

v

5.5

6.0

6.5

(c)

d = 0.66 3

Experiment

d = 0.66 3

Simulation




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0

0 . 2

0 . 4

0 . 6

0 . 8

1.0

0 3 6 9 1 2 1 5 1 8 2 1 2 4 2 7 3 0 Loading cycles N

) %

(

(a)

0

0 . 1

0 . 2

0 . 3

0 . 4

0.5

0 3 6 9 1 2 1 5 1 8 2 1 2 4 2 7 3 0 Loading cycles N

) %

(

(b)

= 0.66 3

= 0.33 3

= 0.66 3

= 0.33 3




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Modeling Cyclic Behavior of Rockfill Materials in a Framework of Generalized Plasticity

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