Computers and Geotechnics 71 (2016) 19–29

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Computers and Geotechnics

journal homepage: www.elsevier .com/locate /compgeo

Research Paper

An evolutive elasto-plastic model for cemented paste backfill

http://dx.doi.org/10.1016/j.compgeo.2015.08.0130266-352X/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +1 613 562 5800x6558; fax: +1 613 562 5173.E-mail address: mfall@uottawa.ca (M. Fall).

Liang Cui, Mamadou Fall ⇑Department of Civil Engineering, University of Ottawa, 161 Colonel By, Ottawa, Ontario K1N 6N5, Canada

a r t i c l e i n f o a b s t r a c t

Article history:Received 28 January 2015Received in revised form 1 July 2015Accepted 30 August 2015

Keywords:Cemented paste backfillTailingsBinder hydrationConstitutive modelElasto-plasticCement

An evolutive elasto-plastic model is developed in this research work to address the vital role of binderhydration in the evolution of the mechanical behavior and properties of cemented paste backfill (CPB).Double hardening/softening parameters, which include effective incremental plastic strain and degreeof binder hydration, are adopted. A non-associated plastic potential function based on the dilation angleis employed to formulate the plastic deformation and dilation of CPB. Mechanical parameters, such ascohesion, internal friction and dilation angles, stiffness, and Poisson’s ratio are expressed as functionsof the degree of binder hydration. The developed model is implemented in a finite element code,COMSOL Multiphysics, and then validated against experimental data obtained from laboratory tests per-formed in this study and by other researchers. The validation results show good agreement between thepredicted and experimental results, thus confirming the capabilities of the new constitutive model towell capture binder hydration induced evolution of the mechanical behavior and properties of CPB.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

As a relatively new technology, cemented paste backfill (CPB, amixture of dewatered tailings, water and binders) has beenincreasingly and intensively used for the backfilling of previouslymined-out underground voids (which are called stopes) in manymines around the world over the past three decades [1–8]. Theapplication of CPB can produce substantial economic and environ-mental benefits [3,7,9–11] while reducing the risks associated withopen stope mining methods, including rock bursts, in undergroundmines. Due to the fact that a large amount of tailings are used asthe key component in the preparation of CPB and then returnedto the stopes, the use of CPB has been proven as an effective tail-ings management approach compared with other surface tailingsdisposal methods. In other words, CPB technology may help inreducing the volume of surface tailings deposits and therefore min-imizes the associated geoenvironmental problems [10,12–14].Moreover, the use of CPB is an effective method for ground support(critical for underground mine work safety) as well as for thereduction of the mining cycle and speeding up of production(increases mine productivity) [7,10,15–17].

The most important design criterion of CPB is mechanical stabil-ity. As a major means of ground support CPB must satisfy themechanical stability requirements to ensure safe undergroundworking conditions for all mining personnel. This is essential

because the failure of CPB not only has substantial financial rami-fications, but can also result in severe injuries and/or fatalities [18].Thus, the understanding and prediction of the mechanical behaviorof CPB at any curing time are of great practical importance.

However, the assessing and predicting of the mechanical stabil-ity of CPB are complex and challenging tasks mainly due to the factthat its mechanical properties evolve with time because of the bin-der hydration process. It is well known that the mechanical behav-ior of any cementitious material is strongly affected by theevolution of the microstructure induced by the progress of cementhydration. Specifically, after mixing the binder and water, thechemical reactions (i.e. binder hydration) start immediately. Asthe curing time or binder hydration progresses, the hydrationproducts, such as calcium silicate hydrate (C–S–H) and calciumhydroxide (CH), gradually form and precipitate into the porousstructure of the CPB [19]. Due to the bonding effect of the hydra-tion products, the microstructure of the CPB will gradually formand change with time, which means that the mechanical proper-ties of the CPB (e.g. strength, cohesion, and stiffness) start todevelop and evolve with time until the binder hydration stops orsignificantly slows down [8,20].

Several experimental studies have been conducted in the pastdecade that investigate the mechanical behavior and propertiesof CPB [2,4,8,19,21,22]. These studies have significantly con-tributed to understanding the mechanical behavior of CPB and itsevolution with time at the experimental level. For example, ithas been experimentally observed that the hardening/softeningbehavior dominates the mechanical response of CPB under static

20 L. Cui, M. Fall / Computers and Geotechnics 71 (2016) 19–29

loading. Moreover, strain hardening/softening behaviors have beenwidely observed and well characterized in laboratory experiments,including through the use of unconfined compression strength(UCS) tests [1,8,19,22–25], as well as triaxial compression tests[4,26–28]. As for the volumetric response, dilation has been docu-mented after the peak stress state in triaxial tests performed onCPB and many other cementitious materials (e.g., concrete, mortarand cemented soils) [29–32].

Despite the tremendous contributions of the previous studiestoward the understanding of the mechanical behavior of CPB andits evolution with time, most of these studies have only dealt withthe experimental characterization of the mechanical behavior ofCPB. However, in backfill engineering practices, success in assessingand predicting the mechanical stability of a CPB structure primarilydepends on the choice and use of an appropriatemechanical consti-tutivemodel. But, a limited number of studies have been performedin the past years to develop constitutive model which enables thedescription and prediction of the mechanical behavior of CPB. Forexample, Pierce [33] used Mohr–Coulomb constitutive relation toanalyze the stability of CPB mass. In this model, constant mechani-cal properties (e.g., elastic modulus, cohesion and internal frictionangle) and perfectly plastic flow are assumed. Helinski et al. [16]developed a structured cam clay model (i.e., cementation is takeninto account) to analyze the consolidation behavior of CPB. In thismodel, the strain-hardening, cementation and possible damageare related to the evolution of yield surface, while the definition ofdegree of maturity does not incorporate the mixing parameters(i.e.,w/c, cement content) and curing condition (i.e., curing temper-ature). Furthermore, the validation of the proposed model in termsof hardening–softening and plastic volumetric expansion behaviorare not performed. Hughes [34] proposed a simple mechanicalmodel (i.e., explicit model) to depict strain-softening behavior ofCPB. A piece-wise function is adopted to control pre- and post-peak stress, which results in a sharp point shown in the stress–strain curve. Moreover, due to the fact that the constitutive relationis not derived from elastoplastic theory, there is no plastic strain inthe model. In addition, the proposed model does not consider theevolution of the mechanical properties (e.g., cohesion, internal fric-tion angle, elastic modulus) with the cement hydration degree. Theaforementioned disadvantages and limitations of themodels devel-oped in the previous works limit their applicability and ability toaccurately predict the mechanical behavior of CPB. Hence, the mainobjective of this study is to develop and validate an evolutive elasto-plastic model for CPB, which is able to describe and predict themechanical response of CPB and its evolutionwithbinder hydration.

2. Mathematical formulation of the constitutive model

In this study, it is assumed that there are small deformationswith respect to the mechanical response. The continuum mechan-ics convention is used; namely, tensile stresses are positive,whereas compressive stresses are negative.

2.1. Binder hydration model

As mentioned in Section 1, the determining factor of the evolu-tion of the mechanical properties of CPB is the binder hydration.Therefore, in order to quantitatively evaluate the impact of binderhydration on the mechanical properties and behaviors, it is moreconvenient to describe binder hydration by means of a normalizedvariable called the degree of binder hydration, n. The degree of bin-der hydration is defined as the cement fraction that has reacted[35]. For cement-based materials, the development of the degreeof binder hydration can be expressed by the following exponentialfunction [36–38]:

nðtÞ ¼ nu � exp � sTt

� �b� �

ð1Þ

where nðtÞ is the degree of hydration, nu is the ultimate degree ofhydration, sT is the hydration time parameter at the temperature ofthe cement-based materials, t is the chronological age, and b repre-sents the hydration shape parameters. The ultimate degree of hydra-tion is determined by thewater–cement ratio andmineral admixture(e.g., fly ash and blast furnace slag) weight ratio in terms of total bin-der content and can be described as a hyperbolic function [39].

nu ¼ 1:031 �w=c0:194þw=c

þ 0:5 � XFA þ 0:30 � Xslag ð2Þ

wherew=c represents the water–cement ratio, and XFA and Xslag rep-resent the weight fraction of fly ash and blast furnace slag withrespect to the total weight of the binder, respectively. If the binderjust consists of cement ðXFA ¼ Xslag ¼ 0Þ; nu is equal to 1 if the w=c isgreater than 6.258 [38]. In the model shown in Eq. (1), there are twounknown parameters, namely, the hydration time parameter, sTand the hydration shape parameter, b. These two parameters willbe determined based on the experimental data of the CPB inSection 3.

2.2. Stress–strain relation of CPB

When plastic strain is considered, the total strain, e, generatedin the CPB is composed of two components: the recoverable (rever-sible) elastic strain, ee, and the irreversible plastic strain, ep, suchthat:

e ¼ ee þ ep ð3ÞDue to the strain history dependence, the deformation model

based on an elasto-plastic framework can be written in terms ofinfinitesimal increments:

dr ¼ Dedee ¼ Deðde� depÞ ð4Þwhere r is the stress vector, and De is the tangent elastic modulusmatrix which is defined by:

De ¼

2lþ k k k 0 0 02lþ k k 0 0 0

2lþ k 0 0 02l 0 0

sym: 2l 02l

2666666664

3777777775

ð5Þ

with k and l as the Lame’s parameters:

k ¼ Etð1þ tÞð1� 2tÞ l ¼ E

2ð1þ tÞ ð6Þ

where E and t are the elastic modulus and Poisson’s ratio respec-tively, which depend on the evolution of the CPB microstructureinduced by the progress of the binder hydration. Therefore, the elas-tic modulus and Poisson’s ratio are no longer constant for CPB.

The precipitation and bonding effect caused by hydration prod-ucts lead to stiffening (elastic aging) of CPB and cause variation inthe elastic modulus, i.e., E = E(n) at the macro-level. In order tofacilitate the implementation of elastic modulus measurement innumerical analysis, the following analytical expression developedin [40] is adopted in this study:

EE1

¼ n� n0nu � n0

� �A

ð7Þ

where E1 and nu are the ultimate elastic modulus and degree ofhydration, n0 refers to the reference degree of hydration, which

(a)

(b)

0

20

40

60

80

100

120

140

160

180

200

220

0 20 40 60 80 100 120 140 160 180 200 220

Pre

dict

ed d

ata

(MP

a)

Experimental data (MPa)

Ghirian and Fall 2014

Fall 2007

Klein and Simmon 2006

1:1 line

R2=0.95

Fig. 1. Evolution of elastic modulus and determination method: (a) predictedvalues vs. measured data; (b) a typical stress–strain curve (secant modulus E50).

0.1

0.2

0.3

0.4

0.5

0.1 0.2 0.3 0.4 0.5

Pre

dict

ed D

ata

Experimental Data

Data

1:1 line

R2=0.97

Fig. 2. Comparison of experimental and predicted values of Poisson’s ratio(experimental data on Poisson’s ratio were obtained by measuring the longitudinaland radial strain of the CPB samples under static loading conditions (triaxial tests);measurements were made by using LVDT (longitudinal strain) and strain gauges(radial strain); range 50 ± 0.25 mm.

L. Cui, M. Fall / Computers and Geotechnics 71 (2016) 19–29 21

means that below a threshold value, n0, no elastic modulus develop-ment occurs. The value of n0 is determined based on the experimen-tal data provided byWang and Fall [41] who conducted a laboratoryCPB study. A is a material constant which depends on the mix com-ponents of CPB. For the evaluation of the parameters A and E1, aregression analysis is performed based on experimental data from[2,4,8]. nu can be calculated by using Eq. (2). The calculation resultsadopted in this study are listed in Table 1. As can be seen in Fig. 1,the calculated data fit well the measured data, and the coefficient ofdetermination, R2, equals to 0.95.

As mentioned in Section 1, the evolution of the mechanicalproperties, including the Poisson’s ratio, is controlled by the pro-gress of the binder hydration. Previous studies conducted on bothCPB [42] and cement paste [43–45] have revealed that, as hydra-tion progresses, the Poisson’s ratio shows a descending trend, rang-ing from 0.5 (which corresponds to fresh paste) to the ultimatePoisson’s ratio at the final degree of hydration. Therefore, the fol-lowing equation is proposed to predict the evolution of the Pois-son’s ratio with the binder hydration.

m ¼ 0:5expðB1nÞ þ B2nB3 expðB4n

B5 Þ ð8Þwhere B1; B2;B3;B4 and B5 are the fitting parameters whichcan be determined through UCS and triaxial compression tests.The measured data reported in [42] are adopted in this study todetermine the five fitting parameters. Based on the results of theregression analysis, B1 ¼ �0:2;B2 ¼ �15;000;B3 ¼ 7;B4 ¼ �10:98and B5 ¼ 0:7. As shown in Fig. 2, the predicted values of thePoisson’s ratio are in good agreement with the measured data,and the coefficient of determination, R2, is equal to 0.97.

2.3. Plastic strain

In order to analyse the plastic deformation of CPB, the incre-mental theory of plasticity is needed. The following section dealswith defining the yield surface (yield criterion) and plastic poten-tial surface (e.g., defining the direction and magnitude of plasticincrements).

2.3.1. Initial yield criterionThe Drucker–Prager (D–P) yield criterion is used, which is a

pressure-sensitive model for determining whether the failure ofthe material has occurred [46]. The D–P criterion, as a generaliza-tion of the von Mises criterion, has a number of advantages. Specif-ically, the influence of the hydrostatic stress on yield surface shapeis defined by an additional term (i.e., first stress invariant I1) in thevon Mises expression [47]. In addition, the material parameters ofthe D–P criterion not only can be directly determined by the testresults, but may also be related to cohesion and the internal fric-tion angle [48]. Moreover, compared with the Mohr–Coulomb yieldcriterion, the D–P criterion considers the effect of intermediateprincipal stress on the yield surface and represents a smooth yieldsurface, which makes it more mathematically convenient to use inthree-dimensional applications [49–52]. Due to the advantages,the D–P criterion is employed to characterize the yield behaviorof CPB in this study. The D–P yield criterion can be expressed as:

f ¼ffiffiffiffiJ2

pþ aðI1 � CÞ ¼ 0 ð9Þ

where I1 is the first stress invariant (i.e., I1 ¼ r1 þ r2 þ r3),ffiffiffiffiJ2

pis

the equivalent deviatoric stress defined by the stress deviator Sij

Table 1Calculated parameters for prediction of elastic modulus.

Parameter n0 nu E1 A

Value 0.09 1 1900 MPa 2.199

(i.e., Sij ¼ rij � I1=3dij; J2 ¼ 12 SijSij), and a and C are the material

parameters of the D–P criterion, which may be expressed by theinternal friction angle, u, and cohesion, c.

a ¼ 2 sinuffiffiffi3

pð3þ sinuÞC ¼ 3c cotu ð10Þ

Due to progress of the binder hydration, hydration products willincrease the cohesion between particles and significantly influencethe microstructure of the CPB material. Consequently, the

22 L. Cui, M. Fall / Computers and Geotechnics 71 (2016) 19–29

evolution of the mechanical parameters (i.e. cohesion and internalfriction angle) in turn determines the variation of the yield surface.Therefore, it is necessary to establish the relationship betweencohesion, c ¼ cðnÞ, internal friction angle, u ¼ uðnÞ and binderhydration. The relationship between the two mechanical parame-ters and binder hydration can be developed based on direct sheartest or triaxial test results. The details of the test procedures andresults are shown in Section 3.

2.3.2. Subsequent yield criterionOnce the stress reaches the initial yield surface, plastic defor-

mation ensues. Under continuous loading conditions, themicrostructure of geo-materials will change with the developmentof plastic deformation, which results in the variation of the mate-rial properties, such as a and C in the D–P yield criterion [53]. Cor-respondingly, the evolution of the yield surface occurs. Theincrease of the yield stress which is needed in order to drive theplastic deformation is defined as strain hardening, whereas thedecrease of the yield stress with plastic deformation after the peakstress is called strain softening [54]. Compared with common geo-materials (e.g. rock and soil), the evolution of the yield surface ofCPB material is not only determined by the plastic strain, but alsocontrolled by the binder hydration which can influence themicrostructure as well. Therefore, the hardening and softeningbehaviors of CPB are composed of two parts: chemical and strainhardening/softening behaviors:

aðn;jÞ ¼ aðjÞ þ aðnÞ ð11Þwhere aðjÞ is the plastic strain hardening/softening parameter andaðnÞ is the chemical hardening/softening parameter. Hence, aðn;jÞcan characterize the chemo-plastic hardening/softening behaviorof CPB. The chemical hardening/softening behavior can be demon-strated by the evolution of two key parameters of the D–P criterionas shown in Eq. (9), which can be rewritten as:

aðnÞ ¼ 2 sinuðnÞffiffiffi3

p½3þ sinuðnÞ�CðnÞ ¼ 3cðnÞ � cotuðnÞ ð12Þ

The plastic-strain hardening/softening process is attributed tothe cumulative plastic strain, j [53]. The cumulative plastic straincan be expressed as follows:

j ¼Z ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

23depdep

rð13Þ

In order to depict plastic strain hardening and softening,namely, the evolution of the yield function from inflation to defla-tion, a combined Weibull–normal distribution expression is used[55]:

aðjÞ ¼ D1f½1� expð�D2jÞ� þ ½D3j expð�D4jÞ�g ð14Þwhere D1;D2;D3 and D4 are the material constants which can bedetermined by using uniaxial or triaxial compression test data.Therefore, the hardening/softening parameter a(n, j) can beexpressed as:

aðn;jÞ¼D1f½1�expð�D2jÞ�þ ½D3jexpð�D4jÞ�gþ 2sinuðnÞffiffiffi3

p½3þsinuðnÞ�

ð15ÞIn order to describe the hardening and softening behaviors of

CPB materials, the isotropic hardening rule is adopted. Hence, thesubsequent yield function can be expressed as follows:

FðI1;ffiffiffiffiJ2

p; n;jÞ ¼

ffiffiffiffiJ2

pþ aðn;jÞ½I1 � CðnÞ� ¼ 0 ð16Þ

Therefore, for the loading function, the parameter aðn;jÞdenotes the chemo-plastic strain hardening/softening parameter.

2.3.3. Plastic potential and plastic flow ruleWhen the stress increases beyond the yield point, the plastic

strain increment is determined by the plastic potential and plasticflow rule. The former gives the direction of the plastic strain incre-ment, while the latter gives its magnitude [56]. The plastic flowrule is defined by the plastic potential. Due to distinctive propertiessuch as shear dilatancy and pressure-sensitive yielding, manygeo-materials demonstrate non-associated behavior whendeforming in the plastic regime [57–59]; namely, the direction ofthe plastic strain increment is not normal to the correspondingsubsequent yield surface. Therefore, the non-associative flow rulehas been commonly utilized for such materials, with the use of aplastic (flow) potential function that is different from the yieldfunction.

dep ¼ dk@g@r

ð17Þ

where k is a non-negative proportionality scalar called the plasticmultiplier. The plastic multiplier can be determined using consis-tency conditions, such that the stress point must lie on the yieldsurface after the initial yield stress has been reached.

dF ¼ @F@r

drþ @F@k

dkþ @F@n

dn ¼ 0 ð18Þ

By combining the flow rule deP ¼ dk @g@r0 and elasto-plastic for-

mula (4), the plastic multiplier can be rewritten as follows:

dk ¼adþ s

2ffiffiffiJ2

p� �

Dedeþ @F@n

dn

adþ s2

ffiffiffiJ2

p� �

De @g@r

� @F@k

@g@r

ð19Þ

For the definition of the plastic potential function, a number ofresearchers have suggested that it is suitable for the plastic func-tion to be related to the dilation angle [59–61]. Therefore, a simpleplastic potential function, g, for the D–P yield surface is considered,which is [62,63]:

g ¼ 2 sinwffiffiffi3

pð3þ sinwÞ I1 þ

ffiffiffiffiJ2

pð20Þ

where w denotes the dilation angle. Similarly, the dilation angle willbe influenced by the binder hydration as well, namely, w ¼ wðnÞ. Inorder to determine the dilation angle, a modified formula proposedin [64] is employed in this study to characterize the evolution of thedilation angle:

sinðwÞ ¼ K1ðnÞ � ln pn

pr

� �þ K2ðnÞ ð21Þ

where pn and pr respectively refer to the normal load stress that actson the CPB sample, and the reference stress, namely the atmospherepressure (101.325 kPa). K1ðnÞ and K2ðnÞ are the material parameterswhich evolve with binder hydration and will be determined fromthe direct shear test data in Section 3.

2.4. Constitutive relation

The incremental form of plastic strain can be expressed basedon the plastic flow rule:

dep ¼adþ s

2ffiffiffiJ2

p� �

Dedeþ @F@n

dn

adþ s2

ffiffiffiJ2

p� �

De � @F@eP

ð22Þ

Then, the constitutive relation can be developed by substitutingEq. (22) into Eq. (4):

L. Cui, M. Fall / Computers and Geotechnics 71 (2016) 19–29 23

dr0 ¼ De �De adþ s

2ffiffiffiJ2

p� �

De

adþ s2

ffiffiffiJ2

p� �

De � @F@eP

8>><>>:

9>>=>>;de ð23Þ

Eq. (23) is the constitutive equation of CPB based on the D–P yieldcriterion.

(a)

(b)

0

50

100

150

200

250

300

350

0 25 50 75 100 125 150

Coh

esio

n (k

Pa)

Curing time (days)

Ghirian and Fall 2014 Test Model

Fig. 3. Evolution of cohesion and determination method: (a) predicted values vs.measured data; (b) a typical shear stress–normal stress graph.

3. Parameter determination, model verification and numericalsimulation of experimental tests

In this section, the identification of the parameters of the pro-posed model is presented. Triaxial compression and direct sheartests were performed and the experimental data were utilized todetermine the material parameters. Moreover, the capabilities ofthe developed model to predict the mechanical behavior of hydrat-ing CPB are tested against well controlled laboratory experiments.These experiments were performed in the present study and otherstudies by other researchers. The procedures of the experimentalmechanical tests performed are briefly described in the nextsubsection.

3.1. Experimental tests

Twomechanical tests, including undrained triaxial compressionand direct shear tests, have been carried out in this study. In orderto prepare the CPB samples, the adopted mixture recipe included4.5% Portland cement type I (PCI) and a water to cement ratio(w/c) of 7.6. The tailings material, cement and water were mixedby using a concrete mixer, and homogenized for about 7 min. Then,the homogeneous CPB paste was poured into plastic cylindermolds. Subsequently, the molds were sealed and cured at roomtemperature. For the triaxial compression tests, the curing timeswere 7, 28 and 90 days, while the curing periods of the CPB sam-ples for the direct shear tests were 1, 3, 7, 28, 60, 90 and 120 days.

The triaxial compression tests were conducted in accordancewith ASTM D4767-02. The deviatoric stress and axial displacementwere measured by using a load cell sensor and a linear variable dis-placement transformer (LVDT), respectively. The experimentaldata were then collected with a computer data acquisition system.The Tri-Flex 2 test system was utilized to measure the volumechange of the CPB samples. A minimum of two samples weretested under each confining pressure.

In accordance with ASTM D5607-08, the direct shear tests wereperformed on CPB samples after different curing times. When theexpected curing time was reached, the samples were removedfrom the molds and trimmed into cube specimens of the requiredsize (60 mm ⁄ 60 mm ⁄ 25 mm). Two LVDTs were installed in thetest system to measure the horizontal and vertical displacements,respectively, and the shear stress was measured through a load cellsensor. The test data were recorded by a computer data acquisitionsystem throughout the testing. During the testing, shear force wasapplied at a rate of 1.0 mm/min. A minimum of three samples weretested under each vertical (normal) load force level.

3.2. Determination of parameters

In order to determine the hydration degree that correspondedto the different curing times, the following values which were col-lected from [65] are utilized in this study: w=c ¼ 7:6; sT ¼ 1:4 day,and b ¼ 0:394. The evolutive formulas of the mechanical propertiescould then be established. For the cohesion of CPB, the experimen-tal results of the direct shear tests showed that there exists anapproximate power relationship between cohesion and the degreeof hydration. Therefore, the following empirical function is pro-

posed in this study to describe the evolution of cohesion (kPa) withthe degree of hydration:

cðnÞ ¼ M1nM2 ð24Þ

whereM1 andM2 are the fitting constants which can be determinedthrough a regression analysis based on the experimental data of thedirect shear tests. In this study, M1 ¼ 478 kPa and M2 ¼ 3:3. Fig. 3shows a comparison of the experimental and predicted data ofthe cohesion. Meanwhile, the predicted data are also comparedwith the experimental data reported in [8]. It can be seen that thevalues of cohesion increase with curing time which contributes tothe growth of bonds between tailings particles with the advanceof binder hydration.

Similarly, the internal friction angle (in degrees) can beexpressed as a function of the degree of hydration through theregression method and the resulting expression can be written asfollows:

uðnÞ ¼ N1nN2 þ N3n ð25Þ

where N1;N2 and N3 are the fitting parameters which can be deter-mined by the experimental data. Based on results of a regressionanalysis on the experimental data, N1 = �176.9�, N2 ¼ 2 andN3 ¼ 174:2�. The evolution of the internal friction angle with curingtime is plotted in Fig. 4. The experimental data reported in [8] arealso collected and compared with the predicted values of the inter-nal friction angle. As can be seen in Fig. 4, there is an opposite

-5

0

5

10

15

20

25

30

35

0 10 20 30 40 50 60 70 80 90 100 110 120

Dila

tion

angl

e (d

eg.)

Curing time (days)

Test-50kPa Test-100kpa Test-200kPa

Predicted-50kPa Predicted-100kPa Predicted-200kPa

Fig. 5. Comparison of test data and prediction of dilation angle with curing time.

24 L. Cui, M. Fall / Computers and Geotechnics 71 (2016) 19–29

variation trend in comparison to cohesion with curing time. Specif-ically, the total trend of the internal friction angle displays adecreasing trend with progress of the binder hydration.

The dilation angle is calculated by using the direct shear testdata. Then, the material parameters K1ðnÞ and K2ðnÞ in the dilationangle expression (Eq. (21)) can be determined through a regressionanalysis. The following fitting functions are proposed to calculateK1ðnÞ and K2ðnÞ:K1ðnÞ ¼ a1nþ a2 ð26Þ

K2ðnÞ ¼ b1n2 þ b2nþ b3 ð27Þ

where a1; a2; b1; b2 and b3 are the fitting parameters. In thisstudy, a1 ¼ �0:2126; a2 ¼ 0:01257; b1 ¼ 1:239; b2 ¼ �0:6461 andb3 ¼ 0:01115 based on the experimental data from the direct sheartests. The results plotted in Fig. 5 are evidence of an upward trendwith curing time and downward trend with normal stress levels.Overall, a good agreement between the test data and model predic-tion is reached.

In order to simulate the hardening/softening behavior of CPB,four unknowns (D1, D2, D3 and D4) that exist in aðjÞ are needed,which can be determined by the UCS or triaxial compression testdata. In this study, the UCS test data reported in [8] are adopted.The following fitting equation is proposed to predict the evolutionof the four unknowns with progress of the binder hydration:

D1 ¼ d1 expðd2nÞ ð28Þ

D2 ¼ d3nd4 þ d5 ð29Þ

D3 ¼ d6nd7 þ d8 ð30Þ

D4 ¼ d9 expðd10nÞ ð31Þwhere d1; d2;d3;d4;d5;d6; d7;d8;d9 and d10 are the fitting parame-ters. Through a regression analysis of the reported experimentaldata in [8], d1 ¼ 3:895, d2 ¼ �6:024, d3 ¼ 272:8, d4 ¼ 6:786,d5 ¼ 78:77, d6 ¼ 45;930, d7 ¼ 22:25, d8 ¼ 143, d9 ¼ 15:87 andd10 ¼ 2:757 are adopted in this study. The fitting results are shownin Fig. 6. It is found that the hardening/softening parameter, a(j),evolves with curing time. Therefore, it can be concluded that theevolution of the hardening/softening parameter, a(j), is consistent

10

15

20

25

30

35

40

45

50

0 20 40 60 80 100 120

Inte

rnal

fric

tion

angl

e (d

eg.)

Curing time (days)

Test Model

Fig. 4. Comparison of test data and prediction of internal friction angle with curingtime.

with the variation of the material properties (i.e. cohesion, internalfriction and dilation angles, stiffness, and Poisson’s ratio), whichdemonstrates the vital role that binder hydration has played inCPB, namely, the evolution of the material properties of CPB is influ-enced by the binder hydration.

3.3. Model verification and numerical simulation of experimental tests

The proposed evolutive elasto-plastic model is implemented ina FEM code, COMSOL Multiphysics� Version 4.4. This code can per-form analysis of diverse categories, including: stationary and time-dependent, linear and nonlinear, eigenfrequency, modal and fre-quency response analyses by using FEM together with adaptivemeshing and error control which employ a variety of numericalsolvers [66]. The capabilities of the developed elasto-plastic modelto predict and capture the time-dependent (induced by the pro-gress of binder hydration) evolution of the mechanical behaviorof CPB are tested against several well controlled laboratory

0.00

0.02

0.04

0.06

0.08

0.10

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0.14

0.16

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

Har

deni

ng/s

ofte

ning

par

amet

er

Effective plastic strain

7-day 28-day 90-day

150-day 7-day predicted 28day-predicted

90day-predicted 150day-predicted

Fig. 6. Comparison of test data and prediction model of hardening/softeningparameters.

0

200

400

600

800

1000

1200

1400

1600

1800

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Axi

al s

tress

(kP

a)

Strain (%)

Test-7 days (Ghirian and Fall 2014) Test-28 days (Ghirian and Fall 2014)

Test-90 days (Ghirian and Fall 2014) Test-150 days (Ghirian and Fall 2014)

Test-180 days (Simon 2005) Test-365 days (Klein and Simon 2006)

Model-7 days Model-28 days

Model-90 days Model-150 days

Model-180 days Model-365 days

Fig. 7. Comparison of numerical prediction and experimental data of UCS tests.

0

250

500

750

1000

1250

1500

1750

2000

0 250 500 750 1000 1250 1500 1750 2000

Pre

dict

ed U

CS

(kP

a)

Measured UCS (kPa)

Ghirian and Fall 2014 Fall et al. 2007

Klein and Simon 2006 Kesimal et al. 2005

Belem et al. 2000 1:1 line

R2=0.98

Fig. 8. Predicted versus measured UCS for various CPB samples.

Table 3Information from reported tests.

Source Belemet al. [26]

Fall et al. [4] Simms andGrabinsky [28]

Binder content (%) 4.5 4.5 5W/B ratio 6.36 7 7.14Water – TW –Binder PCI/Slag at (20/80) PCI/PCV at (50/50) PCITailings Mill tailings Mine tailings Golden giant

tailingsTest UC/TC UC/TC TC

W/B – water to binder ratio; UC – uniaxial compression test; TC – triaxial com-pression test; PC – Portland cement; TW – tap water; FA – fly ash.

L. Cui, M. Fall / Computers and Geotechnics 71 (2016) 19–29 25

mechanical tests performed in this study and by other researchers[1,22,27,28,4,8,26] on CPB of various ages. This model verificationwill provide confidence in its validity and reliability for binderhydration induced evolution of the mechanical behavior of CPB.

3.3.1. Hardening/softening behavior simulationAs mentioned in Section 2, the hardening/softening behavior of

CPB is controlled by both binder hydration and strain develop-ment. To verify the model, numerical simulation of the UCS and tri-axial compression tests was performed and the simulation resultswere compared with numerous experimental data reported in theliterature.

Fig. 7 shows the comparison of the simulation results andreported experimental data of the UCS tests for both short-termcured (7 and 28 days) and long-term cured (more than 28 days)samples. Detailed information on the reported tests is shown inTable 2.

As shown in Fig. 7, the predicted axial stress–strain and UCS val-ues are in good agreement with the experimental results. It isinteresting to notice that the stress–strain curves of the short-term and long-term cured CPB samples illustrate different evolu-tion trends in terms of strain hardening/softening behavior. Forthe short-term (7 and 28 days) cured samples, the strain hardeningbehavior is dominant during uniaxial compression testing, whilefor the long-term (90, 150, 180 and 365 days) cured samples, the

Table 2Information on reported tests.

Source Belem et al. [26] Simon [19] Kesimal et a

Binder content (%) 4.5 5 7W/B ratio 6.36 7.8 4.6Water – DW TWBinder PCI/Slag at (20/80) PC10/FA at (50/50) OPCTailings Mill tailings Mine tailings Mine tailingsTest UC/TC UC UC

W/B – water to binder ratio; UC – uniaxial compression test; TC – triaxial compression tecement type 10 (=PCI); TW – tap water; DW – deionized water; FA – fly ash.

strain hardening/softening behavior becomes obvious withincreasing stiffness. Similar behaviors have also been observed inprevious experimental studies [4,67,68]. In addition, both the lab-oratory and simulation results demonstrate that the stress levelincreases with curing time, which proves that chemical hardeningdoes have a significant impact on the mechanical behavior of CPB.

The simulated UCS values are plotted against the experimentaldata collected from [2,4,8,22,26], see Fig. 8. The simulation resultsshow good agreement with the test data, and the coefficient ofdetermination, R2, is equal to 0.98.

In order to further verify the proposed model under confiningpressure conditions, simulation of the triaxial compression tests

l. [22] Klein and Simon [2] Fall et al. [4] Ghirian and Fall [8]

5 4.5 4.57.8 7 7.6DW TW TWPC10/FA at (70/30) PCI/PCV at (50/50) PCIMine tailings Mine tailings SilicaUC UC/TC UC

st; PCI – Portland cement type I; OPC – ordinary Portland cement; PC10 – Portland

(a)

0

20

40

60

80

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120

140

160

180

0 2 4 6 8 10 12

Dev

iato

ric S

tress

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T20 kPa-3 days (Simms and Grabinsky 2009) Model-T20 kPa

(b)

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200

400

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800

1000

1200

1400

1600

0 0.5 1 1.5 2 2.5 3

Dev

iato

ric S

tress

(kP

a)

Axial Strain (%)

T600-91 days (Fall et al. 2007) T800 kPa-91 days (Fall et al. 2007)

T400 kPa-112 days (Belem et al. 2000) Model-T600 kPa

Model-T800 kPa Model-T400 kPa

Fig. 9. Comparison between numerical prediction and corresponding test data fortriaxial testing on CPB samples: (a) short-term cured, and (b) long-term curedsamples.

(a)

0

100

200

300

400

500

600

-3 -2 -1 0 1 2 3 4 5

Dev

iato

ric S

tress

(kP

a)

Axial Strain (%)

Fig. 10. Comparison of the simulation results and experimental data of triaxial testing opressure – 300 kPa.

26 L. Cui, M. Fall / Computers and Geotechnics 71 (2016) 19–29

was performed and compared with the experimental data reportedin the literature, see [4,26–28]. Table 3 shows detailed informationfrom the reported tests.

The simulation results which are plotted against the reportedexperimental data in [4,26–28] at different confining pressurelevels that range from 20 to 800 kPa are presented in Fig. 9. Itcan be seen that the simulated deviatoric stress–axial strain curveis in good agreement with that of the experimental data. The devi-atoric stress–axial strain curves exhibit a similar developmenttrend to that of the UCS curves. For the short-term cured CPBsamples, the stress–strain relation mainly demonstrates strainhardening behavior. As curing time increases, the strain harden-ing/softening behavior is evident. Similar to the UCS curves, thepeak stress values show a clearly ascending trend with respect tocuring time.

By comparing the numerical simulation results and experimen-tal data (i.e., UCS and triaxial compression test data), it can be con-cluded that the proposed evolutive elasto-plastic model is capableof predicting the strain hardening/softening behavior of CPB andcapturing the influence of binder hydration on the mechanicalbehaviors.

3.3.2. Volumetric response simulationBesides the stress–strain relation, the volumetric behavior of

CPB under loading conditions is also critical. In order to verifythe ability of the model to predict the volumetric response, thenumerical simulation results are compared with the experimentaldata from the triaxial compression tests.

Figs. 10–12 show comparisons of the predicted results andthe experimental data of the volumetric strain in the triaxialcompression tests. The volumetric strain shows a two-stage evo-lution with deviatoric stress development. Firstly, it can beobserved that the volumetric strain of CPB gradually reduceswith deviatoric stress and the volume contraction is maintainedin the elastic regime. Subsequently, around the peak stress, thevolume expansion (also called volume dilation) is initiated andcontinues throughout the softening regime. Based on the exper-imental data, the volume expansion is reduced as the confiningpressure is increased. From a microstructure point of view, dif-ferent mechanisms dominate the two-stage changes of volumet-ric strain. Specifically, the volume contraction is attributed to the

(b)

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100

200

300

400

500

600

700

-3 -2 -1 0 1 2 3 4 5 6 7

Dev

iato

ric S

tress

(kP

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n CPB samples after 7 days of curing: (a) confining pressure – 200 kPa; (b) confining

(a) (b)

0

200

400

600

800

-4 -3 -2 -1 0 1 2 3 4 5

Dev

iato

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tress

(kP

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0

200

400

600

800

1000

-4 -3 -2 -1 0 1 2 3 4 5

Dev

iato

ric S

tress

(kP

a)

Axial Strain (%)

Fig. 11. Comparison of the simulation results and experimental data of triaxial testing on CPB samples after 28 days of curing: (a) confining pressure – 200 kPa; (b) confiningpressure – 300 kPa.

(a)

0

200

400

600

800

1000

1200

-3 -2 -1 0 1 2 3 4

Dev

iato

ric s

tress

(kP

a)

Strain (%)

(b)

0

200

400

600

800

1000

1200

1400

-3 -2 -1 0 1 2 3 4

Dev

iato

ric s

tress

(kP

a)

Strain (%)

Fig. 12. Comparison of the simulation results and experimental data of triaxial testing on CPB samples after 90 days of curing: (a) confining pressure – 400 kPa; (b) confiningpressure – 500 kPa.

L. Cui, M. Fall / Computers and Geotechnics 71 (2016) 19–29 27

compression of pore space which is occupied by pore air. Duringthe curing period, pore water is continuously consumed by bin-der hydration and the former water occupied pore space gradu-ally becomes empty. Therefore, when a compression load isapplied, the pore space will be compacted and the volume ofthe CPB sample will decrease (volume contraction), while vol-ume expansion is induced by the onset, spread and intensifica-tion of microcracks after the yield point [69]. Thecorresponding lateral strain exhibits a similar evolution trend.The simulated lateral strain shows that, during the first stageof the triaxial compression testing, the lateral strain is relativelysmall, then the lateral strain shows a rapidly ascending trendwith deviatoric stress.

According to the comparisons of the numerical simulationresults and numerous experimental data, it can be concluded thatthe developed model is not only capable of predicting the stress–strain and hardening/softening behaviors, but also reproduces wellthe volumetric response of CPB.

4. Summary and conclusion

Based on the results obtained in this study, the following con-clusions can be established.

(i) An evolutive elasto-plastic model based on the D–P yield cri-terion has been developed to capture the mechanical behav-ior of hydrating CPB. The model takes into consideration thedegree of binder hydration and its effect on the evolution ofthe mechanical properties (i.e. cohesion, internal friction anddilation angles, stiffness, and Poisson’s ratio) and mechanicalbehavior (i.e. stress–strain, hardening/softening and volu-metric dilation). Due to the fact that the hardening/softeningbehavior of CPB is controlled by both binder hydration andplastic strain, the model adopts double parameters of hard-ening/softening, including degree of binder hydration andeffective plastic strain. Moreover, non-associative plasticflow is employed to accurately model the plastic behavior.

28 L. Cui, M. Fall / Computers and Geotechnics 71 (2016) 19–29

(ii) The comparison results demonstrate good agreementbetween the model prediction and experimental data, whichverifies the capability of the proposed model to reproducethe mechanical behavior of CPB including volumetricresponse and hardening/softening behavior. Furthermore,the comparison results also show that binder hydrationplays a critical role in the evolution of the mechanical prop-erties of CPB materials and dominates the correspondingmechanical behaviors, which proves the necessity of thedevelopment of an evolutive elasto-plastic mechanicalmodel.

(iii) Due to the fact that the model fully considers the influenceof binder hydration on the material properties and mechan-ical behavior, the developed evolutive elasto-plasticmechanical model is useful for the development of fully cou-pled multiphysics modeling of CPB. In the future, the devel-oped model here can be incorporated into a fully coupledmodel to reveal the interaction of multiphysics processesin CPB, which could optimize CPB designs.

Acknowledgements

The authors would like to thank the Natural Sciences and Engi-neering Research Council of Canada (NSERC) and the University ofOttawa.

Appendix A. Supplementary material

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.compgeo.2015.08.013.

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