as

K.

est

poscrite efaditedasethet

ing the deformation and stability of landlls and any post-closure development structures located on

1. Introduction

Proper assessment of the stability oftremells in00; Staeasonat in or

sition, several studies have attempted to quantify the mechanicalproperties of MSW based on laboratory and eld testing programsas well as back-analysis (or inverse analysis) using landll eldperformance data (Kavazanjian and Matasovic, 1995; Zekkos,2005).

The compressibility and shear strength behavior of MSW hasdrawn the attention of several researchers. For example, Sowers

in terms of primary and secondary coefcients and demonstratedtheir application to post-closure maintenance and developmentplans of landlls.

Singh and Murphy (1990), Jessberger and Kockel (1993), Kava-zanjian and Matasovic (1995), Manassero et al. (1996), SivakumarBabu (1998), Vilar and Carvalho (2002), Zekkos (2005) and Reddyet al. (2009a,b) studied the shear strength response of MSW. Sev-eral investigators evaluated the stability of landll slopes basedon limit equilibrium methods using the drained MohrCoulombshear strength parameters of MSW (e.g., Reddy et al., 1999; Starket al., 2000).

* Corresponding author.E-mail addresses: gls@civil.iisc.ernet.in (G.L. Sivakumar Babu), kreddy@uic.edu

Waste Management 30 (2010) 1122

Contents lists availab

an

els(K.R. Reddy), sandeepkc@civil.iisc.ernet.in (S.K. Chouksey).bility of landll appurtenances (e.g., gas collection pipes, leachatecollection pipes, etc.) and to plan and design post-closure redevel-opment projects. Limit equilibrium methods have been routinelyused to analyze landll slopes and the adopted soil mechanics-based settlement methods have been commonly used to calculatethe settlement of MSW (Sharma and Reddy, 2004). These stabilityand settlement analysis methods require the mechanical proper-ties of MSW such as unit weight, shear strength parameters, andcompressibility parameters (compression and recompression ra-tios, secondary compression ratio). Despite heterogeneous compo-

ques et al. (2003) presented a composite model for compressibilityconsidering three important mechanisms; instantaneous compres-sion in response to applied load, secondary mechanical creep, andtime dependent biological decomposition. They illustrated the useof the model consisting of ve parameters such as compression in-dex and coefcients and rate constants for both mechanical creepand biodegradation, respectively, to predict the settlement ofMSW landll. Sharma and De (2007) presented methods for esti-mating settlements of MSW landlls, including bioreactor landllsinvolving leachate recirculation for enhanced waste degradation,(MSW) landll slopes has become exthe recent failure of several landworldwide (Koerner and Soong, 20and Fourie, 2005). In addition, the rsettlement has also become importan0956-053X/$ - see front matter 2009 Elsevier Ltd. Adoi:10.1016/j.wasman.2009.09.005landlls. 2009 Elsevier Ltd. All rights reserved.

municipal solid wastely important in light ofthe United States andrk et al., 2000; Blightble prediction of MSWder to ascertain the sta-

(1973), Yen and Scanlon (1975), Edil et al. (1990), Landva and Clark(1990), Grisolia et al. (1991), Gabr and Valero (1995), Wall andZeiss (1995), Park and Lee (1997), Ling et al. (1998), SivakumarBabu (1999), El-Fadel and Khoury (2000), Park et al. (2002,2007), Marques et al. (2003), Oweis (2006), Gangathulasi (2008),and Reddy et al. (2009a,b) studied the compression response ofMSW and/or proposed different approaches to predict immediatecompression and time dependent compression under load. Mar-and pore water pressure response of these three types of MSW adequately. The model is useful for assess-Constitutive model for municipal solid wand biodegradation-induced compression

G.L. Sivakumar Babu a,*, Krishna R. Reddy b, SandeepaDepartment of Civil Engineering, Indian Institute of Science, Bangalore 560 012, IndiabDepartment of Civil and Materials Engineering, University of Illinois at Chicago, 842 W

a r t i c l e i n f o

Article history:Accepted 3 September 2009Available online 25 October 2009

a b s t r a c t

A constitutive model is prounder loading using theextended to incorporate thtotal compression under losion and triaxial consolidaobtained from working phof degradation, and (c) syn

Waste M

journal homepage: www.ll rights reserved.te incorporating mechanical creep

Chouksey a

Taylor Street, Chicago, IL 60607, USA

ed to describe the stressstrain behavior of municipal solid waste (MSW)ical state soil mechanics framework. The modied cam clay model isfects of mechanical creep and time dependent biodegradation to calculateng. Model parameters are evaluated based on one-dimensional compres-undrained test series conducted on three types of MSW: (a) fresh MSWof a landll, (b) landlled waste retrieved from a landll after 1.5 yearsic MSWwith controlled composition. The model captures the stressstrain

le at ScienceDirect

agement

evier .com/locate /wasman

The following assumptions have been made in the development

evePeCeBb

t

ste MOverall, the published studies address (1) immediate settlementto applied loading, (2) time dependent mechanical creep, (3) bio-logical decomposition and compression of wastes, and (4) shearstrength and stability of slopes. However, these studies consideredcompression and shear strength behavior of MSW independently.It is well recognized that the compression and shear strength ofMSW are interrelated and there is a need to study the coupledcompression-strength behavior of MSW.

Several researchers have begun to use advanced mathematicalmodels to assess the coupled stability and settlement and strengthof landlls by attempting to account for the complex behavior ofMSW (Reddy et al., 1996; Mahler and Iturri, 1998; McDougall,2007). For example, Reddy et al. (1996) demonstrated that dis-placement-based analysis provides a better understanding of theshear stressdisplacement behavior of MSW and liner systems un-der incremental MSW loading conditions that closely simulate thewaste lling operations in a landll. This study used the DuncanChang hyperbolic model to represent stressstrain behavior ofMSW and the model parameters typical to that of organic soilswere selected to model MSW. The properties of MSW were foundto have a signicant effect on the shear stressdisplacement ofcomposite geosynthetic liner systems. The main drawback of suchadvanced analyses is the use of a constitutive model that does notaccurately describe the coupled compression-strength response ofMSW under drained or undrained conditions. One of the typicalcharacteristics observed in the stressstrain response of MSW isthat the strength of MSW increases continuously with increase in

Notation

MSW municipal solid wastep0 mean effective stressp00 pre-consolidation pressure (kPa)q deviator stress (kPa)Dp0 change in mean effective stressM frictional constantk compression indexj recompression or swelling indexOCR over-consolidation ratioG shear modulusdepv increment in volumetric straindeps increment in plastic shear strain

12 G.L. Sivakumar Babu et al. /Wastrain without showing signs of denite peak or an ultimate value,which is often attributed to the reinforcement effects of some ofthe constituents (e.g., plastics) of MSW (Machado et al., 2002; Red-dy et al., 2009a,b). In addition, the time-dependent secondary com-pression and biodegradation-induced compression also play asignicant role on the stressstrain characteristics of MSW. Thereare several constitutive models developed for soils (Desai and Sir-iwardane, 1984; Reddy and Saxena, 1992; Lade, 2005), but they arenot directly applicable to MSW. Machado et al. (2002) proposed aconstitutive model to simulate the mechanical behavior of MSW,assuming that the behavior is controlled by two distinct parts,the mechanical behavior of the brous material and of the paste,according to a coupled elasto-plastic model. The paste is modeledusing concepts stated in critical state soil mechanics together witha non-associated ow rule. In a follow-up study, Machado et al.(2008) considered the effect of biodegradation of organic matteron the mechanical response of MSW; however, the effect ofmechanical creep is not considered. Dixon et al. (2005) emphasizedthe need for the development of appropriate constitutive models topredict the strength and stiffness of waste body to evaluate thelandll stability and in the analysis of slope failures.of the proposed constitutive model for MSW:

(1) The mechanical behavior follows elasto-plastic behavior inthe framework of critical state soil model with associatedow rule.

(2) The general stressstrain response of MSW is governed bythe brous nature of MSW.

(3) The secondary compression is governed by the time depen-dent phenomenon in exponential function similar to theassumption of Gibson and Los (1961) model, which is givenby

0To date, a simple and complete constitutive model for MSWconsidering stress history, mechanical creep and biodegradationto predict stressstrain response has not been developed. This pa-per presents a constitutive model to describe the stressstrain re-sponse during loading under drained and undrained conditionsbased on a critical state soil mechanics framework. Several seriesof laboratory one-dimensional compression and triaxial consoli-dated undrained shear experiments conducted on three types ofMSW were utilized to determine the model parameters. The appli-cability of the model is assessed by comparing the model resultswith the experimental results.

2. Proposed constitutive model(4)

Leton theisotroMSWtime since placement of the waste in the landllt0 time since application of the stress incrementc the rate constant for mechanical creepEdg total amount of strain that can occur due to biological

decompositiond rate constant for biological decomposition00total volumetric strain for the MSWstrain due to gradually increasing loadtime dependent strain due to mechanical creeptime dependent strain due to biological decompositioncoefcient of mechanical creepanagement 30 (2010) 1122eC bDp01 ect 1where b is the coefcient of mechanical creep; Dp0 is thechange in mean effective stress, c is the rate constant formechanical creep; and t0 is the time since application of thestress increment.The biological composition is related to time and the totalamount of strain that can occur due to biological decompo-sition. The time dependent biological degradation is pro-posed by Park and Lee (1997) and is given by

eb Edg1 edt00 2

where Edg is the total amount of strain that can occur due tobiological decomposition; d is the rate constant for biologicaldecomposition; and t0 0 is the time since placement of thewaste in the landll.

us consider the isotropic loading of MSW as shown in Fig. 1ae ln(p0) plot. If the MSW is normally consolidated at A, thepic loading will follow the path AB. Let us now unload theto the mean effective pressure p0A. Because of the elasto-plas-

bln p0

ste Mtic nature, the unloading path will not follow loading path AB. In-stead, the MSW will follow the path BD upon unloading. Whenthe MSW is reloaded from pressure p0A to p

0B, it will usually fol-

low the same path that indicates the elastic behavior. The slopeof the loading path is denoted by k, and the slope of unloadingreloading path is denoted by j. The vertical distance AD showsthe plastic component in the change in volume, and DE showsthe elastic component of the change in volume. Now we canwrite the total change in void ratio (e) during the loadingunload-ing cycle as follows:

From Fig. 1a, total change in void ratio during loading path AB

e eA eB k ln p0B

p0A

kln p0B ln p0A 3

Change in void ratio in path BD

ee eD eE j ln p0B

p0A

jlnp0B lnp0A 4

Increment in total volumetric strain is given by

dev de k dp0

5

ln p PBPA

ep

ee

A

B

D

E

(

)

0

0ln

ln

ee

p

p

()

0

0

=

e

p

lnln

e

p

=

0e

e

a

Fig. 1. (a) Consolidation behavior in e

G.L. Sivakumar Babu et al. /Wa1 e0 1 e0 p0

Hence, the elastic volumetric strain deev can be written as;

deev dee

1 e j

1 edp0

p06

And, increment in plastic volumetric strain can be written as:

depv k j1 e

dp0

p0 2gdgM2 g2

" #7

The above formulations for increments in volumetric strain due toelastic and plastic are well established in critical state soil mechan-ics literature (Wood, 1990). There is need to extend the elasto-plas-ticity concepts for MSW considering time dependent mechanicalcreep and biological degradation.

In order to consider the compression due to mechanical creepand biological decomposition, the total volumetric strain of theMSW is expressed as:

dev deev depv decv debv 8where deev , de

pv , decv and debv are the increments of volumetric strain

due elastic, plastic, time dependent mechanical creep and biodegra-dation effects.From, Eq. (1) increment in volumetric strain due to creep iswritten as:

decv cbDp0ect0dt0 9

From, Eq. (2) increment in volumetric strain due biodegradation ef-fect is written as:

debv dEdgedt00dt00 10

In the present case t0 time since application of the stress incrementand t0 0 time since placement of the waste in the landll are consid-ered equal to t.

Using Eqs. (6), (7), (9), and (10) and substituting in Eq. (8) totalincrement in volumetric strain is given by

dev j1 edp0

p0 k j

1 e

dp0

p0 2gdgM2 g2

" # cbDrectdt

dEdgedtdt 11On simplication of Eq. (11),

p0 k e e " # 1kj 8< 9=

vuuu

space; and (b) yield locus in pq space.

anagement 30 (2010) 1122 13qMp0 0p0

exp 01e0 bDp

0ectEdgedt 1e 1: ;t12

where p0 is the mean effective stress and p00 is the pre-consolida-tion pressure. A detailed derivation of Eq. (12) is given in appen-dix. The Eq. (12) is the proposed new model for MSW, which isan extended form of modied cam clay model that predicts thestressstrain behavior under loading. In addition to elastic andplastic strains, the total volumetric strain (ev) includes mechanicalcompression under loading as well as mechanical creep and bio-logical degradation effects represented by Eqs. (1) and (2). M isthe frictional constant, e0 is the initial void ratio and e is the voidratio after load increment. The review of the literature (Machadoet al., 2002) indicates that the swelling index (j) is normally varybetween 10% and 20% of the compression index (k). Eq. (12) rep-resents the deviatoric stress for MSW under load consideringmechanical creep and biodegradation effects. The mean stress (p)is given by

p p0initial q3

13

In addition, the pore water response under undrained conditions isgiven by

u p p0 14

3. Experimental program for model validation

To validate the proposed constitutive model, one-dimensionalcompression and triaxial consolidated undrained shear experi-ments were conducted on three types of MSW. These experimentalresults allowed for the determination of the proposed modelparameters. The model results were then compared with theexperimental results.

3.1. MSW samples

Three types of MSW samples were tested in this study: freshMSW, landlled MSW, and synthetic MSW. Fresh MSW and land-lled MSW samples were collected from the Orchard Hills landll(Illinois, USA). Fresh MSW samples were collected from the work-ing phase of this landll. Landlled MSW samples were collectedby drilling a borehole to a depth of 20 m at a capped location onthe landll. Based on the lling records, the landlled MSW sam-ples represent MSW undergone approximately 1.5 years of biodeg-radation. A comparison of the composition of the fresh andlandlled MSW samples is shown in Table 1 (Reddy et al.,2009a,b). For this study, MSW samples collected from the eldwere shredded with a slow-speed, high-Torque shredder (ShredPax Corp., AZ-7H,Wood Dale, IL) to suit small-scale laboratory test-ing. The particle size distribution of MSW samples after shreddingis shown in Fig. 2. These results show that shredding resulted in a

similar size distribution for both fresh and landlled MSW sam-ples. Both fresh and landlled MSW samples had an average in-situmoisture content of about 30% (by wet weight) or 44% (by dryweight). The average organic content was approximately 78% forfresh MSW and 61% for landlled MSW. Degradation during a per-iod of 1.5 years is believed to be the main reason for the lower or-ganic content of landlled MSW.

Field fresh and landlled MSW samples were heterogeneousand it was often difcult to perform replicate testing using thesamples with exactly the same composition. Therefore, syntheticMSW was prepared using different types of materials in propor-tions to create an approximate representation of typical MSW inthe United States (Table 2). For simplicity, 40% nonbiodegradablefraction in MSW was represented by using 20% glacial till soiland 20% clean ne sand instead of using fractions such as metals,plastics, textiles, rubble, glass and miscellaneous inorganic materi-als. Glacial till also represents the typical soil used for daily cover atlandlls in the mid-western United States. Biodegradable fractionswere represented by paper, grass, greens, meat, and bread (Table2). The particle size distribution of synthetic MSW is also shownin Fig. 2. The moisture content and organic content of syntheticMSW samples were about 57% (wet weight basis) or 134% (dryweight basis) and 57.5%, respectively.

3.2. Compressibility testing

Conned compressibility testing was carried out in an oedome-ter to determine the compressibility characteristics of waste sam-ples. Waste was compacted into a 63 mm inside diameter and27 mm thick circular oedometer rings with a tamper. The sample

Table 1Composition of fresh and landll MSW.

Category Components Fresh MSW(% by wet massa)

Landlled MSW(% by wet massa)

Biodegradable Garden waste Grass clippings (size lessVegetable waste Greens (approximately cu

ize

(% Synthetic MSW

14 G.L. Sivakumar Babu et al. /Waste Management 30 (2010) 1122Meat Ground beefCellulose White wheat bread (sEasilybiodegradable

Food waste 6.6 0.5Garden waste 0.3 0.0

Mediumbiodegradable

Paper 8.2 6.6Cardboard 13.3 16.1Food carton 0.0 0.0Sanitary waste 3.1 1.1

Hardlybiodegradable

Textiles 5.8 4.8Nappies 1.7 0.1Wood 11.7 8.5

Inert waste Metal 4.4 4.1Plastic bottles 5.7 5.7Other plastics 5.3 9.7Special waste 0.0 0.0Medical waste 0.1 0.0Other waste 3.5 3.6Inert waste 5.8 5.0Glass 4.4 0.4

Residual nes Fines (

of elapsed time or when compression signicantly decreased, thenormal stress was increased to 96 kPa and compression was mea-

sured at different time intervals. This procedure was repeated fornormal stresses of 192, 383 and 766 kPa to simulate the maximum40 m depth of a landll. Based on the total compression under eachnormal stress, axial strain versus normal pressure was plotted andcompression ratio was calculated. The normal stress versus maxi-mum axial strain is plotted in Fig. 3. The axial strain versus elapsedtime data for all applied stresses for fresh MSW, landll MSW, andsynthetic MSW are shown in Fig. 4ac, respectively. The initialmoisture content, dry unit weight, and specic gravity values forthe three waste types tested are summarized in Table 3. A detailedpresentation and discussion of the results is provided by Reddyet al. (2009ac).

3.3. Consolidated undrained triaxial shear testing

In order to perform consolidated undrained (CU) triaxial sheartesting, the waste was compacted in a cylindrical cell. Tests wereperformed according to the ASTM D4767 (ASTM, 2007) with cylin-drical samples with an average diameter of 70 mm and height of140 mm. For each waste type, three identical samples were pre-pared and then inserted into latex membranes. All samples wereinitially subjected to a conning pressure of 35 kPa and back pres-sure of 21 kPa and were saturated. The samples were then isotrop-

Normal Pressure (kPa)000100101

Axi

al S

trai

n (%

)

0

10

20

30

40

50

60

70

Fresh MSW Landfilled MSW Synthetic MSW

Fig. 3. Compressibility test results for fresh MSW, landlled MSW and syntheticMSW.

0

10

20

30

40

50

Stra

in (%

)

0

10

20

30

40

50

0.0001 0.001 0.01 0.1 1 10

Stra

in (%

)

Time (days) 0.0001 0.001 0.01 0.1 1 10

G.L. Sivakumar Babu et al. /Waste Management 30 (2010) 1122 1560

(a). Fresh MSW

Time (days)

0

10

20

30

40

50

60

70

Stra

in (%

)

(c). Sy

0.001 0.01

Fig. 4. Strain versus elapsed time for fresh MSW, landlled MSW and synthetic MSW undapplied constantly for duration of about 24 h).60(b). Landfilled MSW

Time (days)

nthetic MSW

0.1 1 10er increased normal stresses of 48, 96, 192, 383 and 766 kPa (each normal stress was

Table 3Properties of the three types of MSW tested in this study.

Type of waste Average initial moisture content (%) Average bulk wet density (kg/m3) Average specic gravity

Dry weight basis Wet weight basis

Fresh MSW 43.7 30.4 740 0.85Landll MSW 43.3 30.2 1020 0.97Synthetic MSW 134.6 57.3 1150 1.09

0

300Initial Effective Confining Pressure

Axial Strain (%)0 5 10 15 20 25

160

ained shear test results for landlled MSW.

Table 5Additional parameters to account for mechanical creep and biodegradation in theproposed model.

Type of waste b (m2/kN) c (day1) E d (day1)

16 G.L. Sivakumar Babu et al. /Waste Management 30 (2010) 1122ically consolidated under different conning pressures of 69, 138,and 276 kPa and volume change was measured. The MSW sampleswere nally subjected to shear under undrained condition. Porewater pressures were measured during shearing. To ensure uni-

Axial Strain (%)0 5 10 15 20 25

Fig. 5. Typical triaxial consolidated undrDev

iato

r Str

ess

(kPa

)

50

100

150

200

250 69 kPa 138 kPa 276 kPa form pore pressures throughout the specimen, samples weresheared at a constant strain rate (approximately 1% per minute).Typical results for landlled MSW are shown in Fig. 5. Reddyet al. (2009ac) provide the results for fresh MSW and syntheticMSW and also discuss all of the test results.

4. Results and discussion

4.1. Model parameters

The proposed model requires the same parameters as themodied cam clay model. These parameters are: frictional con-stant (M), compression index (k), recompression or swelling in-dex (j), and over-consolidation ratio, which is dened as theratio of pre-consolidation pressure to mean effective stressOCR p00=p0. In addition to these modied cam clay modelparameters, the implementation of the proposed constitutive

Table 4Modied cam clay model parameters.

Type of MSW Conning press(kPa)

Fresh MSW 69138276

Landlled MSW 69138276

Synthetic MSW 69138276Pore

Pre

ssur

e (k

Pa)

0

20

40

60

80

100

120

140 69 kPa 138 kPa 276 kPa

Initial Effective Confining Pressuremodel requires four additional parameters. These are: twoparameters representing the mechanical creep, namely coef-cient of mechanical creep (b) and rate constant (c); and twoother parameters representing the biodegradation effect givenby total amount of strain (Edg) and rate constant (d), which arefunctions of biodegradation.

The compression and swelling indices (k and j parameters) areobtained from compression test results shown in Fig. 3. The normalpressure vs. axial strain data is replotted as normal pressure vs.void ratio and the slope of the compression line on this plot givesthe value of k. Compression testing did not include an unload andreload cycle; but, the literature shows that recompression index

ure u0 Void ratio(e)

k j G M OCR

16 0.61 0.12 0.012 4274 0.61 116 0.65 0.12 0.012 8779 0.61 116 0.23 0.12 0.012 13206 0.61 1

23 0.46 0.091 0.0091 5092 0.90 123 0.42 0.091 0.0091 10303 0.90 123 0.50 0.091 0.0091 20997 0.90 1

10.3 0.47 0.175 0.0175 2672 0.38 110.3 0.51 0.175 0.0175 8287 0.38 110.3 0.36 0.175 0.0175 14871 0.38 1

dg

Fresh MSW 0.0081 0.0105 0.160 0.0315Landlled MSW 0.0046 0.0127 0.116 0.0243Synthetic MSW 0.0095 0.0420 0.210 0.0600

(j) is 10 to 20% of k. In this study, the value of j is taken as 0.1 k.The over-consolidation ratio (OCR) is taken as 1 (normally consol-idated condition) and frictional constant (M) is calculated using

effective stress friction (u0) based on the following equation appli-cable for triaxial compression tests: M = 6 sin u

0/(3 sin u0). Table

4 summarizes the values of the modied cam clay model parame-ters determined for the three types of MSW.

The mechanical creep and biodegradation parameters(b, c, Edg, d) are determined based on laboratory compression testresults shown in Fig. 4ac for the three types of MSW. It is assumedthat mechanical creep and biodegradation occur simultaneously asfunctions of time, and hence the slope of the strain versus time plotreects the combined effects of c and d. Since the total volumetricstrain is known at the end of testing period of 10 days, EDG is takenas a fraction equal to 40% of biodegradable matter content andthen the rate constant (d) is back-calculated. Knowing d, value ofc is calculated. The term b, the coefcient of mechanical creep, isobtained separately for all increments of load during compressiontest and the average value is taken for this study. Minor adjust-ments to the parameter values are made so that the predictedstress strain data are close to those obtained from experiments. Ta-ble 5 summarizes the values of these parameters for the threetypes of MSW. It should be noted that these parameters are basedon relatively short-term compression experiments. For accuratelong-term predictions, the values of these parameters should beassessed based on long-term large-scale laboratory experimentsand/or eld monitoring data.

0

50

100

150

200

250

300

350

400

0 3 6 9 12 15Strain (%)

Dev

iato

ric s

tres

s (k

Pa)

MCC (69 kPa) MCC (138 kPa) MCC (276 kPa) Predicted (69 kPa) Predicted (138 kPa) Predicted (276 kPa)Experimental (69 kPa) Experimental (138 kPa) Experimental (276 kPa)

Fig. 6. Comparison between the modied cam clay model and the proposed modelfor fresh MSW.

-50

0

50

100

150

200

250

300

q (k

Pa)

Experimental Prediction

(kPa

) -50

0

50

100

150

200

250

300

q (k

Pa)

Experimental Prediction

(kPa

)

G.L. Sivakumar Babu et al. /Waste Management 30 (2010) 1122 17-150

-100

0 3 6 9 12 15

u Strain (%)

(a) Initial Effective Confining Pressure = 69 kPa

-150

-50

50

150

250

350

450

0 3 6

Stra

q (k

Pa)

u (k

Pa)

(c) Initial Effective Con

Fig. 7. Comparison of stressstrain and pore water pressure response based on the proposconning pressures: (a) 69 kPa, (b) 138 kPa and (c) 276 kPa.-150

-100

0 3 6 9 12 15

Strain (%)

u

(b) Initial Effective Confining Pressure = 138 kPa

9 12 15

in (%)

Experimental Prediction

fining Pressure = 276 kPa ed model and triaxial CU experiments for fresh MSW under different initial effective

50

ste M-150

-100

-50

0

50

100

150

200

Strain (%)

q (k

Pa)

Experimental Prediction

u (k

Pa)

(a) Initial Effective Confining Pressure = 69 kPa

0 3 6 9 12 15

500

18 G.L. Sivakumar Babu et al. /Wa4.2. Model predictions

Fig. 6 shows typical results of comparison of the stressstraincurves predicted for the fresh MSW based on the modied camclay (MCC) model and the proposed model Eq. (8) using the param-eter values. The gure shows the variation of deviatoric stress withaxial strain at different conning pressures (69, 138 and 276 kPa)for fresh MSW. From these results, it can be observed that the pre-dicted values of deviatoric stress from the modied cam clay mod-el are lower at the same strain level compared to the experimentalresults as well as the predicted values from the proposed model.The results also show that the values from the proposed modelare in agreement with the experimentally observed behavior offresh MSW. This agreement is attributed to the main feature ofthe proposed constitutive model that the modied cam clay modelhas been extended to include mechanical creep and biological deg-radation effects of MSW. Similar results are obtained for landlledMSW and synthetic MSW. In general, values of deviator stressesobtained from experimental results continuously increase as theaxial strain is increased showing that there is no peak stress andultimate failure. The values in Tables 4 and 5 are used to obtainthe results presented in Figs. 79, which show the variation ofstressstrain and pore water pressure response of the three differ-

-200

-100

0

100

200

300

400

0 3 6Str

q (k

Pa)

u (k

Pa)

(c) Initial Effective Con

Fig. 8. Comparison of stressstrain and pore water pressure response based on the propeffective conning pressures: (a) 69 kPa, (b) 138 kPa and (c) 276 kPa.-150

-100

-50

0

0 3 6 9 12 15Strain (%)

u (k

Pa)

(b) Initial Effective Confining Pressure = 138 kPa 100

150

200

q (k

Pa)

Experimental Prediction

anagement 30 (2010) 1122ent types of wastes (fresh MSW, landll MSW, and syntheticMSW). It can be observed that the proposed model gives a reason-able approximation of the experimental results.

Sivakumar Babu et al. (in press) illustrated the applicability ofthe proposed model to prediction of settlements due to incremen-tal loading of waste with time for typical landll conditions. Thepredicted settlement results using the proposed model are com-pared with the predicted settlement results using fourteen differ-ent reported models. The proposed model predicts the totalsettlement in a range similar to the eld validated models that con-sider all three components (mechanical, creep and biodegradation)of the settlement.

5. Conclusion

A simple constitutive model is proposed for MSW based onan extension of the modied cam clay model to include the ef-fect of time dependent mechanical creep and the time depen-dent biological degradation of MSW. A comparison of stressstrain response between the experimental results and the modelresults for three different types of MSW demonstrated that the

9 12 15ain (%)

Experimental Prediction

fining Pressure = 276 kPa

osed model and triaxial CU experiments for landlled MSW under different initial

ste M-150

-100

-50

0

50

100

150

200

0 3 6 9 12 15Strain (%)

q (k

Pa)

Experimental Prediction

u (k

Pa)

(a) Initial Effective Confining Pressure = 69 kPa

150

200

(kPa

)

G.L. Sivakumar Babu et al. /Waproposed model can capture the stress strain response of MSWadequately.

Acknowledgements

The authors acknowledge the suggestions of the reviewerswhich have been useful in improving the presentation of the re-search work presented in the paper. This project is a collaborativeproject between the Indian Institute of Science, Bangalore and theUniversity of Illinois at Chicago. Partial funding is provided by theUS National Science Foundation Grant (CMMI #0600441), which isgratefully acknowledged.

Appendix A

From Fig. 1a, total change in void ratio during loading path AB

e eA eB k ln p0B

p0A

kln p0B ln p0A A:1

Change in void ratio in path BD

ee eD eE j ln p0B

p0A

jlnp0B lnp0A A:2

Increment in volumetric strain due to elastic and plastic are derivedfrom Fig. 1a.

-150

-100

-50

0

50

100

0 3 6Stra

q u

(kPa

)

(c) Initial Effective Con

Fig. 9. Comparison of stressstrain and pore water pressure response based on the propeffective conning pressures: (a) 69 kPa, (b) 138 kPa and (c) 276 kPa.-150

-100

-50

0

50

100

150

200

0 3 6 9 12 15Strain (%)

q (k

Pa)

Experimental Prediction

u (k

Pa)

(b) Initial Effective Confining Pressure = 138 kPa

Experimental Prediction

anagement 30 (2010) 1122 19de k dp0

p0A:3

dee j dp0

p0A:4

where the superscript e denotes the recoverable elasticcomponent.

The increment in plastic void ratio dep is written as:

dep de dee k j dp0

p0A:5

The total compressive volumetric strain is written as:

dev de1 e0 k

1 e0dp0

p0A:6

Hence, the elastic volumetric strain deev is written as

deev dee

1 e j

1 edp0

p0A:7

In the stressstrain theory based on the critical state concept, it isassumed that there is no recoverable energy associated with sheardistortion (i.e. dees 0). Therefore, at all timesdes deps A:8where des is the shear strain.

9 12 15in (%)fining Pressure = 276 kPa

osed model and triaxial CU experiments for synthetic MSW under different initial

ste MAccording to the normality condition, the incremental plasticstrain vector is normal to the yield surface at any point. With ref-erence to Fig. 1b, this can be expressed as:

depsdepv

dp0

dqA:9

where depv is the volumetric plastic strain. Note that depv versus depsplot is superimposed on the p0 q plot, Fig. 1b.

To derive the equation of the yield locus, let us dene the stressratio

g qp0

A:10

q gp0 A:11

where q is the deviatoric stress and p0 mean stress

dq p0dg gdp0 A:12Let us assume that the slope of the yield curve at any point (p0, q) asshown in Fig. 1b be w. Since q decreases with p0, the sign of w isnegative

dqdp0

w A:13

dq wdp0 A:14Substitution of Eq. (A.12)

p0dg gdp0 wdp0 A:15By rearranging Eq. (A.15)

p0dg wdp0 gdp0 A:16

p0dg w gdp0 A:17

p0

dp0 dgw g 0 A:18

The above equation denes a yield locus. Since for this model thesuccessive yield loci are geometrically similar, w is a function of g(stress ratio) only. Therefore, any yield curve passing through aknown point can be obtained by integrating Eq. (A.18) between(p00; p

0) and (0, g).Z p0p00

dp0

p0Z g0

dgg w 0 A:19

ln p0 ln p00 Z g0

dgg w 0 A:20

Equation (A.20) represents the yield curve passing through (p00, 0).Equation (A.19) can be expressed in the differential form as

dp00p00

dp0

p0 dgg w 0 A:21

dp00p00

dp0

p0 dgg w A:22

The total change in void ratio during loading and unloading

de k dp00

p00A:23

20 G.L. Sivakumar Babu et al. /Wadee j dp00

p00A:24Hence,

dep k jdp00

p00A:25

From Eq. (A.22)

dep k j dp0

p0 dgw g

A:26

Therefore, the plastic volumetric strain can be written as

depv dep

1 e k j1 e

dp0

p0 dgw g

A:27a

The tangential angle w as function of g (stress ratio) is expressedas:

w M2 g22g

Substituting value of w in Eq. (A.27)

depv k j1 e

dp0

p0 2gdgM2 g2

" #A:27b

In addition to elastic and plastic behavior of municipal solid waste,considering compression due to mechanical creep and biologicaldecomposition, the total volumetric strain of the MSW is expressedas:

dev deev depv decv debv A:28where deev , de

pv , decv and debv are the increments of volumetric strain

due elastic, plastic, time dependent mechanical creep and biodegra-dation effects.

The time dependent strain due to mechanical creep is expressedas:

eC bDp01 ect0 A:29where b is the coefcient of mechanical creep; Dp0 is the change inmean effective stress, c is the rate constant for mechanical creep;and t0 is the time since application of the stress increment. Theabove expression is used in the studies of Edil et al. (1990) and isan adaptation of Gibson and Los model. Now, from Eq. (A.29) incre-ment in volumetric strain due to creep effect is expressed as:

decv cbDp0ect0dt0 A:30

The time dependent strain due to biological decomposition is ex-pressed as:

eb Edg1 edt00 A:31

where Edg is the total amount of strain that can occur due to biolog-ical decomposition; d is the rate constant for biological decomposi-tion; and t0 0 is the time since placement of the waste in the landll.Eq. (A.31) was proposed by Park and Lee (1997) to correlate biolog-ical degradation and the associated secondary settlements with thesolubilization rate of degradable matter in solid waste. From Eq.(A.31), total increment in volumetric strain due to biologicaldecomposition is expressed as:

debv dEdgedt00dt00 A:32

In the present case, t0 time since application of the stress incrementand t0 0 time since placement of the waste in the landll are consid-

anagement 30 (2010) 1122ered equal to t.From Eq. (A.28), total increment in volumetric strain due to all

mechanisms is given by

Z e Z p0 0 Z g

ste Me0

de1 e0

k1 e p00

dpp0

k j1 e 0

2gdgM2 g2

cbDrZ t0ectdt dEdg

Z t0edtdt A:36

On simplication of Eq. (A.36)The yield surface is obtained as

qMp0

p00p0

kexp

e0e1e0 bDp

0ectEdgedt

1e " # 1kj

18