Constitutive model for long term municipal solid waste mechanical behavior

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Computers and Geotechnics 35 (2008) 775–790

Constitutive model for long term municipal solid wastemechanical behavior

S.L. Machado a,*, O.M. Vilar b,1, M.F. Carvalho c,2

a Department of Materials Science and Technology, Federal University of Bahia, 02 Aristides Novis Street, Salvador-BA 40210-630, Brazilb Geotechnical Department, University of Sao Paulo, 1465 Dr. Carlos Botelho Avenue, Sao Carlos-SP 13560-250, Brazilc School of Engineering, Catholic University of Salvador, Pinto de Aguiar Avenue, Salvador-BA 40.710-000 2589, Brazil

Received 4 November 2006; received in revised form 16 November 2007; accepted 28 November 2007Available online 30 January 2008

Abstract

This paper presents some improvements in the model proposed by Machado et al. [Machado SL, Carvalho MF, Vilar OM. Consti-tutive model for municipal solid waste. J Geotech Geoenviron Eng ASCE 2002;128(11):940–51] now considering the influence of biodeg-radation of organic matter in the mechanical behavior of municipal solid waste. The original framework considers waste as composed oftwo component groups; fibers and organic paste. The particular laws of behavior are assessed for each component group and then cou-pled to represent waste behavior. The improvements introduced in this paper take into account the changes in the properties of fibers andmass loss due to organic matter depletion over time. Mass loss is indirectly calculated considering the MSW gas generation potentialthrough a first order decay model. It is shown that as the biodegradation process occurs the proportion of fibers increases, however, theyalso undergo a degradation process which tends to reduce their ultimate tensile stress and Young modulus. The way these changes influ-ence the behavior of MSW is incorporated in the final framework which captures the main features of the MSW stress–strain behaviorunder different loading conditions.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Municipal solid waste; Constitutive model; Long term behavior

1. Introduction

As well as so-called ‘end of pipe’ alternatives such asburning, there are many ways to deal with waste, rangingfrom reducing the production, encouraging recycling andre-using municipal solid waste (MSW). However, largeamounts of waste are still disposed in the environmentand the main alternative for the safe deposition of MSWcontinues to be the sanitary landfill. This is an engineeredstructure that comprises many fundamental elements such

0266-352X/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compgeo.2007.11.008

* Corresponding author. Tel.: +55 71 3283 9461/9845; fax: +55 71 32839461.

E-mail addresses: [email protected] (S.L. Machado), [email protected] (O.M. Vilar), [email protected] (M.F. Carvalho).

1 Tel.: +55 16 3373 9489.2 Tel.: +55 71 3322 0748.

as a liner, leachate and gas collection and removal systems,waste cells and temporary and final covers.

As the MSW itself is the main construction materialused to build these engineering structures, its mechanicalbehavior is of paramount importance in obtaining a safeand economical design. MSW is known to be a heteroge-neous material of varying constituent types and dimensionscontaining elements that degrade with time. According toDixon et al. [4], the design of landfills must consider stabil-ity both within and between elements of the lining systemand within the municipal solid waste (MSW) to ensure thatuncontrolled slippage does not occur. However, the designmust also consider the long term integrity of the lining sys-tem. Therefore, stresses and strains in both mineral andgeosynthetic lining materials must be controlled. Assess-ment of integrity requires taking into account the interac-tion between the MSW body and the lining components.

mailto:[email protected]

mailto:orencio@ sc.usp.br

mailto:orencio@ sc.usp.br

mailto:[email protected]

Nomenclature

af rate of increase in the E values as the radialstress increase [–]

Cm organic matter methane specific yield [m3 CH4/dry-Mg]

Ca MSW secondary compression index [–]dx infinitesimal variation of the generic variable

‘‘x”

dxd long term infinitesimal variation of x due tocreep and decomposition

e MSW void ratio [–]E(t = 0), E initial Young’s modulus of the fibers [MPa]E(t) function that relates the changes of E over time

[MPa]ep paste void ratio [–]Eu Young’s modulus of the fibers as measured in a

tension test [MPa]ff fiber yield criterionfm fibers mobilization functionfp paste yield criterionGf shear elastic modulus of the fibers [MPa]Gp shear elastic modulus of the paste [MPa]k methane generation rate constant [year�1]kE constant related to the long term changes in the

fibers Young modulus, E(t) [year�1]kq constant related to the long term changes in the

ultimate tensile stress of the fibers, qmax (t)[year�1]

Lo methane generation potential [m3 CH4/Mg]M slope of the projection of the critical state line

(CSL) on (q;p) plane [–]msf, msp, ms, and m masses of fibers and paste solids,

solids and total [kg]MSW municipal solid wasten shape parameter of the paste yield criterion [–]N specific volume of the paste for a unitary value

of stress [–]p mean normal stress [kPa]Pf ratio between the dry weight of the fibers and

the MSW total dry weight [–]po paste pre-consolidation stress [kPa]Pp ratio between the dry weight of the paste and

the MSW total dry weight [–]q deviator stress [kPa]qf part of deviator stress acting on the fibers [MPa]qmax yielding deviator stress of the fibers [MPa]qmax(t = 0) initial value of the yielding deviator stress of

the fiber [MPa]

qmax (t) function that relates the changes in qmax withtime [MPa]

qp part of deviator stress that acts on the paste[kPa]

t elapsed time [year]Ve ratio between the MSW void ratio and the paste

void ratio [–]Vf ratio between the fiber volume and the MSW to-

tal volume [–]Vp ratio between the paste volume and the MSW

total volume [–]v, vs, vsf and vsp MSW volume, MSW solids volume and

volumes of fibers and paste solids, respectively[m3]

w MSW water content (dry basis) [–]xo initial value of variable x

a(t) parcel of MSW coupled long term compressiongenerated by waste mass loss process [–]

a* rate of increase of the MSW coupled long termcompression [–]

cs specific unit weight of the MSW solid compo-nents [kN/m3]

csf specific unit weight of the fiber components[kN/m3]

csp specific unit weight of the paste solid compo-nents [kN/m3]

ev MSW volumetric strain [–]b parameter of the MSW flow rule [–]es MSW shear strain [–]esf shear strain of the fibers [–]ee

sf elastic shear strain of the fibers [–]esp shear strain of the paste [–]ee

sp elastic shear strain of the paste [–]ep

sp plastic shear strain of the paste [–]ev MSW volumetric strain [–]ee

vp elastic volumetric strain of the paste [–]ep

vp plastic volumetric strains of the paste [–]u paste friction angle at critical state condition

[�]j paste swelling index [–]k paste compression index [–]mf mean Poisson ratio of the fibers [–]mp Poisson ratio of paste [–]ra axial stress [kPa]rr radial stress [kPa]w = n/(1 + n) parameter in the flow rule of the paste

[–]

776 S.L. Machado et al. / Computers and Geotechnics 35 (2008) 775–790

The most appropriate method to assess the interactionbetween the waste and barrier system is the use of numer-ical modeling techniques. These may also be applied tomany other landfill structures, such as gas collecting sys-tems, which are particularly affected by the differential set-tlements of the waste body.

A sanitary landfill is usually conceptualized as a bio-chemical reactor which has waste and water as the maininputs and gas and leachate as major outputs. The genera-tion of gas and leachate is related to organic matter deple-tion in a mass transfer process. The resulting mass lossimplies the generation of new voids in the waste and addi-

Fig. 1. Stress–strain curves: (a) new samples; (b) 4 years old samples.

S.L. Machado et al. / Computers and Geotechnics 35 (2008) 775–790 777

tional settlement which goes on for many years until com-plete organic matter degradation is achieved. The aggressiveenvironment within the landfill is also responsible for mod-ifying the properties of other less degradable componentssuch as plastics, the main component of the fiber fractionof waste. Therefore, one would expect that biodegradationof the organic waste and other physical–chemical processesthat take place during the lifespan of a landfill contributesignificantly to modifying the behavior of the waste overtime, thus influencing the performance of the entire landfill.

The changes in the mechanical properties of MSW overtime as a result of the decomposition process have beenthe focus of study for many researchers working in the area.Unfortunately, very few laboratory and field data are avail-able to confirm and to quantify these variations. Some indi-rect evidence of this variation is embodied in the data ofLandva and Clark [12] who presented results of direct sheartests performed on MSW samples from different landfills inCanada. Values of cohesion intercept between 0 and 23 kPawere found and the friction angle varied between 24� and41�. These variations could be related to differences in wastecomposition, fiber orientation inside the waste and also tothe state of degradation, although no clear evidence wasfound for this. Similar behavior can be found in the datapublished by Jessberger et al. [8] and Konig and Jessberger[11]. The data comes from triaxial tests performed onMSW samples of different ages and they failed to show aclear trend for the influence of waste age on shear strength.Possible reasons for this could be related to the waste com-position that might have changed between the samples ofdifferent ages or to the differences in the degradation processthat developed at different rates in the various samples. Theinfluence of an aggressive environment on material proper-ties can be illustrated by the data of Machado et al. [14].Here a reduction of about 24% in the modulus of elasticityand of about 9% in the ultimate tension strength of plasticmaterial after 4 years since deposition was observed.Fig. 1 compares the behavior of virgin and degraded plastic.

The influence of waste degradation can be more easilyappreciated when the volumetric behavior of the waste isanalyzed. Some data in the technical literature confirmsthe variation of MSW properties with degradation processor at least with time, assuming the degradation to be afunction of time. Mehta et al. [16] have presented field set-tlement results obtained in tests performed on cells withand without leachate recirculation. After 1231 days accu-mulative methane productions reached 63.1 and 27.9 m3

CH4/Mg of wet waste, respectively. The observed averagesettlements were 15.5% of the waste thickness in the caseof the cell with leachate recirculation and 3% in the controlcell. According to El-Fadel et al. [5] the biodegradationprocess weakens the structural resistance of the waste, lead-ing to a substantial loss of volume and settlement. Thus,during the biodegradation process additional immediatesettlements can be expected as a consequence of theMSW structure compression. Hossain et al. [7] have shownthe results of oedometer tests performed on samples at

different stages of decomposition. The state of decomposi-tion was quantified by the methane yield related to the cel-lulose (C) plus hemicellulose (H) to lignin (L) ratio(C + H)/L. It is shown a notable increase in the primarycompression index, Cc, as the decomposition process pro-gressed. Secondary compression was analyzed using amechanical creep index, Ca, and a biological creep index,Cb. The magnitude of the biological index varied with thestate of decomposition. The larger values were associatedto actively decomposing samples. The creep index provedto be independent of the state of the waste decomposition,ranging from 0.02 to 0.03.

In this paper a general framework to describe themechanical behavior of MSW is developed. The frameworkconsiders the MSW as composed of two componentgroups, the paste and the fibers. The fibers are basically dif-ferent types of plastics and other fibrous-like components,such as certain textiles. The paste comprises the otherremaining materials, mainly organic matter. The effect ofbiodegradation and associated mass loss is included inthe model through a first order decay model used to simu-late the gas generation process and a mass balance

Fig. 2. Phase diagram illustrating the effect of the mass loss.

Table 1Some basic relationships, based on the assumed hypothesis, which applyto the fibers and paste component and to the whole MSW

Item Fibers Paste MSW

Phase solids volume vsf vsp vsf + vsp = vs

Phase solids weight msf msp msf + msp = ms

Phase proportion bydry mass

Pf = msf/ms Pp = msp/ms 1

Phase proportion byvolume

Vf = vsf/v Vp = (vsp + vv)/v 1

Phase volume ofvoids

0 vv vv

Mean normal stress p p p ¼ raþ2�rr

3

Deviator stressincrement

dqf ¼ 3�Gf � dee

sf

dqp ¼ 3�Gp � dee

sp

dq = dqp �Vp + dqf � Vf


equation while the degradation of fibers is related to thevariation in fiber properties over time. The framework isintended to be an improvement of the constitutive modeldeveloped by Machado et al. [13] and in this sense it retainsthe major features and assumptions of the base model.

2. Modeling the short term MSW stress–strain behavior

The main features and hypothesis to substantiate thepresent constitutive model for MSW mechanical behaviorhave been proposed by Machado et al. [13] and are repro-duced here:

(1) The entire behavior of the MSW is assumed to becontrolled by: (a) the fibrous materials in the waste(composed basically of plastics) and (b) the MSWpaste, composed of any other non-fibrous material(wood, organic compounds, rubber, glass, water,liquid phases generated during the decompositionprocess, etc.) – thus, the model considers the MSWas a composite of two principal components eachwith its own constitutive model.

(2) The MSW deviator stress, q, is partly supported bythe fibers and partly by the paste. The mean normalstress, p, is taken to be the same for both fibers andpaste. It is assumed that in a shearing plane the pres-ence of the fibers will not affect the normal compo-nents of stress, but they will contribute to theshearing strength.

(3) The strains are related to the fibers and the paste. TheMSW void ratio is attributed to the paste alone. Thevolumetric strains of the fibrous material are ignoredwhich makes the MSW volumetric strains controlledby the paste compression.

The phase relationships of MSW adopted by Machadoet al. [13] are shown in Fig. 2 (before degradation) andTable 1 summarizes some basic relationships that reflectthe aforementioned assumptions. Table 2 summarizes theconstitutive equations used for the main waste compo-nents, the fibers and the paste. The subscripts f and p referto fibers and paste, respectively. The MSW paste behavioris assumed to be predicted by a critical state frameworkwith a non-associated flow rule. The paste swelling andcompression indexes (j and k parameters, respectively)are derived from an hyperbolic relationship between themean normal stress and the paste void ratio (ep). dee

sf and

Table 2Constitutive relationships adopted for paste and fibers

Item Fibers

Shear modulus Gf ¼ E2�ð1þmf Þ

Yield locus ff = qf � qmax = 0

Hardening law (paste only) depvp ¼ ðk� jÞ dpo

po; ep ¼ N

pk � 1

Flow rule (paste only) depvp

depsp¼ 1

1þn

� ��M � ½pn � ðpo � pÞ�ð

Elasticity modulus (fibers only) E = Eu + af � rr

deesp are the elastic shear strain increments of fibers and

paste and depsp and dep

vp are the increments of shear and vol-umetric plastic strains of the paste. M corresponds to theslope of the critical state line (CSL) of paste on (q:p) plane,po corresponds to the paste isotropic pre-consolidationstress and n is a paste yield locus shape parameter, assumedas n = 2. The b parameter in the flow rule makesdep

sp=depvp ¼ 0 for p = po. E refers to the fiber Young mod-

ulus, tp and tf are the paste and fiber Poisson ratios and N

is the paste specific volume for p = 1.

2.1. The influence of the fibers on the MSW mechanical

behavior

The influence of the fibers on the mechanical behavior ofthe waste has been addressed by many authors in the tech-nical literature. It is usually assumed that the MSW frictionangle and the cohesion intercept are mobilized followingdifferent patterns. The friction angle tends to be fully mobi-

Paste

Gp ¼ 3�ð1�2�mpÞ�p2�ð1þmpÞ�j

fp ¼ qp �M � ½pn � ðpo � pÞ�1

1þnð Þ � n 11þnð Þ ¼ 0

11þn�1Þ � ½pn�1 � n � ðpo � pÞ � pn� � 1þ sin p

2�wbppo

� �b� �

� 2 � n� ��1


lized at large strain values, while the MSW cohesion pro-vided by the fibrous material starts to be mobilized atstrains of 10% or higher, and a limit value is usually notobserved [9]. According to Jessberger et al. [8] and Konigand Jessberger [11] the MSW shear strength behaviormay be compared to that found in reinforced soils. Fig. 3was used by Heiss-Ziegler and Fehrer [6] and shows moreclearly how the fiber components sustain shear strains. Itmust be said, however, that apart from the fact that theproportion of fiber components in the MSW body are byfar superior to that found in reinforced soils, the phasesC and D illustrated in the figure are not easily measured,at least when performing conventional direct shearingand triaxial tests as they require very high levels of strainto occur.

The reinforcement effect of fibers is considered stressdependent (see Table 2). This is taken into account by mak-ing the modulus of elasticity, E, dependent on the confiningor radial stress, rr (rr was found by Machado et al. [13] tobe the most important component of stress influencing thevalues of E in the proposed model. For real 3D problems, itis interesting to replace rr by (r2 + r3)/2). Eu is the modu-lus as measured in an unconfined test and af corresponds tothe rate of increase in the E values as the radial stressincreases. The fiber elastic shear strain increments are com-puted as a function of the paste elastic shear strain bymeans of a mobilization function, fm, defined by Eq. (2).The fm function was proposed considering the pattern ofshear strength mobilization reported by several authors inthe literature, which considers that at the beginning ofthe shearing process the fibers have a negligible influenceon the MSW mechanical behavior (see phase A, Fig. 3).As the shearing process continues, the influence of thefibers becomes increasingly important.

deesf ¼ fm � dee

sp ð1Þ

fm ¼2

p� tan�1 q

p

� �2 !

ð2Þ

Fig. 3. Fiber reinforceme

3. Modeling the MSW long term behavior

3.1. The influence of loss of mass on the long term MSW

volumetric strains

The influence of the biodegradation processes and theresulting mass loss on field settlement is assessed consider-ing the phase diagram presented in Fig. 2. In this figure, va,vw, vs correspond to the volumes of air, water, and solids,respectively, and v is the total volume. mw, ms, and m arethe corresponding mass of these phases. Fibers areassumed as having no voids (see Table 1) and the decompo-sition processes are assumed as having no influence on thefiber mass, msf. Solid volume change thus corresponds topaste volume variation (Dvs = Dvsp).

The a parameter, which is identical to that proposed byMcDougall and Pyrah [15], expresses the fact that theadditional volume variation associated with biodegrada-tion will not produce equivalent waste compression, butrather some waste deformation that depends on the rela-tive values of a and eo (Fig. 2). Furthermore, the voidsgenerated by the decomposition process induce modifica-tions in the waste structure which can lead to additionalcompression. As a first qualitative analysis it is worthcommenting that if a is smaller than eo, the MSW voidratio will increase (at least theoretically) leading to alooser waste, whereas a values larger than eo tend toincrease the waste dry density. In the particular case ofa equal to eo, biodegradation volumetric strain will arisebut the relative volume variations of paste and that ofthe waste maintain the same void ratio. Finally, it shouldbe emphasized that some test and field results suggest thatthe a parameter is not a constant value, but a function ofthe MSW biodegradation stage and probably othervariables, such as confining stress, waste composition,etc. For sake of simplicity, the a parameter may beassumed as proportional to the specific total amount ofsolids already decomposed (�Dms/mso). Eq. (3) expresses

nt effect. Kolsch [10].


this assumption mathematically. The a* parameter,presented in Eq. (3), refers to the rate of increment ofa(t) as a function of the specific amount of mass alreadydecomposed.

Mass variation is considered through a first order decaymodel [18] used to simulate the gas generation process anda mass conversion factor Cm (Eqs. (4) and (5)). The maxi-mum amount of decomposable matter can be obtainedmaking t ?1 in Eq. (4). Thus, the maximum a(t) corre-sponds to the product of a* by the maximum specificamount of mass that can be decomposed: a(t)max = a* �Lo � (1 + w)/Cm. In Eqs. (4) and (5) Lo = methane genera-tion potential (m3/Mg of refuse), k = methane generationrate constant (1/year), t = time since sample disposal(year), w is the MSW water content (dry basis) andCm = MSW organic matter methane yield (m3 CH4/dry-Mg). The subscript o refers to initial conditions.

aðtÞ ¼ �a� � Dms

mso

ð3Þ

�Dms

mso

¼ Lo � ð1þ wÞ � f1� e�k�tgCm

ð4Þ

�oms

ot� 1

mso

¼ Lo � k � ð1þ wÞ � e�k�t

Cm

ð5Þ

The following equations can be used to compute MSWvolumetric strains (ev) and void ratio (e) as a function ofthe MSW loss of mass over time. cs, and csp, are the specificweights of MSW and paste particles. A constant paste spe-cific weight, csp, is assumed.

dev ¼ �cso

csp

!� 1

1þ eo

� �� 1� a� � Dms

mso

� �� oms

ot� 1

mso

� dt

ð6Þ

de ¼ cs

csp

!� �a� � Dms

mso

� e� �

� oms

ot� 1

mso � 1þ Dms

mso

� � � dt

ð7Þ

Mechanical creep is described by the MSW coefficient ofsecondary compression, Ca=�De/Dlog(t). Consequently,the total variations of volumetric strain and void ratiocan be expressed by Eqs. (8) and (9), respectively, wherethe subscript d refers to long term variations due to creepand biodegradation processes.

devd ¼ca

ð1þ eoÞ � lnð10Þ � t �cso

csp

!� 1

1þ eo

� �"

� 1� a� � Dms

mso

� �� oms

ot� 1

mso

#� dt ð8Þ

ded ¼cs

csp

!� �a� � Dms

mso

� e� �"

� oms

ot� 1

mso � 1þ Dms

mso

� �� ca

lnð10Þ � t

#� dt ð9Þ

3.2. Variation of waste composition upon biodegradation

Considering the phase diagram of Fig. 2, the followingequations can be derived to calculate Vf and Pf over time.It is worth noting that Vf + Vp = 1 and thereforedVf = �dVp, the same occurring in the case of Pf and Pp.

V f ¼P f � cs

csf � ð1þ eÞ ð10Þ

dP f ¼ �P f �oms

ot� 1

mso � 1þ Dms

mso

� � � dt ð11Þ

The MSW specific weight, cs, may be computed by Eq.(12) as the biodegradation process continues.

cs ¼ V f � ðcsf � cspÞ � ð1þ eÞ þ csp ð12Þ

3.3. The influence of biodegradation on the general MSW

stress–strain behavior

As pointed out before, the influence of the fibrous mate-rial in the MSW stress/strain curves is considered based onthe assumption that part of the deviator stress, q, is carriedout by the fibrous components such as plastic, textiles andvegetable fibers present in the waste body. Thereforechanges in the mechanical properties of the fiber compo-nents will affect the mechanical behavior of the waste asa whole. This is illustrated in Fig. 4. If there is a decreasein the fiber shear modulus over time, part of the deviatorstress supported by the fibers tends to be transferred tothe MSW paste. In Fig. 4 an instantaneous stress path upto point 1 is assumed. As point 1 is reached, the MSWstress state remains the same over time and the changesin the paste stress state over time are only caused by thechanges in the mechanical properties of the fibers. The bio-degradation process can lead the paste from an elastic to anelastoplastic regime as time increases. If the process contin-ues, the paste can even reach a critical state condition. Inthe event that qmax decreases continuously and faster thanqf, a fiber failure can arise. It is worth mentioning that thefiber degradation process does not affect the paste elasticvolumetric strain (there are no variations in p values). Aswill be seen below, however, the degradation process ofthe fibers has an influence on the values of elastic shearstrains and on the values of plastic volumetric and shearstrains of the paste.

As a consequence of the degradation process of the fibers,Eq. (13) (or Eq. (14)) has to be used to calculate a long termincrement in the fiber deviator stress. Eq. (14) should be usedonly if the tensile stress reaches the value of qmax.

dqf ¼ 3 � Gf � deesf þ 3 � oGf

ot� dt � ee

sf ð13Þ

dqf ¼ dqmax ð14Þ

The following equations can be used to take intoaccount the effects of changes in the fibrous componentsand mass loss in the MSW behavior. Mass loss modifiesthe proportion of fibers and paste and the values of void

Fig. 4. Fibers property changes effect in the MSW behavior.


ratio of paste and MSW, thus changing the partition ofdeviator stress between these phases. Thus, the MSW devi-ator stress increment, dq, can be calculated using Eq. (15).This equation was modified slightly from that presented inTable 1 in order to take into account the major variationsin Vf and Vp caused by the process of biodegradation. Thesame occurs with the MSW shearing strain, which is nowcalculated using Eq. (16).

dq ¼ 3 � V p � Gp � deesp þ 3 � V f � fm � Gf � dee

sp

þ qf � dV f þ qp � dV p þ 3 � V f �oGf

ot� dt � ee

sf ð15Þ

des ¼ desp � V p þ desf � V f þ dV f � ðdesf � despÞ ð16ÞEq. (15) can be rearranged in order to compute a long termincrement in the elastic shear strain of the paste. This isshown in Eq. (17). If Eq. (14) prevails, Eq. (17) assumesa more simplified form (Eq. (18)).

deesp ¼

dq�½qf�qp ��dV f

3�V f� oGf

ot � dt � eesf

Gp�V p

V fþ Gf � fm

ð17Þ

deesp ¼

dq� dqmax � V f � ½qf � qp� � dV f

3 � Gp � V p

ð18Þ

The MSW volumetric strain increment, dev, can be ob-tained using Eq. (19). In this equation, Ve corresponds tothe ratio between e and ep (Eq. (20)) and devp is the pastetotal volumetric strain.

dev ¼ devd þ devp � V e �1þ ep

1þ V e � ep

� �ð19Þ

V e ¼eep

¼P p

csp

P p

c þP f

c

0@

1A ð20Þ

sp sf

The paste yield locus and hardening law may now beused to compute the remaining long term increments ofthe paste volumetric strains. For illustrative purposes,Eq. (21) presents the calculus of the long term incrementof plastic volumetric strain of the paste considering thelong term changes in Gf, q and Vf.

depvp ¼

3 � Gp � ðk� jÞðnþ 1Þ � p�po

p

� � nnþ1

po �M � nn

nþ1

��oGf

ot � dt � eesf þ

dq�½qf�qp��dV f

3�V f

Gp �V p

Vf þ Gf � fm

" #ð21Þ

The long term increments in the plastic shear strainof the paste, dep

sp, derived from loading or due to changesin the mechanical properties of the fibers, are calculatedusing the paste flow rule.

Despite the volumetric strains due to creep and loss ofmass both being plastic, the treatment given in terms ofthe associated plastic shear strains is different. It is assumedthat the MSW plastic shear strains associated with creepvolumetric strains can be calculated using the paste flowlaw. In the case of the MSW volumetric strains due to lossof mass, no associated plastic shear strain is considered. Itis believed that, in this case, the plastic strain are isotropic,as the loss of mass tends to act in the same way in all direc-tions, regardless of the stress state. No aging effect is con-sidered in either cases.

3.4. The influence of degradation on fiber properties

The effect of the biodegradation process on the mechan-ical characteristics of the fibers can be evaluated through


changes in the values of qmax and E over time. The follow-ing equations attempt to do this. These equations considerthat after initial conditions that represent the maximumvalues of the elasticity modulus and deviator stress thatcan be reached by the fibers, there is a degradation of bothvalues represented by a first order decay model, similar tothat used in organic matter depletion. In these equations,kE is the constant related to the rate of changes in the fibersyoung modulus (1/year) and kq is the constant related tothe rate of changes in the ultimate tensile stress of the fibers(1/year).

EðtÞ ¼ Eðt¼0Þ � e�kE�t ð22ÞqmaxðtÞ ¼ qmaxðt¼0Þ � e�kq�t ð23Þ

4. Qualitative model performance in simulating the MSW

behavior under different stress-time paths

In order to test the ability of the model to predict theMSW behavior under different loading conditions, somegeneric simulations of the MSW behavior over time wereperformed. The following parameters were adopted:Pfo = 0.218, Vfo = 0.13, csp = 22.1 kN/m3, csf = 10 kN/m3,Eu = 82.4 MPa, af = 10,000, qmax (t = 0) = 7.7 MPa, mf =0.25, k = 0.16, j = 0.007, N = 8.26, eo = 2.82, / = 22�,mp = 0.36. These values were obtained using the MSW sam-ples tested by Carvalho [3]. The parameters relating to lossof mass and its effects and the changes in the fiber proper-

Fig. 5. Stress-time paths followed

ties over time were: Lo = 85 m3 CH4/Mg, w = 0.8,Cm = 420 m3 CH4/dry-Mg, k = 0.21 year�1, Ca = 0.04,a* = 20, kq = 0.024 year�1 and kE = 0.067 year�1. Cm is atypical value considered in methane generation studiesand can be estimated from waste composition. Lo, k andCa are values measured in MSW from Salvador, Brazil[2]. The values of kq and kE were estimated using resultsof simple tension tests performed by Machado et al. [14]on samples of plastic fibers of different ages. The value ofthe a* parameter was chosen modeling the experimentalresults published by Mehta et al. [16], Carvalho [3] andOlivier and Gourc [17].

Two sets of simulations were performed. In the firstset, only the changes in the fiber parameters were evalu-ated, the effect of mass loss and creep were ignored. Inthe second set of simulations, the coupled effects of massloss, creep and the changes in the mechanical propertiesof the fibers were assessed. Fig. 5 presents the stress-timepaths followed by each set of simulations performed.Sample 1 was considered as instantaneously loaded att = 0 (fresh MSW). The initial state of stress was thenmaintained for 6 years, allowing for the occurrence ofdegradation. Sample 2 was loaded under a constant rateof stress over 6 years to reflect the influence of stressand degradation variation. Finally, sample 3 was allowedto degrade for 6 years and was subsequently instanta-neously loaded to the target stress. All the samplesstarted with the same isotropic stress of 100 kPa, andreached a value of q = 160 kPa after 6 years of degrada-

in the performed simulations.


tion, but following different time and stress paths. InFig. 5 there are discontinuities in the stress-time paths 1and 3 which were named d1 and d3, respectively. Forthe sake of simplicity, Figs. 7 and 8 show the effect of

Fig. 6. Vf and Pf var

Fig. 7. Pred. MSW behavior. Mas

these discontinuities on the long term mechanical behav-ior of the waste.

Fig. 6 shows the variation of Pf and Vf over time andconsiders the three conditions analyzed. As it was assumed

iations over time.

s loss and creep not included.


that the specific unit weight of fibers and paste particlesremain constant, when mass loss and creep are not consid-ered the values of Pf remain constant over time. However,as degradation takes place, Pf and Vf increase according topatterns governed by the way the degradation processtakes place. The increase in the Pf values is independentof the stress path followed and depends only on theamount of mass loss that has occurred in the consideredtime span. Vf also increases with time, but now the rateof increase depends on the stress path followed.

Fig. 7 shows the results obtained when the MSW loss ofmass and creep are not taken into account. Fig. 7a presentsthe paste stress-time paths followed in qp � p plane. Thestress-time path 1 leads to lower values of paste deviatorstress, qp during the loading process. This can be explainedby the fact that initially the fibers present their highestvalue of Young modulus which implies a faster rate oftransference of MSW deviator stress to the fibers. Thisbehavior can be confirmed by Eq. (17) which shows thatthe higher the value in Gf, the lower the paste elastic shear

Fig. 8. Pred. MSW behavior. M

strain increment. Once the MSW is left to decompose, qp

increases at constant p values, reaching a final value thatis close to the values reached in the other trajectories.The increase in the qp values is related to the decrease inthe qf values, as illustrated in Fig. 7d.

Another interesting feature is that stress-time path 1leads to lower values of axial strain when compared tothe other stress-time paths. Fig. 7b and c illustrate thisstrain response that seems to be controlled by the pasteflow law. This flow law implies that the lower the qp/p val-ues, the lower the values of the ratio dep

sp=depvp. The same

reasoning can be used to explain the behavior presentedby the stress-time path 2. In this case there is a gradualchange in the properties of the fibers and the obtainedresults are a rough average among the three trajectories.

Fig. 8 shows the results obtained when the MSW loss ofmass and creep are considered. In Fig. 8a the qp reached inthe other trajectories is very close to the values obtainedduring the loading process followed by the stress-time path1. This occurs because of the effect of the loss of stiffness of

ass loss and creep included.

Table 4Paste volatile solid and organic matter contents obtained

Component content (%, dry basis) Fresh waste 4 years old waste

FG&O volatile solid 36.1 50.2Paste volatile solid 58.0 19.8MSW FG&O organic matter 20.8 9.9


the fibers is counterbalanced by the increase in the Vf andPf values over time (see Figs. 6 and 8d).

Fig. 8b and c demonstrates that when loss of mass isconsidered, the axial strain shows a behavior that is oppo-site to that presented in Fig. 7. The stress-time path 1reaches the largest axial strain. This is also in response tothe paste flow law: in this case the creep volumetric strainsoccur at higher values of qp/p and after the loading processhave taken place. In the stress-time path 3, the creep volu-metric strains occur before loading, with qp = 0 and theassociated plastic shear strains are zero. The stress-timepath 2 continues to be an intermediate case.

5. Validation of the proposed model

To validate the proposed model it will be tested againstdata from triaxial laboratory tests performed using recon-stituted samples of fresh and 4 years old MSW. Thereforesamples tested may be non representative of the field wastestructure and do not reflect all the changes brought aboutby the decomposition process after deposition. It isbelieved, however, that the material used is representativeof each waste component separately considering the caseof fresh samples and 4 years old samples and the changesobserved in the triaxial tests results may be helpful tounderstand long term MSW behavior. The waste studiedcomes from metropolitan center sanitary landfill locatedin the city of Salvador, Bahia, Brazil. Table 3 shows thephysical composition of the collected waste samples (drybasis).

A reduction in almost all components is observed overtime, apart from the FG&O fraction which tends toincrease with time. The increase in FG&O is dictated bythe large amount of other components that are incorpo-rated into it after deposition, making component segrega-tion difficult. Thus, parts of the cover soil, paper, glassand textiles are incorporated into the FG&O during landfilloperation and after deposition and was computed as thatin the 4 years old samples. Indeed, when the FG&O’sorganic content is evaluated a decrease over time can beobserved, as reported in Table 4. An average volatile con-

Table 3MSW physical composition

Component Percentage – dry basis (%)

New MSW – average 4 years old MSW

Wood 4.0 5.6Stone/ceramic 10.5 13.4Textile 2.6 2.5Rubber 0.48 0.20Plastic 23.3 13.8Glass 3.5 4.1Metal 2.40 5.0Paper/cardboard 17.1 5.2FG&Oa 36.1 50.2

a FG&O – food and garden waste and other non-easily separablematerials.

tent of only 19.8% was obtained for 4 years old samples,much less than the 58.0% obtained for the fresh ones.FG&O organic matter related to the whole waste alsodecreases from about 20.8% to about 9.9% after 4 years.

The specific unit weight of the MSW solid particles, cs,was determined using applicable standards [1]. The averagevalue of cs obtained was cs = 17.5 kN/m3 considering thenew samples of waste tested.

Saturated and drained triaxial compression tests (CD)were carried out on statically compacted MSW sampleswith nominal size of 20 cm � 40 cm. MSW samples weremolded with an unit weight of about 10 kN/m3, whichmeans a dry unit weight of about 5 kN/m3, consideringan average water content of 100% (dry basis). The strainrate was 0.8 mm/min and effective confining stresses of50, 100 and 200 kPa were used.

Two sets of conventional triaxial compression tests fol-lowing stress path dq:dp of 1:3 were performed using freshand 4 years old samples. The paste component of the freshMSW samples was also tested. These tests were to verifythe influence of the fibrous material on the mechanicalbehavior of the MSW. Fig. 9 shows a comparison betweenthe results of fresh waste and paste, that is, the waste afterthe removal of the fibrous material. As expected, the wholefresh waste shows larger shear strength, thus suggestingthat the influence of eventual changes in the mechanicalproperties of the fibrous material can not be ignored. Thereis also a clear decrease in the volumetric strains when thefibrous material is removed.

The 4 years old samples were also tested following thestress paths of 1:1 (increasing radial stress), and 1:50(decreasing radial stress). Fig. 10 shows a comparisonbetween the test results of fresh and 4 years old waste. Bothtests results show a similar behavior and the volumetricstrains measured in the test with old sample and confiningstress of 50 kPa were relatively low and probably do notrepresent a trend. Fig. 11 shows a comparison betweenthe results obtained in the stress paths of 1:1, 1:3 and1:50. As shown in this figure, the stress path 1:1 has a morepronounced concavity, whereas the stress path 1:50 pre-sents a soil-like behavior.

Two confined compression tests were carried out in alarge odometer with 548.3 mm in diameter and 496.8 mmin height. Only the fresh waste was tested. It was placedin the odometer in layers (four layers in all) and was lightlypressed to reach the initial unit weights of 9.4 and 7.1 kN/m3. Each loading stage lasted about 30 days, althoughsome stages were extended for more days in order to definemore precisely the long term behavior of the waste. Fig. 12

Fig. 10. Fresh and 4 years old MSW triaxial tests results.Fig. 9. Fresh paste and MSW results.


shows the results of confined tests. In these curves, it can beobserved that after about 10 min the strains are becomeslinear, suggesting a predominance of the creep compressionprocess. These curves were used to determine the MSW sec-ondary compression index, Ca = De/Dlog(t) and the sec-ondary compression coefficient C0a ¼ Ca=ð1þ eoÞ for eachstage of loading. An average Ca value of 0.058 wasobtained, whereas the average value of C0a was 0.021. Thestandard deviation was about 44% of the average valuefor both indexes.

In order to obtain the paste compression indexes, theconfined compression curves obtained considering thewaste as a whole were converted into paste compressioncurves. This was done using the parameter Ve (Eq. (20)).MSW void ratio values corresponding to an elapsed timeof 10 min were used. Paste void ratio values, ep, wereobtained considering an initial value of Ve = 0.62 (corre-sponding to values of Pf = 21.8%, cs = 17.5 kN/m3 andcsf = 10.0 kN/m3). The values of Pf and cs were correctedin order to take into account the loss of mass accordingto Eqs. (11) and (12). The values of Lo and k used were

Lo = 85 m3 CH4/Mg of refuse and k = 0.21 year�1. The fit-ting of the experimental data presented in Fig. 13 producedvalues of k = 0.23, N = 12.8 and j = 0.013. As can be seen,only a small part of the volumetric strains are recoveredunder unloading.

Fig. 14 shows the experimental and the modeled resultsof the triaxial tests performed with the paste of fresh waste.Values of / = 24� and mp = 0.36 were adopted. Modeledresults fit the experimental data fairly well, specially theq � ea curves. However, in the case of the ev � ea curves,the model overestimates the values of volumetric strain.

The derived parameters from MSW confined compres-sion and paste triaxial tests were used to model test results.Fig. 15 presents the MSW confined compression curvesand the results predicted by the model. A value ofKo = 1 � sin(/) was used for the at rest earth pressure coef-ficient. As this figure demonstrates, there is a good agree-ment between experimental and predicted results.

Triaxial test results of fresh waste were used in order toassess the values of Eu and af. Initial values of mf = 0.25,Pf = 21.8%, cs = 17.5 kN/m3 and csf = 10.0 kN/m3 were

Fig. 12. MSW long term confined compression curves.

Fig. 13. Paste confined compression curves.

Fig. 11. Stress path performed on 4 years old samples.


considered which leads to an initial value of Vf = 0.12, con-sidering the test performed with a confining stress of100 kPa. For each conventional triaxial test performed,the strategy adopted consisted of making af = 0 and settingthe best value of E (the value of E that leads to the bettermodel performance). After that, a linear adjustment wasmade between the E values and the values of confiningpressure used, allowing the acquiring of Eu and af. Valuesof Eu = 160 MPa and af = 3500 were found following thisprocedure. Fig. 16 presents the experimental results andthe results predicted by the model. As can be seen, thereis a good match between predicted and experimental val-ues. Although the values of volumetric strains remain over-estimated, there is now an improvement in the modelperformance compared to the tests performed using onlyMSW paste.

As far as the 4 years old MSW is concerned, the vari-ables related to the biodecomposition process (Cm, Lo, k,a*, kE and kq) have to be defined. The value of Cm maybe estimated from waste composition but a value ofCm = 420 m3 CH4/dry-Mg may be used for practical pur-

poses. Lo and k can be obtained either from field gas yieldmeasurements or laboratory characterization tests per-formed on samples of varying ages. In the case of the wastesamples tested, values of Lo = 85 m3 CH4/Mg of refuse andk = 0.21 year�1 can be considered acceptable [2].

Values of kE = 0.067 year�1 and kq = 0.024 year�1 canbe derived from data published by Machado et al. [14].These values must be considered as a first approach andadditional simple tension tests must be performed usingfibrous material of different ages to better understand the

Fig. 15. Exp. and predicted results. Confined comp. tests.

Fig. 16. Exp. and predicted results. Triaxial tests. New MSW.

Fig. 14. Exp. and predicted results. Paste of new waste.


effect of the biodecomposition process in the mechanicalproperties of the fibers and validate the use of Eqs. (22)and (23).

Triaxial conventional tests results obtained using 4 yearsold samples were used in order to estimate the a* value. Avalue of a* = 20 was used. This value is close of those pre-sented in the qualitative model performance assessment. Itmust be said, however, that a* values must be calculatedusing preferentially results of confined compression testswith mass loss measurements or coupled field settlementsand gas production data. Complementary triaxial testsmust be performed on samples of different ages in orderto validate model abilities. Fig. 17 shows the experimentalresults and the results predicted by the model in the case ofthe conventional triaxial tests performed on 4 years oldsamples. As can be observed, the model was able to fairlyreproduce the behavior of the MSW. There is an evenbetter reproduction of the ev � ea curves. In the case ofthe samples of 4 years old waste, the value of Pf calculatedby Eq. (11) increased to Pf = 27.4% and the initial values

Fig. 17. Triaxial exp. and predicted results. 4 years old MSW.Fig. 18. Stress paths exp. and pred. results. 4 years old MSW.


calculated to Ve (Eq. (20)) and Vf (Eq. (10)) were 0.54 and0.16, respectively, considering the test performed with aconfining stress of 100 kPa. The value of cs was about16.5 kN/m3. On the other hand, there is a decrease of about24% in the Young modulus of the fibrous material whichtends to counterbalance the effect generated by the increasein the Pf and Vf values. The decrease in the fibers tensilestrength had no influence on the modeling, since fibers ten-sile stress during tests modeling are always relatively smallcompared to qmax. For comparative purposes, all thestrains that occurred prior shearing, due to creep and lossof mass, were ignored in Fig. 17 and assumed as an initialstate of strain. Fig. 18 shows the obtained and predictedresults in the case of the stress paths 1:1 and 1:50. As canbe seen, experimental results of the stress path 1:50 are bet-ter reproduced by the model.

6. Conclusion

A framework to reproduce the mechanical behavior ofMSW has been proposed. The framework improves the

original proposed by Machado et al. [13] as it includesthe effect of degradation of waste on the mechanical prop-erties of waste components and on the volumetric deforma-tion that occurs as a consequence of organic matterdepletion and creep. The model considers that as the bio-degradation process progresses, the proportion of fibersincreases, though their ultimate tension strength and mod-ulus of deformation is reduced. The potentialities of themodel are first shown by simulating different stress-timepaths where certain aspects related to the degradationcan be appreciated.

A comparison between the experimental resultsobtained using samples of fresh waste with and withoutfibers showed that the MSW fibrous components play avery important role in its mechanical behavior. There is anotable decrease in the MSW shear strength as the fibersare removed. Therefore more attention should be paid tothe long term behavior of the fibrous components of thewaste. When the results obtained using fresh and 4 yearsold waste are compared, similar results are obtained.


Despite the influence of aspects such as the changesimposed in the field waste structure by the decompositionprocess, it is possible that the decrease in the fibers Youngmodulus is counterbalanced to some degree by the increasein the proportion of mass and volume of fibers (Pf and Vf)leading to similar stress–strain curves. This is captured bythe model that was able to reproduce the stress–straincurves of MSW tested under two different loading condi-tions and different stages of organic matter depletion.

The model requires a relatively large set of parametersconsidering the many processes it encompasses. However,parameter estimation rests on relatively simple tests, suchas waste characterization tests and common soil mechanicslaboratory tests, namely confined compression and triaxialcompression tests. Perhaps the most difficult parameter toobtain is a* as it demands tests using MSW of differentcompositions and ages or the execution of confined com-pression tests with mass loss and settlement measurements.At the moment, a value of a* = 20 seems to be a good firstapproach considering the few tests available in the litera-ture and the tests performed by the authors. As more testresults become available, a database of a* parameter couldbe assessed to help this model parameter estimate.

References

[1] ABNT NBR 6508. Determination of the specific density of solidparticles; 1984 [in Portuguese].

[2] Britto MLCPS. Biogas production rate and MSW biodegradationparameters in the Metropolitan Center Landfill. Msc Thesis. Salva-dor, Brazil: Federal University of Bahia; 2006 [in Portuguese].

[3] Carvalho MF. Mechanical behavior of municipal solid waste. PhDThesis. Sao Carlos, SP, Brazil: University of Sao Paulo; 1999 [inPortuguese].

[4] Dixon N, Zhang B, El-Hamalawi A. Constitutive model ofMSW. In: Proceedings Sardinia 2005, Tenth International WasteManagement and Landfill Symposium. Cagliari, Italy; S. Mar-gherita di Pula; 2005.

[5] El-Fadel M, Shazak S, Saliby E, Leckie J. Comparative assessment ofsettlement models for municipal solid waste applications. WasteManage Res 1999;17(5):347–68.

[6] Heiss-Ziegler C, Fehrer K. Geotechnical behavior of mechanically–biologically pre-treated municipal solid waste (MSW). In: Proceed-ings Sardinia 2003, ninth international waste management and landfillsymposium. Cagliari, Italy: S. Margherita di Pula, 2003.

[7] Hossain MS, Gabr MA, Barlaz MA. Relationship of compressibilityparameters to municipal solid waste decomposition. J GeotechnGeoenviron Eng ASCE 2003;129(12):1151–8.

[8] Jessberger HL, Syllwasschy O, Kockel R. Investigation of wastebody- behavior and waste-structure-interaction. In: Proceedingssardinia 95, fifth international landfill symposium. Cagliari, Italy: S.Margherita di Pula; October 1995. p. 731–43.

[9] Kockel R, Jessberger HL. Stability evaluation of municipal solidwaste slopes. In: Proceedings of the 11th ECSMFE, vol. 2. Copen-hagen, Denmark: Danish Geotechnical Society; 1995.

[10] Kolsch F. Material values for some mechanical properties of domesticwaste. In: Proceedings sardinia 95, fifth international landfill sympo-sium. Cagliari, Italy: S. Margherita di Pula; October 1995. p. 711–29.

[11] Konig D, Jessberger HL. Waste mechanics. In: ISSMFE technicalcommittee TC5 on environmental geotechnics; 1997. p. 35–76.

[12] Landva AO, Clark JI. Geotechnics of waste fill. In: Landva A, DavidKnowles G, editors. Geotechnics of waste fills – theory and practiceASTM STP 1070. Philadelphia: American Society for Testing andMaterials; 1990. p. 86–103.

[13] Machado SL, Carvalho MF, Vilar OM. Constitutive model formunicipal solid waste. J Geotechn Geoenviron Eng ASCE2002;128(11):940–51.

[14] Machado SL, Carvalho MF, Nascimento, Dourado KA. Aging effecton MSW mechanical behaviour. In: Proceedings of the VI ICEG sixthinternational congress on environmental geotechnics, vol. 2. Cardiff,UK: Thomas Telford; 2006. p. 1439–46.

[15] McDougall JR, Pyrah IC. Phase relations for decomposable soils.Geotechnique 2004;54(7):487–93.

[16] Mehta R, Barlaz MA, Yazdani R, Augenstein D, Bryars M,Sinderson L. Refuse decomposition in the presence and absence ofleachate recirculation. J Environ Eng 2002;128(3):228–36.

[17] Olivier F, Gourc JP. Hydro-mechanical behavior of municipal solidwaste subject to leachate recirculation in a large-scale compressionreactor cell. Waste Manage 2007;27:44–58.

[18] USEPA. Landfill air emissions estimation model (version 2.01), EPA68-D1- 0117, EPA 68-D3-0033; 1998.

Constitutive model for long term municipal solid waste mechanical behavior

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