An elasto-plastic constitutive model foragricultural cohesive soil

L. CHI2, R.L. KUSHWAHA1 and J. SHEN1

Agricultural and Bioresource Engineering Department, University ofSaskatchewan, Saskatoon, SK, Canada S7N 0W0; andnow at Departement de genie rural, Universite Laval, Ste-Foy, PQ, Canada GIK 7P4. ASAE Paper No. 90-1084 Received28 May1992; accepted 30 August1993.

Chi,L., Kushwaha, R.L.and Shen,J. 1993. An elasto-plastic constitutive model for agricultural cohesive soil. Can. Agric. Eng.35:245-251. An elasto-plastic model has been developed for cohesive soils which reflects both the elastic and plastic stress-strainbehaviour of soils. Two yield surfaces were used to calculate theplastic strain underhydrostaticcompressionand plastic shear strainassociated withthefailuresurface. Themodelshowed a goodprediction accuracy underboth uniaxialand hydrostatic compressions.Keywords: soil stress-strain, elasto-plastic model, constitutivemodel, finite element method

Un modele elasto-plastique a ete developpe pour les sols co-hesives. Ce modele montre la conduite des tensions elastiques etplastiques des sols. Deux surfaces debits ont ete utilisepour faire lacalculation de la tension plastique sous une compression hydrosta-tique et celle associee avec la surface faillite. Le modele a demontrerunebonnepredictiond'exactitude sous les compressionsuniaxialeethydrostatique.

INTRODUCTION

Finite element analysis has been widely used in simulatingsoil cutting (Yong and Hanna 1977; Chi and Kushwaha1989), soil compaction (Pollock et al. 1986), and traction(Yong et al. 1978). A stress-strain relationship, generallyreferred to as a constitutive relationship, is an essential component of the finite element stress analysis technique. Theprediction accuracy of a finite element model depends directly on the accuracy of the constitutive equation of the soil.

The stress-strain behaviour of soils is complex and difficult to describe with a simple relationship. Since soil is anon-linear material, its failure with the action of tillage toolsis elastic as well as plastic. Most of the present constitutivemodelsfor soils can be divided into two categories,nonlinearelastic and elasto-plastic models. A nonlinear hyperbolicstress-strain model developed by Kondner and Zelasko(1963) and later improved by Duncanand Chang (1970)hasbeen used widely in finite element analysis (Chi and Kushwaha 1989; Pollock et al. 1986; Yong and Hanna 1977;Chang and Duncan 1970; Clough and Duncan 1969; Kul-hawy et al. 1969, 1970). Some useful and satisfactorypredictions havebeenmadeby usingthis model.Baileyet al.(1984) developed a nonlinear exponential soil compactionmodel which showed an adequate accuracy for hydrostaticcompression of the soil. In these nonlinear elastic models, thedeformation was assumed to be totally recoverable. However, soil undergoes a considerable amount of plasticdeformation during loading. Therefore, an elasto-plasticmodel is needed to describesoil behaviourmore precisely.

Drucker (1950) first proposed an elasto-plastic model forsoil in which a linear elastic model was assumed before soilfailure. After the stress reached the failure state, it was assumed that plastic deformation occurred. In fact, the plasticdeformation also occurred before the soil failure. As a result,Drucker's model predicted a low value of displacement (De-sai and Siriwardane 1984). The critical state model, alsoreferred to as the Cam Clay model (Schofield and Wroth1968), and the DiMaggio-Sandler cap model (DiMaggio andSandler 1971) were later developed to account for shear andvolumetric plastic deformation before and after soil failure.Both of these models used a cap yield function to calculatethe plastic deformation before soil failure. In these models,two or three constitutive relationships were proposed. Therefore, a complicated computational procedure is requiredduring finite element analysis to identify when the stresslevel has passed a certain yield surface and to assign a different constitutive relationship for each state (Siriwardane1984).

Lade (1977) proposed another model developed for cohe-sionless soil based on the results of carefully conductedexperiments under various stress paths in cubical triaxialequipment. In the model, two work hardening theories wereused, one for a cap type yield surface and the other for aconical yield surface. The major difference between Lade'smodel, the critical state model, the cap model and othertraditional plastic models is that in Lade's model, both elasticand plastic strains are assumed to occur from the beginningof loading (Lade and Duncan 1975). Therefore, a singleelasto-plastic constitutive model is applied throughout theloading process. The computational algorithms of Lade'smodel are simpler than those of the critical state and capmodels. Lade's model was also the only model which couldsimulate the work-softening behaviour of soils (Lade 1977).However, Lade's model is valid only for cohesionless soilsuch as sand.

All of the above elasto-plastic models were originallydeveloped for civil engineering applications. The objectiveof this study was to develop an elasto-plastic model forcohesive agricultural soils.

MODEL DEVELOPMENT

A stress-strain model was developed based upon the following objectives:

CANADIAN AGRICULTURAL ENGINEERING Vol. 35, No. 4, October/November/December 1993 245

1. A model to account for elasto-plastic behaviour of soil,and

2. A model which is convenient to apply in a finite elementanalysis.

Since considerable plastic strain occurs during triaxial aswell as hydrostatic loading, it is essential to formulate anelasto-plastic model to predict stress-strain relationship.Based upon the two objectives established above, the modelwas developed using the principles of Lade's model (Ladeand Duncan 1975). Since Lade's model was developed foronly cohesionless soil, the yield surfaces were modified toaccount for the effect of soil cohesion.

Elastic and plastic strains

In the development of the model, it was assumed that bothelastic and plastic strain occurred throughout the loadingprocess from the beginning. The total plastic strain was divided into two parts: (1) the plastic strain associated with acap-hardening yield surface, and (2) the plastic strain associated with a conical yield surface. The total strain incrementcan be written as:

{de} ={dze} +{dec} +{de/} (1)

where:

dz = total strain increment vector,

dee = total elastic strain increment vector,dec = plastic strain incrementvector associated with

cap-type yield surface, andd^ - plastic strain increment vector associated with

conical yield surface.

The three strain components in Eq. 1 were calculatedseparately, the elastic strain dze by Hooke's law; dzc by aplastic stress-straintheory involvinga cap-typeyield surface;and the def by a plastic stress-strain theory involving thecohesional-frictional failure surface.

Yield functions

Two most commonly used failure criteria for cohesive soilsare: (1) the Mohr-Coulomb model, which states that soil failsunder a shear stress dependent on cohesion and internalfriction, and (2) the Drucker-Prager model (Drucker andPrager 1952), which states that soil fails under maximumdistortion energy. Since the Mohr-Coulomb model is not asmooth continuous function in three-dimensional cases(Nayak and Zienkiewicz 1972), it is difficult to employ forconstructing a plastic stress-strain relationship. Therefore,the Drucker-Prager model was used as the failure criterion,as given by:

l\ :0 (2)

where:

Ij = an, first stress invariant (kPa),J2 = 0.5 SijSij, second deviatoric stress invariant (kPa ),k = a parameter determined from test data (kPa), anda = a dimensionless parameter determined from test data.

The Drucker-Prager criterion provides a smooth conicalfailure surface in principle stress space as shown in Fig. 1,

246

Failure surface Conical yield surfaces

Hydrostatic axis

(Ji —dz =(73

Fig. 1. The Drucker-Prager soil failure model in principalstress space.

and is described by:

//=h

(all - k)2ff=l, atfailure (3)

whereof = conical yield function.Plastic strain was also found during hydrostatic loading in

which the conical yield function Eq. 3 was kept constant onthe hydrostatic axis of Fig. 1. A cap-type yield criterion wasproposed to account for the plastic strain d£c (Eq. 1) occurring during hydrostatic compression. The cap yield surfacewas assumed as a sphere with the centre at the tip of theconical failure surface (O" in Fig. 2) and thus the cap yieldsurface is orthogonal to the conical yielding surfaces. Therefore, an increase in stress state can be considered as acombination of the stress changes due to an increase in theconical yield function and in the cap yield function. Thecap-type yield function is given by:

/c=/12 +2J2 +^/12*

3a(4)

where:

fc = cap yield function (kPa2), and12 =0.5 (Iil-OijGjj), the second stress invariant (kPa2).

Elastic constitutive matrix

The Elastic strain in Eq. 1 was calculated by using Hooke'slaw. For three dimensional stresses, the elastic constitutivematrix is given by:

^0"

Fig. 2. Cap yield surface.

Conical yield surface

Hydrostatic axis

cr, = crP = cr3

Cap yield surface

cr2 , cr3

KUSHWAHA, CHI, and SHEN

1 - v v

1 - v v 0 0

[Ce] = -(l+v)(l-2v)

v 1 -v 0 0

l-2v0 0 0

0 0 0

0 0 0

2- 0 0

0 i^ „

0 o 1-2v

(5)

where:

EUr = elastic modulus (kPa),v = Poisson's ratio, and[Ce] = elasticity matrix.

Janbu (1963) proposed a model to describe the modulus asa function of confining pressure by using:

^ur ~ KuGrt

(6)

where:

Eur

C3

Pa

= unloading and reloading elastic modulus (kPa),= confining pressure (kPa),= atmospheric pressure (kPa), and

KUr, n= dimensionless parameters determined fromtest data.

Poisson's ratio was determined using (Zhang et al. 1986):

1R -Fu "ur '-'urV =

6B„ (7)

where Bur = unloading and reloading bulk modulus (kPa).

Plastic strain associated with the cap-type yield surface

By using the associated flow rule, the plastic strain incrementassociated with the cap yield surface is given by:

\dfcw-yu (8)

where Xc = a positive scalar factor.The work hardening law was applied to determine the

scaler ^c in Eq. 8. Lade (1977) developed a power functionto derive a working equation for calculatingthe total plasticwork required to produce the collapse strains. However, itwas found that the Lade's equation did not fit the data obtained from a hydrostatic loading test of cohesive soil,especially when the volumetric strain was greater than 10percent. An exponential equation proposed by Bailey et al.(1984) for soil compaction showed good accuracy over awide range of volumetric strain. Therefore, an exponentialfunction was proposed to describe the relationship betweenthe total plastic work required to produce the plastic straindec andthedegree of hardening expressed by/c, as given by:

Wc =C^[l-exp(-pfc/pl)] (9)

where:1CIC.

Wc = plastic work associated with cap yield surface doneper unit volume (kPa), and

C,p = dimensionless parameters determined from testdata.

Plastic strain associated with the conical yield surface

By using the associated flow rule, the plastic strain associatedwith the conical yield surface is given by:

w-vfil (10)

where Xf= a positive scalar factor.The work hardening law was applied to determine the

scaler fyin Eq. 10. Lade's model included a work hardeninglaw with the conical yield surface which could be used tosimulate both work hardening and work softening behaviourof materials. Lade's work hardening and softening model canbe expressed by:

//= Ff(Wf) =-- a e~bwf-w£Pa

\ eP" 1<7

a =

"fpeak

b =a Wfpeak

Wfpealc-®Pa

<*3

(Ha)

(lib)

(He)

(lid)

(He)<7 =Y+P

where:

Wf

Wfpeaka, b, q

= plastic work associated with conical yieldsurface done per unit volume (kPa),

= plastic work atff= 1,= intermediate variables, and

y, |3, 6, A, = dimensionless work hardening parameters.

Complete elasto-plastic constitutive matrix

The complete elasto-plastic constitutive relationship includesan elastic componentand two work hardeningplastic components. Lade and Nelson (1984) developed a procedure toconstruct an incremental elasto-plastic model with multi-intersecting yield surfaces. By using this Lade-Nelson method,the complete elasto-plastic model was derived as:

{da} = [Cep][de]epi

[Cepl = [Ce]-[Ce] dF

9a[LY

[L] =

-JdF

da[Ce]

dF

da+ [D]

dF

8a[Ce]

(12a)

(12b)

CANADIAN AGRICULTURALENGINEERING Vol. 35,No.4, October/November/December 1993 247

where:

dF

da

[D] =

f

(12c)

(12d)

In Eq. 12, [CJ,/c(o), Wc(fc\ff(c) and f/Wf) are determined by Eqs. 5, 4, 9, 3a, and 11a, respectively. Therefore,theelasto-plastic constitutive matrix [Cep] canbedeterminedif a stress state is known.

TEST PROCEDURE

To investigate the mechanical behaviour of agricultural soils,axial compression tests were conducted in a conventionaltriaxial test apparatus. The confining pressure was applied bypressurized water. The volume change in soil samples wasmeasured by a volumeter developed by Chi and Kushwaha(1989). Axial load, axial displacement, confining pressure, andchange in volume were measured during tests. The data weresampled by a 21X Campbell Scientific Micro-Logger (Campbell Scientific, Logan, UT) and then transferred to a computerfor analysis.

The test soil was a clay loam. Remolded soil samples 50mm in diameter and 100 mm in length were used in the tests.A split compaction mould was used to prepare soil specimensand all specimens were compacted in 8 layers at a density of1434kg/m3 and a moisture contentof 14.8%.

An unloading and reloading procedure was applied duringthe triaxial tests (Fig. 3). The stress-strain curve indicates thatsoil undergoes a considerable amount of plastic strain duringthe loading. The elastic modulus of unloading and reloading, AXIAL STRAIN e (%)Eur was used in the elastic constitutive relationship. Three Fig. 4. Stress-strain curves under different loadingseries of tests were conducted in calibrating the model. The speeds with a confining pressure 50 kPa.first series of triaxial tests was carried out with five different

248

0 ' i i i i i | i i i i i i ifi i i i i • i i • ' i ' ' -' ' ' I ' ' ' ' ' I ' ' ' ' ' I ' ' ' '0 5 10 15 20 25 30 35 40

AXIAL STRAIN £ (%)

Fig. 3. Soil stress-strain curve with unloading andreloading.

axial strain rates of 0.00014, 0.007, 0.036, 0.18 and 0.7/minunder a confining pressure of 50 kPa. The results from thisseries of tests (Fig. 4) show that the stress-strain relationshipis sensitive to loading speeds greater than 0.036/min. Thesecond series of triaxial tests was conducted under four con

fining pressures of 25, 50, 75 and 100 kPa with a strain rateof 0.007/min (Fig. 5). In both the first and second series oftests, the unloading and reloading procedure was applied.However, for clarity, the unloading and reloading curves arenot plotted in Fig. 4 and 5. The final series of tests wasconducted under hydrostatic loading (0\ = G2 = G3). Theresults (Fig. 6) indicate that plastic strain occurs under hydrostatic loading. Bur in Fig. 6 indicates a bulk elastic modulusof unloading and reloading. All tests were repeated threetimes.

RESULTS

Soil parameters

The complete elasto-plastic model includes 11 soil parameters: three elastic parameters (KUr, n and v), two soil failure

500

400

CONFINING PRESSURE: 100 kPa

CONFINING PRESSURE: 75 kPa

10 15 20 25

AXIAL STRAIN e (%)

Fig. 5. Stress-strain curves under different confiningpressures.

KUSHWAHA, CHI, and SHEN

The Drucker-Prager failure criterion was applied in themodel to account for the effect of cohesion. The model

included two yield surfaces: a cap yield surface to calculatethe plastic strain caused by the hydrostatic compression, anda conical yield function to calculate the plastic strain associated with the Drucker-Prager yield function. In the model, anew work hardening function was proposed for the cap yieldsurface, which is more suitable for cohesive soils.

The developed model showed a good curve-fitting accuracy for both axial and hydrostatic compressions. Allparameters can be determined from conventional triaxialtests. The model provided a single form of constitutive relationship during the entire loading period. Therefore, themodel can be more easily implemented in finite elementanalyses compared with the previous critical state and capmodels.

The elastic modulus of soil was affected by loading speed.However, more investigations are needed to develop a modelto include the dynamic effect of loading speed.

ACKNOWLEDGEMENTS

The authors acknowledge the financial support provided bythe Natural Sciences and Engineering Research Council ofCanada and Agriculture Canada under the University-Industry program.

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