International Journal of Damage Mechanics 2003 Al Shayea 305 29

8/10/2019 International Journal of Damage Mechanics 2003 Al Shayea 305 29

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A Plastic-damage Model for StressStrain

Behavior of Soils

N. A. AL-SHAYEA,* K. R. MOHIB AND M. H. BALUCH

Civil Engineering Department,

King Fahd University of Petroleum & Minerals,

Box 368, Dhahran 31261, Saudi Arabia

ABSTRACT: This paper presents a constitutive model for soil, which combineselements of plasticity with damage mechanics to simulate the stressstrain behavior.The model is primarily suitable for soil types that exhibit a postpeak strain-softeningbehavior, such as dense sand and stiff clay. The postpeak stress drop is captured bythe elasto-damage formulation, while the plasticity is superimposed beyond theelastic range. The total strain increment is composed of an elasto-damage strainincrement and a plastic strain increment. The elasto-damage strain increment isfound using the elasto-damage formulation, while the plastic strain increment isfound using either the DruckerPrager classical plasticity model or as a function ofdamage strain.

To implement this model, an experimental program was conducted on local

cohesive and cohesionless soils. Various physical and mechanical properties of thesesoils were determined. Both triaxial tests and hydrostatic tests were performed underdifferent confining pressures, in order to obtain the model parameters. Theseparameters were used to calibrate the model, which was coded in computer programsto simulate the stressstrain behavior of soils. The model was verified and found tobe a good predictor of the geomaterial response for the selected stress path.

KEY WORDS: damage mechanics, soil plasticity, strain softening, confiningpressure.

INTRODUCTION

THE DAMAGE MECHANICS technique has been applied successfully

to various materials such as metals and concrete. It was assumed that

a purely brittle material unloads in an elasto-damage manner, with

nonlinearity caused by degradation in the modulus of elasticity due to

damage. However, a purely elasto-plastic material will always show

International Journal ofDAMAGEMECHANICS, Vol. 12October 2003 305

1056-7895/03/04 030525 $10.00/0 DOI: 10.1177/105678903036224 2003Sage Publications

*Author to whom correspondence should be addressed.

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permanent deformation when unloaded after yielding. In the case of soils,

plasticity plays a significant role in addition to damage. The actual soil

behavior is neither of a purely elasto-damage type nor of a purely elasto-

plastic type, but is assumed to be a combination of both.In this paper, a plasto-damage model was developed for the stressstrain

behavior of cohesive and noncohesive soils (marl and sand). The model was

customized for the application of conventional triaxial compression (CTC),

because of the abundant use of this test for the determination of soil

strength. The dense soils investigated in this work have a damage effect,

which is the degradation in material compliance along with the plasticity

effect, which is the irrecoverable permanent deformation. The strain-

softening behavior is considered to be a combination of elasto-damage and

plastic behavior. Figure 1 represents schematically the various straincomponents. The plasticity is assumed to cause no degradation in material

compliance, so a perfectly elasto-plastic material will always unload at the

same modulus as the initial modulus, but along a different path after

yielding has taken place (Path 1), which results in a permanent strain

following complete unloading. On the other hand, the elasto-damage strain

Eo

Stress

Strain

EoE3E2

Path 2

Path 1

E2< E3< Eo

Path 3

p

d

e

Figure 1. Schematic unloading paths with ideal plastic deformation (path 1), perfectly brittlebehavior (path 2), and combined plasticity and damage (path 3) [Abu-Labdeh and Voyiadjis(1993)].

306 N. A. AL-SHAYEA ET AL.

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is assumed to be completely recoverable and is composed of elastic and

damage parts. Such a material unloads with a degraded modulus of

elasticity after the occurrence of damage, and returns to a zero stress and

strain state, as shown by Path 2 in Figure 1. The damage part of the strain isassumed to occur due to an increase in compliance caused by the damage.

The actual soil behavior is neither of these two types.

The actual soil material is assumed to follow Path 3, as shown in Figure 1,

so that there will be elastic strain, damage strain due to degradation in the

initial modulus of elasticity, and plastic strain due to dislocation and

permanent readjustment of the soil particles.

At a particular value of the stress increment, the total strain increment

was obtained by adding the elasto-damage strain increment and the plastic

strain increment, as found from the respective formulations. The total strainincrement (d") is composed of the elastic strain increment (d"e), the damage

strain increment (d"d), and the plastic strain increment (d" p); i.e.,

d"d"e d"d d"p 1

In this work, the elasto-damage component of the strain was obtained

from the elasto-damage formulation derived for the CTC test. The postpeak

stress-drop was also obtained by the same elasto-damage formulation. The

plastic strain was found by two different methods. In Method 1, the plasticstrain was taken as some factor times the damage strain. In Method 2, the

DruckerPrager model was derived for the conventional triaxial test case,

from which the plastic strain was calculated.

A rigorous analytical formulation was carried out to obtain the plasto-

damage stressstrain behavior by the two methods. Also, an extensive

experimental program was conducted on the two different types of soils

(sand and marl) at high density, exhibiting postpeak strain-softening

behavior, to find model parameters. In addition, FORTRAN programs

were developed based on the analytical formulations to be used forpredicting the soil behavior, and compared with the experimental results.

LITERATURE REVIEW

The theory of continuum damage mechanics (CDM) proposed by

Kachanov (1958) has been applied successfully to different materials, and

is used to predict both strain-hardening and strain-softening types of

behavior. The damage phenomenon and damage variable can be explained

by considering a damaged body, with (S) being the overall cross sectionalarea defined by the normal (n). Within the same cross section the cracks and

cavities have intersections of different shapes, with their area (SD). These

StressStrain Behavior of Soils 307



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micro cracks and cavities cannot resist the externally applied load. The

effective area ( ~SS) is obtained by subtracting the area of the voids (SD) from

the total area (S).

The damage variable (!n), associated with the normal (n), represents thearea of the cracks and cavities per unit surface in a plane perpendicular to

the normal (n) and is given as:

!nS ~SSS

SDS

2

Therefore, !n 0 corresponds to the undamaged state, while !n 1corresponds to the rupture of the element in two parts; a value between 0

and 1 characterizes the occurrence of damage. The effective area gives rise tothe effective stress concept in damage mechanics as given by Kachanov

(1958), as the mean density of the forces acting on the elementary surface

that effectively resists the applied load.

A practical soil model should be such that it involves a minimum possible

number of parameters and is still able to predict the behavior of the material.

The hyperbolic model (Kondner, 1963) is the frequently used model for

predicting the stressstrain behavior of soil. The limitation of the model lies

in its inability to predict the post peak stress-drop; i.e., the strain softening.

Chow and Wang (1987) presented the development of an anisotropicelastic damage theory. They presented a modified damage effect tensor~MMD for the effective stress equations to take into account the effect of

anisotropic material damage. The reduction of the proposed tensor to a

scalar for isotropic damage state occurred not only in the principal stress

direction, but also in any arbitrary coordinate system. The model was

applied to metals.

To account for the plasticity effect, Chow and Wang (1988) further

developed a generalized damage characteristic tensor ( ~JJ), to characterize

anisotropic damage evolution in combination with plasticity. The tensorialequation of the damage evolution for tensor ( ~JJ) reduced to a scalar if the

damage state was isotropic.

Following the developments in metals, Sauris et al. (1990) presented a

damage model for monotonic and cyclic loading of concrete. The

compliance tensor presented for the stressstrain relationship was dependent

on accumulated damage. Damage evolution was obtained by using a

loading surface (f) and bounding surface (F), along with a limit fracture

surface (fo) defined in terms of the thermodynamic force conjugates of the

damage variables, Figure 2. The model was applied to the case wherethe principal stresses and the strain axes coincided and did not rotate as the

material deformed.




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Abu-Labdeh and Voyiadjis (1993) presented a plasticity damage model

for concrete under cyclic multi axial loading. The model was based on thebounding surface concept and combined plastic deformations with

deformations due to damage. The total strain increment (d") was divided

into an elastic strain increment (d"e ), a damage strain increment (d"d) and a

plastic strain increment (d"p ) at a particular value of stress.

Crouch and Wolf (1995) presented a constitutive model which treated

clays, silts, and sand in their loose and dense state under both monotonic

and complex stress paths within a single framework.

Karr et al. (1996) presented a brittle damage model for rock. In their

model the effect of material degradation on failure mechanism of brittledamage materials was investigated.

Gupta and Bergstrom (1997) presented a model for the compressive

failure of rocks via a process of shear faulting. The model presented the

progressive growth of damage that led to the formation of a critical fault

nucleus, which grew unstably into its own plane by fracturing the grain

boundaries.

Khan et al. (1998) proposed a constitutive model for concrete based on

CDM, using the damage-effect tensor ( ~MM) with a damage magnification

factor and for tension and compression, respectively. The boundingsurface concept, as introduced by Dafalias and Popov (1977), was used for

the constitutive relationship and evolution of damage. The proposed model

Triaxial compression pathR3

Rc

Ro

b = b = 1R2

F

f

fo

in

ii nRf =

45

Figure 2. Limit fracture, loading and bounding surfaces in strain energy release space afterSuaris et al. (1998).




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was able to identify the degradation in elastic properties, strain softening,

the gain in strength under increasing confinement and different behaviors in

tension and compression.

ANALYTICAL BACKGROUND

The soil model proposed in this paper is primarily suitable for isotropic

soil types that exhibit a postpeak strain-softening behavior, under triaxial

loading condition. This model combines existing elements of damage

mechanics with plasticity to simulate the stressstrain behavior. It uses the

concepts given by Suaris et al. (1990) and the formulation developed by

Khan et al. (1998) for the damage part. The formulation utilized a damage

magnification factor results in a critical damage at !n 1/rather than aspreviously stated. The plastic part is added to the damage part by two

different methods. In Method 1, the plastic strain is taken as a constant

factor times the damage part. The constant factor does not result in a

constant plastic part, because the damage part is nonlinear. In Method 2,

the plastic strain is calculated using the DruckerPrager model for the

plastic part. The proposed model combines the damage mechanics and the

plasticity formulations by programming them to find the contribution of

each component to stress and strain. The equations are not combined

together mathematically in a closed form.

Damage Formulation

For the case of CTC test, 23 cell constant, and the deviator stressin the axial direction (1) is changing. This case reduces to a uniaxial case.

The deviator part of the stress vector, causing the shearing failure in the

cylindrical soil specimen, is given as:

deviator 1 0 0

T 3

For the CTC test the compliance and stiffness terms for ij 1, and with!1 0 and !2!3!, can be shown to be:

~CC11 1Eo 1! 4

, or ~DD11Eo 1! 4 4

where, Eo is the initial modulus of elasticity, and is the peak strength

factor or damage magnification factor.




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The strain energy density (W) for the deviator part of the CTC test,

which is to be predicted by the formulation, is given as:

W 12 if gT ~CCij

h i j 1

221

~CC21112

21

Eo1!1

2

1!2 2 1!3 2

5

where, ( ~CCij) is the effective compliance matrix, the inverse of the effective

stiffness matrix, ( ~DDij). The relationship between axial stress (1), strain

("1), and the strain energy (W) gives:

1"1Eo 1!2 2 1!3 2

1

!1

2

6

Substituting Equation (6) into Equation (5) gives:

W 12"21Eo

1!2 2 1!3 21!1 2

7

The thermodynamic force conjugates (Ri) can be obtained by taking the

negative derivative of the strain energy density with respect to the damage

variables (!n

) as:

R10 8R2R3"21Eo 1! 3 9

R1 is set to zero since negative values for Riare inadmissible in view of the

non-negativity of the thermodynamic conditions. This further implies that

damage in direction 1 is also zero, i.e., !1 0. For the CTC test,!2

!3

!, due to symmetry. The final expression for strain energy release

rate at onset of damage (Ro), for the CTC test, is given as:

Roffiffiffi

2p

2

Eo10

where, is some percentage of the peak deviator strength (1,p).

The final expression for critical value of strain energy release rate at

failure (Rc), for the CTC test, is given as:

Rcffiffiffi

2p 21,pEo 1! 5

11




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For the deviatoric stress increment in axial direction is:

d1 ~

DD11d"1"1@ ~DD11

@!k

@f

@Rk d 12

where, d 2ffiffiffi

2p

"1Eo 1! 3d"1H32"21Eo 1! 2 , 13

H Din

h i damage modulus,

is the normalized distance between f and F surfaces, Figure 2, in is the

maximum distance between F and fo surfaces, Figure 2, and D is

the parameter used to control the shape of the peak.

Substituting all the terms in Equation (12), the final form of the

incremental stress for a strain-controlled formulation becomes:

d1

Eo 1

!

4

82"21E2o 1! 6

H32

"

2

1Eo 1! 2" #d"1

14

Equation (14) gives the stressstrain relationship in an incremental form for

the conventional triaxial test case. Notice that d"0 in Equation (14) is theelasto-damage strain, which is equal to d"e d"d. For the input of elasto-damage strain increments on the right hand side of the equation, the

corresponding elasto-damage stress increments can be determined. These

stress and strain increments are added to provide the predicted behavior.

The elasto-damage behavior alone is not able to predict the actual behavior,

due to the missing plastic strain component.For the CTC, the incremental form of the damage variable (!) is given as

d!d=ffiffiffi

2p

: The d is a function of damage modulus (H), which in-turn

depends on the normalized distances between the limit fracture, loading, and

bounding surfaces. The incremental expressions are programmed to get the

predicted stressstrain curves. The parameters, which are constant for a

particular prediction such as, cell pressure (), peak stress (p), initial

modulus of elasticity (Eo), peak shape factor (D), peak deviator stress factor

(), and the strain increment (d"1) are given as input. The strain ("1),

the deviator stress (), and the damage variable (!) are initialized to zero.The current level of the strain energy release rate (R) corresponding to the

loading surface (f) is calculated from the expression [ffiffiffi

2p

Eo"21 1! 3].




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The value of the strain-energy-release rate at the onset of damage ( Ro) is

calculated using Equation (11). The comparison of the two values decides

weather the damage has started or not. IfR is less thanRothe damage is not

occurring, and the value of damage increment (d!) and the total damage (!)are zero. The corresponding stress is within the linear elastic range, and is

calculated as given by Equation (6) with (!1 0 and !2!3! 0). Thecurrent stress is obtained by adding the stress increment to the initial stress

value (i.e., 11 d1). For the first iteration in the program 0.The current strain value is updated as (" " d"). For the first iteration"1 d"1. This process is repeated with updated values of deviator stress andaxial strain (1and "1). For each iteration, the value ofR is calculated and

compared with Ro. As soon as Rbecomes greater than Ro, the value ofd,

the increment in damage (d!) are calculated. The evolution of damage takesplace in the subsequent iterations, asdis a function of the damage modulus

(H), which in turn is a function of the normalized distance ( and in)

between the three surfaces shown in Figure 2. Once the damage increment

(d!) is evaluated, the damage ! is calculated as !! d!. Notice that! 0 for the iterations before R becomes equal to Ro. After the values ofdamage have been updated, the increment of the stress (d1) is now

calculated from Equation (14) instead of Equation (6). The updated values

of stress (1) and strain ("1) are calculated. The program runs till the value

ofR reaches Rc, which is calculated from Equation (11).

Plastic Strain (Method 1)

The total strain is assumed to be composed of elastic, damage, and plastic

strains. In terms of principal strains in axial direction,"e1 is the elastic strain;

"d1 is the damage strain resulting from the damage growth by the purely

brittle elasto-damage theory, and"p1is the plastic strain component that is to

be added. The plastic strain is a nonlinear irrecoverable part, and is taken as

a factor times the damage strain so that:

"p1Fp"d1 15

where Fp is a constant factor, which is determined by trial and error using

experimental results.

At any stress value found by the elasto-damage formulation, the purely

elastic part of the strain is found as:

"e1

1=Eo

16

whereEois the initial modulus of elasticity value, which is free from damage

effect.




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DruckerPrager Model

The DruckerPrager model was used for finding the plastic strain by

method 2. The DruckerPrager failure criteria is given as:

fffiffiffiffiffiffiffiffiffiffi

J2Dzp

J1k 17where, is the positive material parameter defining the yield criteria in J1versus

ffiffiffiffiffiffiffiJ2D

p space, and represents the slope of the straight line fit,k is the

intercept of the yield envelope, J1 is the first invariant of the stress tensor,

and J2D is the second invariant of the deviatoric stress tensor.

As long as the material is within the linear elastic range, the governing

stressstrain equation for a general case is given by Desai and Siriwardane

(1984), and it has the following form for the CTC test case:

d1 4Go3Ko Go3Ko 3Ko2Go 2Go3Ko

3 3KoGo d"1 18

where,Gois the initial shear modulus, and Kois the initial bulk modulus of

elasticity.

When the stress state reaches the yield envelope, the governing stress

strain equation for a general case is given by Desai and Siriwardane (1984),

and it has the following form for the CTC test case in the axial direction:

d1 2Go 1T11R1 12T2cell2R2 2 T1cellR1 T21R2 12T2cell2R2

d"1

19where, Ti A Ci, and Ri AiB, with i 1, 2.

A and B are the factors depending on material parameters determined

from the test results, and are given as:

A hp0k

20

p0ffiffiffiffiffiffiffi

J2Dp

k 19

2Ko

Go

21

hk p01

6


p 3Ko2Go

J16


p

22

B 2h2

1 92Ko=Go3Ko

Eo, and 23




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EXPERIMENTAL PROGRAM

In order to implement the model and to determine its parameters, an

extensive experimental program was conducted. Two different types of soilmaterial were used, i.e., cohesive and noncohesive soil (marl and sand). The

marl had a specific gravity of 2.85, a liquid limit of 55.8, a plastic limit of

37.3, and a plasticity index of 18.49. Particle size distribution for marl was

obtained using sieve and hydrometer analyses, which indicated that marl

contained about 74% fines most of which are of clay size. The marl was

classified according to USCS as an organic clayey soil with high plasticity

(OH). The marl was tested at a maximum value of dry density of 1.526 g/cm3

and at optimum moisture content of 27.12% obtained from the Modified

Proctor compaction test.The sand used was medium in size with Cu 2.342 and Cc 1.124. The

sand had a specific gravity of 2.64, and was classified according to USCS as

poorly graded medium sand (SP). The minimum and maximum densities of

the sand were found to be 1.86 and 1.609 g/cm3, respectively.

The marl samples were prepared at the maximum density and optimum

moisture content using five layers under static compaction, to obtain a

uniform sample. The sand was tested at a density of 1.853 g/cm3, which

corresponds to a relative density of 98%. The pluviation technique was used

for preparing the sand samples.Unconsolidated drained triaxial compression tests were performed at cell

pressures of 100, 200, 300, 400, and 500 kPa, for both soils. A hydrostatic

compression test was conducted on both soils to obtain the initial bulk

modulus of elasticity (Ko). Isotropic cell pressures used were 25, 50, 100,

200, 400, 800, 1000 kPa for loading; and 800, 400, 200, 100, 50, 25 kPa for

unloading. All items of apparatus were calibrated to ensure accurate

measurements of stresses and strains.

RESULTS AND DISCUSSIONS

Elasto-damage Model Parameters

The expressions for the damage model parameters were determined from

the experimental results and then used in the computer program. These

parameters are presented as follows:

INITIAL MODULUS OF ELASTICITY (EO)

Eo was obtained from the results of the triaxial tests, as the slope of theinitial straight-line portion of the deviator stress versus axial strain curve,

for various values of cell pressure (3). Normalized values of Eo versus 3




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with respect to atmospheric pressure (Pa 101.325 kPa) are shown in Figure 5for both marl and sand. The expressions for Eo in kPa are:

EoaPa3

Pa

b25

where a and b are fitting parameters shown in Table 1.

PEAK DEVIATOR STRESS AND SHEAR STRENGTH PARAMETERS

The shear strength parameters, cohesion (c) and angle of internal friction

(), were determined by plotting the Mohr stress circles and MohrCoulomb

failure envelopes for the peak deviator stress value (1,p) at different cell

pressure values. For marl 33.73 and c 335 kPa. For sand 42.96and c 0. Using these values, the value of1 can be calculated as:

1,p2 c cos3sin

1sin 26

8 9 2 3 4 5 61

(3/ Pa)

2

4

6

8

2

4

6

8

1000

10000

(Eo

/Pa

)

Soil type

Sand

Marl

Figure 5. Normalized plots of initial modulus of elasticity for marl and sand.




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PEAK SHAPE FACTOR (D)This parameter is used to define the shape of the peak of the predicted

stressstrain curves. The value of this factor for both the marl and sand was

found, by trial and error, to be 0.0001 kPa.

STRAIN ENERGY RELEASE RATE AT ONSET OF DAMAGE (RO)

The value of Ro corresponds to the stress level at which the damage

starts. The stress level at which the damage starts is taken as some

percentage of the peak deviator stress. Ro was calculated using Equation

(10) at 20 and 30% of the peak deviator stress value for marl and sand,respectively; which was decided on the basis of the experimental

stressstrain data.

CRITICAL STRAIN ENERGY RELEASE RATE (RC)

The values ofRcwere calculated using Equation (11), and found to be 0.1

and 0.2 kN m/m3 for marl and sand, respectively. These are the minimum

values ofRc calculated at different cell pressures.

PEAK DEVIATOR STRESS FACTOR ()The parameter is used to match the peak of the predicted stressstrain

curve with the experimental curve. The value of was calculated by trial and

Table 1. Parameters for fitting expressions.

Soil type

Parameter

Marl Sand

a 682.5 1176.06

b 0.238156 0.862364

R2 0.885948 0.986754

c 0.5697 0.87782

d 0.916701 0.785059R2 0.984115 0.993890

e 1.187 104 3.055f 1.694 0.1845

R2 0.928568 0.810207

g 3.2557 104 8.0804 105

h 0 0.075R2 0.964011 0.871921

i 2.8754 2.5230

j 518.542 0

R2 0.945357 0.996425

k 101.23 52.76

l 1.6295 0.7781

R2 0.810456 0.679902




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error. Figure 6 shows the variation of versus peak deviator stress (1,p)

for marl and sand, with the following equations of the fit:

c 1,p d 27

where c and dare fitting parameters shown in Table 1.

Plastic Strain by Method 1

The plastic strain in Method 1 was calculated as a factor (Fp) times

the damage strain, Equation (15). The damage strain was calculated and then

multiplied with the plastic strain factor (Fp) to obtain the plastic strain. The

total strain was obtained by adding the plastic strain to the elasto-damage

strain. Figure 7 shows the variation of the plastic strain factor (Fp) with cell

pressure (3) for marl and sand, with the expressions of the fit:

Fpe 3 f 28where e and fare fitting parameters shown in Table 1.

3 4 5 6 7 8 9 321000

1, p (KPa)

2

3

4

5

67

89

2

3

4

5

6

789

0.0010

0.0100

Soil type

Sand

Marl

Figure 6. vs. peak deviator stress for marl and sand.




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Figure 8 shows a comparison between the experimental stressstraincurve for sand at 3 200 kPa and the stressstrain prediction by elasto-damage formulation. It is clear that without the plastic strain component,

the two curves are far apart from each other on the strain axis. After adding

the plastic strain, the elasto-plasto-damage prediction matches the experi-

mental curve very closely.

The plasto-damage strain when plotted with the stress values gives the

stressstrain curves predicted by this method. Figure 9 shows a comparison

between the experimental and the predicted stressstrain curves by this

method for different cell pressures for sand. Similar curves were alsoobtained for marl.

Plastic Strain by Method 2

The expression for the initial modulus of elasticity (Eo) is the same as that

given by Equation (25). Other parameters required for this method are

briefly explained as follows:

INITIAL BULK MODULUS (KO)Kovalues were found from the results of the hydrostatic tests, as the slope

of the reloading portion of the hydrostatic tests, as the slope of the reloading

8 9 2 3 4 5 6100

3 (KPa)

2

4

6

8

2

4

6

8

2

0.10

1.00

10.00

Fp

Soil type

Sand

Marl

Figure 7. Plastic strain factor vs. cell pressure for marl and sand.




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0.00 0.04 0.08 0.12 0.16

1

0

200

400

600

800

1000

1200

1400

1600

1800

2000

2200

1

(KPa)

Experimental

Prediction

100 KPa

200 KPa

300 KPa

400 KPa

500 KPa

Cell pressure

Figure 9. Comparison between experimental and predicted stressstrain curves for sand bymethod 1.

0.00 0.04 0.08 0.12

1

0

200

400

600

800

1000

1

(KPa)

Experimental

without plastic strain

Elasto-plasto-damage (1)

Elasto-plasto-damage (2)

Figure 8. Comparison between experimental and predicted stressstrain curves for sand at3 200 kPa, with and without plastic strain by the two different methods.




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portion of the hydrostatic stress versus volumetric strain curves. The average

Ko values found for marl and sand were 84,580 and 260,252 kPa,

respectively.

POISSONS RATIO () AND INITIAL SHEAR MODULUS (GO)

The values ofEo and Ko were used to determine the values of Poissons

ratio () and initial shear modulus (Go) by the following elastic relationships:

12

3KoEoKo

29

Go Eo

2 1 30

YIELD ENVELOPE PARAMETERS () AND (K)

The yield envelope drawn between the first invariant of the stress tensor

(J1) and the square root of the second invariant of the deviatoric stress

tensor (ffiffiffiffiffiffiffi

J2Dp

) is a straight-line fit. Figure 10 shows the DruckerPrager yield

envelopes for marl and sand. For marl, 0.2612 and k 397.7 kPa, whilefor sand 0.3355 and k 9.477 kPa.

TOTAL ELASTO-PLASTIC STRAIN ("t)

One of the inputs used for finding plastic strains by the DruckerPrager

model is the total elasto-plastic strain ("t). Its values were found by trial and

error at different cell pressures. The value of"t is used for determining the

appropriate plastic strain component to be added to the elasto-damage

strain component to obtain the total strain. The value of"twas obtained by

trial and error by comparing the experimental stressstrain curve with the

analytical one. Figure 11 shows plots of "t versus 3 for marl and sand,

whose fits have the following expressions:

"tg 3 h 31

where g and h are fitting parameters shown in Table 1.

ELASTO-PLASTIC RESIDUAL STRESS (r)

This is used as an input for the determination of the plastic strains by the

DruckerPrager model. Figure 12 shows plots ofr versus 3 for marl and

sand, whose fits have the following expressions:

ri3j 32where iand jare fitting parameters shown in Table 1.




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Using the above expressions for different parameters, the FORTRAN

program was run and the predicted curves by Method 2 are comparedwith experimental ones in Figure 13 sand. Similar plots were obtained

for marl.

0 200 400 600

3 (KPa)

0.00

0.10

0.20

t

Soil type

Sand

Marl

Figure 11. Total elasto-plastic strain vs. cell pressure for marl and sand.

1000 2000 3000 4000 5000

J1 (KPa)

200

400

600

800

1000

1200

1400

(J2D

)

(KPa)

Soil type

Sand

Marl

Figure 10. DruckerPrager yield envelopes for marl and sand.




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0.00 0.04 0.08 0.12 0.16

1

0

200

400

600

800

1000

1200

1400

1600

1800

2000

2200

1

(KPa)

Experimental

Prediction

100 KPa

200 KPa

300 KPa

400 KPa

500 KPa

Cell pressure

Figure 13. Comparison between experimental and predicted stressstrain curves for sandby method 2.

0 200 400 600

3 (KPa)

0

500

1000

1500

2000

2500

r

(KPa)

Soil type

SandMarl

Figure 12. Residual stress vs. cell pressure for marl and sand.




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Comparison Between Methods 1 and 2

In both methods the elastic and damage strain components are found

using the elasto-damage formulation. The plastic strain is found as Fptimesthe damage strain in Method 1, while it is calculated from the elasto-plastic

DruckerPrager model in Method 2. Figure 8 also shows the comparison

between the experimental behavior and predictions by Methods 1 and 2, at

3 200 kPa for sand. Predictions by Methods 1 and 2 are very close to eachother and to the experimental behavior for sand. The difference between the

two methods is that in the case of Method 2 there is a singularity or sudden

change in slope when plastic strains are added to the elasto-damage strains

because of the behavior of the DruckerPrager model for CTC. It can also

be observed that the prediction by Method 1 is slightly better than that byMethod 2. Method 2 on the other hand is theoretically more sound, because

the plastic strain is evaluated from a rigorous elasto-plastic model. Both

methods validate each other.

Figure 14 shows the plots between Fp and "t for marl and sand. The

expressions between Fp and "t are given as:

Fpk "t l 33where k and lare fitting parameters shown in Table 1.

2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9

0.01 0.10 1.00t

2

4

6

8

2

4

6

8

2

0

1

10

Fp

Soil type

Sand

Marl

Figure 14. Variation of Fp with "t.




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The fact that Fp can be expressed in terms of"t validates the assumption

that plastic strains can be taken as a factor of damage strains, and while

each one was calibrated separately and was obtained using different models,

a relationship still exists between them.

CONCLUSIONS

A new plasto-damage model is presented for the stressstrain behavior of

dense soils. The model is suitable for simulating post-peak strain-softening

stressstrain behavior of soil. The model combines elasto-damage strain and

plastic strain to determine the total strain. Two different methods were used

for finding the plastic strain. The predictions by both methods are excellent

and are able to identify all the features of the stressstrain behavior. It isobserved that the model is good for both cohesive and noncohesive types of

soils. Predictions by Method 1 are smoother than by Method 2. For a

conventional triaxial test case, the DruckerPrager model prediction by

Method 2 has a sudden change in slope of the predicted stressstrain curves

at transition from elastic to elasto-plasto-damage behavior. Method 1 is

very simple and needs fewer parameters to be calibrated than Method 2.

Only conventional triaxial test data is required, with no additional

experimental data required for the calculation of plastic strain. In the case

of Method 2, isotropic compression test data is also required in addition toconventional triaxial test data for the calculation of plastic strain by the

DruckerPrager model.

LIST OF SYMBOLS

~SS effective area~CCij

effective compliance matrix

~DDijeffective stiffness matrix~ effective stressc cohesion

CDM continuum damage mechanicsD peak shape factor

d proportionality factor defining relationship between damageincrement and loading surface

Eo initial value of modulus of elasticityF bounding surface (damage model)f loading surface function (damage model)f yield function for DruckerPrager model

fo limit fracture surface (damage model)




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Fp plastic strain factor (Method 1)Go initial shear modulusH

damage modulus

J1 first invariant of stress tensorJ2D second invariant of deviatoric stress tensorKo initial value of bulk modulus of elasticity

k intercept of DruckerPrager yield functionPa atmospheric pressure in kPaRc strain energy release rate at failureRi thermodynamic force conjugates in tensor notation

Ro strain energy release rate at onset of damageS

overall area of cross section

SD area of micro cracks and discontinuities Poissons ratio

1,p peak deviator stress slope of DruckerPrager yield function peak stress factor for damage model normalized distance between the loading surface and bounding

surface

in normalized distance between the limit fracture surface andbounding surface

"ij strain tensor"t elasto-plastic total strain (Method 2) angle of internal friction density3 cell pressurer elasto-plastic residual stress! elasto-plastic residual stress

ACKNOWLEDGMENTS

The authors acknowledge the support of King Fahd University of

Petroleum and Minerals for providing computing and laboratory facilities.

The assistance of Dr. Asad-ur-Rehman Khan is greatly appreciated.

REFERENCES

Abu-Lebdeh, T.M. and Voyiadjis, G.Z. (1993). Plasticity Damage Model for Concrete underCyclic Multi-axial Loading, Journal of Engineering Mechanics, ASCE,119(7): 14651484.

Chow, C.L. and Wang, J. (1987). An Anisotropic Theory of Elasticity for Continuum Damage

Mechanics, International Journal of Fracture, 33: 316.

Chow, C.L. and Wang, J. (1988). A Finite Element Analysis of Continuum Damage Mechanicsfor Ductile Fracture, International Journal of Fracture, 38: 83102.




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Crouch, R.S. and Wolf J.P. (1995). On a Three-Dimensional Anisotropic Plasticity Model forSoil,Geotechnique, 45(2): 301305.

Dafalias, Y.F. and Popov, E.P. (1977). Cyclic Loading for Material with a Vanishing ElasticRegion,Nuclear Engineering Design, 41: 283302.

Desai, C.S. and Siriwardane, H.J. (1984). Constitutive Laws for Engineering Materials withEmphasis on Geologic Materials, p. 468, Prentice Hall, Inc., Englewood Cliffs, New Jersey.

Gupta, V. and Bergstrom, J.S. (1997). Compressive Failure of Rocks, International Journal ofRock Mechanics and Mining Science, 34(34): 376378.

Kachanov, L.M. (1958). Time of the Rupture Process under Creep condition, Izv Akad Nauk,U.S.S.R, Otd. Tekh. Nauk, 8: 2631.

Karr, D.G., Wimmer, S.A. and Sun, X. (1996). Shear Band Initiation of Brittle DamageMaterials,International Journal of Damage Mechanics, 5: 403421.

Khan, A.R., Al-Gadhib, A.H. and Baluch, M.H. (March 1998). An Elasto DamageConstitutive Model for High Strength Concrete, In: Proceedings of the Euro-C 1998Conference on Computational Modelling of Concrete Structures, pp. 133142, Austria.

Kondner, R.L. (January 1963). Hyperbolic StressStrain Response: Cohesive Soils, Journal ofSoil Mechanics and Foundation Division, 89(SM1): 115143.

Sauris, W., Ouyang, C. and Ferdenando, V.M. (May 1990). Damage Model for Cyclic Loadingof Concrete, Journal of Engineering Mechanics, ASCE, 116(5): 10201035.


International Journal of Damage Mechanics 2003 Al Shayea 305 29

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