1

New Strain-Energy Based Coupled Elastoplastic Damage-Healing

Formulations Accounting for Effect of Matric Suction

During Earth Moving Processes

K.Y. Yuan1 and J.W. Ju

2

Department of Civil and Environmental Engineering,

University of California, Los Angeles, CA 90095-1593, USA

ABSTRACT

Innovative initial elastic strain energy based coupled elastoplastic hybrid isotropic damage

and healing models for partially saturated soils have been developed and implemented for

numerical simulation of 2-D earth pushing processes. A class of elastoplastic constitutive

damage-healing models, based on a continuum thermodynamic framework, is proposed within

an initial elastic strain energy based formulation. In particular, change of effective stress due to

matric suction in the formulation is considered and the governing incremental damage and

healing evolutions are coupled and characterized through the effective stress concept in

conjunction with the hypothesis of strain equivalence. Further, plastic flow is introduced by

means of an additive split of the stress tensor. In this innovative formulation, two characteristic

energy norms of the tensile and compressive strain tensors, respectively, are introduced for the

corresponding damage and healing mechanisms.

By incorporating a micromechanics-motivated damage characterization ( P ) and a healing

characterization ( P ), the proposed model and computational algorithms have been implemented

to demonstrate the significant flexibility on numerical simulation of earth pushing processes.

Completely new computational algorithms are systematically developed based on the two-step

operator splitting methodology. The elastic-damage-healing predictor and the plastic corrector

are implemented within the RKPM (Reproducing Kernel Particle Method) meshfree codes. A

numerical example under earth pushing is presented to illustrate the effect of matric suction for

partially saturated soils.

Key Words: Damage mechanics; geomaterials; coupled elastoplastic damage and healing

models; matrix suction effect; earth-moving processes.

ASCE Journal of Engineering Mechanics (revised on October 18, 2011)

1 Post doctoral scholar and lecturer. Email: yuanky@ucla.edu

2 Distinguished Professor and correspondence author. E-mail: juj@ucla.edu. Web: http://www.cee.ucla.edu/faculty/ju/profile

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INTRODUCTION

In the early 1980s, the development of constitutive equations for saturated soils has

involved three main concepts, including the concept of effective stress from soil mechanics, the

existence of coupled elastoplastic damage models from damage mechanics, and the theory of

two-phase mixtures for a solid skeleton and a fluid. Some engineering applications and

numerical demonstrations by finite element analysis and meshfree method were reported in the

literature; cf. Sanavia et al. (2006), Wu et al. (2001) and Murakami et al. (2005).

On the other hand, an unsaturated soil is commonly defined as having three phases: 1)

solids, 2) water, and 3) air. However, it may be more correct to recognize the existence of a

fourth phase, namely, the air-water interface or contractile skin (Fredlund and Morgenstern,

1978). From the standpoint of the volume-mass relations for an unsaturated soil, it is acceptable

to consider the soil as a three-phase system since the volume of the contractile skin is small and

its mass can be considered as part of the mass of water. Some rigorous framework to define the

constitutive behavior of unsaturated soils for three phases has been developed by Alonso et al.

(1990), Wheeler and Sivakumar (1995), Wheeler et al. (2002, 2003), Bolzon et al. (1996), Borja

(2004), and Georgiadis et al. (2005). For the stress analysis of a multiphase continuum, it should

be realized that the air-water interface behaves as an independent phase (Fredlund and Rahardio,

1993). A comprehensive framework to define the constitutive behavior of unsaturated soils for

four phases was proposed by Loret and Khalili (2000).

It is widely accepted that theories for correctly describing the states of stress and failure

in unsaturated soils require two fundamental considerations. Firstly, both the net stress and

matric suction in Bishop’s effective stress need to be considered independently; cf. Bishop

(1959), Bishop et al. (1960), Fredlund et al. (1978). Fredlund (1979), Gallipoli et al. (2003a,

2003b), Gens and Alonso (1992), and Escario and Saez (1986). Secondly, plasticity or failure

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models such as the Mohr-Coulomb (Fredlund et al., 1978), the Cam-Clay (Alonso et al., 1990) or

the Cap models (Simo et al. 1988, Kohler and Hofstetter, 2008) must be modified for unsaturated

(partially saturated) soils due to the matric suction. Further, to tackle different engineering

problems, material variables (e.g., grain size and grain size distribution), state variables (e.g.,

degree of saturation) and the consequent inter-particle forces (suction-induced effective stress or

suction stress) need to be considered as well for more comprehensive soil models.

In this paper, innovative initial elastic strain energy based hybrid isotropic elastoplastic

damage-healing formulations accounting for the effect of matric suction are developed and

implemented for 2-D earth moving simulation. In particular, since we are mainly interested in

simulating rapid earth-pushing activities and the qualitative features of our approach, the

following assumptions are made to simplify both modeling and computation efforts:

1. The air is assumed to remain at atmospheric pressure during the earth moving processes.

2. The suction refers here to the matric suction which is the difference between the air and water

pressures induced by the capillary tension. The difference in pressures induced by osmotic

effects; i.e., ionic disequilibrium, is not considered here.

3. The matric suction and the parameter in Bishop’s effective stress are only functions of

saturation which can be obtained by numerical equations (based on experimental observations)

for different types of soils. The effects of grain size or grain size distribution on the matric

suction and the parameter are not considered here.

4. For soils, the wetting (saturation changing from low to high) or drying (saturation changing

from high to low) process will induce different soil-water characteristic curve. This soil-water

hysteresis is not considered here.

5. During the rapid earth moving processes, the water is assumed undrained. Therefore, the

gravimetric water content can be considered as a constant but the degree of saturation varies due

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to void ratio change when soil particles undergo complex gliding and rolling motions. Unlike the

typical geotechnical problems such as bearing capacity, seepage and flow net, or slope stability,

Darcy’s law or Navier-Stokes equations are not applicable here for our specific earth moving

processes. Instead, a numerical equation based on experimental data is used to calculate the

degree of saturation at different stage during the earth moving activities.

6. The mass transfer due to vaporization and condensation is neglected during the rapid earth

moving processes, although vaporization certainly occurs in actual experiments where water and

air are in contact over a long period of time.

The remaining part of this paper is organized as follows. The change of effective stress

due to the matric suction and the Bishop’s effective stress for unsaturated soils are introduced.

The experimental data and some numerical equations which are employed to evaluate the

parameters of Bishop’s effective stress and matric suction from literature are systematically

presented. Subsequently, the coupled elastoplastic hybrid isotropic damage-healing formulations

featuring the effect of matric suction for unsaturated soils are developed. The corresponding

efficient two-step operator splitting methodology is rendered. Finally, numerical simulation of a

2-D earth-pushing process is demonstrated and discussed in detail. Conclusions and future work

are addressed as well.

STRAIN-ENERGY BASED COUPLED ELASTOPLASTIC HYBRID ISOTROPIC

DAMAGE-HEALING MODELS WITH MATRIC SUCTION EFFECT

Change of Effective Stress due to Matric Suction

The effective stress approach by Bishop (1959) for unsaturated soil expanded Terzaghi’s

classic effective stress equation as follows:

' a a wσ σ u u u (1)

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The difference aσ u is referred to as the net normal stress, the difference

a wu u is the matric

suction, and the effective stress parameter is a material variable that is generally considered to

vary between zero and unity. For 0 , it corresponds to completely dry soil; for 1 , it

corresponds to fully saturated soil, which leads to Terzaghi’s classic effective stress equation for

saturated soil. Notice that the pore water pressure, wu , in saturated soil is generally compressive

and isotropic. On the other hand, the pore water pressure in unsaturated soil is generally tensile

and matric suction is always positive.

Furthermore, cohesion is the component of shear strength of soil that is independent of

inter-particle friction. For partially saturated soil, in addition to the electrostatic forces and

cementing by minerals (e.g., 2 3 3Fe O , CaCO , NaCl, etc.), the partial loss or recovery of cohesion

between granular soil particles due to water menisci is given by the change of matric suction. For

instance, for loose, uncemented sand that is either completely dry or completely saturated, the

cohesion in the Mohr-Coulomb failure criterion may be considered essentially equal to zero.

However, if some water is added to the completely dry sand, considerable cohesive strength may

exist due to matric suction. This apparent cohesion unique to unsaturated soil arises from

negative pore water pressure and surface tension effects occurring at the interface of the pore

water, pore air and soil solids among the unsaturated soil grains.

Matric Suction

The soil suction is commonly termed “total suction” which has two components, namely,

the matric suction and the osmotic suction. However, it is primarily the matric suction

component, a wu u that governs the engineering behavior of unsaturated soils in the lower

suction range encountered in most field situations (Vanapalli et al., 1996). Moreover, laboratory

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data indicate that a change in total suction is essentially equivalent to a change in matric suction

where the water contents are less than the residual value (Krahn and Fredlund, 1972). Some

typical values of matric suction for different types of soils can be found in the literature; cf.

Fredlund and Rahardio (1993), Lu and Likos (2004), and Mitchell and Soga (2005). In general,

the latter values range from 0 to 1,000 KPa.

Disregarding pore air pressure, the contribution of pore water pressure to total stress

depends on the degree of saturation and pore size distribution. Based on the studies on granular

physics, four states of saturation can be characterized as follows (Mitarai and Nori, 2006):

1. Pendular state: Soil particles are held together by water bridge at their contact points.

2. Funicular state: Some pores are fully saturated by water, but there still remain voids filled

with air.

3. Capillary state: All voids between soil particles are filled with water, but the surface water is

drawn back into the pore under capillary action.

4. Slurry state: Soil particles are fully immersed in water and the surface of water is convex; i.e.,

no capillary action at the surface.

We refer to Mitarai and Nori (2006) for the schematic diagram and physical description

of these four regimes. To macroscopically estimate the matric suction in different saturation

states, it is necessary to introduce the soil-water characteristic curve (SWCC). The SWCC is a

fundamental constitutive relationship in unsaturated soil mechanics, which can be employed to

predict unsaturated soil properties such as the hydraulic conductivity, diffusivity, adsorption,

shear strength and volume change. In general terms, the SWCC describes the relationship

between the matric suction and water content (or degree of saturation).

Numerous mathematical models were proposed for the soil-water characteristic curve,

and various key equations were summarized by Zapata et al. (2000) and Nishimura et al. (2006).

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In particular, Fredlund and Xing (1994) proposed the following relationship between the

volumetric water content and the matric suction:

1

ln /

m

s ne s a

(2)

where is the volumetric water content, s represents the saturated volumetric water content, s

is the matric suction, e is the exponential number (approximately 2.718281828), and (m, n, a) are

material parameters. Figure 1, Figure 2 and Figure 3 exhibit the plots of Eq. (2) with different

sets of parameters (m, n, a).

Parameter in Bishop’s Effective Stress

Loret and Khalili (2000) stated that a properly defined effective stress is an efficient tool

in the qualitative and quantitative descriptions of the behavior of unsaturated soils subject to the

following two provisos: (1) that the parameter is evaluated with sufficient accuracy, and (2)

that the observed stiffening effect due to suction is also accounted for. The material variable in

Eq. (1) is captured by its strong dependency on the degree of pore water saturation. Moreover,

the nature of can be described systematically from both the microscopic and macroscopic

perspective; cf. Lu and Likos (2004).

The validity of several forms of as a function of the degree of saturation was also

examined by Vanapalli and Fredlund (2000) using a series of shear strength test results for

statically compacted mixtures of clay, silt and sand from Escario et al. (1989). For the matric

suction ranging between 0 to 1,500 KPa, the following two forms showed good fit to the

experimental results. The first form reads

8

k

k

s

S

(3)

where S is the degree of saturation, is the volumetric water content, s is the saturated

volumetric water content, and k is a fitting parameter optimized to obtain the best fit between

the measured and predicted values. The second form reads

1

r r

r s r

S S

S

(4)

where r is the residual volumetric water content and

rS is the residual degree of saturation. The

nature of Eqs. (3) and (4) is illustrated in Figure 4 for several values of k and rS .

Determination of Saturation

From previous sections, we have shown that the matric suction and Bishop’s effective

stress parameter can be represented by the degree of saturation. However, as mentioned earlier,

the continuity equation of pore water with Darcy’s law or Navier-Stokes equations for

unsaturated soils are not applicable here due to the instantaneous and complex rolling and gliding

motions of soil particles during earth pushing processes. In this section, we have to make

physically reasonable assumptions which are motivated by the literatures to calculate the degree

of saturation at different time step for our specific earth-pushing problems.

Gallipoli et al. (2003b) suggested an improved form of expression for the variation of

degree of saturation, accounting for the influence of changes in void ratio. The experimental data

exhibits that for certain matric suction, the degree of saturation is monotonically increasing with

the mean net stress. Motivated by these experimental observations, we propose the following

assumptions to simplify modeling and computation efforts:

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1. The degree of saturation is a function of the mean net stress, gravimetric water content and

soil types.

2. For a certain gravimetric water content and soil type, the degree of saturation is monotonically

increasing (linearly or nonlinearly depending on the soil types) with respect to the mean net

stress.

For simplicity, we assume that the relationship between the mean net stress and the

degree of saturation is linear as exhibited in Figure 5. Here, in Figure 5, iniS is the assumed

initial degree of saturation. When the mean net stress (compression) is increased, the degree of

saturation increases linearly. Once the assumed critical compressive mean net stress cσ is

reached, we assume that the saturation will reach a maximum value, maxS . On the other hand, if

the mean net stress is zero or tensile, the soil will reach a minimum constant value of saturation,

minS , and the effect of matric suction will be eliminated once the tensile mean net stress is greater

than the maximum matric suction, tσ .

A Coupled Hybrid Isotropic Formulation for Partially Saturated Soils

The proposed initial elastic strain energy based hybrid isotropic damage-healing model

for partially saturated soils is under the hypothesis that incremental damage and healing in the

soils are directly linked to the history of total strains. The notion of effective stress along with

the hypothesis of strain equivalence then follows from the assumed form of free energy.

Attention is focused on the modification of effective stress due to the matric suction.

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The Additive Stress Split

The plastic flow is introduced by means of an additive split of the stress tensor into the

initial and inelastic parts

0

a a wu S u u

pε

σ σ 1ε

(5)

where σ is the effective (undamaged) stress, ε is the total strain, 0ε is the initial elastic

stored energy function of the undamaged soil, pσ is the effective (undamaged) plastic relaxation

stress, au represents the pore air pressure, and

wu is the pore water pressure. Moreover, S is

a function of the degree of saturation and represents the Bishop’s effective stress parameter, and

1 denotes the second-order identity tensor. For the linear elasticity,

0 01: :

2ε ε C ε (6)

where 0C denotes the linear elasticity tensor. Eq. (5) then can be written as

0 : a a wu S u u p

σ C ε σ 1 (7)

Thermodynamic Basis

In order to introduce both the net effect of damage-healing and plastic flow processes, a

free energy potential of the following form is proposed:

0, , , 1 : ,p p pε σ q ε ε σ q σ

net netd d (8)

where ε denotes the total strain tensor, pσ is the plastic relaxation stress tensor, and q is a

suitable set of internal (plastic) variables. Here, netd is the isotropic scalar variable of net

(combined) effect of damage-healing which depends on the damage variable d , the incremental

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damage variable d , and the incremental healing variable R , all varying between 0 and 1 in

numerical values. 0ε is the initial elastic stored energy function of the undamaged material

and , pq σ is the plastic potential function.

Confining our attention to the purely mechanical theory, the Clausius-Duhem (reduced

dissipation) inequality (Coleman and Gurtin, 1967) (for purely mechanical isothermal theory)

takes the form for any admissible process

: 0σ ε (9)

By taking the time derivative of Eq. (8), substituting it into Eq. (9), and making use of standard

arguments (the Gurtin-Coleman argument) along with the additional assumption that the net

effect of damage-healing and plastic unloading are elastic processes, we can obtain the stress-

strain constitutive law

0

1 1net net

a a wd d u S u u

pε ε

σ σ σ 1ε ε

(10)

and the dissipative inequalities

: 0

p

p q ε σ

q σ

(11)

0 : 0netd

p

pε q ε σ

q σ

(12)

It follows from Eq. (10) that within the present strain space formulation, the stress tensor is split

into the elastic-damage-healing with suction effect and the plastic relaxation parts. From Eqs. (11)

and (12), it shows that the dissipative energy by the plasticity itself is positive and if damage-

healing effect with suction effect is involved, the sum of them (the damage-healing effect and the

plasticity) is also positive. It is also clear from Eqs. (10), (11) and (12) that the present

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framework is capable of accommodating general (nonlinear) elastic response and general plastic

response.

The potential , pq σ is linked to the plastic dissipation. Its role is such that inequality

(11) is satisfied for arbitrary processes. Note that we have assumed ( , )pq σ independent of

netd . From Eq. (8), it then follows that

0, , ,p

ε σ qε

net

net

dY

d

(13)

Hence, the initial (undamaged) elastic strain energy 0ε is the thermodynamic force Y

conjugate to the net damage-healing variable netd .

Characterization of Initial Elastic Strain Energy Based Hybrid Isotropic Damage Evolution

We first characterize the progressive degradation of mechanical properties of soils due to

damage by means of a simple isotropic damage mechanism. To this effect, we ascribe the notion

of equivalent tensile strain as the (undamaged) energy norm of the tensile strain tensor. The

isotropic damage mechanism is called “hybrid”, since the computation of this equivalent tensile

strain involves the principle tensile direction of total strain. Accordingly, motivated by Eq. (13),

we set

0 01: :

2ε ε C ε (14)

where :ε P ε . The ε denotes the total strain and the fourth order tensor P

denotes the

mode I positive (tensile) projection tensor with components

1

2P ε Q Q Q Qijkl ik jl il jk

(15)

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where 2 2

1 1

ˆ ; 1Q ε p p ε ε p p , pi i i i i i i

i i

H

(for 2-D simulations in this paper).

Here, εi is the i-th principal strain, pi

is the i-th corresponding unit vector in the principal

direction and ˆ εi

H is the smoothed Heaviside function.

We then characterize the state of damage in soils by means of a damage criterion

, 0d

t tg , formulated in the strain space, with the following functional form:

, 0 d

t t t tg g t R

(16)

Here, the subscript t refers to a value at the current time t R , and tg is the damage threshold

at the current time t . Note that tg will be numerically lowered due to the potential incremental

healing from the previous time step. If 0g denotes the initial damage threshold before any

loading is applied, we must have 0tg g . Here,

0g is considered to be a characteristic material

property. Condition (16) then states that damage in soils is initiated when the energy norm of the

tensile strain tensor t

exceeds the initial damage threshold 0g . For the isotropic damage case,

we define the evolution of the damage variable d and the damage threshold tg by the rate

equations

,t t td H d (17)

g (18)

where 0 is a damage consistency parameter that defines damage loading/unloading

conditions according to the Kuhn-Tucker relations

0, , 0, , 0 d d

t t t tg g (19)

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Moreover, H is Eq. (17) signifies the damage hardening function. The conditions (19) are

standard for problems involving unilateral constraint. If , 0d

t tg , the damage criterion is

not satisfied and by conditions (19), 0 ; hence, the damage rule (19) implies that 0d and

no further damage takes place. If, on the other hand, 0 ; that is, further damage (“tensile

loading”) is taking place, conditions (19) implies that , 0d

t tg . In this event, the value of

is determined by the damage consistency condition; i.e.,

, , 0 d d

t t t tg g (20)

so that tg is defined by the expression

0 , 1( , )

max , maxt s h ts t

g g g

(21)

where 1h t

g

is the reduced value for the current damage threshold due to the incremental

healing (if any) from the previous time step.

If ,t tH d in condition (17) is independent of td , the above formulation may be

rephrased as follows: let :G R R be such that /t t tH G . We shall assume that

G is monotonic. A damage criterion entirely equivalent to condition (16) is given by

, 0d

t t t tg G G g . The flow rule and loading/unloading conditions then become

,

,

d

t t

t t

t

gd g

(22)

0, , 0, , 0 d d

t t t tg g (23)

Conditions (22) and (23) are simply the Kuhn-Tucker optimality conditions of a principle of

maximum damage dissipation.

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Characterization of Initial Elastic Strain Energy Based Hybrid Isotropic Healing Evolution

Similar to the characterization of damage in previous section, we characterize the

progressive recovery of mechanical properties of soils due to healing by means of a simple

isotropic healing mechanism. We use the notion of equivalent compressive strain as the

energy norm of the compressive strain tensor. Accordingly, we set

0 01: :

2ε ε C ε

(24)

where :ε P ε . The fourth order tensor P

denotes the Mode I negative (compressive)

projection tensor with components

1

2P ε Q Q Q Qijkl ik jl il jk

(25)

where Q 1 Q .

We also characterize the state of healing in soils by using a healing criterion , 0h

t tr ,

formulated in the strain space, with the following functional form:

, 0 h

t t t tr r t R

(26)

Here, tr is the healing threshold at the current time t . Note that

tr will be numerically lowered

due to the incremental damage from the previous time step. If 0r denotes the initial healing

threshold before any loading is applied, we must have 0tr r . Further, 0r is considered as a

material property. Condition (26) then states that healing in the material is initiated when the

energy norm of the compressive strain tensor t

exceeds the initial healing threshold 0r . For the

isotropic healing case, we define the evolution of the healing variable R and the healing

threshold by the rate equations

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,t t tR Z R (27)

r (28)

where 0 is a healing consistency parameter that defines healing loading/unloading

conditions according to the Kuhn-Tucker relations

0, , 0, , 0 h h

t t t tr r (29)

Further, Z in Eq. (27) renders the healing hardening function. Conditions (29) are standard for

problems involving unilateral constraint. If , 0h

t tr , the healing criterion is not satisfied

and by conditions (29), 0 ; hence, the healing rule (27) implies that 0R and no further

healing takes place. If, on the other hand, 0 ; that is, further healing (“compressive loading”)

is taking place, conditions (29) implies that , 0h

t tr . In this event, the value of is

determined by the healing consistency condition; i.e.,

, , 0h h

t t t tr r (30)

so that tr is given by the expression

0 , 1( , )

max , maxt s d ts t

r r r

(31)

where , 1d t

r

is the reduced value for the current healing threshold due to the incremental damage

(if any) from the previous time step.

If ,t tZ R in condition (27) is independent of tR , the above formulation may be

rephrased as follows: let * :G R R be such that * /t t tZ G . We shall assume that

*G is monotonic. A healing criterion entirely equivalent to conditions (26) is given by

* *, 0h

t t t tr G G r . The flow rule and loading/unloading conditions then become

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,

,

h

t t

t t

t

rR r

(32)

0, , 0, , 0 h h

t t t tr r (33)

Net (Combined) Effect of the Hybrid Isotropic Damage and Healing

We have illustrated in the previous work (Ju et al., 2011a, 2011b, 2011c) that the

following equation for the net effect of damage and healing is physically incorrect:

1netd d R (34)

The net effect of damage and healing mechanism for new energy based model is identical to the

micromechanics-motivated scalar incremental form as expressed in Eq. (33) of previous work

(Ju et al., 2011a, 2011b).

Modification of the Drucker-Prager Model

For saturated soil, shear strength is commonly described by the Mohr-Coulomb failure

criterion, which defines shear strength in terms of the material variables ' and 'c as

' tan 'f wτ c σ uf

(35)

where fτ is the shear stress on the failure plane, 'c is the effective cohesion, wσ u is the

effective normal stress and ' is the effective angle of internal friction.

For unsaturated soil, modern experimental studies regarding the shear strength date back

to 1950s and 1960s. Inspection of Blight’s triaxial testing results (Blight, 1967) and Escario’s

direct shear test results (Escario, 1980) demonstrates two general trends in the shear strength

behavior of unsaturated soil. Firstly, the shear strength of unsaturated soil generally increases as

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net normal stress increase. Secondly, emerging from the triaxial and direct shear testing results is

that shear strength increases as applied matric suction increases. Fredlund et al. (1978)

formulated an extended Mohr-Coulomb criterion to describe the shear strength behavior of

unsaturated soil. The failure envelop is a planar surface in the stress space of the stress state

variables aσ u and

a wu u ; the shear stress τ may be written as (Lu and Likos, 2004)

' tan ' tanf a a wτ c σ u u ub

f (36)

where 'c is the cohesion at zero matric suction and zero net normal stress, au is the pore air

pressure, wu is the pore water pressure, ' is the angle of internal friction associated with the net

normal stress variable, and b is an internal friction angle associated with the matric suction that

describes the rate of increase in shear strength relative to matric suction. Some experimental data

of ', ',cb for a wide variety of soil types can be found in the literature; cf. Fredlund and

Rahardio (1993), and Lu and Likos (2004).

Furthermore, the Drucker-Prager criterion can be written in the form

1 2 2 1( , ) 0I If J J k (37)

where 1I is the first invariant of the Cauchy stress tensor, 2J is the second invariant of the

deviatoric stress tensor, and the two parameters and k are (positive) material constants.

Motivated by the discussion of Mohr-Coulomb failure criterion for unsaturated soil proposed by

Fredlund et al. (1978), Kohler and Hofstetter (2008) proposed the extension of a cap model to

describe the material behavior of partially saturated soils. Ignoring the effect of the third

invariant of the deviatoric stress tensor, the shear failure surface (Drucker-Prager model) can be

describe as follows

' '

1 2 2 1( , ) ' ' 0a wI I u uf J J k (38)

19

where '

1I is the first invariant of the net stress tensor, au is the pore air pressure,

wu is the pore

water pressure, and ( ', ', )k are material constants.

COMPUTATIONAL ALGORITHMS: TWO-STEP OPERATOR SPLITTING

In the previous section, we developed initial elastic strain energy based hybrid isotropic

damage-healing formulations based on the effective stress concept for partially saturated soil. In

this section, we focus, in detail, on the computational aspects of the proposed models within the

context of numerical method. More precisely, the attention is focused on the following local

elastoplastic-damage-healing rate constitutive equations:

,

0, , 0, , 0

,

s

t t t

d d

t t t t

t t t

t

d H d

g

g g

R Z R

ε u

0

0, , 0, , 0

ˆ

h h

t t t t

net

a a

r

r r

d d R d

d du S u

dt dt

p

εσ σ

ε

0

0

0

( ), ( )

( ), ( )

( ), 0

w

a a w

a a w

a a w

u

fu S u u associative flow rule

q u S u u plastic hardening law

f u S u u

p p

p

p

1

εσ σ 1 q

ε ε

εh σ 1 q

ε

εσ 1 q

ε

( )yield condition

(39)

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From an algorithmic standpoint, the problem of integrating the evolution equations (39)

reduces to updating the basic variables , , , , , , ,p

a wσ σ q u unetd S in a manner consistent with the

constitutive model. It is essential to realize that in this computational process the history of

strains ε ust t is assumed to be given.

Equations of evolution are to be solved incrementally over a sequence of given time steps

1, , 0,1,2...n nt t R n . Thus, the initial conditions for equations are

( ) ( ), , , , , , , | , , , , , , ,p p

a w a wσ σ q u u σ σ q u un

net net

t t n n n n n n n nd S d S (40)

In accordance with the notion of operator split, we consider the following additive

decomposition of problem of evolution into elastic-damage-healing and plastic parts.

0

0

0

0

0

0

s

d d

t t

d d

t t

h h

t t

t

H iffd

otherwise

iffg

otherwise

Z iffR

otherwise

r

1. Elastic-damage-healing part:

ε u

0

0

0

ˆ

( )

h h

t t

net

a a w

iff

otherwise

d d Rd

du S u u

dt

p

εσ 1

ε

σ 0

q 0

(41)

21

0

0

0

0

( ),

( ),

net

a a w

a a w

d

r

f du S u u

dt

du S u u

dt

p

p p

p

2. Plastic part:

ε 0

σ σ

εσ σ 1 q

ε ε

εq h σ 1 q

ε

(42)

It is noted that Eqs. (41) and (42) do indeed add up to Eq. (39) in agreement with the

notion of operator split. The formulation of an algorithm consistent with equations is based on

the following fundamental result concerning operator or split method. Given the two algorithms,

the first one is consistent with problem (41) (elastic-damage-healing predictor) and the second

one is consistent with problem (42) (return mapping corrector) In turn, the product algorithm

obtained by successive application of these two algorithms is consistent with the original

problem.

The Elastic-Damage-Healing Predictor

An algorithm consistent with problem (39), referred to as the elastic-damage- healing

predictor in the sequel, is given by the following step-by-step procedure.

Step 1: Strain update: Given the incremental displacement field 1un , the strain tensor is updated

as

1 1ε ε us

n n n (43)

22

Step 2: Compute the Mode I positive 4th

rank projection operator based on the total strain

tensor 1εn

1

1

2P ε Q Q Q Qijkl n ik jl il jk

(44)

where 2 2

1 1

1 1

ˆ ;Q ε p p ε ε p p n i i i n i i i

i i

H

(for 2-D simulations in this paper)

and ˆ εi

H is the smoothed Heaviside function.

Step 3: Compute the negative 4th

rank projection operator based on the total strain tensor

1εn

1

1

2P ε Q Q Q Qijkl n ik jl il jk

(45)

where 1 1n n

Q 1 Q

Step 4: Compute the initial (undamaged) elastic tensile and compressive strain energies,

and .

0 0

1 1 1 1

1: : ,

2ε ε C εn n n n if linear elastcity

(46)

0 0

1 1 1 1

1: : ,

2ε ε C εn n n n if linear elastcity

(47)

where :ε P ε and :ε P ε

Step 5: Update the scalar damage parameter threshold 1nd (a history variable, initially 0).

1nd denotes the net amount of scalar damage threshold after an incremental healing

takes place from the previous time step (if any).

If 0nR and 0nd , then update (lower) the scalar damage parameter threshold

n+1P

n+1P

23

1

1.0n n n

d d R (48)

Otherwise, set

1n nd d

(49)

Step 6: Compute the scalar damage predictor 1nd based on 1n

.

If 1n

at current time step is less than the initial damage threshold, no further damage is

generated. Set 1

0n

d . If 1n

is larger than the given initial damage threshold,

compute the scalar damage predictor 1nd

by using one of the following nonlinear

damage functional evaluation:

1

1 1

1

c n i

n n

n c i

dk k

k k

(50)

where c

k and i

k are material constants for damage evolution.

or

1 1 1

1

11 exp

n n n

n

A BAd B

(51)

where A and B are material constants for damage evolution.

Step 7: Check the incremental scalar damage criterion 1nd .

Compute 1 1 1n n nd d d .

1

1

1 1

, 0

;

0,

0,

Step 9

n

n

n n

no further damage d Go to

further damage Set d continued

dIf

Step 8: Compute the incremental hybrid isotropic damage predictor tensor 1Dn and the

updated hybrid isotropic non-symmetric interim damage tensor 1D̂n

1 1n nd D I

24

1 1D̂ D Dnet

n n n (53)

where I denotes the fourth-order identity tensor.

Step 9: Update the scalar healing parameter threshold 1nR : (a history variable, initially 0)

1nR denotes the net amount of healing sustained after damage takes place. If 1 0nd

and 0nR , then update the (lower) healing parameter threshold for the next time step:

1 11.0

n n nR R d

(54)

Otherwise, set

1n nR R

(55)

Step 10: Compute the scalar healing predictor 1nR based on 1n

.

If 1n

at current time step is less than the initial healing threshold, no further healing

occurs. Set 1 0nR . If 1n

is larger than the given initial healing threshold, compute

the scalar healing predictor 1nR by using one of the following nonlinear healing

functional evaluation:

1

1 1

1

c n i

n n

n c i

Rh h

h h

(56)

where c

h and i

h are material constants for healing evolution.

or

1 1 1

1

11 exp

n n n

n

A BR AB

(57)

where A and B are material constants for healing evolution.

Step 11: Check the incremental scalar healing criterion 1nR

Compute 1 1 1n n nR R R

.

25

1

1

1 1

, 0

ˆ ;

0,

0,

Step 14

n

n

n n

no further healing R Go to

further healing Set R R continueIf R

Step 12: Compute the hybrid isotropic incremental healing tensor 1Rn

1 1n nR R I

Step 13: Compute the incremental healing corrector, and update the hybrid isotropic net

(combined) damage-healing tensor (true damage measure, a history variable)

1 1 1

1 1 1

ˆ •

ˆ

H

n n n

net H

n n n

D D R

D D D

(59)

Step 14: Calculate the mean net stress and compute the associated degree of saturation based

on the previous discussions. Once the degree of saturation is known, the corresponding

Bishop’s effective stress parameter and matric suction can be obtained.

Trial (predictor) stress: By mere substitution into the potential for the stress tensor, we

obtain

0

10

1

0

1 1

1

n

n a a w

trial

n n n

trial

n n

u S u u

p

εσ 1

ε

σ σ σ

q q

(60)

The Effective Plastic Return Mapping Corrector

To develop an algorithm consistent with the plastic part of the operator split, one first

checks the loading/unloading conditions.

Step 15: Check for yielding. The algorithmic counterpart of the Kuhn-Tucker conditions is

trivially implemented in terms of the elastic-damage trial stress. One simply checks

26

1 1

0( , )

0

trial trial

n n

elastic damage healing predictor final statef

plastic return mapping

σ q

(61)

Multi-surface plasticity: In the case of plastic loading, it is necessary to determine the active

plastic surface for Drucker-Prager criterion.

Step 16: Plastic return mapping corrector. In the case of plastic loading, the predictor stresses

and internal variables are “returned back” to the yield surface along the algorithmic counterpart

of the flow generated by Eq. (42). The algorithmic construction of this flow follows a proposed

procedure which was inspired by a form of Kelley’s convex cutting plane method for non-linear

optimization, with its basic structure inherited from Newton’s method. Two fundamental

advantages of this procedure are (a) the quadratic rate of convergence towards the yield surface

and (b) the need for computing the gradient of the flow rule and hardening law are entirely

bypassed.

Step 17: Update the homogenized (nominal) stress 1σn:

1 1 1:net

n n n a a wu S u u σ I D σ 1 (62)

NUMERICAL SIMULATIONS

To demonstrate the effect of matric suction, an earth pushing process is performed. In the

numerical simulation, the modified Drucker-Prager associative multi-surface plasticity

formulation as expressed in Eq. (38) is employed to model the soil behavior for the sake of

simplicity. The associated soil properties and assumed numerical material constants are listed in

Table 1. The NMAP (Nonlinear Meshfree Analysis Program) meshfree codes are provided by

Prof. J.S. Chen’s group at UCLA.

27

The initial configuration and discretization of the earth pushing process is rendered in

Figure 6. The blade of the bulldozer is treated as a rigid body and thus represented by two

contact surfaces in black color. To model the contact between soil particles and blade surfaces,

the traditional penalty formulation is employed. A layer of soil with dimension 5 0.5 (meter) is

discretized into 101 11 1111 uniformly distributed particles. The blade as exhibited in Figure

6 is controlled to move horizontally from right to left for 5 m.

The initial degree of saturation is assumed 20% and varies linearly when the mean net

stress of soil particles is changed as discussed. The corresponding matric suction then follows the

soil water characteristic curves as presented in Figure 7 which are obtained from Eq. (2). By

setting m = 1, n = 2 and allowing varying a, three curves are generated with the matric suction

ranging from 0 ~ 440 (KPa), 0 ~ 890 (KPa) and 0 ~ 1,330 (KPa).

The initial elastic strain energy based couple elastoplastic hybrid isotropic damage-

healing model with three different ranges of matric suction is then applied to generate the earth

pushing simulations as demonstrated in Figure 8.

The histories of horizontal resulting force on bulldozer blade by considering three

different ranges of matric suction are plotted in Figure 9 and their corresponding maximum

values are listed in Table 2.

From Figure 9 and Table 2, the numerical results exhibit the trend of the effect of matric

suction. Namely, for partially saturated soils, matric suction is an important factor to influence its

mechanical behavior. More specifically, the effective stress is increased and soils become more

stiffened when the matric suction increases.

In the absence of associated experimental data, the proposed initial elastic strain energy

based coupled elastoplastic hybrid isotropic damage-healing formulations with the consideration

of matric suction for partially saturated soil models only demonstrate the salient feature of the

28

effect of matric suction. Additional future efforts are required to obtain the validations to the

numerical materials constants and parameters in the proposed formulations.

CONCLUSIONS

In this study, new initial elastic strain energy based coupled elastoplastic hybrid isotropic

damage-healing formulations have been presented for partially saturated soils in the specific

simulation of earth pushing processes. The equivalent strain based on +P and P

for the energy

criteria of damage and healing is used for the proposed models. New two-step operator split

algorithms (featuring the elastic-damage-healing predictors and the effective plastic return

mapping corrector) are presented. In addition, the effect of matric suction and the coupling of

damage and healing are proposed in an incremental form by using the predictor formula and

corrector formula as introduced. In the absence of associate experimental data, the proposed

innovative formulations and algorithms for partially saturated soils only demonstrate the effect of

matric suction. Further validations to the material parameters in each damage and healing

evolutions will be performed once the associated experiment data become available in the near

future.

ACKNOWLEDGMENTS

This work was in part sponsored by the Caterpillar Corp. under grant number UBJQ

86701:08, in part by the Faculty Research Grant of the Academic Senate of UCLA under fund

number 4-592565-19914 and in part by Bellagio Engineering. We express our gratitude to Dr.

Pai-Chen Guan for his assistance in implementing the proposed model into the NMAP

(Nonlinear Meshfree Analysis Program) meshfree code.

29

REFERENCES

[1] Alonso, E.E., Gens, A., and Josa, A. (1990). “A Constitutive Model for Partially Saturated

Soils.” Geotechnique, 40(3), 405-430.

[2] Bishop, A.W. (1959) “The principle of effective stress.” Technisk Ukeflad, No. 39.

[3] Bishop, A.W., Alpan, I., Blight, G.E., and Donald, I.B. (1960). “Factors controlling the

shear strength of partly saturated cohesive soils.” ASCE Research Conference on Shear

Strength of Cohesive Soils, University of Colorado, Boulder, CO.

[4] Blight, G.E. (1967). “Effective stress evaluation for unsaturated soils.” ASCE Journal of the

Soil Mechanics and Foundations Division, 93 (SM2), 125-148.

[5] Bolzon, G., Schrefler, B.A., and Zienkiewicz, O.C. (1996). “Elastoplastic soil constitutive

laws generalized to partially saturated states.” Geotechnique, 46(2), 279-289.

[6] Borja, R.I. (2004). “Cam-Clay plasticity. Part V: A mathematical framework for three-phase

deformation and strain localization analyses of partially saturated porous media.” Computer

Methods in Applied Mechanics and Engineering, 193(48-51), 5301-5338.

[7] Coleman, B.D., and Gurtin, M.E. (1967). "Thermodynamics with internal state variables."

Journal of Chemical Physics, 47(2), 597-613.

[8] Escario, V. (1980). “Suction-controlled penetration and shear tests.” In Proceedings of the

4th

International Conference on Expansive Soils, Denver, CO, 781-787.

[9] Escario, V., and Saez, J. (1986). “The Shear-Strength of Partly Saturated Soils.”

Geotechnique, 36(3), 453-456.

[10] Escario, V., Juca, J., and Coppe, M.S. (1989). “Strength and deformation of partly saturated

soils.” In Proceedings of the 12th

International Conference on Soil Mechanics and

Foundation Engineering, Vol. 3, Rio de Janeiro, 43-46.

[11] Fredlund, D.G., and Morgenstern, N.R. (1978). “Stress State Variables for Unsaturated

Soils.” Journal of the Geotechnical Engineering Division-ASCE, 104(11), 1415-1416.

[12] Fredlund, D.G., Morgenstern, N.R., and Widger, R.A. et al. (1978). “Shear-Strength of

Unsaturated Soils.” Canadian Geotechnical Journal, 15(3), 313-321.

[13] Fredlund, D.G. (1979). “2nd Canadian Geotechnical Colloquium - Appropriate Concepts

and Technology for Unsaturated Soils.” Canadian Geotechnical Journal, 16(1), 121-139.

[14] Fredlund, D.G., and Rahardio, H. (1993) “Soil Mechanics for Unsaturated Soils.” John

Wiley & Sons.

[15] Fredlund, D.G., and Xing, A.Q. (1994). “Equations for the Soil-Water Characteristic

Curve.¨ Canadian Geotechnical Journal, 31(4), 521-532.

30

[16] Gallipoli, D., Gens, A., Sharma, R., and Vaunat, J. (2003a). “An elasto-plastic model for

unsaturated soil incorporating the effects of suction and degree of saturation on mechanical

behaviour.” Geotechnique, 53(1), 123-135.

[17] Gallipoli, D., Wheeler, S.J., and Karstunen, M. (2003b). “Modelling the variation of degree

of saturation in a deformable unsaturated soil.” Geotechnique, 53(1), 105-112.

[18] Gens, A., and Alonso, E.E. (1992). “A Framework for the Behavior of Unsaturated

Expansive Clays.” Canadian Geotechnical Journal, 29(6), 1013-1032.

[19] Georgiadis, K., Potts, D.M. and Zdravkovic, L. (2005). “Three-Dimensional Constitutive

Model for Partially and Fully Saturated Soils.” International Journal of Geomechanics, 5(3),

244-255.

[20] Ju, J.W., Yuan, K.Y., and Kuo, A.W. (2011a). “Novel strain energy based coupled

elastoplastic damage and healing models for geomaterials – Part I: Formulations.” Int. J. of

Damage Mechanics, 1056789511407359, first published on August 17, 2011 as doi:

10.1177/1056789511407359.

[21] Ju, J.W., Yuan, K.Y., Kuo, A.W., and Chen, J.S. (2011b). “Novel strain energy based

coupled elastoplastic damage and healing models for geomaterials – Part II: Computational

Aspects.” Int. J. of Damage Mechanics, 1056789511407360, first published on August 17,

2011 as doi: 10.1177/1056789511407360.

[22] Ju, J.W., and Yuan, K.Y. (2011c) “New strain energy based coupled elastoplastic two-

parameter damage and healing models for earth moving processes.” Int. J. of Damage

Mechanics, accepted.

[23] Kohler, R., and Hofstetter, G. (2008). “A cap model for partially saturated soils.”

International Journal for Numerical and Analytical Methods in Geomechanics, 32(8), 981-

1004.

[24] Krahn, J., and Fredlund, D.G. (1972). “Total, Matric and Osmotic Suction.” Soil Science,

114(5), 339-348.

[25] Loret, B., and Khalili, N. (2000). “A three-phase model for unsaturated soils.” International

Journal for Numerical and Analytical Methods in Geomechanics, 24(11), 893-927.

[26] Lu, N., and Likos, W. (2004) “Unsaturated Soil Mechanics.” John Wiley & Sons.

[27] Mitarai, N., and Nori, F. (2006). “Wet granular materials.” Advances in Physics, 55(1-2). 1-

45.

[28] Mitchell, J.K., and Soga, K. (2005). “Fundamentals of Soil Behavior (Third Edition).” John

Wiley & Sons.

[29] Murakami, A., Setsuyasu, T., and Arimoto, S. (2005). “Mesh-free method for soil-water

31

coupled problem within finite strain and its numerical validity.” Soils and Foundations,

45(2), 145-154.

[30] Nishimura, T., Murasawa, Y., and Okami, T. (2006). “Estimating air-water hydraulic

conductivity using soil-water characteristic curve.” Proceeding s of the fourth international

conference on unsaturated soils, geotechnical special publication, 147(2), 1595-1606.

[31] Sanavia, L., Pesavento, F., and Schrefler, B.A. (2006). “Finite element analysis of non-

isothermal multiphase geomaterials with application to strain localization simulation.”

Computational Mechanics, 37(4), 331-348.

[32] Simo, J.C., Ju, J.W., Pister, K.S., and Taylor, R.L. (1988), “An assessment of the cap model:

consistent return algorithms and rate-dependent extension”, J. of Eng. Mech., ASCE, 114(2),

191-218.

[33] Vanapalli, S.K., Fredlund, D.G., Pufahl, D.E., and Clifton, A.W. (1996). “Model for the

prediction of shear strength with respect to soil suction.” Canadian Geotechnical Journal,

33(3), 379-392.

[34] Vanapalli, S.K., and Fredlund, D.G. (2000). “Comparison of empirical procedures to predict

the shear strength of unsaturated soils using the soil-water characteristic curve.” In

Advances in Unsaturated Geotechnics, Shackelford, C.D. Houston, S.L., and Chang, N.Y.,

eds., GSP No. 99, ASCE, Reston, VA, 195-209.

[35] Wheeler, S.J., and Sivakumar, V. (1995). “An elasto-plastic critical state framework for

unsaturated soil.” Geotechnique, 45(1), 35-53.

[36] Wheeler, S.J., Gallipoli, D., and Karstunen, M. (2002). “Comments on use of the Barcelona

Basic Model for unsaturated soils.” International Journal for Numerical and Analytical

Methods in Geomechanics, 26(15), 1561-1571.

[37] Wheeler, S.J., Sharma, R.S., and Buisson, M.S.R. (2003). “Coupling of hydraulic hysteresis

and stress-strain behaviour in unsaturated soils.” Geotechnique, 53(1), 41-54.

[38] Wu, C. T., Chen, J.S., Chi, L., and Huck, F. (2001). "Lagrangian meshfree formulation for

analysis of geotechnical materials." Journal of Engineering Mechanics, 127, 440-449.

[39] Zapata, C.E., Houston, W.N., Houston, S.L. and Walsh, K.D. (2000) “Soil-water

characteristic curve variability.” Advanced in Unsaturated Geotechnics, geotechnical

special publication No.99, 84-124.

32

Table Captions

Table 1. The associated soil properties and assumed numerical material constants

Table 2. The maximum horizontal resulting force on the bulldozer blade

Figure Captions

Figure 1. The sample plots of Eq. (2) with n = 2 and m = 1 (with varying a)

Figure 2. The sample plots of Eq. (2) with a = 100 and m = 1 (with varying n)

Figure 3. The sample plots of Eq. (2) with a = 100 and n = 2 (with varying m)

Figure 4. Various forms for the effective stress parameter as functions of the degree of

saturation

Figure 5. A schematic plot of the relationship between the degree of saturation and the mean net

stress

Figure 6. The initial configuration and discretization of the earth pushing processes

Figure 7. Three soil-water characteristic curves for the earth pushing simulations

Figure 8. The damage contours and soil deformations for earth pushing simulations

Figure 9. The horizontal resulting forces on the bulldozer blade for different ranges of the matric

suction

33

Table 1. The associated soil properties and assumed numerical material constants

Young’s modulus 24.7 (MPa)

Poisson’s ratio 0.35

Density 3 31.88 10 (kg / m )

Cohesion 0.19 (MPa)

Lame constants 7

7

2.13457 10 (Pa)

9.14815 10 (Pa)

Yield stress 51.88312 10 (Pa)

S in Eq. (1) 1k in Eq.(3)

(m, n, a) in Eq. (2) m = 1, n = 2, a varies so that the matric suction

ranges from 0~440 KPa, 0~890 KPa

and 0~1130 KPa

c ini max min,S ,S ,Sσ c ini max min2000 KPa , S 0.2, S 1.0, S 0.1σ

in Eq. (38) 0.5

Table 2. The maximum horizontal resulting force on bulldozer blade

Initial elastic

energy based

hybrid isotropic

damage-healing

models

No matric suction

effect

Matric suction

ranges from

0 ~ 440 (KPa)

Matric suction

ranges from

0 ~ 890 (KPa)

Matric suction

ranges from

0 ~ 1130 (KPa)

Max. horizontal

resulting force on

bulldozer blade

(N)

3.5681e+004

4.7764e+004

5.3514e+004

5.1145e+004

Percentage 100% 133.86% 149.98% 143.34%

34

Figure 1. The sample plots of Eq. (2) with n = 2 and m = 1 (with varying a )

Figure 2. The sample plots of Eq. (2) with a = 100 and m = 1 (with varying n)

35

Figure 3. The sample plots of Eq. (2) with a = 100 and n = 2 (with varying m)

Figure 4. Various forms for the effective stress parameter as functions of the degree of

saturation

36

Mean net stress

Tensile

Degree of Saturation

Smax

Sini

Smin

0Compressivec t

Figure 5. A schematic plot of the relationship between the degree of saturation and the mean net

stress

Figure 6. The initial configuration and discretization of the earth pushing processes

37

Figure 7. Three soil-water characteristic curves for the earth pushing simulations

38

Figure 8. The damage contours and soil deformations for earth pushing simulations

39

Figure 9. The horizontal resulting forces on the bulldozer blade for different ranges of the matric

suction