Elasto-Plastic Modeling of Saturated
and Non-Saturated Residual Soil with
Parameters Optimization
João Paulo Laquini
Department of Civil Engineering, Federal University of
Viçosa, Minas Gerais, Brazil
[email protected]
and
Roberto Francisco de Azevedo
Professor, Department of Civil Engineering, Federal
University of Viçosa, Minas Gerais, Brazil
[email protected]
and
Rodrigo Martins dos Reis
Department of Civil Engineering, Federal University of
Viçosa, Minas Gerais, Brazil
[email protected]
ABSTRACT
This paper deals with constitutive modeling of soil in
saturated and non-saturated conditions, including the
volumetric decrease during wetting stress-paths (collapse).
It presents an extended effective stress principle that,
together with small adaptations made in an elasto-plastic
model developed for dry and saturated conditions, allows
for modeling soil behavior in non-saturated conditions.
Besides briefly discussing theoretical aspects, the paper
presents results of triaxial tests performed on a residual soil
in saturated and non-saturated conditions, shows how to
obtain new parameters included in the model, presents a
method to find optimum parameters and shows
comparisons between experimental and analytical results
obtained with the model.
KEYWORDS: Elasto-plastic modeling; collapse; extended
effective stress principle;
parameters optimization
non-saturated
condition;
INTRODUCTION
Saturated soil mechanics is based on Terzaghi’s effective stress principle. The
extension of this principle to non-saturated soils was formulated by Bishop
(1959),
' = ( - ua) +            (1)
where ' is the effective stress,  is the total stress,  = ua – uw is the matric
suction, ua is the air pore-pressure, uw is the water pore pressure and  is a
material parameter.
However, the extended effective stress principle, equation (1), does not describe
satisfactory the volume change behavior due to wetting (collapse phenomenon)
(Jennings et al., 1962). Due to this limitation, most of the constitutive models
developed to represent the behavior of non-saturated soil do not use effective
stresses (Wheeler et al., 1995). However, effective stresses are not abandoned
when they alone are not able to model dilatancy of soils in dry or saturated
conditions, which turn out to be feasible when effective stresses are used
together with elasto-plasticity theories.
Kogho et al. (1993) and Moderassi et al. (1995) formulated extended effective
stress concepts that, together with elasto-plasticity theories, are able to model
important aspects of non-saturated soil behavior, including the collapse
phenomenon. The approach is very interesting because it allows modeling soil
behavior in any conditions (dry, non-saturated and saturated) within the same
framework of the effective stress concept.
This paper, based on the similar ideas, presents results of an elasto-plastic model
adapted to represent the behavior of saturated and non-saturated soils using an
extended effective stress concept.
MATERIAL AND METHODS
Material
The soil was selected from a weathering profile commonly encountered in the
region of Viçosa city, state of Minas Gerais, Brazil. Its profile has an A horizon
approximately 0,5m deep and a red-yellow latosoil layer, B horizon, with a
thickness of approximately 6m. The B horizon corresponds to the limits of the
mature residual soil that does not keep traces with the parent rock and,
consequently, has a homogeneous and isotropic appearance. The C horizon, a
young gneiss residual soil which is different from the mature soil, keeps the
characteristics of the parent rock. This layer is very heterogeneous and
anisotropic (Reis, 2004).
The soil chosen for this study was collected from the C horizon 12m below the
slope surface. Its main characteristics are presented in Table 1. Samples of the
young residual soil clearly present foliation planes similar to the ones of the
parent rock (Reis, 2004).
Table 1. Characterization test results.
Grain-Size (%)
Sand
(kN/m³)
Silt
Coarse
Medium
6
24
Clay
LP
LL
w
(%)
(%)
(%)
s
Fine
20
45
5
38
23
17
2.67
The laboratory testing program consisted of drained triaxial tests and
conventional hydrostatic compression tests on saturated and non-saturated
samples following conventional and non-conventional stress paths (Reis, 2004).
Drained conventional triaxial compression tests on saturated samples were done,
shearing samples in three orthogonal directions: vertical, parallel and
perpendicular to the foliation planes. Tests were executed with confining
pressure varying from 50 to 400 kPa. In addition, conventional hydrostatic
compression tests were performed on saturated and non-saturated samples (Reis,
2004).
This paper presents results of experiments carried out with the young residual
soil sheared in the direction vertical to the foliation planes (Reis, 2004).
Methods
The proposed extended effective stress concept is:
' = - ueq
(2)
In this expression, ueq is the equivalent pore-pressure defined as:
u eq  u w ,



,
u eq  
a  b

if    e
(3)
if    e
where e is the air entry suction and a and b are material parameters. Figure 1
shows the variation of ueq with uw for a and b equal to 0.419 and 0.00403,
respectively, and e equal to 50kPa, values corresponding to the ones of the
young residual soil.
The hyperbolic function used to relate ueq and  was selected due to simplicity
and accuracy to fit the soil modeled in this paper. However, other functions may
be utilized, depending on the soil behavior (Reis, 2004, and Kogho et al., 1993).
Equivalent water pore-pressure (kPa)
1000
-1000
800
600
400
200
0
-500
0
500
1000
-200
-400
Water pore-pressure (kPa)
Figure 1. Equivalent pore-pressures values.
The model described in this paper presents modifications in Lade and Kim
(1988a,b,c) elasto-plastic model to rend it able to deal with the behavior of non
saturated soil by using the extended effective stress concept. Besides being a
model that incorporates effects of the three principal stresses, Lade and Kim
model has being demonstrating to model reasonably well important aspects of
different soil behavior in dry and saturated conditions (Azevedo et al., 1996,
Lade and Kim, 1995, and Lade, 1990). However, the extended effective stress
concept and the ideas described herein can be used with other constitutive
models.
As it is usual in elasto-plasticity with hardening, Lade and Kim model divides
the increment of strains, {d}, in two parts:
{d }  {d e }  {d p }          (4)
in which {de} and {dp} are the increment of elastic and plastic strains,
respectively.
The increment of elastic strain is calculated using Hooke’s law with Young’s
modulus E calculated by:
 I '  2
J' 
E  M pa  1   R 22 
 pa 
pa 



(5)
where pa is the atmospheric pressure, I’1 is the first invariant of the stress tensor,
J’2 is the second invariant of the deviatoric stress tensor, M and  are material
parameters, and
R6
1
1  2
(6)
In above expression, υ is the Poisson’s ratio, also a material parameter.
The increments of plastic strain are calculated by the non-associated flow rule:
 g p 
{d p }  d 

  ' 
(7)
in which, gp is the plastic potential function:
 I '3 I ' 2
  I'
g p ({ '})   1 1  1   2   1
 I'
 p
I '2
3

 a





(8)
Figure 2 shows aspects of the plastic potential function for the young residual
soil.
gp = 150
1
1
1
2
gp = 100


gp = 50
3
Hydrostatic axis
2 2
√
2

(a) Octahedral plane view.
(b) Contours in Rendulic plane.
Figure 2. Plastic potential surface for the young residual soil.
Since this function is homogeneous of order  (Wylie, 1980), then:

 g p 

{ '}T 
   gp


'




(9)
Multiplying both sides of the flow rule by { '}T and substituting this expression,
the result is:
d 
dW p
(10)
 gp
In the above expressions, I’2 and I’3 are the second and third invariants of the
stress tensor, Wp is the plastic work, 2 and  are material parameters. 1 is
calculated by:
 1  0.00155 m1.27
(11)
where m is a material parameter defined by the failure criteria.
The Khun-Tucker conditions are:
F ({’}, Wp) ≤ 0
(12)
d  0
(13)
F  d  0
(14)
According to these conditions, if F ({’}, Wp) < 0, d must be equal to zero,
therefore only elastic strains will occur. And, if F ({’}, Wp) = 0, d will be
greater than zero and plastic strains will happen. So, when plastic strains occur,
F ({’}, Wp) must be always equal to zero, thus dF will also be null.
The yield function, loccii of points with the same values of plastic work, is given
by:
F {' }, W p ,   f ' ({' })  fWp , (Wp, ) 
 W p  k 

 f ' ({' })  

 D pa 
1

0
(15)
and its aspects for the young residual soil are shown in Figure 3.
500
fWp, = 20
 1 400
1
fWp, =15
300
fWp, =10
200
2
3
Hydrostatic
axis
100
0
0
100
200
300
400
500
600
√2  3
(a) Octahedral plane view.
(b) Contours in Rendulic plane.
Figure 3. Yield surface for the young residual soil.
In above expression:
 I '3 I ' 2   I '
f ' ({ '})   1 1  1   1
 I'
I ' 2   pa
3

h

  eq ;


(16)
k is a material parameter;

q
p
;
h
(17)
S
;
1  (1   )S
(18)
and
S
fr
1
.
(19)
In the above expression, S is the stress level which is equal to zero at the
hydrostatic axis and equal to one at failure.
 I '3

f r   1  27 
 I'

 3

 I '1

p
 a
m

  1


(20)
defines the failure criteria.
In equation (18), q has the same limits of S. On other hand, p,  and 1 are
material parameters.
At the hydrostatic axis, equation (15) becomes:
 I'
(27 1  3)  1
 pa
 W  k

   p
 Dp
a


h




1

(21)
Supposing that during hydrostatic compression:
 I' 
W p  k  C pa  1 
 pa 
p
(22)
Combining these two last equations leads to:
D
C
(23)
(27 1  3) 
where C is a material parameter.
From equation (15), the plastic work may be written as:

W p  f ({ '}) D pa  k
'
(24)
Therefore, the plastic work increment is equal to:
dW p 
W p
f 
df  
W p

  D p f ({'})
a '
d 
1
(25)
df   k d
Substituting this result in equation (10):
d 
 D p a f ' ({ '})
 1
 gp
df  k d
(26)
During a wetting stress path, df = 0. However, d  0 because d  0 . Thus,
depending on the soil behavior and, consequently, the soil parameters, the
volumetric plastic strain may model a collapsive behavior.
In summary, the proposed elasto-plastic constitutive model needs the parameters
shown in Table 2 to deal with soils in any degree of saturation. Besides a and b
that are in fact failure parameters, only one new parameter (k) was incorporated
to model the non-saturated behavior.
Model Calibration
To obtain the model parameters for saturated or dry conditions (conventional
parameters) a minimum of three drained triaxial conventional tests and a
hydrostatic compression test performed with saturated or dry soil samples is
needed. The procedure is straightforward and well described previously by Lade
and Kim (1988a,b,c). Therefore, this paper focuses on how to obtain the
additional “non-saturated parameters”.
Parameters a and b are obtained by the following procedure. Knowing failure
parameters m and 1 and the total stress at failure, 1rup, 2rup,3rup, the failure
criteria, equation (20), is re-written as equation (22) bellow and numerically
solved to obtain ueq.


( 1rup   2rup   3rup  3u eq ) 3

 27               
rup
rup
rup
 (

 1  3u eq )( 2  3u eq )( 3  3u eq )

  1rup   2rup   3rup  3u eq
.

pa

m

   0
1


(27)
Using, at least, two different triaxial shearing tests with different suction values,
it is possible to obtain two suction values corresponding to two ueq values that
satisfy equation (27). Consequently, values of parameters a and b are obtained
using equation (3) twice.
The values of parameters C and p are obtained with the hydrostatic compression
test results for the soil in saturated conditions. With these parameters known, any
point of an unsaturated hydrostatic compression test with a constant suction
value, , may be used to find parameter k with the equation (28):
p
k
 I '1 
 W
p

p
 a
C pa 

(28)
Single optimization method
After the material parameters have been found from model calibration,
comparisons between experimental and analytical results are made to evaluate its
agreement. However, the material parameters obtained do not guarantee the best
adjustment between experimental and analytical results. It is possible to improve
adjustment by using an optimization method which tries to find the best material
parameters.
As optimization methods are usually difficult to be implemented and sometimes
avoided, an expeditious and single optimization method was developed into
Excel© software.
The first step is to determine an objective function that will be minimized. The
objective function used here is:
 np 
2
exp
exp 

(
P
,

)




d
j
1
j
d
j


nt 


 (P)    i 1

np
2
i 1
  exp 
 dj

i 1

exp
exp
  v j (P, 1 j )   v j  

 i 1

np
2
  exp 

 vj 

i 1
np
(29)
2
where P = {, m, 1, , 2, C, p, h, , a, b, k} is the material parameters vector,
nt is the number of tests utilized, np is the number of points in each test, dj
(P,1jexp) and vj(P,Ijexp) are the calculated deviatoric stress and volumetric strain,
respectively, djexp, Ijexp and vjexp are the experimental deviatoric stress, axial
strain and volumetric strain, respectively.
The optimization procedure requires a number of changes to be made within the
selected material parameter, keeping the other parameters constant, until the
minimum objective function result is found. This procedure can be done within
the Excel© program using the function goal seek. A goal value is informed to the
objective function result. A material parameter is chosen and the function goal
seek changes its value until the goal value is achieved. The procedure is applied
to other material parameters until all of them are optimized. The optimization is
guaranteed when the objective function result converges. This optimization
procedure reveals the sensitive material parameters.
RESULTS AND DISCUSSIONS
Table 2 presents the model parameters obtained with the methodology discussed
previously for the young gneiss residual soil (Reis, 2004). It also shows the
optimum parameters which were found after the optimization method was
performed.
The parameter ψ2 did not change with the optimization. The irreversibility
conditions  > 0 and 2 > - (271 + 3) was satisfied. The objective function
result was reduced by 62% when the parameters were optimized. The elastic
parameter M was the most sensitive, reducing by 58.6% the objective function
result. The parameters a, , m and h reduced the objective function by 2.5%,
1.4%, 1.3% and 1.0%, respectively. The other parameters produced a reduction
smaller than 1.0% in the objective function result.
Table 2. Model parameters.
Parameters
Optimization____
Before
M
217.270

0.238

0.200
m
0.309
1
31.623
After
140.197
Elastic
0.248
0.192
Failure
0.302
Saturated or
32.287
Dry Conditions
1.735

Plastic
Plastic
1.731
Potential
2
-2.90
Hardening
C
0.000620
Function
p
1.754
Yield
h
0.784
Function

0.2046
a
0.597
b
0.00407
k
0.038
-2.90
0.000617
1.745
0.754
0.2047
Non-Saturated
0.419
Failure
Conditions
0.00403
Plastic Work
0.049
Figure 4 presents comparisons between experimental and optimum analytical
results for hydrostatic compression tests performed on saturated and unsaturated
samples with suction equal to 80 kPa and 320 kPa. The agreement was
reasonable. However, the model response of this test was strongly influenced by
the stress chosen at the start of the model analysis.
Figure 5 presents comparisons between experimental and analytical results for
hydrostatic compression tests on an unsaturated sample with suction equal to 320
kPa. Firstly, net hydrostatic stress was increased till 600 kPa. At this point, a
wetting path was followed in which the net hydrostatic stress was kept constant
while suction decreased from 320 kPa till zero (saturation). Figure 5 shows that
the analytical model predicted a decrease in volume (collapse) similar to the
experimental one.
1200
Lab 
p = 80 kPa
Lab 
p = 320 kPa
1000
 - ua (kPa)
Lab 
p = 0 kPa
Model 
p = 0 kPa
Model 
p = 80 kPa
800
Model 
p = 320 kPa
600
400
200
0
0
2
4
6
8
 v (%) 10
Figure 4. Saturated and unsaturated hydrostatic compression tests.
1200
Lab 
p = 320 kPa
Lab 
p = 0 kPa
1000
Model 
p = 0 kPa
 - ua (kPa)
800
Model p
= 320 kPa
and Collapse
600
400
200
0
0
2
4
6
8
 v (%) 10
Figure 5. Hydrostatic compression tests plus collapse.
Figure 6 presents comparisons between experimental and analytical results for
drained conventional triaxial tests on saturated samples, with confining pressure
equal to 50 and 200 kPa. The agreement was strong.
450
400
 1 -  3 (kPa)
350
300
250
200
Lab 50 kPa
150
Lab 200 kPa
100
Model 50 kPa
50
Model 200 kPa
0
0
2
4
6
8
 1(%)
10
0
2
4
6
8
10
0.0
 v (%)
0.5
1.0
1.5
2.0
2.5
Figure 6. Drained conventional triaxial compression tests on saturated samples.
Figure 7 presents comparisons between experimental and analytical results for
drained conventional triaxial tests on saturated and unsaturated samples with
confining pressure equal to 200 kPa. Suctions on the unsaturated tests were equal
320 kPa. Although at the beginning, the model behaved stiffer than the
experimental results, in general the agreement can be considered good,
especially towards the end of the tests, close to failure.
700
 1 - 3 (kPa)
600
500
400
300
Lab 
p = 0 kPa
200
Lab 
p = 320 kPa
Model 
p = 0 kPa
100
Model 
p = 320 kPa
0
0
2
4
6
8
 1 (%) 10
0
2
4
6
8
10
0.0
0.5
 v (%)
1.0
1.5
2.0
2.5
3.0
3.5
Figure 7. Suction controlled conventional triaxial compression tests with total
confining pressure equal to 200 kPa on non-saturated samples.
CONCLUSIONS AND DISCUSSION
This paper dealt with a unified approach to model dry, non-saturated and
saturated soil behavior using an extension of the effective stress principle and an
elasto-plastic model. Further, it presents an expeditious and single optimization
method which can be easily implemented into Excel© program. The main
conclusions are:
1. The agreement between the experimental and numerical results given by the
constitutive model was strong in saturated, non-saturated and during collapse.
2. The parameters were mostly obtained with saturated test results. Only three
new parameters were included in the elasto-plastic model to account for nonsaturated behavior.
3. The unified approach allows the use of the effective stress concept for any soil
conditions (dry, unsaturated and saturated).
4. The function used to model hardening for suction was simple, as it was linear.
Other functions and elasto-plastic constitutive models may be utilized in
conjunction with the extended effective stress concept presented herein.
5. The optimization method developed was easily implemented and calculated
the best parameters for adjusting experimental and analytical results in an
expeditious way.
ACKNOWLEDGEMENTS
The authors would like to thank the CAPES for the doctorate fellowship given to
the first author and the support received from Post Graduate Program of Civil
Engineering Department of Federal University of Viçosa.
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