1

Explicit Formulation of At-Rest Coefficient and Its Role in 1

Calibrating Elasto-plastic Models for Unsaturated Soils 2

3

Xiong Zhang1, Eduardo E. Alonso

2, and Francesca Casini

3 4

5

ABSTRACT 6

Normally, suction-controlled triaxial tests are used to characterize soil behavior in 7

constitutive modeling of unsaturated soils. However, this type of tests requires sophisticated 8

equipment and is time-consuming. This has been one of the major obstacles to the 9

implementation and dissemination of unsaturated soil mechanics beyond the research context. 10

In contrast to suction-controlled triaxial tests, the suction-controlled oedometer test 11

requires simpler equipment and a shorter testing period. Oedometer tests represent the at-rest 12

earth pressure (K0) condition, which is an important stress state in any simulation. The major 13

disadvantage of the oedometer test is that its lateral stress is controlled by the condition of zero 14

lateral strain and remains unknown during the testing process. At present, no well-established, 15

simple, and objective methods are available that take advantage of oedometer test results for 16

constitutive modeling purposes. 17

This paper derives an explicit formulation of the at-rest coefficient for unsaturated soils 18

and develops an optimization approach for simple and objective identification of material 19

parameters in elasto-plastic models for unsaturated soils using the results from suction-controlled 20

oedometer tests. This is achieved by combining a modified state surface approach (MSSA), 21

1 Associate Professor, Department of Civil and Environmental Engineering, University of Alaska Fairbanks,

Fairbanks, AK 99775, USA (xzhang11@alaska.edu) 2 Professor, Department of Geotechnical Engineering and Geosciences, Building D2, Technical University of

Catalunya, Jordi Girona 1-3, 08034 Barcelona, Spain 3 Assistant Professor, DICII Dipartimento di Ingegneria Civile e Ingegneria Informatica, Università di Roma Tor

Vergata, 00133 Roma, Italy (francesca.casini@uniroma2.it)

2

recently proposed to model the elasto-plastic behavior of unsaturated soils, with the quasi-1

Newton method to simultaneously calibrate all parameters governing virgin behavior in elasto-2

plastic models. The Barcelona Basic Model (BBM) is used to demonstrate the application of the 3

proposed explicit formulation and calibration method. Results predicted using obtained 4

parameters are compared with laboratory test results for the same stress paths in order to evaluate 5

the simplicity and objectivity of the proposed method. 6

7

Keywords: Unsaturated soils, elasto-plastic, suction-controlled oedometer tests, K0 conditions, 8

state boundary surface, Barcelona Basic Model 9

10

3

INTRODUCTION 1

Unsaturated soils often exhibit irrecoverable (elasto-plastic) behavior when subjected to 2

loading and wetting or drying cycles. Alonso et al. [1] proposed the first elasto-plastic model for 3

unsaturated soils, which later was called the Barcelona Basic Model (BBM) [2]. The BBM 4

successfully explained many features of unsaturated soils and received extensive acceptance. 5

Since the 1990s, many elasto-plastic models have been developed [3–26, 64–70]. Review of 6

elasto-plastic models for unsaturated soils can be found in several references [7, 10, 27, 28, and 7

29]. 8

At present, most researchers use the results from suction-controlled triaxial tests to 9

develop and calibrate their models. However, suction-controlled triaxial tests require 10

sophisticated and therefore costly testing equipment to accurately measure the volume change of 11

unsaturated soil specimens during triaxial testing. Often a double-cell system proposed by 12

Bishop and Donald [30] is used, although other methods are also available. Such a system 13

typically costs more than $120,000. Due to the requirement for relatively large specimen heights 14

and low unsaturated permeability, it is time-consuming to perform a suction-controlled triaxial 15

test. A single constant suction triaxial test often takes one to three months, and it is not 16

uncommon to spend two to three years in full characterization of the elasto-plastic behavior of an 17

unsaturated soil (e.g., [31–33]). As a result, implementation of the principles of unsaturated soil 18

mechanics cannot be justified for routine engineering projects. This has been one of the major 19

obstacles to the application and dissemination of unsaturated soil mechanics beyond the research 20

context. 21

The oedometer test is classical in soil mechanics to simulate the in-situ at-rest stress state 22

for obtaining parameters for the calculation of consolidation settlements and for assessing the 23

4

stress history of soils. Compared with suction-controlled triaxial tests, the suction-controlled 1

oedometer test requires much simpler equipment and is much easier to perform [34–37]. Since 2

no lateral deformation occurs, the volume change of the soil specimen can be directly derived 3

from the vertical displacement, which can be easily measured using a linear variable differential 4

transformer (LVDT). Due to the smaller thickness of the specimen (normally around 25 mm), the 5

time needed for completing a test is much less. For single-sided drainage, the oedometer test is 6

expected to be at least nine times more efficient than the suction-controlled triaxial test (which 7

typically has heights of 75 mm or more). 8

The major disadvantage of the oedometer test is that its lateral stress is controlled by the 9

condition of zero lateral strain and remains unknown during the testing process. At present, there 10

are no well-established, simple, and objective methods for directly using the results from 11

oedometer tests for constitutive modeling purposes. Most researchers only use oedometer test 12

results to validate their models, not to develop constitutive models. Since the oedometer test 13

represents the at-rest earth pressure K0 (defined as the ratio of horizontal to vertical stress) 14

condition, which is an important stress state in any simulation (e.g., [54]), it is highly desirable to 15

have an approach that takes advantage of oedometer test results at the model development stage. 16

This paper derives an explicit formulation of the at-rest coefficient for unsaturated soils. 17

Based on the explicit formulation, a method is developed to analyze the results from suction-18

controlled (constant suction) oedometer tests for constitutive modeling purposes. This is 19

achieved by combining a modified state surface approach (MSSA), recently proposed to model 20

the elasto-plastic behavior of unsaturated soils, with the quasi-Newton method to simultaneously 21

calibrate all parameters governing virgin behavior in elasto-plastic models. The BBM is used as 22

an example to demonstrate the role of the proposed explicit formulation in model calibration, 23

5

although the proposed method can be used for calibrating other constitutive models as well. 1

Results predicted using obtained parameters are compared with laboratory test results for the 2

same stress paths to evaluate the simplicity and objectivity of the proposed method. 3

4

BACERLONA BASIC MODEL: AN OVERVIEW 5

Although since the 1990s, many elasto-plastic models have been developed for 6

unsaturated soil, Sheng et al. [21] concluded that, from the elasto-plastic theory point of view, all 7

existing elasto-plastic models have a similar framework, and can be considered variants of the 8

BBM. Gens et al. [29] concluded that most existing elasto-plastic models have kept the same 9

core of basic assumptions as the BBM, and sought to improve some of the BBM’s limitations. 10

The BBM is one of the most widely used elasto-plastic models for unsaturated soils [38]. This is 11

the reason why in this paper, the BBM is selected to demonstrate the application of the method 12

proposed in this paper. 13

The BBM is developed using two independent stress state variables with the following 14

definitions: 15

1 3 1 3

2 2 =

3 3

a a

a

u up u

(1) 16

1 3 1 3 a aq u u (2) 17

a ws u u (3) 18

where 1, 2, and 3 are the total principal stresses, ua is the pore air pressure, uw is the pore 19

water pressure, s is matric suction, p is mean net stress, and q is deviator stress. 20

The total strain is the sum of the elastic and plastic strain. The volumetric and shear strain 21

increments can be calculated as follows: 22

6

e p

v v vd d d (4) 1

e p

q q qd d d (5) 2

where stands for strain, the superscripts e and p represent elastic and plastic strains, 3

respectively, and the subscripts v and q stand for volumetric and deviator strains, respectively. 4

In the elastic zone, the volumetric and shear strains are calculated as follows [1]: 5

( )

e sv

at

dp dsd

v p v s p

(6) 6

2(1 )

3 9(1 2 )

e

q

dqd dq

G vp

(7) 7

where v = specific volume of soil; = slope of the unloading-reloading line associated with 8

changes in the mean net stress; s = slope of the unloading-reloading line associated with 9

changes in soil suction; = Poisson’s ratio, and G = shear modulus of the soil. 10

For isotropic compression tests at constant suction, s, the BBM adopted the following 11

equations to predict soil behavior at virgin states: 12

( ) lnc

pv N s s

p (8) 13

where 0 1 exps r s r , and pc = reference stress. 14

A loading collapse (LC) yield curve, which is the boundary between elastic and elasto-15

plastic behavior under isotropic conditions, was derived as follows: 16

(0) / ( )*

0 0

s

c c

p p

p p

(9) 17

7

where *

0p = preconsolidation pressure in saturated conditions, p0 = apparent preconsolidation net 1

mean stress at a certain suction, r = parameter controlling the slope of the virgin compression 2

line, and = parameter that controls the slope of the virgin compression line for s0. 3

The above equations were used to describe soil behavior under isotropic stress 4

conditions. The BBM was then extended into triaxial stress states by assuming that the yield 5

curve at constant suction s is an ellipse in the p-q plane as follows: 6

2 2

0( )( ) 0q M p ks p p (10) 7

where M = slope of theoretical critical state line, and k = parameter describing the increase in 8

cohesion with suction. The hardening law of the BBM is given as follows: 9

*

0

*

0

0p

v

dpd

v p

. (11) 10

A non-associated flow (dilatancy) rule was used to calculate the deviatoric plastic strains 11

as follows: 12

2

0

2 1

(2 )

p

s

p

v

d q

d M p ks p d

(12) 13

where d is the dilatancy. In the original BBM [1], it was suggested that is chosen in such a way 14

that the flow rule predicts zero lateral strain for stress states corresponding to Jaky’s [39] K0 15

values 0 1 sin ' 6 2 / 6K M M . Alonso et al. [1] derived the following expression 16

for 17

9 3 1

9 6 1 / 0

M M M

M

. (13) 18

8

Two assumptions are made to derive Equation (13): (1) the elastic shear strain is small 1

and negligible, and (2) for saturated and unsaturated conditions, the K0 consolidation line shares 2

the same slope and is a constant, as shown in Figure 1a. As is evident from the following 3

discussions, neither of the two assumptions holds true for unsaturated soils. Figure 1b is used to 4

demonstrate why assumption (2) is problematic. Since the equation used to calculate deviatoric 5

strain in the elastic zone (Equation 7) is different from the equation used for the plastic zone 6

(elastic and plastic components given by Equations 7 and 12 should be added), it is expected that 7

the K0 consolidation line in the elastic zone (AB in Figure 1b) has a slope different from that in 8

the plastic zone (CD in Figure 1b). It is also expected that soil behavior changes continuously. 9

As a result, C to D should be a smooth curve instead of a straight line, with constant slope. 10

Furthermore, it is difficult to imagine that one can start an oedometer test from point A in Figure 11

1b, which implies tensile stress. Instead, an oedometer test most likely starts from point A’ with 12

zero confining pressure. The corresponding stress path should be schematically shown by 13

A’B’C’D’, as shown in Figure 1b and as depicted by Alonso et al. [40]. Consequently, Equation 14

(13) is not used in this paper. Instead, it is proposed that be used as an additional constant for 15

the “modified” BBM. 16

17

EXPLICIT FORMULATION OF THE AT-REST COEFFICIENT OF EARTH 18

PRESSURE FOR UNSATURATED SOILS 19

This section derives the explicit formulation of the at-rest coefficient of earth pressure 20

(K0) for unsaturated soils. Under K0 conditions, lateral strain is equal to zero, which means that 21

in an oedometer test, for any increase in vertical stress, 22

2 3 0d d . (14) 23

9

As a result, total strain is equal to vertical strain: 1

1 2 3 1vd d d d d . (15) 2

The deviatoric strain is defined as follows: 3

1 3 1

2 2 2

3 3 3s vd d d d d . (16) 4

Thus, one has 5

0 0

3

2

e p

v v v

e p

s s sK K

d d d

d d d

. (17) 6

Under elastic conditions, the plastic components are equal to zero. For the BBM, 7

e sv v

at

dp dsd d

v p v s p

(18) 8

2(1 )

3 9(1 2 )

e

s s

dqd d dq

G vp

. (19) 9

Substituting Equations (18) and (19) into equation (17), one has 10

(1 )

3(1 2 )

s

at

dq dp dsp p s p

. (20) 11

Substituting Equations (1) and (2) for Equation (20) and expressing 3 au in terms of 12

1 au and suction s, one has [55] 13

3 1

1 2

1 1

sa a

at

pd u d u ds

s p

. (21) 14

In the elasto-plastic region, for the BBM, the plastic volumetric strain is given by the 15

hardening laws, which is related to the expansion of the yield surface as follows: 16

10

*

0

*

0

0 1p

v

dp f f fd dp dq ds

v p v p q s

, (22) 1

where f is related to the yield surface in the BBM as expressed in Equation (10) and 2

2

2ln ln 0 lnc c

qf s p p p

M p ks

. 3

Accordingly, 4

22 2

2 2

M p ks qsfAB

p p ksq M p ks p

(23) 5

2 22 2

sfq Aq

q q M p ks p

(24) 6

2 2

2 22 2

2

2

0 1 exp ln ln

0 1 exp ln ln

c

c

fC

s

s q k qr s p p

M p ksq p ks M p ks p

sAq kr s p

p ks AM p ks

(25) 7

where

2 2

sA

q M p ks p

and

22 2M p ks q

Bp ks

. 8

In the elasto-plastic region, for the BBM, the plastic deviatoric strain is given by the 9

dilatancy rule (d) in Equation (12). Substituting Equations (1), (2), (12), (18), (19), and (22) into 10

Equation (17) and expressing 3 au in terms of 1 au and suction s, one has 11

11

1

3

2 2

3 3

22

3 3

sa

at

a

EAD EB d u EC ds

p s pd u

EAD EB

p

, (25) 1

where 2(1 )

3(1 2 )D

p

and

22 2

6 2

q p ksE

M p ks q

. 2

Equation (25) indicates the relationship between the changes in vertical stress and suction 3

and the lateral stress under K0 conditions. For constant suction oedometer tests, ds = 0, Equation 4

(25) becomes: 5

6

3 1 0 1

2

3 3

22

3 3

a a a

EAD EB

pd u d u K d u

EAD EB

p

(26) 7

or 8

3

0

1

2

3 3

22

3 3

a

a

EAD EB

pd uK

d u EAD EB

p

. (27) 9

Equation (27) is the explicit formulation of the at-rest (K0) coefficient for the BBM under 10

constant suction level. It is worth noting that Equation (27) is an explicit formulation derived 11

solely upon the definition of the K0 condition (lateral strain is zero) with no additional 12

assumptions. 13

14

12

MODIFIED STATE SURFACE APPROACH 1

One important step in the development of constitutive models is to calibrate the model 2

parameters, which yields the overall best model predictions for all available experimental results. 3

An objective function is needed, therefore, to measure the differences between the model 4

predictions and the experimental results. A corresponding optimization strategy is also needed to 5

search the minimum of the objective function, which represents the overall best model 6

predictions. 7

Nearly all existing constitutive models including the BBM are proposed in incremental 8

forms according to theories of soil plasticity. Unsaturated soil behavior is complicated. 9

Consequently, in these models, multiple constitutive relations were proposed in which individual 10

aspects of soil behavior are controlled by multiple parameters, while at the same time, more than 11

one aspect of soil behavior is controlled by a single parameter. For example, the BBM has 13 12

model parameters and 28 constitutive relations. Hence, formulating an objective function itself is 13

challenging and often requires an in-depth understanding of every detail of the model. In the 14

meantime, advanced numerical methods in engineering optimizations and substantial personal 15

judgment in data preprocessing are needed to solve the complicated objective function. Although 16

some researchers developed various search algorithms for the calibration of elasto-plastic soil 17

models involving relatively large numbers of parameters (e.g., [52, 53]), these search algorithms 18

can only deal with experimental tests with known simple stress paths, such as suction-controlled 19

triaxial tests. The search algorithms cannot be used to calibrate model parameters using results 20

from oedometer tests in which the stress paths remain unknown. 21

In this paper, a modified state surface approach (MSSA) [43–46], recently proposed to 22

model the elasto-plastic behavior of unsaturated soils, is used to facilitate the model parameter 23

13

calibration for the BBM. Zhang and Lytton [43–45] developed a MSSA to explain the elasto-1

plastic behavior of unsaturated soils under isotropic stress conditions. The approach was then 2

extended to triaxial stress conditions in Zhang [46]. According to the MSSA, under the triaxial 3

stress state, the volume change of an unsaturated soil in the BBM can be represented by an 4

elastic surface with the following expression in the elastic region: 5

1 ln lne

s atv C p s p (28) 6

where C1 = constant related to initial specific volume of the soil and a plastic hyper-surface in 7

the elasto-plastic region, as follows: 8

2

2(0) ln ln ln ln cat

sc

at

p s p qv N s p p

p p M p ks

. (29) 9

Similar to the isotropic conditions [43–45], in the v-p-q-s space, the following criteria can 10

be determined for elastic and plastic surfaces [46]: 11

1. The plastic hyper-surface as expressed by Equation (29) is always fixed for soil in the v-12

p-q-s space. Plastic loading only changes the range of the plastic hyper-surface. 13

2. During an elastic loading or unloading process, the shape and position of the unloading-14

reloading elastic surface and the plastic surface remain unchanged in the v-p-q-s space. 15

The volume change of any isotropic elastic loading or unloading stress path must fall on 16

the elastic surface in the v-p-q-s space. Under elastic conditions, shear stress will not 17

cause a volume change in soil. So the elastic surface is the same as Equation (28) for the 18

two-dimensional condition in the v-p-s space [43–45]. 19

3. During a plastic loading process, the shape of the unloading-reloading elastic surface 20

remains unchanged ( and s are constants), but its position will change. Specifically, the 21

unloading-reloading elastic surface will move downward in parallel with the original 22

14

unloading-reloading elastic surface (i.e., C1 in Equation 28 decreases). The volume 1

change of any isotropic plastic loading stress path must fall on the plastic surface in the v-2

p-s space. 3

4. The intercept of the unloading-reloading elastic surface and the plastic surface hyper-4

surface becomes a yield surface, as expressed by Equation (10). 5

6

USE OF THE K0 EXPLICIT FORMULATION TO CALIBRATE MODEL 7

PARAMETERS IN THE BBM 8

Formulation of the Objective Function 9

The MSSA summarized the BBM under triaxial stress conditions into two concise 10

surfaces: the elastic surface (Equation 28) and the plastic surface (Equation 29). While the elastic 11

surface is easy to determine (C1 in Equation 28 can be determined by the initial condition of the 12

soil), calibration of the BBM under triaxial stress conditions is simplified to the following 13

problem: find a combination of the model parameters (0), 0 , , , , , ,and cN r p M k that 14

best fits the oedometer test results for the K0 stress paths defined by Equation (27) at different 15

suction levels. Since all eight model parameters have physical meanings, there are some 16

constraints, as follows: 17

(0) 0, 0 0, 0, 0, 0, 0, 0, and 0cN r p M k . (30) 18

Mathematically, the problem can be described as follows: Calibration of the BBM under 19

K0 stress conditions finds an appropriate combination of (0), 0 , , , , , ,and cN r p M k that 20

minimizes the overall difference between the experimental data at virgin states and the 21

theoretical results (specific volume predicted by Equation 29) for all oedometer test stress paths 22

15

as defined by Equation (27) under the constraints of Equation (30). The corresponding objective 1

function can be expressed as follows using the Euclidean norm and a set of independent state 2

variables (0), 0 , , , , , ,and cN r p M k for all experimental points at virgin states: 3

2

1

22

21

ˆ

0 ln 0 1 exp ln ln ln

n

i i i

i

nci at i

i i s i i

i at i i

F X w v v

s p qw v N r s r p p

p M p ks

. (31) 4

If all of the experimental results have the same weight wi of 1, the above objective 5

function is actually the least-squares method, in which the objective function is defined as the 6

sum of the squares of the difference between the experimental values and the theoretical values 7

predicted by Equation (29) (i.e., the sum of the squares of the residuals). The “best-fit” is defined 8

as a combination of model parameters that results in the least error between results from the 9

oedometer tests and the predicted values using Equation (29). The least-squares method finds its 10

optimum when the sum of squared residuals, F(X), is the minimum subject to constraints in 11

Equations (27) and (30). According to Criterion 1 of the MSSA, the shape and position of the 12

plastic surface are always the same for soil in the v-p-s-q space, and plastic yielding can only 13

change the range of the plastic surface. Consequently, even if the soil specimens used in the 14

oedometer tests at different suction levels have different stress histories, the results can still be 15

used to calibrate model parameters for the BBM. Under this condition, the accurate LC yield 16

surface can be obtained using Criterion 4 of the MSSA, after the elastic and plastic surfaces are 17

determined: that is, the yield surface is the interception of the unloading-reloading elastic surface 18

and the plastic surface. 19

16

The Optimization Strategy 1

One of the contributions made in this paper is the combining of a derived explicit 2

formulation (Equation 27) with the analytical solution of the BBM (Equation 29) and the MSSA 3

to simplify the complicated model parameter calibration process as a classical multidimensional 4

optimization problem (Equation 31). This problem has been studied by mathematicians for many 5

years [47, 56]. Newton’s method is considered one of the most successful and efficient 6

algorithms for this type of optimization [56]. It has been proven that Newton’s method has at 7

least a quadratic order of convergence [47]. For specific application in this paper, Equation (29) 8

is at least twice-continuously differentiable. Thus, solving the above problem is even easier. 9

Newton’s method, although very efficient, requires calculation of the inverse of the 10

Hessian matrix, which increases computation efforts [41]. The Broydon–Fletcher–Goldfarb–11

Shanno (BFGS) method [47], which is a quasi-Newton method, is used to perform the 12

optimization. The BFGS method is a popular numerical method used for engineering 13

optimization. This method has been programmed not only into sophisticated commercial 14

statistical software such as SAS®, but also into many routine office software programs such as 15

Microsoft Excel® and MatLab

®. As a result, even inexperienced users can easily use the BFGS 16

method without any need for programming. 17

18

EXAMPLE ANALYSIS AND DISCUSSIONS 19

The procedures described in the previous sections are applied here to demonstrate their 20

simplicity and objectivity for calibration of model parameters in the BBM. 21

17

Data Preparation and Procedures 1

The following example is based on the experimental data set from suction-controlled 2

oedometer tests conducted by Rampino et al. [48] on dynamically compacted silty sand. Figure 2 3

shows the experimental data used in the study. The yield stresses reported by Rampino et al. [48] 4

for each suction-controlled oedometer test are listed in Table 1, which also lists the slopes and 5

intercepts of the void ratio versus natural log of the vertical net stress curves beyond the yield 6

stressed (in the virgin compression stages for each test). The elastic parameters and s are 7

considered equal to 0.00524 and 0.0043, respectively, according to Rampino et al. [49]. 8

Poisson’s ratio of the soil is assumed as = 0.35. To calibrate the parameters 9

(0), 0 , , , , , ,and cN r p M k in the BBM, only the data points at the virgin states are used. 10

Without knowing the actual intervals for all the data points, the test results in Rampino et al. [48] 11

were sampled according to a logarithmically spaced interval of 0.05 to demonstrate the proposed 12

approach. This interval leads to a total of (log10(higher vertical stress limit)-log10 (lower vertical 13

stress limit))/0.05+1 data points for each test, as shown in Table 1. For the analysis, 140 data 14

points were generated. The Solver in Microsoft Excel®, which is based on the BFGS method, 15

was used to solve the optimization problem described above. The procedure is as follows: 16

1. Start at the yield point with the known vertical stress. Prepare the experimental data from 17

each test. Each data point has three entries: (1-ua)i, si, and measured specific volume vmi. 18

2. Find the lateral stress at the initial yield point using Equation (21), with consideration of 19

ds = 0: 20

3 10 01a au u

. 21

Calculate p0 and q0 for the same point using Equations (1) and (2). 22

18

3. Assume some initial values for (0) 0 T

cX N r p M k . 1

4. Assume a small increase in the vertical stress 1 au and calculate the increase in 2

lateral stress 3 au using Equation (27), based on the assumed 3

(0) 0 T

cX N r p M k . 4

5. Update the stress state variables: 5

1 1 11 1a a ai iu u u

6

3 3 31 1a a ai iu u u

7

1i i ip p p 8

1i i iq q q . 9

6. Repeat Steps 4 and 5 to calculate the stress path for the entire K0 loading condition. 10

7. Repeat Steps 1 through 6 for all five constant suction oedometer tests. 11

8. Calculate the void ratio based on the exact solutions (Equation 29). 12

9. Calculate the square of the difference between the predicted values and measured values. 13

10. Sum all the squares of the difference between the predicted values and measured values. 14

11. Use the Solver in Microsoft Excel® to search the minimum of the sum of the squares of 15

the difference between the measured and predicted results (Equation 27) by changing 16

(0) 0 T

cX N r p M k under the constraints of Equation (30). The stop 17

criterion is set so that variations between the current and previous iterations for each of 18

the model parameters are less than 0.001. 19

19

Initial Guesses, Convergence, and Uniqueness of the Solution 1

Newton’s method or the quasi-Newton method uses the gradient approach. The two 2

methods require assignment of an initial guess to start the iteration. Accordingly, there are two 3

potential concerns: (1) whether the given initial guess and the proposed optimization algorithm 4

can lead to a converged solution; and (2) whether the converged solution is the true solution. For 5

example, Cekerevac et al. [52] combined the quasi-Newton method with stochastic methods to 6

calibrate a multi-mechanism elasto-plastic constitutive model to ensure that the global minimum 7

was reached. A sensitivity analysis was conducted as well to explore the convergence of the 8

optimizations. 9

The experience of Zhang and Xiao [41] and the example problem described in this paper 10

indicate that the two concerns are not an issue for the following reasons: (1) The BFGS method 11

has been proven to be a power tool for the classical multidimensional optimization problem, as 12

defined by Equation (31); (2) The analytical solution of the BBM (Equation 29) is “smooth” and 13

at least twice-continuously differentiable; (3) The soil behavior is continuous without rapid 14

changes at different suction and loading levels, as shown in Figure 2; and (4) Most of the model 15

parameters in the BBM have physical meanings, and their ranges can be easily confined to 16

specific ranges. 17

Equation (30) gives some general constraints. Zhang and Xiao [41] proposed further 18

guidelines for selection of initial values for the model parameters, which can be used for this 19

study also. The guidelines significantly reduce the possible ranges of the model parameters and 20

make the calibration a constrained optimization problem instead of an unconstrained problem. 21

For example, (0) and 0 N can be taken as the intercept and slope of the oedometer test at 22

zero suction, that is, 1.292 and 0.024 in Table 1, respectively. An alternative is to use the 23

20

averaged intercepts and slopes of the virgin compression segments of each oedometer test in 1

Table 1. As shown in Table 1, the slopes of the virgin compression lines decrease with increasing 2

suction. According to Wheeler et al. [63], r should be taken as less than 1, for example, r = 0.5. 3

Table 1 shows the yield stresses for each oedometer test from which the range of pc can be 4

determined, since pc< *

0p [63]. Although these initial guesses are very crude when compared with 5

the calibrated final results (e.g., the intercepts and slopes in Table 1 are not the same as 6

( ) and N s s ), mathematically they are sufficiently close to the true solution of 7

(0) 0 T

cN r p M k . As long as the initial guess is sufficiently close to the true 8

solution, Newton’s method or the quasi-Newton method can quickly converge, with at least a 9

quadratic order of convergence [47]. Zhang and Xiao [41] calibrated the model parameters in the 10

BBM using isotropic compression test results with known stress paths using the same approach, 11

and found that if the initial guesses are arbitrarily taken as 1, the calibration can quickly 12

converge to R-square of 99.4% in less than 50 iterations. Another sensitivity test indicated that 13

when different initial guesses are used, the calibrated results finally converge to the same results. 14

More discussions regarding selection of initial guess, convergence, and uniqueness of the 15

solution can be found in Zhang and Xiao [41]. 16

Calculated K0 Stress Paths and Accuracy 17

Column 3 in Table 2 shows the calibrated model parameters for the BBM based on the 18

method described in the previous sections using the oedometer tests performed by Rampino et al. 19

[48] and the constraints in Equation (30). Figure 3 shows the calculated K0 stress paths for the 20

oedometer tests at different suction levels in the p-q plane. The stress paths are calculated using 21

Equations (21) and (27) for the elastic and elasto-plastic zones, respectively. As expected, these 22

21

stress paths are not straight lines. Instead, all of them consist of two straight lines connected by a 1

transition curve. In the elastic zone, deviatoric stress increases with net mean stress at a constant 2

slope of 0.667, which corresponds to a Poisson’s ratio of 0.35. Once the mean net stress reaches 3

the yield stress, further increase in the mean net stress will cause the deviatoric stress to decrease 4

temporarily, then smoothly transit to increase, and finally reach slightly different slopes for 5

different suction levels. The range of transition zones seems to increase with an increase in 6

suction. Figure 3b is the K0 stress path in the vertical-lateral stress plane. Traditionally, K0 is 7

considered a constant, which is the slope of the horizontal and vertical stress curves (K0 = 8

0.5385). As shown in Figure 3b, K0 is not a constant. 9

Figure 4 presents the calculated K0 values for oedometer tests at different suction levels. 10

As shown in Figure 4, the K0 values are constant at 0.5385 in the elastic zone. Once entering the 11

elasto-plastic zone, K0 values rapidly increase to peak values and then slowly decrease to 12

constants when the applied vertical stresses are large. For oedometer tests at suction 0, 100, 200, 13

300, and 400 kPa, the peak K0 values are 1.2291, 1.2254, 1.1880, 1.1687, and 1.1547, 14

respectively. The corresponding final K0 values are 0.8464, 0.8408, 0.8375, 0.8358, and 0.8352, 15

respectively. These analysis results are similar to the previous studies on K0 coefficients for 16

saturated soils [57–61]. It is well established that for normally consolidated saturated soils, K0 is 17

less than 1 [39, 57–58], while for over-consolidated soils, K0 could be greater than 1 [57–61]. 18

The soils tested in the present study were over-consolidated unsaturated soils, since initially all 19

soil specimens were loaded in the elastic zones (see Figure 2). It is normal for the K0 in soils to 20

initially be greater than 1 in the over-consolidated stage and then decrease to less than 1 when 21

the applied stress is large and the soil becomes normally consolidated, as shown in Figure 4. This 22

change is consistent with the results of previous studies [60, 61]. Figure 4 explains the K0 stress 23

22

paths in Figure 3. Since K0 is greater than 1 at the initial plastic loading stage and then decreases 1

to less than 1, increases in lateral stress are larger than increases in vertical stress, and then the 2

trends reverse when the applied vertical loads are large, as shown in Figure 3b. This results in a 3

decrease in deviatoric stress with an increase in mean net stress (shown in Figure 3a) at the initial 4

stage of plastic loadings, and the trends are opposite as mean net stress increases. 5

Figure 5 shows the calculated lateral strain increments at different vertical stresses for 6

oedometer tests at different suction levels. Note that the lateral strain increments were computed 7

using the calibrated model parameters, as shown in the third column of Table 2 and the K0 stress 8

paths in Figure 3a. The increments predicted by Equation (27) vary from -5.0 × 10-18

to 1.2 × 10-

9

17 and are essentially zero for all oedometer tests, indicating that the predicted K0 stress paths 10

according to the proposed approach are highly accurate. This conclusion is supported by Figure 11

6, which is a comparison between the experimental specific volumes and the predicted results. 12

The R-square for the regression is found to be 99.8%, and the standard deviation is 0.023. This 13

indicates a strong correlation between the independent variables and the dependent variable. 14

Figure 7 shows a comparison between predicted and experimental specific volumes and 15

vertical net stress for each oedometer test at different suction levels. The specific volumes at the 16

virgin states are calculated with Equation (29) using the corresponding suction for each 17

oedometer test under the K0 stress path specified by Equation (27), as shown in Figure 3. For the 18

elastic loading and unloading parts, C1 in Equation (28) is also selected in order to minimize the 19

sum of the squares of the difference between the experimental specific volumes and the 20

theoretical values predicted by Equation (28) under the K0 stress path specified by Equation (15). 21

It was found that initial C1 in Equation (28) is 1.3324, 1.3176, 1.3218, 1.3232, and 1.3242 for the 22

oedometer test at suctions of 0, 100, 200, 300, and 400 kPa, respectively, as shown in the sixth 23

23

column of Table 1. A similar approach is used to calculate C1 in Equation (28) when there is 1

elastic unloading or reloading. Inspection of Figures 7a through 7e indicates that the predictions 2

match the experimental results well over the entire experimental stress range. Note that the 3

hysteresis in the specific volume is not considered according to the elasto-plastic framework 4

adopted in the BBM. Note, too, that although there are decreases in deviatoric stress at the initial 5

plastic loading stage, as shown in Figure 3a, the predicted void ratio decreases with the 6

combined applications of mean net stress and deviatoric stress, indicating that these are plastic 7

hardening processes as shown in Figure 7. 8

Calculation of Yield Surface, LC Yield Curve, and Preconsolidation Stress 9

Figure 8 shows predicted yield surfaces in the p-s-q space and the LC yield curve in the 10

s-p plane using the final results of the model parameters in the third column of Table 2. Since the 11

BBM with these model parameters provides a good representation of soil behavior, an accurate 12

prediction of the constant suction oedometer tests will automatically result in a correct estimate 13

of the yield surface shape and the hardening parameters [38]. With the known initial C1 in 14

Equation (28), the initial shapes of the yield surfaces for specimens in each oedometer test can be 15

calculated. The expression of the yield surface is obtained by letting Equation (28) equal 16

Equation (29) according to Criterion 4 in the MSSA: 17

21

2

(0) ln lnexp

c

s atcN C p pq

p pM p ks s

. (32) 18

The size of the yield surface is controlled by C1 in Equation (32). Figure 8a shows the 19

initial shapes of the yield surfaces for specimens in each oedometer test. Figure 8a is plotted 20

using the model parameters in the third column of Table 2 and C1 in Table 1. The K0 stress paths 21

shown in Figure 3a are also plotted with the yield surfaces in the p-q-s space in Figure 8a. The 22

24

LC yield curve for soil in the p-s plane, which can be obtained by letting q = 0 in Equation (32), 1

has the following expression: 2

10 ln ln

exp

c

s atcN C p p

p ps

. (33) 3

Figure 8b shows the initial shapes of the yield curves for specimens in each oedometer 4

test with different C1. The preconsolidation stress can be further calculated by letting s = 0 in 5

Equation (33): 6

10 ln ln

exp0

c

s atcN C p p

p p

. (34) 7

The seventh column in Table 1 shows the yield stress for specimens in each oedometer 8

test. It was found that the corresponding *

0p is 66.3 kPa, 145.3 kPa, 116.3 kPa, 107.9kPa, and 9

102.3 kPa for the oedometer tests at suction 0 kPa, 100 kPa, 200 kPa, 300 kPa, and 400 kPa, 10

respectively. If the specimens used in the oedometer tests at five different suction levels have the 11

same stress history, the preconsolidation stress for all the oedometer tests will be the same. There 12

will be only one initial yield surface and one yield curve in Figures 8a and 8b, respectively, for 13

all five oedometer tests. These results indicate that the specimens used in the five oedometer tests 14

have different stress histories. Such tests should be considered “problematic” in terms of 15

determination of yield curve/surface and constitutive modeling, which is the reason why in 16

unsaturated soil testing, great care has always been taken to carefully prepare the soil specimens 17

with the same stress histories [31–33, 48–49, 55]. Using the proposed approach, however, a 18

difference in stress histories of the soil specimens can be automatically handled and is no longer 19

an issue. 20

25

In the original BBM [1], it is assumed that elastic shear strain is small and negligible 1

compared with plastic shear strain. This assumption is also extensively used in critical state soil 2

mechanics when dealing with K0 conditions. Figure 9 shows the ratio of elastic shear strain to 3

plastic shear strain with the vertical net stress for all oedometer tests. It was found that initially 4

the ratios of elastic shear strain to plastic shear strain were negative for all oedometer tests, 5

varying from -12% to -18%. This finding was mainly attributed to the initial decrease in 6

deviatoric stress on the K0 stress path, as shown in Figure 3a. The ratios then increase to positive 7

values and stabilize to 5.9%, 7.7%, 8.8%, 9.4%, and 9.7% for oedometer tests at suction 0, 100, 8

200, 300, and 400 kPa, respectively. The results indicate that elastic shear strains should not be 9

neglected. 10

11

DISCUSSIONS 12

In the previous sections, it has been demonstrated that, based on the derived explicit K0 13

formulation and the MSSA, a single optimization process can be used to simultaneously 14

determine the eight model parameters in the BBM ( (0), 0 , , , , , , and cN r p M k ). 15

Guidelines were provided to facilitate selection of the initial guess so that calibration can quickly 16

converge to the true solution. The procedure is straightforward and theoretically sound. The 17

calibration can be handled easily by an inexperienced user, and associated computational efforts 18

are small. The method is much simpler and faster than conventional methods (such as those 19

proposed in [52] and [53]). 20

However, care should be taken regarding the application scope of the proposed method. 21

Worth noting is that the oedometer tests are used to represent the at-rest earth pressure conditions 22

in which the shear strength of a soil is not mobilized, while in the BBM, M and k are related to 23

26

the shear strength and tensile strength of soil, respectively. Consequently, M and k values 1

obtained from the proposed method are extrapolations of the test results to failure conditions and 2

may be subject to significant error. 3

The following example can be used to demonstrate the previous comments. The first two 4

columns in Table 3 show five pairs of measured x and y values. Assume that the model used for 5

the measured data is a quadratic equation as follows: 6

2y ax bx c , (35) 7

where a, b, and c are coefficients of the quadratic equation. 8

When only the first four pairs of x and y values are used to perform a least-squares 9

regression to calibrate the model parameters a, b, and c, the following results are obtained: 10

2 21.300 2.620 0.080 (R =99.15%)y x x . (36) 11

The third and fifth columns in Table 3 show the predicted y values using Equation (36) 12

and the associated absolute errors, respectively. It is found that for x ranging from 0 to 3, the 13

differences between the measured and predicted values are small (<0.24). However, the 14

predicted y value for x = 5 is 19.48, much larger than the measured y value at x = 5, which is 15

reasonable considering that only the first four pairs of data are used to calibrate the values of a, 16

b, and c. The predicted y values with x ranging from 0 to 3 are interpolated with higher 17

accuracies, while the predicted y value at x = 5 is extrapolated with lower accuracy. 18

Similarly, when all five pairs of x and y values are used to calibrate the a, b, and c values 19

for the quadratic equation, the following results are obtained: 20

2 20.9371 1.6237 0.1773 (R =99.60%)y x x . (37) 21

27

The fourth and sixth columns in Table 3 show the predicted y values using Equation (37) 1

and the associated absolute errors, respectively. It can be seen that the predicted y values match 2

the measured data reasonably well with x ranging from 0 to 5, although the overall absolute 3

errors for the first four points become larger, which is reasonable since all the predictions made 4

are interpolated values. 5

The two calibrations are both theoretically correct. If the interested x varies from 0 to 3, 6

Equation (36) provides better predictions. However, if the interested x has a range of 0 to 5, 7

Equation (37) provides better overall predictions. Figure 10 shows a comparison of the 8

calibration processes. 9

The oedometer test results used in the calibration are similar to the first four pairs of x 10

and y values in Table 3, while the compressive and tensile shear strengths of the soil (related to 11

M and k values) are similar to the fifth point at x = 5. For example, M as shown in the third 12

column in Table 2 is 0.580. For the same soil, Rampino et al. [48] reported an M value of 1.50 13

for saturated conditions. While the actual M value for unsaturated conditions is not available, 14

0.580 is too small to be realistic for soil under unsaturated conditions. 15

In order to overcome this limitation, other information regarding the shear strength 16

should be included in accurately determining the model parameters of the soil. It is suggested 17

that the proposed approach be used, with “external” information for M and k values as additional 18

constraints. Two additional calibrations were performed using the same oedometer test data as 19

shown in Figure 2: One calibration included an additional constraint of M = 1.3, and the other 20

calibration included the additional constraints of M = 1.3 and k = 1.0. The fourth and fifth 21

columns in Table 2 list calibration results with different constraints. As shown in Table 2, with 22

additional constraints, the parameters (0), 0 , , , , and cN r p , which are mainly 23

28

related to the compressive behavior of an unsaturated soil in the BBM, do not change much 1

(similar to the second calibration for the quadratic equation). The associated R-squares are high 2

(99.8%). Since oedometer tests are mainly related to the compression behavior of unsaturated 3

soils, the results indicate that the proposed method of analyzing oedometer tests is robust and can 4

accurately identify the parameters closely related to the compression behavior of unsaturated 5

soils. 6

Ideally, when calibrating a constitutive model for unsaturated soils, it is best to include all 7

types of soil tests such as oedometer tests, isotropic compression tests, triaxial shearing with 8

different stress ratios (including failure and non-failure tests), and tests for measuring tensile 9

strength. The idea is to include all types of stress paths and cover maximum possible stress 10

ranges so that the predictions will be made based upon interpolations instead of extrapolations. 11

In reality, only limited tests can be performed when investigating unsaturated soil 12

behavior because of the extremely long testing time required and the various constraints involved 13

with soil testing. For example, it is difficult, if not impossible, to measure the tensile strength of 14

unsaturated soils. Consequently, development of a constitutive model for unsaturated soils more 15

or less unavoidably includes an extrapolation process. Regardless of this limitation, it is of 16

significant importance to have the ability to use results from oedometer tests for constitutive 17

modeling purposes, since (1) the K0 condition is an important stress state that cannot be avoided, 18

and (2) an oedometer test has fewer requirements in terms of equipment and time. Note that the 19

method proposed here can be used with other methods (see [41]) to include more test results for 20

model parameter calibration. Since the focus of this paper is to use the results from oedometer 21

tests to directly calibrate model parameters in the BBM in a simple and objective way, no more 22

29

effort was invested in combining results from oedometer tests with other triaxial tests to calibrate 1

model parameters. This topic is important for future research, however. 2

The BBM was used in this study as an example to demonstrate the application of the 3

proposed method. Note, though, that the proposed approach can be used for other constitutive 4

models as well. The reasoning is as follows: Equation (27) was derived based on the definition of 5

the at-rest earth pressure condition with no additional assumption; that is, the lateral strain is 6

equal to zero. Any constitutive model must provide an answer to the following question: For any 7

combined small changes in the net normal stress dp, deviatoric stress dq, and suction ds, what are 8

the corresponding changes in the volumetric (v ) and deviatoric (

s ) strains? 9

1 2 3v f dp f dq f ds (38) 10

1 2 3s g dp g dq g ds (39) 11

where f1, f2, f3, g1, g2, and g3 are material parameters related to model parameters and soil stress 12

states. If Equations (1), (2), (38), and (39) are plugged into Equation (17), an explicit formulation 13

similar to Equation (27) for the K0 condition can always be obtained. In addition, the basic 14

assumption for any constitutive model in critical state soil mechanics is that there is a unique 15

state-boundary surface for the soil, as experimentally verified by Wheeler and Sivakumar [8]. 16

Accordingly, the volume change in the elasto-plastic zone of any constitutive model must be able 17

to be expressed as a hyper-surface either explicitly or implicitly (similar to Equation 29). Hence, 18

the approach proposed in this paper is a general approach that can be applied to other constitutive 19

models. 20

21

30

CONCLUSIONS 1

This paper derives an explicit formulation of an at-rest coefficient for unsaturated soils. 2

An optimization approach was especially developed for simple and objective identification of 3

material parameters in elasto-plastic models for unsaturated soils using the results from suction-4

controlled oedometer tests. This is achieved by combining a modified state surface approach 5

(MSSA), recently proposed to model the elasto-plastic behavior of unsaturated soils under 6

isotropic stress conditions, with the quasi-Newton method to simultaneously determine all 7

parameters governing virgin behavior in elasto-plastic models. The BBM was used to 8

demonstrate the application of the proposed method, although a similar approach could be 9

applied to other elasto-plastic models. The following conclusions have been obtained: 10

1. In the oedometer tests, is not a constant at the transition of elastic to elasto-plastic 11

conditions. 12

2. Elastic shear strain should not be ignored under K0 conditions. 13

3. The proposed method relies on the definition of the K0 conditions only, without using any 14

additional assumption. 15

4. The proposed method can be used to calibrate model parameters using oedometer tests 16

with different stress histories. 17

5. The comparison between predictions using the proposed method and laboratory test 18

results indicated that the proposed method is simple and objective. 19

6. In oedometer tests, the shear strength of a soil is not mobilized. Consequently, shear 20

strength parameters obtained from the proposed method are an extrapolation of the test 21

results to a failure condition and may be subject to significant error. It is suggested that 22

31

the proposed approach be used by introducing “external” information for M and k values 1

to overcome the limitation. 2

7. The K0 condition represented by oedometer tests is an important stress condition that 3

cannot be avoided in the numerical simulation. Compared with suction-controlled triaxial 4

tests, much simpler equipment is needed for suction-controlled oedometer tests, and the 5

duration of the testing period is about 10 times less. 6

7

32

REFERENCES 1

1. Alonso EE, Gens A, Josa A. A constitutive model for partially saturated soils. Géotechnique, 2

1990, 40(3): 405–430. 3

2. Alonso EE, Vaunat J, Gens A. Modeling the mechanical behavior of expansive clays. 4

Engineering Geology, 1999, 54(1–2): 173–183. 5

3. Gens A, Alonso EE. A framework for the behavior of unsaturated expansive clays. Canadian 6

Geotechnical Journal, 1992, 29: 1013–1032. 7

4. Alonso EE, Gens A, Gehling, WYY. Elasto-plastic model for unsaturated expansive soils. 8

Numerical models in geotechnical engineering. Balkema, 1999, pp. 11–18. 9

5. Cui YJ, Delage P. Yielding and plastic behaviour of an unsaturated compacted silt. 10

Geotechnique, 1996, 46(2): 291–311. 11

6. Bolzon G, Schrefler BA, Zienkiewicz OC. Elastoplastic soil constitutive laws generalised to 12

partially saturated states. Geotechnique, 1996, 46: 279–289. 13

7. Gens A. Constitutive modelling: Application to compacted soil. In: Proc. of 1st International 14

Conference on Unsaturated Soils, Paris, 1995, 3, pp. 1179–1200. 15

8. Wheeler SJ, Sivakumar V. An elasto-plastic critical state framework for unsaturated soil. 16

Géotechnique, 1995, 45(1): 35–53. 17

9. Wheeler SJ. Inclusion of specific water volume within an elastoplastic model for unsaturated 18

soil. Can. Geotech. J., 1996, 33: 42–57. 19

10. Wheeler SJ, Karube D. State of the art report-constitutive modeling. In: Proc. of 1st 20

International Conference on Unsaturated Soils, Paris, 1995, 3, pp. 1323–1356. 21

33

11. Dangla OL, Malinsky L, Coussy O. Plasticity and imbibition-drainage curves for unsaturated 1

soils: a unified approach. In: 6th International Conference on Numerical Models in 2

Geomechanics, Montreal. Rotterdam: Balkema, 1997, pp. 141–146. 3

12. Geiser F, Laloui L, Vulliet L. Modelling the behaviour of unsaturated silt. In: Experimental 4

Evidence and Theoretical Approaches in Unsaturated Soils. Rotterdam: Balkema, 2000, pp. 5

155–175. 6

13. Jommi C. Remarks on the constitutive modelling of unsaturated soils. In: Experimental 7

Evidence and Theoretical Approaches in Unsaturated Soils. Tarantino A, Mancuso C (Eds.). 8

In: Proc. of International Workshop on Unsaturated Soils, Trento. Rotterdam: Balkema, 9

2000, pp. 139–153. 10

14. Vaunat J, Romero E, Jommi C. An elastoplastic hydro-mechanical model for unsaturated 11

soils. In: Experimental Evidence and Theoretical Approaches in Unsaturated Soils. 12

Rotterdam: Balkema, 2000, pp. 121–138. 13

15. Wheeler SJ, Sharma RS, Buisson MSR. Coupling of hydraulic hysteresis and stress-strain 14

behavior in unsaturated soils. Geotechnique, 2003, 53(1): 41–54. 15

16. Gallipoli D, Gens A, Sharma R, Vaunat J. An elastoplastic model for unsaturated soil 16

incorporating the effects of suction and degree of saturation on mechanical behaviour. 17

Geotechnique, 2003, 53: 123–135. 18

17. Gallipoli D, Wheeler SJ, Karstunen M. Modeling the variation of degree of saturation in a 19

deformable unsaturated soil. Geotechnique, 2003, 53(1): 105–112. 20

18. Sheng D, Sloan SW, Gens A. A constitutive model for unsaturated soils: Thermomechanical 21

and computational aspects. Computational Mechanics, 2004, 33(6): 453–465. 22

34

19. Sheng D, Fredlund DG, Gens A. A new modelling approach for unsaturated soils using 1

independent stress variables. Canad. Geotech. J., 2008, 45(4): 511–534. 2

20. Sheng D, Gens A, Fredlund DG, Sloan SW. Unsaturated soils: from constitutive modelling to 3

numerical algorithms. Computers and Geotechnics, 2008, 35(6): 810–824. 4

21. Sheng D, Sloan SW, Gens A. A Constitutive model for unsaturated soils: thermomechanical 5

and computational aspects. Computational Mechanics, 2004, 33(6): 453–465. 6

22. Sheng D, Sloan SW, Gens A, Smith DW. Finite element formulation and algorithms for 7

unsaturated soils. Part I: Theory. Int. J. Numer. Anal. Meth. Geomech., 2003, 27: 745–765. 8

23. Sheng D, Smith DW, Sloan SW, Gens A. Finite element formulation and algorithms for 9

unsaturated soils. Part II: Verification and Application. Int. J. Numer. Anal. Meth. Geomech. 10

2003, 27: 767–790. 11

24. Sun DA, Sheng DC, Xiang L, Sloan SW. Elastoplastic prediction of hydro-mechanical 12

behaviour of unsaturated soils under undrained conditions. Computers and Geotechnics, 13

2008, 35: 845–852. 14

25. Tamagnini R. An extended Cam-clay model for unsaturated soils with hydraulic hysteresis. 15

Géotechnique, 2004, 54(3): 223–228. 16

26. Tamagnini R. On the effective stress principle in unsaturated soil. Rivista Italiana di 17

Geotecnica, 2011, 3/11: 25–40. 18

27. Delage P, Graham I. State of the art report – Understanding the behavior of unsaturated soils 19

requires reliable conceptual models. In: Proc. of 1st International Conference on Unsaturated 20

Soils, Paris, 1995, 3, pp. 1323–1356. 21

35

28. Vaunat J. Elastoplastic framework to model unsaturated materials. 1 st. MUSE School 1

“Fundamentals of Unsaturated Soils.” Barcelona, Spain, 1–2 June 2005. 2

http://muse.dur.ac.uk/ 3

29. Gens A, Sánchez M, Sheng D. On constitutive modelling of unsaturated soils. Acta 4

Geotechnica, 2006, 1(3): 137–147. 5

30. Bishop AW, Donald IB. The experimental study of partly saturated soil in the triaxial 6

apparatus. In: Proc. of 5th International Conference on Soil Mechanics, 1961, 1, pp. 13–21. 7

31. Sivakumar V. A critical state framework for unsaturated soil. Ph.D thesis, University of 8

Sheffield, UK, 1993. 9

32. Hoyos LR. Experimental and computational modeling of unsaturated soil behavior under true 10

triaxial stress states. Ph.D. dissertation, Georgia Institute of Technology, Atlanta, 1998. 11

33. Sharma RS. Mechanical behaviour of unsaturated highly expansive clays. Ph.D. thesis, 12

University of Oxford, UK, 1998. 13

34. Maswoswe J. Stress paths for a compacted soil during collapse due to wetting. Ph.D. thesis, 14

University of London, 1985. 15

35. Romero E, Lloret A, Gens A. Development of a new suction and temperature controlled cell. 16

In: 1st International Conference on Unsaturated Soil, Paris, France, Alonso E, Delage P 17

(Eds.), 1995, 2, pp. 553–559, 18

36. Kassif G, Ben Shalom A. Experimental relationship between swell pressure and suction. 19

Geotechnique, 1971, 21(3): 245–255. 20

37. Dineen K, Burland JB. A new approach to osmotically controlled oedometer testing. In: 21

Proc. of 1st Conference on Unsaturated Soils (UNSAT’95), Paris, Alonso EE, Delage P 22

36

(Eds.), 6–8 September 1995, Rotterdam, The Netherlands: A.A. Balkema, Vol. 2, pp. 459–1

465. 2

38. Gallipoli D, D’Onza F, Wheeler SJ. A sequential method for selecting parameter values in 3

the Barcelona Basic Model. Canadian Geotechnical Journal, 2010, 47: 1175–1186. 4

39. Jaky J. Pressure in silos. In: Proc. of 2nd International Conference on Soil Mechanics and 5

Foundation Engineering, London, 1948, Vol. 1, pp. 103–107. 6

40. Alonso EE, Gens A, Hight DW. Special problem soils. General report. In: Proc. of 9th 7

European Conference on Soil Mechanics and Foundation Engineering, Dublin, 1987, Vol. 3, 8

pp. 1087–1146. 9

41. Zhang X, Xiao M. Using modified state surface approach to select parameter values in the 10

Barcelona Basic Model. International Journal for Numerical and Analytical Methods in 11

Geomechanics, 2013, 37: 1847–1866. 12

42. Gallipoli D, D’Onza F. Calibration of elasto-plastic models for unsaturated soils under 13

isotropic stresses. Engineering Geology, 2013, 165: 64–72. 14

43. Zhang X, Lytton RL. A modified state surface approach on unsaturated soil behavior study 15

(I): Basic concept. Canadian Geotechnical Journal, 2009, 46(5): 536–552. 16

44. Zhang X, Lytton RL. A modified state surface approach on unsaturated soil behavior study 17

(II): General formulation. Canadian Geotechnical Journal, 2009, 46(5): 553–570. 18

45. Zhang X, Lytton RL. A modified state surface approach on unsaturated soil behavior study 19

(III) Modeling of coupled hydro-mechanical effect. Canadian Geotechnical Journal, 2012, 20

49: 98–120. 21

37

46. Zhang X. Analytical Solution of the Barcelona Basic Model. GeoShanghai 2010 1

International Conference, 3–5 June 2010, Shanghai, China, ASCE Special Publication 376: 2

Experimental and Applied Modeling of Unsaturated Soils, 2010, pp. 96–103. 3

47. Rao SS. Engineering Optimization, Theory and Practice, 3rd ed., John Wiley & Sons, 1996. 4

48. Rampino C, Mancuso C, Vinale F. Laboratory testing on an unsaturated soil: Equipment, 5

procedures, and first experimental results. Canadian Geotechnical Journal, 1999, 36: 1–12. 6

49. Rampino C, Mancuso C, Vinale F. Experimental behaviour and modelling of an unsaturated 7

compacted soil. Canadian Geotechnical Journal, 2000, 37: 748–763. 8

50. Boyd JL, Sivakumar V. Experimental observations of the stress regime in unsaturated 9

compacted clay when laterally confined. Géotechnique, 2011, 61(4): 345–363. 10

51. Casagrande A. The Determination of the Preconsolidation Load and Its Practical 11

Significance. In: Proc. of 1st International Conference on Soil Mechanics and Foundation 12

Engineering, Cambridge, Mass. Rotterdam, The Netherlands: A.A. Balkema, 1936, Vol. 3, 13

pp. 60–64. 14

52. Cekerevac C, Girardin S, Klubertanz G, Laloui L. Calibration of an elasto-plastic constitutive 15

model by a constrained optimisation procedure. Computers and Geotechnics, 2006, 33(8): 16

432–443. 17

53. Mattsson H, Klisinski M, Axelsson K. Optimisation routine for identification of model 18

parameters in soil plasticity. International Journal for Numerical and Analytical Methods in 19

Geomechanics, 2001, 25(5): 435–472. 20

54. Thomas HR, He Y. Modelling the behaviour of unsaturated soil using an elastoplastic 21

constitutive model. Geotechnique, 1998, 48(5): 589–603. 22

38

55. Fredlund DG, Rahardjo H. Soil Mechanics for Unsaturated Soils. New York: John Wiley & 1

Sons, 1993. 2

56. Chong EKP, Zak SH. An Introduction to Optimisation. John Wiley & Sons, 1996. 3

57. Holtz RD, Kovacs WD. An Introduction to Geotechnical Engineering. 1981. Englewood 4

Cliffs, NJ: Prentice-Hall. 5

58. Das BM. Fundamentals of Geotechnical Engineering, 4th ed., 2013. Cengage Learning. 6

ISBN-13: 978-1111576752. 7

59. Schmidt B. 1966. Earth pressures at rest related to stress history. Discussion. Can. Geotech. 8

J., 1966, 34: 239–242. 9

60. Mayne PW, Kulhawy FH. KO-OCR relationships in soil. ASCE Journal of Geotechnical 10

Engineering Division, 1982, 108(GT6): 851–872. 11

61. Mesri G, Hayat TM. The coefficient of earth pressure at rest. Can. Geotech. J., 1993, 30: 12

647–666. 13

62. Zhang X, Liu J, Li P. A new method to determine the shapes of yield curves for unsaturated 14

soils. ASCE Journal of Geotechnical and Geoenvironmental Engineering, 2010, 136(1): 239–15

247. 16

63. Wheeler SJ, Gallipoli D, Karstunen M. Comments on use of the Barcelona Basic Model for 17

unsaturated soils. International Journal for Numerical and Analytical Methods in 18

Geomechanics, 2002, 26(15): 1561–1571. 19

64. Khalili N, Geiser F. Blight GE. Effective stress in unsaturated soils: Review with new 20

evidence. Int. J. Geomech., 2004, 4(2): 115–126. 21

65. Borja RI. On the mechanical energy and effective stress in saturated and unsaturated porous 22

continua. International Journal of Solids and Structures, 2006, 43(6): 1764–1786. 23

39

66. Nuth M, Laloui L. Effective stress concept in unsaturated soils: Clarification and validation 1

of a unified framework. International Journal for Numerical and Analytical Methods in 2

Geomechanics, 2008, 32(7): 771–801. 3

67. Li XS. Thermodynamics-based constitutive framework for unsaturated soils. 1: Theory. 4

Géotechnique, 2007, 57(5): 411–422. 5

68. Li XS. Thermodynamics-based constitutive framework for unsaturated soils. 2: A basic 6

triaxial model. Géotechnique, 2007, 57(5): 423–435. 7

69. Rojas E. Equivalent stress equation for unsaturated soils. I: Equivalent stress. Int. J. 8

Geomech., 2008, 8(5): 285–290. 9

70. Rojas E. Equivalent stress equation for unsaturated soils. II: Solid-porous model. Int. J. 10

Geomech., 2008, 8(5): 291–299. 11

12

40

1

LIST OF TABLES 2

Table 1. Parameters for constant suction oedometer compression tests (data from Rampino et al. 3

1999) 4

Table 2. Calibrated model parameters for the BBM with different constraints 5

Table 3. Calibrations of model parameters for a quadratic equation 6

7

8

41

Table 1. Parameters for constant suction oedometer compression tests (data from Rampino et al. 1

1999) 2

3

4

Table 2. Calibrated model parameters for the BBM with different constraints 5

6

7

42

1

Table 3. Calibrations of model parameters for a quadratic equation 2

3

4 5

6

7

43

LIST OF FIGURES 1

FIGURE 1. Stress Path for K0 Consolidation Test 2

FIGURE 2. Constant Suction Oedometer Compression Test Results Used in this Study 3

(Modified from Rampino et al. 1999). 4

FIGURE 3. Stress Paths of the Oedometer Tests 5

FIGURE 4. Calculated K0 Values for Oedometer Tests at Different Suction Levels 6

FIGURE 5. Calculated Lateral Strain Increments at Different Vertical Stress 7

FIGURE 6. Comparison between Measured and Predicted Results 8

FIGURE 7. Comparison between Predictions and Experiments for Oedometer Test at (a) s = 0 9

kPa, (b) s = 100 kPa, (c) s = 200 kPa, and (d) s = 300 kPa, and (e) s = 400 kPa. 10

FIGURE 8. Initial Shapes of the Yield Surfaces and Yield Curves for Specimens in Each 11

Oedometer Tests. (a) Yield Surface, (b) Yield Curve. 12

FIGURE 9. Ratio between Elastic and Plastic Shear Strains at Different Vertical Stress Levels. 13

FIGURE 10. Comparison between Two Calibrations Made for a Quadratic Equation. 14

15

44

1

2

(a) K0 conditions in the (p,q) stress plane as suggested in Alonso et al. (1990) 3

p

q

K0 line

K0 lines

p0

ps

*

0p

A

B CD

A’

B’ C’D’

4

(b) Possible stress path for K0 consolidation test for unsaturated soils 5

FIGURE 1. Stress Path for K0 Consolidation Test 6

7

8

9

45

s = 400 kPa, WNS4

s = 300 kPa, WNS3

s = 200 kPa, WNS2

s = 100 kPa, WNS1

s = 0 kPa, WS

0.001 0.01 0.1 1 10

Vertical net stress, (v-ua) (MPa)

Spec

ific

volu

me,

v

1.2

4 1

.28 1

.32 1.3

6 1.4

0

1

FIGURE 2. Constant Suction Oedometer Compression Test Results Used in this Study 2

(Modified from Rampino et al. 1999). 3

4

46

1

2 3

(a) K0 stress paths in the p-q plane 4

5 6

(b) K0 stress paths in the vertical-lateral net stress plane 7 8 9

FIGURE 3. Stress Paths of the Oedometer Tests 10

11

47

1

FIGURE 4. Calculated K'0 Values for Oedometer Tests at Different Suction Levels 2

3

48

1

FIGURE 5. Calculated Lateral Strain Increments at Different Vertical Stress 2

3

49

1

FIGURE 6. Comparison between Measured and Predicted Results 2

3

4

y = 1x

R² = 0.9980

1.24

1.27

1.30

1.33

1.36

1.39

1.24 1.27 1.30 1.33 1.36 1.39

Pre

dic

ted

Sp

ecif

ic V

olu

me,

v

Measured Specific Volume, v

50

1

(a) Oedometer test at s = 0 kPa 2

3

(b) Oedometer test at s = 100 kPa 4

1.20

1.25

1.30

1.35

1.40

0.001 0.01 0.1 1 10

Sp

ecif

ic v

olu

me,

v

Vertical net stress , v-ua (MPa)

Experimental (s=0 kPa)

Predicted (s=0 KPa)

1.26

1.28

1.30

1.32

1.34

0.01 0.1 1 10

Sp

ecif

ic v

olu

me,

v

Vertical net stress, v-ua (MPa)

Experimental (s=100kPa)

Predicted (s=100 kPa)

51

1

2

(c) Oedometer test at s = 200 kPa 3

4

(d) Oedometer test at s = 300 kPa 5

1.28

1.30

1.32

1.34

1.36

0.01 0.1 1 10

Sp

ecif

ic v

olu

me,

v

Vertical net stress, v-ua, (MPa)

Experimental (s=200 kPa)

Predicted (s=200 kPa)

52

Spec

ific

volu

me,

v

Vertical net stress, (v-ua), (MPa)

1

(e) Oedometer test at s = 400 kPa 2

3

FIGURE 7. Comparison between Predictions and Experiments for Oedometer Test at 4

(a) s = 0 kPa, (b) s = 100 kPa, (c) s = 200 kPa, and (d) s = 300 kPa, and (e) s = 400 kPa. 5

6

53

1

2

FIGURE 8. Initial Shapes of the Yield Surfaces and Yield Curves for Specimens in Each 3

Oedometer Tests. (a) Yield Surface, (b) Yield Curve. 4

0

0.1

0.2

0.3

0.4

0.5

0 0.2 0.4 0.6 0.8 1

Matr

ic s

uct

ion

, s

(M

Pa)

Mean net stress, p (MPa)

s = 0 kPa

s = 100 kPa

s = 200 kPa

s = 300 kPa

s = 400 kPa

54

1

2

3 4 FIGURE 9. Ratio between Elastic and Plastic Shear Strains at Different Vertical Stress 5

Levels. 6

7

-0.20

-0.10

0.00

0.10

0.20

0 1 2 3 4 5

s

e/ s

p

Vertical net stress, v -ua (MPa)

s=400 kPa (9.7%)

s=0 kPa (5.9%)

s=100 kPa (7.7%)

s=200 kPa (8.8%)

s=300 kPa (9.4%)

55

1 FIGURE 10. Comparison between Two Calibrations Made for a Quadratic Equation. 2

3 4

5

6

7

8

56

1 Notation 2

3

The following symbols are used in this paper: 4

C1, C2, C3, and C’ = constants 5

e = void ratio 6

e0 = initial void ratio 7

ee = void ratio constitutive surface in the elastic zone 8

G = shear modulus 9

l = parameter that relates cohesion and suction 10

M = slope of theoretical critical state line 11

k = parameter describing the increase in cohesion with suction 12

N(s) = specific volume for p=pc 13

p = m−ua = net mean stress 14

pat = atmospheric pressure 15

pc= reference stress 16

m = total mean stress 17

p0 = apparent preconsolidation pressure at a certain suction 18

*

0p = preconsolidation pressure in saturated conditions 19

1 3q = deviatoric stress 20

r = parameter controlling the slope of the virgin compression line 21

s = ua−uw = soil suction 22

0s = maximum historical suction applied to the soil 23

57

ua = air pressure 1

uw = water pressure 2

v = specific volume 3

= parameter that controls the non-associated flow rule 4

= parameter that controls the slope of the virgin compression line for s0 5

e

qd = deviatoric strain increment 6

v = volumetric strain 7

vd = volumetric strain increment 8

e

v = elastic strain 9

p

v = plastic strain 10

q = deviatoric strain 11

p

v = plastic volumetric strain 12

p

vp = plastic volumetric strain generated by a mean net stress increment 13

p

vs = plastic volumetric strain generated by a suction increment 14

= slope of the unloading-reloading line associated with the mean net stress 15

s = slope of the unloading-reloading line associated with soil suction 16

s = slope of the virgin expansion line associated with the mean net stress 17

s = slope of the virgin compression line associated with soil suction 18

s = slope of the virgin compression line associated with the mean net stress 19

for 0s 20

58

0 = slope of the virgin compression line associated with the mean net stress 1

for 0s 2

e

vd = elastic volumetric strain 3

vd = total volumetric strain, 0 01

vv

dV ded

V e

4

p

vd = plastic volumetric strain 5

pd = irrecoverable “plastic” volumetric water content variation 6

X = a vector containing the model parameters 7

F(X) = Objective function, measures the difference between the theoretical and 8

experimental results 9

m and n = lower and upper limits of X 10

wi= weight of each experimental data point 11

vi = experimental specific volume at virgin states for a stress status of (pi, si) 12

iv = predicted specific volume at virgin states for a stress status of (pi, si) 13

14