- 1675 -

Improved Hyperbolic Model for Harden/soften Stress-strain Curve of

Yangtze River Soil

Jinjun Guo

Department of Civil Engineering, Luoyang Institute of Science and Technology, Luoyang, China

Wei Wang *

Department of Civil Engineering, Shaoxing University, Shaoxing, China * corresponding author, e-mail: wellsking.wang@gmail.com

Xiaoni Wang

Zijin College, Nanjing University of Science and Technology, Nanjing, China

ABSTRACT Mathematical model for soil stress-strain curve is very important to theory study of soil mechanics and numerical simulation of corresponding geotechnical engineering; however it is not well solved. Based on experimental data of Yangtze River soil, shortcoming of traditional hyperbolic stress-strain model is pointed out. Composite two-segment hyperbolic model (CTH model) and composite hyperbolic-line model (CHL model) are presented to describe harden type and soften type stress-strain curves, respectively. These two improved models both can be simplified into hyperbolic model with their special parameters. Finally, one harden type and two soften type curves are put forwarded in order to validate the accuracy of CTH model and CHL model, and good agreements have been found between experimental curves and fitting results of the improved models. The results of this study put good foundation for numerical simulation and design of corresponding geotechnical engineering.

KEYWORDS: mathematical model, stress, strain, Yangtze River soil

INTRODUCTION With the rapid economic growth, more and more engineering structures are built on

foundations nearby rivers. Good geotechnical design should be made before these structures construction, which includes field investigation, foundation numerical simulation, et al. Accuracy of geotechnical engineering foundation numerical simulation depends strongly on soil constitutive law and mechanical parameters used during the simulation process. Understanding properly of mathematical description of stress-strain relationship is necessary to construct soil constitutive law, and great effort has been devoted to it recently (He and YANG, 2002; Al-Shayea, et al. 2003; Wang, et al. 2004; Xu, et al. 2011). Several empirical models based on

Vol. 17 [2012], Bund. L 1676 experimental dada have been proposed to describe the relationship (Habibagahi and Mokhberi, 1998; Li and Ding, 2002; Ramu and Madhira, 2010). The work of Duacan and Chang is important for understanding the stress-strain behavior (Duncan and Chang, 2002). They proposed a hyperbolic model for it. At the same time, it had been found that stress-strain relationship can be divided into harden type and soften type. Hyperbolic model is available for not soften type but only harden type stress-strain relationship (Mitaim and Detournay, 2005; Wang, et al. 2010; Wang, 2012; Zhang, 2010). In order to rationally describe both harden type and soften type stress-strain relationship, some more general models are needed. Object of this study is to properly understand the stress-strain behavior and to establish mathematical models to well describe harden/soften type stress-strain curves of Yangtze River soil.

HYPERBOLIC MODEL AND TYPICAL p-ε CURVES

Hyperbolic Model

According to the experimental results, hyperbolic model is proposed by Duacan and Chang to describe relationship between the axial strain, ε, and the deviator stress q=(σ1 −σ3) of the soil during shearing process, where σ1 is maximum principal stress and σ3 is minimum principal stress. This model can be written as:

q

a b

εε

=+

(1)

where a and b are two undetermined soil parameters. It is obviously that in Eq. 1, there is a limit to the deviator stress, and the limiting deviator stress is 1/b, shown as Fig. 1.

Figure 1: Hyperbolic model of stress-strain relationship

Differentiating Eq. 1 with ε, the tangent modulus, Et can be obtained:

22

1(1 )

( )t

aE bq

a b aε= = −

+ (2)

q

0 ε

Et

Ei1

q= a+bε1

1bε

Vol. 17 [2012], Bund. L 1677

When q=0, the initial tangent modulus Ei=1/a. Thus the two parameters a and b are directly related to the limiting deviator stress and initial tangent modulus, respectively, shown as Fig. 1.

Given p=ε/q, Eq. 1 can be converted as a linear expression:

p a bε= + (3)

where a is intercept and b is slope coefficient of the p-ε line, shown as Fig. 2. So we can easily determine the two parameters from Eq. 3 based on experimental data, and then this model is extensively used both to theory analysis and to engineering numerical simulation.

Figure 2: p-ε curve of hyperbolic model

Typical p-ε Curves

Many experimental data show that hyperbolic model can not accurately fit the investigated q-ε data of Yangtze River soil. Sometimes, the fitting error is considerable. In practical geotechnical engineering design, simulating initial deformation using hyperbolic model is often bigger than that of field investigations. This error is perhaps originated from the shortcomings of hyperbolic model itself. In order to properly understand the q-ε behavior of Yangtze River soil, triaxial tests of 25 set specimen are conducted by us. The tested data provide that p-ε curve of Yangtze River soil is not one strait line like Fig. 2 but one two-segment line, which may be the key source of hyperbolic model simulating error. In order to conveniently discuss the p-ε curves, we only take two typical specimens as examples, and they are named as No. 1 and No. 2 specimen, denoting soften and harden type q-ε curves respectively, shown as Fig. 3 and Fig. 4.

Fig. 3-4 demonstrate that p-ε curves are not only one strait line as Eq. 3, but one convex or concave two-segment line with one switch point. In fact, the convex two-segment line denotes the harden type q-ε curve and the concave one denotes the soften type q-ε curve. It is necessary to improve the p-ε description in Eq. 3.

It is very important to determine the switch point, (εs, qs) or (εs, ps) of p-ε curve. According to the experimental data, we investigated that:

(1) For convex type p-ε curve, εs is depended strongly on soil failure strain εf, and value εs/εf varies with a small area from 0.25 to 0.30.

b

ε0

p=a+b

a

1

ε

p

Vol. 17 [2012], Bund. L 1678

(2) Concave type p-ε curve is combined soil dilatancy with peak point (εp, εv) of ε-εv curve, shown as Fig. 5, where εv is volume strain. Further study find that εs of concave type p-ε curve is almost equal to εp of εv-ε curve.

Figure 3: Convex type p-ε curve of No. 1 soil specimen

Figure 4: Concave type p-ε curve of No. 2 soil specimen

Figure 5: ε-εv curve of of No. 2 soil specimen

0

5

10

15

20

25

30

35

40

0 2 4 6 8 10ε / %

p /

MP

a-1

100 kPa 200 kPa

300 kPa 400 kPa

0

10

20

30

40

50

60

0 2 4 6 8 10 12 14 16ε / %

p /

MP

a-1

100 kPa 200 kPa

300 kPa 400 kPa

-6

-4

-2

0

2

0 5 10 15 20

ε / %

ε v /%

100 kPa 200 kPa

300 kPa 400 kPa

Vol. 17 [2012], Bund. L 1679

NEW MODEL FOR HAEDEN TYPE q-ε CURVE

Expression of New Model

Considering the switch point (εs, ps), we suggest following two-segment line model to describe the concave p-ε curve:

1 1

2 2

s

s

p a b

p a b

ε ε εε ε ε

= + ≤ = + >

(4)

In Eq. 4, a1, b1, a2 and b2 are four undetermined soil parameters with similar physical meanings and calculating ways to those of parameters a and b, shown as Fig, 6. They have following relationship, a1<a, b1>b and a<a2, b>b2. So the q-ε model can be expressed:

1 1

2 2

s

s

qa b

qa b

ε ε εε

ε ε εε

= ≤ + = > +

(5)

When a1=a2 and b1=b2, Eq. 5 is degraded to Eq. 1, and the new model is degraded to conventional hyperbolic model. Because of its two parts, we name it as composite two-segment hyperbolic model (CTH model).

Figure 6: Parameters meaning of new convex type model

Tangent Modulus of CTH Model

At the same time, tangent modulus of Yangtze River soil can be obtained as two parts:

a2b1

b2

ε

a1

0

( , p )p

ssε1

1

Vol. 17 [2012], Bund. L 1680

2

11

22

2

1(1 )

1(1 )

t s

t s

E b q q qa

E b q q qa

= − ≤ = − >

(6)

It is obvious that when a1=a2 and b1=b2, Eq. 6 is degraded to Eq. 2.

Given two deviator stress q, one is less than qs and the other is bigger than qs, tangent modulus with various minimum principal stresses (σ3) of No. 1 soil specimen calculated by Eq. 2 and Eq. 5 are list in table 1. From this table we can make following conclusions:

(1) As to this specimen, when q is less than qs, tangent module resulted from CTH model are bigger than that from conventional hyperbolic model.

(2) When q is bigger than qs, tangent modulus of this specimen resulted from CTH model are similar to that from conventional hyperbolic model.

These two properties enable CTH model to overcome the above mentioned shortcoming of hyperbolic model that simulating initial deformation is often bigger than that of field investigations.

Table 1: Tangent modulus of No. 1 specimen calculated by two models /kPa

σ3 qs q CTH model Hyperbolic model

1/a 1/b Et 1/a 1/b Et

100 90 20 157 120 109 95 189 76

150 68 213 6 95 189 4

200 135 20 146 236 122 104 416 94

250 88 451 17 104 416 17

300 159 20 145 382 130 123 592 114

300 111 623 30 123 592 30

400 370 20 160 657 150 147 764 139

450 117 873 27 147 764 25

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NEW MODEL FOR SOFTEN TYPE q-ε CURVE Comparing Fig. 3 and Fig. 4, one new formula is built to express concave type p-ε curve for

soften type q-ε curve, which is proposed as:

1 ( )

a bp

k a b

εε

+=− +

(7)

where k is a positive undetermined parameter. Because of (1-k(a+bε))<1, Eq. 7 would appear concave shape like Fig. 3. If k=0, then Eq. 7 is degraded to Eq. 2 as a straight line.

Then, one composite hyperbolic-line model (CHL model) can be obtained:

-q k

a b

ε εε

=+

(8)

It is evident that CHL model is degraded to conventional hyperbolic model with k=0.

Taking specimen 2 with two typical confining pressure σ3=100kPa and σ3=200kPa as example, its tested and fitted p-ε curves are presented in Fig. 7. This figure shows that fitted p-ε curves of CHL model are more closely to tested data than that of hyperbolic model.

(a) Confining pressure σ3=100kPa

0

10

20

30

40

50

60

70

0 2 4 6 8 10 12 14 16

ε / %

p /

MP

a-1

tested data

hyperbolic model

CHL model

Vol. 17 [2012], Bund. L 1682

(b) Confining pressure σ3=200kPa

Figure 7: Tested and fitted p-ε curves of No. 2 soil specimen

Differentiating Eq. 8 with ε, the tangent modulus Et of CHL model can be deduced:

2

-( )t

aE k

a bε=

+ (9)

Because CTH model and CHL model are both built based on hyperbolic model, they two can be considered as improved hyperbolic models.

COMPARISON WITH LABORATORY TESTS In order to verify applicable ability of the two presented models, comparisons among

laboratory test, CTH model and CHL model fitting are conducted, and results are shown as Fig. 8 and Fig. 9.

0

5

10

15

20

25

30

0 2 4 6 8 10 12 14 16

ε / %

p / M

Pa-1

tested data

hyperbolic model

CHL model

Vol. 17 [2012], Bund. L 1683

Figure 8: Tested and fitted q-ε curves of No. 1 soil specimen with σ3=100kPa

(a) Confining pressure σ3=100kPa

(b) Confining pressure σ3=200kPa

Figure 9: Tested and fitted q-ε curves of No. 2 soil specimen

0

50

100

150

200

0 2 4 6 8 10 12

ε / %

q /

kP

a

tested data

hyperbolic model

CTH model

0

100

200

300

400

500

0 2 4 6 8 10 12 14 16

ε / %

q /

kP

a

tested data

hyperbolic model

CHL model

0

150

300

450

600

750

900

0 2 4 6 8 10 12 14 16

ε / %

q /

kP

a

tested data

hyperbolic model

CHL model

Vol. 17 [2012], Bund. L 1684

These two figures demonstrate that CTH model and CHL model can offer better agreements with tests data than that of hyperbolic model.

CONCLUSIONS AND DISCUSSIONS Model for stress-strain relationship of Yangtze River soil is studied in details, and basic work

and conclusions are as fellows.

(1) Triaxial tests provide that, p-ε curve of Yangtze River soil is not one strait line but two-segment line which includes convex type, concave type. The convex p-ε curve denotes the harden type q-ε curve and the concave one denotes the soften type q-ε curve. Conventional hyperbolic model can not well simulate them.

(2) Based on hyperbolic model, CTH model is established and proposed to describe harden type q-ε curve, and CHL model is proposed to describe soften type one. Good agreements have been found between CTH model, CHL model fitting and tested data.

It is worth emphasizing that this article just discusses the q-ε curve of Yangtze River soil from the point of mathematical behavior. In fact, the q-ε curve is affected by multiple factors. It is an important and interesting task to investigate the curve from the point of soil structure and stress history in the future study.

ACKNOWLEDGMENTS The authors thank the reviewers who gave a through and careful reading to the original

manuscript. Their comments are greatly appreciated and have help to improve the quality of this paper. This work is supported in part by the Key Project of Chinese Ministry of Education (NO. 211068), and by the Nature Science Foundation of Zhejiang Province (NO. Y1080839). Some Laboratory tests were carried out by Dr. Zhang in Hohai University.

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