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Method of Numerical Modeling for Constitutive Relations of Clay
Journal o f China University of Geosciences , Vol. 1 7 , No. 4 , p. 355 - 360 , December 2006 Printed in China
ISSN 1002 - 0705
Method of Numerical Modeling for Constitutive Relations of Clay Zhou Baochun'
(%@%)
School of C i v i l Engineering and Mechanics, Huazhong University of Science and T e c h n o l o g y , Wuhan 430074, China ; Department of Architectural Engineering, X i n y a n g Normal U n i v e r s i t y , X i n y a n g 464000, China
Wang Jingtao
(Z$$&) Wei Jun
(ZF)
School of C i v i l Engineering and Mechanics, Huazhong University of Science and T e c h n o l o g y , Wuhan 430074, China ABSTRACT: In order to study the method of numerical modeling for constitutive relations of clay, on the basis of the principle of interaction between plastic volumetric strain and plastic generalized shear strain, the two constitutive functionals that include the function of stress path were used as the basic framework of the constitutive model, which are able to demonstrate the dependence of stress path. The two partial differential cross terms appear in the expression of stress-strain increment relation, which are used to demonstrate the interaction between plastic volumetric strain and plastic generalized shear strain. The elasoplastic constitutive models of clay under two kinds of stress paths, CTC and TC, have been constructed using the triaxial test results. The three basic characteristics of deformation of soils, pressure sensitivity, dilatancy, and dependence of stress path, are well explained using these two models. Using visualization, the three-dimensional surfaces of shear and volume strains in the whole stress field under stress paths of CTC and TC are given. In addition, the two families of shear and volumetric yield loci under CTC and TC paths are plotted respectively. By comparing the results of deformation under these two stress paths, it has been found that, there are obvious differences in the strain peaks, the shapes of strain surfaces, and the trends of variation of volumetric yield loci, however both families of shear yield loci are similar. These results demonstrate that the influences of stress path on the constitutive relations of clay are considerably large and not negligible. The numerical modeling method that can sufficiently reflect the dependence of stpath is superior to the traditional one. KEY WORDS: numerical modeling, clay, constitutive relations, stress path.
stress path. Wang (2004) proposed a principle of the INTRODUCTION T h e mechanical responses of soils are more com- interaction between plastic volume and shear strains, plicated compared with metals. By comparing the that is, there are two relatively independent strains : physical and mechanical properties of the metals with volume and shear strains in the plastic deformation of soils, Lade (1988) found that there are 17 different rocks and soils. Also there are complex and nonlinear points between the metals and soils, which differ interactions in rocks and soils, which generates comfrom metals in three basic mechanical characteristics: plexity and variety of deformations in rocks and pressure sensitivity, dilatancy , and dependence of soils, and this interaction is called the principle of .e:.ez interaction. In the plastic deformation of metals, This paper is supported by the Natural Science Foundation of Henan there is no interaction between plastic volume and Province (No. 0511045200) and the Youth Science Foundation of Xinshear strains because the plastic volume strain is apyang Normal University (No. 20050107). proximately equal t o zero, which is only a single * Corresponding author: zhoubaochun@ smail. hust. edu. cn process of shear deformation. However, large plastic Manuscript received July 12, 2006. volume strain appears in the deformation of rocks and Manuscript accepted September 25, 2006.
Zhou Baochun, Wang Jingtao and Wei Jun
356
soils. T h e interaction between volume and shear strains is of significant importance. T h e pressure sensitivity and dilatancy are the behaviors of this interaction. It has been demonstrated theoretically as well as experimentally that the dependence of constitutive relations on the stress path is the composite behavior of this interaction that not only includes pressure Sensitivity, but also dilatancy effects. Wang (2002) proposed a method of numerical modeling of constitutive relations in rocks and soils. On the basis of the inverse problem theory (Tarantola, 1987), the four basic steps of numerical modeling were obtained. T h e objective of modeling constitutive relations in soils is to describe the basic properties of deformation of soil completely and accurately. According to the principle of €:-€: interaction, t o describe the three basic characteristics of deformation of soil, it is necessary that the interaction between E: and E! should be considered in the numerical modeling. In numerical modeling, to show the dependence of stress path, the two constitutive functionals that include the functions of stress path are chosen a s the basic framework of model, from which the expressions of stress-strain increment relation are derived, which involve the interactions between E! and €,P. The triaxial tests of clay under two kinds of stress paths, CTC and T C , were carried out. According to the results of triaxial tests, the two sets of elastoplastic constitutive equations have been established. A stress-strain computer program has been developed to calculate the deformation of the samples of triaxial tests. T h e computation results coincide well with the test curves, which demonstrates that the numerical modeling method is feasible and effective. Using visualization, the three-dimensional surfaces of shear and volume strains in the whole stress field ( p , q ) under stress paths of CTC and T C are given. In addition, the two families of shear and volumetric yield loci under CTC and T C are plotted. By comparing these results, it is found that there are obvious differences between the three-dimensional surfaces of deformation and yield loci under the two kinds of stress paths. T h e above results show that
the constitutive models constructed for clay are able to describe the basic properties of deformation in soils, especially the influence of stress path. T h e dependence of constitutive relations of clay on the stress path is considerably large and not negligible. Also it is shown that the numerical modeling method that can sufficiently reflect the dependence of stress path is superior to the traditional one.
BASIC FRAMEWORK OF NUMERICAL MODELING In the numerical method for modeling the constitutive relations in rocks and soils (Wang, 2002) , the incremental stress-strain relation of soil is expressed as follows (Ren and Wang , 2005) dr! = ( a f i / a p ) d p (afJaq)dq (1) dd' = ( a f i / a p ) d p (afi/aq)dq (2) where E,P is the plastic volumetric strain, s.Sp is the plastic generalized shear strain, and p and q are the mean normal stress and the generalized shear stress respectively, fland fi are the two constitutive functionals which reflect the interaction between E: and €I). T o determine the coefficient functions in equations ( 1 ) and ( 2 ) , the stress-strain relation in the whole stress field is calculated from the triaxial test data using a highly precise fitting approach, from which the four coefficient functions can be derived, which are expressed as a series of Gauss function
+ +
N
f(p,
4 ) = xw(k)exp(k=i
d(P
-
pk>z
+ (4 -
(3) where e x p ( - - a ( ( p - p P 6 ) 2 + ( q - q k ) 2 ) ) is the Gauss function, w ( K ) is the weight of the Gauss central point, p , and q b are the coordinates of the Gauss central point in the stress field ( p , q ) , N is the number of the Gauss central points. qP>'>>
PREPARATION OF SOIL SAMPLES T h e soil samples were collected from a buildingsite. T h e soil samples were prepared strictly according t o the specification of soil test (Nanjing Hydraulic Research Institute, 1999). Atterberg limits test and specific gravity test are carried out. T h e parameters of physical properties are given in Table 1.
Table 1 The parameters of physical properties 17 mm liquid limit ( % )
limit ( % )
(%>
48. 19
42.67
29.52
10
mm
liquid
Plastic limit
Plasticity index 18.67
Water ratio
Dry density
Specific
(%)
(P;/cm3)
gravity
33. 4
1. 44
2. 73
Method of Numerical Modeling for Constitutive Relations of Clay
DESIGN OF TRIAXIAL TESTS Triaxial compression tests adopt drained loading. The soil samples are divided into two groups, each containing four samples. After installed on the tester, back pressure saturation is carried out. T h e samples of each group are consolidated under confining pressures of a3 = 100, Z O O , 300, 400 kPa after complete saturation. After consolidation, drained triaxial compression test is carried out until the axial strain reaches ZO%, where the axial strain rate is 0.014 ydmin. In the tests, the first group followed increasing p stress path, that is the path of conventional triaxial compression (CTC) : two of the principal stresses (az and a,) are kept constant while the principle stress a1 is increased, hence the increment in p is given by Ap= ( Aal f 2 A a 3 )/3>0. The second group followed constant p ( triaxial compression: T C ) path: a1 is increased by AD,, whereas both az and a3 are decreased such that Aaz = Aas = - AD,/ 2 , and p always remains constant. Isotropic consolidation test is carried out to find the elastic bulk modu-
357
lus K . T h e elastic shear modulus G is measured using the drained triaxial compression test with loading-unloading cycle.
STRESS-STRAIN CURVE a1-a3-~1 curves (el is axial strain) and E " - E ~ curves ( E , is volumetric strain) under CTC and T C stress paths are shown in Fig. 1. It is clear that all the soil samples under CTC and T C stress paths demonstrate the characteristic of strain hardening, but the stress under the CTC path is much higher compared with the TC path. Furthermore, Fig. 1 shows that the volumetric strains under CTC and TC stress paths are purely compressive, and increase with the confining pressures. T h e volumetric strain peaks under the CTC path are approximately two times larger compared with the peaks under the TC path. T h e large difference between the stress-strain relations under CTC and TC stress paths demonstrates that the dependence of constitutive relations of clay on the stress path is considerably large.
600 V
h
P I
% .
V
g
x&@(
urcr#x p(300kPa)
400
kPa)
* 7 2 0 0
b
200
p ( 100 kPa)
0
20
10
0
30
20
10
30
El(%)
El(%)
I
I
2.0
0
10
20
30 El(%)
El(%)
Figure 1. u1-a3-el and YIELD SURFACES T h e theory of two yield surfaces is used in this article. E! and E: are respectively chosen as the hardening parameters of shear and volumetric yield loci, which can be directly plotted using the triaxial test
curves.
data ( H u a n g , 1983). T h e elastic bulk modulus K is calculated by the isotropic consolidation test with loading-unloading cycle, and the elastic shear modulus G is calculated by the conventional triaxial compression tests with loading-unloading cycle ( L u et
Zhou Baochun, Wang Jingtao and Wei Jun
358
al. , 1984) K = 158. 161p, ( ' p pa)'.
(4)
730
G=327. 11p.((p-d3)/p.)o.888i (5) where p,=lOl. 3 k f a . The two families of shear and volumetric yield loci under CTC and T C stress paths are plotted using the method mentioned above and are shown in Fig.
2. It is found that both families of shear yield loci are similar, but the shear strains for TC rise rapidly. T h e volumetric yield loci for CTC first turn right , and then slowly turn left. However the volumetric yield loci for TC turn left monotonously. These facts demonstrate the dependence of deformation process in soils on the stress path.
400-
TC
&.'(18.5%)
16.6
3 300a
h
c
200100 -
500
800 E'.(
18.8%) c
400-
TC
d(17.1%) 16.3
h
I
NUMERICAL MODELING FOR CLAY Stress-strain relations in the whole stress field
and q remains constant respectively, are considerably large.
( p , q ) under CTC and T C stress paths are calculated from triaxial test data using a highly precision fitting method, and the threedimensional surfaces of E ~ - (p , q ) and E , - ( P , q ) relations under CTC and T C stress paths are shown in Fig. 3. It is found by the comparison of two kinds of the surfaces that there are obvious differences in rising slopes and stress scopes between the shear and volume surfaces. For the volume surfaces, the peak value under CTC stress path is much higher compared with the peak value under TC. Rising slopes under T C are steeper compared with the rising slopes under CTC in either shear or volume surface, which are consistent with the results of yield loci. These results further demonstrate that the influence of stress path on the constitutive relations is significant, and the differences between the effects of the two stress paths, in which p increases
VERIFICATION OF CONSTITUTIVE MODEL T o examine the accuracies of the constitutive models constructed in this article, stress-strain computer programs which fit increment expressions t o stress-strain relationships are developed according to the stress paths occurring in the triaxial compression tests. On the basis of the elastoplastic constitutive equations proposed in this article, the deformations of two samples for aa=300 kPa under CTC and for p = 300 kPa under T C are calculated. T h e computed and the experimental o , - u - E , and E,-E, curves under CTC and TC stress paths are shown in Fig. 4. T h e computation results coincide well with the results of test curves, and this demonstrates that the numerical modeling method is effective and accurate.
Method of Numerical Modeling for Constitutive Relations of Clay
359
Figure 3. Relationship between gS-(p , q ) and E"-( p , q ) .
Y
?
300x test-computation
l3 200-
x test-computation
I
0
20
10
30
0
10
20
30
El(%)
El(%)
p(300 kPa) 2.0 n
s Y
1.5
x test-computation
x test-computation 0.5 0.
0
20
10 &I(%)
30
0
20
10
30
EI(%)
Figure 4. The computation and test curves.
CONCLUSIONS tutive model, which are able to demonstrate the deOn the basis of the principle of E ~ - E , P interaction, pendence of the stress path. The two partial differenthe method of numerical modeling for constitutive re- tial cross terms in the expression of stress-strain inlations of clay has been studied. T h e two constitutive crement relation are used to reflect the interaction befunctionals that include the function of the stress tween E: and E ! . T h e elastoplastic constitutive models path were used as the basic framework of this consti- of clay under these two stress paths, C T C and T C ,
360
have been constructed using numerical modeling method. The three basic characteristics of deformstion for soils, pressure sensitivity dilatancy , and dependence of stress path, are satisfactorily described by these models. Therefore, the numerical modeling method is considered t o be superior t o the traditional one. Using visualization, the three-dimensional surfaces of shear and volume strains in the whole stress field under stress paths of CTC and TC are given. By the comparison of these two surfaces, it is found that there are obvious differences in slope and scope between the two shear surfaces and also between the two volume surfaces. These results further demonstrate that the influence of stress path on the constitutive relations is significant especially the differences between the effects of the two stress paths, in which p increases and q remains constant respectively, are considerably large. By the comparison of the two families of shear and volumetric yield loci under CTC and T C paths, it is found that both families of shear yield loci are similar, but the shear strains for TC rise rapidly. T h e volumetric yield loci for CTC first slightly turn right, and slowly turn left. However, the volumetric yield loci for T C turn left monotonously. These facts demonstrate that the deformation of soils strongly depends on the stress path.
Zhou Baochun? Wang Jingtao and Wei Jun REFERENCES CITED Huang, w. x . , 1983. Engineering Characteristics of soil. Hydraulic and Electric Press, Beijing (in Chinese) Lade, P. V. , 1988. Effects of Voids and Volume Changes on the Behavior of Frictional Materials. International Journal f o r Numerical and Analytical Methods in Geomechanics , (12) : 351-370 Lu, P. Y.,Chen, S. Y . , Xiong, L. Z., et al., 1984. Elastoplastic Constitutive Equation for Hydraulic-Fill Soil of Slurry-Fall Dam. Chinese Journal of Geotechnical Engineering, 6 ( 2 ) : 23 - 39 ( in Chinese with English Abst rac t ) Nanjing Hydraulic Research Institute, 1999. Specification of Soil Test. Water Resources and Electric Power Press, Beijing (in Chinese) Ren, Q. Y. , Wang, J. T . , 2005. Numerical Method for Modeling the Constitutive Relationship of Sand under Different Stress Paths. Journal of China University of Geosciences, 16(3) : 268-270 Tarantola, A. , 1987. Inverse Problem Theory. Elsevier Science Publishers, Amsterdam. Wang, J. T. , 2002. Numerical Method in Modeling the Constitutive Relations of Rock and Soil. Journal of Huazhong University o f Science and Technology ( Urban Science E d i t i o n ) , 19( 1 ) : 44-47 (in Chinese with English Abstract) Wang, J. T. , 2004. Particularity and Unity of the Constitutive Relations for Rock and Soil. Journal of Huazhong University of Science and Technology (Urban Science Edit i o n ) , 21(4) : 5-8 (in Chinese with English Abstract)
ISSN 1002 - 0705
Method of Numerical Modeling for Constitutive Relations of Clay Zhou Baochun'
(%@%)
School of C i v i l Engineering and Mechanics, Huazhong University of Science and T e c h n o l o g y , Wuhan 430074, China ; Department of Architectural Engineering, X i n y a n g Normal U n i v e r s i t y , X i n y a n g 464000, China
Wang Jingtao
(Z$$&) Wei Jun
(ZF)
School of C i v i l Engineering and Mechanics, Huazhong University of Science and T e c h n o l o g y , Wuhan 430074, China ABSTRACT: In order to study the method of numerical modeling for constitutive relations of clay, on the basis of the principle of interaction between plastic volumetric strain and plastic generalized shear strain, the two constitutive functionals that include the function of stress path were used as the basic framework of the constitutive model, which are able to demonstrate the dependence of stress path. The two partial differential cross terms appear in the expression of stress-strain increment relation, which are used to demonstrate the interaction between plastic volumetric strain and plastic generalized shear strain. The elasoplastic constitutive models of clay under two kinds of stress paths, CTC and TC, have been constructed using the triaxial test results. The three basic characteristics of deformation of soils, pressure sensitivity, dilatancy, and dependence of stress path, are well explained using these two models. Using visualization, the three-dimensional surfaces of shear and volume strains in the whole stress field under stress paths of CTC and TC are given. In addition, the two families of shear and volumetric yield loci under CTC and TC paths are plotted respectively. By comparing the results of deformation under these two stress paths, it has been found that, there are obvious differences in the strain peaks, the shapes of strain surfaces, and the trends of variation of volumetric yield loci, however both families of shear yield loci are similar. These results demonstrate that the influences of stress path on the constitutive relations of clay are considerably large and not negligible. The numerical modeling method that can sufficiently reflect the dependence of stpath is superior to the traditional one. KEY WORDS: numerical modeling, clay, constitutive relations, stress path.
stress path. Wang (2004) proposed a principle of the INTRODUCTION T h e mechanical responses of soils are more com- interaction between plastic volume and shear strains, plicated compared with metals. By comparing the that is, there are two relatively independent strains : physical and mechanical properties of the metals with volume and shear strains in the plastic deformation of soils, Lade (1988) found that there are 17 different rocks and soils. Also there are complex and nonlinear points between the metals and soils, which differ interactions in rocks and soils, which generates comfrom metals in three basic mechanical characteristics: plexity and variety of deformations in rocks and pressure sensitivity, dilatancy , and dependence of soils, and this interaction is called the principle of .e:.ez interaction. In the plastic deformation of metals, This paper is supported by the Natural Science Foundation of Henan there is no interaction between plastic volume and Province (No. 0511045200) and the Youth Science Foundation of Xinshear strains because the plastic volume strain is apyang Normal University (No. 20050107). proximately equal t o zero, which is only a single * Corresponding author: zhoubaochun@ smail. hust. edu. cn process of shear deformation. However, large plastic Manuscript received July 12, 2006. volume strain appears in the deformation of rocks and Manuscript accepted September 25, 2006.
Zhou Baochun, Wang Jingtao and Wei Jun
356
soils. T h e interaction between volume and shear strains is of significant importance. T h e pressure sensitivity and dilatancy are the behaviors of this interaction. It has been demonstrated theoretically as well as experimentally that the dependence of constitutive relations on the stress path is the composite behavior of this interaction that not only includes pressure Sensitivity, but also dilatancy effects. Wang (2002) proposed a method of numerical modeling of constitutive relations in rocks and soils. On the basis of the inverse problem theory (Tarantola, 1987), the four basic steps of numerical modeling were obtained. T h e objective of modeling constitutive relations in soils is to describe the basic properties of deformation of soil completely and accurately. According to the principle of €:-€: interaction, t o describe the three basic characteristics of deformation of soil, it is necessary that the interaction between E: and E! should be considered in the numerical modeling. In numerical modeling, to show the dependence of stress path, the two constitutive functionals that include the functions of stress path are chosen a s the basic framework of model, from which the expressions of stress-strain increment relation are derived, which involve the interactions between E! and €,P. The triaxial tests of clay under two kinds of stress paths, CTC and T C , were carried out. According to the results of triaxial tests, the two sets of elastoplastic constitutive equations have been established. A stress-strain computer program has been developed to calculate the deformation of the samples of triaxial tests. T h e computation results coincide well with the test curves, which demonstrates that the numerical modeling method is feasible and effective. Using visualization, the three-dimensional surfaces of shear and volume strains in the whole stress field ( p , q ) under stress paths of CTC and T C are given. In addition, the two families of shear and volumetric yield loci under CTC and T C are plotted. By comparing these results, it is found that there are obvious differences between the three-dimensional surfaces of deformation and yield loci under the two kinds of stress paths. T h e above results show that
the constitutive models constructed for clay are able to describe the basic properties of deformation in soils, especially the influence of stress path. T h e dependence of constitutive relations of clay on the stress path is considerably large and not negligible. Also it is shown that the numerical modeling method that can sufficiently reflect the dependence of stress path is superior to the traditional one.
BASIC FRAMEWORK OF NUMERICAL MODELING In the numerical method for modeling the constitutive relations in rocks and soils (Wang, 2002) , the incremental stress-strain relation of soil is expressed as follows (Ren and Wang , 2005) dr! = ( a f i / a p ) d p (afJaq)dq (1) dd' = ( a f i / a p ) d p (afi/aq)dq (2) where E,P is the plastic volumetric strain, s.Sp is the plastic generalized shear strain, and p and q are the mean normal stress and the generalized shear stress respectively, fland fi are the two constitutive functionals which reflect the interaction between E: and €I). T o determine the coefficient functions in equations ( 1 ) and ( 2 ) , the stress-strain relation in the whole stress field is calculated from the triaxial test data using a highly precise fitting approach, from which the four coefficient functions can be derived, which are expressed as a series of Gauss function
+ +
N
f(p,
4 ) = xw(k)exp(k=i
d(P
-
pk>z
+ (4 -
(3) where e x p ( - - a ( ( p - p P 6 ) 2 + ( q - q k ) 2 ) ) is the Gauss function, w ( K ) is the weight of the Gauss central point, p , and q b are the coordinates of the Gauss central point in the stress field ( p , q ) , N is the number of the Gauss central points. qP>'>>
PREPARATION OF SOIL SAMPLES T h e soil samples were collected from a buildingsite. T h e soil samples were prepared strictly according t o the specification of soil test (Nanjing Hydraulic Research Institute, 1999). Atterberg limits test and specific gravity test are carried out. T h e parameters of physical properties are given in Table 1.
Table 1 The parameters of physical properties 17 mm liquid limit ( % )
limit ( % )
(%>
48. 19
42.67
29.52
10
mm
liquid
Plastic limit
Plasticity index 18.67
Water ratio
Dry density
Specific
(%)
(P;/cm3)
gravity
33. 4
1. 44
2. 73
Method of Numerical Modeling for Constitutive Relations of Clay
DESIGN OF TRIAXIAL TESTS Triaxial compression tests adopt drained loading. The soil samples are divided into two groups, each containing four samples. After installed on the tester, back pressure saturation is carried out. T h e samples of each group are consolidated under confining pressures of a3 = 100, Z O O , 300, 400 kPa after complete saturation. After consolidation, drained triaxial compression test is carried out until the axial strain reaches ZO%, where the axial strain rate is 0.014 ydmin. In the tests, the first group followed increasing p stress path, that is the path of conventional triaxial compression (CTC) : two of the principal stresses (az and a,) are kept constant while the principle stress a1 is increased, hence the increment in p is given by Ap= ( Aal f 2 A a 3 )/3>0. The second group followed constant p ( triaxial compression: T C ) path: a1 is increased by AD,, whereas both az and a3 are decreased such that Aaz = Aas = - AD,/ 2 , and p always remains constant. Isotropic consolidation test is carried out to find the elastic bulk modu-
357
lus K . T h e elastic shear modulus G is measured using the drained triaxial compression test with loading-unloading cycle.
STRESS-STRAIN CURVE a1-a3-~1 curves (el is axial strain) and E " - E ~ curves ( E , is volumetric strain) under CTC and T C stress paths are shown in Fig. 1. It is clear that all the soil samples under CTC and T C stress paths demonstrate the characteristic of strain hardening, but the stress under the CTC path is much higher compared with the TC path. Furthermore, Fig. 1 shows that the volumetric strains under CTC and TC stress paths are purely compressive, and increase with the confining pressures. T h e volumetric strain peaks under the CTC path are approximately two times larger compared with the peaks under the TC path. T h e large difference between the stress-strain relations under CTC and TC stress paths demonstrates that the dependence of constitutive relations of clay on the stress path is considerably large.
600 V
h
P I
% .
V
g
x&@(
urcr#x p(300kPa)
400
kPa)
* 7 2 0 0
b
200
p ( 100 kPa)
0
20
10
0
30
20
10
30
El(%)
El(%)
I
I
2.0
0
10
20
30 El(%)
El(%)
Figure 1. u1-a3-el and YIELD SURFACES T h e theory of two yield surfaces is used in this article. E! and E: are respectively chosen as the hardening parameters of shear and volumetric yield loci, which can be directly plotted using the triaxial test
curves.
data ( H u a n g , 1983). T h e elastic bulk modulus K is calculated by the isotropic consolidation test with loading-unloading cycle, and the elastic shear modulus G is calculated by the conventional triaxial compression tests with loading-unloading cycle ( L u et
Zhou Baochun, Wang Jingtao and Wei Jun
358
al. , 1984) K = 158. 161p, ( ' p pa)'.
(4)
730
G=327. 11p.((p-d3)/p.)o.888i (5) where p,=lOl. 3 k f a . The two families of shear and volumetric yield loci under CTC and T C stress paths are plotted using the method mentioned above and are shown in Fig.
2. It is found that both families of shear yield loci are similar, but the shear strains for TC rise rapidly. T h e volumetric yield loci for CTC first turn right , and then slowly turn left. However the volumetric yield loci for TC turn left monotonously. These facts demonstrate the dependence of deformation process in soils on the stress path.
400-
TC
&.'(18.5%)
16.6
3 300a
h
c
200100 -
500
800 E'.(
18.8%) c
400-
TC
d(17.1%) 16.3
h
I
NUMERICAL MODELING FOR CLAY Stress-strain relations in the whole stress field
and q remains constant respectively, are considerably large.
( p , q ) under CTC and T C stress paths are calculated from triaxial test data using a highly precision fitting method, and the threedimensional surfaces of E ~ - (p , q ) and E , - ( P , q ) relations under CTC and T C stress paths are shown in Fig. 3. It is found by the comparison of two kinds of the surfaces that there are obvious differences in rising slopes and stress scopes between the shear and volume surfaces. For the volume surfaces, the peak value under CTC stress path is much higher compared with the peak value under TC. Rising slopes under T C are steeper compared with the rising slopes under CTC in either shear or volume surface, which are consistent with the results of yield loci. These results further demonstrate that the influence of stress path on the constitutive relations is significant, and the differences between the effects of the two stress paths, in which p increases
VERIFICATION OF CONSTITUTIVE MODEL T o examine the accuracies of the constitutive models constructed in this article, stress-strain computer programs which fit increment expressions t o stress-strain relationships are developed according to the stress paths occurring in the triaxial compression tests. On the basis of the elastoplastic constitutive equations proposed in this article, the deformations of two samples for aa=300 kPa under CTC and for p = 300 kPa under T C are calculated. T h e computed and the experimental o , - u - E , and E,-E, curves under CTC and TC stress paths are shown in Fig. 4. T h e computation results coincide well with the results of test curves, and this demonstrates that the numerical modeling method is effective and accurate.
Method of Numerical Modeling for Constitutive Relations of Clay
359
Figure 3. Relationship between gS-(p , q ) and E"-( p , q ) .
Y
?
300x test-computation
l3 200-
x test-computation
I
0
20
10
30
0
10
20
30
El(%)
El(%)
p(300 kPa) 2.0 n
s Y
1.5
x test-computation
x test-computation 0.5 0.
0
20
10 &I(%)
30
0
20
10
30
EI(%)
Figure 4. The computation and test curves.
CONCLUSIONS tutive model, which are able to demonstrate the deOn the basis of the principle of E ~ - E , P interaction, pendence of the stress path. The two partial differenthe method of numerical modeling for constitutive re- tial cross terms in the expression of stress-strain inlations of clay has been studied. T h e two constitutive crement relation are used to reflect the interaction befunctionals that include the function of the stress tween E: and E ! . T h e elastoplastic constitutive models path were used as the basic framework of this consti- of clay under these two stress paths, C T C and T C ,
360
have been constructed using numerical modeling method. The three basic characteristics of deformstion for soils, pressure sensitivity dilatancy , and dependence of stress path, are satisfactorily described by these models. Therefore, the numerical modeling method is considered t o be superior t o the traditional one. Using visualization, the three-dimensional surfaces of shear and volume strains in the whole stress field under stress paths of CTC and TC are given. By the comparison of these two surfaces, it is found that there are obvious differences in slope and scope between the two shear surfaces and also between the two volume surfaces. These results further demonstrate that the influence of stress path on the constitutive relations is significant especially the differences between the effects of the two stress paths, in which p increases and q remains constant respectively, are considerably large. By the comparison of the two families of shear and volumetric yield loci under CTC and T C paths, it is found that both families of shear yield loci are similar, but the shear strains for TC rise rapidly. T h e volumetric yield loci for CTC first slightly turn right, and slowly turn left. However, the volumetric yield loci for T C turn left monotonously. These facts demonstrate that the deformation of soils strongly depends on the stress path.
Zhou Baochun? Wang Jingtao and Wei Jun REFERENCES CITED Huang, w. x . , 1983. Engineering Characteristics of soil. Hydraulic and Electric Press, Beijing (in Chinese) Lade, P. V. , 1988. Effects of Voids and Volume Changes on the Behavior of Frictional Materials. International Journal f o r Numerical and Analytical Methods in Geomechanics , (12) : 351-370 Lu, P. Y.,Chen, S. Y . , Xiong, L. Z., et al., 1984. Elastoplastic Constitutive Equation for Hydraulic-Fill Soil of Slurry-Fall Dam. Chinese Journal of Geotechnical Engineering, 6 ( 2 ) : 23 - 39 ( in Chinese with English Abst rac t ) Nanjing Hydraulic Research Institute, 1999. Specification of Soil Test. Water Resources and Electric Power Press, Beijing (in Chinese) Ren, Q. Y. , Wang, J. T . , 2005. Numerical Method for Modeling the Constitutive Relationship of Sand under Different Stress Paths. Journal of China University of Geosciences, 16(3) : 268-270 Tarantola, A. , 1987. Inverse Problem Theory. Elsevier Science Publishers, Amsterdam. Wang, J. T. , 2002. Numerical Method in Modeling the Constitutive Relations of Rock and Soil. Journal of Huazhong University o f Science and Technology ( Urban Science E d i t i o n ) , 19( 1 ) : 44-47 (in Chinese with English Abstract) Wang, J. T. , 2004. Particularity and Unity of the Constitutive Relations for Rock and Soil. Journal of Huazhong University of Science and Technology (Urban Science Edit i o n ) , 21(4) : 5-8 (in Chinese with English Abstract)