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A Critical State Bounding Surface Model for Sands
SECTION
ii.ii
A Critical State Bounding Surface Model for Sands M A J I D T. MANZARI 1 a n d YANNIS E DAFALIAS 2
1Civil and Environmental Engineering, The George Washington University, Washington, D.C., USA 2Department of Mechanics, National Technical University of Athens, 15773, Hellas, and Civil and Environmental Engineering, University of California, Davis, California, USA
Contents 11.11.1 Triaxial Space Formulation . . . . . . . . . . 1164 11.11.1.1 Basic Equations . . . . . . . . . . . . 1165 11.11.1.2 Critical State . . . . . . . . . . . . . . 1166 11.11.2 Multiaxial Stress Space Generalization 1167 11.11.3 Implementation and Model Constants 1169 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . 1170 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1170
11.11.1 TRIAXIAL SPACE FORMULATION The following constitutive model applies to sandy soils at different densities and pressures which do not cause crushing of the grains. It is a general purpose model for multiaxial, drained, undrained, monotonic, and cyclic loading conditions, within the general framework of critical state soil mechanics and bounding surface plasticity. The presentation is a direct derivative of the work by Manzari and Dafalias [1] and includes some additional expressions. The basic concepts and related equations of the sand plasticity model will first be presented in the classical triaxial space where q = or1 - ~r3, p : ( 1 / 3 ) ( ~ r l + 2cr3), ~q = ( 2 / 3 ) ( e l - ~3), and ~v = ~1 + 2~3, w i t h ~i and ~i (i = 1,3) being the p r i n c i p a l stress and strains (or2 = or3,
82
=
1164
83). Handbookof MaterialsBehaviorModels.ISBN0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
11.11 A Critical State Bounding Surface Model for Sands
1165
11.11.1.1 BASICEQUATIONS The elastic relation will be assumed hypoelastic for simplicity, given in terms of the r a t e s / / = dq/dt and ib = dp/dt as 9e _ // eq 3 6 O - Go
.e _ /5 ~
(1)
/ < - Ko
(2)
where G and K are the elastic shear and bulk moduli, respectively, Pat is the atmospheric pressure, and the exponent a is usually given a default value of a =0.5. The elastic range is represented by the shaded wedge shown in Figure 11.11.1 in the q, p space, whose straight line boundaries Oc and Oe constitute the yield surface described analytically by f=~/-~:Fm=O
(3)
where the stress ratio r / = q/p and the dimensionless (stress ratio type) quantities c~ and m are shown in Figure 11.11.1. The c~-line is the bisector of the wedge angle, and 2 mp measures the wedge "opening." Equation 3 implies that upon constant r/ loading no plastic loading occurs, which is approximately correct if the p is not high enough to cause crushing of the sand grains and/or the sand sample is not very loose. Based on Eq. 3, plastic deformation occurs only when r/is on f = 0 and there is a change dq = Odt pointing outwards f = O. In this case the plastic rate equations are given by
O
(4/
FIGURE 11.11.1 Schematicrepresentation of the yield, critical, dilatancy, and bounding lines in
q, p space.
1166
Manzari
and Dafalias
where Kp and D are the plastic modulus and dilatancy, respectively. Since .e + ~v 9p, it is a trivial exercise to combine Eqs. 1 and 3 in ~q = ' e~q+ ~ a n d ~ v = ~v order to express ~q and ~v in terms o f / / ( o r ~) and/~, and vice versa. There remains the very important task of specifying Kp and D. In reference to Figure 11.11.1, assume that the line shown with a slope Mcb represents a peak stress ratio for a given state. Henceforth, subscripts c and e imply association of a quantity with triaxial compression and extension, respectively. Such value of Mcb is a bound for ~/; hence, within the framework of bounding surface plasticity one can write a stress ratio "distance"-dependent plastic modulus expression such as K~ -- h ( M ~ - ~)
(5)
in terms of a model parameter h. Similarly, the line shown with a slope M~ in Figure 11.11.1, represents the phase transformation line, or for better naming, the dilation line. According to standard dilatancy theory, one can write
D -- A ( M ~ - ~1)
(6)
with A another model parameter. Hence, Eqs. 5 and 6 determine Kp and D via M~, Mca, h and A, and the model is complete.
11.11.1.2 CRITICALSTATE The critical state in soil mechanics is defined as a triplet of qc, Pc, and void ratio ec values, at which unlimited plastic deviatoric strain occurs at zero volumetric strain rate. Such a critical state is defined simultaneously in the q,p space by t/c--qc/pc=Mc, and in the e - p space by ec = (ec)ref- 2ln(pc/pref), where Me, 2, and (ec)ref (for a chosen pref) are standard soil constants. The Mc is related to friction angle, and its corresponding visualization in q, p space is shown as a line of slope Mc in Figure 11.11.1. If left as is, the previous formulation will not meet the critical state requirements. For example, it follows from Eqs. 4, 5, and 6 that as ~/ approaches M~, which may be assumed to be equal to Me, Kp~ 0 and ~ ~ 0 while ~ > 0 and ~v p < 0 since M~ > M~. This implies unlimited dilation, contrary to physical expectation (negative volumetric strain rate means dilation). Furthermore, a fixed M~ does not allow for the softening response in drained loading observed in many dense sand samples. The remedy is to consider variable M~ and Mca, such that at critical state Mcb = Mca = Me. If the so-called state parameter ~ = e - ec [2], is used as a measure of "distance" from critical state in the e, p space (e and ec refer to the same p), the idea put forth by Wood et al. [3] and supplemented by Manzari
11.11 A Critical State Bounding Surface Model for Sands
1167
and Dafalias [1] in regards to the concept of 0~ in Eq. 3 can be expressed by M~-- (zc + m = Mc - kbc~
(7)
where kcb is a material constant and ~cb is the "bound" for c~ in the same sense that M~ is the bound for r/. The second important modification refers to the variation of Mca and was proposed in Reference [1] as = o~c 4- m -- Mc 4-
(8)
where k~ is a material constant and a~a is a back-stress dilatancy ratio corresponding to the stress dilatancy ratio Mca. Observe that for ~ < 0 (denser than critical), M a c < M c < M ~ , while for ~ > 0 (looser than critical), Mc~
(9)
which defines the critical back-stress ratio ~ in terms of m and the critical stress ratio Mc. Equations 1-8 provide a complete constitutive model in triaxial space. It only requires the specification of Me, k~, and kea in triaxial extension as well to describe reverse and cyclic loading. In such case, Eqs. 5 and 6 utilize the ensuing values of M~ and M~ in extension, following from Eqs. 7 and 8.
11.11.2 MULTIAXIAL GENERALIZATION
STRESS SPACE
The multiaxial stress generalization of the model follows standard procedures [1]. Equation 1 becomes ~=
s 2--G
P ~ - K
(10)
where bold-face characters imply tensor quantities and e e and s are the deviatoric elastic strain and stress tensors, respectively. Equation 2 remains as is. Equation 3 generalizes to f -- [(s - pat):(s - pot)] 1/2 - X//-i/3 m p -- 0
(11)
where ~ of Eq. 3 generalizes to the back-stress-ratio tensor at and : implies the trace of the product of two tensors. The plastic strain rate equations are expressed in the following in terms of i', where the deviatoric stress ratio tensor r - s/p is the generalization of r/. After some algebra which involves the O f / O ~ = n - (1/3)(n : r)I with I the identity tensor and n given in following text, one has for the deviatoric and
1168
Manzari
and Dafalias
volumetric plastic strain rates, respectively, ~P -- < L > n - - < L > ( 2 / 3 ) - 1 / 2 ( r - a t ) L
1
~P -- < L > D
2Gn" ~ - K(n'r)~v -
+
2O
-
(12a)
(12b)
D(n
Equations 12 are the generalizationof Eqs. 4 involving the plasticmodulus Kp and dilatancy D, with the added explicit feature of the loading index L expressed either in terms of/" or in terms of ~ and ev. A combination of Eqs. 10 and 12 together with ~ - he + ~P, ev - ~e + ~v p yields 6 -- E ep" /;-- [Ee -
(2Gn + K D I ) | ( 2 G n - K(n 9 r)I)] Kp 4- 2 G - KD(n i~) _ "~
(13)
for the total effective stress rate 8 in terms of the total strain rate ~ and the elastoplastic tangent stiffness moduli E~. The latter is given explicitly by the bracketed quantity of the last member of Eq. 13, where | means tensor product and E ~ is the well known isotropic elastic moduli tensor which in component form is given by E~jkz -- 2 G ( 6 i k 6 j l - (1/3)cSijcSkz) + K6ij6kz. The Kp and D in the muhiaxial space will be obtained by generalization of corresponding triaxial concepts and equations. First, the bounding, dilatancy, b ~c, d and ~cc in Eqs. 7 , 8, and 9, and critical triaxial back-stress ratios c~c, correspondingly, are generalized as the bounding, dilatancy, and critical surfaces whose traces on the principal stress ratio ri = s i / p 7z-plane are shown in Figure 11.11.2. The To-plane trace of the yield surface with its center a, Eq. 1 l, is also shown in Figure 11.11.2 as a circle. For a stress ratio r and
/~.~
~ t~,~_. ~
./ // ,t~'~~ ,,t ,, / / ' n'~J "
r2 = ---~
Yield Surface
i~'.. ) BoundingSurface r - ' . ~ , . - ~ CriticalSurface ~ ancy Surface
...................
-
tl~+~
_
s3
r3----~
FIGURE 11.11.2 Schematic representation of the yield, critical, dilatancy, and bounding surfaces. (Reproduced with permission from Manzari, M. T., and Dafalias Y. E (1997). A critical state two=surface plasticity model for stress. Geotechnique47: 255-272.)
11.11 A Critical State Bounding Surface Model for Sands
1169
associated n, the "image" back-stress ratio tensors ~ , ~d, and ~ are defined as the intersection of the n direction emanating from the origin with the foregoing three surfaces. Their scalar-valued norms are analytically given by ~bo - - g ( O ) M c
-
gb(0)kcbff -- m
(14a) (14b)
m
~co - - g ( O ) M c -
(14c)
in terms of a third stress invariant, the Lode angle 0 as shown in Figure 11.11.2, entering the interpolation functions g, gb, and gd in order to account for the variation from triaxial compression to triaxial extension. Observe that for ~ - 0, ~ - ~ - ~, and the three surfaces collapse into the critical one. The corresponding image tensor quantities follow according to ~ - x/~~n, with a = b, d, or c (one must distinguish between the tensor a~ and its norm ~). The kinematic hardening is given by & - < L > h • (~b0 - ~ ) in terms of a model parameter h, which, together with the consistency condition f = 0 applied to Eq. 11, yields for t/l = 0 the value of the plastic modulus as a)'n
(15)
D -- A ( ~ - ~): n
(16)
K p - - ph(ctbo --
Similarly, the dilatancy D is given by
Observe that a combination of Eqs. 3, 5-8 yields for the triaxial case that K p - h ( ~ - ~) and D - a ( ~ - ~ ) , hence, Eqs. 15 and 16 are their direct generalization. In applications, h and A may be constant or functions of the corresponding distances b - ( a ~ - ~ ) ' n and d - ( ~ - ~)" n, respectively, Figure 11.11.2. In Reference [1] the expression h = h o ] b ] / ( b r e f - ]b[) was used in terms of a model constant h0. The dependence of Kp, and by extension of D, on a distance between a stress-type quantity ~ and its "image" ~b0 on a surface is the classical constitutive feature of bounding surface plasticity.
11.11.3
IMPLEMENTATION
AND
MODEL
CONSTANTS The model is a usual bounding surface plasticity model, and its implementation follows standard procedures. The reader is referred to Manzari and Prachathananukit [4] for details of a fully implicit implementation. The model constants are summarized and divided in categories in Table 11.11.1, together with a set of typical values in parentheses employed in Reference [1].
1170
Manzari and Dahlias
TABLE 11.11.1 Model Constants Elastic
Critical state
State parameter
Hardening
Dilatancy
Go (3.14 x 104) K0 (3.14 x 104) a (0.6)
Mc (1.14)
kcb (3.97) k~a (4.20)
h0 (1200)
A (0.79)
2 (0.025) (ec)ref (0.80)
The most peculiar to the model among the foregoing constants are the kcb and kca (and their corresponding value keb, kea in extension). The kcb can be obtained from Eq. 7 and the experimentally observed values of the peak stress ratio Mcb and state parameter ft. Similarly, the kca can be obtained from the observed value of Mca when consolidation changes to dilation together with the corresponding value of ft. These presuppose knowledge of the critical state line in e-p space. For different Mcb, Mca, and ~ts, different k~ and kca may be determined. It is hoped that these values do not differ a lot, and then an average value is the overall best choice. The h0 and A are obtained by trial and error (there are some direct methods also). All constants can be determined by standard triaxial experiments.
ACKNOWLEDGEMENTS M.T. Manzari would like to acknowledge partial support by the NSF grant CMS-9802287, and Y.E Dafalias by the NSF grant CMS-9800330.
REFERENCES 1. Manzari, M.T., and Dafalias, Y.E (1997). A critical state two-surface plasticity model for sands. Geotechnique 47: 255-272. 2. Been, K., and Jefferies, M.G. (1985). A state parameter for sands. Geotechnique 35: 99-112. 3. Wood, D.M., Belkheir, K., and Liu, D.E (1994). Strain softening and state parameter for sand modelling. Geotechnique 44: 335-339. 4. Manzari, M.T., and Prachathananukit, R. (2001). On integration of a cyclic soil plasticity model. Int. Journal for Numerical and Analytical Methods in Geomechanics, 25: 525-549.
ii.ii
A Critical State Bounding Surface Model for Sands M A J I D T. MANZARI 1 a n d YANNIS E DAFALIAS 2
1Civil and Environmental Engineering, The George Washington University, Washington, D.C., USA 2Department of Mechanics, National Technical University of Athens, 15773, Hellas, and Civil and Environmental Engineering, University of California, Davis, California, USA
Contents 11.11.1 Triaxial Space Formulation . . . . . . . . . . 1164 11.11.1.1 Basic Equations . . . . . . . . . . . . 1165 11.11.1.2 Critical State . . . . . . . . . . . . . . 1166 11.11.2 Multiaxial Stress Space Generalization 1167 11.11.3 Implementation and Model Constants 1169 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . 1170 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1170
11.11.1 TRIAXIAL SPACE FORMULATION The following constitutive model applies to sandy soils at different densities and pressures which do not cause crushing of the grains. It is a general purpose model for multiaxial, drained, undrained, monotonic, and cyclic loading conditions, within the general framework of critical state soil mechanics and bounding surface plasticity. The presentation is a direct derivative of the work by Manzari and Dafalias [1] and includes some additional expressions. The basic concepts and related equations of the sand plasticity model will first be presented in the classical triaxial space where q = or1 - ~r3, p : ( 1 / 3 ) ( ~ r l + 2cr3), ~q = ( 2 / 3 ) ( e l - ~3), and ~v = ~1 + 2~3, w i t h ~i and ~i (i = 1,3) being the p r i n c i p a l stress and strains (or2 = or3,
82
=
1164
83). Handbookof MaterialsBehaviorModels.ISBN0-12-443341-3. Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved.
11.11 A Critical State Bounding Surface Model for Sands
1165
11.11.1.1 BASICEQUATIONS The elastic relation will be assumed hypoelastic for simplicity, given in terms of the r a t e s / / = dq/dt and ib = dp/dt as 9e _ // eq 3 6 O - Go
.e _ /5 ~
(1)
/ < - Ko
(2)
where G and K are the elastic shear and bulk moduli, respectively, Pat is the atmospheric pressure, and the exponent a is usually given a default value of a =0.5. The elastic range is represented by the shaded wedge shown in Figure 11.11.1 in the q, p space, whose straight line boundaries Oc and Oe constitute the yield surface described analytically by f=~/-~:Fm=O
(3)
where the stress ratio r / = q/p and the dimensionless (stress ratio type) quantities c~ and m are shown in Figure 11.11.1. The c~-line is the bisector of the wedge angle, and 2 mp measures the wedge "opening." Equation 3 implies that upon constant r/ loading no plastic loading occurs, which is approximately correct if the p is not high enough to cause crushing of the sand grains and/or the sand sample is not very loose. Based on Eq. 3, plastic deformation occurs only when r/is on f = 0 and there is a change dq = Odt pointing outwards f = O. In this case the plastic rate equations are given by
O
(4/
FIGURE 11.11.1 Schematicrepresentation of the yield, critical, dilatancy, and bounding lines in
q, p space.
1166
Manzari
and Dafalias
where Kp and D are the plastic modulus and dilatancy, respectively. Since .e + ~v 9p, it is a trivial exercise to combine Eqs. 1 and 3 in ~q = ' e~q+ ~ a n d ~ v = ~v order to express ~q and ~v in terms o f / / ( o r ~) and/~, and vice versa. There remains the very important task of specifying Kp and D. In reference to Figure 11.11.1, assume that the line shown with a slope Mcb represents a peak stress ratio for a given state. Henceforth, subscripts c and e imply association of a quantity with triaxial compression and extension, respectively. Such value of Mcb is a bound for ~/; hence, within the framework of bounding surface plasticity one can write a stress ratio "distance"-dependent plastic modulus expression such as K~ -- h ( M ~ - ~)
(5)
in terms of a model parameter h. Similarly, the line shown with a slope M~ in Figure 11.11.1, represents the phase transformation line, or for better naming, the dilation line. According to standard dilatancy theory, one can write
D -- A ( M ~ - ~1)
(6)
with A another model parameter. Hence, Eqs. 5 and 6 determine Kp and D via M~, Mca, h and A, and the model is complete.
11.11.1.2 CRITICALSTATE The critical state in soil mechanics is defined as a triplet of qc, Pc, and void ratio ec values, at which unlimited plastic deviatoric strain occurs at zero volumetric strain rate. Such a critical state is defined simultaneously in the q,p space by t/c--qc/pc=Mc, and in the e - p space by ec = (ec)ref- 2ln(pc/pref), where Me, 2, and (ec)ref (for a chosen pref) are standard soil constants. The Mc is related to friction angle, and its corresponding visualization in q, p space is shown as a line of slope Mc in Figure 11.11.1. If left as is, the previous formulation will not meet the critical state requirements. For example, it follows from Eqs. 4, 5, and 6 that as ~/ approaches M~, which may be assumed to be equal to Me, Kp~ 0 and ~ ~ 0 while ~ > 0 and ~v p < 0 since M~ > M~. This implies unlimited dilation, contrary to physical expectation (negative volumetric strain rate means dilation). Furthermore, a fixed M~ does not allow for the softening response in drained loading observed in many dense sand samples. The remedy is to consider variable M~ and Mca, such that at critical state Mcb = Mca = Me. If the so-called state parameter ~ = e - ec [2], is used as a measure of "distance" from critical state in the e, p space (e and ec refer to the same p), the idea put forth by Wood et al. [3] and supplemented by Manzari
11.11 A Critical State Bounding Surface Model for Sands
1167
and Dafalias [1] in regards to the concept of 0~ in Eq. 3 can be expressed by M~-- (zc + m = Mc - kbc~
(7)
where kcb is a material constant and ~cb is the "bound" for c~ in the same sense that M~ is the bound for r/. The second important modification refers to the variation of Mca and was proposed in Reference [1] as = o~c 4- m -- Mc 4-
(8)
where k~ is a material constant and a~a is a back-stress dilatancy ratio corresponding to the stress dilatancy ratio Mca. Observe that for ~ < 0 (denser than critical), M a c < M c < M ~ , while for ~ > 0 (looser than critical), Mc~
(9)
which defines the critical back-stress ratio ~ in terms of m and the critical stress ratio Mc. Equations 1-8 provide a complete constitutive model in triaxial space. It only requires the specification of Me, k~, and kea in triaxial extension as well to describe reverse and cyclic loading. In such case, Eqs. 5 and 6 utilize the ensuing values of M~ and M~ in extension, following from Eqs. 7 and 8.
11.11.2 MULTIAXIAL GENERALIZATION
STRESS SPACE
The multiaxial stress generalization of the model follows standard procedures [1]. Equation 1 becomes ~=
s 2--G
P ~ - K
(10)
where bold-face characters imply tensor quantities and e e and s are the deviatoric elastic strain and stress tensors, respectively. Equation 2 remains as is. Equation 3 generalizes to f -- [(s - pat):(s - pot)] 1/2 - X//-i/3 m p -- 0
(11)
where ~ of Eq. 3 generalizes to the back-stress-ratio tensor at and : implies the trace of the product of two tensors. The plastic strain rate equations are expressed in the following in terms of i', where the deviatoric stress ratio tensor r - s/p is the generalization of r/. After some algebra which involves the O f / O ~ = n - (1/3)(n : r)I with I the identity tensor and n given in following text, one has for the deviatoric and
1168
Manzari
and Dafalias
volumetric plastic strain rates, respectively, ~P -- < L > n - - < L > ( 2 / 3 ) - 1 / 2 ( r - a t ) L
1
~P -- < L > D
2Gn" ~ - K(n'r)~v -
+
2O
-
(12a)
(12b)
D(n
Equations 12 are the generalizationof Eqs. 4 involving the plasticmodulus Kp and dilatancy D, with the added explicit feature of the loading index L expressed either in terms of/" or in terms of ~ and ev. A combination of Eqs. 10 and 12 together with ~ - he + ~P, ev - ~e + ~v p yields 6 -- E ep" /;-- [Ee -
(2Gn + K D I ) | ( 2 G n - K(n 9 r)I)] Kp 4- 2 G - KD(n i~) _ "~
(13)
for the total effective stress rate 8 in terms of the total strain rate ~ and the elastoplastic tangent stiffness moduli E~. The latter is given explicitly by the bracketed quantity of the last member of Eq. 13, where | means tensor product and E ~ is the well known isotropic elastic moduli tensor which in component form is given by E~jkz -- 2 G ( 6 i k 6 j l - (1/3)cSijcSkz) + K6ij6kz. The Kp and D in the muhiaxial space will be obtained by generalization of corresponding triaxial concepts and equations. First, the bounding, dilatancy, b ~c, d and ~cc in Eqs. 7 , 8, and 9, and critical triaxial back-stress ratios c~c, correspondingly, are generalized as the bounding, dilatancy, and critical surfaces whose traces on the principal stress ratio ri = s i / p 7z-plane are shown in Figure 11.11.2. The To-plane trace of the yield surface with its center a, Eq. 1 l, is also shown in Figure 11.11.2 as a circle. For a stress ratio r and
/~.~
~ t~,~_. ~
./ // ,t~'~~ ,,t ,, / / ' n'~J "
r2 = ---~
Yield Surface
i~'.. ) BoundingSurface r - ' . ~ , . - ~ CriticalSurface ~ ancy Surface
...................
-
tl~+~
_
s3
r3----~
FIGURE 11.11.2 Schematic representation of the yield, critical, dilatancy, and bounding surfaces. (Reproduced with permission from Manzari, M. T., and Dafalias Y. E (1997). A critical state two=surface plasticity model for stress. Geotechnique47: 255-272.)
11.11 A Critical State Bounding Surface Model for Sands
1169
associated n, the "image" back-stress ratio tensors ~ , ~d, and ~ are defined as the intersection of the n direction emanating from the origin with the foregoing three surfaces. Their scalar-valued norms are analytically given by ~bo - - g ( O ) M c
-
gb(0)kcbff -- m
(14a) (14b)
m
~co - - g ( O ) M c -
(14c)
in terms of a third stress invariant, the Lode angle 0 as shown in Figure 11.11.2, entering the interpolation functions g, gb, and gd in order to account for the variation from triaxial compression to triaxial extension. Observe that for ~ - 0, ~ - ~ - ~, and the three surfaces collapse into the critical one. The corresponding image tensor quantities follow according to ~ - x/~~n, with a = b, d, or c (one must distinguish between the tensor a~ and its norm ~). The kinematic hardening is given by & - < L > h • (~b0 - ~ ) in terms of a model parameter h, which, together with the consistency condition f = 0 applied to Eq. 11, yields for t/l = 0 the value of the plastic modulus as a)'n
(15)
D -- A ( ~ - ~): n
(16)
K p - - ph(ctbo --
Similarly, the dilatancy D is given by
Observe that a combination of Eqs. 3, 5-8 yields for the triaxial case that K p - h ( ~ - ~) and D - a ( ~ - ~ ) , hence, Eqs. 15 and 16 are their direct generalization. In applications, h and A may be constant or functions of the corresponding distances b - ( a ~ - ~ ) ' n and d - ( ~ - ~)" n, respectively, Figure 11.11.2. In Reference [1] the expression h = h o ] b ] / ( b r e f - ]b[) was used in terms of a model constant h0. The dependence of Kp, and by extension of D, on a distance between a stress-type quantity ~ and its "image" ~b0 on a surface is the classical constitutive feature of bounding surface plasticity.
11.11.3
IMPLEMENTATION
AND
MODEL
CONSTANTS The model is a usual bounding surface plasticity model, and its implementation follows standard procedures. The reader is referred to Manzari and Prachathananukit [4] for details of a fully implicit implementation. The model constants are summarized and divided in categories in Table 11.11.1, together with a set of typical values in parentheses employed in Reference [1].
1170
Manzari and Dahlias
TABLE 11.11.1 Model Constants Elastic
Critical state
State parameter
Hardening
Dilatancy
Go (3.14 x 104) K0 (3.14 x 104) a (0.6)
Mc (1.14)
kcb (3.97) k~a (4.20)
h0 (1200)
A (0.79)
2 (0.025) (ec)ref (0.80)
The most peculiar to the model among the foregoing constants are the kcb and kca (and their corresponding value keb, kea in extension). The kcb can be obtained from Eq. 7 and the experimentally observed values of the peak stress ratio Mcb and state parameter ft. Similarly, the kca can be obtained from the observed value of Mca when consolidation changes to dilation together with the corresponding value of ft. These presuppose knowledge of the critical state line in e-p space. For different Mcb, Mca, and ~ts, different k~ and kca may be determined. It is hoped that these values do not differ a lot, and then an average value is the overall best choice. The h0 and A are obtained by trial and error (there are some direct methods also). All constants can be determined by standard triaxial experiments.
ACKNOWLEDGEMENTS M.T. Manzari would like to acknowledge partial support by the NSF grant CMS-9802287, and Y.E Dafalias by the NSF grant CMS-9800330.
REFERENCES 1. Manzari, M.T., and Dafalias, Y.E (1997). A critical state two-surface plasticity model for sands. Geotechnique 47: 255-272. 2. Been, K., and Jefferies, M.G. (1985). A state parameter for sands. Geotechnique 35: 99-112. 3. Wood, D.M., Belkheir, K., and Liu, D.E (1994). Strain softening and state parameter for sand modelling. Geotechnique 44: 335-339. 4. Manzari, M.T., and Prachathananukit, R. (2001). On integration of a cyclic soil plasticity model. Int. Journal for Numerical and Analytical Methods in Geomechanics, 25: 525-549.