A Bounding Surface Plasticity Model for Soils with Stress Increment Direction Dependent Plastic Potential

The 12th International Conference of
International Association for Computer Methods and Advances in Geomechanics (IACMAG)
1-6 October, 2008
Goa, India
A Bounding Surface Plasticity Model for Soils with Stress
Increment Direction Dependent Plastic Potential
N. Khalili, M. D. Liu
Dept. of Civil Engineering, The University of New South Wales, Sydney, NSW, 2052, Australia
Keywords: bounding surface, constitutive equations, critical state, dilatancy, plasticity, soils.
ABSTRACT: A new, continuous stress direction dependent plastic potential is proposed for the stress strain
behaviour of soils. The resulting flow rule is incorporated into a bounding surface plasticity model within a
critical state framework. The hypoplastic bounding surface model thus formulated is employed to predict both
the compression and shearing behaviour of soil for various stress paths in the p¢ - q plane. It has been
demonstrated that the proposed model describes successfully many important features of the behaviour of
soils for loading along complicated stress paths and has improved significantly the performance of bounding
surface models with a single set of yield surfaces.
1 Introduction
Many constitutive models have appeared in the literature for the stress-strain behaviour of soils. They range
from simple elasto-plastic models, involving a single yield surface and a single plastic potential, to
sophisticated constitutive models based on the sub-yielding of the material within the virgin yield surface,
multiple yield surfaces, kinematic hardening and bounding surface plasticity among others (e.g., Proc. of 16th
International Conference of Soil Mechanics and Goetechnical Engineering, 2005). The conventional elastoplastic
models form the basis for a majority of the contemporary models proposed in the literature. The more
advanced models are primarily used where the non-linearity and/or the cyclic response of the soil are
concerned. Common to a majority of these constitutive models is the use of a stress direction independent
plastic potential. This has rendered their numerical prediction for stress-strain behaviour of soils less than
reliable particularly along complex non-proportional loading paths.
In this work a new, continuous stress direction dependent plastic potential is proposed for the stress strain
behaviour of soils. The resulting flow rule is incorporated into a bounding surface plasticity model within a
critical state framework. The validity and the accuracy of the hypoplastic bonding surface model are
demonstrated using experimental data from the literature. It has been shown that the model describes
successfully many important features of the behaviour of soils and improves significantly the performance of
conventional bounding surface plasticity models for proportional and non-proportional loading paths.
2 Preliminaries
2.1 Notation
For simplicity, stress and strain states of axisymmetrical conditions are considered in this paper. s¢1 (or e1) and
s¢3 (or e3) are the axial effective stress (strain), and the radial effective stress (strain) respectively. The mean
effective stress p¢ and shear stress q are given by
( 1 3 ) 2
3
p¢ = 1 s¢ + s¢ , ( ) 1 3 q = s¢ - s¢ (1)
The corresponding volumetric strain increment, p e& , and shear strain increment, q e& , are defined by
1 3 e& = e& + 2e& p , ( ) 1 3 3
e& = 2 e& - e& q
(2)
16
2.2 Critical state line and limiting isotropic compression line
After reviewing several experimental data over a wide range of stresses, Russell and Khalili (2004) suggested
the critical state of soils can be represented using three linear segments (Fig. 1). More specifically, six
parameters are used to define the critical state line (CSL) in the u - lnp¢ plane: l0, G0, ucr, lcr, uf and lf. The
parameters l0 and G0 are the slope of the initial portion of the CSL and its specific volume at p¢ = 1kPa,
respectively; ucr is the specific volume at the onset of particle crushing; lcr is the slope during the particle
crushing stage; and uf and lf are the specific volume at the end of crushing and the slope of the CSL at
extremely high stresses, respectively.
Figure 1. Critical State Line (CSL) and Limiting Isotropic Compression Line (LICL)
Similar to the critical state line, LICL is a reference line in the u - lnp¢ plane where the stress state approaches
with increasing isotropic compression. The isotropic compression line can also be regarded as the locus of
the loosest possible state that a soil can achieve for a given mean effective stress. In the present
investigation, the isotropic compression line is taken as parallel to the critical state line and at a constant shift
along the k line from the CSL in the u - lnp¢ plane. A graphical depiction of LICL is shown in Fig. 1.
Based on the definition of the shear stress q, the critical state stress ratio is positive for stress states on the
compression side and negative on the extension side. In this paper, the critical state friction angle is allowed
to be dependent on the stress state. Thus,
ï ïî
ï ïí
ì
< - - >
-
=
for 0
3 sin
6 sin
for 0
3 sin
6 sin
q
q
M
ex
ex
cm
cm
f
f
f
f
(3)
fcm and fex are the critical state friction angles measured from the triaxial compression tests and extension
tests respectively.
2.3 Stress and strain relationship
The total strain increment is made up of elastic and plastic parts. For simplicity, elastic deformation of soil is
assumed to be described by Hooke’s law. In bounding surface plasticity, the plastic strain increment can be
expressed as follows in terms of the loading surface f (p ¢, q), plastic potential g(p¢,q), and hardening
modulus h (e.g., Dafalias, 1986; Khalili et al, 2005),
( ) p q p p
p
p n p n q m L m
h
& = 1 & ¢ + & = e , ( ) p q q q
p
q n p n q m L m
h
& = 1 & ¢ + & = e (4)
Mean effective stress p'
Specific volume u
CSL
LICL
l0
lcr
lf
p¢=1
0 G
cr u
f u
17
2 2
÷ ÷ø
ö
ç çè
æ
¶
¶
+ ÷ ÷
ø
ö
ç çè
æ
¶ ¢
¶
¶ ¢
¶
=
q
f
p
f
p
f
n p
,
2 2
÷ ÷ø
ö
ç çè
æ
¶
¶
+ ÷ ÷
ø
ö
ç çè
æ
¶ ¢
¶
¶
¶
=
q
f
p
f
q
f
nq
(5)
2 2
÷ ÷ø
ö
ç çè
æ
¶
¶
+ ÷ ÷
ø
ö
ç çè
æ
¶ ¢
¶
¶ ¢
¶
=
q
g
p
g
p
g
mp
,
2 2
÷÷ø
ö
ççè
æ
¶
¶ + ÷÷
ø
ö
ççè
æ
¶ ¢
¶
¶
¶
=
q
g
p
g
q
g
mq
(6)
(n p n q)
h
L p q = 1 & ¢ + & (7)
The bracket stands for = a for a > 0 and = 0 for a < 0. 3 Proposed bounding surface model 3.1 Bounding surface, loading surface and mapping rule A bounding surface defines the permissible area for the current soil state in the stress space, and is denoted as ( ¢, , ¢ ) = 0 c F p q p , where the super imposed bar indicates the stress states on the bounding surface. Based on an examination of a large body of experimental data from conventional triaxial tests (e.g., Ishihara, 1993; Montrasio and Nova, 1997; Dafalias and Manzari, 2004), Khalili et al (2005) proposed the following bounding surface equation for soil: ( ) ln 0 ln 1 , , = ÷ ÷ ø ö ç çè æ ¢ ¢ ÷ø ö çè æ - ÷ ÷ ø ö ç çè æ ¢ ¢ ¢ = p p Mp R q F p q p c N c (8) c p¢ defines the size of the bounding surface. A sketch for the bounding surface is shown in Fig. 2a. Figure 2. Bounding surface, loading surface and mapping rule Loading surface is the surface on which the current stress state always stays. In bounding surface theory plastic deformation is associated with the change of the loading surface. For first time loading, the loading and bounding surfaces are assumed of the same shape and homologous about the origin of stresses (Fig. 2a). Thus, the loading surface can be written as follows ( ) ln 0 ln 1 , , = ÷ ÷ ø ö ç çè æ ¢ ¢ ÷ø ö çè æ - ÷ ÷ ø ö ç çè æ ¢ ¢ ¢ = p p Mp R q f p q p c N c (9) p¢c is an isotropic hardening parameter controlling the size of the loading surface and is illustrated in Fig. 2a. The behaviour of soil is linked to the bounding surface by an image stress state. The image stress state for the current stress state s¢ is determined according to an assumed mapping rule. The adopted mapping rule is that the normal at the image stress state on the bounding surface s ¢ and that at the current stress state on Mean effective stress p' Shear stress q M CSL p¢c Loading surface A: s¢ Radial mapping rule cs c p p R ¢ ¢ = c cs p¢ p¢ A: s ¢ Bounding surface Mean effective stress p' Shear stress q A: s ¢ A: s¢ c p¢ Bounding surface Current loading surface Historical loading surface c ( ) pˆ¢ a a p¢ ,q (a) First loading (b) Cyclic loading 18 the loading surface s¢ are kept the same. For first loading, both the bounding surface and the loading surface are of the nature of homogenous function. Consequently, the image stress state is the point where a radial line, passing the origin of the stress coordinates and the current stress state, meets the bounding surface (Fig. 2a). For unloading and reloading, the centre of homology moves to the last point of stress reversal. In this case, the maximum loading surface (historical loading surface) through the point of stress reversal serves as a local bounding surface for the loading surfaces within the surface (Fig. 2b). The loading surfaces undergo kinematic hardening during loading and unloading such that they are tangent to the historical loading surface at the centre of homology. The image point for cyclic loading is located sequentially by projecting the stress point onto a series of intermediate image points on successive local bounding surfaces passing through each point of stress reversal. The loading history of the soil is captured through the stress reversal points and the corresponding maximum loading surfaces. Hence, the loading surface for unloading and reloading takes the form ( ) 0 ˆ ˆ ln ln 1 ˆ ˆ ˆ , ˆ , ˆ = ÷ ÷ ø ö ç çè æ ¢ ¢ ÷ø ö çè æ - ÷ ÷ ø ö ç çè æ ¢ ¢ ¢ = p p Mp R q f p q p c N c (10) and î í ì = - ¢ = ¢ - ¢ a a q q q p p p ˆ ˆ (11) ( ) a a p¢ , q describes the kinematic transition of the loading surface, and pˆ ¢c is the size of the current loading surface. ( ) a a p¢ , q is determined by enforcing the constraint that the loading surface must be tangent to the local bounding surface at the centre of homology. Figure 3. Flow rule for stress state A 3.2 Flow rule It has long been observed that the response of soil to proportional loading and unloading is fundamental different from that to cyclic shearing. Darwin in 1883 noted that the volume of loose granular materials can not be reduced effectively by compression but rather by shaking (or cyclic shearing). Modern tests reveal that monotonic volumetric compression is induced for cyclic shearing on loose sand meanwhile compression is induced by proportional loading and expansion is induced by proportional unloading (e.g., El-Sohby, 1969; Sarsby, 1978; Wood, 1982). If a model with a single set of yield surfaces is to represent reliably the behaviour of soil under both stress level yielding and stress ratio yielding, this fundamental difference must be reconciled rationally. In order to overcome this problem, a new type of plastic flow rule, which is dependent on both the stress state and the direction of the stress increment, is proposed as follows, ( )( ) * * sin sin cos sin a a a a h e e + + M - = c c A p q p p & & (12) where A and c are material parameters. The value of c is much smaller than 1; usually c = 0.1 is assigned. As shown in Fig. 3, a is the angle between the op¢ axis and the stress increment vector ( p& ¢, q& ), measured anticlock- wisely. a* is angle between the stress vector (p¢, q) and the stress increment vector ( p& ¢, q& ), measured anti-clock-wisely. Considering that softening should be treated as loading, not unloading, a consistent definition for angle a is A: (p ¢,q) p' q a a* h b I II III IV (p& ¢, q& ) 19 suggested. The signs for sina and cosa are linked to the sign of the hardening modulus h as well. Thus, ( )2 ( )2 ( )2 ( )2 sin sign( ) , cos sign( ) dp dq dq h dp dq dp h ¢ + = ´ ¢ + ¢´ a = a (13) 2 1 2 1 , cos 1 sin h b h h b + = + = (14) a* = a - b (15) The direction of the plastic strain increment vector is examined here. Soil response is divided approximately into four regions by two lines, i.e., the proportional loading line and the line perpendicular to it. Considering that the value of c is usually much smaller than 1, the signs of plastic volumetric and distortional deformation for various loadings in the p¢ - q space are shown in Table 1. There is obviously a smooth transition between proportional loading and the shearing loading. Because the value of c is much smaller than 1, the corresponding transition region is also small. Table 1. Signs of plastic deformation for general loading in the p¢ - q space for h < M. Loading type h=ct, p&¢> 0, q& >0
h=ct,
p&¢ < 0, q& < 0 Region I q& > 0, h& > 0
Region II
p& < 0, h& > 0
Region III
q¢ < 0, h& < 0 Region IV p& > 0, h& < 0 pp e& + - + + + + p &eq + - + + - - 3.3 Hardening modulus Following a common approach in bounding surface plasticity, the hardening modulus h is divided into two components as follows b f h = h + h (16) where hb is the plastic modulus at s ¢ on the bounding surface, and hf is the modulus simulates the plastic deformation for loading inside the bounding surface. Modulus hb can be derived from the consistency condition of the bounding surface. Hence, 2 2 1 ÷÷ø ö ççè æ ¶ ¶ + ÷÷ ø ö ççè æ ¶ ¢ ¶ ¶ ¢ ¢ ¶ ÷ø ö çè æ l - k = - + q F p F p F m p e h c p c b (17) l is the current slope of the LICL in the u - lnp¢ space. The magnitude of hf is dependent on the closeness between the bounding surface and the loading surface, and it should provide a smooth transition for subyielding behaviour for stress excursions inside the bounding surface and virgin yielding behaviour for stress state on the bounding surface. The following equation for the hardening modulus hf is proposed though trial and errors ( ) ÷ ÷ ø ö ç çè æ - ¢ ¢ - ¢ = 1 ˆc c f m p p p h k l k u (18) 4 Model parameters The model parameters are studied here. The two parameters for describing the elastic deformation are k and n. M, lo, Go, ucr, lcr, uf and lf are to define the critical state line; N and R to define the shape of the bounding surface; km to calibrate the hardening modulus; c and A to define the stress-dilatancy relationship. The elastic parameters k and n with the critical state constants M, lo, Go, ucr, lcr, uf and lf can be determined from triaxial tests using conventional procedures. lo and lcr and lf are the slopes of the initial, particle crushing and final segments of the critical state line, respectively. Go is the reference specific volume at the critical state at a unit confining pressure. ucr and uf are specific volumes on the critical state line at the start and end of the particle crushing phase, respectively. The parameters N and R can be determined by fitting the equation of the bounding surface to the effective stress path of undrained deviatoric response on very loose samples. R is the distance between the CSL and LICL along the k-line, and can be determined from isotropic consolidation test data. 20 It is suggested that c = 0.1 be assumed unless there is reliable experimental data for determining the variation of the flow rule with the direction of the stress increment. Then, assuming elastic strains are negligible in comparison to plastic strain. A is determined by plotting the stress ratio h against the measured total dilatancy in standard drained triaxial compression tests. km is usually obtained by fitting, using the initial slope of drained deviatoric loading and unloading responses in the q ~ eq plane. Figure 4. Stress paths for the tests 5 Model validation In this section, the proposed hypoplastic bounding surface model is employed to simulate the behavior of reconstituted Beaucaire silty clay for loading along various stress paths in p¢ - q the plane. The tests were performed by Costanzo et al (2006) with a conventional triaxial apparatus. The detailed information about the material properties and the testing of the soil can be found in a paper by Costanzo et al (2006). The soil was loaded isotropically to the 147.5 kPa and with the initial void ratio being 0.746. One isotropic test and six drained conventional triaxial tests were carried out. The stress paths for triaxial tests are shown in Fig. 4 and Table 2. Figure 5. Behaviour of Beaucaire clay under triaxial compression tests (Test data after Costanzo et al, 2006) Table 2. Stress paths for tests on Beaucaire clay. Test T121 T126 T123 T124 T125 T122 Type of test Compression Compression Compression Extension Extension Extension Stress path p q &¢ & 3 +¥ -1.5 -1.5 -¥ 2.9 Values of the model parameters used for simulations are listed in Table 3. For the range of stress level -200 -100 0 100 200 300 0 100 200 300 Mean effective stress p ' (kPa) Shear stress q (kPa) T126 T121 T123 T122 T125 T124 Initial stress state 0 100 200 300 400 0 0.1 0.2 0.3 Shear strain eq Shear stress q (kPa) Test 121 Test 126 Test 123 -0.025 0 0.025 0.05 0.075 0 0.1 0.2 0.3 Shear strain eq Volumetric strain e p Test 121 Test 126 Test 123 21 tested, the critical state line is linear. A comparison between the simulations and the experimental data is shown in Figs 5, 6 and 7. As can be seen in Fig. 7 for the isotropic test, the virgin yield stress for this reconstituted clay after previously loading to 147.5 kPa is increased to 178.5 kPa for reloading because of the effect of creep and ageing. This type of the enlargement of the yield surface by creep is well reported (e.g., Leroueil and Vaughan, 1990; Burghignoli, 1998). In the simulation, this effect has been considered by selecting the LICL corresponding to the one with creep and ageing effect. It is seen that the overall response of the Beaucaire silty clay for loading exploring different directions in the p¢ - q plane is described highly satisfactorily by the proposed model. Table 3. Model parameters for Beaucaire clay. k l fcm fcm N R km A c CSL 0.005 0.25 33° 33° 3 1.6 3 2.4 0.1 u=2.2-0.096lnp¢ For tests T121, T126, T124, and T125, all the stress paths point towards the outside of the historical maximum loading surface, the soil experiences first loading. During the loading, the shear stress ratio increases steadily and the soil fails at a critical state of deformation. According to the proposed flow rule, the plastic volumetric deformation remains compressive during the entire process. The plastic distortional strain is positive for the tests with positive shear stress and negative for ones with negative shear stress. The behaviour of soil simulated is in consistent with experimental data. Figure 6. Behaviour of Beaucaire clay under triaxial extension tests (Test data after Costanzo et al, 2006) Figure 7. Behaviour of Beaucaire clay under isotropic tests (Test data after Costanzo et al, 2006) For tests T123 and T122, the stress path turns inside the historical maximum loading surface. Thus during the 1.6 1.65 1.7 1.75 1.8 1.85 1.9 10 100 1000 Mean effective stress p ' (kPa) Specific volume u Initial state Isotropic unloading Isotropic loading -200 -100 0 -0.2 -0.15 -0.1 -0.05 0 Shear strain eq Shear stress q (kPa) Test 124 Test 125 Test 122 -0.02 0 0.02 0.04 0.06 -0.2 -0.15 -0.1 -0.05 0 Shear strain eq Volumetric strain ep Test 124 Test 125 Test 122 22 test the loading surface expands from the initial size of zero. For the two tests, it is found that soil can reach a peak strength state (within the bounding surface) and then softens to a critical state of deformation. The plastic volumetric deformation is initially compressive and becomes expansive during the softening process. This pattern of soil behaviour is clearly seen in test T123. For test T122, though not dramatic, some reduction in volumetric deformation can still be seen at p 0.08 q e > .
The behaviour of the clay for an isotropic unloading test and an isotropic loading test is shown in Fig. 7. The
initial state of the two tests is the same and marked in the figure by a black dot. For isotropic unloading, the
plastic volumetric deformation predicted by the proposed model is expansive, which is confirmed by the
experimental data. A very special characteristic of the behaviour of noncohesive soil during isotropic unloading
is observed here. The specific volume increases rapidly with the reduction of the mean effective stress,
particularly at near zero stress state. This important feature of the behaviour of non-cohesive materials at near
zero stress state has been widely observed for both constant stress ratio loading and shearing loading (e.g.,
Wood, 1982; Picarelli, 1991). The proposed model has improved significantly the performance of bounding
surfaces models in predicting the volumetric deformation of soils for loading along complicated stress paths
and for isotropic unloading.
6 Conclusion
A bounding surface model for soils with stress increment direction dependent plastic potential is formulated in
order to describe reliably soil deformation for loadings along complicated stress paths with a single set of
yield surfaces. The proposed hypoplastic bounding surface model has been used to predict both the
compression and the shearing behaviour of soil. The simulations cover different stress paths in the p¢ - q
plane. It has been demonstrated that the proposed model describes successfully many important features of
the behaviour of soils and has improved significantly the performance of bounding surfaces models with a
single set of yielding surfaces.
7 References
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Mechanics, ACME, 112(12), 1263-1291
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