Comparison of model predictions of the anisotropic plasticity of Lower Cromer Till

Computers and Geotechnics 69 (2015) 365–377

Contents lists available at ScienceDirect

Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo

Research Paper

Comparison of model predictions of the anisotropic plasticity of Lower Cromer Till Chao Yang ⇑, John P. Carter, Shengbing Yu Centre of Excellence for Geotechnical Science and Engineering, The University of Newcastle, Callaghan, NSW 2308, Australia

a r t i c l e

i n f o

Article history: Received 9 March 2015 Received in revised form 9 June 2015 Accepted 10 June 2015 Available online 22 June 2015 Keywords: Anisotropy Plasticity Fabric Rotational hardening Constitutive relations Clays

a b s t r a c t This paper compares predictions, made using selected soil constitutive models, of the anisotropic plastic response of a sandy silty-clay, viz., Lower Cromer Till (LCT). The performance of four elastoplastic models, designated as MCC (Roscoe and Burland, 1968), S-CLAY1 (Wheeler et al., 2003), SANICLAY14 (Dafalias and Taiebat, 2014) and YANG2015 (Yang et al., 2015), are systematically evaluated based on a series of drained triaxial stress path tests, including virgin constant-stress-ratio (CSR) compression tests, probing stress path tests on initially K0 consolidated samples, and also various transitional CSR tests. Comparison of the various predictions shows that the isotropic MCC model cannot properly describe the mechanical behaviour of LCT due to its neglect of fabric anisotropy. The other three anisotropic models differ in their definition of the rotational hardening laws, particularly in the description of the equilibrium state of fabric anisotropy achieved under CSR loading. While significant improvements in model predictions can be observed from the three anisotropic models, for LCT S-CLAY1 generally tends to underestimate the volumetric deformation and both S-CLAY1 and SANICLAY14 are likely to overestimate the ratio of the deviatoric and volumetric strains for more anisotropic stress states. YANG2015 exhibits the most consistent performance in reproducing the mechanical behaviour of LCT among the four models under comparison. The importance of the virgin CSR tests to properly understanding the plastic anisotropy of soil fabric is highlighted. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction The importance of considering anisotropic plasticity of soils in order to obtain accurate predictions of their mechanical behaviour has long been recognised (e.g., [10,4,28,13,36,1,3,15,11,34,19]). Various propositions have been made for incorporating the concept of rotational hardening into the isotropic Critical State constitutive models, particularly the Modified Cam Clay (MCC) model [24]. Some of the representative works can be found in Newson and Davies [20], Pestana and Whittle [21], Wheeler et al. [30], Dafalias and Taiebat [8] and Yang et al. [33], among others. Some common assumptions can be summarised from those previous works: (1) the fabric anisotropy can be described by the inclination of the yield and plastic potential surfaces in the traditional triaxial stress space; (2) a change of fabric anisotropy is only induced by plastic straining; and (3) an equilibrium state of fabric anisotropy can be established, either explicitly or implicitly, by virgin consolidation at constant stress ratio (CSR).

⇑ Corresponding author. Tel.: +61 2 49854996; fax: +61 2 49215500. E-mail address: [email protected] (C. Yang). http://dx.doi.org/10.1016/j.compgeo.2015.06.009 0266-352X/Ó 2015 Elsevier Ltd. All rights reserved.

Key differences between existing models can be identified. For instance, some involve an associated plastic flow rule, often for simplicity or otherwise to avoid the difficulty of accurately determining the plastic potential surface in experiments, e.g., Dafalias [5], Wheeler et al. [30], Sun et al. [28], and Yang et al. [32]; whereas others have included a non-associated flow rule, e.g., Newson and Davies [20], Pestana and Whittle [21], Dafalias and Taiebat [8] and Yang et al. [33]. Some attribute the variation of fabric anisotropy merely to the volumetric component of plastic straining, e.g., Dafalias [5], some consider the different contributions from both the volumetric and deviatoric plastic strain components, e.g., Pestana and Whittle [21] and Wheeler et al. [30], while others use the total plastic strain to quantify the change of fabric anisotropy, e.g., Dafalias and Taiebat [8] and Yang et al. [32,33]. Another difference between the various models lies in their description of the equilibrium state of fabric anisotropy. Newson and Davies [20] and Pestana and Whittle [21] assumed that at the equilibrium state of soil fabric the yield surface is aligned with the imposed CSR loading path. Dafalias [5] considered that the degree of inclination of the yield surface is a fixed fraction of the deviation of the CSR loading path away from the hydrostatic state.

366

C. Yang et al. / Computers and Geotechnics 69 (2015) 365–377

This concept has also been adopted by Newson and Davies [20] and Dafalias et al. [6] to describe the orientation of the plastic potential surface. Later, Wheeler et al. [30] and Dafalias and Taiebat [8] found that a non-linear relationship better describes the variation of the inclination of the yield surface at the equilibrium state with the imposed virgin CSR. Yang et al. [32,33] defined the equilibrium state of fabric anisotropy in terms of the inclination of both the yield and plastic potential surfaces, based on the available experimental evidence. Evolutionary improvements in the description of fabric anisotropy for clays can therefore be traced from the works of Dafalias [5], Newson and Davies [20], Pestana and Whittle [21], Wheeler et al. [30], Dafalias et al. [6], Dafalias and Taiebat [8] and Yang et al. [33]. Conventional drained and undrained triaxial tests have been commonly adopted to evaluate the performance of the proposed anisotropic models [20,22,35,8]. The stress path imposed in those conventional triaxial tests for normally consolidated samples covers a continuous variation of stress ratios (g = q/p0 ), which causes simultaneous changes in the plastic anisotropy of the clay fabric. Note that in the current treatment p0 is the mean effective stress and q is the deviatoric stress. These conventional triaxial tests may be useful to generally validate any proposed constitutive model, but they cannot provide definitive tracking of the change of fabric anisotropy along the imposed stress paths, and thus may be inefficient and of limited value when used to validate the proposed rotational hardening rules. Therefore, laboratory tests that can provide explicit information on fabric anisotropy are preferred. For instance, a series of CSR tests would be appropriate due to the fact that a unique fabric anisotropy can be achieved in each CSR test [32,33]. Simple examples include the K0 consolidation experienced by naturally deposited soils and the isotropic consolidation widely investigated in laboratory tests. By shifting the value of the constant stress ratio applied in these CSR tests, the validity of the proposed rotational hardening laws can be unambiguously examined, as indicated by Wheeler et al. [30], Karstunen and Koskinen [14], Belokas and Kavvadas [2] and Yang et al. [32,33]. Some natural clays, like those Scandinavian clays studied by Toivanen [29], Koskinen et al. [16], Karstunen and Koskinen [14] and others, may be significantly heterogeneous in terms of their mineral composition and may also exhibit a time-dependent response to loading. They can also be sensitive to disturbance. The coupling or co-existence of fabric anisotropy, soil structure and also time-dependency makes it difficult for a clear-cut evaluation of the single aspect of anisotropic plasticity in proposed models. Therefore, the most appropriate soils, that will allow this aspect of soil behaviour to be revealed, would be those with a relatively homogeneous mineral composition, which are time-independent and display no discernible effect of structure. Of the data available in the literature, the series of tests on reconstituted Lower Cromer Till (LCT) conducted by Gens [9] is one of the best candidates to investigate the anisotropic plasticity inherent in soil. This systematic testing program on LCT provides a relatively complete data set for the validation of a typical critical state elastoplastic constitutive model. The various CSR and stress path probing tests can be used to provide a clearer understanding of the mechanical effect of fabric anisotropy, as suggested by Yang et al. [33]. In the following, the performance of four constitutive models, namely the Modified Cam Clay (MCC) model [24], the S-CLAY1 model [30], the recently modified SANICLAY model [8], and another model proposed by Yang et al. [33] will be compared though their numerical predictions of various virgin and transitional CSR tests on Lower Cromer Till. This work will be presented with first a concise description of these four models, then a systematic comparison of the model predictions, and finally some discussion and conclusions.

2. Model descriptions The four models listed above will be briefly introduced in this section. For convenience, the model suggested by Dafalias and Taiebat [8] is denoted as SANICLAY14, whereas the model proposed by Yang et al. [33] is denoted here as YANG2015. 2.1. Plastic potential surface and yield surface S-CLAY1, SANICLAY2014 and YANG2015 all employ a rotated and distorted ellipse to describe the plastic potential surface (PPS), which in the traditional p0  q stress plane is given as

  2 g ¼ ðq  ag p0 Þ  M 2g  a2g p0 ðp0m;g  p0 Þ

ð1Þ

where Mg is the critical state stress ratio; and p0m;g and ag are the internal hardening parameters controlling the size and the inclination of the PPS. Note that Mg can be Lode-angle dependent and will acquire different values for the compression and extension states of stress, i.e., Mg,c and Mg,e [26], where the subscripts, c and e, denote the compression and extension stress state, respectively. The corresponding yield surface (YS) can be expressed in a similar elliptical function as

  2 f ¼ ðq  af p0 Þ  M 2f  a2f p0 ðp0m;f  p0 Þ

ð2Þ

where p0m;f and af, similar to p0m;g and ag, are the internal hardening parameters for the YS; whereas Mf is effectively the shape factor of the YS. Eqs. (1) and (2) meet the various requirements for all four models, the specific configurations of which can be found in Table 1. For LCT samples subjected to K0 (=0.5) consolidation to p0 = 233.3 kPa, both the PPS and YS predicted by Eqs. (1) and (2) for all four models are depicted in Fig. 1. It can be seen that S-CLAY1 has a single inclined surface (Fig. 1b), whereas SANICLAY14 and YANG2015 employed two different inclined surfaces (Fig. 1c and d). As indicated in Table 1, SANICALY14 assumes the same size and inclination for both the YS and PPS (p0m;f ¼ p0m;g and af = ag) but different shape parameters (Mf – Mg), which was suggested by Jiang and Ling [12]. This configuration leads to the plastic potential surface in SANICLAY14 not crossing the current stress state on the yield surface (Point A in Fig. 1c). Dafalias and Taiebat [8] then referred to the outward normal to the PPS at a conjugate point A0 along the same stress ratio to specify the dilatancy at the current stress point A on the YS (see Fig. 1c). Once anisotropy is absent the surfaces for S-CLAY1, SANICLAY14 and YANG2015 automatically degenerate to that of MCC, whose principal direction is constant and aligned with the hydrostatic axis (Fig. 1a). 2.2. Flow rule   The plastic strain rate vector e_ p ¼ e_ pp ; e_ pq can be expressed by _ i.e., _ qÞ, the flow rule as a function of the stress rate vector r_ ¼ ðp;

e_ p ¼ ng

nTf r_ HM

ð3Þ

where HM is the plastic modulus, and nf and ng are the unit vectors on the yield and the plastic potential surfaces, respectively. The Table 1 Configuration of YS and PPS via Eqs. (1) and (2) for the four models. Models

f vs. g

MCC

f=g

S-CLAY1

f=g

SANICLAY14

f–g

YANG2015

f–g

af vs. ag

p0m;f vs. p0m;g

Mf vs. Mg

af = ag af = ag af = ag af – a g

p0m;f p0m;f p0m;f p0m;f

Mf = Mg

¼ ¼ ¼ –

p0m;g p0m;g p0m;g p0m;g

Mf = Mg Mf – Mg Mf = Mg

C. Yang et al. / Computers and Geotechnics 69 (2015) 365–377

367

(c)

(a)

(b)

(d)

Fig. 1. Plastic potential surfaces and yield surfaces of the four studied constitutive models.

superscripts, p and q, denote the volumetric and deviatoric components of the strain rate. 2.3. Isotropic volumetric hardening All models considered here assume the same volumetric hardening law to describe the change of the size of YS, which can be derived from the isotropic normal compression line and written as,

p_ 0m;f p0m;f

¼

v  e_ pp kj

ð4Þ

where v is the specific volume, and k and j are the slopes of normal compression lines (NCL) and unloading–reloading lines (URL) in the v – ln p0 plane, respectively. 2.4. Anisotropic rotational hardening The definition of the rotational hardening laws is unique to each of the anisotropic models studied here. A brief introduction of the related rotational hardening laws is now provided. 2.4.1. MCC The MCC model is an isotropic elastoplastic constitutive model [24]. As such,

ag ¼ af ¼ 0 a_ g ¼ a_ f ¼ 0

ð5Þ

2.4.2. S-CLAY1 Wheeler et al. [30] analysed a series of transitional CSR tests on Otaniemi soft clay, and summarised an empirical relationship of

the inclination of YS (af) with the imposed stress ratio (g) for a given CSR. They proposed the following rotational hardening law,

    3 1 g  a he_ pp i þ b g  a je_ pq j 4 3

a_ g ¼ a_ f ¼ a_ ¼ u

ð6Þ

where the soil constant u controls the absolute rotation rate and b describes the weight of deviatoric component of the plastic strain rate on the rotation rate of the YS. An equilibrium state of fabric anisotropy, which corresponds to a constant form of anisotropic fabric and thus a unique value of aE for each constant g loading, can be achieved by setting a_ ¼ 0 in Eq. (6). Note the subscript, E, denotes the equilibrium state of fabric anisotropy maintained under any virgin CSR loading test. Combining this condition with the flow rule (Eq. (3)) renders a quadratic equation for aE as a function of g [30],

3ð3g  4aE ÞðM 2  g2 Þ  8bð3aE  gÞðg  aE Þ ¼ 0

ð7Þ

2.4.3. SANICLAY14 The latest in a series of rotational hardening laws elaborated by Dafalias and Taiebat [8] will be considered here. It has the following form,

a_ g ¼ a_ f ¼ a_ ¼ hLicpat

p0 ½aE ðgÞ  a p0m

ð8Þ

where the soil constant c, equivalent to u in Eq. (6), controls the absolute rotation rate of the YS and PPS, and pat is the atmospheric pressure. L in the Macauley brackets ‘h i’ is the plastic multiplier, which is given as

368

C. Yang et al. / Computers and Geotechnics 69 (2015) 365–377

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u  p 2  p 2 u e_ p þ e_ q hLi ¼ u t 2  2 @g þ @g @p0 @q

The plastic strain variable e_ pr is given as

ð9Þ _ pr

e

The equilibrium fabric anisotropy aE(g) was directly proposed by Dafalias and Taiebat [8] as an exponential function,



aE ðgÞ ¼ g

  n  jgj þm 1 Mg Mg

ac

ð10Þ

where m and n are positive soil constants, and ac is the soil constant describing the fabric anisotropy at critical state. 2.4.4. YANG2015 A non-associated flow rule is assumed in YANG2015 and also different definitions of aE,g and aE,f [33]. Thus two sets of rotational hardening laws are required, but they can be expressed in a uniform style as

a_ g=f ¼ xg=f ðgÞðaE;g=f ðgÞ  ag=f Þe_ pr

ð11Þ

where the subscripts, g and f, denote the variables or constants for the PPS and YS, respectively. For instance, xg/f means xg or xf, which control the absolute rotation rates of the PPS and YS, respectively. The quantity e_ pr is the plastic strain rate driving the evolution of fabric anisotropy. Yang et al. [33] found that a series of CSR consolidation tests is crucial to an accurate description of fabric anisotropy of soils. From the parallel NCLs and the uniqueness of the dilatancy (dg) at each CSR loading, the inclinations of both the YS and PPS can be rigorously achieved. For instance, the equilibrium fabric anisotropy in terms of PPS can be derived from Eqs. (1) and (3) as,

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  p 2  p 2 e_ p þ e_ q ¼ 2

ð17Þ

As seen, e_ pr fulfils the same role as hLi in Eq. (9). It should be noted that all the above listed rotational hardening laws (Eqs. (6), (8) and (11)) are only effective on the ‘wet’ side of the Critical State Line with |g/Mg| < 1. Combining both the volumetric and the rotational hardening laws together with the flow rule into the consistency condition on the YS of Eq. (2) provides the value of the plastic modulus, HM. Substitution of HM back into the flow rule (Eq. (3)) renders the values of the plastic strain rates, e_ p . For simplicity, all four models adopt isotropic elasticity, for which the volumetric and deviatoric elastic strain rates are _ The bulk modulus K is stress-state depene_ ep ¼ p_ 0 =K and e_ eq ¼ q=3G. dent and defined as K = vp0 /j. With a constant Poisson’s ratio l being assumed, the total strain rate can then be additively obtained from its elastic and plastic components. 2.5. Summary of model parameters

where ff and vf are parameters controlling the spacing ratio with stress ratio. In addition, a limit of aE;f 6 aE;f;max has to be imposed on the inclination of the YS as experiments [30,33] indicate that a more subtle change of fabric anisotropy would be expected when loading at high stress ratios. Defined as functions of the current and equilibrium values of fabric anisotropy, xg/f can be further given as

As previously indicated, all three anisotropic models have been established within the framework of Critical State Soil Mechanics and can be considered to be extensions of the MCC model. So these four models share the same critical state soil constants, i.e., Mg, N, k, j and l. Values of these soils constants can be determined via routine laboratory tests [9,31] and are listed in Table 2 for LCT. Note that the soil constant N is the specific volume at p0 = 1 kPa on the isotropic normal compression line. S-CLAY1, as shown in Table 3, has two additional explicit parameters (l and b), as well as two implicit parameters 3  and 13 . The soil constant l defines the rate at which a 4 approaches its equilibrium state. As suggested by Wheeler et al. [30] and Karstunen and Koskinen [14], the value of l can be determined by best-fitting experiments involving significant rotation of the YS. An empirical formulation by Zentar et al. [37], 10=k 6 l 6 20=k, which has been successfully applied to Otaniemi clay [30,14], cannot be used for LCT, as an extremely large value of l would be predicted. A typical trial and error method is used to give l = 50 for LCT. The soil constant b of S-CLAY1 controls the relative weight of plastic deviatoric strains and plastic volumetric strains in rotating the YS. Wheeler et al. [30] suggested a procedure to determine the value of b according to the equilibrium state of soil fabric achieved under one-dimensional loading on K0 normally consolidated samples. An estimate of b = 0.91 is obtained for LCT. It has to be admitted that application of such a value of b to describe other equilibrium states of soil fabric may require caution since a different weighting of the plastic deviatoric and volumetric strains in rotating the YS can be expected. SANICLAY14 requires an additional six parameters, c, m, n, ac, Mf,c and Mf,e, as revealed in Table 3. The two soil constants, Mf,c and Mf,e, defining the shape of the YS, can be determined by best-predicting the ultimate shear strength in conventional undrained triaxial tests. The soil constant c, which controls the absolute rotation rate, is obtained by trial and error [7]. The soil constant ac which describes the ultimate inclination of the YS at critical state is determined by an empirical relationship, i.e.,

 1  gaE;g=f  Mg=f   a  a g=f E;g=f  C xg=f ðgÞ ¼ exp B @kg=f  A  M

Table 2 Values of MCC-based critical state model parameters for LCT.

 2 ! Mg g  dg ¼  1 Mg Mg 2 Mg

aE;g

g

ð12Þ

where dg is the experimentally measured dilatancy which is found to be a function of g:

dg ¼



e_ q 1 g ¼  ln 1  fg e_ p Mg

 ð13Þ

with the parameter fg describing the variation of dg with g. Similarly, the equilibrium fabric anisotropy in terms of the YS, aE,f, can be obtained as one root of the following quadratic equation, rewritten from Eq. (2),



g

Mf



aE;f Mf

2

 1



aE;f

 2 ! 0 pm;f

Mf

p0

 1 ¼0

ð14Þ

where p0m;f =p0 , quantifying the spacing ratio of the NCLs, can be determined from experiments, and is given as

p0m;f p0

¼1

  vf  1 g ln 1  ff Mf

ð15Þ

0

ð16Þ

g=f

with kg/f defining the possible maximum rotation rates of the PPS and YS.

Mg,c

Mg,e

N

k

j

l

1.18

0.86

1.790

0.066

0.009

0.258

369

C. Yang et al. / Computers and Geotechnics 69 (2015) 365–377 Table 3 Values of additional anisotropy-related model parameters for LCT. Models

Constant 1

S-CLAY1 SANICLAY14 YANG2015

u = 50 c = 250 ff = 2.69

Constant 2 b = 0.91 m = 0.412 fg = 1.46

Constant 3

Constant 4

⁄

Constant 5

Constant 6

Mf,c = 0.90 kg = 5.36

Mf,e = 0.66

⁄

1/3 n=9 vf = 1.82

3/4 ac = 0.23 kf = 4.6

aE;f;max M

¼ 1=6

⁄

Note: The symbol, , indicates the implicit model parameters in S-CLAY1.

ac ¼ 0:5aE;K 0 , where aE;K 0 quantifies the equilibrium inclination of

p

dg  dg ¼

e_ pq 2ðg  aE;g Þ ¼ e_ pp M 2g  g2

ð18Þ

The value of p0m;f =p0 can be obtained from Eq. (2) and written as,

p0m;f p0K

¼1þ

ðg  aE;f Þ2 M 2f  a2E;f

ð19Þ

1

Yang2014 YANG2015

(a)

SANICLAY2014 SANICLAY14 S-CLAY1 S-CLAY1

0.8

α E,g /Mg,c

MCC MCC

0.6 0.4 0.2 0

0

0.2

0.4

0.6

0.8

1

η /M 1

Yang2014 YANG2015

(b)

SANICLAY2014 SANICLAY14

0.8

S-CLAY1 S-CLAY1

MCC MCC

α E,f / M g,c

the YS under virgin K0 consolidation [8]. The pair of soil constants, m and n, being used to define the value of aE, have to satisfy certain inequalities given in Dafalias and Taiebat [8], and their values are determined by best-fitting conventional undrained triaxial tests. Values of the six additional parameters for SANICLAY14 provided in Dafalias and Taiebat [8] to represent the behaviour of the same LCT [9] are adopted here and listed in Table 3. YANG2015 also needs another six anisotropy-related model parameters, namely fg, ff, vf, kf, kg and aE,f,max, all of which can be properly determined from CSR loading tests. Stress path probing tests on initially K0 consolidated samples can provide information on the size and shape of the yield surface and also of the dilatancy along a K0 loading path, which can be combined with Eqs. (13) and (15) to derive the values of fg, ff and vf. The soil constants, kf and kg, which control the absolute rotation rate of the YS and PPS, can be obtained by best-fitting one transitional CSR loading test, for example, an isotropic loading path on initially K0 consolidated samples. The soil constant aE,f,max, which defines the maximum possible inclination of the YS for CSR tests with g 6 M, can be approximated by a CSR test with g > gK 0 . A series of virgin and transitional CSR tests can theoretically provide a more accurate description of all these extra anisotropic model parameters, the values of which for LCT are listed in Table 3. With the values of all parameters determined, the equilibrium fabric anisotropy of LCT can be described in the four models by aE,g and aE,f from Eqs. (5), (7), (10), (12) and (14), and the results are depicted in Fig. 2. It can be seen in Fig. 2a that S-CLAY1 and SANICLAY14 predict similar values of aE,g which reach a peak value at g < Mg, indicating that increasing the stress ratio for the virgin CSR consolidation can induce a decrease in the degree of fabric anisotropy. YANG2015 predicts a concave-up increasing aE,g, which suggests a continuously increasing degree of fabric anisotropy with increasing stress ratio. MCC assumes an isotropic fabric and thus has a constant value of aE,g = 0. Similar trends can also be perceived in terms of aE,f for the YS from Fig. 2b, where YANG2015 predicts a relatively stable equilibrium state of fabric anisotropy for high values of stress ratio, while S-CLAY1 and SANICLAY14 still maintain the same parabolic form for aE,f (=aE,g), and MCC has aE,f = 0. The difference inherent in the definitions of aE,g and aE,f for these four models will dominate their different predictions, which will be disclosed subsequently. The determination of the equilibrium fabric anisotropy via aE,g and aE,f, provides a chance to understand the variation of dilatancy (dg) and the spacing ratio of the NCLs (p0m;f =p0 ) with stress ratio (g), described in each of those four models. For soft clays where plastic deformation dominates, the value of dg can be approximated by the p plastic dilatancy (dg ), which can be derived from the combination of Eqs. (1) and (3),

0.6 0.4 0.2 0

0

0.2

0.4

0.6

0.8

1

η/M Fig. 2. The equilibrium state of fabric anisotropy in terms of aE,g and aE,f for the four models.

By substituting the values of aE,g and aE,f depicted in Fig. 2 into Eqs. (18) and (19) the values of dg and p0m;f =p0 can be obtained for each model, which can be compared in a straightforward manner with the experimental data. It can be observed in Fig. 3a that YANG2015 can accurately reproduce the dilatancy of LCT, while S-CLAY1 and SANICLAY14 significantly overestimate the dilatancy for LCT samples consolidated at larger values of g (g/Mg > 0.6). The isotropic MCC model apparently overestimates the dilatancy for the whole range of g. The predicted values of p0m;f =p0 from all four models are provided in Fig. 3b. S-CLAY1 gives relatively lower values compared with the test data. SANICLAY14 underestimates the values of p0m;f =p0 for lower values of g but tends to give better predictions near critical state. In contrast, YANG2015 can successfully reproduce p0m;f =p0 for most values of g but appears less satisfactory near critical state. MCC obviously overestimates p0m;f =p0 for all the stress ratio range. Different performance can thus be anticipated between the four models.

370

C. Yang et al. / Computers and Geotechnics 69 (2015) 365–377

1.0

1.55

(a)

1

0.8

0.667

0.6

Test Test Yang2014 YANG2015 SANICALY2014 SANICLAY14 S-CLAY1 S-CLAY1

0.4

0.4

0.34

1.45

1.40

MCC MCC

0.2

0.5

K= 1 0.8 0.667 0.5 0.4 0.34

1.50

v

η / M g,c

0.8

K Sym

0

(a) 0.0

0

1

2

3

CSL

1.35 50

4

100

p' (kPa)

dg -0.3

1.0

200

(b)

K = 0.4

(b)

K = 0.5

0.8

Test Test Yang2014 YANG2015 SANICLAY2014 SANICLAY14 S-CLAY1 S-CLAY1

0.4

K = 0.667

-0.1

K = 0.8

MCC MCC

0.2

0.0

ε a-ε a,ref

η /Mg,c

-0.2 0.6

K= 1

1

1.5

2

2.5

3

p'm / p' Fig. 3. Prediction of dg and p0m =p0 in virgin CSR tests by the four models.

3. Comparison of model predictions The capability of MCC, S-CLAY1, SANICLAY14 and YANG2015 in describing the anisotropic plasticity of soils will be evaluated in this section for the LCT samples. The first prediction scenario is the series of virgin CSR consolidation tests at different values of   stress ratio K ¼ r0r =r0a ¼ ð3  gÞ=ð3 þ 2gÞ . The subscripts, a and r, denote the axial and radial directions of the cylindrical triaxial samples. Next, the prediction for conventional K0 consolidated samples under various probing stress paths will be conducted. Last, transitional CRS tests will be predicted to reveal the effectiveness of the proposed rotational hardening laws. All model predictions are depicted by solid lines in the figures that follow, unless specified otherwise. It should also be noted that the explicit modified Euler method with automatic error control proposed by Sloan et al. [27] was used to integrate the stress rates in all four models. 3.1. Prediction of virgin CSR tests Virgin consolidation tests with various stress ratios (K = 1, 0.8, 0.667, 0.5, and 0.4) are simulated and the results are presented in terms of NCLs in the v  ln p0 plane and also in terms of relative strain rates in the ea  ep plane. The reference state for strain rates is taken at p0 = 233.3 kPa. The critical state line (K = 0.34) on the compression plane is also provided for all models. The numerical predictions are shown in Figs. 4–7. It can be seen that MCC overestimates the volumetric compression for CSR tests (Fig. 4a) while S-CLAY1 underestimates it (Fig. 5a).

0.0 0.00

-0.05

-0.10

-0.15

-0.20

εp-εp, ref Fig. 4. Model predictions in virgin CSR consolidation tests by MCC.

Improvements can be observed in SANICLAY14 (Fig. 6a) and greater accuracy is achieved in the predictions of YANG2015 (Fig. 7a). MCC and SANICLAY14 predict the critical state line more accurately for LCT in the semi-log compression plane, which can be attributed to the relatively flatter shape of the YS due to the absence of rotation of the YS in MCC and the introduction of smaller values of the shape parameter Mf (=0.9) for the YS in SANICLAY14. These observations are consistent with the predicted spacing ratio shown for LCT in Fig. 3b. In Fig. 4b to Fig. 7b, a linear ea:ep relationship is obtained, the slope of which is correlated with the dilatancy by e_ a =e_ p ¼ dg þ 13. MCC clearly overpredicts the value of dg for all CSRs except K = 1, which corresponds to isotropic consolidation (Fig. 4b). S-CLAY1 and SANICLAY14, by introducing the rotational hardening laws, can reasonably predict dg for less anisotropic stress states (K = 1– 0.5), but significantly overpredict dg for more anisotropic stress states (K = 0.4), as shown in Figs. 5a and 6b. However, YANG2015 is capable of providing a very accurate prediction of dg for all stress states (Fig. 7b). All these differences in the predicted values of dg actually originate from the different description of the equilibrium fabric anisotropy (see Fig. 2) and also the different choices of the PPS (see Fig. 1). 3.2. Prediction of probing stress path tests on K0 consolidated samples Natural soils generally experience a one-dimensional sedimentary process [10,31,30]. So it is desirable that any proposed constitutive model should have the ability to reproduce the mechanical

371

C. Yang et al. / Computers and Geotechnics 69 (2015) 365–377

1.55

K Sym

1

0.8

0.667

0.5

1.55

0.34

K=1 0.8 0.667 0.5 0.4 0.34

1.50

1.45

1.40

1

0.8

0.667

0.4

0.34

K= 1 0.8 0.667 0.5 0.4 0.34

1.45

0

CSL

(a)

CSL

(a) 100

1.35 50

200

100

-0.3

-0.3 K = 0.4

(b)

200

p' (kPa)

p' (kPa)

K = 0.4

(b) -0.2 K = 0.5

εa-ε a,ref

ε a-ε a,ref

-0.2

K = 0.5

-0.1

-0.1

K = 0.667 K = 0.8

K = 0.667 K = 0.8 K=1

0.0 0.00

0.5

1.40

0

1.35 50

K Sym

1.50

v

v

0.4

-0.05

-0.10

-0.15

-0.20

ε p-ε p, ref Fig. 5. Model predictions in virgin CSR consolidation tests by S-CLAY1.

behaviour of K0 consolidated samples. In this section, a series of drained shearing tests on K0 consolidated LCT samples, as shown in Fig. 8, will be used to evaluate the performance of the four models. The predictions of shear strength and volume change from all the models are provided in Figs. 9–12. Note that normalisation of the values of p0 and q has been performed using r0a;max , i.e., the maximum axial effective stress experienced during each virgin consolidation. As seen in Fig. 9a to Fig. 12a, the shear strength in tests AR1, AR2, AR9 and AD1 is generally well captured by all four models, even though a different rotational hardening law is employed by each model. MCC seems less effective in predicting the shear strength of AR3 and AR10, the latter with a continuous K0 consolidation. In terms of volumetric behaviour, the performance of these models differs significantly. MCC and S-CLAY1 both cannot produce well the volume change for all the probing stress paths on LCT (Figs. 9b and 10b), due to their inability to reproduce the virgin CSR tests mentioned previously. The predictions of volumetric deformation by SANICLAY14 in tests AR1, AR2 and AD1 agree very well with experiments. However, the volumetric deformation in the rest of the tests seems to be underestimated (Fig. 11b). As indicated in Fig. 12b, YANG2015 provides overall satisfactory predictions of the volumetric changes for all of these tests on LCT. 3.3. Prediction of transitional CSR tests Transitional CSR tests, with stress ratios shifting from one value to another during the loading, are suitable for the validation of the

K=1

0.0 0.00

-0.05

-0.10

-0.15

-0.20

ε p-ε p, ref Fig. 6. Model predictions in virgin CSR consolidation tests by SANICLAY14.

proposed rotational hardening laws [30,14,33]. A very comprehensive set of transitional CSR tests has been conducted by Gens [9] on LCT samples with various initial consolidation histories. Some selected numerical predictions from all four models will be presented below. (1) Transitional CSR tests on initially isotropically consolidated LCT samples. Samples in tests IN1, IN2 and IN3, were first isotropically consolidated to p0 = 233.3 kPa, and were then subjected to undrained shearing to three designated CSRs, K = 0.4, 0.5 and 0.667. Each sample thereafter experienced the CSR test at its corresponding stress ratio (Fig. 13). The model predictions are depicted in Figs. 14–17, with the reference state set to the start of each CSR reloading. It can be seen that MCC, SANICLAY14 and YANG2015 provide reasonably close predictions of the volumetric change (Figs. 14a, 16a, and 17a), but S-CLAY1 gives some lower predictions (Fig. 15a). As for the strain rate ratio, MCC predicts constant values of e_ a =e_ p (Fig. 14b), since no rotation of the PPS is taken into account. S-CLAY1 and SANICLAY14 deliver similar changes of e_ a =e_ p , with good agreement for K = 0.667 and 0.5 but significant overpredictions of e_ a =e_ p for K = 0.4 (Figs. 15b and 16b). This is consistent with the previous observations that S-CLAY1 and SANICLAY14 tend to overestimate the dilatancy of LCT at the more anisotropic stress states. YANG2015 can satisfactorily reproduce the rapid change of e_ a =e_ p at various CSRs (Fig. 17b). It is worth mentioning that the predicted undrained stress paths depicted in

372

C. Yang et al. / Computers and Geotechnics 69 (2015) 365–377

K Sym

1

0.8

0.667

0.5

1.45

1.2 1.0 0.8 0.6 0.4 0.2 0.0 AR9 -0.2 -0.01

0.34

K= 1 0.8 0.667 0.5 0.4 0.34

1.50

v

0.4

q / σa,max

1.55

1.40

0

200

AR4

AR1

(a) 0.04

0.09

ε 11

0.14

(b)

AR1

AR4

v

100

AD1 AR2

AR9

1.40

1.35 50

AR10

1.42

CSL

(a)

AR3

p' (kPa)

1.38

AR2

AR10 AD1

-0.3

1.36 0.5

(b)

AR3

1

p' / σa,max K = 0.4 Fig. 9. Model predictions in tests on K0 consolidated samples by MCC.

ε a-ε a,ref

-0.2 K = 0.5

-0.1

K = 0.667

q /σ a,max

K = 0.8 K= 1

0.0 0.00

-0.05

-0.10

-0.15

-0.20

ε p-ε p, ref

1.2 AR3 AR10 1.0 0.8 0.6 AR4 0.4 0.2 0.0 AR9 -0.2 -0.01 0.04

Fig. 7. Model predictions in virgin CSR consolidation tests by YANG2015.

1.43 1.2

q /σa,max

0.8 0.6

AR10

AR1

AR3 AR4

0.4 0.2 0.0 0.0

ε 11

0.09

AR9

0.14

(b) AR4

1.39 1.37 0.5

AR2

AR10 AD1

AR3

1

p' / σ a,max

AR9

0.5

(a)

v

AR2

AR1

1.41

AD1

1.0

AR1

AD1 AR2

1.0

1.5

p' / σ a,max

Fig. 10. Model predictions in tests on K0 consolidated samples by S-CLAY1.

Fig. 8. Probing stress paths in K0 consolidated LCT samples.

Fig. 13 are different for each model due to the different configurations of the YS, and SANICLAY14 comparatively gives the closest simulation of the undrained stress path. (2) Transitional CSR tests on initially K0 consolidated LCT samples. Two samples, AN2 and AX3, initially K0 consolidated to p0 = 233.3 kPa, were reconsolidated at CSRs of K = 0.4 and 0.667 following the prescribed unloading and reloading stress paths, as revealed in Fig. 18. The numerical predictions from all four models are provided in Figs. 19–22. MCC, as shown in Fig. 19, seems to

accurately capture the volume change for both tests, but clearly overpredicts the strain rate ratios due to its inability to describe anisotropic plasticity. S-CLAY1 overpredicts the volume change for AX3 (Fig. 20a) but gives a good estimation of the corresponding strain rate ratio (Fig. 20b). However, with respect to AN2, S-CLAY1 predicts well the volume change but significantly overestimates the strain rate ratio, as indicated in Fig. 20. SANICLAY14, similar to S-CLAY1, overestimates the value of Dv for AX3 but predicts well for AN2 (Fig. 21a). An overprediction of the value of e_ a =e_ p by SANICLAY14 is obvious for AX3 and AN2 (Fig. 21b). YANG2015 seems to perform the best among the four models, as indicated in Fig. 22, the predictions of which agree very well with the experimental data for LCT.

373

2.5 AD1

AR1

0.09

ε 11

1.0

YANG2015 & MCC S-CLAY1

0.0 0.0

IN2 = 0.5

IN3

= 0.667

0.5

1.0

1.5

2.0

Fig. 13. Stress paths of transitional CSR tests, IN1, IN2 and IN3.

v

AR4

-0.02 AR10

AR2

AR3

1.37 0.5

(a)

0.00

AD1

0.02

1

p' / σ a,max

0.04 IN3 IN2 IN1

0.06 Fig. 11. Model predictions in tests on K0 consolidated samples by SANICLAY14.

1.2 AR3 AR10 1.0 0.8 0.6 AR4 0.4 0.2 0.0 AR9 -0.2 -0.01 0.04

0.08 0.4

AD1

6

AR2

5

0.8

1.6

(b) IN1

3 2

(a) 0.09

ε 11

3.2

p' / σ a,max

4 AR1

IN2 IN3

1 0.14

0 0.4

0.8

1.6

3.2

p' / σ a,max

1.43

(b)

AR9

1.41

2.5

p' / σ a,max

AR1

1.39

Fig. 14. Model predictions in tests IN1, IN2 and IN3 by MCC.

AR1

v

AR4

1.39

AR10

AR2 AD1

1.37 0.5

-0.02 AR3

1

p' / σ a,max Fig. 12. Model predictions in tests on K0 consolidated samples by YANG2015.

4. Discussion Extensive laboratory tests on reconstituted Lower Cromer Till reveal that fabric anisotropy is a characteristic of this clay and macroscopically any plastic straining along stress paths with changing stress ratios can modify the degree of fabric anisotropy [9,33]. Four elastoplastic constitutive models, MCC [24], S-CLAY1 [30], SANICLAY14 [8], and YANG2015 [33], were compared at element level based on benchmark experimental data for LCT [9]. MCC is an isotropic elastoplastic constitutive model. S-CLAY1, SANICLAY14 and YANG2015 are anisotropic critical state models, all of which fall into the same framework of rotational hardening. The focus of this paper is to investigate the effect of variously defined rotational hardening laws on the proper description of the anisotropic plasticity of soils in general and LCT in particular.

(a)

0.00 0.02

Δv

q / σa,max

IN1

= 0.4

0.5

0.14

(b)

AR9

SANICLAY14

1.5

(a)

1.43 1.41

2.0

AR2

q / σa,max

1.2 AR3 AR10 1.0 0.8 0.6 AR4 0.4 0.2 0.0 AR9 -0.2 -0.01 0.04

Δv

q / σa,max

C. Yang et al. / Computers and Geotechnics 69 (2015) 365–377

0.04 IN3 IN2 IN1

0.06 0.08 0.4

0.8

1.6

3.2

p' / σa,max 4

(b)

3 IN1

2 1 0 0.4

IN2 IN3

0.8

1.6

3.2

p' / σa,max Fig. 15. Model predictions in tests IN1, IN2 and IN3 by S-CLAY1.

374

C. Yang et al. / Computers and Geotechnics 69 (2015) 365–377

-0.02

0.00

(a)

0.00

0.02

0.04

0.8

1.6

0.06 0.2

3.2

p' / σa,max

4

AN2

0.04

IN3 IN2 IN1

0.06

0.4

6

(b)

5

AN2

2

1

IN2

1

IN3

0.8

1.6

AX3

0 0.2

3.2

0.4

0.00

(a)

0.00

3.2

(a)

0.02

Δv

0.02

Δv

1.6

Fig. 19. Model predictions in tests AX3 and AN2 by MCC.

Fig. 16. Model predictions in tests IN1, IN2 and IN3 by SANICLAY14.

-0.02

0.8

p' / σ a,max

p' / σa,max

0.04

0.04 IN3 IN2 IN1

0.06 0.08 0.4

0.8

1.6

AN2 AX3

0.06 0.2

3.2

0.4

4

5

(b)

4

0.8

1.6

3.2

p' / σa,max

p' / σa,max

(b)

3

3

AN2

2

2

IN1

1

1

IN2 IN3

0.8

1.6

0 0.2

3.2

Fig. 17. Model predictions in tests IN1, IN2 and IN3 by YANG2015.

AX3

0.4

= 0.667 0.5

1.0

0.8

1.6

3.2

Fig. 20. Model predictions in tests AX3 and AN2 by S-CLAY1.

4.1. MCC

= 0.4

0.8

0.4

The main features, capabilities and limitations of each model will be briefly summarised in the following with respect to the comparisons made with the observed behaviour of LCT.

AN2

1.2

AX3

p' / σa,max

p' / σ a,max

q / σa,max

3.2

3

2

0.0 0.0

1.6

(b)

4

IN1

0 0.4

0.8

p' / σa,max

3

0 0.4

AX3

Δv

Δv

0.02

0.08 0.4

(a)

1.5

2.0

p' / σ a,max Fig. 18. Stress paths of transitional CSR tests, AX3 and AN2.

MCC, as an isotropic model, can be considered to be a special case of the other three anisotropic models, as described by Eq. (5) and depicted in Fig. 2. The yield surface, as well as the plastic potential surface, have their principal directions aligned with the hydrostatic state, as shown in Fig. 1a. The predicted equilibrium fabric anisotropy, expressed in terms of dg and p0m;f =p0 in Fig. 3, implies that MCC overestimates both the dilatancy and the volumetric compression of LCT for all the anisotropic stress states,

C. Yang et al. / Computers and Geotechnics 69 (2015) 365–377

0.00

(a) Δv

0.02 AN2

0.04 0.06 0.2

AX3

0.4

0.8

1.6

3.2

p' / σa,max 4

375

derived from the rotational hardening law, which is described by Eq. (7) and depicted in Fig. 2. A parabolic curve is obtained for aE,f (=aE,g due to the associated flow rule), indicating the maximum degree of fabric anisotropy is achieved at |g/Mg| < 1. This further induces a significant overestimation of the dilatancy dg (Fig. 3a) at high values of g, at least for LCT. Meanwhile, the prediction of smaller spacing ratio p0m;f =p0 in Fig. 3b demonstrates a more compact set of parallel NCLs than the experimental measured curves for LCT (Fig. 5a). It can be seen that S-CLAY1 generally tends to underestimate the volumetric deformation and overestimate the dilatancy of LCT in more anisotropic stress states, as illustrated in Figs. 10, 15 and 20.

(b)

3

4.3. SANICLAY14 AN2

2 1

AX3

0 0.2

0.4

0.8

1.6

3.2

p' / σa,max Fig. 21. Model predictions in tests AX3 and AN2 by SANICLAY14.

0.00

(a)

Δv

0.02 AN2

0.04

AX3

0.06 0.2

0.4

0.8

1.6

3.2

p' / σa,max 4

(b)

4.4. YANG2015

3 2

AN2

1 0 0.2

SANICLAY14 introduces an exponential function (Eq. (10)) to describe the equilibrium fabric anisotropy (aE), which resembles that proposed by S-CLAY1, as shown in Fig. 2. However, the yield surface in SANICLAY14 is modified by choosing smaller values for Mf, thus leading to a much flatter shape of the YS (Fig. 1c). A non-associated flow rule is required, though the same inclination and the same size are still imposed on both the YS and the PPS. The value of dg predicted by SANICLAY14 is similar to that of S-CLAY1 (Fig. 3a), and so an overestimation of the dilatancy of LCT is apparent for stress states with g > gK 0 (Fig. 6b). The value of p0m;f =p0 predicted by SANICLAY14 agrees better with the experimental data for LCT than that predicted by S-CLAY1, thus leading to a more accurate prediction of the volumetric behaviour of the studied LCT samples (see Fig. 6a). Model predictions of the stress path probing tests on K0 consolidated samples (Fig. 11) and also those of the transitional CSR tests (Figs. 16 and 21) consistently demonstrate the less accurate description of the dilatancy of LCT in SANICLAY14 in the more anisotropic stress states. It should be noted that the introduction of a flatter yield surface can improve the description of the volumetric behaviour near critical state. However, some abnormal stress paths may be thus introduced in undrained triaxial tests particularly for overconsolidated LCT samples, as shown by Dafalias and Taiebat [8].

AX3

0.4

0.8

1.6

3.2

p' / σa,max Fig. 22. Model predictions in tests AX3 and AN2 by YANG2015.

which has been clearly illustrated in Fig. 4 for the virgin CSR consolidation tests. All these eventually lead to the incapacity of MCC to reproduce satisfactorily the probing stress path tests on K0 consolidated samples (Fig. 9) and also the various transitional CSR tests (Figs. 14 and 19). As such, the neglect of fabric anisotropy renders MCC as limited in its description of the mechanical behaviour of natural clays. 4.2. S-CLAY1 S-CLAY1 adopts a rotational hardening law, which was developed based on many transitional CSR tests on one specific soft clay [30], not LCT. The equilibrium fabric anisotropy can be indirectly

YANG2015 differs from S-CLAY1 and SANICLAY14 particularly in the description of the equilibrium fabric anisotropy. Yang et al. [33] illustrated that a unique fabric anisotropy can be established for each single virgin CSR loading, which can be quantified together by dg and p0m;f =p0 (Eqs. (13) and (15)). Both of them can be further combined with the plastic potential surface and the yield surface to determine the values of aE,g and aE,f (Eqs. (12) and (14)). A non-associated flow rule is automatically obtained. The rotational hardening law defined in Eq. (11), together with the volumetric hardening laws in Eq. (4), make YANG2015 a good model to describe the anisotropic plasticity of LCT. For instance, the shear strength and compression behaviour of LCT in the virgin CSR consolidation tests, the various probing tests on initially K0 consolidated samples, and the series of transition CSR tests, can be satisfactorily captured by YANG2015, as depicted in Figs. 7, 12, 17 and 22. It is also worth mentioning that the three anisotropic models, and even the isotropic MCC model, can all predict closely the shear strength of K0 consolidated LCT for most probing stress paths (see Figs. 8–12). Meanwhile, S-CLAY1, SANICLAY14 and YANG2015 have been found to offer similar predictions in the conventional undrained triaxial tests on LCT [7,8,33]. This implies that merely a good prediction of the shear strength obtained in traditional drained or undrained triaxial tests might not provide a sufficient evaluation of a proposed model, particularly with regard to the

376

C. Yang et al. / Computers and Geotechnics 69 (2015) 365–377

proposed rotational hardening laws. On the other hand, the conventional drained and/or undrained triaxial tests cannot provide a definitive description of the change of fabric anisotropy along the imposed stress paths. Therefore, some more effective experimental schemes are required to reveal the effect of fabric anisotropy and validate the anisotropic models. Virgin and transitional CSR tests, as indicated from the above analysis, prove to be feasible in the laboratory (i.e., the isotropic or one-dimensional consolidation tests) and also efficient in the interpretation of the effects of fabric anisotropy. Particularly the stress path probing tests on initially K0 consolidated samples, which resemble the various possible stress conditions encountered in natural soils, provide useful guidance on the applicability of any constitutive model in geotechnical engineering practice. The conventional drained and/or undrained triaxial tests can serve as a complementary approach to validate the overall performance of the proposed anisotropic models. The above analysis of the performance of MCC, S-CLAY1, SANICLAY14, and YANG2015 has been conducted with respect to one specific clay, LCT, due to its stable mechanical properties and also the availability of a systematic and complete experimental investigation. However, an indirect but less authoritative comparison of the four models can also be made for another very soft clay, Otaniemi clay, by reference to the works of Wheeler et al. [30], Karstunen and Koskinen [14], Dafalias and Taiebat [7] and Yang et al. [32]. Wheeler et al. [30] and Dafalias and Taiebat [7] simulated some transitional CSR tests on natural Otaniemi clay samples, while Karstunen and Koskinen [14] and Yang et al. [32] worked with reconstituted samples. Although Dafalias and Taiebat [7] and Yang et al. [32] used their earlier versions of SANICALY14 and YANG2015 in their studies, it is clear that similar conclusions can be summarised, based on the common simulations of transitional CSR tests conducted on natural and reconstituted Otaniemi clay samples. MCC fails to capture the effects of fabric anisotropy of Otaniemi clay [30]. S-CLAY1 and SANICLAY14 perform well qualitatively, by introducing the rotational hardening concept, but perhaps still require significant quantitative improvements, at least for Otaniemi clay [30,14,8]. YANG2015 seems to provide promising performance in reproducing the mechanical behaviour of Otaniemi clay [32], which indicates the feasibility of its proposed methodology for describing the effects of fabric anisotropy of soils. However, it should also be borne in mind that Otaniemi clay has a non-homogeneous mineral composition, is very sensitive in terms of its soil structure and displays strong time-dependency which, taken together, imply that it may not be an ideal candidate to investigate the single effect of fabric anisotropy in soils. Nevertheless, the importance of the compression behaviour revealed in CSR tests for properly understanding the plastic anisotropy of soil fabric can be observed in Otaniemi clay, as well as in many other soils, such as those studied by Roscoe and Poorooshasb [25], Le Lievre [17], Lewin and Burland [18], Wood [31], Rampello et al. [23] and Belokas and Kavvadas [2]. It has been suggested that any increase in accuracy of the model predictions brings with it an increasing complexity of the model formulation, as indicated by the number of additional parameters required for the three anisotropic models considered here (Table 3). However, the adoption of more parameters does not necessarily imply a cumbersome constitutive model, especially if every model parameter has its own physical meaning and can be determined via practical laboratory tests. For example, for YANG2015, constant stress ratio loading tests provide the experimental basis for the constitutive model formulation and thus only routine laboratory experiments, such as oedometer and isotropic compression tests, need be conducted to determine the extra model parameters.

5. Conclusion Comparisons of the predictions of the four elastoplastic constitutive models, MCC, S-CLAY1, SANICLAY14 and YANG2015, demonstrate that the isotropic MCC model cannot provide accurate predictions of the mechanical behaviour of LCT and fabric isotropy is merely a special case of the more general fabric anisotropy. Consideration of plastic anisotropy of soil fabric can significantly improve the accuracy of constitutive modelling at a macroscopic level. Comparison of model predictions in various tests indicates that YANG2015 provides a more consistent and better performance than S-CLAY1 and SANICLAY14, at least for LCT. S-CLAY1 underestimates the volumetric behaviour while both S-CLAY1 and SANICLAY14 overestimate the dilatancy of LCT in more anisotropic stress states. The key difference between the three anisotropic models lies in their description of the equilibrium fabric anisotropy. Virgin CSR consolidation tests can be combined with the properly chosen functions for the plastic potential surface and yield surface to describe this equilibrium state of fabric anisotropy. The results of transitional CSR tests provide a more appropriate and more rigorous means of validating the proposed rotational hardening law than the conventional drained or undrained triaxial tests. Although the comparisons presented here are only for one particular soil, viz., LCT, the main features, including the capabilities and limitations, of each model have been revealed in some detail, which may be of assistance in choosing constitutive models for engineering practice. Acknowledgements Support for this work from the Australian Research Council Centre of Excellence for Geotechnical Science and Engineering, hosted at the University of Newcastle, Australia, is gratefully acknowledged. References [1] Belokas G, Kavvadas M. An anisotropic model for structured soils. Part I: theory. Comput Geotech 2010;37(6):737–47. [2] Belokas G, Kavvadas M. An intrinsic compressibility framework for clayey soils. Geotech Geol Eng 2011;29(5):855–71. [3] Chen L, Shao JF, Huang HW. Coupled elastoplastic damage modeling of anisotropic rocks. Comput Geotech 2010;37(1–2):187–94. [4] Cudny M, Vermeer PA. On the modelling of anisotropy and destructuration of soft clays within the multi-laminate framework. Comput Geotech 2004;31(1):1–22. [5] Dafalias YF. An anisotropic critical state soil plasticity model. Mech Res Commun 1986;13(6):341–7. [6] Dafalias YF, Manzari MT, Papadimitriou AG. SANICLAY: simple anisotropic clay plasticity model. Int J Numer Anal Meth Geomech 2006;30(12):1231–57. [7] Dafalias YF, Taiebat M. Anatomy of rotational hardening in clay plasticity. Géotechnique 2013;63:1406–18. [8] Dafalias YF, Taiebat M. Rotational hardening with and without anisotropic fabric at critical state. Géotechnique 2014;64:507–11. [9] Gens A. Stress–strain and strength of a low plasticity clay. Ph.D. University of London; 1982. [10] Graham J, Noonan ML, Lew KV. Yield states and stress–strain relationships in a natural plastic clay. Can Geotech J 1983;20(3):502–16. [11] Hu C, Liu H, Huang W. Anisotropic bounding-surface plasticity model for the cyclic shakedown and degradation of saturated clay. Comput Geotech 2012;44:34–47. [12] Jiang J, Ling HI. A framework of an anisotropic elastoplastic model for clays. Mech Res Commun 2010;37(4):394–8. [13] Karstunen M, Wiltafsky C, Krenn H, Scharinger F, Schweiger HF. Modelling the behaviour of an embankment on soft clay with different constitutive models. Int J Numer Anal Meth Geomech 2006;30(10):953–82. [14] Karstunen M, Koskinen M. Plastic anisotropy of soft reconstituted clays. Can Geotech J 2008;45(3):314–28. [15] Karstunen M, Yin ZY. Modelling time-dependent behaviour of Murro test embankment. Géotechnique 2010;60(10):735–49. [16] Koskinen M, Zentar R, Karstunen M. Anisotropy of reconstituted POKO clay. Numerical models in geomechanics. Taylor & Francis; 2002. p. 99–105. [17] Le Lievre B. The yielding and flow of cohesive soils in triaxial compression. Ph.D. Waterloo University; 1967.

C. Yang et al. / Computers and Geotechnics 69 (2015) 365–377 [18] Lewin PI, Burland JB. Stress-probe experiments on saturated normally consolidated clay. Géotechnique 1970;20:38–56. [19] Liu W, Shi M, Miao L, Xu L, Zhang D. Constitutive modeling of the destructuration and anisotropy of natural soft clay. Comput Geotech 2013;51:24–41. [20] Newson TA, Davies MCR. A rotational hardening constitutive model for anisotropically consolidated clay. J Jpn Geotech Soc: Soils Found 1996;36(3):13–20. [21] Pestana JM, Whittle AJ. Formulation of a unified constitutive model for clays and sands. Int J Numer Anal Meth Geomech 1999;23(12):1215–43. [22] Pestana JM, Whittle AJ, Gens A. Evaluation of a constitutive model for clays and sands: Part II – clay behaviour. Int J Numer Anal Meth Geomech 2002;26(11):1123–46. [23] Rampello S, Viggiani GMB, Amorosi A. Small-strain stiffness of reconstituted clay compressed along constant triaxial effective stress ratio paths. Géotechnique 1997;47:475–89. [24] Roscoe KH, Burland JB. On the generalised stress–strain behaviour of ‘‘wet’’ clay. Eng Plast 1968:535–609. [25] Roscoe KH, Poorooshasb HB. A theoretical and experimental study of strains in triaxial compression tests on normally consolidated clays. Geotechnique 1964;13(1):12–38. [26] Sheng D, Sloan SW, Yu HS. Aspects of finite element implementation of critical state models. Comput Mech 2000;26(2):185–96. [27] Sloan SW, Abbo AJ, Sheng D. Refined explicit integration of elastoplastic models with automatic error control. Eng Comput 2001;18(1–2):121–54 [Erratum: Engineering Computations (2002), 19(5/6), 594–594].

377

[28] Sun DA, Matsuoka H, Yao YP, Ishii H. An anisotropic hardening elastoplastic model for clays and sands and its application to FE analysis. Comput Geotech 2004;31(1):37–46. [29] Toivanen TL. Modelling the anisotropy of Otaniemi clay. Master’s thesis, Helsinki University of Technology, Finland; 1999. [30] Wheeler SJ, Näätänen A, Karstunen M, Lojander M. An anisotropic elastoplastic model for soft clays. Can Geotech J 2003;40(2):403–18. [31] Wood DM. Soil behaviour and critical state soil mechanics. Cambridge University Press; 1990. [32] Yang C, Sheng D, Carter J, Pineda J, Kelly R. From compression behavior to plastic anisotropy of reconstituted soft soils. Soil Behav Geomech 2014:658–67. [33] Yang C, Sheng D, Carter JP, Sloan SW. Modelling the plastic anisotropy of Lower Cromer Till. Comput Geotech 2015;69:22–37. [34] Ye B, Ye G, Zhang F. Numerical modeling of changes in anisotropy during liquefaction using a generalized constitutive model. Comput Geotech 2012;42:62–72. [35] Yin Z-Y, Chang CS. Microstructural modelling of stress-dependent behaviour of clay. Int J Solids Struct 2009;46(6):1373–88. [36] Yin Z-Y, Chang CS, Hicher P-Y, Karstunen M. Micromechanical analysis of kinematic hardening in natural clay. Int J Plast 2009;25(8):1413–35. [37] Zentar R, Karstunen M, Wiltafsky C, Schweiger H, Koskinen M. Comparison of two approaches for modelling anisotropy of soft clays. Numerical models in geomechanics. Taylor & Francis; 2002. p. 115–21.