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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Monotonic and Cyclic Analysis of Granular Soils N. Khalili *, M. A. Habte, S. Valliappan School of Civil and Environmental Engineering, The University of New South Wales, Sydney, NSW, 2052, Australia Email: n.khalili@unsw.edu.au Abstract A simple yet rigorous bounding surface plasticity model is developed to describe the stress-strain behaviour of variably saturated granular soils subjected to monotonic and cyclic loading. The model is formulated incrementally within the critical state framework using the effective stress approach. Both plastic volumetric strain and matric suction are introduced as hardening parameters. Cyclic behaviour is captured through a radial mapping rule in which the point of stress reversal is taken as the centre of projection. The effect of particle crushing at high stresses is considered through a three-segmented critical state and isotropic compression lines. A non-associative flow rule is employed to generalise application of the model to all soil types. The model is validated using experimental data from the literature. Key words: bounding surface plasticity, effective stress, granular soils, cyclic loading, suction hardening INTRODUCTION

Response of granular materials to monotonic and cyclic loadings is complex due to the pressure and specific volume dependency of the stress-stain relationship and the highly nonlinear behaviour of the soil matrix. This is particularly the case under undrained conditions in which repeated loading and unloading can lead to a substantial rise in pore water pressure and a sudden loss in the shear strength and the stiffness of the soil. In fact, the study of loading and unloading response in soils and development of relationships for its prediction in natural formations and engineered materials has been a major area of research in modern geomechanics. Concerted effort has been made to develop predictive capabilities associated with such topics as earthquake engineering, soil-structure interaction, soil liquefaction, off-shore engineering, etc. The objective of this paper is to present a comprehensive and unified elasto-plastic constitutive model for monotonic and cyclic loading of granular soils. The model is formulated using the bounding surface plasticity theory within a critical state framework. A new bounding surface along with a radial mapping rule is introduced to obtain a more realistic response under monotonic and cyclic loading. Other crucial aspects taken into account are the non- associativity of the flow rule, and the crushing of the soil particles at high stresses. Crushing of the particles is captured through the introduction of a three-segmented critical state and limiting isotropic compression lines. The model is developed within the context of both fully and partially saturated soils. Unsaturated behaviour is taken into account using the effective stress principle, and introducing suction as a hardening parameter. PRELIMINARIES

1. Notation The material behaviour is assumed isotropic and rate independent in both elastic and elastic-plastic responses. For simplicity, triaxial stress notation ~q p′ is adopted throughout; 1

1 33 ( 2 )p σ σ′ ′ ′= + is the mean effective stress and )( 31 σσ ′−′=q is the deviator stress, where 1σ ′ and 3σ ′ are the axial and radial stresses respectively. The corresponding work conjugate strain variables are the soil skeleton volumetric strain 31 2εεε +=p

and shear strain ( )3132 εεε −=q . The pairs of stresses and strains are abbreviated in the vector form as [ ]T,qp′=′σ

and [ ]T, qp εε=ε . Following soil mechanics convention, compression is considered positive and tension is negative.

2. The Effective Stress In its incremental form, the effective mean stress is expressed as [1]

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( )a a w netp p p p p p sψ ψ′ = − + − = + (1)

Where, a superimposed dot indicates the rate of change, p′ is the effective mean stress, p is the total mean stress,

wp is the pore water pressure, ap is the pore air pressure, net ap p p= − is the net mean stress, wa pps −= is the matric suction, and ψ is the incremental effective stress parameter, attaining a value of one for saturated soils and zero for dry soils. The effective stress parameter describes the contribution of suction to the effective stress. It is a scaling factor that averages matric suction from the pore-scale level to a macroscopic level over a representative elementary volume. In this formulation, the effective stress parameter is defined using an experimentally obtained correlation between soil suction and the effective stress [1-3],

⎪⎪⎩

⎪⎪⎨

⎧

>

≤<⎟⎠⎞

⎜⎝⎛

≤

=−

e

eee

e

ssfor

sssforss

ssfor

250

2545.0

155.0

ψ

(2)

where es is the suction value separating saturated from unsaturated states. For wetting processes, exe ss = , and for drying processes aee ss = , in which exs is the air expulsion value and aes is the air entry value. For fully saturated soils,

1=ψ and equation (1) reduces to wppp &&& −=′ .

3. The Critical State The critical state (CS) is an ultimate condition towards which all states approach with increasing deviatoric shear strain. To take into account crushing of soil particles at high stresses [4-6], the critical state line in the

p′ln~υ plane is assumed to take the form of three linear segments [4].

Figure 1: Illustration of critical state line (CSL), limiting isotropic compression line (LICL),

state parameter (ξ) and elastic unload/reload line on υ ~ lnp′ plane

As shown in Figure 1, six parameters are used to define the critical state line (CSL) in the p′ln~υ plane: λ0, Γ0, υcr, λcr, υf and λf. The parameters λ0 and Γ0 are the slope of the initial portion of the CSL and its specific volume at p′ = 1kPa, respectively; υcr is the specific volume at the onset of particle crushing; λcr is the slope during the particle crushing stage; and υf and λf are the specific volume at the end of crushing and the slope of the CSL at extremely high stresses, respectively. The CSL in the pq ′~ plane is defined using a straight line passing through the origin. The slope of the CSL, denoted by csM , is linked to the critical state friction angle ( csφ′ ) through,

6sin3 sin

cscs

cs

Mt

φφ′

=′− (3)

with t = +1 for compression and t = -1 for extension. In addition, certain features of the model are linked to the state parameter (ξ), which is a dimensionless parameter defined as the vertical distance between the current state and the CSL in the υ ~ln p′ plane (Fig. 1).

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4. Limiting Isotropic Compression Line Similar to the critical state line, the limiting isotropic compression line (LICL) is a reference line in the p′ln~υ plane where the stress state approaches with increasing isotropic compression. For granular soils, the limiting isotropic compression line can also be regarded as the locus of the loosest possible state that a soil can achieve for a given effective mean stress. In the present investigation, the LICL is taken as parallel to CSL at a constant shift along the κ line in the p′ln~υ plane (Figure 1). For partially saturated soils, the stiffening effect of suction on the soil response is manifested through its effect on the position of the LICL. In general, a suction increase has a dual effect of increasing the slope and vertical positioning of the LICL [12]. STRESS-STRAIN RELATIONSHIP

The total strain rate is decomposed into elastic and plastic parts according to, e p= +ε ε ε& & & (4)

where superscripts e and p denote the elastic and plastic components, respectively. Incremental elastic strains are linked to the incremental stresses through,

eeεDσ && = (5)

where eD is the elastic property matrix. The incremental plastic strain-stress relationship is written as,

p 1h

′=ε σ& &mn (6)

where n is the unit vector normal to the loading surface at the current stress state σ′ , m is the unit direction of plastic flow at σ′ , and h is the hardening modulus. Combining (4), (5) and (6) yields the elasto-plastic stress-strain relationship,

e T ee

T eh⎛ ⎞

′ = −⎜ ⎟+⎝ ⎠

D mn Dσ D εn D m

&& (7)

ELASTIC BEHAVIOUR

For pq ′~ formulation the elastic property matrix eD is defined as,

00 3

e KG

⎡ ⎤= ⎢ ⎥

⎣ ⎦D (8)

in which K and G are the drained tangential bulk and shear moduli, respectively. These tangential elastic moduli are calculated assuming that unloading/reloading occurs along a κ line in the ~ ln pυ ′ plane,

κ

υpK′

= and ( )( ) κ

υνν pG

′+

−=

12213

(9)

where υ is the specific volume and ν is the Poisson’s ratio. In the bounding surface theory, a purely elastic region is often assumed, bounded by a loading surface. Elastic behaviour occurs while the current stress state σ′ lies inside the loading surface. First yield occurs when σ′ intercepts the loading surface, beyond which the loading surface moves in the stress space such that σ′ remains on it at all times. A purely elastic region is omitted in this investigation, such that all deformation is elastic-plastic. This accords with the observation that truly elastic behaviour in sands ends at shear strains in the order of 0.00001 [7]. PLASTIC BEHAVIOUR

The elastic-plastic behaviour is captured using the theory of bounding surface plasticity [8]. In this approach, plastic straining occurs when the stress state σ′ lies on or within the bounding surface. This is achieved by defining the hardening modulus h as a decreasing function of the distance between σ′ and an “image point”, σ′ , on the bounding surface. The image point is selected using a mapping rule such that the unit normal vectors to the loading surface and the bounding surface at points σ′ and σ′ are the same (Fig. 2).

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Figure 2: Schematic illustration of principles of bounding surface plasticity

1. Bounding Surface The bounding surface is the limit of admissible stress states. The shape of the bounding surface is selected from the undrained response of the material at its loosest state. The function (F) below was found to best fit the experimental data [9],

( ) 0ln

ln),,( =′′

−⎟⎟⎠

⎞⎜⎜⎝

⎛′

=′′R

pppM

qpqpF c

N

csc (10)

where the superimposed bar denotes the stress condition on the bounding surface, csM is the slope of the CSL in the pq ′~ plane, cp′ is the parameter controlling the size of F and is equal to the isotropic preconsolidation stress. For

partially saturated soils, cp′ is a function of suction and plastic volumetric strain. R and N are material constants. The bounding surface has a tear drop shape (Figure 3) and remains smooth and continuous as it intersects the p′ axis at right angles when 0=′p and cpp ′=′ for N > 1, allowing yielding to occur during isotropic loading. This enables prediction of plastic strains during isotropic loading, a feature lacking from most of the existing models for granular soils ( see [10] for example). The nearly straight line shape of the bounding surface on the dense side of the critical state ( Rpp c /′<′ ) allows for a realistic prediction of the peak strength of granular soils which is over-estimated using elliptical yield surfaces. The existence of a peak deviatoric stress above the critical state value on the loose side ( R/pp c′>′ ) is a key feature in the undrained behaviour of loose samples and is captured using the proposed bounding surface. Strain softening due to shearing is an area of significant interest in static liquefaction analysis of loose sands.

Figure 3: Bounding surface, loading surface and mapping rule for first time loading

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2. Loading Surface Implicit in the bounding surface formulation proposed is the existence of a loading surface on which the current stress state lies. The loading surface is of the same shape and homologous to the bounding surface with respect to the centre of homology. For virgin loading, the centre of homology is taken at the origin of coordinate system in the pq ′~ plane and the image point is located using a radial mapping rule (Figure 3). For unloading and reloading, the centre of homology is taken as the last point of stress reversal (Figure 4). In this case, the maximum loading surface through the point of stress reversal serves as a local bounding surface for the loading surfaces within the maximum loading surface. To maintain similarity with the bounding surface, the loading surfaces undergo kinematic hardening during loading and unloading such that they remain tangent to the maximum 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 modified mapping rule incorporates the loading history of the soil in the cyclic stress-strain analysis using the stress reversal points and the corresponding maximum loading surfaces. In general, the loading surface (f) takes the form

( ) 0ln

ˆˆlnˆ

ˆ)ˆ,ˆ,ˆ( =

′′−⎟

⎠

⎞⎜⎝

⎛′

=′′R

pppM

qpqpf cN

csc (11)

where ppp α−′=′ˆ , qqq α−=ˆ , pcc pp α−′=′ˆ . pα and qα are the components of the kinematic hardening vector

)( qp α,αα (Fig. 4). cp̂′ is the isotropic hardening parameter controlling the size of the loading surface. The loading surface for first time loading is obtained when 0α qp == α .

Figure 4: Mapping rule for unloading and reloading

The unit normal vector [ ]T, qp nn=n at the image point defining the direction of loading is given using the general equation,

σσ

σσn

′∂∂′∂∂

=′∂∂′∂∂

=FF

ff

(12)

3. Plastic Potential The plastic potential (g=0) defines the ratio between the incremental plastic volumetric strain and the incremental plastic shear strain. The plastic flow rule is generally defined using the relationship between plastic dilatancy ( pp

qvd εε &&= ) and the corresponding stress ratio pq ′=η . The stress-dilatancy relationship adopted in this model is,

( ) ( ) ( )η−=′∂∂∂∂= csMApgqgd (13)

where A is a material constant accounting for the mechanism of plastic energy dissipation. Equation (13) is similar to Rowe’s original stress-dilatancy relationship derived from minimum energy considerations of particle sliding [11].

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The expression for the plastic potential (g) is obtained by integrating equation (13) with respect to p′ and q,

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ′−′

−−

′−=′

−1

11),,(

A

o

cso p

pA

pA

pAMqtpqpg (14)

Typical shapes of the plastic potential are shown in Figure 5. Notice that two families of curves are identified: +csM for

compressive loading ( 0q > ) and −csM for extensive loading ( 0q < ). Specifically, at any stress point,σ′ , we identify

two vectors of plastic flow, one for compressive and the other for extensive loading. In this case, the components of [ ]T, qp mm=m at σ′ are defined in the general form,

+ +=

′∂∂′∂∂

=21 d

dtg

pgmp σ and

+ +=

′∂∂∂∂

=21

1d

tg

qgmq σ (15)

with t = +1 for compressive loading ( 0q > ) and t = -1 for extensive loading ( 0q < ). Notice that for components of m , the direction of loading and thus the sign of t , is determined based on the sign of deviatoric stress at σ′ rather than at σ′ . Determination of the direction of the plastic flow vector based on the direction of loading enables prediction of continual volumetric contraction observed in drained loose sands during cyclic loading as well as the predominant volumetric contraction accompanied by volumetric expansion observed in dense sands during cyclic loading. This feature lacks from the existing models of plasticity proposed for the cyclic analysis of granular materials.

pqε&q

p′, ppε&

η −m

+m

σ′

+csM

−csM

Figure 5: Vectors of plastic potential at σ′ for compressive and extensive loading

4. Hardening Modulus Following the usual approach in the bounding surface plasticity, the hardening modulus h is divided into two components fb hhh += , where bh is the plastic modulus at σ′ on the bounding surface, and fh is

some arbitrary modulus at σ′ , defined as a decreasing function of the distance between σ′ and σ′ . Applying the consistency condition at the bounding surface taking into account the hardening effects of both plastic volumetric strain and matric suction, the expression for bh becomes,

σ ′∂∂′∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂′∂

+∂

′∂′∂

∂−=

Fpgs

spp

pFh p

p

cp

v

c

cb εε &

& (16)

The modulus fh is defined such that it is zero on the bounding surface and infinity at the point of stress reversal. Following [3], [4] and [9], fh is assumed to be of the form,

)(1ˆ

ηηεε

−⎥⎦

⎤⎢⎣

⎡−

′′

′′

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂′∂

+∂

′∂= pm

c

c

cpp

cp

v

cf k

pp

pps

spp

th&

& (17)

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where cp′ and cp̂′ define the sizes of the bounding and loading surfaces, respectively, mk is a material parameter controlling the steepness of the response in the qq ε~ plane, csp M)k1( ξη −= is the slope of the peak strength line

in the pq ′~ plane, and k is a constant defining the relationship between density and peak strength.

5. Suction Hardening In addition to plastic volumetric hardening, partially saturated soils may undergo additional hardening due to matric suction. The effect of suction on the hardening parameter can either be coupled or decoupled with the plastic volumetric hardening [3]. The coupled influence is expressed using a multiplicative function whereas the decoupled effect is expressed using an additive function. The general expression for the hardening rule is written as [3],

( ) ( ) ( )( )

( )ss

spspp

vcc γ

κλεΔυγ ˆexpˆ̂0 +⎟⎟

⎠

⎞⎜⎜⎝

⎛−

′=′ (18)

where ( )spc′ and ( )0cp′ are the respective values of the hardening parameter at suction s and at full saturation,

respectively. The functions ( )sγ̂ and ( )sγ̂̂ represent the additive and multiplicative effects of suction hardening. Following [12], only the coupled influence of suction hardening is considered in this investigation. That is, ( ) 0ˆ =sγ

and ( )sˆ̂γ is given by,

( ) ( ) ( )( )

( ) ( )( ) ( )[ ]⎟⎟

⎠

⎞⎜⎜⎝

⎛ ′−

−−

−−

= 0ln00expˆ̂cp

ss

sss

κλλλ

κλΝΝγ (19)

where ( )0Ν and ( )sΝ are intercepts of the LICL at 1=′p kPa for the saturated and unsaturated states respectively, similarly ( )0λ and ( )sλ are slopes of the saturated and unsaturated LICL.

6. Model Parameters The constitutive parameters used in the model are: κ and ν to describe the elastic behaviour; csM , 0λ , 0Γ , crυ , crλ , fυ and fλ to define the critical state; N and R to define the shape of the bounding surface;

k to define the peak strength line; mk to calibrate the hardening modulus; and A to define the stress-dilatancy relationship. The elastic parameters κ and ν with the critical state constants csM , 0λ , 0Γ , crυ , crλ , fυ and fλ can be determined from triaxial tests using conventional procedures. κ is the slope of the elastic unload reload line on

p′ln~υ plane, ν is the Poisson’s ratio, csM is the slope of the critical state line on the triaxial pq ′~ plane. 0λ ,

crλ and fλ are the slopes of the initial, particle crushing and final segments of the critical state line, respectively. 0Γ is the reference specific volume at the critical state at a unit confining pressure. crυ and fυ are specific volumes on the critical state line at the start and end of the particle crushing phase, respectively. The parameters N and R are 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 κ -line, and is determined from isotropic consolidation test data. N and R are considered to be independent of suction. Assuming elastic strains are negligible in comparison to plastic strains, A is determined by plotting the stress ratio η against the measured total dilatancy in standard drained triaxial compression tests. The value of k is obtained from the slope of csp Mη versus ξ . In general, k = 2.0 for granular soils [10]. mk is strongly influenced by the initial state

parameter of the soil oξ and the loading direction. It is best obtained by fitting, using the initial slope of drained deviatoric loading and unloading responses in the q~q ε plane.

APPLICATION TO MONOTONIC LOADING

Application of the proposed model is demonstrated using results of monotonic and cyclic loading tests on fully and partially saturated soils. A number of drained and undrained triaxial tests from the literature are analysed. In all the cases analysed, the model predictions are shown using continuous solid lines and the experimental results using discrete symbols. 1. Drained Triaxial Tests on Saturated Hostun Sand Figure 6 shows results of model simulation for drained triaxial compression tests on loose and dense samples of Hostun sand. The experimental data were taken from the group of results compiled in [11]. This data set consists of results of several classical triaxial tests on Hostun sand conducted in

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different laboratories on 70 mm and 100 mm diameter samples using triaxial apparatus with lubricated and unlubricated ends and miscellaneous methods of sample preparation. The results in Fig. 6 show that the model captures both 1~ εq and pε ~ 1ε responses with acceptable accuracy. The model was able to simulate the hardening and plastic contraction observed in the loose sample, and the softening and plastic dilation observed in the dense sample. The initial conditions for the samples used in the analyses were: p′ = 300 kPa and e = 0.945 for the loose sample, and p′ = 300 kPa and e = 0.574 for the dense sample. The material constants used in the simulations were: κ = 0.003, ν = 0.1, Mcs = 1.24, λ0 = 0.028, Γ0 = 2.037, λcr = 0.24, N = 2.3, R = 7.5, k = 2.0, with A = 1.0 and km = 0.0 for the loose sample while A = 0.86 and km = 3.0 for the dense sample.

0

200

400

600

800

1000

0 0.04 0.08 0.12 0.16 0.2ε1

q(k

Pa)

p' = 300, e = 0.574p' = 300, e = 0.945

Experimental Data Model Simulation

-0.12

-0.08

-0.04

0

0.04

0.080 0.04 0.08 0.12 0.16 0.2ε1

εp

p' = 300, e = 0.574p' = 300, e = 0.945

Experimental Data Model Simulation

Figure 6: Model simulations for drained compression tests on saturated samples of Hostun sand

2. Undrained Triaxial Tests on Saturated Ottawa Sand Figure 7 shows results of model simulation for undrained triaxial compression tests on loose samples of Ottawa sand. The test data were obtained from conventional triaxial tests on samples 63 mm in diameter and 126 mm in height, prepared using moist-tamping technique [13]. The soil sample was made up of uniform, medium to coarse grained, rounded to sub rounded quartz grains. The results in Figure 7 show that the model captures the undrained response within reasonable tolerance. The effective stress path and the 1~ εq plots for the loose samples show that q increased initially, reaching a peak value before markedly decreasing towards the critical state. The results confirm capability of the model to simulate static liquefaction phenomenon observed in undrained loading of very loose sands, which is entirely controlled by the shape of the bounding surface. The initial conditions of the samples were: p′ = 348 kPa and e = 0.793 for the first sample, and p′ = 475 kPa and e = 0.793 for the second sample. The model parameters used in the simulations were: κ = 0.005, ν = 0.3, Mcs= 1.19, λ0 = 0.0168, Γ0 = 1.864, N = 2.5, R = 69.2, k = 2.0, A = 1.0 and km= 0.3 for both samples.

0

100

200

300

0 100 200 300 400 500p' (kPa)

q (k

Pa)

p' = 348, e = 0.793p' = 475, e = 0.793

Experimental Data Model Simulation

0

50

100

150

200

250

0 0.01 0.02 0.03 0.04 0.05 ε1

q(k

Pa)

p' = 348, e = 0.793p' = 475, e = 0.793

Experimental Data Model Simulation

Figure 7: Model simulations for undrained compression tests on saturated samples of Ottawa sand

3. Drained Triaxial Tests on Unsaturated Kurnell Sand Application of the proposed model to monotonic loading of unsaturated soils is demonstrated using results of drained (constant suction) and undrained (constant water content) triaxial tests on Kurnell sand [3]. These tests were conducted on samples of 50 mm in diameter and 51 mm in height prepared by pulviation of dry soil. Suction values varying from 50 to 410 kPa were applied by keeping the pore air

— 76 —

pressure constant and varying the pore water pressure. Results for drained tests at constant suction values 100 and 200 kPa are shown in Figures 8 and 9 respectively. The initial conditions for these tests were: pnet = 50 kPa and e = 0.763, pnet = 100 kPa and e = 0.687 for the samples tested at s = 100 kPa; pnet = 51 kPa and e = 0.780, pnet = 101 kPa and e = 0.697 for the samples tested at s = 200 kPa. The model parameters used in the simulations were: κ = 0.006, v = 0.3, Mcs = 1.475, λ0 = 0.0284, Γ0 = 2.0373, υcr = 1.835, λcr = 0.195, υf = 1.25, λf = 0.04, N = 3, R = 7.3, k = 2.0, A = 0.7 and km = 0.1 for all the samples. In addition, the air entry value was sae = 6.0 kPa. Figures 8 and 9 show a very good match between the model simulations and experimental data. All the samples were initially denser than critical and hence exhibited a classic stress-strain behaviour of a strain softening material. Specifically, hardening occurred up to a peak in the shear resistance, accompanied by initial volumetric contraction followed by volumetric expansion. Softening towards the critical state line was observed after reaching the peak and was accompanied by volumetric expansion.

Figure 8: Model simulations for drained triaxial tests on unsaturated samples of Kurnell sand at s = 100 kPa

Figure 9: Model simulations for drained triaxial tests on unsaturated samples of Kurnell sand at s = 200 kPa

Figure 10: Model simulations for undrained triaxial tests on unsaturated samples of Kurnell sand at s = 100 kPa

4. Undrained Triaxial Tests on Unsaturated Kurnell Sand Figure 10 shows comparison between model simulations and experimental results for unsaturated undrained (constant water content) triaxial tests on Kurnell sand

— 77 —

at a suction value of s = 100 kPa. The initial conditions of the samples were: pnet = 49 kPa and e = 0.777, and pnet = 98 kPa and e = 0.674. The model parameters used in the undrained analyses were the same as the parameters used in the drained analyses above. Here also, the model captured both the qq ε~ and pε ~ qε responses with remarkable accuracy. These results illustrated capability of the model to capture both the deviatoric and volumetric responses of undrained loading of variably saturated soils. Due to compressibility of the air phase in the unsaturated samples, the results of undrained unsaturated tests show the same behaviour as saturated drained tests. APPLICATION TO CYCLIC LOADING

1. Drained Cyclic Tests on Saturated Toyoura Sand Figures 11 and 12 show results of cyclic, constant p′, drained tests on loose and dense samples of Toyoura sand. The tests were conducted on 75mm diameter and 150mm high specimens of Toyoura sand, consisting of mainly angular to sub-angular quartz grains, and prepared by pulviating air dried samples [14]. The initial conditions were: p′ = 98.1 kPa and e = 0.845 for the loose sample, p′ = 98.1 kPa and e = 0.653 for the dense sample. The basic material constants for both samples were: κ = 0.001, ν = 0.30, Mcs = 1.24, λ0 = 0.030, Γ0 = 1.969, λcr = 0.24, N = 3.0, R = 5.6, k = 2.0 and A = 1.0. The parameter km for the loose sample was 1.0 for first loading and 3.0 for unloading/reloading; its corresponding values for the dense sample were 10.0 and 18.0. The increase in the value of km for the dense sample is due to the change in the value of oξ . The model simulations capture the stress-strain and the associated volumetric responses during the loading and unloading stages of both samples. In particular, the model captures the contractive response and the progressive stiffening of the loose sample. For the dense sample, the model predicts the initial compression followed by expansion and successive softening of the sample with the progress of cyclic loading.

-2.0

-1.0

0.0

1.0

2.0

-0.02 -0.01 0 0.01 0.02εq

η

Experimental DataModel Simulation

0

0.01

0.02

0.03

-0.02 -0.01 0 0.01 0.02εq

εp

Experimental DataModel Simulation

Figure 11: Model simulations for drained cyclic test on saturated samples of loose Toyoura sand

-2.0

-1.0

0.0

1.0

2.0

-0.02 -0.01 0 0.01 0.02εq

η

Experimental DataModel Simulation

-0.01

-0.005

0

0.005

0.01

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0η

εp

Experimental DataModel Simulation

Figure 12: Model simulations for drained cyclic test on saturated samples of dense Toyoura sand

2. Undrained Cyclic Tests on Saturated Niigata Sand Figure 13 shows the results of constant stress amplitude, cyclic undrained tests on loose Niigata sand [15]. The initial conditions of the test were p′ = 212.6 kPa and e = 0.737.

— 78 —

The basic material parameters were: κ = 0.01, ν = 0.3, Mcs = 1.48, λ0 = 0.032, Γ0 = 1.87, λcr= 0.21, N = 3.0, R = 6.2, k = 2.0 and A = 1.0; with km = 0.14 for first loading, and km = 18.0 for unloading and reloading. The simulated behaviour is in good agreement with the experimental data. The model captures the failure of the sample by liquefaction in which the effective normal stress decreases progressively until the material becomes unstable at the critical state.

-100

-50

0

50

100

0 50 100 150 200 250p' (kPa)

q (k

pa)

Experimental DataModel Simulation

-100

-50

0

50

100

-0.02 -0.015 -0.01 -0.005 0 0.005 0.01εq

q (k

Pa)

Experimental DataModel Simulation

Figure 13: Model simulations for undrained cyclic test on saturated samples of loose Niigata sand

3. Undrained Cyclic Tests on unsaturated Texas Silty Sand Lastly, application of the model to cyclic loading of unsaturated soils is demonstrated using undrained (constant water volume) cyclic triaxial tests on lightly compacted, partially saturated samples of silty sand [16]. In these tests, cyclic loading was applied through axial strain increments while controlling the peak deviatoric stress to a predetermined cyclic stress ratio. Results of the test conducted at a suction value of 40 kPa with cyclic stress ratio of 0.35 is presented in Figure 14. This figure compares 1~ εq and

1~ εε p responses obtained from the model simulation against the experimental data. In the results shown, the model simulations matched the experimental data reasonably well. The model captures the successive stiffening of the sample with the progress of the cyclic loading and the mechanical hysteresis between unloading and reloading. It is also interesting to note the similarity between the undrained cyclic response of unsaturated soils and the cyclic drained response of saturated soils. The initial conditions for this test were: pnet = 120 kPa and e = 0.898. The model parameters used in the simulations were: κ = 0.004, v = 0.3, Mcs = 1.32, λ0 = 0.08, Γ0 = 2.142, N = 3, R = 7.5, k = 2.0, A = 1.0 for all the samples. In addition, the air entry value was sae = 10.0 kPa. Parameter km which controls the additive hardening modulus was taken as 120 for unloading and 160 for reloading.

0

20

40

60

80

0 0.005 0.01 0.015 0.02ε1

q (k

Pa)

Experimental DataModel Simulation

0

0.004

0.008

0.012

0.0160 0.005 0.01 0.015 0.02ε1

εp

Experimental DataModel Simulation

Figure 14. Model simulations for undrained cyclic test on unsaturated sample of Texas silty sand.

CONCLUSIONS

A unified bounding surface plasticity model is developed to describe the stress-strain behaviour of the granular soils subjected to monotonic and cyclic loading. The model is formulated incrementally within the critical state framework

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using the effective stress approach. Elastic behaviour is captured through isotropic elasticity, and plastic behaviour is captured through the hardening effects of plastic volumetric strain and matric suction. The shape of the bounding surface is selected from experimental observations of undrained stress paths for loose samples. Kinematic hardening is incorporated implicitly through the position of the loading surface. The loading surface is assumed to have the same shape as the bounding surface passing through the current stress point. A radial mapping rule is adopted to capture the response during loading, unloading and reloading, with the last point of stress reversal taken as the centre of homology. The proposed mapping rule facilitated realistic prediction of cyclic response by incorporating loading history through points of stress reversal. The effect of particle crushing at high stresses is considered through three-segmented critical state and isotropic compression lines. A single set of material parameters is introduced for the complete characterization of the constitutive model. Essential elements of the proposed model are validated using experimental data from the literature. It is shown that numerical predictions match experimental results well for both saturated and unsaturated soils subjected to monotonic and cyclic loadings in drained and undrained conditions.

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