Geotech Geol Eng DOI 10.1007/s10706-012-9582-6
ORIGINAL PAPER
A Strain-Rate Dependent Clay Constitutive Model with Parametric Sensitivity and Uncertainty Quantification H. Martindale • T. Chakraborty • D. Basu
Received: 15 October 2011 / Accepted: 20 October 2012 Ó Springer Science+Business Media Dordrecht 2012
Abstract This paper presents a strain-rate dependent plastic constitutive model for clays. Based on the concepts of critical-state soil mechanics and bounding surface plasticity theory, the model reproduces the mechanical response of clays under triaxial and simple shear loading conditions. The model parameters are determined for Boston Blue Clay, London Clay and Kaolin Clay, and the performance of the model in simulating the mechanical response of these clays is demonstrated for low to medium strain rates. The sensitivity of each model parameter is checked by perturbing the calibrated values by ±20 %. Subsequently, a probabilistic analysis using Monte Carlo simulations is performed by treating the model parameters as random variables and the impact of
H. Martindale Department of Civil and Environmental Engineering, University of Connecticut, 261 Glenbrook Road, Unit 2037, Storrs, CT, USA e-mail:
[email protected] T. Chakraborty Department of Civil Engineering, Indian Institute of Technology, Delhi, India e-mail:
[email protected] D. Basu (&) Department of Civil and Environmental Engineering, University of Waterloo, 200 University Avenue West, Waterloo, ON, Canada e-mail:
[email protected];
[email protected]
the statistics of the parameters on the undrained shear strength is investigated. Keywords Clay Constitutive relations Plasticity Rate-dependence Sensitivity analysis Probabilistic analysis Monte Carlo simulations
1 Introduction The mechanical behavior of clayey soils is affected by the rate of induced strains (Kavazanjian and Mitchell 1980; Sorensen et al. 2007; Sheahan 1991, 2005; Dı´azRodrı´guez et al. 2009; Matesˇic´ and Vucetic 2003). Some examples of practical problems where ratedependent behavior of clay is important are landslides, pile penetration, earthquakes and wave loads on offshore foundations, where the rate of strain may reach up to 103/h (Ngo et al. 2007). It has been observed in the laboratory triaxial compression tests that, for low to medium applied strain rates of 10-2– 102 %/h (&10-7–10-4/s), the undrained shear strength increases by 5–20 % per log-cycle increase in the strain rate. The initial shear modulus increases at a rate of about 10 % per log-cycle increase in the applied strain rate (Matesˇic´ and Vucetic 2003). The critical-state strength of clays, however, remains rather unaffected by the rate of induced strains under low to medium strain rates (Sheahan et al. 1996; Sorensen et al. 2007; Dı´az-Rodrı´guez et al. 2009). The overconsolidation ratio (OCR) plays an important role
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in the rate-dependent mechanical response of clay. For a particular OCR, the deviatoric stress attains its peak at approximately the same strain level for different strain rates. However, the strain at which the peak occurs increases with increasing OCR. Significant post-peak softening is observed for low OCR of 1 and 2 due to the generation of positive excess pore pressure while, for OCR of 4 or greater, the post-peak softening is relatively small. The rate dependent behavior is negligible for strain rates below 0.05 %/h (Sheahan 1991). Rate-dependence is also observed in field vane shear tests (Biscontin and Pestana 2001; Leroueil and Marques 1996)—the strain rate in these tests may reach up to 100 %/h (Randolph 2004). Schlue et al. (2010) showed that the undrained peak shear strength of harbor mud measured using vane shear test increases by about 30 % per ten-fold increase in the peripheral velocity. Rate dependence of clay has been mostly investigated for creep and stress relaxation, and elastoviscoplastic constitutive models based on overstress theory have been developed to simulate them (Perzyna 1963, 1966; Zienkiewicz and Cormeau 1974; Adachi and Oka 1982; Hinchberger and Rowe 1998). Adachi and Oka (1982) proposed an elasto-viscoplastic constitutive model based on the original Cam-clay model to capture creep, stress relaxation and secondary consolidation. Hinchberger and Rowe (1998) incorporated Perzyna’s overstress theory in elliptical Drucker–Prager cap model to capture the secondary consolidation behavior of clay. Viscoplasticity has been combined with the bounding surface plasticity theory as well to simulate creep, stress relaxation and secondary compression of normally-consolidated (NC) and overconsolidated (OC) clays (Dafalias 1982; Kaliakin and Dafalias 1990a, b). The behavior of clay due to applied rate-dependent loading has been mostly modeled using explicit strainrate dependent equations with one notable exception of Hinchberger et al. (2010) who used the viscoplastic overstress theory with the Drucker–Prager model. In the models with explicit rate dependent equations, the relevant clay properties (e.g., peak undrained shear strength) are explicitly expressed as functions of the applied strain rates where the applied strain rate is an input, and the rate-independent plasticity theory is used to simulate the rate-dependent behavior (Jung and Biscontin 2006; Zhou and Randolph 2007; Chakraborty 2009). These constitutive models do not
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involve numerically expensive viscoplastic stress– strain integration scheme and can predict the strainrate dependent behavior with reasonable accuracy. This paper presents a strain-rate dependent constitutive model and examines the sensitivity of its parameters and the associated uncertainties through a systematic probabilistic study. The model is an extension of a bounding surface plasticity model developed by Manzari and Dafalias (1997), which was later modified by Li and Dafalias (2000), Papadimitriou and Bouckovalas (2002), Dafalias et al. (2004), Loukidis and Salgado (2009) and Chakraborty (2009). The current model simulates the mechanical response of clay subjected to strains applied with a rate of up to 50 %/h (&1 9 10-4/s). Based on the concepts of critical state soil mechanics, the model incorporates the rate effects under various loading conditions in the triaxial stress space. The model parameters are determined for Boston Blue Clay (BBC), London Clay (LC) and Kaolin Clay (KC) by comparing the simulation results with the experimental data available in the literature. These parameters are determined from the strain-rate dependent triaxial compression tests, one-dimensional and isotropic consolidation tests, resonant column tests, triaxial compression and extension tests, and simple shear tests following a hierarchical process. A sensitivity analysis is performed by varying the input model parameters by ±20 % of the calibrated values. Finally, a probabilistic analysis is performed by treating the input model parameters as random variables and performing Monte Carlo simulations, and the statistics of the output undrained shear strength is investigated.
2 Constitutive Model 2.1 Model Surfaces in Stress Space The basic, rate-independent part of the model consists of yield, dilatancy, and critical-state surfaces that are made up of two distinct geometrical surfaces: a cone with straight surfaces in the meridional plane and apex at the origin, and a bounding flat cap on the critical state surface. The model formulation is done in terms of stress ratios, i.e., stresses normalized with respect to the effective mean stress p0 (=r0 kk/3, where r0 ij is the effective Cauchy stress tensor). Figure 1a shows the projection of the yield, dilatancy and critical-state
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surfaces on the p-plane of the principal deviatoric stress ratio space s1/p0 - s2/p0 - s3/p0 [sij (=r0 ij dijr0 kk/3) is the deviatoric stress tensor in which dij denotes the Kronecker’s delta]. Figure 1b shows the projection of the model surfaces on the longitudinal q - p0 plane (q = r0 1 - r0 3 is the deviatoric stress in the triaxial stress space). The yield surface is expressed in terms of the deviatoric stress ratio tensor rij (=sij/p0 ) as (Manzari and Dafalias 1997) f ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi ðrij aij Þðrij aij Þ 2=3m ¼ 0
ð1Þ
which can be visualized as a cone in the principal deviatoric stress space intersecting the p-plane as a pffiffiffiffiffiffiffiffi circle with radius 2=3 m and center aij, which is the kinematic hardening tensor. The yield surface cannot harden isotropically (i.e., m stays constant in the model) but can harden kinematically through the evolution of aij given by (Manzari and Dafalias 1997)
Md ¼ gðhÞ 2Mcc þ
kd ¼
ns 1=n
1
1c1
s
θ = 0o
Triaxial Compression
s1 p′ r −α r
Yield Surface
s3 p′
α
θ
Triaxial Extension
θ = 60o
d c n
2 3M c
2 3M d
Dilatancy Surface
s2 p′
Critical State Surface
b
3
6 7 ns 7 Mc ¼ gðhÞMcc ¼ 6 4 5Mcc 1=ns 1c1 1 1=n cos 3h s 1=n 1þc1 s
ð5Þ
in which k and j are the slopes of the linear normalconsolidation and overconsolidation lines in the e-ln(p0 ) space and q (=p0 c/p0 CS) is the ratio of the preconsolidation pressure p0 c to the critical-state pressure p0 CS along the same overconsolidation line in the e-ln(p0 ) space (Fig. 2). The dilatancy surface hardens isotropically as the state parameter n changes due to a change in the stress state.
where k_ shear is the shearing-related plastic multiplier, Hs is the plastic modulus controlling the development of the plastic shear strain (the equations of k_ shear and Hs are given later), nij h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii deter¼ sij p0 aij = ðskl p0 akl Þðskl p0 akl Þ
2
ð4Þ
Mcc ðk jÞ ln q
ð2Þ
mines the direction of the projection of the current stress on the critical-state and dilatancy surfaces (i.e., the mapping rule) and Mc denotes the critical-state surface defined by (Dafalias and Manzari 2004)
where n = e - eCS is the state parameter (Been and Jefferies 1985) in which e and eCS are the current and critical-state void ratios at the same mean stress (Fig. 2), OCR is the overconsolidation ratio and kd is given by
a
! rffiffiffi Hs 2 _ ðMc mÞnij aij a_ ij ¼ kshear 0 3 p ! , rffiffiffi 2 ðMc mÞ aij nij 3
Mcc kd n OCR 1 expðkd n OCRÞ
Bounding Flat Cap
ð3Þ
1þc1
Cap Hardening
0
where Mcc is the stress ratio q/p at the critical state (CS) under triaxial compression, g(h) is a function of the Lode’s angle h and determines the shape of the critical state surface in the deviatoric stress space and c1 = 3/ (Mcc ? 3). The parameter ns takes a constant value of 0.2 for all the clays considered in this study. The dilatancy surface is defined by (Chakraborty 2009)
Preconsolidation Pressure
Fig. 1 Plastic constitutive model plotted in a the deviatoric plane and b the meridional plane
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Γ
λ
λ
ξ κ
Fig. 2 Locus of the normal consolidation line and critical state line in e-ln(p0 ) space Fig. 3 Preconsolidation pressure as a function of strain rate
The flat cap to the critical-state surface helps in capturing the yielding and development of plastic strains under pure compression. The bounding flat cap is perpendicular to the hydrostatic (mean stress) axis and intersects the hydrostatic axis at p0 = p0 c (Fig. 1b). It is given by (Vermeer 1978; Ohmaki 1979; Wang et al. 1990; Li 2002) Fc ¼ p0 p0c ¼ 0
ð6Þ
Flat caps have been used by many researchers, both in classical plasticity (e.g., the double hardening models of Vermeer 1978; Ohmaki 1979) and bounding surface plasticity constitutive models (e.g., Wang et al. 1990; Li 2002). The movement of the cap along the meanstress axis signifies the increase in the preconsolidation pressure along the normal consolidation line (NCL) in the e-ln(p0 ) space. The consistency condition is not applied to the cap, and hence, stress states marginally outside the cap are possible. The normal consolidation line is given by (Fig. 2) eNC ¼ N k ln
0 0 p p ¼ N k ln c pa pa
ð7Þ
where eNC is the normal consolidation void ratio and N is the void ratio at the reference mean stress pa (=100 kPa). The critical state line (CSL) follows the same slope as that of the normal consolidation line in the e-ln(p0 ) space, and is given by
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eCS ¼ C0 k ln
0 p pa
¼ N ðk jÞ ln q k ln
0 p pa
ð8Þ
where C0 is the critical state void ratio at the reference mean stress pa under rate-independent loading. Note that the rate-independent loading is considered to be a loading with a strain rate of 0.05 %/h or lower for which no effect of strain rate on the stress–strain response is observed (Sheahan 1991). It has been observed in experiments that the NCL of clays under rate-independent loading lies to the left of the rate-dependent NCL in the e-ln(p0 ) space, and the rightward shift of the NCL happens with increase in the rate of the applied strain (Leroueil et al. 1985; Sheahan 2005). This movement of NCL as a function of strain rate signifies an apparent increase in the preconsolidation pressure p0 c with increase in the strain rate (Silvestri et al. 1986). Based on the available experimental results, the rate dependent preconsolidation pressure p0 c,rd is assumed to be a function of the applied strain rate e_ ij and can be expressed in terms of pffiffiffiffiffiffiffiffi an equivalent strain rate e_ eq ¼ e_ ij e_ ij as (Fig. 3)
ð9Þ p0c;rd ¼ p0c;ri 1:3_e0:05 eq where p0 c,ri is the rate-independent value of the preconsolidation pressure and e_ eq is expressed in %/h. Under rate-dependent loading, the p0c in Eq. (7)
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becomes the same as p0c;rd which causes the rightward shift of the NCL. Experimental studies also show that the peak undrained strength increases with increase in strain rate. It is hypothesized that the increase in peak shear strength can be captured by translating the critical state line to the right in the e-ln(p0 ) space as the applied strain rate increases (Chakraborty 2009). Mathematically, the movement of CSL is achieved by replacing C0 in the equation of critical state line [Eq. (8)] with C given by
C ¼ C0 1 þ C0 ln 0:1C0 e_ eq þ 1 flnðdc þ 1Þg ð10Þ where C0 is a model parameter that captures the rate dependence as a function of OCR (as explained later), dc denotes a distance in the three-dimensional void ratio-mean stress-deviatoric stress space between the current soil state and the critical state and e_ eq is expressed in %/h. Thus, C = C0 when e_ eq = 0. Equation (10) has been adopted from Chakraborty (2009) and modified to reduce the number of parameters. In this model, the NCL [Eq. (9)] and CSL [Eq. (10)] move independently as the strain rate increases, but the CSL does not cross the NCL. The distance dc in Eq. (10) is defined as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i 2 ð11Þ dc ¼ C0 n2ini þ ðdÞ where nini is the initial value of the state parameter and d is a normalized distance of the current stress state from the critical-state surface in the deviatoric plane. d is given by qffiffi D E pffiffiffiffiffiffiffiffi Mc 32 rij rij d ¼ ð12Þ Mc where hi represents a Macaulay bracket.
The rightward shift of the CSL due to increase in strain rate leads to an increase in C, a decrease in n, and, consequently, shrinking of the dilatancy surface. Because of this, the soil becomes more dilatant with increase in the strain rate and exhibits greater stiffness and peak undrained shear strength, as observed in real tests. The parameter C0 controls the increase in soil stiffness and peak shear strength, which happens at smaller strain level. With further shearing after peak, clay softens towards the critical state, and dc and C approaches zero and C0, respectively. Hence, the effect of strain rate disappears and the deviatoric stress q decreases to the critical-state value.
2.2 Elastic Shear and Bulk Moduli The stress–strain (rij0 - eij) relationship is given by
2 r_0 ij ¼ 2G e_ ij e_ pij þ K G e_ kk e_ pkk dij 3
ð13Þ
where the dot () above the effective stress (rij0 ) and strain (eij) tensors denotes time rate of change, the total strain rate e_ ij has an elastic (_eeij ) and a plastic (_epij ) component [_eij ¼ e_ eij þ e_ pij ], G and K are the shear and bulk moduli, respectively, and dij is the Kronecker’s delta. When the stress state is entirely within the yield surface, there is no plastic strain in the soil. Since the yield surface is very small in this model, the plastic process is prevalent for almost the entire loading duration. G and K are assumed to decrease exponentially with increasing shear and/or mean stresses from the initial values Gmax and Kmax until certain minimal values Gmin and Kmin are reached (Chakraborty 2009)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 ffi13 pffiffiffiffiffiffiffiffi p0 p0 3=2 rij aij;ini rij aij;ini ini A5 G ¼ Gmin þ ðGmax Gmin Þ exp4f@ þ p0c Mcc 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 ffi13 pffiffiffiffiffiffiffiffi p0 p0 3=2 rij aij;ini rij aij;ini ini A5 K ¼ Kmin þ ðKmax Kmin Þ exp4f@ þ p0c Mcc
ð14Þ
2
ð15Þ
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where p0 ini is the initial mean stress, f is a model parameter and aij,ini is the initial value of kinematic hardening tensor. The small-strain shear modulus Gmax is given by (Hardin 1978) Gmax ¼ Cg
ð2:97 eÞ2 pffiffiffiffiffiffiffiffi p0 pa ðOCRÞ0:2 1þe
ð16Þ
where Cg is a material parameter. The initial bulk modulus Kmax is related to the small-strain shear modulus Gmax through a constant Poisson’s ratio m as Kmax ¼ Gmax
2ð1 þ mÞ 3ð1 2mÞ
ð17Þ
The minimal values of the bulk modulus Kmin is directly related to the inclination j of the unloading or reloading overconsolidation lines observed in onedimensional compression tests, and is given by Kmin ¼
p0 ð1 þ eÞ j
ð18Þ
This bulk modulus value is much smaller than the small-strain bulk modulus Kmax (i.e., the value corresponding to Gmax, as measured in resonant column or bender element tests, and a realistic value of Poisson’s ratio). This is because the bulk modulus determined from one-dimensional compression test data corresponds to intermediate to large strains. We define a minimal value of the shear modulus Gmin corresponding to Kmin, given by Gmin ¼
3ð1 2mÞ Kmin 2ð1 þ mÞ
ð19Þ
The above formulation is capable of predicting the hysteresis in the consolidation and shear unloading– reloading loops even when the loops lie entirely inside the yield surface. As the loading direction changes from unloading to reloading, the elastic moduli take new values of Gmax and Kmax corresponding to the current values of p0 and aij (i.e., p0 ini and aij,ini are set to the current values of p0 and aij). Hence, the simulated material behavior inside the yield surface is not elastic but non-linear elastic, as done in the paraelastic approach of Hueckel and Nova (1979). 2.3 Flow Rule The plastic strain tensor epij has two components: epij;shear and epij;cap . The component epij;shear is related to
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the conical yield, dilatancy and critical state surfaces, and its rate is given by 1 ð20Þ e_ pij;shear ¼ k_ shear Rij ¼ k_ shear R0ij þ Ddij 3 The gradient Rij of the plastic potential in the stress space is assumed to consist of a deviatoric component R0 ij [given in Loukidis and Salgado (2009) and Chakraborty (2009)], which expresses the direction of the deviatoric plastic strain rate e_ pij;shear , and a mean component related directly to the dilatancy D that controls the shearinduced plastic volumetric strain rate e_ pkk;shear . The dilatancy D, as modified from Chakraborty (2009), is given by "rffiffiffi # d0 2 D¼ ðMd mÞ aij nij 3 Mcc OCR
1 þ ln e_ eq þ 1 exp½d1 ð1 OCRÞ ð21Þ pffiffiffiffiffiffiffiffi and it depends on the ‘‘distance’’ d ð¼ 2=3ðMd mÞ aij nij Þ between the current and projected stress states on the dilatancy surface (Fig. 1a) and on the strain rate. The unit vector nij is parallel to the vector connecting the center of the yield surface in the p-plane (i.e., the axis of the yield surface) to the current stress point on the yield surface (Fig. 1a). nij determines the image stresses on the critical-state and dilatancy surfaces, and hence, determines the direction of projection of the current stress on the critical-state and dilatancy surfaces. The parameter d0 controls the development of D with stress ratio. The parameter d1 controls the dependence of dilatancy on OCR. The plastic multiplier k_ shear for yielding in the shearing mode is obtained by satisfying the consistency condition for the conical yield surface and is given by (Manzari and Dafalias 1997) 1 of _0 1 1 _ 0ij k_ shear ¼ ðn ¼ n r Þd ð22Þ r ij kl kl ij r ij Hs or0ij Hs 3 The plastic modulus Hs in the above equation, controlling the development of epij;shear , is given by (Manzari and Dafalias 1997) OCR Hs ¼ h0 G hqffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 3 2 ðrij aij;ini Þðrij aij;ini Þ ! rffiffiffi rffiffiffi 2 2 ðMc mÞ aij nij 3 3
ð23Þ
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Hs depends on the distance between the current stress state and the image stress state on the critical-state surface (represented by c in Fig. 1a). The second component of plastic strain, epij;cap , is given by k_ cap 1 p 0 e_ ij;cap ¼ Rij þ dij D ð24Þ 3 D where D is the cap related dilatancy and k_ cap is the cap related plastic multiplier (Chakraborty 2009). The equation of D was obtained by Chakraborty (2009) as 0 1 1 1K0;NC 1þ2K 1 k j 0;NC Þ @3ð pffiffiffiffiffiffiffiffi A D ¼ 0 p 1þe rij rij ,2 1K0;NC 3 1 4 1 þ 1 k j 3ð1þ2K0;NC Þ5 ð25Þ K p0 1 þ e 2G where K0,NC is the coefficient of earth pressure at rest for normally consolidated clay. k_ cap is given by 1 0 k_ cap ¼ hp_ i Hc
ð26Þ
in which Hc is the plastic modulus controlling the development of epij;cap and is given by 0 1þe p p0 exp f c 0 H c ¼ p0 ð27Þ kj p Hc is very high for stress states far from the cap but decreases exponentially with decrease in p0 c - p0 . In Eq. (26), the Macaulay brackets indicate that hp_ 0 i is positive only when p_ 0 0 and is equal to zero when p_ 0 \0. Thus, during unloading along a recompression line, p_ 0c ¼ 0. The rate of hardening increases exponentially as the distance p0 c - p0 of the stress state from the cap decreases. The rate of hardening is controlled by the model parameter f. The hardening of the cap depends on the plastic volumetric strain e_ pkk , which is the summation of the plastic volumetric strain e_ pkk;shear caused by the shear-induced dilatancy (i.e., by the kinematic hardening of the conical yield surface) and the plastic volumetric strain e_ pkk;cap associated with the bounding flat cap. Thus, when the stress state is on the conical yield surface, both the shear and the cap hardening mechanisms remain active and the bounding cap hardens even without the stress state being on
it. This allows for the simulation of the increase of OCR that is observed during the shearing of an initially NC clay. The constitutive model presented in this paper is an extension of previously developed constitutive model by Manzari and Dafalias (1997), Dafalias et al. (2004), Loukidis and Salgado (2009) and Chakraborty (2009). The unique features of the constitutive model presented herein are (1) a new equation for the strain-rate dependent preconsolidation pressure, (2) use of a single parameter C0 to capture the rate dependence as a function of OCR, (3) variable distance between CSL and NCL in the e-ln(p0 ) space and (4) rate dependent dilatancy equation.
2.4 Stress–Strain Integration The stress–strain integration of the constitutive model is performed using an elastic predictor-plastic corrector algorithm illustrated in Fig. 4 (Ortiz and Simo 1986). In this algorithm, at a certain time step t = ti, the stresses r0 ij, kinematic hardening variable aij and void ratio e are the input parameters. The elastic predictor stress is calculated from the input stresses using the strain increment Dekl in the iteration step. The elastic stiffness matrix Del ijkl used in the stress– strain relation is the shear modulus when deviatoric stress is calculated from the deviatoric strain and is the bulk modulus when mean stress is calculated from the volumetric strain. Once the elastic predictor stresses are calculated, the value of the yield function f is checked with the yield surface error tolerance FTOL (=10-9). When f \ FTOL, it signifies that the stress state is within the elastic range and the iteration loop completes and the next strain increment starts. If f [ FTOL, then the stress state is outside the yield surface, and the plastic correction starts. The plastic multiplier k_ is calculated to perform the plastic correction. The shear induced plastic multiplier k_ shear is calculated for the deviatoric stress correction and the cap induced plastic multiplier k_ cap is calculated for the cap induced volumetric stress correction. In the correction step, the total incremental strain Dekl for that particular increment remains constant. After the stress correction is done, stresses, kinematic hardening variable and void ratio are updated and the yield stress
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Set iteration time t= t i, Input σ ij′t = ti = σ ij′ti , e t=ti = e ti , α ijt=ti = α ijti Perform Elastic Predictor Calculation e i+1 = e i + (1 + e i )Δε kk t
σ′
el.pr. ij
= σ ′ + Δσ ′ ti ij
el.pr. ij
=σ′ + D
Stress state inside yield surface
ti ij
el ijkl
yes
t
( Δε kl ) ,
f (σ
t
el.pr ij
,α
ti ij
)= f
(t i )
Check if f (ti ) ≤ − FTOL , FTOL= 10
σ ij′t = σ ij′el.pr. , α ijti+1 = α ijti i+1
−9
no Set iter = 0, k = 1
corr.( k )
Store σ ij′
= σ ij′corr.(iter ) = σ ij′el.pr. , α ijcorr.(k ) = α ijcorr.(iter ) = α ijti , f (k ) = f ( ti )
Calculate plastic multipliers, update stresses and hardening variables, calculate updated yield surface magnitude
λ shear =
1 ∂f 1 ⎛ 1 ⎞ σ ij′ = ⎜ nij − (nkl rkl )δ ij ⎟ σ ij′ , 3 H s ∂σ ij′ Hs ⎝ ⎠
el Rkl , σ ij′corr.(k+1) = σ ij′corr.(k ) − λ Dijkl
(
)
1 p′ λ = f λ shear , λ cap Hc , ⎛ 2 ⎞ ( M c − m ) nij − α ij ⎟ ⎜ 3 H ⎠ α ijcorr.(k+1) = α ijcorr.(k ) + λ s ⎝ p′ ⎛ 2 ⎞ ( M c − m ) − α ijnij ⎟ ⎜ ⎝ 3 ⎠
(
corr.( k+1)
Calculate f σ ij′
λ cap =
)
,α ijcorr.(k+1) = f (k+1)
Check if f (k+1) < − FTOL
no
Set iter= iter+1, k = k+1
yes Store
σ ij′ti+1 = σ ij′corr.(k+1) α ijt = α ijcorr.( k+1) i+1
End
Fig. 4 Flowchart for the elastic predictor-plastic corrector algorithm used for stress–strain integration of the constitutive model
value is again checked with FTOL. Further plastic correction is performed before starting a new strain increment if f [ FTOL. For the simulation of the rate-independent, single element triaxial test, one-step plastic correction iteration is generally sufficient when the incremental strain Dekl is sufficiently small. However, for simulating rate-dependent behavior, 3–5 correction iterations are necessary depending on the applied rate of strain.
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3 Determination of Model Parameters The constitutive model has 15 parameters. These parameters were determined based on one-dimensional consolidation, triaxial compression and extension, simple shear, bender element and resonant column tests. The model parameters are determined for Boston Blue Clay (BBC), Kaolin Clay (KC), and London Clay (LC). BBC is low-plasticity marine clay, composed of illite and quartz (Terzaghi et al. 1996).
Geotech Geol Eng Table 1 Index properties of Boston Blue Clay, London Clay, and Kaolin Clay Clay type
Liquid limit (%)
Plastic limit (%)
Classification
Reference
Boston Blue Clay
32.6
19.5
Inorganic Clay or silt of low to medium plasticity—CL (USCS)
Ladd and Varallyay (1965)
London Clay
69.6
26.2
High plasticity stiff clay
Nishimura (2005)
Kaolin Clay
62.0
30.0
Low Compressibility—CL/ML (USCS)
Prashant and Penumadu (2005)
Table 2 Constitutive model parameters for Boston Blue Clay (BBC), London Clay (LC) and Kaolin Clay (KC) Model relationships
BBC
LC
KC
Small-strain (elastic) poisson’s ratio m
0.25
0.2
Table 3 Parameter C0 for Boston Blue Clay OCR
Strain-rate (%/h)
C0
1
50
0.6
5
1.1
0.5
2.0
0.25
Small-strain shear modulus parameter Cg
250
100
120
Elastic moduli with degradation f
5
10
5
j
0.036
0.064
0.033
Normal consolidation line N 1.138 k
0.187
1.07
0.984
0.168
0.18
2
0.05
2.0
50
0.55
5
1.05
0.5
2.0
0.05
2.0
4
0.05–50
0.1
8
0.05–50
0.1
Critical state surface
0.2
0.2
0.2
d0
1.0
0.24
0.8
c2
0.95
0.95
0.95
ns
0.2
0.2
0.2
1.1
1.1
1.1
0.1–2.0
2.0
0.05
Dilatancy surface 1.02
Line
Flow rule
datio
d1
1.1
doli
2.7
C on
1.18
2.5
m al
0.827
2.2
N or
1.305
q
te Sta ical Crit
Mcc
n Lin e
0.94
Hardening h0 Rate dependence C0
0.86 Oedometer Test (BBC) Test Data: Papadimitriou et al., 2005 Simulation of Loading-Unloading-Reloading
LC contains illite, kaolinite, smectite and quartz (Gasparre et al. 2007a, b). KC mainly contains kaolinite. Table 1 shows the index properties of BBC, LC and KC. The calibrated values of the model parameters are given in Table 2. The experimental data for BBC used in this study are obtained from Papadimitriou et al. (2005), Pestana et al. (2002) and Ling et al. (2002) [the experiments were originally performed by Ladd and Varallyay (1965), Ladd and Edgers (1972) and Sheahan (1991)]. The data for LC are obtained from Gasparre (2005), Gasparre et al.
0.78 10
100
1000
σ Fig. 5 One-dimensional consolidation behavior of Boston Blue Clay
(2007a, b) and Hight et al. (2003). The experimental data for KC are obtained from Ling et al. (2002). The model was calibrated in a hierarchical manner by curve fitting over given sets of experimental data points, and only a few model parameters required trial-
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and-error simulations of element tests. The parameters N, k (describing NCL) and j (describing overconsolidation lines) are determined first using the data available in the literature of one-dimensional or isotropic consolidation tests for which the stress levels are much greater than the preconsolidation pressure and for which accurate measurement of the volumetric strains during unloading–reloading cycles are available. The small-strain Poisson’s ratio m, considered a constant in this model, is determined next using the data of triaxial (TX) tests on K0 consolidated specimens and sheared at different OCR values, as available in the literature. The fact that m controls the slope of the stress path during uniaxial (elastic) unloading in K0 consolidation test of NC clay in triaxial apparatus is Fig. 6 Rate-independent, K0 consolidated triaxial compression test results for Boston Blue Clay. a Stress– strain plot and b stress path plot (test data from Pestana et al. 2002)
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used to determine m using trial and error by simulating one dimensional unloading paths and matching the locus of the initial states of K0 TX tests (Dafalias et al. 2006). This procedure was followed to obtain the values of m for BBC. For LC, the values of m are available in the literature (Gasparre 2005). For KC, m = 0.25 was assumed. The spacing ratio q, which controls the position of the CSL relative to the NCL, was determined next. Given that the parameters defining the NCL are determined first, q is obtained by fitting Eq. (8) through the experimental values, at large shear strains, of void ratio as a function of effective mean stress obtained from undrained triaxial and undrained simple shear tests. The critical-state stress ratio Mcc
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is subsequently determined based on end-state data from triaxial tests. The values of Mcc were collected from the values given in the literature. In order to determine the parameter Cg in the equation of Gmax, the values of Gmax obtained using Eq. (16) are compared with the triaxial compression test data of Gmax for BBC obtained by Santagata et al. (2005, 2007) and with the bender element test data of Gasparre (2005) for LC. For KC, a reasonable match with the experimental data was obtained by assuming Cg = 120. The parameter f describing the degradation of the shear modulus (G) is determined through curve fitting on data for the degradation of G with increasing shear strain. For BBC, the data is obtained from
consolidated undrained triaxial compression tests performed by Santagata et al. (2007). For LC, triaxial compression test data reported by Gasparre et al. (2007a, b) is used. For KC, the value of f was obtained by trial and error so that the simulated shear stress– strain response matched the experimental data with reasonable accuracy. The parameters d0 and d1 describing the dilatancy, and the parameter h0 describing the hardening are determined through trial and error by simulating triaxial compression tests at different OCR and at different strain rates. The parameter ns controls the shape and convexity of the critical state and dilatancy surfaces (through g(h)), and hence, controls the
Fig. 7 Rate-independent, isotropically consolidated triaxial compression test results for London clay. a Stress strain plot and b stress path plot (test data from Gasparre 2005)
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friction angle for conditions other than triaxial compression and extension. It was found that ns = 0.2 for BBC, KC and LC guaranteed convex dilatancy and critical state surfaces, and ensured a difference of 3°– 4° between the critical state friction angles under plain strain and triaxial compression loading, as observed in experiments on clays (Prashant and Penumadu 2005). The parameter c2 controls the value of intermediate principal stress ratio b [= (r0 2 - r0 3)/(r0 1 - r0 3)] and it is set to a value of 0.95, which makes b = 0.43 at the end of simple shear tests. The choice of c2 = 0.95 is consistent with the available literature where it has been reported that b lies between 0.4 and 0.5 in undrained plane-strain compression tests (Randolph and Wroth 1981), that b & 0.3 for plane-strain tests (Hambly 1972; Vaid and Campanella 1974) and that
Fig. 8 Rate-independent, isotropically consolidated triaxial compression test results for Kaolin clay. a Stress strain plot and b stress path plot (test data from Ling et al. 2002)
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the analytical value of b is 0.5 (Nakai and Matsuoka 1986). It was found that the value of m does not affect the simulation results much but affect the stability of the numerical simulations. Assigning a small value to m helps avoid numerical instabilities. It was observed that a value of m = 0.05 produced effective and stable solutions. The parameter C0 was finally determined by comparing the simulation results with the rate-dependent triaxial compression data of Sheahan (1991) and Sheahan et al. (1996) for BBC, of Sorensen et al. (2007) for LC, and of Mukabi and Tatsuoka (1999) for KC. It was observed for BBC that assuming one value of C0 for different OCRs did not produce good match with the experimental results. A similar observation
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was made by Ha´jek et al. (2009) who found that the constitutive equations that captured the behavior of normally consolidated clays did not capture the behavior of overconsolidated clays well. Therefore, Ha´jek et al. (2009) considered an OCR-dependent model calibration process. Following a similar approach, C0 is assumed to be OCR dependent in this study. It was observed that only BBC required an OCR dependent calibration for C0 the values of which are given in Table 3. For LC and KC, C0 was found not to vary with OCR (Table 2). Fig. 9 Rate-independent K0 consolidated undrained simple shear test results for Boston Blue Clay with OCR = 1, 2, 4 and 8. a Stress–strain response and b stress path
4 Model Simulations 4.1 Undrained Rate-Independent Behavior Figures 5 and 6 show the rate-independent response of BBC in one-dimensional consolidation test and undrained K0-consolidated triaxial test, respectively. The simulation results match the K0 consolidation data very well in Fig. 5. Figure 6a compares the model predictions with the experimental data of the deviatoric stress q as a function of axial strain ea
a
b
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while Fig. 6b shows the comparisons for the corresponding stress path (deviatoric stress versus mean stress) plots. The stress values are normalized with respect to the vertical preconsolidation pressure ra,max. In the simulations, the same Mcc value is used for both isotropic and K0 consolidation cases. This causes under prediction of stresses in Fig. 6 at all OCR values. Overall, the simulations match the experimental results reasonably well. Figures 7 and 8 compare the model predictions with the rate-independent experimental data of undrained triaxial compression tests performed on isotropically-consolidated specimens of LC and KC respectively. The normalized deviatoric stress-axial strain (Figs. 7a, 8a) and the normalized deviatoric stress-normalized mean stress plots (Figs. 7b, 8b) show a reasonable match between the simulation results and experimental data. Figures 9a and b show the shear stress-shear strain (s - c) and stress path plots of BBC in rate-independent undrained simple shear test. The stresses are normalized with respect to the vertical preconsolidation pressure ra,max. The simulation results match the experimental data reasonably well.
4.2 Undrained Rate-Dependent Behavior Figures 10 and 11 show the comparison between the simulation and experimental results of rate-dependent, K0-consolidated triaxial compression tests on BBC with the applied strain rates of 5 and 50 %/h respectively. The stresses are normalized with respect to the vertical preconsolidation pressure ra,max. The stress– strain (Figs. 10a, 11a) and stress path plots (Figs. 10b, 11b) demonstrate the ability of the constitutive model to capture the mechanical response of clays under strain rate-dependent loading. The model captures the peak undrained strength su,peak as a function of strain rate reasonably well for the range of OCR considered in the study. The post-peak shear strength is, however, under predicted. The stress paths are also captured with reasonable accuracy. Similar comparisons for BBC for other strain-rate values are given in Martindale (2011). Figure 12 shows the deviatoric stress versus axial strain plots of rate-dependent, isotropically consolidated triaxial tests performed on LC samples with OCR = 1 and 5 at different strain rates. The simulated
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Fig. 10 Rate-dependent, K0 consolidated triaxial compression test results for Boston Blue Clay (applied strain rate = 5 %/h). a Stress strain plot and b stress path plot (test data from Sheahan et al. 1996)
plots are in reasonable agreement with the experimental plots. 4.3 Parametric Sensitivity Study The sensitivity of each model parameter was checked for BBC, LC and KC. For the sensitivity study, the model parameters were perturbed by ±20 % of their calibrated values one at a time. An average error Eaverage was calculated for each parameter as
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3 2P jqbase;i qvar;i j ns 100 qbase;i 6 i¼1 7 7 Eaverage ð%Þ ¼ 6 4 5 ns
ð28Þ
where qbase,i is the value of deviatoric stress obtained from model simulation at ith strain increment using the calibrated model parameters, qvar,i is the corresponding value of deviatoric stress obtained from model simulation when a parameter is perturbed, and ns is the total number of strain values at which the error calculations are performed. The error E = |qbase,i - qvar,i|/qbase,i is calculated at different strain values and then the total accumulated error for
all the strain values is divided by the number of calculations to obtain Eaverage. Eaverage calculated for all the parameters of BBC for rate-independent, K0 consolidated triaxial test simulations are shown in Table 4 for different values of OCR. The values corresponding to OCR = 2 and ?20 % perturbation are shown in Fig. 13. It is evident that Mcc, q, OCR and K0 are the most sensitive parameters. For normally consolidated clays, k and j are relatively more sensitive than the remaining parameters while, for overconsolidated clays, the dilatancy parameters d0 and d1 are relatively more sensitive. Similar trends were observed for LC and KC as well (Martindale 2011).
Fig. 11 Rate-dependent, K0 consolidated triaxial compression test results for Boston Blue Clay (applied strain rate = 50 %/h). a Stress strain plot and b stress path plot (test data from Sheahan et al. 1996)
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Fig. 12 Stress strain plots for rate-dependent, isotropically consolidated triaxial compression tests performed on London clay (test data from Sorensen et al. 2007)
4.4 Uncertainties in Model Parameters The uncertainties associated with the estimation of the model parameters is investigated probabilistically by considering the model parameters as random variables and performing Monte Carlo (M-C) simulations of the mechanical response of BBC. The study was done
assuming that the random variables (model parameters) follow normal and uniform probability distributions. The calibrated deterministic values were assumed to be the means l of the random parameters. The standard deviations r were calculated with the assumption that the ±20 % scatter about the mean (deterministic) values correspond to ±3r. Thus, the coefficient of variation COV (=r/l) for all the parameters is 0.067. The same mean and standard deviation values were used for normal and uniform distributions in the M-C simulations. Representative histograms of peak undrained shear strength su,peak of BBC considering normal and uniform probability distribution functions are shown in Fig. 14. These results were obtained for triaxial simulations at 50%/h strain rate on K0 consolidated specimens with OCR = 2. The nature of the distributions of su,peak is approximately the same for both the normal and uniform probability distributions of the input parameters. The difference in the mean values of su,peak obtained for normally and uniformly distributed input parameters is only 0.21 %. A similar trend was observed for the undrained shear strength su,CS at the critical state (Martindale 2011). Figure 15 shows the mean and COV of su,peak of K0 consolidated BBC at OCR = 8 as a function of strain rate. The su,peak values are obtained deterministically
Table 4 Average parameter sensitivity error [Eaverage (%)] for Boston Blue Clay Parameter
?20 % variation OCR = 1
OCR = 2
-20 % variation OCR = 4
OCR = 8
OCR = 1
OCR = 2
OCR = 4
OCR = 8
Cg
0.15
0.11
0.50
0.82
0.19
0.14
0.60
0.96
d0 d1
0.83 0
0.25 0.06
0.51 0.39
2.64 4.27
1.33 0
0.39 0.06
0.75 0.35
3.39 1.15
Mcc
17.89
18.54
19.46
22.64
18.68
19.01
19.87
22.67
h0
0.39
0.39
0.48
0.27
0.55
0.56
0.73
0.39
j
2.94
0.73
1.58
4.21
3.01
0.77
1.82
5.30
k
2.09
0.61
0.58
1.09
3.26
0.93
0.85
1.73
N
0.11
0.17
0.53
1.38
0.12
0.16
0.54
1.36
q
11.22
11.802
11.75
12.38
17.03
18.08
18.18
20.89
m
0
0
0
0
0
0
0
0
f
0.39
0.22
1.25
2.26
0.56
0.25
1.69
3.76
K0
9.14
2.27
2.62
3.72
10.14
2.52
2.70
3.26
OCR
4.33
4.18
5.77
11.89
5.29
6.44
12.51
m
0.01
0
0
0
0
0
0
0
n1
0.40
0.24
0.08
0.11
0.48
0.28
0.09
0.12
123
–
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Fig. 13 Average normalized cumulative error Eaverage for ?20 % variation of model parameters of Boston Blue Clay for rate-independent, K0 consolidated triaxial simulations at OCR = 2. a Parameters with low sensitivity and b parameters with high sensitivity
and probabilistically using normally and uniformly distributed inputs. The mean values for all the three cases match very well. The mean (or deterministic) undrained shear strength increases while the COV decreases with increase in the applied strain rate. Figure 15a shows that the deterministic and mean values are the same for practical purposes. Figure 15b indicates that, in most likelihood, the magnitude of error in the estimation of the undrained shear strength due to erroneous model parameter estimations will not be significant.
Fig. 14 Histogram of su,peak obtained using a normal and b uniform probability distribution functions for the model parameters
5 Conclusions The paper presents a rate-dependent plastic constitutive model for clays and examines the sensitivity of the model parameters and the effect of uncertainties associated with the estimation of the parameters. The model consists of conical yield, dilatancy and critical state surfaces with a flat cap on the critical state surface. The model is capable of simulating clay behavior for both isotropic and anisotropic initial
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simulates the rate-dependent clay response without an expensive numerical algorithm. The sensitivity of each model parameter was checked by perturbing the calibrated values by ±20 % one at a time and the most sensitive parameters are identified. Subsequently, the uncertainties associated with the estimation of the model parameters was investigated probabilistically by considering the model parameters as random variables that follow normal or uniform probability distributions. Monte Carlo simulations were performed and the statistics of the undrained shear strength of BBC was investigated. The undrained shear strength values obtained deterministically and probabilistically matched very well. The coefficients of variation of the undrained shear strength were found to be not more than 12 %, which suggests that the magnitude of error in the estimation of the undrained shear strength using the constitutive model, due to erroneous model parameter estimations, will not be significant. Acknowledgments This material is based upon work supported by the U.S. Department of Homeland Security under Grant Award Number 2008-ST-061-TS002. The views and conclusions contained in this document are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied, of the U.S. Department of Homeland Security.
References
Fig. 15 Variation of a mean and deterministic su,peak and b COV of su,peak with applied strain rate for OCR = 8
stress state and for loading paths that are more general than triaxial compression/extension. The proposed model has 15 parameters. The parameters were determined for Boston Blue Clay, London Clay and Kaolin Clay following a hierarchical manner. The model considers OCR dependent model calibration process for the strain-rate dependent parameter. The proposed constitutive model captures adequately the rate-independent and rate-dependent response of clay behavior under isotropic and K0consolidated triaxial compression conditions. The model retains the rate-independent formulation and
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