Model Prediction of Static Liquefaction: Influenceof the Initial State on Potential InstabilitiesGiuseppe Buscarnera, Aff.M.ASCE1; and Andrew J. Whittle, M.ASCE2
The static liquefaction of sands is a form of instability affectinggranular materials and having major implications in a variety ofunderwater slope failures (Castro and Poulos 1977; Sladen et al.1985; Seed et al. 1988; Hight et al. 1999). The close link betweensand liquefaction and flow slides has motivated research interest inthis topic, and improved engineering predictions are nowpossible bycombining mechanical theories with advanced constitutive modelsfor sands (Nova 1989; Lade 1992;Darve 1994; Borja 2006;Andrade2009). The predictive capabilities of the constitutive model arecrucial in this application. It is well know that the undrained behaviorof sands is affected by stress and density conditions (Been andJefferies 1985; Ishihara 1993), as well as by other state variables thatcan evolve during consolidation and loading, such as fabric an-isotropy (Kramer and Seed 1988; Kato et al. 2001). Even minorchanges in the initial state (e.g., owing to deposition processes orchanges in drainage conditions) can alter the expected undrainedresponse, either favoring or preventing instability, and are crucial indeveloping a predictive framework.
This paper focuses on a modeling strategy that is able to capturethe influence of the initial state on the tendency to undergo lique-faction. The consequences of modeling assumptions will be
discussed, stressing their effects on predictive capabilities. For thispurpose, well-established theoretical concepts are used as a tool fordisclosing the more subtle aspects of liquefaction phenomena.The goal is to shed light on the fundamental features of liquefaction,the concept of latent (or potential) instability, and the role of in situconditions.
The main questions this paper is intended to answer are (1) howdo we quantify the available shear resistance capacity as a functionof the current state? And (2) how are stability conditions affected bya change in drainage conditions, and how is it possible to keep trackof these changes? In answering these questions, we show that it ispossible to set a framework for the future application of soil modelsfor assessing flow slide susceptibility. These issues are addressedfrom a constitutive modeling perspective using the theory of strain-hardening elastoplasticity. Although most of the reasoning is basedon numerical simulations with a particular constitutive model (MIT-S1; Pestana andWhittle 1999), the theoretical framework is general,and the conclusions can be applied to an entire class of elastoplasticsoil models.
Influence of Density and Stress State on theSusceptibility for Instability in Undrained Shearing
This section uses laboratory data for Toyoura sand (Verdugo 1992;Ishihara 1993) to illustrate the influence of stress state and densityon the undrained response of cohesionless soils. Results of somekey laboratory shear tests are compared with predictions of theMIT-S1 model (Pestana and Whittle 1999; Pestana et al. 2002).MIT-S1 is a generalized elastoplastic effective-stress soil modelthat was developed to predict the rate-independent anisotropic be-havior of a broad range of soils. The features of the model that aremost relevant for the purpose of this analysis are the incorporationof effective stress and void ratio as independent state variablescontrolling the mechanical response (so-called barotropic and
1Assistant Professor, Dept. of Civil and Environmental Engineering,Northwestern Univ., Evanston, IL 60208; formerly, Research Assistant,Politecnico di Milano, 20133 Milan, Italy (corresponding author). E-mail:[email protected]
2Professor, Dept. of Civil and Environmental Engineering, Massachu-setts Institute of Technology, Cambridge, MA 02139.
pycnotropic effects) and representation of the directions of an-isotropy through the initial orientation of the bounding surface andits evolution with rotational hardening.
The inception of static liquefaction implies an abrupt increase inpore water pressure and a dramatic loss of shear resistance. Fig. 1compares experimental data from undrained shearing of very looseToyoura sand (e0 5 0:912 0:93) in triaxial compression (Verdugo1992), with corresponding numerical simulations obtained usingMIT-S1 with model input parameters calibrated for Toyoura sand(Pestana 1994). The figure shows how a small perturbation in voidratio can cause large changes in postpeak behavior and undrainedstrength at large strains.
Fig. 2 shows further comparisons of undrained shear behaviorfor medium-dense Toyoura sand (e0 5 0:7352 0:833) at differentlevels of hydrostatic consolidation stress. Tests performed at thesame preshear void ratio develop higher excess pore pressures inundrained shearing at higher confining stress [Test #3, versus Test#2, versus Test #1; Fig. 2(a)] and exhibit instability when shearedbeyond peak resistance [points I1 and I2 in Figs. 2(a and b)]. Incontrast, an increase in preshear density (e.g., reducing e0 from
Test #3 to Test #4) can alter the potential for instability. The un-drained response in Test #4 (e0 5 0:735) shows a continuous in-crease in stress deviator, whereas Test #3 (e0 5 0:833), at the sameconfining pressure, becomes unstable at an axial strain ɛ1 5 2% andcollapses to steady-state conditions. These considerations are of thesame kind as those that inspired the steady/critical-state frameworkfor sands (Poulos 1981; Poulos et al. 1985; Been et al. 1991;Verdugo and Ishihara 1996) and suggest the existence of a limitlocus in the void ratio–mean effective stress space toward which thestate of the material evolves at large strains.
Such a limit locus is often used to distinguish between loosestates (with net contraction expected on shearing) and dense states(net expansion) and gives some insight into the tendency of thematerial to undergo liquefaction. Constitutive models for sandsshould reflect in their mathematical structure the role of the currentstate, appreciating the dependency of the steady-state strength on thevoid ratio and the tendency of the state to evolve toward a limit locus.However, these features are not sufficient to capture soil instabilityand liquefaction processes. Looking at Fig. 2, mechanical instabilityin Tests #2 and #3 (at I1 and I2, respectively) occurs at stress ratios
Fig. 1. (a) Undrained stress-strain response of loose Toyoura sand and (b) corresponding stress paths (circles represent the initiation of staticliquefaction)
Fig. 2. Effect of changes in the initial state: experimental evidence against model predictions (data from Verdugo 1992; numerical simulations fromPestana et al. 2002); (a) stress paths; (b) stress-strain response.
below critical state, whereas no instabilities are predicted in Tests#1 and #4.
The difference between initiation of liquefaction and critical stateis clarified further in Fig. 3(a), where MIT-S1 simulations are usedto illustrate how sand specimens with different formation densitiesapproach critical-state conditions for undrained shearing to largeshear strains. The figure confirms that in very loose specimens (e.g.,e0 5 0:93) the inception of instability anticipates critical-stateconditions (that are eventually approached at large deformations).In contrast, denser specimens (e0 5 0:90; Fig. 3) can return toa stable response when sheared beyond the quasi–steady state(Ishihara 1993). In these examples, the peak shear stress coincideswith a proper mechanical instability, and the location of deviatoricpeaks is a feature of the undrained response that depends on theinitial state and its evolution.
Theoretical Interpretation of Static Liquefaction
The preceding section shows the remarkable variability of lique-faction scenarios depending on initial conditions. This sectiondescribes a theoretical approach for predicting the occurrence ofthese changes. Some key theoretical questions involved in this topicare as follows: (1) how is it possible to identify instabilities usingelastoplastic predictive models? And (2) what are the analyticalconditions that correspond to the inception of liquefaction?
For the particular case of elastoplastic models, Klisinski et al.(1992) showed that in some circumstances, uniqueness and exis-tence of the predicted response could not be guaranteed for mixedstress-strain control conditions. In other words, when mixed testboundary conditions are imposed, the mathematical theory of plas-ticity suggests that a material can be even more susceptible to suffergeneralized failure modes than under stress-controlled loading. Thismarked dependency of the mechanical response of geomaterials onthe control conditions led to development of the concept of con-trollability, which reflects the role of control conditions on the onsetof failure (Nova 1994; Imposimato and Nova 1998). This conceptcan be expressed in mathematical form as follows:
f_ 5Xc_ ð1aÞ
where X is the control matrix,
f_ ¼(s_0a
ɛ_ b
)ð1bÞ
is the control vector, and
c_ ¼(ɛ_ a_s 0b
)ð1cÞ
is the associated response vector. The control vector f_ collects thevariables that are governed during the loading process and mustbe work-conjugate to c_ in the second-order work equation:
Fig. 3. Evolution of the state of stability on triaxial compression.MIT-S1 simulations showing (a) path in the e2 log p0 plane and pre-dicted Critical State Line (CSL) and (b) the stress-strain response forloose Toyoura sand with preshear void ratios e0 5 0:90 and 0:93
Fig. 4. MIT-S1 simulations for loose Toyoura sand; stress-strain re-sponse and evolution of the stability index LLIQ as a function of theaxial strain
wherematrix notation has been used and the superposedT stands fortransposed. Loss of controllability of the incremental response isachieved when
detX ¼ 0 ð3Þ
In loading paths conducted by controlling the parameters in Eq. (1),the second-order work vanishes when condition (3) is satisfied(Imposimato and Nova 1998), thus suggesting a correspondencebetween the mathematical concept of controllability and the fun-damental notion of material stability (Hill 1958).
Within an elastoplastic constitutive framework, themathematicalcondition expressed by Eq. (3) can be reformulated in a moreconvenient form. It can be proved, in fact, that Eq. (3) is satisfiedwhen the hardening modulus takes a specific value. These criticalvalues of hardening modulus depend on control conditions and canbe evaluated for any mixed stress-strain incremental perturbation(Buscarnera et al. 2011). Undrained loading is commonly modeledas a particular case of mixed stress-strain control in which at afirst approximation volumetric strains are held constant while
independently imposing the shear stresses. If _ɛv 5 0 is the onlykinematic constraint featuring the loading path, the onset of un-drained instability and the initiation of liquefaction can be foundfrom the following critical hardening modulus:
Fig. 5. Stability analysis of the undrained response of Toyoura sand:MIT-S1 simulation for (a)medium-dense sand (e0 5 0:833; p00 5 2MPa);(b) dense sand (e0 5 0:735; p00 5 3 MPa)
Fig. 6. Evolution of the predicted stability index LLIQ for varyingdensity conditions: (a) loose sand (e0 5 0:906; p00 5 1 MPa); (b)medium-dense sand (e0 5 0:833; p00 5 2 MPa); (c) dense sand(e0 5 0:735; p00 5 3 MPa)
with f being the current yield surface, g the plastic potential, andK the elastic bulk modulus [see the Appendix for the analyticalderivation of Eq. (4)].
Buscarnera et al. (2011) recently showed that the definition ofcritical hardening modulus makes it possible to associate control-lability, uniqueness, and existence of the incremental response withthe sign of a suitable scalar index. Such a scalar stability index forliquefaction can be defined as follows:
LLIQ ¼ H2HLIQ ð5Þ
where H is the hardening modulus. Eqs. (4) and (5) have to becalculated in accordance with the derivatives of yield function andplastic potential (see Appendix). Magnitudes and units of the sta-bility indices thus will depend on the analytical expressions of theconstitutive functions.
WheneverH coincides with the critical hardeningmodulusHLIQ,the model predicts a generalized failure mode and the possible onsetof liquefaction. Following Buscarnera et al. (2011), a positive valuefor the index (5) (LLIQ . 0) is associated with a unique incrementalresponse (i.e., it is still possible to apply additional shear stresses),whereas LLIQ # 0 is associated with a loss of existence/uniquenessof the undrained response. The relation between the sign of thestability index and the incremental response of the material providesa convenient way to assess how the state of stability is affected bychanges in control conditions and can be used to differentiate themathematical conditions associated with the initiation of instability(i.e., the attainment of a peak in the stress deviator), the quasi–steadystate (i.e., the minimum deviator attained on undrained shearing;Ishihara 1993), and the steady/critical state (achieved only at verylarge shear strains). The onset of an undrained instability implies
LLIQ ¼ 0 and _LLIQ , 0 ð6Þ
whereas the quasi–steady state is associated with
LLIQ ¼ 0 and _LLIQ . 0 ð7Þ
The value of LLIQ governs the stationary points for the stress de-viator. Conditions with LLIQ 5 0 correspond to loss of uniquenessfor incremental undrained loading, whereas the sign of _LLIQ definesthe consequences of such a loss of uniqueness. Eq. (6) defines theentrance to a stress domain in which undrained loading is not ad-missible, whereas Eq. (7) is associated with an increase in LLIQ
corresponding to a region of stable undrained response. Critical-stateconditions occur when
LLIQ ¼ 0 and _LLIQ ¼ 0 ð8Þ
Figs. 3(a and b) relate the analytical conditions given by Eqs. (6)–(8)to the undrained response predicted by MIT-S1 for two differentinitial void ratios, showing that the most relevant characteristics ofthe predicted stress-strain response are captured by theory.
The notion of critical hardening modulus provides a simple andeffective tool that already has been used successfully for the study ofliquefaction processes during undrained loading (di Prisco and Nova1994; Andrade 2009). The conditionLLIQ 5 0 reflects the existenceof an infinity of solutions activated only by a specific loading pro-gram (i.e., undrained shearing). By analogy with theory of struc-tures, we will refer to this loss of uniqueness as latent instability(Ziegler 1968), and we will use LLIQ to assess the unexpressedpotential for liquefaction (Nova 2003). This is achieved by moni-toring LLIQ during both undrained and drained loading paths anddemonstrating the ability to promote liquefaction by reducing theavailable undrained strength resistance.
Model Prediction of Undrained Monotonic Loading
The numerical simulations of Toyoura sand (Figs. 1 and 2) now canbe reinterpreted in light of appropriate theoretical concepts. For thesake of simplicity, the constitutive model has been used consideringstress states always located on the bounding surface and inhibitingchanges in the stress reversal point that controls nonlinear elasticity(Pestana and Whittle 1999).
Fig. 4 replots the stress-strain response of loose Toyoura sandwith preshear void ratios e0 5 0:9062 0:933 and p00 5 1MPa(Fig. 1). The peak deviator stress in each case coincides with
Fig. 7. MIT-S1 prediction for Toyoura sand; hardening response for drained shearing from the peak of the undrained stress path
LLIQ 5 0 (i.e., loss of controllability) and corresponds to a statewhere an increment in undrained shear stress is capable of triggeringa catastrophic flow failure (with complete loss of undrained shearresistance).
Similar considerations can be drawn for more general initialstates. Fig. 5 provides a theoretical interpretation for two testsreported previously (Tests #2 and #4 in Fig. 2). The evolution ofstability conditions can be studied by tracking the stabilityindex LLIQ associated with MIT-S1 predictions. Fig. 5(a) refers toa test on a specimen of medium-dense Toyoura sand (e0 5 0:833,p00 5 2MPa) characterized by partial liquefaction (i.e., an instabilityfollowed by a transition to a stable response during shearing). Thepredicted response is reflected by the evolution of LLIQ that ischaracterized by two roots. The first root (P1) marks the initiation ofinstability, whereas the second root (P2) corresponds to the phasetransition after which the undrained response becomes stableagain owing to the tendency to dilate at high stress ratios. Fig. 5(b)illustrates the theoretical interpretation of a test on dense Toyourasand (e0 5 0:735, p00 5 3 MPa). In this case, the simulation shows noinstability (LLIQ . 0 throughout) and tends asymptotically to criticalstate at large strains.
These examples point out the remarkable role of thematerial statein the assessment of stability conditions. Most important, they il-lustrate the complex evolution of stability conditions under externalperturbations and the critical role of the soil model in describingbarotropic and pycnotropic effects.
Modeling Latent Instability: Effect of a Changein Drainage Conditions
When the constitutive model is employed to predict the undrainedresponse under monotonic loading, the stable/unstable nature ofthe predicted response is immediately evident. As a result, themathematical indices of stability are not essential to disclose crit-ical conditions, and the role of the theory is purely explanatory. Bycontrast, the added value of the theory is evident in nonstandardsimulations, such as those characterized by multiple changes indrainage conditions. To clarify this statement, let us considerFig. 6, which illustrates some important features of the numericalsimulations presented in Figs. 4 and 5. The two evolving com-ponents of LLIQ (i.e., the hardening modulus H and the criticalhardening modulus HLIQ) are plotted separately. Since all thesimulations in Fig. 4 have the same qualitative characteristics,only one test is commented on in detail (e0 5 0:906). The figureillustrates the evolution of the hardening modulus during un-drained shearing (H. 0 is required for stability in stress-controlleddrained loading). Figs. 6(a and b) show that LLIQ vanishes forthe first time when the hardening modulus is still positive (H. 0).By contrast, Fig. 6(c) shows that the model prediction for thedense sample is stable (i.e., H.HLIQ and LLIQ . 0), even thoughat large strains the stress state enters within a region of strainsoftening (H, 0, and stress-controlled drained loading is no longeradmissible).
These considerations inspire a conceptual experiment that canshed light on some subtle features of liquefaction phenomena heredisclosed by the model. Fig. 7 shows MIT-S1 simulations for a test(on loose Toyoura sand) characterized by a passage from undrainedto drained shearing. The change in shearing mode is imposed at thepeak deviator stress in undrained shearing. Although the undrainedtest was clearly passing across an unstable condition, the drained
path exhibits a hardening stress-strain response for shearing tocritical state. Such a particular feature of static liquefaction has beendemonstrated experimentally by di Prisco et al. (1995) (for looseHostun sand).
The example illustrates certain fundamental mechanical featuresof static liquefaction that require a reevaluation of the classicalnotions of hardening and softening. According tomodel predictions,the admissibility of drained and undrained loading are independentconditions, and the existence of two independent contributions inLLIQ is crucial. The first contribution (H) reflects either the ability ofthe material to sustain further loading by expanding the domainof admissible stresses (hardening, H. 0) or the loss of this capa-bility (softening,H, 0). The second term (HLIQ) reflects the role ofthe isochoric kinematics imposed by the interstitial fluid. As a result,drained and undrained failures are governed by different quantities,and it is possible to come across critical instability conditions
Fig. 8. MIT-S1 simulations for Toyoura sand; shearing at constantmean effective stress for two formation densities (e0 5 0:94 ande0 5 0:90)
even during apparently safe stress paths (without clear signs ofincipient risk). This circumstance is an example of latent instabilitybecause the potential for collapse is contingent on particularboundary conditions. Given the crucial role of undrained kinematicconstraints in promoting static liquefaction, the example shown inFig. 7 suggests that portraying liquefaction processes as strain-softening mechanisms can be misleading, especially in the case ofimposed changes in drainage conditions. As a result, the termspseudosoftening [undrained failure taking place while H. 0, as inFig. 6(a)] and pseudohardening [stable undrained response pre-dicted with H, 0, as in Fig. 6(c)] are here preferred and will bedistinguished through the sign of LLIQ. In the following it willbe shown that potential instabilities can be identified beforehandby using the index LLIQ.
Effects of Transitions from Drained toUndrained Loading
Model predictions are used hereafter to disclose how the passagefrom drained to undrained shearing can activate liquefaction insta-bilities otherwise present only in potential form. These simulationsillustrate the concept of latent instability and offer ways to explorethe initiation of liquefaction in laboratory tests.
Fig. 8 shows simulations of the drained shearing of two loosespecimens of Toyoura sand (e0 5 0:94 and 0:90) at constant meaneffective stress. Neither case shows signs of instability for shearingto critical state. Nevertheless, the evolution of stability conditionscan be examined by tracking the variation in LLIQ. Fig. 9(b)illustrates the evolution of the stability index LLIQ for the loosest
Fig. 9.Latent instability analysis (loose Toyoura sand, e0 5 0:94): (a) stress paths for the numerical simulations (p0-constant drained shearing followedby undrained shearing; O is the initial stress state; I marks the initiation of instability for the first simulation; P1, P2, and P3 are the points at whichdrainage conditions are modified); (b) evolution of the stability index LLIQ for increasing stress deviator during p0-constant drained shearing
Fig. 10.MIT-S1 simulation illustrating the effect of a change in drainage conditions (e0 5 0:90): (a) stress paths; (b) stress-strain response for a changeof control imposed at three different values of deviatoric stress
specimen (e0 5 0:94) and also shows that LLIQ 5 0 at q5 180 kPa.Fig. 9(a) shows the simulated undrained stress paths for cases wherethere is a switch to undrained shearing at stress levels P1, P2, and P3
(these conditions are also shown in Fig. 8).The stability index is positive (LLIQ . 0) for the case of un-
drained shearing at P1. Case P2 corresponds to incipient instability(LLIQ 5 0), whereasLLIQ is negative atP3. The undrained responsesfor P2 and P3 are characterized by spontaneous collapse (decreasingdeviatoric stress with increasing shear strain), whereas a reserveshear resistance Dq1 is available for Sample P1 (increment in de-viator stress from P1 to the instability point I). On the basis of thevalues taken by LLIQ during drained shearing, it is possible toidentify two intervals of stress deviator [Fig. 9(a)]: (1) LLIQ . 0,where the material has a reserve of undrained resistance, and (2)LLIQ , 0, where there is incipient instability. The example showsthat the potential for liquefaction is not overlooked even if it is notimmediately apparent from the predicted drained behavior.
Results for the second case (e0 5 0:90) are shown in Fig. 10. Thiscase includes three distinct scenarios for undrained shear responses(delimited by values of LLIQ). The stability index vanishes at twodeviatoric stress levels, both anticipating critical-state conditions.There is a transition from stable to unstable states (Zone 1 to Zone 2)and a subsequent return to a stable condition (Zone 2 to Zone 3). Therange of deviatoric stresses at which latent instability is predicted(Zone 2) is smaller than it was for the looser specimen. In this case,a change in control within Zone 2 can produce a sudden drop indeviatoric stress (point Q2 in Fig. 10), whereas a change in controlbeyond this zone implies undrained stability owing to the tendencyof the system to dilate (point Q3 in Fig. 10). Both series of simu-lations are reported in Fig. 11, showing the effects of a change incontrol in the e2 log p0 plane and relating these changes to thesimulated critical-state locus.
The concepts illustrated in this section also apply to other formsof drained preloading (e.g.,K0 consolidation, radial consolidation,drained triaxial compression, etc.). These loading paths reproducestress conditions of practical relevance. A notable example withinthe class of triaxial tests is the constant shear drained test (CSDtest), which has often been used in the literature as a tool forstudying the onset of shallow landslides induced by hydrolog-ic perturbations (Anderson and Sitar 1995; Chu et al. 2003;Buscarnera and Nova 2011; Wan et al. 2011). The evolution of thestability conditions for these tests is investigated by numericalsimulations consisting of three phases: (1) p0-constant drainedshearing up to a prescribed deviatoric stress, (2) drained unloadingat constant q (effective stress p0 is progressively reduced), and (3)a change in control conditions passing from drained to undrainedshearing. Examples of CSD simulations are reported in Fig. 12 fortwo values of initial void ratio (e0 5 0:94 and 0:90), whereas theevolution of the void ratio during the CSD unloading stage is givenin Fig. 13. There is no apparent instability in the simulated stresspaths until reaching a stress state close to the CSL. The indexLLIQ
vanishes atQ1 [e0 5 0:94; Fig. 12(a)] and atQ2 andQ3 [for the testwith e0 5 0:90; Fig. 12(b)]. The range of unstable states of stresscan be indentified by means of LLIQ, and density conditions affectthe extent of this range, which becomes smaller with increasingdensity [Fig. 12(b)]. Figs. 12(c and e) show two simulations fore0 5 0:94, where undrained shearing is imposed at two states: P1,where there is a small residual shear resistance (LLIQ . 0), and P2,where incipient instability is predicted (LLIQ , 0). Figs. 12(d and f)illustrate similar simulations for the slightly denser initial con-dition (e0 5 0:90). As the stress ratio increases, the specimenpasses from a stable state [LLIQ . 0 at P3; Fig. 12(d)] to partialliquefaction [LLIQ , 0 at P4; Fig. 12(f)] and finally to stableconditions at high stress ratios [LLIQ . 0 at P5; Fig. 12(d)].
Inspection of the simulations in the e2 log p0 plane clarifies theseconcepts (Fig. 13). All points associated with LLIQ 5 0 (Q1, Q2,and Q3) are characterized by _e5 0. Points Q1 and Q2 are precededby swelling ( _e. 0) and followed by compaction ( _e, 0). Asa consequence, the predicted second-order work d2W [Eq. (2)]vanishes when _e5 0 and then becomes negative. While there isa remarkable tendency for volumetric compaction predicted forthe loose sample (e0 5 0:94), there is much less contraction forthe case with e0 5 0:90 [Q22Q3; Fig. 13(a)], followed by dilationafter point Q3. The simulations can be conducted while stresscontrol is admissible (i.e., H. 0). For the denser sample, this
Fig. 11. MIT-S1 simulations illustrating the effect of a change indrainage conditions in the e2 log p0 plane: (a) (e0 5 0:94); (b)(e0 5 0:90); the arrows along the e2 log p0 curves indicate the sign ofvolumetric deformations at the end of the simulations (arrows point tothe CSL)
condition is violated before critical state, with breakdown instress control at H5 0.
These considerations illustrate the relation between values of theindex LLIQ and the second-order work and have been used in theliterature to explain the occurrence of drained collapses of loosesamples (Daouadji et al. 2011). By contrast, there are no similaranalyses illustrating the effects of changes in drainage conditions formedium-dense sands. Thus the simulations on the denser sample canbe seen as model predictions that provide conceptual guidance onhow the state of density can affect the evolution of stability con-ditions during CSD tests. The changes in drainage numericallyimposed after the q-constant unloading phase disclose the tendencyfor undrained collapse (reflected by the sign of LLIQ). In fact, thetheory requires an external trigger to produce undrained collapses,and significant deviations from the elastoplasticity frameworkwould be needed to reproduce a spontaneous transitions to undrainedconditions (di Prisco et al. 2000). Although the proposed meth-odology does not reproduce all the aspects of spontaneous drainedcollapses, the examples shown in this section point out that the index
LLIQ is able to locate the possibility of undrained failure and can beused to quantify the increasing risk of liquefaction during drainedloading paths.
Role of Initial Anisotropy Prior to Undrained Loading
Drained stress paths such as those discussed earlier bring aboutsignificant changes in anisotropic properties that are reflected byrotations of the bounding surface in models such as MIT-S1. Hencethe mechanical response in undrained perturbations (and predictedinstability conditions) will depend on the specific prior stress paths.The importance of initial anisotropy on the susceptibility to staticliquefaction is well known in the literature (Castro and Poulos 1977;Kramer and Seed 1988). Fig. 14 compares MIT-S1 model pre-dictions and measured data for an undrained triaxial compressiontest performed on K0-consolidated loose Toyoura sand. The sim-ulation assumes full reorientation of the yield surface along thedirection of consolidation (i.e., the preshear stress state is at the tip ofthe yield surface). This choice provides a satisfactory prediction in
Fig. 12.Model predictions for change of control during q-constant unloading (Toyoura sand): drained path up to CSL: (a) e0 5 0:94; (b) e0 5 0:90;stable response (LLIQ . 0) on change of control: (c) e0 5 0:94; (d) e0 5 0:90; unstable response (LLIQ , 0) on change of control: (e) e0 5 0:94;(f) e0 5 0:90
terms of undrained response. Fig. 14 also shows a simulation wherethere is no initial anisotropy. In this case, the yield surface is orientedalong the hydrostatic axis. The latter assumption significantlyoverestimates the susceptibility to liquefaction compared with theassumption of full reorientation.
Fig. 15 illustrates further the role of yield-surface orientation onthe state of stability that is achieved after anisotropic consolidation.For this purpose, different yield surfaces are considered beforeundrained shearing. All surfaces pass through the same anisotropicstress state (s0
v0 5 588 kPa and K0 5 0:5) but have different ori-entations, characterized by the obliquity ratio b (Pestana andWhittle1999). Full reorientation of the yield surface along the direction ofthe applied stresses is associated with b5 0:75 (maximum value
shown in Fig. 15). Stability conditions are reflected by the indexLLIQ, which is plotted for the given initial stress and void ratios. Themodel can predict either incipient instability in undrained loading(LLIQ # 0) or initially stable response (LLIQ . 0) depending on theinitial anisotropy. For the set of void ratios in Fig. 15(b), the value ofb at which the stress state lies on the top of the yield surface(i.e., ∂f =∂p0 5 0) ranges between 0.52 and 0.62. Thefigure illustrateshow density conditions can alter the location of incipient instability(LLIQ 5 0), showing that for e0 5 0:95 the initial stress statebecomes unstable for b# 0:4, whereas denser specimens exhibitonly latent instability at smaller values of b. The simulations alsoillustrate that while latent instabilities can be predicted for loosespecimens, these circumstances are unlikely for denser conditions,suggesting that both density and induced anisotropy are criticalfactors for a reliable assessment of the liquefaction resistance.
Conclusions
This paper investigated some theoretical concepts concerning thephenomenon of static liquefaction. In particular, the influence ofinitial and current states (seen as a combination of preshear anisot-ropy, stress state, and void ratio) on the assessment of the suscep-tibility to liquefaction has been expounded, illustrating its engineeringimplications. The theoretical procedure outlined in this paper con-sists of (1) a mathematical criterion of stability and (2) a phenome-nological model capable of introducing into the formulation thefeatures of the soils encountered in situ. While the theoretical ap-proach for identifying instability conditions provides generality tothe formulation, adequate predictive capabilities are guaranteedonly by using a constitutive model that can describe realistically thefundamental mechanical aspects of the problem.
Even though similar methods have been used in the past, thisstudy is distinct from other stability analyses in that the concept oflatent instability is fully recognized, and the evolution of stabilityindices for undrained shearing is explored for drained stress paths. Inthese circumstances, all the components concurring for the defini-tion of the current state evolve, and unstable conditions may not beimmediately evident. Also in these cases, this theory can account forthe occurrence of catastrophic instabilities caused by undrainedshear perturbations.
A number of model simulations have been presented that displaypotentially unstable responses (i.e., responses that are activated onlywhen specific boundary conditions are imposed). It has been shownthat a change in the current state is reflected by an appropriate sta-bility index, which describes a mechanical response that can bestable or unstable depending on current anisotropy, stress, anddensity conditions. These conditional forms of instability have beentermed latent instabilities and this paper shows the possibility ofpredicting their occurrence. Latent instabilities result in fact fromalteration of the initial state caused by external perturbations, and thecapability of themodel to reflect these changes is critical. This aspectwas shown numerically by means of simulated changes in drainageconditions (one of the main causes of sudden underwater collapses).The simulations show that the ability of a constitutive model toreproduce the effect of the undrained kinematics on material failure(as distinct from the phenomena governing drained failure) is fun-damental in predicting potential instabilities arising from changesin drainage conditions and to quantify the stress states at whichthe margin of safety from liquefaction vanishes.
The modeling strategy used in this paper provides an example ofa general approach that can substitute earlier methods based onthe concept of steady state strength. The paper suggests a novelapproach for exploring liquefaction which is inspired by the
Fig. 13. (a) Simulated changes in void ratio induced by q-constantunloading; (b) locations of the points at which changes in drainageconditions are numerically imposed; the arrows along the e2 log p0
curves indicate the sign of volumetric deformations at the end of thesimulations (arrows point to the CSL)
concept of latent instability. In doing this it provides conceptualtools that can be used to conceive laboratory experiments and assessthe predictive capabilities of constitutive models. In addition, itprovides mathematical support to distinguish the initiation of liq-uefaction from critical state conditions and its combination with andappropriate constitutive model enables to evaluate as a function ofthe current state how drained loading paths can promote the initia-tion of liquefaction by reducing the available undrained strengthcapacity.
The approach enables analyses of triggering and propagation ofsoil instabilities, and its extension to more realistic field conditionsis straightforward because it is possible to cope with depositionprocesses, include different kinematic constraints, and reproducethe in situ state in shallow slopes. Under this perspective, this papersets a vision for the application of the theory of material stability tothe quantitative assessment of flow slide susceptibility (Buscarneraand Whittle 2012).
Fig. 14. Anisotropic consolidation before shearing and effect of initial directions of anisotropy; undrained triaxial compression on loose Toyourasand (e0 5 0:926): laboratory data and predicted mechanical response
Fig. 15. (a) Sketch of yield surfaces for different initial orientations ofthe anisotropy tensor (btop indicates the obliquity ratio atwhich the stressstate is located at the top of the yield surface); (b) dependency of thestability index LLIQ (prior to shearing) on fabric anisotropy
Appendix. Derivation of the Stability Index
The analytical expression of the stability index LLIQ can be derivedfrom themathematical formalism of strain-hardening elastoplasticity.This appendix provides a particular elaboration for axisymmetric/isochoric conditions, whereas the general procedure to cope withother kinematic constraints is outlined by Buscarnera et al. (2011).As an outcome of an incremental loading process, the magnitude ofplastic strains can be quantified from the consistency condition, hereexpressed as
∂f∂p0
_p0 þ ∂f∂q
_q þ ∂f∂J
TJ_ ¼ 0 ð9Þ
where _p0 and _q are the increments in mean effective stress anddeviatoric stress (whose increments fully define an axisymmetricincremental loading process), whereas J represents the tensor ofinternal state variables (function of the plastic strains via hardeninglaws). Eq. (9) can be reformulated as
The first author gratefully acknowledges the Rocca Fellowshipprogram, which provided support for his research studies at MIT.The authors are also grateful to Professor Roberto Nova for usefulsuggestions during the editing of the paper and to anonymousreviewers for their thoughtful comments.
Notation
The following symbols are used in this paper:b 5 orientation tensor for the yield surface;b 5 obliquity ratio (orientation of the yield surface
HLIQ 5 critical hardening modulus for liquefaction;K 5 elastic bulk modulus;K0 5 lateral earth pressure coefficient for zero lateral
strain;m 5 material parameter describing slenderness of the
bounding surface;p 5 material parameter describing change of bounding
surface shape as a function of current void ratio;p0 5 current mean effective stress;p00 5 initial mean effective stress;q 5 deviatoric stress;a0 5 size of the yield surface;ɛ 5 strain vector (matrix notation);ɛ1 5 axial strain;ɛv 5 volumetric strain;ɛev 5 elastic volumetric strain;ɛvp 5 plastic volumetric strain;
ɛ_a 5 partition of the strain rate vector;ɛ_b 5 partition of the strain rate vector;h 5 deviatoric stress tensor;z 5 scalar parameter describing current aperture of the
bounding surface;_l 5 scalar plastic multiplier controlling magnitude of
plastic strain increments;LLIQ 5 stability index for liquefaction;
J 5 tensor of internal state variables;s0 5 effective stress vector (matrix notation);s_ 0a 5 partition of the effective stress rate vector;s_ 0b 5 partition of the effective stress rate vector;s0v0 5 initial vertical effective stress;
fmr 5 material parameter defining maximum frictionangle;
f 5 vector of control variables;X 5 constitutive control matrix; andc 5 vector of response variables.
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