mathematical identiﬁcation of diffuse and localized instabilities … · 2015. 4. 6. ·...

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech. (2013)Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nag.2196

Mathematical identification of diffuse and localized instabilities influid-saturated sands

Constance Mihalache and Giuseppe Buscarnera*,†

Northwestern University, Department of Civil and Environmental Engineering, Sheridan Road, 2145, Evanston, USA

SUMMARY

The mathematical properties of diffuse and localized failure modes in fluid-saturated sands are investigated.The granular medium is modeled as an elastoplastic solid, and a recently proposed set of scalar indices, herereferred to as moduli of instability, is used to identify the onset of potential bifurcations of the incrementalresponse. First, the analytical properties of these moduli are discussed, stressing their dependence on thekinematic constraints associated with the imposed deformation modes. Then, by using an elastoplasticmodel for sands, drained and undrained loading paths are simulated under axisymmetric, plane-strain andsimple shear conditions. For each deformation mode, the instability moduli are computed and monitoredthroughout the simulations, with the purpose of elucidating the consequences of changes in controlconditions. In addition, it is illustrated that suitable linear transformations allow the same strategy to be usedto perform drained or undrained shear band analyses and predict the interval of possible band inclinations.The final comparison against literature experiments on loose Hostun sand shows that the instability moduliare indicators of the loss of resistance against specific modes of deformation. As a result, they can be used toidentify and explain a number of failure mechanisms that can be commonly observed in experiments.Copyright © 2013 John Wiley & Sons, Ltd.

Received 10 November 2012; Revised 6 April 2013; Accepted 14 April 2013

KEY WORDS: plasticity; material stability; sand; liquefaction; shear bands

1. INTRODUCTION

The evaluation of safety conditions in the natural and engineered systems that interact with (or are madeof) earthen materials is a primary focus of applied geomechanics [1, 2]. Indeed, geomechanical modelingis a powerful tool for elucidating the peculiar implications of material properties and field conditions. Inparticular, the inspection of numerical predictions from simulated specimen-scale tests [3, 4] orfield-scale boundary value analyses [5, 6] enables the implications of material instability to beexplored. The formulation of material models, for example, is inspired by laboratory tests that arecarried out under highly controlled static/kinematic conditions. Such conditions have a remarkable roleon the stresses that originate failure [7, 8] and their effects must be captured by mathematical models.Similarly, numerical analyses combine the use of constitutive models with specific geometricassumptions (e.g. plane-strain kinematics) [4, 9]. In these cases, material models are challenged toreproduce failure under a kinematics that is often different from that used in experiments, andtheoretical interpretations are essential to assess the accuracy of the results. Another intriguing set ofproblems derives from the porous and multiphase nature of geomaterials. Drainage conditions andfluid properties can indeed alter the mode of deformation of the medium [10–13] and represent yetanother source of kinematic constraints that affect the limit resistance.

*Correspondence to: Giuseppe Buscarnera, Northwestern University, Department of Civil and Environmental Engineering,Sheridan Road, 2145, Evanston, USA.†E-mail: [email protected]

Copyright © 2013 John Wiley & Sons, Ltd.

C. MIHALACHE AND G. BUSCARNERA

It is therefore crucial to combine advanced constitutive models with theories that reflect the interplaybetween kinematic conditions, pore-fluid constraints and failure mechanisms. For this purpose, thispaper aims at implementing a mathematical strategy that can cope with general deformation modesand inspect the role of the abovementioned factors. The study relies on a recently proposedapproach for elastoplastic constitutive laws [14], in which the effect of kinematic conditions isclearly isolated in the definition of a scalar quantity (hereafter referred to as modulus of instability).While Buscarnera, Dattola and di Prisco [14] provided a general analytical formulation forelastoplastic solids, the purpose of this contribution is to compute the instability moduli in light of aconstitutive model, track their value during specific loading scenarios and identify the onset ofdiffuse and localized failure modes in fluid-saturated granular media. The theory is linked to anelastoplastic constitutive model for geomaterials, with the purpose of discussing its predictivecapabilities. The analyses are then specialized to loose sands, a particular class of geomaterialsprone to different forms of unstable mechanisms. The connection with the concept of controllabilityis discussed [3], and comparisons with experiments are used to illustrate the versatility of theapproach originally suggested by [14]. In the next sections, after a brief description of the theoreticalbackground, material point analyses will be presented, with the main goal of providing a unifiedstrategy for tackling both diffuse and localized failure modes.

2. THEORETICAL BACKGROUND

2.1. Review of the available approaches

Since the pioneering works by Drucker [15, 16] and Hill [17], a variety of methods have been proposedto identify loss of uniqueness in mechanical analyses and discuss its relation with the physical notion ofstability. In the domain of geomechanics, significant advances have been inspired by the furtherdevelopment of the concept of second-order work:

d2W ¼ dsijdeij (1)

which was introduced by Hill’s contributions and established a conventional criterion of stability.Indeed, the positive definiteness of the quantity in (1) upon any incremental stress–strain pathrepresents a sufficient condition for the stability of the medium and the uniqueness of theincremental solution. By postulating the validity of this concept, Maier and Hueckel [18] clarifiedthat, within an elastoplastic framework, it was possible to relate the loss of positive definiteness ofthe second-order work to critical values of the hardening modulus. This logic can be found in avariety of other interpretations of bifurcation and failure. For instance, the loss of uniqueness and/orexistence of the elastoplastic solution was also addressed under mixed control conditions [19], whilespecific critical values of the hardening modulus have been defined for shear strain localization[20, 21], compaction banding [22] and static liquefaction [23–25].

A significant step forward in the mathematical and physical understanding of Hill’s condition hasbeen achieved through the concepts of controllability [3] and sustainability [26]. While the formerenabled the vanishing of the second-order work to be related with the selection of particular controlconditions, the latter interpreted the same circumstance as a sudden transition from a static to adynamic regime of deformation. In both cases, the vanishing of the second-order work was relatedwith a proper mathematical bifurcation, which underpins the physical occurrence of failure.

For these reasons, in the following, the term failure will be used to indicate the inability of thematerial to sustain additional perturbations. From a mathematical point of view, these events will beassociated with the lack of uniqueness and/or existence of the predicted response.

Along these lines, Buscarnera et al. [14] recently suggested a reinterpretation of the concept ofcontrollability that, while being linked with the aforementioned theories, establishes a direct connectionwith the idea of a critical hardening modulus. The method proposed by [14] is indeed equivalent to theuse of the principal minors of the constitutive tangent operator [3, 27], in that both techniques are inagreement with the second-order work criterion [26]. In addition, it links some properties of the

Copyright © 2013 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2013)DOI: 10.1002/nag

DIFFUSE AND LOCALIZED INSTABILITIES IN FLUID-SATURATED SANDS

elastoplastic response (e.g. the current value of the hardening modulus) to potential loss in controllability,thus providing advantages to cope with both virgin and preloaded soils, as well as to capture instabilitiesassociated with the sharp transition from elastic to elastoplastic conditions [14].

2.2. The instability modulus: definition and mathematical properties

Buscarnera et al. [14] have recently suggested a scalar measure of the potential for loss ofcontrollability, which has been originally termed modulus of instability (HIN). This quantity hasbeen defined based on the framework of elastoplasticity and can be expressed as

HIN ¼ H � Hw (2)

where H is the plastic hardening modulus and the term Hw is a control-dependent quantity, which playsthe role of critical hardening modulus for the considered control conditions. For this reason, Hw is afunction of the parameters selected to impose the perturbation, as follows

Hw ¼ � @fT

@s0bDebb � Deba Deaa

� ��1Deab

� � @g@s0b

(3)

In the above expression, the apex T indicates transposed, f is the yield function, g is the plasticpotential and s0 is the effective stress vector. The subscripts a and b refer to the selected controlvariables, while Deij are suitable partitions of the elastic stiffness matrix (see [14] for more details).

Vanishing or negative values of (2) identify the inadmissibility of the considered incrementalperturbation and the possible bifurcation of the elastoplastic response. As is readily apparent from itsmathematical definition, HIN is a function of both the current state of stress (through H) and thecontrol conditions imposed on the medium (through the control-dependent term Hw). This formalismis particularly convenient for isolating the effect of specific kinematic constraints, thus making it aversatile index for exploring a wide range of testing conditions and capturing instability modes thatare typically overlooked by failure limit criteria (e.g. static liquefaction, shear banding, etc.).

2.3. Evaluation of the instability modulus for general kinematic conditions

In many practical circumstances, the control conditions imposed on specimens are defined as linearcombinations of stress–strain measures [3]. Generalized variables must be defined to accommodatesuch possibility, and they can be represented through linear transformations, as follows:

_j ¼ Ts _s0 (4a)

_h ¼ T« _« (4b)where Ts and T« are suitable transformation matrices. Using this formalism, a constitutive relationcan be written,

_j ¼ TsDTTs� �

_h ¼ Δ _h (5)

whereΔ is a transformed constitutive matrix. As expounded in [14], these transformations allow oneto rewrite (2) as follows

HIN ¼ H � Hw ¼ H þ @f@jb

T

Δebb �Δeba Δeaa� ��1Δeab� � @g@jb (6)

where _ja and _hb are the variables being controlled, while Δe ¼ TsDeTTs is the transformed elasticconstitutive matrix.



In this paper, the modulus of instability has been computed for a number of constraints, includingdrained and undrained triaxial (TXD, TXU), plane-strain (PSD, PSU) and simple shear tests (SSD,SSU). The expression of the linear transformations necessary to impose the kinematic constraints forthe above mentioned tests is detailed in the Appendix. As will be shown in the following sections, theformalism encapsulated in (4)–(6) is particularly convenient to address potential instabilities (i.e. failuremodes that can be caused by sharp changes in control conditions), as well as to detect possible shearbanding processes. In the following, the activation of such particular mechanisms will be assessed byincorporating appropriate constraints into the transformation matrix, Ts, and monitoring the resultingvalues of the modulus (2) during the simulations.

3. MODEL SIMULATION OF DRAINED AND UNDRAINED INSTABILITIES UNDERDIFFERENT MODES OF DEFORMATION

3.1. Elastoplastic modeling of loose sand response

In order to illustrate the performance of the theory, the instability moduli discussed in the previoussection have been evaluated on the basis of the constitutive law proposed by Nova et al. [28]. Thismodel combines the standard features of critical state plasticity with specific enhancements thatallow soil instabilities to be captured (e.g. non-associated flow rule, versatile shapes of the yieldsurface and plastic potential, etc.). While these choices allow a sufficiently simple mathematicalstructure of the model, they also provide modeling capabilities for capturing the response of loosesands upon drained and undrained loading paths. The expressions of the constitutive functions thatdefine the model, as well as the mechanical meaning of the material constants that characterize theirexpressions, are detailed in the Appendix.

The model was calibrated for saturated Hostun sand, a fine and uniform granular material that hasoften been used in the literature to investigate mechanical instabilities [1, 29, 30]. The calibrationwas performed for loose sand samples, with void ratios ranging from 0.9 to 1.1 and for a range ofconfinement pressures between 100 kPa and 800 kPa. The set of parameters that has been used forthe following simulations is given in Table I. The same set of calibrated parameters is usedthroughout the paper to compare obtained stability predictions, both quantitatively and qualitatively,to those observed in typical experiments.

3.2. Mathematical capture of diffuse instabilities

The first form of geomaterial instability that has been mathematically inspected through modelsimulations is the so-called static liquefaction [10, 11, 31], a process in which the interactionbetween the pore fluid and the solid skeleton causes pore pressure build-up, loss of resistance andchaotic deformation [6, 32]. In a contractive granular medium, liquefaction is usually promoted by

Table I. List of model parameters calibrated for 1 loose Hostun sand.

Type of parameter Symbol Loose Hostun sand

Elastic constants k̂ 0.0046a 0G0 14000 kPapr 1 kPa

Hardening constants Bp 0.0074xs 0

Yield function af 0.9906mf 1.146Mcf 0.52Mef 0.442

Plastic potential ag 0.051mg 0.98Mcg 1.28Meg 1.0



lack of fluid drainage, which, under the conventional assumptions of incompressible fluid and solidparticles, implies an isochoric deformation mode. In this section, undrained loading has beensimulated under axisymmetric, plane-strain and simple shear conditions (labeled as TXU, PSU andSSU, respectively) and has been compared with the corresponding drained counterparts (TXD, PSDand SSD, respectively). Figure 1 shows the results from simulated undrained and drained triaxialcompression paths starting at 300 kPa isotropic confinement. Throughout each simulation, thesecond-order work was computed incrementally (Figure 1c), predicting that d2W vanishes at thedeviatoric peak of the undrained simulation (point P), while it maintains a positive value throughoutdrained shearing. Such predictions can be compared with the value of the instability moduluscomputed at any step of each simulation (HTXUIN and H

TXDIN in Figure 1c, respectively). While H

TXUIN is

calculated for mixed stress–strain control conditions (the requirement of constant volume being theonly kinematic constraint), its drained counterpart HTXDIN is related with pure stress control, thereforecoinciding with the hardening modulus (the mathematical definition of these moduli is given in theAppendix). Figure 1c shows that the modulus of instability associated with the undrained constraint,HTXUIN , vanishes at point P, thus indicating loss of existence/uniqueness of the incremental solution.By contrast, the modulus HTXDIN associated with drained shearing is positive throughout the TXDsimulation. These examples illustrate the correspondence between the predictions obtained fromthe modulus of instability and the second-order work. Most importantly, these analyses providea strategy to associate the vanishing of the second-order work with a precise scenario of lossof controllability. Similar conclusions can be derived by comparing drained and undrainedsimulations of plane-strain and simple shear loading (Figures 2 and 3). It is worth noting, however,that the instability moduli for the abovementioned shearing modes (HPSUIN , H

PSDIN and H

SSUIN , H

SSDIN in

Figures 2c and 3c, respectively) do not coincide with those calculated for axisymmetric conditions,

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Figure 1. Model simulations of drained and undrained triaxial compression tests: (a) effective stress paths,(b) evolution of deviatoric stress with axial strain, (c) second-order work and modulus of instability. Point

P marks the onset of undrained instability.


Plane StrainCompression

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IN (kPa)HIN

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PSD

(c)

Figure 2. Model simulations of drained and undrained plane-strain tests: (a) effective stress paths, (b) evo-lution of deviatoric stress with axial strain, (c) second-order work and modulus of instability. Point P marks

the onset of undrained instability.


as they all include additional kinematic constraints. In other words, regardless of the drainageconditions, the simulations in Figures 2 and 3 entail a mixed stress–strain loading program.

3.3. Model prediction of latent instability

Until this point, the analyses have focused on the evaluation of the instability moduli associated with thetest currently being run. This strategy has elucidated the correspondence between the second-order workcriteria and the modulus of instability. However, during a simulated loading process the susceptibility ofthe material to specific modes of failure can change. This implies that the potential for failure can beexacerbated by a change in static-kinematic conditions (i.e. the onset of failure depends on the way inwhich the test is controlled). Consequently, a thorough stability analysis should also inspect theevolution of the instability moduli that correspond to other kinematic conditions (that differ from theimposed loading mode). This strategy allows one to study the consequences of a potential alteration inconstraints, and hereafter it will be referred to as latent instability analysis.

The potential for latent instability can be explored for each of the conditions previously discussed.Indeed, the latent instability moduli can be computed by using different transformation matrices, Ts, inexpressions (4)–(5) and, hence, in the computation of the modulus (6). The remaining component ofthe modulus of instability (i.e. the hardening modulus, H), is instead evaluated from the current stateof the simulations. In this way, it is possible to track the susceptibility of the material to severalmodes of failure and define the conditions for their activation. In other words, if an instabilitymodulus vanishes and its kinematic constraints do not coincide with those associated with thesimulation, the condition HIN= 0 predicts that the material has lost its strength capacity against thedeformation mode considered for the computation of HIN. Such predicted events are termed ‘latent’


Simple Shear Test

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Shear Strain, γ12

(c)

Figure 3. Model simulations of drained and undrained simple shear tests: (a) effective stress paths, (b)evolution of shear stress with shear strain, (c) second-order work and modulus of instability. Point P marks

the onset of undrained instability.


because their activation requires specific changes in the control parameters, which in actual experimentscan either be externally imposed or spontaneously generated by boundary and/or dynamics effects.

Figure 4 presents latent instability results for the undrained and drained simulations previously discussed.During each simulation, latent instability was checked for the remaining five types of constraints. By meansof this strategy, interesting considerations can be drawn about the passage from drained to undrainedconditions and vice versa. Indeed, regardless of the shearing mode, all the simulated drained pathsproduced a latent instability condition for the corresponding undrained counterpart (see details inFigures 4a, 4c and 4e). This is captured by HTXUIN , H

PSUIN and H

SSUIN , respectively, which assume a zero or

negative value as the strain increases. In all cases, a negative instability modulus at a particular stage ofthe simulation indicates that a switch to undrained loading would cause an unstable deformation mode,whereas a positive value of HIN indicates that the resistance capability has not yet disappeared.

By contrast, the scenario associated with undrained shearing depends remarkably on thedeformation mode (Figures 4b, 4d and 4f). For example, from Figure 4b, it can be seen thatthroughout the TXU simulation, a positive value of the drained counterpart (HTXDIN ) is obtained. Inthis specific case, no latent instabilities are predicted as all the other moduli are positive (with theonly exception of HTXUIN itself, which vanishes at around 0.5% axial strain).

Plane-strain and simple shear deformation modes are instead characterized by a different scenario.Although in both cases, undrained failure is the first instability mode achieved, the correspondingdrained moduli (HPSDIN and H

SSDIN , respectively) vanish upon continued deformation. In other words,

the undrained paths inspected in Figures 4d and 4f lead the stress state into a condition which isunstable under both drainage conditions. The simulations show that drained and undrainedbifurcation mechanisms are generally not mutually exclusive and can be predicted to occur within


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Figure 4. Evolution of the instability modulus computed for different types of kinematic constraints. Sixsimulations are reported: drained and undrained triaxial compression (a–b), drained and undrainedplane-strain compression (c–d), drained and undrained simple shear tests (e–f). For each simulation, theinstability modulus computed for the actual simulation constraints describes the stability of the simulation,whereas the other computed moduli describe potential latent instabilities. Points C1 and C2 indicate changes

in control conditions, which are relevant for the processes illustrated in Figures 5 and 6.


the same region of the stress space. This prediction is found to occur within the hardening regime(i.e.HTXDIN ¼ H > 0when eitherHPSDIN orHSSDIN vanish) and can be understood as a direct consequence ofthe mixed stress–strain connotation of PSD and SSD deformation modes [3].

Further inspection allows other aspects to be noted. From Figure 4b, it is evident that HTXUIN is theonly modulus that captures the onset of the predicted instability obtained from the TXU simulation.



At variance with HTXUIN , in fact, the other moduli for undrained shearing modes (HPSUIN and H

SSUIN )

combine the isochoric condition with different constraints (see the Appendix for their definition).The inspection of Figures 4d and 4f, however, shows that when undrained paths are simulated underplane-strain or simple shear deformation modes, the moduli HPSUIN and H

SSUIN tend to approach the

value of HTXUIN . In these cases, even if the tests are simulated by imposing additional conditionsalongside the isochoric constraint, the modulus HTXUIN approximates closely the onset of failure in alltypes of undrained simulations (Figures 4d and 4f). These results suggest that, regardless of thedeformation mode, the isochoric constraint plays a dominant role on the potential for diffuseundrained failure compared to other conditions simultaneously imposed.

Similar comments apply also to the evolution of the undrained moduli (HTXUIN ;HPSUIN ;H

SSUIN ) upon

drained loading paths (Figures 4a, 4c and 4e). While the three moduli are substantially different uponaxisymmetric drained loading (i.e. axisymmetric undrained bifurcation is captured only by HTXUIN ;Figure 4a), the simulation of drained conditions for plane-strain (Figure 4c) and simple shear modes(Figure 4e) causes the moduli HPSUIN and H

SSUIN , respectively, to approach the value of H

TXUIN . This

aspect is remarkable for plane-strain conditions, for which HTXUIN is practically superposed with HPSUIN

and approximates with great accuracy the onset of undrained bifurcation under plane-strainkinematics. Non-negligible differences can instead be noted between the values of HSSUIN and H

TXUIN

computed during the simulation of drained simple shear deformation. In the latter case, the modulusHTXUIN provides a prediction of latent instability that does not correspond to that given by H

SSUIN (and,

hence, would not be appropriate for simple shear deformation). As a result, although there arequantitative similarities among the moduli computed by considering an isochoric constraint, theseanalogies should be considered to be path specific. Indeed, differences associated with the presence ofother kinematic constraints can be non-negligible, especially upon drained paths (in which isochoricconditions do not hold), and may have an impact in the quantitative assessment of latent instabilities.

3.4. Model simulation of changes in control conditions

This section illustrates the consequences of changes in control conditions, as they relate to latentinstability predictions; examples of passages from drained to undrained constraints (TXDTXU) andfrom undrained to drained constraints (PSUPSD) are both considered.

The TXDTXU simulation consists of an initial drained triaxial shearing (TXD) step, followed by asubsequent switch to undrained triaxial loading (TXU) at an axial strain of e1 = 1.75%. As can beobserved from Figure 4a, for a TXD simulation, immediate instability is predicted once the drainageis blocked off at sufficiently high strains, as reflected by the predicted modulus of instability forlatent TXU conditions. For the particular case of the TXDTXU simulation, instability is predictedsince HTXUIN is negative at the change of control (point C1 in Figures 4a and 5). Figure 5 shows theeffective stress path, stress–strain curve and second-order work results from the TXDTXUsimulation. As expected, a negative value of HIN at the change of control is associated with a step-wisechange of the second-order work, which becomes negative at that point (Figure 5c).

The PSUPSD simulation illustrates the effects of a passage from undrained to drained constraints; inthis scenario, loading is applied under plane-strain conditions, with the change in constraints imposed

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Figure 5. Results from the TXDTXU simulation: (a) effective stress path, (b) evolution of deviatoric stresswith axial strain, (c) second-order work, indicating instability at e1 = 1.75%.



at an axial strain of e1 = 0.8%. Although the release of drainage occurs after the PSU simulation hasalready become unstable, the switch to PSD constraints at 0.8% strain is predicted to be stable(point C2 in Figures 4d and 6). In other words, the instability modulus for plane-strain drainedshearing indicates a stable predicted response right after the change of constraints (i.e. HPSDIN at C2).The stable connotation of the incremental response is also reflected by a step-wise change insecond-order work, which in this case, goes back to positive values (Figure 6c).

Latent instability techniques can also provide insight for other simulated stress paths, such as the drainedconstant shear test [29]. These experiments are useful to model the effects of rainfall events, during whichpore water pressure is increased slowly enough to consider the process as drained. Numerous experimentshave shown that, even though the tests are initially conducted under drained conditions, undrained failurecan occur during the fluid-pressurization stage because of an unstable mechanism. Such instabilities aregenerally observed to occur when the measured second-order work takes negative values [29]. Here, aseries of model simulations has been performed by decreasing the mean effective pressure under imposedconstant shear (equivalent to increasing pore water pressure). The simulations are composed of two parts:first drained shearing (with TXD, PSD or SSD constraints) is applied, and then a subsequent drainedconstant-shear path is imposed (during which the effective confinement is reduced). Constant-sheardrained paths have been simulated under triaxial (CSD-TX), plane-strain (CSD-PS) and simple shearconstraints (CSD-SS), with the purpose of stressing that the notion of latent instability hinges with thepredicted evolution of the second-order work and can provide an interpretation of the evidence.

The effective stress path from the CSD-TX simulation is shown in Figure 7a, and the stress–strainresponse during constant shear unloading is shown in Figure 7b. The evolution of second-order workshows the passage from positive to negative values during the unloading stage (Figure 7c). Although themodulus of instability under TXD conditions is never negative (Figure 7d), the modulus of instabilitycomputed for undrained failure (i.e. TXU latent instability) does suggest the onset of a possible undrainedcollapse. This circumstance coincides again with the moment at which the second-order work vanishesand indicates that the material has lost residual strength capabilities against an undrained deformationmechanism. This result supports the idea that the instabilities observed in constant shear tests are relatedwith the buildup of pore water pressure. Similar trends also appear in plane-strain and simple shearsimulations (CSD-PS and CSD-SS, respectively). Figure 8 shows the results obtained from the CSD-PSsimulation. Once again, the PSU latent instability results match the instability points that would bepredicted by the second-order work criteria. Likewise, for the CSD-SS simulation (Figure 9), the SSUlatent instability coincides with the instability detected from the CSD-SS second-order work plot. In thelatter figure, a comparison with the other undrained moduli is shown, illustrating that only HSSUIN matchesthe evolution of the second-order work and hence the possibility for a spontaneous collapse. The examplecorroborates the notion that only an appropriate evaluation of the kinematic constraints used to computethe instability moduli yields conclusions that can capture the mechanics of the problem at stake.

3.5. Mathematical capture of localized failure modes

The mathematical strategy used for diffuse instabilities can be adapted to also cope with localizedfailure modes. In particular, this section focuses on the prediction of shear banding, i.e. the

0

50

100

150

(' 1

- ' 2

) / 2

, (

kPa)

( + '1

'2) / 2, (kPa)

Change of constraints

C

(a)

0

50

100

150

(' 1

- ' 2

) / 2

, (

kPa)

Axial Strain,

PSUPSD

PSU

PSU PSD

C

(b)

0

1 10-4

2 10-4

3 10-4

4 10-4

0 75 150 225 300 0 0.01 0.02 0.03 0.04 0 0.01 0.02 0.03

Sec

on

d-O

rder

Wo

rk, d

2 W

Axial Strain,

PSU PSD(c)

Figure 6. Results from the PSUPSD simulation: (a) stress path, (b) evolution of deviatoric stress with axialstrain, (c) second-order work, indicating passage to a stable response at e1 = 0.8%.


0

50

100

150

200

250

300

350

400

0 50 100 150 200 250 300 350 400

Dev

iato

ric

Str

ess,

q (

kPa)

Mean Effective Stress, p (kPa)

Failure limit criterion

Onset of CSD

Onset of latent instability (H

IN

TXU=0)

O

(a)

150 200 250 300 350 400

Vo

lum

etri

c S

trai

n,

vol


O

(b)

-8 10-4

-7 10-4

-6 10-4

-5 10-4

-4 10-4

-3 10-4

-2 10-4

-1 10-4

0

150 200 250 300 350 400

Sec

on

d-O

rder

Wo

rk, d

2 W


O

(c)

-1 105

-5 104

0

5 104

1 105

1.5 105

2 105

150 200 250 300 350 400

Mo

du

lus

of

Inst

abili

ty, H

IN (

kPa)


HIN

TXD

HIN

TXU

O

(d)

0.03

0

0.005

0.01

0.015

0.02

0.025

'

' '

I

I

I

I

Figure 7. Model simulation of a CSD-TX test. Effective stress path of the simulation (a); behaviorduring constant-shear unloading, stress–strain response (b) and second-order work (c); modulus ofinstability computed for TXD constraints (that reflect the simulated drainage conditions) and TXU constraints(that reflect the potential for latent undrained failure) (d). Point O indicates the onset of constant shear

unloading and point I indicates the onset of instability.


concentration of shear strains within a narrow zone of intense shearing [20, 21]. Shear banding isusually treated as a form of bifurcation at which a non-homogeneous mode of deformation becomespossible without violating equilibrium or compatibility [33]. Hereafter, this process is addressedwith reference to drained [30, 34, 35] and undrained conditions [36–39].

Although onset and evolution of shear bands are intrinsically boundary value problems, materialpoint analyses can be used to identify the conditions at which a localized deformation becomespossible. In other words, though this paper focuses on instabilities at the material point level (oftenreferred to as homogeneous bifurcations, [40]), the modulus of instability can be used to predict thepotential occurrence of shear band localization by selecting appropriate kinematic constraints thatreflect the deformation mechanisms within a band. Such an approach is equivalent to otherwell-established methods for detecting deformation banding [20], in that it captures the conditionsfulfilled by the material at the point in which the acoustic tensor exhibits singularity.

Whether drained or undrained, shear bands are characterized by intense localized shearing within athin domain. As a consequence, the kinematics of deformation inside the process zone can beconveniently described through simple shear constraints. In order to perform a complete shear bandanalysis, careful consideration should be given to the inclination of the localization band in which shearstrains concentrate (Figure 10). For this purpose, the potential occurrence of localized shearing along aspecified direction can be studied by evaluating the moduli associated with the SSD and SSUtransformation matrices in a rotated reference system. The angle of rotation is defined in accordance withthe inclination of the band, and the mathematical formalism used to impose such a rotation is similar to


0

20

40

60

80

100

120

140

160

(' 1

- ' 2

) / 2

, (k

Pa)

( '1 + '

2) / 2, (kPa)

Onset of CSD

Onset of latent instability

I O

(a)

0.01

0.012

0.014

0.016

0.018

0.02

400 450 500 550 600 650 700

Axi

al S

trai

n,

I O

(b)

-1.5 10-4

-1 10-4

-5 10-5

0

5 10-5

Sec

ond

-Ord

er W

ork

, d2 W

O

I

(c)

-3 105

-2 105

-1 105

0

1 105

2 105

260 280 300 320 340 360 380 400 420

( '1 + '

2) / 2, (kPa) ( '

1 + '

2) / 2, (kPa)

( '1 + '

2) / 2, (kPa)

260 280 300 320 340 360 380 400 420 250 300 350 400

Mo

du

lus

of

Inst

abili

ty, H

IN (

kPa)

HIN

PSD

HIN

PSU

I

O

(d)

Figure 8. Model simulation of a CSD-PS test. Effective stress path of the simulation (a); behaviorduring constant-shear unloading, stress–strain response (b) and second-order work (c); modulus ofinstability computed for PSD constraints (that reflect the simulated drainage conditions) and PSU constraints(that reflect the potential for latent undrained failure) (d). Point O indicates the onset of constant shear

unloading and point I indicates the onset of instability.


the linear transformations that specify the kinematic constraints (4). As a consequence, the analysispresented here studies the process of potential shear banding through the latent instability approach.

Using a directional cosine matrix, TR (see the Appendix), it is indeed possible to express the stressmeasures in a rotated reference system (Figure 10), as follows:

s0R ¼ TRs0 (7)

In this way, the analysis of stability conditions with respect to shearing modes along an inclined bandrequires a chain of two linear transformations: a first one converting the stresses into a rotated referencesystem with axes normal or tangential to the band (through the matrix TR), and a second one thatincorporates the simple-shear constraints describing the kinematics inside the band (through a matrix Ts).More specifically, the matrix TR can be incorporated into the modulus of instability, (2)–(3) as follows,

DeR ¼ TRDeTTR (8a)

@f

@jR¼ TTR

� ��1 @f@j

;@g

@jR¼ TTR

� ��1 @g@j

(8b)

in which the elastic constitutive matrix and the gradients of the plastic functions are rewritten in the rotatedreference system. Using this procedure, it is possible to address the potential onset of shear bands withinclinations ranging from θ=0� to θ=180�.


0

20

40

60

80

100

Sh

ear

Str

ess,

, (

kPa)

Normal Effective Stress, '1, (kPa)

Onset of CSD

Onset of instability

I O

(a)

0.014

0.015

0.016

0.017

0.018

0.019

0.02

0.021

Sh

ear

Str

ain

,


IO

(b)

-7 10-5

-6 10-5

-5 10-5

-4 10-5

-3 10-5

-2 10-5

-1 10-5

0

1 10-5

Sec

on

d-O

rder

Wo

rk, d

2 W


I

O (c)

-1 105

-5 104

0

5 104

200 220 240 260 280 300

230 240 250 260 270 280 290 300 310

230 240 250 260 270 280 290 300 310 240 250 260 270 280 290 300 310

Mo

du

lus

of

Inst

abili

ty, H

IN (

kPa)


HIN

TXU

HIN

PSU

HIN

SSU

HIN

SSD

I

O

(d)

Figure 9. Model simulation of a CSD-SS test. Effective stress path of the simulation (a); behavior duringconstant-shear unloading, stress–strain response (b) and second-order work (c); modulus of instabilitycomputed for SSD constraints (that reflect the simulated drainage conditions) and TXU, PSU and SSUconstraints (that reflect three cases of latent undrained failure) (d). Point O indicates the onset of constant

shear unloading and point I indicates the onset of instability.

SHEAR

LOCALIZATION

SHEAR DEFORMATION

INSIDE THE BAND

Figure 10. Rotated reference system for identifying the onset of shear instabilities along inclined bands oflocalization. A schematic representation of the simple shear kinematics inside the band is shown. Rotationis applied clockwise around the 3

⇀axis by angle, θ (the normal to the shear band is assumed to be always

contained in the 1⇀ � 2⇀ plane).




3.6. Model simulation of shear banding in fluid-saturated sand

In granular media, shear banding is usually associated with a dilative response near the peak of theapplied deviatoric stresses. Earlier findings by Desrues and coworkers [30, 39], however, suggest thatsuch processes are possible also in loose contractive sands. Based on this consideration, the currentsection uses the constitutive model and the set of parameters given in Table I, combining them with(8). This strategy can be used to perform shear band analyses for the all the test conditions simulatedpreviously. Shear banding potential, however, is usually exacerbated by plane-strain conditions both inexperiments [41] and model simulations [42]. For this reason, this section focuses on the analysis ofsimulated plane-strain tests. An example of this strategy is given in Figure 11, in which a simulationof drained plane-strain compression from the previous section is checked for drained localization. Thevalues of the modulus of instability for drained shear banding can be plotted versus the angle of bandinclination for selected strain values (Figure 11b). Zero or negative values of HIN therefore indicatethat the mechanical conditions for localized shear deformation are fulfilled at some point of thesimulation. Initially, only one angle of undrained localization is predicted to be possible (θ=49� atabout 5% strain). As the deformation proceeds, a wider range of potential angles of localizationbecomes feasible (Figure 11c). These features are also evident by plotting the modulus of instabilityfor selected band inclinations (Figure 11d).

Interesting properties emerge from the simultaneous analysis of drained and undrained shearbanding potential (Figure 12). From Figure 12c (PSD simulation), the drained shear band analysisindicates that model predictions fulfill the conditions for the formation of a shear band. Thepossibility of drained shear banding is predicted for band inclinations within the rangeθ = 45∘� 60∘. The model predictions in Figure 12e indicate that latent undrained shear banding ismathematically plausible even before drained localization, and its associated bands are predicted tobe possible over a wider range of inclinations. It must be noted, however, that the latter instabilitiesapply to undrained conditions, and require the trigger of an undrained process to be activated. As aresult, the moduli for drained shear banding should be compared to those associated with a diffuse/drained plane-strain instability (modulus HPSDIN ; Figure 13a). From such a comparison, it is readilyapparent that drained shear banding is the first potential form of drained failure promoted by theconsidered deformation conditions (for a band inclination θ = 49�).

Figures 12d and f illustrate the shear band analysis for the PSU simulation and show that also in thiscase, both drained and undrained bands are possible. The condition for undrained localization ishowever fulfilled slightly earlier. Figure 13b compares drained and undrained shear banding withdiffuse PSU instability. It is interesting to note that the instability modulus for undrained plane-strain failure coincides with that of undrained localization calculated for a band inclination ofθ = 45∘. Since this inclination is the first one for which undrained banding is possible (Figure 12f),the two modes of failure are predicted to become simultaneously possible at the peak of the stresspath. This property is related with the fact that isochoric plane-strain deformation modes arekinematically equivalent to an undrained pure shear deformation in a reference system rotated by45∘. As a result, they reflect the same mode of deformation and are characterized by the same valueof Hw (i.e. the two instability moduli coincide). In this case, model predictions alone do not providea basis for distinguishing whether undrained failure is more likely to occur in a localized or adiffuse manner, but they do indicate that both localized and diffuse failure modes are feasible atthat point.

4. INTERPRETATION OF LABORATORY TESTS

4.1. Drained and undrained triaxial compression tests

In this section, the values of the instability modulus predicted from model simulations are discussed inconjunction with laboratory data, with the aim of explaining observed instabilities in loose Hostunsand. Figure 14 shows the results from TXU simulations run at 100 kPa, 200 kPa and 300 kPa, aswell as their comparison against undrained triaxial compression data [1, 29]. The computed valuesof the instability modulus capture the onset of instability at peak deviatoric stress (Figure 14a and


0

50

100

150

200

250

300

350

400

(' 1

- ' 2

) / 2

(kP

a)

Axial Strain,

P3 ( =8%)

P2 ( =5%)

P1 ( =2%)

(a)

-1 106

0

1 106

2 106

3 106

4 106

5 106

Band Inclination, (deg)

=8%

=2%

=5%

Mo

du

lus

of I

nst

abili

ty, H

IN (

kPa)

Range of angles at = 8% (b)

Range of angles at

1 = 8%

0

40

80

120

160

Ban

d In

clin

atio

n,

(d

eg)

Axial Strain,

(c)

-1 106

-5 105

0

5 105

1 106

1.5 106

2 106

2.5 106

3 106

0 0.02 0.04 0.06 0.08 0.1 0 50 100 150 200

0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1

Mo

du

lus

of I

nst

abili

ty, H

IN (

kPa) = 0o

Axial Strain,

= 50o

= 25o

= 65o

= 45o

(d)

Figure 11. Drained shear band analysis for the PSD simulation: (a) deviatoric stress–strain plot; (b)dependence on band inclination of the instability modulus for undrained shear banding (latent instabilitypotential); (c) range of potential shear band inclinations; (d) evolution of the modulus for undrained shear

banding computed for different band inclinations.


14b). As expected, the modulusHTXDIN for latent drained instability is positive throughout the simulation(Figure 14d). This reflects the notion that failure criteria based on drained experiments can severelyoverlook the onset of static liquefaction.

Similarly, experimental TXD results from di Prisco et al. [1, 29] for loose Hostun sand can bequantitatively compared to results from TXD simulations run at 100 kPa and 200 kPa confiningpressure (Figures 15a and 15b). While TXD simulations do not achieve instability within therange of applied strains, undrained instabilities are predicted to be possible in a latent form.Plotting the modulus of instability for latent undrained stability, HTXUIN (Figure 15c), it canindeed be observed that latent undrained instability is possible after 1.2% and 1.4% axial strainfor the 100 kPa and 200 kPa simulations, respectively. The possibility of activating anundrained failure upon passage to undrained shearing was actually observed in experiments by diPrisco et al. [1].

4.2. Drained constant shear tests under axisymmetric conditions

Daouadji et al., [29] conducted a number of CSD-TX experiments on loose Hostun sand and observedthe occurrence of instability modes prior to frictional failure. Given the structure of the constitutivemodel used in our analyses, q-constant paths tended to be characterized by an elastic unloading (theonly exception being the simulation CSD-TX3, which was run at a sufficiently high stress deviatorto cause plastic deformation throughout the unloading stage). Then, in the most general case,stability conditions along paths that involve elastic unloading must be evaluated right at the yieldingpoint (i.e. when the inelastic response can promote bifurcation modes; Figure 16a).


0

50

100

150

200

250

300

350

400

(' 1

- ' 2

) / 2

, (k

Pa)

(' 1

- ' 2

) / 2

, (k

Pa)

Axial Strain,

(a)

0

10

20

30

40

50

60

70 (b)

0

40

80

120

160

Ban

d In

clin

atio

n,

(d

eg)

0

40

80

120

160

Ban

d In

clin

atio

n,

(d

eg)

0

40

80

120

160

Ban

d In

clin

atio

n,

(d

eg)

(c) (d)

0

40

80

120

160

Ban

d In

clin

atio

n,

(d

eg)

(e)

0 0.02 0.04 0.06 0.08 0.1

Axial Strain, 0 0.02 0.04 0.06 0.08 0.1

Axial Strain, 0 0.02 0.04 0.06 0.08 0.1

Axial Strain, 0 0.02 0.04 0.06 0.08 0.1

Axial Strain, 0 0.02 0.04 0.06 0.08 0.1

Axial Strain, 0 0.02 0.04 0.06 0.08 0.1

(f)

Simulation of Drained Plane Strain Test

Str

ess-

Str

ain

Dra

ined

Sh

ear

Ban

dU

nd

rain

ed S

hea

r B

and

Simulation of UndrainedPlane Strain Test

Figure 12. Drained and undrained shear band analysis for the PSD and PSU simulations: deviatoric stress–strain plots for both PSD (a) and PSU (b) simulations, range of potential localization angles for drained shear

banding (c,d) and undrained shear banding (e,f).


Though all CSD-TX tests were run as drained in the original experiments, Daouadji et al. [29]observed spontaneous bursts in pore pressure when the specimens become unstable, as is typical ofundrained failure. In addition, water-volume injection was also observed to produce a similarresponse, where in that case, the burst in pore pressure was promoted directly by a mixed stress–strain control setup [43]. Consequently, this behavior motivates the use of TXU constraints tointerpret the observed unstable mechanisms. In other words, the specific stress path of theseexperiments promotes a spontaneous trigger of undrained conditions, making the index HTXUINrelevant for their interpretation as latent instability processes. Figure 16b compares the experimentalresults with the predictions provided by the instability modulus computed for undrained triaxiallatent instability. Although the selected constitutive framework cannot naturally reproduce a


-4 105

-2 105

0

2 105

4 105

0 0.02 0.04 0.06

Axial Strain,

Mo

du

lus

of

Inst

abili

ty, H

IN (

kPa)

for = 45o

for = 49o

HIN

PSD

Undrained Localization

Drained Localization

from = 35o to 55oUndrained Localization Range

from = 44o to 54oDrained Localization Range

(a)

Localization for = 45oH & Undrained

-2.5 105

-2 105

-1.5 105

-1 105

-5 104

0

5 104

1 105

1.5 105

0 0.005 0.01 0.015 0.02

Axial Strain,

Mo

du

lus

of

Inst

abili

ty, H

IN (

kPa)

Drained Localization

HIN

PSD

Undrained LocalizationRange for = 40o to 50o

Range for = 40o to 50oDrained Localization

for = 45o

(b)

Figure 13. Comparison between instability moduli for diffuse and localized failure (drained and undrained)for the PSD simulation (a) and PSU simulation (b). Dotted lines indicate the instability modulus for drainedand undrained shear banding computed for band inclinations between 0� and 180�; the values of selected

band inclinations are indicated with solid lines.

0

20

40

60

80

100

120

140

0 0.02 0.04 0.06 0.08 0.1

Dev

iato

ric

Str

ess,

q (

kPa)

Axial Strain,

P1

P2

P3

(a)

0

50

100

150

200

250

300

0 50 100 150 200 250 300

Dev

iato

ric

Str

ess,

q (

kPa)


Failure Limit Criteria

P1

P2

P3

(b)

-1 104

0

1 104

2 104

3 104

4 104

5 104

0 0.01 0.02 0.03 0.04 0.05

Mo

du

lus

of

Inst

abili

ty, H

IN

TX

D (

kPa)

Axial Strain,

(d)

-6 105

-5 105

-4 105

-3 105

-2 105

-1 105

0

1 105

2 105

0 0.01 0.02 0.03 0.04 0.05

Mo

du

lus

of

Inst

abili

ty, H

IN

TX

U (

kPa)

Axial Strain,


p0 = 100kPa

p0 = 200kPa

p0 = 300kPa

(c)

Figure 14. Theoretical interpretation of undrained triaxial compression tests on loose Hostun sand [tests after1, 29]: computed vs measured stress–strain response (a) and computed vs measured stress paths (b); evolutionof the computed values of HTXUIN (c) and H

TXDIN (which reflects the potential for latent drained failure) (d).


spontaneous transition from drained to undrained conditions, the modulus of instability captures thepotential for the onset of an undrained failure, further suggesting that undrained deformationmechanisms were likely to be involved in the collapses reported in [29]. Indeed, the trend ofvariation of HIN reflects the evolution of the predicted second-order work (Figure 17) and providesan interpretation of the unstable passage to an undrained deformation mechanism.


Figure 15. Theoretical interpretation of drained triaxial compression tests on loose Hostun sand (tests after [1]):(a) stress–strain response and identification of the region of potential undrained instability; (b) evolution of vol-umetric strains; (c) evolution of HTXDIN and H

TXUIN (which reflects the potential for latent undrained failure).

CSD-TX Stress Paths

Yield Surface at H =0

Intermediate Yield Surface (at CSD onset)

0

100

200

300

400

500

600

700

Dev

iato

ric

Str

ess,

q (

kPa)


CSD-TX2

CSD-TX3P3

P2

(a)

0

50

100

150

200

250

300

350

400

Dev

iato

ric

Str

ess,

q (

kPa)


CSD-TX1 CSD-TX2

CSD-TX3Failure Limit Line(Daouadji et al, 2010)

Exp. Instability Line(Daouadji et al, 2010)

OA

OB

OC P

2

P3

P1

(b)

0

40

0 100 200 300 400 500 600 700 0 50 100 150 200 250 300 350 400

0 40 80 120

CSD-TX1

P1

Figure 16. Theoretical interpretation of CSD-TX tests (data after [29]): (a) simulated stress paths and yieldsurfaces at the onset of the instability (the inset shows a detail for the test at the lowest effective confine-ment); (b) computed vs measured stress paths. Circle points mark the predicted onset of latent instability.


It is worth noting that the mathematical properties of the instability modulus HIN allow one toestablish a direct relation between its sign and the controllability of the incremental response right atthe yielding point. This is an advantage compared to prior versions of the theory, which wereoriginally focused on the stability of ‘virgin’ sands (i.e. the detection of singularities of the controlmatrix only along a continuously plastic loading path).



4.3. Drained and undrained simple shear tests

To the authors’ knowledge, there were no simple shear experiments on loose Hostun sand from whichlatent instability modes could be clearly inspected. For this reason, this section discusses a qualitativecomparison against simple shear tests performed on a highly liquefiable micaceous sand [44], using thesame calibration as for the previous Hostun sand examples (Section 3.1). The comparison is illustratedin Figures 18 and 19, and refers to simple shear stress paths consisting of two stages: (i) a drained pathat constant normal stress and (ii) a subsequent undrained shearing imposed at different levels of shearstress. It is readily evident that the drained path can promote the complete loss of residual strengthcapacity against undrained shearing. The simulated results for Hostun sand show a similar trend interms of both stress paths (Figure 18b) and stress–strain response (Figure 19b). Most importantly,the instability modulus HSSUIN from the simulated tests offers a mechanical explanation for theseresults. Figure 20 shows, indeed, that the achievement of the limit undrained strength coincides withthe condition HSSUIN ¼ 0. This notion is particularly important to quantify the tolerance to undraineddisturbances (Δtu) as a function of the shear stresses originated by the prior drained shearing(Figure 21). In particular, if the drained shearing path achieves sufficiently high stresses, a latentundrained instability condition is fulfilled at which undrained collapse can be activated even byinfinitesimal undrained perturbations. This example elucidates the close link between the instabilitymodulus HSSUIN and the risk assessment for flow slide triggering, as it allows the loss of undrainedresistance capabilities to be identified from drained or undrained numerical simulations based on soilmodels calibrated for site specific properties [45].

4.4. Shear band analysis of drained and undrained plane-strain tests

This section presents a series of stability analyses of plane-strain tests on loose Hostun sand. Aspreviously mentioned, earlier experiments by Desrues and coworkers [30, 39] showed that shearbands can also be obtained in loose/contractive sands. Therefore, such experiments offer anexcellent proving ground for the evaluation of shear band characteristics by means of the proceduredetailed in Section 3.5.

Figure 22b shows a series of PSD simulations at confining pressures of 100 kPa, 200 kPa, 400 kPaand 800 kPa. While the model simulations exhibit a continuous and stable stress–strain response, theexperimental data from global measurements are featured by a marked discontinuity (associated withthe formation of a persistent shear band within the specimens). The possibility of shear localizationcan be identified from model predictions by plotting the modulus of drained shear banding for arange of band inclinations (Figures 23, 24). The first inclinations for which drained shear bands are

CSD-TX1

-4 10-5

-2 10-5

0

2 10-5

4 10-5

6 10-5

8 10-5

50 100 150 200 250 300 350

Sec

on

d-O

rder

Wo

rk, d

2 W


CSD-TX2

CSD-TX3


P1

P2

P3

(a)Elastic Region(H

S undefined) O

A

-1.5 105

-1 105

-5 104

0

5 104

1 105

1.5 105

50 100 150 200 250 300 350

Mo

du

lus

of

Inst

abili

ty, H

IN (

kPa)


CSD-TX3

OB

OC

P2

P3

P1

CSD-TX2

CSD-TX1

(b)

' '

Figure 17. Theoretical interpretation of model predictions for CSD-TX simulations: (a) evolution of second-order work with mean effective stress, (b) variation of the instability modulus,HTXUIN (the index is not definedduring elastic unloading). OA marks the beginning of the CSD-TX2 simulation, OB marks the onset of the

constant shear stage, and OC marks the end of the simulation.


Change of control toundrained behavior

ci

0

20

40

60

80

100

Sh

ear

Str

ess,

(

kPa)


Undrained

Drained

24kPa

Failure Line

c1

c2

c3

c4

36kPa

12kPa

~0kPa

τ0 =

(a)

0

20

40

60

80

100

0 20 40 60 80 100 0 20 40 60 80 100

Sh

ear

Str

ess,

(

kPa)


c'1

c'2

c'3

c'4

Failure Line

(b)

Figure 18. Qualitative comparison between (a) simple shear stress paths experimentally obtained by Hightet al., [44] on micaceous sands and (b) numerical simulations obtained through the constitutive model

calibrated for loose Hostun sand.

0

10

20

30

40

50

60

0 0.05 0.1 0.15 0.2 0.25

Sh

ear

Str

ess,

(

kPa)

Shear Strain,

Undrained ( ~0kPa)

Drained

= 12kPa

= 24kPa

= 36kPa

(a)

0

10

20

30

40

50

0 0.01 0.02 0.03 0.04 0.05

Sh

ear

Str

ess,

(

kPa)

Shear Strain,

SSU

SSD

Increasing 0

(b)

Figure 19. Qualitative comparison between (a) stress–strain response measured by Hight et al., [44] forloose micaceous sand and (b) numerical simulations obtained through the constitutive model calibrated

for loose Hostun sand. Circle markers indicate changes of control conditions to undrained shearing.

Figure 20. Definition of the tolerance to undrained disturbance (Δtu) from model simulations of simple sheartests (t0 indicates the initial shear stress prior to undrained shearing)


theoretically possible are reported in Figures 24 and 22b (diamond symbols). Upon increasing strains,shear bands become possible for a wider interval of inclinations (Figure 24).

The observations of band inclinations reported by Desrues and Hammad [30] are included inFigure 24 (circle points) and compared with the first values of band inclinations predicted by the


-3 105

-2.5 105

-2 105

-1.5 105

-1 105

-5 104

0

5 104

0 10 20 30 40Shear Stress, (kPa)

SSU

SSD

Increasing 0

Mo

du

lus

of

Inst

abili

ty

for

SS

U C

on

stra

ints

, HIN

SS

U (

kPa)

(a)

0

5

10

15

20

25

0 5 10 15 20 25 30 35 40

To

lera

nce

to

un

dra

ined

d

istu

rban

ce,

u (

kPa)

Initial Shear Stress, (kPa)

SSU

SSD

(b)

Figure 21. (a) Modulus of instability results for undrained simple shear constraints for a range of model sim-ulations, where the circle markers indicate changes of control conditions; (b) dependence of the tolerance to

undrained disturbance as a function of the initial shear stress.


instability modulus (diamonds in Figure 24). For each simulation, the first prediction of potential shearbanding anticipates the onset of persistent shear bands within the specimens. Therefore, the simulationssuggest that, by the time persistent shear bands were observed, the material had already lost residualresistance against certain forms of localized shearing mechanisms. In addition, although the modeldoes not capture all the quantitative aspects of the localization process, the band inclinationsobserved in the experiments fall very close to the possible range of band inclinations predicted bythe theory (Figure 24). Indeed, the formation of a shear band is a gradual process during which thelocalization domain spreads over the specimen [39, 41]. Material point simulations should thereforebe interpreted as a homogeneous approximation of a complex process and are valid only untilsample homogeneity is lost. From this point of view, the theory can elucidate the effects of thisparticular loss of material strength capacity. For this purpose, it is useful to recall the notion oflatent instability and impose a convenient change in control conditions. This is illustrated inFigure 25a, where, at the point in which the instability modulus for drained shear banding vanishedfor the first time, new control parameters have been imposed. The new control condition imposes adrained shear deformation oriented as the band for which the shear localization modulus had alreadyvanished. This procedure engages the resources of material resistance along the weaker direction ofdeformation, identified by the instability moduli associated with the constitutive predictions. Indeed,the application of a change in control produces a marked variation in the evolution of the simulatedstress state with strain. This example provides a particular representation of the loss of strength

0

500

1000

1500

2000

2500

3000

Axi

al E

ffec

tive

Str

ess,

' 1

(kP

a)

Axial Strain, 1

p'0 = 800 kPa

p'0 = 400 kPa

p'0 = 200 kPa

p'0 = 100 kPa

(a)

0

500

1000

1500

2000

2500

3000

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Axial Strain, 1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Axi

al E

ffec

tive

Str

ess,

' 1

(kP

a)

p'0 = 800 kPa

p'0 = 400 kPa

p'0 = 200 kPa

p'0 = 100 kPa

(b)

Figure 22. Comparison between stress–strain curves from drained plane-strain tests by Desrues and Hammad[30] (a) and model simulations (b). The diamonds represent the first potential onset of localization as predictedby a latent instability analysis, while the circles mark the onset of a persistent shear band in the experiments.


-1 105

0

1 105

2 105

3 105

4 105

5 105

6 105

0 50 100 150 200

p0 = 200 kPa

1 = 5.0%

1 = 4.1%

1 = 2.0%

(b)

-2 106-1 106

0

1 1062 1063 1064 1065 1066 106

0 50 100 150 200

p0 = 800 kPa

1 = 12.0%

1 = 10.2%

1 = 5.0%

(d)

-1 105

0

1 105

2 105

3 105

4 105

5 105

6 105

0 50 100 150 200

p'0 = 100 kPa

1 = 4.5%

1 = 3.0%

1 = 1.5%

(a)

-6 105

-4 105

-2 105

0

2 105

4 105

6 105

8 105

0 50 100 150 200

p0 = 400 kPa

1 = 6.3%

1 = 3.0%

1 = 9.0%

(c)

Mo

du

lus

of

Inst

abili

ty, H

IN (

kPa)

Angle of Band Inclination, (deg)

'

' '

'

Figure 23. Dependence of the modulus for drained shear banding on the angle of band inclination. Resultsfrom PSD simulations at initial confinement stresses of (a) 100 kPa, (b) 200 kPa, (c) 400 kPa and (d) 800 kPa

at different strains.


capability predicted by the instability modulus. As shown in Figure 25b, similar results are obtained ifthe same procedure is repeated at larger strains and for band inclinations within the range shown inFigure 24. It is worth noting that the change of control parameters alters the characteristics of thesimulation. In other words, from that point onwards, the simulation has no direct relation with thereal experiment, but rather has only the purpose of explaining the mechanisms by which the materialweakened and failed in a localized manner.

Figure 26 shows a similar comparison between an undrained plane-strain test on loose Hostun sand[39] and its correspondent model simulation. Given the lack of dilative potential in the model setup, thesimulation does not capture the final portion of the stress path. A direct comparison between theory andmodel is therefore possible only until the tendency to dilate does not significantly affect themacroscopic response.

The range of possible drained and undrained shear band inclinations predicted by the model isshown in Figure 27. It is worth noting that, even without dilative potential, the theory predicts thepossibility of both drained and undrained localization processes. In particular, undrained shear bandsare predicted to be possible slightly before drained localization. Although, Mokni [46] reported theformation of two conjugated shear bands at the end of the test, the deformation of the sample wasobserved to be homogeneous also after the peak in deviatoric stresses. This observation iscompatible with the predictions obtained from the instability modulus. Indeed, at the peak, themodulus for diffuse plane-strain failure (HPSUIN ) and that for undrained shear banding at θ = 45

o

vanish simultaneously for the first time. Hence, according to the model, both diffuse and localizedundrained failure are possible at that point. Slightly after the peak, the constitutive predictionssuggest that drained shear banding is also mechanically feasible if the undrained constraint isreleased. This is in accordance with earlier hypotheses by Vardoulakis [47], who suggested that,


40

60

80

100

120

140

0 0.05 0.1 0.15

p0 = 200 kPa (b)

40

60

80

100

120

140

0 0.05 0.1 0.15

p0 = 800 kPa (d)

40

60

80

100

120

140

0 0.05 0.1 0.15

p0 = 100 kPa (a)

40

60

80

100

120

140

0 0.05 0.1 0.15

p0 = 400 kPa (c)

An

gle

of

Ban

d In

clin

atio

n,

(d

eg)

Axial Strain, 1

'

' '

'

Figure 24. Predicted range of potential shear band inclinations for the four PSD simulations. The firstpredictions of possible shear banding provided by the model (diamonds) and the moment of observed per-

sistent localization bands (circles) are both reported.

0

500

1000

1500

2000

2500

3000

Axi

al E

ffec

tive

Str

ess,

' 1

(kP

a)

Axial Strain, 1

p'0 = 800 kPaθ = 51o

p'0 = 400 kPaθ = 49.5o

p'0 = 200 kPaθ = 47.5o

p'0 = 100 kPaθ = 45.5o

(a)

53

4348

5450

0

500

1000

1500

2000

2500

3000

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14Axial Strain,

1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

p'0 = 800 kPa

p'0 = 400 kPa

p'0 = 200 kPa

p'0 = 100 kPa

46

5651

46

5652

48

(b)

Axi

al E

ffec

tive

Str

ess,

' 1

(kP

a)

Figure 25. Model simulations of the effect of latent shear band instability: (a) shearing imposed at the firstpotential onset of drained localization (dotted lines show the original PSD simulation with no change in

control conditions) and (b) shear deformation imposed for various potential shear band inclinations.


even during globally undrained deformation, the process of shear banding can be an outcome of alocally drained process.

It is worth noting that a simultaneous prediction of multiple forms of failure cannot be resolved byrelying only on material point analyses. Indeed, the eventual occurrence of either diffuse or localizedfailure depends on complex hydro-mechanical considerations at a scale of boundary value problems(e.g. material heterogeneities, boundary imperfections and the physical capability of sustaining localsingularities in pore water pressure [47]). The model predictions commented here, however, provide


0

100

200

300

400

0 100 200 300 400

(' 1

- ' 2)

/ 2

(kP

a)

( '1 + '

2) / 2 (kPa)

0 100 200 300 400

( '1 + '

2) / 2 (kPa)

MeaningSymbolPSU Simulation ResultsPSU Experimental DataOnset of latent instability for undrained localization and diffuse PSUOnset of latent instability for drained localizationLoss of homogeneity

(a)

-6 105

-5 105

-4 105

-3 105

-2 105

-1 105

0

1 105

2 105

Mo

du

lus

of

Inst

abili

ty, H

IN (

kPa)

Localization for = 45oH

IN

PSU & Undrained

Drained Localization for = 45o(b)

Figure 26. Theoretical interpretation of the undrained plane-strain test on loose Hostun sand by Mokni andDesrues [39]. (a) measured vs predicted stress paths; (b) computed values of instability moduli associated

with the simulation. Open symbols indicate experimental data after loss of homogeneity.

0

40

80

120

160

0 0.02 0.04 0.06 0.08 0.1

(a)

0

40

80

120

160

0 0.02 0.04 0.06 0.08 0.1

(b)

Onset of potential undrained localization

Onset of potential drained localization

Figure 27. Prediction of the range of potential shear band inclinations for (a) undrained shear banding and(b) drained shear banding for a PSU simulation run at 400 kPa confining pressure.


an overview of the different instability modes that are feasible at a given state of stress. As a result, theyshow the ability of the theory to reflect the richness of failure mechanisms that are possible for agiven material.



5. CONCLUSIONS

This paper has focused on themathematical capture of failure in fluid-saturated sands. Failure mechanismshave been described as material bifurcation modes, and their activation has been identified throughsuitably defined instability moduli. Each modulus has been evaluated for a given set of kinematicconstraints and their vanishing (or negative) values have been associated with the loss of residualresistance against a specific mode of deformation. In particular, once the control conditions at stake aredefined, the methodology can be used for interpreting both diffuse and localized failure.

The analyses have focused on the combination of three modes of deformation (axisymmetric, plane-strain and simple shear) and two drainage conditions for an incompressible pore fluid (drained andundrained). To illustrate the capabilities of the theory, a non-associated constitutive law has beencalibrated for loose Hostun sand, computing the evolution of the moduli and back-analyzing theresults of prior experiments. The main remarks stemming from the study are summarized as follows:

• It has been shown that the key features of drained and undrained tests are reflected by the instability

moduli computed for the kinematic conditions being imposed. The moduli calculated for other modesof deformation (i.e. for latent instability), however, disclosed two forms of latent bifurcation: oneprovoked by the blockage of drainage and the other by the release of the undrained constraint.Regardless of the deformation mode, drained paths have been shown to promote the loss ofresidual undrained shearing resistance. By contrast, undrained paths have been shown to be apossible cause of latent drained bifurcation only for plane-strain and simple shear modes. While inthese cases drained failure never anticipated the loss of stability of the undrained test currentlybeing run, the analyses showed that, even in the hardening regime, certain stress paths can promoteconditions simultaneously susceptible to both drained and undrained instabilities.

• The inspection of different indicators of undrained instability has pointed out that the moduluscomputed by considering only the isochoric constraint approximates the onset of failure in alltypes of undrained simulations. This result suggests that, regardless of the deformation mode,the isochoric constraint has a dominant influence on the potential for undrained bifurcation.Drained simulations, however, illustrated that such characteristics depend on the stress path cur-rently being imposed. For example, the simulations of constant shear tests highlighted non-negligible differences among the different indicators, which may have crucial mechanical impli-cations when they are used for capturing latent undrained instabilities.

• A mathematical strategy has been outlined that allows shear band analyses to be performed withinthe same mathematical framework. This was possible by describing the kinematics inside the bandas a simple shear mode of deformation, while the role of the band inclination was embedded in theinstability moduli through a rotation matrix. It has been shown that this logic enables the compu-tation of the moduli associated with drained and undrained shear banding, as well as the definitionof the range of possible band inclinations.

• Finally, it has been shown that latent instability analyses can be used for interpreting several forms

Copy

of diffuse and localized failure mechanisms observed in experiments. Indeed, vanishing ornegative values of the instability moduli indicate the loss of strength capacity against theirassociated mode of deformation and allowed us to back-analyse different forms of materialinstabilities (e.g. static liquefaction, undrained collapse, drained and undrained shear banding, etc.).

APPENDIX

Description of control conditions

The following section describes the procedure used to adapt the modulus of instability (6) to the maincontrol conditions considered in this paper. More information about applying the generalized modulusof instability expression can be found in [14].

The stress–strain variables controlled upon drained triaxial compression (TXD) allow astraightforward definition of generalized stress and strain measures, which in this case coincide with

right © 2013 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2013)DOI: 10.1002/nag


those used in the definition of the constitutive law, i.e.

_j ¼ _s0 and _h ¼ _« (A1)This implies that both transformation matrices,TTXDs andT

TXD« , equal the identity matrix. In addition,

stress-controlled drained triaxial compression implies pure stress control, with

_f ¼ _j ¼ _s0 (A2a)

_c ¼ _h ¼ _« (A2b)

As a result, for stress-controlled conditions, the partition which describes the controlled generalized

stresses ( _ja) coincides with the entire generalized stress vector ( _j), while the generalized stress response

term ( _jb ) degenerates to an empty vector. Examining (6) with these conditions indicates that for TXDconstraints,

Hw ¼ 0 (A3)

In other words, for TXD controls, the modulus of instability (HIN) coincides with the hardeningmodulus (H).

By contrast, the control conditions for undrained triaxial compression (TXU) imply mixed stress–strain relations, which in the reference system in Figure 10 can be described by using the followingcontrol variables

_x1 ¼ _s01 � _s02 (A4a)

_x2 ¼ _s02 � _s03 (A4b)

_�3 ¼ _ev ¼ _e1 þ _e2 þ _e3 (A4c)where _x1 and _x2 are used to impose the deviatoric stress components, while _�3 is the generalized strainvariable necessary to impose isochoric conditions (i.e. _�3=0). These equations can be incorporated inlinear transformations to define Ts and T«

_j ¼_x1_x2_x3

8<:

9=; ¼

1 �1 00 1 �11=3 1=3 1=3

24

35 _s01_s02

_s03

8<:

9=; ¼ TTXUs _s0 (A5a)

_h ¼_�1_�2_�3

8<:

9=; ¼

2=3 �1=3 �1=31=3 1=3 �2=31 1 1

24

35 _e1_e2

_e3

8<:

9=; ¼ TTXU« _« (A5b)

For this set of conditions, the control vector can be written as,

_f ¼ _ja_hb

� �(A6)

wh

mathematical identiﬁcation of diffuse and localized instabilities … · 2015. 4. 6. ·...

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