A unified method to predict diffuse and localized instabilities in sands

Transcription

1 Geomechanics and Geoengineering: An International Journal, 3 Vol. 8, No., 65 75, htt://dx.doi.org/.8/ A unified method to redict diffuse and localized instabilities in sands WaiChing Sun* Mechanics of Materials, Sandia National Laboratories, Mail Sto 99, 7 East Avenue, Livermore, CA 9455, USA (Received 6 October ; final version received 4 May ) A simlified method to analyse diffuse and localized bifurcations of sand under drained and undrained conditions is resented in this aer. This method utilizes results from bifurcation analysis and critical state lasticity theory to detect the onset of ure and dilatant shear band formation, static uefaction and drained shear failures systematically. To cature the soil collase observed in exeriments, the instability state line concet originated by Chu, Lo and Lee in 993 is adoted. Emhasis is given to examine how the resence of ore-fluid may facilitate or delay instability after yielding occurs. The redictions of instabilities are comared with exerimental data from triaxial comression tests on Toyoura and Changi sands. Downloaded by [WaiChing Sun] at :54 4 July 3 Keywords: bifurcations; static uefaction; diffuse instability; shear band; instability state line. Introduction The stability of fluid-infiltrated orous solids, such as sand, rock, concrete and bone, is governed by constitutive resonses of the solids and the interactions among solid grains and orefluid. Bifurcation analyses conducted by Rudnicki and Rice (975) and Rudnicki (9) have revealed that substantial lastic deformation of a orous solid may lead to the develoment of inhomogeneous strain concentrated in a narrow zone as an alternative to homogeneous deformation. Recent advances in exerimental techniques such as article image velocimetry, however, have shown that shear bands are not the only bifurcation modes observable in the laboratory (Ikeda et al. 6). Instead, a diffuse, convection-like barrel-shaed bifurcation mode that receded the formation of shear bands was observed in soil. Desite the difference in kinematics features, Borja (6a,b) has shown that both localized and diffuse instability can be redicted via bifurcation analysis, rovided that the kinematic features of the deformation mode (diffuse and localized) are taken into account roerly. Prediction of both the onset and tye of instability is critical for many geotechnical and geomechanical alications, such as sloe stability analysis, ground imrovement for uefiable deosit and underground CO storage. For instance, Terzaghi (943) estimates the bearing caability of foundations by assuming soil deosits behave as rigid blocks with localized failure surfaces. This assumtion, however, may not be relevant if the soil deosit underneath the foundations fails in a diffusive mode under static uefaction. Predicting both the onset and tyes of failure at both the drained and undrained limits is not a trivial task, rimarily because the rate-deendence and ore-ressure-gradientdeendence introduced by the couling between inelastic deformation and ore-fluid flow makes local, homogeneous analysis invalid. However, if the material of interest is at either the fully drained or undrained limit, then the resence of ore-fluid becomes only a volumetric constraint to orous solid, which legitimatizes the bifurcation analysis of instantaneous material roerties at the material oint level. By taking full advantage of this simlification at drainage limits, we establish a simle and unified bifurcation analysis aroach to assess stability and redict the instability modes of sands, without sacrificing the rigour of correct hysics. Our goal here is to achieve a balance between simlicity and sohistication of concets related to instabilities and bifurcations such that the analytical framework can be eventually useful for ractising engineers. The two major building blocks of this unified aroach are the critical state lasticity model resonsible for relicating constitutive resonses of sand (Manzari and Dafalias 997, Dafalias and Manzari 4) and the bifurcation criteria detecting localized and diffuse tyes of instability modes from loss of uniqueness (Hill 958, Rudnicki and Rice 975, Raniecki 979, Runesson et al. 996, Rudnicki 4, Borja 6a,b, Darve et al. 7, Andrade 9, Rudnicki 9). To simlify the analysis further, we assume that lastic yielding is rimarily a frictional mechanism and that an increase in stress under a constant stress ratio is assumed to cause wsun@sandia.gov 3Taylor&Francis

2 66 W. Sun Downloaded by [WaiChing Sun] at :54 4 July 3 only elastic strain (Dafalias and Manzari 4). As a result, any ca-surface yielding mechanism and the corresonding instability modes, such as comaction banding (Issen and Rudnicki, Issen, Rudnicki 4) and cataclastic flow ((Borja 6a, 7), are not considered here. As a result of this assumtion, the ressure-sensitive lasticity sand model develoed by Dafalias and Manzari (4) becomes our obvious choice. While not accounting for any ca-surface yielding behaviour, this model incororates concets from critical state soil mechanics (Schofield and Wroth 968) to derive akinematichardeningrulecaableofrelicatingstress strain resonses remarkably well and yet maintains its simlicity by using the common two-invariant Drucker Prager yield surface to redict the onset of yielding. The usage of the two-invariant Drucker Prager yield surface makes it ossible to describe incremental elasto-lastic resonses and the instability criteria with only five indeendent scale arameters (i.e. bulk modulus K, shearmodulusg, hardeningmodulus,frictionalcoefficient µ and dilatancy factor β) andthelasticflowdirection. Since the instability criteria are functions of material arameters that can all be measured and extracted in conventional exerimental settings (e.g. triaxial, biaxial, simle shear, direct shear tests), we can redict diffuse and localized instabilities without further emirical interretations. As noted earlier, a more comrehensive stability analysis requires examination of the rate-deendent and non-local constitutive resonses as well as the heterogeneous nature of the materials. These elements are beyond the scoe of this aer. Nevertheless, simlified analyses of stability at drainage limits are useful for giving guidances for a fuller analysis and insights for conservative engineering designs. In the following section, we rovide an overview of limit conditions that trigger diffuse and localized instabilities under fully drained and undrained conditions. While the accuracy of the instability redictions strongly deends on the erformance of the constitutive model (Issen ), these limit conditions are valid for any two-invariant model without ca surface.. Drained resonses The fully drained condition is achieved if the time scale of the loading is much larger than that of the ore-fluid flow. In this case, the influence of the ore-fluid flow on the stability of the solid skeleton is negligible. Hence, the constitutive relation is assumed to be rate-indeendent and local, without the need for introducing any oro-elasticity arameter.. Onset of the drained diffuse collase Assume that the material deforms homogeneously rior to reaching the instability oints and that the elastic resonse is isotroic. Under axisymmetrical loading, the linearized relation between stress and total strain increments can be sufficiently exressed by a elasto-lastic tangential oerator with scalar deviatoric and volumetric comonents, i.e. [ ] ṗ = [ ] [ ] χk K µβ 3GKβ ϵv q χ 3GKµ 3χG 9G ϵ s }{{}}{{}}{{} σ C e ϵ where χ = + 3G + Kβµ. Thedeterminantoftheelastolastic oerator C e can be exressed in terms of the hardening modulus,thefrictionalcoefficientµ, thelasticdilatancy factor β,shearmodulusg and bulk modulus K (Raniecki 979, Willam, Borja 6a), i.e. det(c e ) = 3KG χ where the eigenvalues of C e are and 3GK( + βµ)/χ when =. The non-trivial eigen-strain associated with the zero eigenvalue of the elasto-lastic tensor takes the form () () [[ ϵ]] = 3 βi + 3 n (3) where n is the sectral direction of the deviatoric stress increment (same as R in Dafalias and Manzari 4). The hysical consequence of C e having a zero eigenvalue is that the material will not exerience extra stress if a strain erturbation linearly roortional to the eigen-strain in Equation (3) is alied. In other words, the material loses controllability at = underthedrainedcondition.. Onset of shear banding Rudnicki and Rice (975) and Rudnicki (9) have shown that any constitutive resonses simulated by a two-invariant lasticity model has localized bifurcation oints at which heterogeneous deformation may relace the initially homogeneous deformation while maintaining a uniform traction field normal to the band. The instability mode is localized and corresonds to a rank-one eigen-strain (strain that is the dyadic roduct of two indeendent vectors) that leads to zero stress traction across anarrowzone,i.e. n C e : (m n) = n [[ σ ]] = (4) where n is a unit vector normal to the lanar band. A non-trivial solution for m is ossible only if the drained acoustic tensor becomes singular, i.e. det n i C e ijkl n l = (5) This critical condition that triggers strain localizations can be exressed as a function of the hardening modulus H, frictional coefficient µ, lastic dilatancy β and the rincial deviatoric stress, i.e.

3 Geomechanics and Geoengineering: An International Journal 67 G [ K(β K 3 6N µ) + (β + µ) = 4G + 3K 4(G + 3K) ] (6) [ ] ṗ [ K K µβ/χ 3GKβ/χ B ] ][ ϵv q = 3GKµ/χ 3G 9G /χ ϵ s B /M ṗ f (8) Downloaded by [WaiChing Sun] at :54 4 July 3 where the scalar N is the ratio between the intermediate rincial deviatoric stress and the Euclidean norm of the rincial deviatoric stresses. For material that does not exhibit ca-tye lastic yielding, the lane of localization is orthogonal to the direction of the rincial intermediate stress (Rudnicki and Rice 975, Perrin and Leblond 993, Issen, Rudnicki 4, 9). Moreover, shear band formation recedes drained diffuse collase (i.e. shear bands form in the hardening regime) if the following relation is satisfied: (β µ) 4G + 3K ( 3 6N + 3β + µ) > (7) 4(G + 3K) Clearly, this is not ossible if the flow rule is associative (β = µ). 3. Undrained resonses Under the undrained condition, ore-fluid remains traed inside the ores during the loading cycle and hence constrains the volumetric deformation of the solid skeleton. This volumetric constraint may facilitate or delay instability. Since orefluid flow within ores is negligible, the constitutive resonse is aroximately rate-indeendent and ore-ressure-gradientindeendent (Rudnicki 9). The constitutive resonses of the solids are nevertheless affecting how ore ressure builds u, and vice versa. Constitutive relations for undrained resonses can be exressed in the same form as Equation () by relacing the drained lasticity arameters with their undrained counterarts if (i) the elastic resonse remains isotroic, i.e. there is no couling between the elastic deviatoric and volumetric resonses and (ii) the effective stress is in Terzaghi form (σ ij + δ ij ) (Terzaghi 943) or in Biot form (σ ij + Bδ ij )(Biot94)and (iii) the inelastic increment in the aarent void volume fraction is equal to the inelastic volume strain increment (Rudnicki 9). 3. Onset of static uefaction The couled macroscoic constitutive resonses of a orous solid at the undrained limit can be described by the elastolastic material arameters augmented with the oro-elasticity arameters to relicate the couling between the solid skeleton and ore-fluid. The momentum and mass balance of the uniform undrained solid subjected to axisymmetric setting is (Nur and Byerlee 97, Rice and Cleary 976, Borja 6b) where B is the Skemton ore ressure coefficient, M is Biot s modulus and ṗ f is the rate of change of ore ressure, for which the following relations hold (Coussy 4): B = K K s ; M = B φ + φ (9) K s K f where φ is the orosity; K s and K f are the bulk modulus of the solid grains that form the solid skeleton and ore-fluid traed inside. B is introduced to account for the influence of excess ore ressure on effective stress when both the solid skeleton and the solid grains are comressible (i.e. K/K s = ). For soil, B, but it can be as low as.5 for concrete or rock (Zienkiewicz et al. 999). On the other hand, Biot s modulus M is introduced to account for the additional volume stored by the comression of ore-fluid (φṗ f /K f )andthecomressionof grains [(B φ)ṗ f /K s ]duetooreressureincrease.interested readers should refer to Zienkiewicz et al. (999) and Coussy (4) for the details. Condensing Equation (8) leads to the following relation: [ ] ṗ = [ ] [ ] χ(k + BM ) K µβ 3GKβ ϵv q χ 3GKµ 3χG 9G ϵ s }{{}}{{}}{{} σ C und ϵ () The undrained tangential constitutive resonse exressed in Equation (8) ceases to remain stable if C und becomes singular. This haens if the hardening modulus satisfies the following relation: K = B M Kµβ () B M + K This result is identical to the static uefaction criterion in Andrade (9) ( = Kµβ) ifbothconstituentsareincomressible. Under axisymmetric loading, the non-trivial strain increment can be exressed as a tensor whose volumetric and deviatoric comonents maintain the following relation: [[ ϵ]] = Kβ 3 B M + K I + 3 n () In cases where the solid grains and ore-fluid are nearly incomressible (comared to the orous solid), M, B = andtheeigen-strainbecomesisochoric.thisimliesthatthe orous solid can maintain the same deviatoric stress if a nontrivial, ure shear strain erturbation is alied. In other words, the material loses controllability at = B MKµβ/(B M + K) Kµβ under undrained conditions. Since the shear strength is lost at this oint, the material would behave like a frictional fluid and hence can be considered to be uefied.

4 68 W. Sun Downloaded by [WaiChing Sun] at :54 4 July Onset of the ure shear band Following the rocedures in Rudnicki and Rice (975) augmented with the ore-fluid mass balance law, one can derive the undrained limiting hardening modulus, which reads K G = (Kund BM) (β µ) K und (4G + 3K und ) [3 6K und N + (K und BM)(β + µ)] 4K und (G + 3K und ) B MKµβ GK und (3) where K und = K + B M is the undrained bulk modulus of the orous media. If both the ore-fluid and the solid grains are incomressible such that M and B =, then Equation (3) can be simlified into K G = 9 N Kµβ G (4) where the band angle is always 45 to the rincial stress regardless of the adoted yield criterion (Runesson et al. 996, Rudnicki 9). By comaring Equation (4) with Equation (), we redict that undrained shear banding in materials comosed of incomressible constituents always occurs in the softening regime and after static uefaction for both associative and non-associative flow rules. 4. Critical state lasticity model for sands We incororate the instability state line concet (Chu et al. 993, Lade 993, Chu et al. 3) into a critical state lasticity model. The framework of the critical state lasticity theory used here was develoed by Manzari and Dafalias (997) and Dafalias and Manzari (4) (referred to as the MD model hereinafter). This model incororates the bounding surface model concets to maniulate volumetric and deviatoric hardening such that the simulated elasto-lastic constitutive resonse is consistent with critical state theory (Schofield and Wroth 968). Interested readers should refer to the original aer cited above for details. The MD model includes an emirical elasticity model that reads ϵ ij = 3K(e, )ṗδ ij + (5) G(e, )ṡij where s ij = σ ij /3δ ij is the deviatoric art of the Cauchy stress. The bulk modulus K(e, ) and the shear modulus G(e, ) arefunctionsofcurrentmeanstress and void ratio e, i.e. (.97 e) ( ) / K(e, ) = K o ; + e at ( ) (.97 e) / (6) G(e, ) = G o +e at where at denotes the atmosheric ressure. The elastic domain is enclosed by a urely kinematic hardening Drucker Prager yield surface that reads f (s ij, α ij ) = [(s ij α ij )(s ji α ji )] / m = (7) 3 where m is a constant and α ij is the back stress. The hardening rule of the material is modelled based on critical state theory (Schofield and Wroth 968). In critical state theory, it is assumed that there exists a unique critical state under which materials exerience large lastic shear without volume change. In the MD model, the critical state is reached if both the stress ratio (deviatoric over hydrostatic) q/ and void ratio e reach their corresonding critical values, i.e. q = ( ) ( ) q ξ = M c ; e c = e = e c o λ c (8) c at where M c, e o and ξ are scalar arameters determined by interolating the void ratio e, mean stress and deviatoric stress q = 3/ s ij s ij at the observed critical state. To make the hardening and lastic dilatancy consistent with critical state theory, Dafalias and Manzari (4) introduce two isotroic limit surfaces and use their distances from the critical state to maniulate the hardening modulus and lastic dilatancy. These two limit surfaces read as follows: q M b = ; M b = M c ex[ n b (e e c )] (hardening limit surface) q M d = ; M d = M c ex[ n d (e e c )] (dilatancy limit surface) (9) () where n d and n b are scalar constants that control the size of the limit surfaces. They can be determined from the initial yielding oints. The hardening modulus and lastic dilatancy factor β are set to be continuous interolation functions which deend on (i) the distance between limit and critical surfaces and (ii) other interolation functions added to imrove the accuracy of the MD model. In axisymmetrical settings, the hardening modulus and the dilatancy factor β are = h 3 ( M b q ) ; h = G oh o ( c h e) ( ) q at q ( β = A o (+ < z ij n ji >) M d q ) ; ż ij = c z < ϵ v > (z max n ij + z ij ) in () () 3 n ij = m (s ij α ij ) (3)

5 Geomechanics and Geoengineering: An International Journal 69 Downloaded by [WaiChing Sun] at :54 4 July 3 where n ij is the deviatoric art of the yield surface gradient. h and A d are functions introduced to imrove interolation of the hardening and dilatancy from exerimental observations. h o, c h, c z and z max are ositive scalar material arameters obtained by trial and error. < x > is the Macaulay bracket, which is equal to x if x > andifx <. (q/) in is the stress ratio at the beginning of the loading rocess. The tensor z ij is called the fabric-dilatancy tensor (Dafalias and Manzari 4). It is introduced to take account of the content-normal orientation distribution changes during forward and reverse shearing. Furthermore, the frictional coefficient µ can be obtained from the volumetric comonent of the yield surface, i.e. 3 µ = m (s ijs ji s ij α ji ) (4) With the indeendent scale arameters (i.e. bulk modulus K, shear modulus G, hardeningmodulus,frictionalcoefficient µ and dilatancy factor β) determinedfromthemdmodel,we have sufficient information to relicate the incremental linear tangential resonses of the sands and assess the stability of the resonses. 4. Incororating shear failure behaviour The constitutive relation develoed so far is caable of relicating many imortant henomenological characteristics of granular materials (Manzari and Dafalias 997, Dafalias and Manzari 4, Jefferies and Been 6). However, the constitutive models described above redict a erfectly lastic resonse if e = /c h,regardlessoftheamountofconfining ressure to which the material is subjected. This is consistent with the fact that the instability line is defined by linking the to of the undrained aths, while assuming that it is indeendent of the confining ressure within a limited range (Lade and Pradel 99, Chu et al. 993, Imam et al., Chu et al. 3, Wanatowski and Chu 7a,b). However, if one wishes to model materials under a wide range of confining ressure with the same material arameter, then a henomenological amendment can be made by rewriting c h as a function of effective hydrostatic ressure. Inthatcase,theexression( c h )in Equation () is relaced by the following relations: = h 3 c h () = ( M b q ) ; h = G oh o ( c h ()e) ( ) ; q at q e s o λ ( at ) ξ in (5) where e s o, s = L, D are the zero intercets of the dilatant ( ) e D o and contractive ( ) e L o instability state lines in e sace. The instability state line is underneath the critical state line in the e sace if the material is dilatant and above the critical state line if the material is contractive. 5. Numerical redictions In this section, we resent fully drained and undrained numerical simulations at the material oint level. The urose of erforming these simulations is twofold (i) to comare instability redictions with exerimental observations (ii) to analyse how the resence of ore-fluid affects stability. As shown in these numerical examles, the modified Manzari Dafalias model is caable of relicating realistic constitutive resonses of sand. This feature is imortant to ensure the accuracy of the instability redictions. Both drained and undrained triaxial test simulations are conducted with the material arameters resented in Table. The material arameters are calibrated with the drained triaxial comression tests conducted by Chu et al. (3) (Changi sand) and undrained and drained triaxial comression tests by Verdugo and Ishihara (996) (Toyoura sand). 5. Examle : Collases in drained contractive Changi sand This simulation is conducted to relicate the contractive constitutive resonses of the DR39 constant shear test reorted in figure of Chu et al. (3). The drained triaxial test was conducted on Changi sand with initial void ratio =.95. A5kPahydrostaticloadisfirstaliedontheChangisand secimen, then the secimen is loaded in triaxial comression until the confining ressure reaches around 8 kpa as shown in Figure a. Following this ste, the confining ressure decreases under constant deviatoric stress. This decrease in confining ressure causes further axial strain in the secimen and makes the stress state closer to the critical state line as shown in Figures a c. Our result shows that the simulated contractive resonse is in general in agreement with the exeriments. In articular, our simulation is consistent with the observation that the loose, contractive secimen collases at a stress state below the critical state line in q sace. As deicted in Figures d and, the contractive secimen exhibits erfectly lastic behaviour as both the lastic dilatancy Table. Summary of material arameters used for numerical redictions Symbol Parameter Changi sand Toyoura sand G o Shear coefficient 5 5 ν Poisson ratio.5.5 M c Critical stress ratio.35.5 e c o Critical void ratio arameter λ c Critical state arameter.4.9 ξ Critical state arameter.4.7 e D o Instability state arameter.7.85 e L o Instability state arameter.98.8 h o Hardening arameter.8 7. A o Dilatancy arameter z max Dilatancy arameter c z Dilatancy arameter 6 6 n b Limit surface size arameter.. n d Limit surface size arameter.8 3.5

6 7 W. Sun Downloaded by [WaiChing Sun] at :54 4 July 3 void ratio 4 3 (a) 4.8 (c) IL.6 ε a K (b) Exeriment Simulation 4 x 4 band (d) Drained Collase Figure. Numerical simulation of drained triaxial comression test on Changi sand with initial void ratio =.95 (red): (a) simulated deviatoric stress, (b) axial strain, (c) void ratio, and (d) generalized hardening modulus versus effective hydrostatic ressure. and the hardening modulus reduce to zero at q/ = M b = M d (see Equations and ). As a result, a significant amount of shear strain develos without lastic dilatancy and thus leads to the collase of the secimen. This unstable resonse relicated by the MD model is consistent with the observed raid increase in axial strain of the secimen occurring before the stress asses through the critical state line as deicted in figure 7 of Chu et al. (3). To quantify the asymtotically worst case of how much strain may develo in roortion to small changes in stress, we comute the condition number of the tangential tensor C e,whichisdefinedastheratiobetween the largest singular value and the smallest singular value (Horn and Johnson 985). A large condition number is an indication that the constitutive resonse of the secimen becomes unstable with resect to erturbation. A salient observation shown in Figure 3 is that the condition number of the tangent stiffness tensor actually increases much more raidly when the generalized hardening modulus is aroaching zero. Due to the deterioration in stability signalled by the increased condition number, a small stress erturbation might cause severe strain accumulation if the secimen is close to the instability state line. Since the hardening modulus remains close to zero in between the instability state line and the critical state line, this allows significant axial strain to develo, even though the.5 8 q/.5 Drained Collase when β = M b M c M d q/ Figure. The stress ratio q/ and the sizes of the hardening (M b )and dilatancy (M d )limitsurfacesversustheeffectivehydrostaticressureofthe contractive Changi sand. Condition number Figure 3. Condition number of C e versus effective hydrostatic ressure of the contractive Changi sand.

7 Geomechanics and Geoengineering: An International Journal 7 Downloaded by [WaiChing Sun] at :54 4 July 3 µ µ β Figure 4. Plastic dilatancy β and frictional coefficient µ versus effective hydrostatic ressure of the contractive Changi sand. effective stress change is small. In this case, drained collase may occur at the instability state line without a shear band being formed, as shown in Figure d. The MD model redicts that both the lastic dilatancy β and the frictional coefficient µ are ositive after yielding, thus suggesting a contractive lastic behaviour as shown in Figure Examle : Formation of shear band in drained dilatant Changi sand This simulation is conducted to relicate the dilatant constitutive resonses of the DR39 constant shear test reorted in figure of Chu et al. (3). A triaxial load is alied under 5 kpa confining ressure on Changi sand with a.657 initial void ratio 4 3 (a) (c).7 IL.6 ε a q/ K (b) Exeriment Simulation x 4 (d) Drained Collase when β band Shear Band Shear Band Drained Collase M b M c M d q/ Figure 6. The stress ratio q/ and the sizes of the hardening (M b )and dilatancy (M d )limitsurfacesversuseffectivehydrostaticressureofthedilative Changi sand. void ratio. Figure 5a deicts the stress ath. The secimen is first sheared until the deviatoric stress reaches 3 kpa. Then the deviatoric stress remains constant while the hydrostatic stress is decreasing. Our result shows that the simulated dilatant resonse is in general in agreement with the exeriments. By monitoring the generalized hardening modulus and its limit value for shear banding, we found that shear bands may form in the hardening regime when the effective hydrostatic ressure of the dilatant secimen is decreasing under constant deviatoric stress. The generalized hardening modulus remains ositive but its magnitude decreases as the axial strain develos. This henomenon is catured in the MD model by shortening the distance between the current stress and the instability line (denoted as IL in Figure 5c) and the distance between the current stress and the isotroic hardening limit surface as illustrated in Figure 6. Figure 5. Numerical simulation of drained triaxial comression test on Changi sand with initial void ratio =.657 (red): (a) simulated deviatoric stress, (b) axial strain, (c) void ratio, and (d) generalized hardening modulus versus effective hydrostatic ressure are lotted (red line) and comare with exerimental observations (blue dots).

8 7 W. Sun Condition number is aroximately one order smaller than that of the loose sand secimen, thus suggesting that the constitutive resonse of the dense Changi sand in Examle is more stable than the loose Changi sand counterart in Examle when both are at the instability state line. Figure 8 shows the evolution of lastic dilatancy and frictional coefficient redicted by the MD model during the drained triaxial comression test. The lastic dilatancy is found to be negative after the onset of shear bands redicted via Equation (6), thus suggesting the formation of a dilatant shear band. Downloaded by [WaiChing Sun] at :54 4 July Figure 7. Condition number of C e versus effective hydrostatic ressure of the dilatant Changi sand. µ, β µ β Figure 8. Plastic dilatancy β and frictional coefficient µ versus effective hydrostatic ressure of the dilative Changi sand. At the onset of shear band formation, the material exhibits lastic dilatancy (because q/ < M d ). Hence, the strain localization zone may exhibit both shear and dilatancy. Following the onset of shear banding, the modified MD model redicts that the material would become very sensitive to erturbation as the resonse becomes more and more close to erfectly lastic as deicted in Figures 5b and 5d. Notice that the dense, dilatant secimen collases at a stress state above the critical state line in q sace when the current stress reaches the hardening limit surface, i.e. q/ = M b as deicted in Figure 6. This rediction is consistent with the observed raid increase in axial strain of the secimen occurring after the stress asses through the critical state line described in figure 8 of Chu et al. (3). To assess the stability of the constitutive resonse, we comute the condition number of C e,which is shown in Figure 7. By comaring the condition number of the loose and dense secimen at the instability state line, we found that the condition number of C e of the dense secimen 5.3 Examle 3: Static uefaction of undrained Toyoura sand in loose state Three monotonic undrained triaxial comression tests on loose Toyoura sand are simulated and comared with the exeriments conducted by Verdugo and Ishihara (996). Figures 9a and 9b comare the simulated undrained constitutive resonse with the exeriments. As demonstrated in Figure 9, the simulated constitutive resonses are in good agreement with the exeriments. Moreover, we found that softening only occurs if the loose Toyoura sand is first consolidated under sufficient confining ressure. In the monotonic undrained triaxial comression test with confining ressure = kpa, both exeriential and simulated constitutive resonses exhibit only hardening. Figures a and b comare the hardening modulus redicted by the modified MD model with the threshold values for static uefaction (K )andshearbanding(k ). Interestingly, we found that the erfectly incomressible constraint imosed by the traed ore-fluid may stabilize the material and revent the formation of shear bands. Furthermore, this comarison also reveals that static uefaction may occur either (i) near the critical state line or (ii) at the eak of the deviatoric stress. The former case is very similar to a critical state in the drained condition in which the hardening modulus, frictional coefficient and lastic dilatancy are very small and lead to materials shearing as a frictional fluid at constant volume and thus make material very sensitive to effective stress erturbation as demonstrated by the high condition number of C e.inthelattercase,whichiscoined undrainedinstability in Andrade (9), the material become unstable and very sensitive to the total stress erturbation due to the singularity of C und.noticethatstrainsofteningoccurswhen material is between (i) and (ii). As ointed out by Verdugo and Ishihara (996), the deviatoric stress at the onset of (ii) strongly deends on the initial confining ressure, whereas the onset of (i) does not. One ossible exlanation is that the stability triggered by eak deviatoric stress is usually associated with non-zero hardening modulus and non-zero frictional coefficient β as indicated in Figure. This makes the onset of (ii) highly deendent of the initial confining ressure. Using the localization criteria exressed in Equation (4), we confirm that strain softening does not necessarily guarantee strain localization, which is consistent with the exerimental observations of sand made by Chu et al. (993, 996).

9 Geomechanics and Geoengineering: An International Journal Static Liquefaction 5 Static Liquefaction ε a Figure 9. Numerical simulations of undrained triaxial comression tests on Toyoura sand with initial void ratio =.97: (a) simulated deviatoric stress versus effective hydrostatic ressure and (b) deviatoric stress versus axial strain. The confining ressures of the three simulations are kpa (blue), kpa (red) and kpa (green). Exerimental data from Verdugo and Ishihara (996) used to calibrate the material arameters are lotted in dots. Downloaded by [WaiChing Sun] at :54 4 July 3 K.5 x 4 x 4 x K K 3 4 K Figure. Simulated evolution of hardening modulus (,inbluecolour)andthecorresondingthresholdvaluesforureshearband(k,inredcolour)and static uefaction (K, in green colour) of Toyoura sand with initial void ratio =.97 and confining ressures equal to (a) kpa, (b) kpa, and (c) kpa Shear failure near ε a Figure. Numerical simulations of undrained triaxial comression tests on Toyoura sand with initial void ratio =.735: (a) simulated deviatoric stress versus effective hydrostatic ressure and (b) deviatoric stress versus axial strain. The confining ressures of the three simulations are kpa (blue), kpa (red), kpa (green) and 3 kpa (cyan). Exerimental data from Verdugo and Ishihara (996) used to calibrate the material arameters are lotted in dots.

10 74 W. Sun x x x 5 x K K K K Downloaded by [WaiChing Sun] at :54 4 July Figure. Simulated evolution of hardening modulus (,inbluecolour)andthecorresondingthresholdvaluesforureshearband(k,inredcolour)and static uefaction (K, in green colour) of Toyoura sand with initial void ratio =.735 and confining ressures equal to (a) kpa, (b) kpa, (c) kpa, and (d) 3 kpa. 5.4 Examle 4: Stable constitutive resonse of undrained Toyoura sand in dense state We simulate an undrained triaxial comression test on dense Toyoura sand with confining ressures of,, and 3 kpa. The initial void ratio of this simulation is.734. As demonstrated in Figures a and b, the simulated constitutive resonse matches closely with the exerimental data in Verdugo and Ishihara (996). Unlike loose Toyoura sand, dense Toyoura sand does not exhibit eak deviatoric stress. As a result, shear failures only occur near the critical state line. Figure comares the generalized hardening modulus with the static uefaction and shear band threshold values. The simulation result demonstrated in Figure confirms that there is no shear band formed in dense Toyoura sand under undrained conditions. This observation is consistent with the homogeneous deformation modes observed by Verdugo and Ishihara (996). of ore-fluid may facilitate or delay instabilities and how contractive/dilatant states affect both the onsets and modes of failure. Using exerimental data available in the literature, we comare the simulated and observed constitutive resonses as well as the redicted and actual onsets of instability. Our findings confirm that the framework is able to relicate robustly both the constitutive resonse and the onset of various instability modes observed in exeriments. Acknowledgements The author thanks Dr Dariusz Wanatowski of the University of Nottingham for roviding exerimental data relating to drained triaxial comression tests on Changi sand. Financial suort through a graduate fellowshi from Northwestern University is gratefully acknowledged. The exert review from the anonymous reviewer is also gratefully acknowledged. Sandia National Laboratories is a multi-rogram laboratory managed and oerated by the Sandia Cororation, a wholly owned subsidiary of the Lockheed Martin Cororation, for the U.S. Deartment of Energy s National Nuclear Security Administration under contract DE-AC4-94AL Conclusions We roose a simle and yet unified method to simulate and redict the onsets of drained collase, static uefaction and the formation of deformation bands under drained and undrained conditions. This method is based on bifurcation analyses and atwo-invariantcriticalstatelasticitytheory.bycomaring numerical simulations with exerimental data, not only do we show that the instability criteria are caable of delivering redictions consistent with exerimental observations, but we also rovide a hysical interretation of why the resence References Andrade, J.E., 9. A redictive framework for uefaction instability. Géotechnique, 59(8), Biot, M.A., 94. General theory of three-dimensional consolidation. Journal of Alied Physics,(),55 6. Borja, R.I., 6a. Conditions for instabilities in collasible solids including volume imlosion and comaction banding. Acta Geotechnica, (6),7. Borja, R.I., 6b. Conditions for uefaction instability in fluidsaturated granular soils. Acta Geotechnica, (4), 4.

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