Flow liquefaction instability prediction using finite elements

Transcription

1 Acta Geotechnica (215) 1:83 1 DOI 1.17/s z RESEARCH PAPER Flow instability rediction using finite elements Toktam Mohammadnejad José E. Andrade Received: 14 November 213 / Acceted: 6 June 214 / Published online: 3 July 214 Ó Sringer-Verlag Berlin Heidelberg 214 Abstract In this aer, a mathematical criterion based on bifurcation theory is develoed to redict the onset of instability in fully saturated orous media under static and dynamic loading conditions. The roosed criterion is general and can be alied to any elastolastic constitutive model. Since the criterion is only as accurate as the underlying constitutive model utilized, the modified Manzari Dafalias model is chosen for its accuracy, relative simlicity and elegance. Moreover, a fully imlicit return maing algorithm is develoed for the numerical imlementation of the Manzari Dafalias model, and a consistent tangent oerator is derived to obtain otimal convergence with finite elements. The accuracy of the imlementation is benchmarked against laboratory exeriments under monotonic and cyclic loading conditions, and a qualitative boundary value roblem. The framework is exected to serve as a tool to enable rediction of occurrence in the field under general static and dynamic conditions. Further, the methodology can hel advance our understanding of the henomenon in the field as it can clearly differentiate between unstable behavior, such as flow, and material failure, such as cyclic mobility. Keywords Finite element analysis Fully imlicit return maing algorithm Granular materials Liquefaction instability Manzari Dafalias lasticity model Static and dynamic T. Mohammadnejad J. E. Andrade (&) Mechanical and Civil Engineering, California Institute of Technology, Pasadena, CA, USA jandrade@caltech.edu 1 Introduction Liquefaction as an instability henomenon is erhas one of the most devastating yet elusive concets in geotechnical engineering. Abundant field evidence shows a sudden loss of shear strength tyically associated with an aarent build-u of ore water ressure, usually observed in relatively loose cohesionless soil formations. Several case studies from earthquakes around the globe have shown the world the devastating ower of : Niigata 1964, Kobe 1995, Haiti 21, and Christchurch 211, among others. Yet, very little is known about the mechanisms controlling the henomenon to the extent that redictive models have remained elusive. In fact, for many years, the henomenon was associated solely as a result of earthquake (cyclic) loading and confounded with cyclic mobility. It is now widely acceted that: (1) is an instability (usually termed flow, see [14]) and (2) it can occur under static or dynamic conditions (the former is tyically called static ). Several works in the field have shown the unstable behavior of the material in the laboratory under static and cyclic loading conditions [7, 15, 27]. Based on Hill s instability criterion [11], Lade [15] showed that there exists an instability line that connects the oints at which the material loses stability in the laboratory under undrained conditions and quasistatic loading. Borja [5] ostulated a condition for instability, and Andrade [1] simlified the latter condition and alied it to elastolastic models, extracting in this way a simle critical hardening modulus inointing the onset of, which coincides with the instability line of Lade [15] and the concet of loss of controllability ostulated by Nova [22]. Later, Andrade et al. [3] built on the concet of instability and exanded the

2 84 Acta Geotechnica (215) 1:83 1 criterion to dynamic (cyclic) loading. In their work, it was shown that flow (or simly ) is an instability, whereas cyclic mobility is a failure henomenon resulting from material resonse, but it is not an instability. In this way, they showed the fundamental difference between (as an instability) and cyclic mobility. Most, if not all, the aforementioned works are based on laboratory scale observations. Once the jum to the field scale, most of the works become observational, with otential failure mechanisms ostulated to reconstruct the observed failure atterns. The classic examle is the failure of the lower San Fernando dam in 1971, which was remarkably reconstructed by Seed et al. [25]. In an attemt to rovide a quantitative and rational aroach, ienkiewicz et al. [3] develoed a finite element model based on elastolasticity, where they showed some qualitative and limited quantitative agreement with the observations made for the lower San Fernando dam. Most remarkable is their rediction of the deformation attern as a result of the earthquake excitation and caturing of the delay in failure, occurring aroximately 6 s after the earthquake. Nevertheless, the field-scale aroach based on finite elements has lacked a roer criterion for and corresondingly robust constitutive models to make the redictions lausible. As a first attemt to ma instabilities, Ellison and Andrade [1] develoed a finite element model based on the mixed formulation to simulate static and were able to reroduce a classic examle reviously ostulated by Lade [15] for submarine levees. However, the solution is quasi-static, and it is not caable to shed full light into the dynamic regime. This work is based on the criterion of Andrade [1] and the mixed finite element for saturated granular materials under static conditions roosed by Andrade and Borja [2]. In this contribution, the criterion is further generalized from that of Andrade et al. [3] to general loading conditions (as oosed to triaxial (axisymmetric) loading). In that way, the criterion roosed herein is closer in hilosohy to the one obtained by Borja [5] under static conditions. The roosed criterion is alied to the most recent Dafalias and Manzari model for sands [9], which catures accurately the cyclic behavior of the material invoking elastolasticity. The Manzari Dafalias lasticity model with recent modifications is numerically integrated using a fully imlicit return maing algorithm. The imlicit integration of the original model or slight variations of the model can be found in [8, 16, 21]. Finally, the model and corresonding criterion are cast into a dynamic finite element code caable of simulating couled soil deformation with fluid flow using the classic u- formulation. In this way, this aer resents the fully imlicit imlementation of the Manzari Dafalias model within a finite element code with the caability of maing the onset of instability under dynamic conditions. The accuracy of the imlementation is benchmarked against laboratory exeriments under monotonic and cyclic loading conditions, and a boundary value roblem under dynamic loading conditions. The framework is exected to serve as a tool to enable rediction of occurrence in the field under general loading conditions. As such, the current aer should not be interreted to alying uniquely to the Manzari Dafalias model, but can be used with any other elastolasticity model in the field. The authors have chosen the aforementioned model for its caabilities, relative simlicity, and elegance, but this is a ersonal choice. The structure of the aer is as follows. In Sect. 2, the equations governing the roblem are summarized. In Sect. 3, the Manzari Dafalias lasticity model based on its recent modifications is briefly described. In Sect. 4, a fully imlicit integration scheme is develoed for the numerical imlementation of the Manzari Dafalias lasticity model, and a consistent tangent oerator relevant to the roosed return maing algorithm is resented. In Sect. 5, a general, mathematical criterion for the onset of instability is obtained. In Sect. 6, the caability of the roosed criterion in caturing the onset of instability under monotonic and cyclic loading conditions is demonstrated by solving a number of triaxial tests, and its alicability to boundary value roblems under dynamic loading conditions is shown through a fully couled dynamic analysis of a lane strain test. Finally, conclusion is made in the last section. 2 Governing equations In this section, governing equations of fully saturated orous media are resented. These governing equations are based on two basic laws of force equilibrium and mass conservation. For numerical solution of the governing equations, finite element method is emloyed to discretize the weak form of the governing equations in sace together with the generalized Newmark scheme for time domain discretization. 2.1 Strong form of the governing equations The artial differential equations governing the solid hase deformation and ore fluid flow through the orous medium are based on the balance equations of linear momentum and mass. The linear momentum balance equation for a fully saturated orous medium can be written as [17]

3 Acta Geotechnica (215) 1: rr þ qb q u ¼ ð1þ where r is the total stress tensor defined as r ¼ r a w I; with r the modified effective stress tensor, a the Biot constant, w the ore fluid ressure, and I the second-order identity tensor, b is the body force vector, u is the acceleration vector of the solid hase, q is the average density of the whole mixture defined as q ¼ ð1 nþq s þ nq w ; in which n stands for the orosity of the orous medium and q s and q w are densities of solid and fluid hases, resectively, and the symbol r denotes the vector gradient oerator. Throughout this aer, the mechanics convention is used, where stress is considered ositive in tension, while the ore fluid ressure is considered as comression ositive. The continuity equation for ore fluid flow through a fully saturated orous medium can be written as [17] a n þ n _ K s K w þ ar _u þr _w w ¼ ð2þ w where K s and K w are bulk moduli of solid and fluid hases, resectively, _u is the velocity vector of the solid hase, and _w w is the Darcy s velocity vector of the ore fluid given by _w w ¼ ðk=l w Þ½r w þ q w ðb u ÞŠ with k the intrinsic ermeability matrix of the orous medium, simly relaced by a scalar value k for an isotroic medium, and l w the dynamic viscosity of the ore fluid. 2.2 Discrete form of the governing equations The finite element aroximation of the rimary unknown variables can be written as [17] u h ðx; tþ ¼ N u ðxþuðþ t h w ðx; tþ ¼ N ð3þ w ðxþp w ðþ t where N u ðxþ and N w ðxþ are matrices of finite element shae functions and U(t) and P w ðþ t are vectors of dislacement and ressure degrees of freedom, resectively. It is worth noting that if the incomressible and undrained limit state is never aroached, equal-order shae functions can be used for the aroximation of both solid and fluid variables. Otherwise, it is necessary that the dislacement field be aroximated by olynomial shae functions one order higher than those used for the aroximation of the ressure field [17]. Different order of shae functions revents surious oscillations and locking in the ressure field in the limit of nearly incomressible and undrained state. Following the Bubnov Galerkin technique, the discretized form of the governing Eqs. (1) and (2) is obtained as M u U þ X B T r dx Q w ¼ F u M w U þ Q T _U þ C _ w þ H w ¼ F w ð4þ where B is the strain dislacement matrix. The definition of the coefficient matrices and the force and flux vectors is given in Aendix 1. The above system of equations is then discretized in time following the line of the wellknown Newmark scheme. The resulting system of fully couled nonlinear equations is solved using the unconditionally stable direct time-steing rocedure combined with the Newton Rahson iterative rocess. As a result, the main unknowns are obtained simultaneously, leading to the full solution of the roblem. For a detailed resentation of the solution rocedure, see [13]. 3 Constitutive model In order to simulate the mechanical behavior of the granular material, the Manzari Dafalias lasticity model is used. This model was originally roosed in [2] and then was extended to account for the effect of fabric changes during the dilative hase of deformation on the subsequent contractive hase uon loading reversal [9]. The Manzari Dafalias constitutive model is framed within the context of the critical state soil mechanics [24], which is based on the theory that for a given soil, a unique critical state line (CSL) exists. In the following, the Manzari Dafalias lasticity model [9] is briefly described. In Manzari Dafalias lasticity model, the nonlinear elastic resonse of the material is described by the hyoelastic formulation. The volumetric and deviatoric art of the elastic strain rate tensor are given by _e e v ¼ _ K ; _ee ¼ _s ð5þ 2G where _e e v ¼ tr _ee ; _e e ¼ _e e _e e v =3I; is the mean effective stress defined as ¼tr r =3; s is the deviatoric effective stress tensor defined as s ¼ r þ I; and K and G are the elastic bulk and shear moduli, resectively, which are the functions of the mean effective stress and void ratio [18, 23] K ¼ 21þ ð tþ 31 ð 2tÞ G; G ¼ G ð2:97 eþ 2 1=2 at ð6þ ð1 þ eþ at where t is the Poisson s ratio, G is a material constant, at is the atmosheric ressure and e is the void ratio whose evolution is given by _e ¼ ð1 þ eþ_e v ; with _e v ¼ tr _e the volumetric strain rate.

4 86 Acta Geotechnica (215) 1:83 1 The yield surface is defined by f r ; a ¼ ½ðs aþ: ðs aþš 1=2 ffiffiffiffiffiffiffi 2=3 m ¼ ð7þ which geometrically reresents a circular cone whose aex is at the origin in the effective stress sace. The yield surface determines the limit of the elastic domain. In the above relation, a is the deviatoric back stress-ratio tensor defining the orientation of the cone, m is a constant defining the size of the cone, and the symbol : denotes the inner roduct of two second-order tensors. The evolution of a is governed by the kinematic hardening law as _a ¼ _L ð2=3þhb ð8þ where L is the loading index, the symbol hi denotes the Macaulay brackets, and h is the hardening coefficient, which is ositive, and given by b h ¼ ða a in Þ:n ; b 1=2 ¼ G h ð1 c h eþ ð9þ at In the above relation, a in is the initial value of a at the initiation of a new loading rocess and is udated when ða a in Þ:n becomes negative, n is the unit deviatoric tensor, which is equal to the deviatoric art of the normal to the yield surface, and h and c h are ositive constants. The unit deviatoric tensor n is defined as n ¼ ðs aþ= ks ak; in which the symbol kk denotes the L 2 norm of a tensor. It is obvious that the unit deviatoric tensor n has the following roerties: tr n ¼ and n:n ¼ 1: In Eq. (8), b is given by b ¼ ffiffiffiffiffiffiffi 2=3 a b h n a; a b h ¼ M ex nb w m ð1þ where n b is a ositive material constant, w ¼ e e c is the state arameter, in which e c is the critical void ratio corresonding to the existing mean effective stress comuted from the ower relation roosed by Li and Wang [19] e c ¼ e c k c ð= at Þ n ð11þ with e c ; k c and n constants. In Eq. (1), M is the critical state stress ratio obtained by 2c M ¼ gðh; cþm c ; gðh; cþ ¼ ð12þ ð1 þ cþð1 cþcos3h where h is an effective Lode angle whose value varies from to/3 defined by cos3h ¼ ffiffi 6 trn 3 and c ¼ M e =M c with M c and M e the triaxial comression and extension critical state stress ratios, resectively. The lastic strain rate tensor is given by _e ¼h_LiR; R ¼ Bn þ C n I 1 3 DI ð13þ where R determines the direction of lastic flow, B ¼ 1þ ffiffiffiffiffiffiffi 31 ð cþ= ð2cþgðh; cþcos3h; C ¼ 3 3=2 ð1 cþ=cgðh; cþ and D is the dilatancy defined by D ¼ A d d:n; A d ¼ A ð1 þhz:niþ ð14þ where A is a ositive constant and z is the so-called fabricdilatancy tensor whose evolution is governed by _z ¼c z _L hdiðz max n þ zþ ð15þ with c z and z max material constants. The above equation imlies that the evolution of z occurs during the dilative hase of deformation. In Eq. (14), d is given by d ¼ ffiffiffiffiffiffiffi 2=3 a d h n a; a d h ¼ M ex nd w m ð16þ where n d is a ositive material constant. 4 Numerical imlementation In this section, a fully imlicit return maing algorithm is develoed for the most recent lasticity model for sands by Dafalias and Manzari [9], in which the Newton Rahson scheme is emloyed to iteratively arrive at a solution satisfying all the constitutive equations at the local level. Furthermore, a consistent tangent oerator relevant to the roosed return maing algorithm is derived by exloiting the converged residual vector. In what follows, Voigt notation is used. Following this, a second-order tensor is reduced to a six-dimensional vector and a forth-order tensor is reduced to a matrix. 4.1 Imlicit return maing algorithm Assuming that the strain increment at time t nþ1 ; that is De; is given, the current strain can be directly comuted from e ¼ e n þ De; in which e n is the strain vector at time t n which is known, and the current void ratio can be obtained by integrating its evolution equation given in the revious section, i.e., e ¼ ð1 þ e n ÞexðDe v Þ1; with De v ¼ tr De the volumetric strain increment. It is noted that throughout this section, the subscrit n þ 1 is assumed for variables without subscrit. The objective is to find the stress vector r corresonding to the given strain increment De: Initially, i.e., at iteration i ¼ ; the material resonse is assumed to be elastic (De e; ¼ De), that is e e; ¼ e e;tr ¼ e e n þ De a ¼ a n ð17þ z ¼ z n DL ¼

5 Acta Geotechnica (215) 1: where e e;tr is the trial elastic strain. The trial stress is obtained by integrating the rate equation _r ¼ C e _e e using the Euler scheme r ; ¼ r ;tr ¼ r n þ Ce; De ð18þ where r n is the stress vector at time t n which is known, and C e; is the trial elasticity matrix, in which the trial bulk and shear moduli are evaluated from Eq. (6) as follows K ¼ 21þ ð tþ 31 ð 2tÞ G ; G ð2:97 eþ 2 1=2 ¼ G at ð1 þ eþ at ð19þ where is the trial mean effective stress obtained by integrating the volumetric art of the elastic relation given in Eq. (5)! ¼ n 1=2 ð1 þ tþ 31 ð 2tÞ G 1=2 ð2:97 eþ 2 2 at De v ð2þ ð1 þ eþ If the trial stress state is located inside or on the yield surface, that is f r ; ; a ¼ s a s a 1=2 ffiffiffiffiffiffiffi 2=3 m ð21þ elastic rediction is correct. If the trial stress state obtained using the elastic rediction is located outside the yield surface, that is f r ; ; a [ ; the trial state is corrected by simultaneously satisfying the following equations, which are generated by the strain increment De 8 >< r ¼ >: ¼ r e e r a r z r DL 9 8 e >= e e e;tr 9 þ DLR >< a a n DLð2=3Þhb >= ¼ z z n þ c z DLhDiðz max n þ zþ >; >: ½ðs aþðs aþš 1=2 ffiffiffiffiffiffiffi >; 2=3 m ð22þ These equations involve the additive decomosition of the strain rate tensor into elastic and lastic arts, i.e., _e ¼ _e e þ _e ; the evolution of the deviatoric back stressratio tensor a [Eq. (8)], the evolution of the fabricdilatancy tensor z [Eq. (15)], and the yield condition [Eq. (7)]. In order to solve the above system of equations, the Newton Rahson iterative scheme is imlemented to linearize the above system of equations. By exanding the local residual equations with the first-order truncated Taylor series, the following linear aroximation is obtained for the nonlinear system to be solved 8 >< >: r iþ1 e e r iþ1 a r iþ1 z r iþ1 DL 9 8 >= >< ¼ >; >: r i e e r i a r i z r i DL 9 >= 8 >< þ J i >; >: which can be rewritten as de e;iþ1 da iþ1 dz iþ1 ddl iþ1 9 >= ¼ >; ð23þ r iþ1 ¼ r i þ J i dx iþ1 ¼ ð24þ where X ¼ ðe e Þ T a T z T T DL denotes the vector of local unknowns, inferring that r ¼ rx ð Þ; and J i is the local Jacobian matrix at iteration i of time t nþ1 defined as J i ¼ or i =ox i ; that is 2 3 or i e e=oee;i or i e e=oai or i e e=ozi or i e e=odli J i or ¼ i a =oee;i or i a =oai or i a =ozi or i a =odli or i z =oee;i or i z =oai or i z =ozi or i z =odli 5 or i DL =oee;i or i DL =oai or i DL =ozi or i DL =odli ð25þ which imlies that the local Jacobian matrix is udated at each iteration. The comonents of the local Jacobian matrix are given in Aendix 2. By finding the solution of the linearized system of Eq. (24), that is the increment of the elastic strain vector, the deviatoric back stress-ratio vector, the fabric-dilatancy vector, and the loading index ðdx iþ1 ¼ J i 1r i Þ; the vector of local unknowns is udated using the following relation X iþ1 ¼ X i þ dx iþ1 and the corresonding stress is comuted from r ;iþ1 ¼ r n þ Ce;iþ1 De e;iþ1 ð26þ ð27þ where De e;iþ1 ¼ e e;iþ1 e e n ; and Ce;iþ1 is the elasticity matrix at iteration i? 1oftimet nþ1 ; which is udated using K iþ1 ¼ 21þ ð tþ 31 ð 2tÞ Giþ1 ; G iþ1 ð2:97 eþ 2 iþ1 1=2 ð28þ ¼ G at ð1 þ eþ at in which iþ1 is given by iþ1 ¼ n 1=2 ð1 þ tþ 31 ð 2tÞ G 1=2 ð2:97 eþ 2 at ð1 þ eþ De e;iþ1 v! 2 ð29þ where De e;iþ1 v ¼ trde e;iþ1 : Hence, for the given strain increment De; a sequence of linearized system of equations is solved at each Gauss integration oint until the iteration convergence is achieved. That is, the iterative rocess continues until the residual vector r iþ1 vanishes within the

6 88 Acta Geotechnica (215) 1:83 1 given tolerance, i.e., r iþ1 Tol: A summary of the return maing algorithm is given in Table Consistent tangent oerator It is well known that when a tangent oerator consistent with the integration method emloyed for the numerical imlementation of the elastolastic constitutive model is used in the global Newton Rahson algorithm to solve the system of equations resulting from the governing equations, the rate of convergence of the global iterative rocess is imroved. In the following, a consistent tangent oerator relevant to the return maing algorithm resented in the foregoing subsection is derived. The consistent tangent oerator at time t nþ1 is defined as C ¼ dr =de: In order to obtain a closed-form exression for the tangent oerator consistent with the roosed return maing algorithm, the chain rule is used C ¼ dr =de ¼ dr =de e de e =de ð3þ substituting dr =de e ¼ C e into the above relation yields C ¼ C e de e =de ð31þ To derive a closed-form exression for de e =de; the local residual vector at the converged state, r ¼ ; is differentiated with resect to e. It is noted that at the global level r ¼ rx; ð eþ; so its differentiation with resect to e gives dr=de ¼ or=oe þ or=ox dx=de ¼ ð32þ Table 1 Summary of the return maing algorithm for the Manzari Dafalias lasticity model 1. Begin elastic redictor hase in which we set e e; ¼ e e;tr ¼ e e n þ De a ¼ a n z ¼ z n DL ¼ : 2. Comute stress r ; ¼ r ;tr ¼ r n þ Ce; De. 3. Comute yield function f r ; ; a : If f r ; ; a ; exit. Otherwise, set iteration counter i = and continue. 4. Solve J i dx iþ1 ¼r i. 5. Udate the vector of local unknowns using X iþ1 ¼ X i þ dx iþ1 : 6. Comute stress r ;iþ1 ¼ r n þ Ce;iþ1 De e;iþ1 : 7. Evaluate residual vector r iþ1 : If r iþ1 Tol; exit. Otherwise, set i = i? 1 and go to ste 4. in which or=ox is the local Jacobian matrix J evaluated at the locally converged state. Rearranging the above relation yields dx=de ¼J 1 or=oe ð33þ from which a closed-form exression for de e =de is obtained as follows de e =de ¼ ½1 ŠdX=de ¼½1 ŠJ 1 or=oe ð34þ in which 1 is the identity matrix, is the zero matrix, ½1 ŠJ 1 is equivalent to a submatrix corresonding to the first 6 rows of the inverse of the local Jacobian matrix and or=oe ¼ ðor e e=oeþ T ðor a =oeþ T ðor z =oeþ T ðor DL =oeþ T Š T whose comonents are given in Aendix 3. Based on the above relation, a closed-form exression for the consistent tangent oerator is obtained as C ¼C e ½1 ŠJ 1 or=oe ð35þ 5 Liquefaction criterion In this section, a general, mathematical criterion for the onset of flow instability is derived. The criterion resented herein is on the basis of the ioneering work of Borja [5, 6], which is based uon the bifurcation theory. According to this theory, instabilities develo when the loss of equilibrium results in multile feasible solutions [11]. Hill s loss of uniqueness or stability in the infinitesimal deformations case can be exressed as ½ _r ŠŠ: ½½Š _e Š ¼ ð36þ where ½½Š _e Š ¼ _e _e is the jum in the strain rate tensor due to otentially dulicate solutions for the velocity field and ½ _r Š ¼ _r _r is the jum in the stress rate tensor. Hill s instability condition imlies that uniqueness is lost or instability occurs when the jum in the stress rate tensor vanishes, that is ½ _r Š ¼ : This means that the continuity of the stress rate tensor for a nonzero jum in the strain rate tensor results in the loss of uniqueness or stability. Using the definition of the total stress tensor and the constitutive equation _r ¼C:_e; the instability condition for the fully saturated case becomes C: ½½Š _e Ša ½ _ w ŠŠI ¼ ð37þ where C ¼ dr =de is the consistent tangent and ½ _ w Š is the jum in the rate of the ore fluid ressure. It has been shown in [4] that the consistent tangent can be used instead of the continuum elastolastic tangent C e for detecting the material instability. The instability is catured by the loss of uniqueness conditioned to the undrained bifurcation. To

7 Acta Geotechnica (215) 1: imose this constraint, the jum in the Darcy s velocity vector of the fluid hase is forced to vanish, i.e., ½ _w w Š ¼ : It is noted that the constraint of undrained bifurcation is not equivalent to the locally undrained condition, in which _w w ¼ : That is, this criterion is exected to detect instability not only in low ermeability orous media, in which no significant drainage occurs, but also when the rate of loading is so raid that excess ore water ressure develos before the ore water can drain away. In general, field-scale boundary value roblems, all otential instabilities, e.g. instabilities in the form of strain localization (shear band) or drained diffuse instabilities, which are induced by kinematic constraints different from that observed in instability, must be checked. Setting ½ _w w Š ¼ in the continuity equation of flow [Eq. (2)] yields the equivalent constraint for the instability a n þ n ½ _ K s K w ŠŠþa ½ _e v Š ¼ ð38þ w where ½ _e v Š is the jum in the volumetric strain rate. The loss of uniqueness [Eq. (37)] subject to the constraint obtained in Eq. (38) leads to C am ½½Š _e Š am T an K s þ n ¼ ð39þ K w ½ _ w Š where C is the consistent tangent oerator obtained in the revious section and m is the identity vector defined as m ¼ ½1 1 1 Š T : For a nontrivial solution to exist, the determinant of the matrix, which is referred to as the matrix L, must vanish. Hence, instability occurs when det L ¼ ð4þ The above condition imlies that instability is redicted as a function of the state of the material, rather than the material roerty. The criterion given above is general in the sense that it was derived without using any secific assumtion. Moreover, it can be alied to any elastolastic constitutive model, but it should be taken into account that the accuracy of the rediction is highly deendent on the constitutive model used to simulate the mechanical behavior of the material. In the resent work, the Manzari Dafalias lasticity model is utilized, which is caable of simulating the behavior of granular materials under monotonic and cyclic loading conditions. Moreover, the recision with which the constitutive model is integrated has a significant effect on the accuracy of the rediction. If the algorithm used for the numerical integration of the constitutive model involves numerical errors, the rediction of instability will be as oor as the rediction of material behavior. If the solid and fluid hases are incomressible, the condition given in Eq. (39) becomes C m ½½Š _e Š m T ¼ ð41þ ½ _ w Š where the last row of the matrix imlies the incomressible bifurcation, i.e., ½ _e v Š ¼ : This means that for the incomressible solid and fluid hases, the constraint of undrained bifurcation ð½ _w w Š ¼ Þ is equivalent to the constraint of incomressible bifurcation. The consequence of this constraint is that at the onset of instability, the material deforms in the isochoric manner. The condition for instability derived here is more general comared to that obtained by Andrade et al. [3], in which instability is attained when the hardening modulus reaches the critical hardening modulus, in the sense that it was derived under conditions of triaxial (axisymmetric) loading, and incomressible solid and fluid hases. In general, it is well known that the instability criteria based on the constitutive tangent oerator and critical hardening modulus are equivalent. 6 Numerical simulations In this section, a number of examles are resented to verify the imlicit return maing algorithm develoed for the Manzari Dafalias lasticity model and to illustrate the redictive caability of the roosed criterion under different loading conditions. 6.1 Monotonic undrained and drained triaxial tests on Toyoura sand In order to verify the roosed numerical algorithm, a series of triaxial comression tests conducted by Verdugo and Ishihara [28] on Toyoura sand samles are simulated. These tests have been considered by various researchers to verify their numerical algorithm [9, 12]. In these test series, samles are loaded monotonically under undrained and drained conditions after consolidated isotroically, and then unloaded. Numerical simulation of these exeriments is erformed using the material subroutine inside the finite element code as well as the finite element analysis. Material arameters of the Manzari Dafalias lasticity model used in the simulations for Toyoura sand are listed in Table 2. Figure 1a, b shows the numerical and exerimental results for a series of undrained triaxial comression tests conducted on isotroically consolidated samles of loose Toyoura sand (e =.97) with initial confining ressures ranging from = 1 to 2, kpa. It is observed that when sheared under undrained conditions, loose Toyoura

8 9 Acta Geotechnica (215) 1:83 1 Table 2 Material arameters of the Manzari Dafalias lasticity model Toyoura sand Elasticity t.5.5 G Critical state M c e c k c n.7.28 Yield surface m.1.1 Plastic modulus h c h n b Dilatancy A n d Fabric-dilatancy tensor c z 6 5 z max 4 1 Nevada silty sand sand samles exhibit contractive behavior together with softening. As exected for the undrained case and incomressible solid and fluid hases, void ratio remains constant during the simulation. Figure 1c, d shows the results for dense Toyoura sand samles with void ratio of e =.735 under undrained triaxial loading conditions. The initial confining ressure varies from = 1 to 3, kpa. As exected for dense sand, the material undergoes the socalled hase transformation, which is a transition from contractive behavior to dilative behavior without softening. Figure 1e, f shows undrained triaxial test results for the case where the void ratio is equal to e =.833, and the initial confining ressure varies from = 1 to 3, kpa. It is observed that for intermediate sand, deending on the initial confining ressure resonse of the same material varies from contractive resonse with softening to dilative resonse without softening, demonstrating the ressure-deendent behavior of sand. As exected, for undrained samles with the same void ratio, steady state is reached at the same oint. Steady state is the state at large strains where continued straining does not roduce further change in the stress state of the material. Figure 2a, b shows the numerical and exerimental results for a series of drained triaxial comression tests conducted on Toyoura sand samles with initial void ratios of e =.831,.917,.996. In all these cases, the initial confining ressure is equal to = 1 kpa. As can be seen, deending on the state of the sand reflected in the state arameter samles shows different resonses. In Fig. 2c, d, numerical results are indicated for a drained triaxial test series with a higher initial confining ressure of = 5 kpa and different initial void ratios of e =.81,.886,.96. As can be seen, a similar trend is observed as with the revious test series uon reversal of loading in the sense that samles show contractive resonse during unloading. Comressing samles under drained conditions causes the void ratio to change, unlike the undrained case in which the void ratio remains constant. For both undrained and drained triaxial test series, it is observed that the obtained numerical results are in good agreement with the exerimental ones, verifying our numerical imlementation and showing the caability of the adoted constitutive model in caturing the main features of sand behavior. In order to illustrate the caability of the resented criterion in caturing the onset of instability, the sequence of undrained triaxial comression tests conducted on loose Toyoura sand samles (e =.97) with different initial confining ressures is considered. The deviatoric stress-axial strain diagrams and the effective stress aths are lotted in Fig. 3a, b, resectively. According to the exerimental observations, instability is attained at the maximum deviatoric stress oint, marked by the circle symbol in Fig. 3a, b. It is observed that only samles with initial confining ressures of = 1, and 2, kpa liquefy, and the samle with a relatively low confinement never liquefies. Indeed, two different tyes of behavior are observed. Loose samles with high confinement dislay high contractive tendency, causing the ore water ressure to increase continuously. The continuous increase in the ore water ressure is evidenced by the continuously decreasing mean effective stress. In this case, the samles undergo contractive behavior and deviatoric stress increases until instability occurs at the eak oint. After reaching the eak oint, the behavior of the material becomes unstable in the sense that the deviatoric stress-axial strain diagrams and the effective stress aths exhibit a dro in the deviatoric stress. This unstable behavior continues until failure oint or steady-state oint is reached. At the failure oint, the state of stress remains constant and does not change with further shearing. In contrast, the loose samle with low confinement encounters the so-called hase transformation and does not liquefy. In other words, at low confinement, the samle dislays a reversal from contractive behavior to dilative behavior due to the low contractive tendency of sand at low confining ressures, directing the effective stress ath to the right toward the failure oint. In this case, the behavior of the material is always stable in the sense that the deviatoric

9 Acta Geotechnica (215) 1: (a) 1 (b) P=1 kpa (Simulation) P=1 kpa (Simulation) P=2 kpa (Simulation) P=1 kpa (Exeriment) P=1 kpa (Exeriment) P=2 kpa (Exeriment) Mean effective stress, (kpa) (c) 5 (d) P=1 kpa (Simulation) P=1 kpa (Simulation) P=2 kpa (Simulation) P=3 kpa (Simulation) P=1 kpa (Exeriment) P=1 kpa (Exeriment) P=2 kpa (Exeriment) P=3 kpa (Exeriment) Mean effective stress, (kpa) (e) 2 (f) P=1 kpa (Simulation) P=1 kpa (Simulation) P=2 kpa (Simulation) P=3 kpa (Simulation) P=1 kpa (Exeriment) P=1 kpa (Exeriment) P=2 kpa (Exeriment) P=3 kpa (Exeriment) Mean effective stress, (kpa) Fig. 1 Comarison between numerical and exerimental results for the undrained triaxial comression test on samles of Toyoura sand with void ratios of a, b e =.97, c, d e =.735 and e, f e =.833

10 92 Acta Geotechnica (215) 1:83 1 (a) 35 (b) e=.831 (Simulation) e=.917 (Simulation) e=.996 (Simulation) e=.831 (Exeriment) e=.917 (Exeriment) e=.996 (Exeriment) Void ratio, e (c) 16 (d) e=.81 (Simulation) e=.886 (Simulation) e=.96 (Simulation) e=.81 (Exeriment) e=.886 (Exeriment) e=.96 (Exeriment) Void ratio, e Fig. 2 Comarison between numerical and exerimental results for the drained triaxial comression test on samles of Toyoura sand with initial confining ressures of a, b = 1 kpa and c, d = 5 kpa stress-axial strain diagram and the effective stress ath do not indicate a dro in the deviatoric stress. This stable behavior continues until failure occurs. Figure 3c shows the evolution of the determinant of the matrix. The onset of instability is detected at the oint where the zero axis is crossed. As can be seen, for the samles with high initial confinement the determinant of the matrix obtained from Eq. (41) becomes equal to zero at the oint corresonding to the eak oint in the deviatoric stress-axial strain diagram and the effective stress ath. For the samle with the initial confining ressure of = 1 kpa, the determinant of the matrix is entirely negative and never reaches zero. That is, the criterion is never satisfied in this case, in accordance with the exerimental observations. These results clearly demonstrate the efficiency of the resented criterion in caturing the instability under monotonic loading conditions. To assess the erformance of the roosed imlicit integration technique along with the consistent tangent oerator, the local and global convergence rofiles obtained from the finite element analysis of undrained triaxial comression test on Toyoura sand samle with initial confining ressure of = 1, kpa and void ratio of e =.97 and drained triaxial comression test on Toyoura sand samle with initial confining ressure of = 5 kpa and void ratio of e =.96 are resented. It is observed from Fig. 4a, c that for the given local tolerance (in these examles 1-12 ), a few iterations are required to achieve the local convergence, and an asymtotic quadratic rate of convergence is obtained. As shown in Fig. 4b, d, using the consistent tangent oerator, the

11 Acta Geotechnica (215) 1: (a) P=1 kpa P=1 kpa P=2 kpa (b) Mean effective stress, (kpa) (c) 3E+14 2E+14 1E+14 det L -1E+14 3E+12 det L -2E+14-3E E Fig. 3 Numerical results of the undrained triaxial comression test on samles of loose Toyoura sand with void ratio of e =.97 global residual vector vanishes within the given tolerance (in these examles 1-13 ) after a few iterations. Moreover, the asymtotic quadratic convergence rate of the global Newton Rahson iterative algorithm is reserved, demonstrating the comutational efficiency of the roosed return maing algorithm along with the consistent tangent oerator in the solution of boundary value roblems using finite elements. It is worth mentioning that several monotonic undrained and drained triaxial tests were simulated using a broad range of increments to evaluate the erformance of imlicit and exlicit integration schemes. It was observed that when very large increments are used, the imlicit algorithm still rovides solutions with reasonable accuracy, while the exlicit algorithm fails to converge from the beginning when the increment size exceeds a certain intermediate value, indicating the accuracy of the imlicit algorithm in a relatively wide range of increments. It was also observed that for the same increment size, the exlicit algorithm requires less CPU time than the imlicit algorithm. This is exected because of the heavy comutational load required for the imlicit algorithm. However, regarding that the exlicit algorithm requires smaller increments comared to the imlicit algorithm, the required CPU time of the two algorithms is comarable. 6.2 Cyclic undrained triaxial test on Toyoura sand In this examle, a comarison is made with the results reorted in [9] for the cyclic undrained triaxial test conducted on a Toyoura sand samle with void ratio of e =.88. This examle in fact illustrates the difference between instability and cyclic mobility. First, the samle is consolidated isotroically to the confining ressure of = 294 kpa (oint A in Fig. 5a). Then, it is loaded cyclically under undrained conditions with the shear stress difference of kpa. Fig. 5a shows the resulting effective stress ath, in which the well-known butterfly

12 94 Acta Geotechnica (215) 1:83 1 (a) L2 norm of local residual (b) L2 norm of global residual Iteration number Iteration number (c) L2 norm of local residual (d) L2 norm of global residual Iteration number Iteration number Fig. 4 Local and global convergence results at different axial strains for a, b undrained triaxial comression test on Toyoura sand samle with initial confining ressure of = 1, kpa and void ratio of e =.97 and c, d drained triaxial comression test on Toyoura sand samle with initial confining ressure of = 5 kpa and void ratio of e =.96 shae is formed. It is observed that there is a good agreement between the numerical results of ours and those reorted in [9], verifying our numerical algorithm for cyclic loading. It can be seen from the undrained effective stress ath deicted in Fig. 5a that the sand behavior is always stable in the sense that the sand can sustain the constant deviatoric stress difference during the whole loading until the occurrence of failure. Consequently, the determinant of the matrix obtained from Eq. (41) is always negative and never crosses the zero axis. That is, the criterion is never satisfied, as shown in Fig. 5c. In fact, instability never aears in this case although there is a decrease in the mean effective stress. This behavior is a tyical behavior observed in exeriments where cyclic mobility occurs, in which ore water ressure increases gradually with cyclic loading without exhibiting noticeable sudden changes in ore water ressure, and loss of controllability or instability is never observed [26]. This behavior contrasts with the characteristic behavior of flow, which is associated with a sudden ore water ressure build-u and an occurrence of instability. In exeriments exhibiting, the samle dislays an unstable behavior and failure follows, while in exeriments with cyclic mobility the behavior is entirely stable until failure is reached. This examle clearly shows that the roosed criterion can distinguish between instability and cyclic mobility. Figure 5d shows the local convergence results of the roosed imlicit integration scheme at several oints, i.e., the loading reversal oints

13 Acta Geotechnica (215) 1: (a) 6 4 H F D Simulation Ref. [14] B (b) 6 4 Shear stress, q/2 (kpa) 2 I A Shear stress, q/2 (kpa) G E C Mean effective stress, (kpa) -6 det L (c) -1E+15-2E (d) L2 norm of local residual Point B C D E F G H I -3E Load ste Iteration number Fig. 5 Numerical results of the cyclic undrained triaxial test on Toyoura sand samle with void ratio of e =.88, initial confining ressure of = 294 kpa and shear stress difference of kpa for the case where the fabric-dilatancy variable is considered (z max = 4) and a stress oint very close to the origin. As can be seen, a few iterations are required to achieve the local convergence within the given tolerance (in this examle 1-12 ). It is imortant to note that in the very low mean effective stress region, which is challenging from a numerical oint of view, the resented imlicit method is stable, and even at stress oint I, the rate of convergence is asymtotically quadratic. This examle was also simulated using the fully exlicit method to test its stability. It was observed that the fully exlicit method is unstable, and reducing the increment size does not revent numerical instability. 6.3 Cyclic undrained triaxial test on Nevada silty sand In this examle, the cyclic undrained triaxial test conducted by Yamamuro and Covert [29] on Nevada silty sand is simulated. The material arameters of the Manzari Dafalias lasticity model for the Nevada silty sand used in this numerical simulation are summarized in Table 2. Initially, the secimen is consolidated isotroically until the effective confining ressure of = 225 kpa is reached (oint A in Fig. 6a). After isotroic consolidation, the secimen is sheared under drained conditions from the isotroic confining ressure of = 225 kpa to the confining ressure of = 25 kpa, which corresonds to the deviatoric stress of q = 75 kpa, i.e., from oint A to oint B. Along this ortion, the effective stress ath has a sloe of 3, as exected for the drained triaxial comression test. Then, the secimen is turned undrained, and the cyclic undrained triaxial test is erformed. During the cyclic loading, the mean effective stress decreases until the effective stress ath reaches the oint marked by the red

14 96 Acta Geotechnica (215) 1:83 1 (a) C CSL IL Simulation Exeriment B det L (b) -1E+15-2E E+13 det L A Mean effective stress, (kpa) -3E E Load ste Load ste Fig. 6 Numerical results of the cyclic undrained triaxial test on Nevada silty sand samle with initial confining ressure of = 225 kpa circle in Fig. 6a. At this oint, instability occurs. After reaching this oint, the constant deviatoric stress difference cannot be maintained, and the behavior of the material becomes unstable in the sense that the material cannot resist the constant deviatoric stress difference. Corresondingly, the determinant of the matrix obtained from Eq. (41) becomes equal to zero at the oint where instability occurs as shown in Fig. 6b, demonstrating the caability of the criterion resented herein in detecting the onset of instability under cyclic loading conditions. These results in fact show that the roosed criterion makes no distinction between monotonic and cyclic loading, imlying that the instability is a function of the state of the material, rather than the state of loading. Figure 6a also shows the CSL or the failure line as well as the instability line, which is the locus of oints at which instability is initiated under monotonic undrained triaxial loading conditions. The instability line is built by joining the eak oints derived from a series of monotonic undrained triaxial comression tests with the same void ratio and different initial confining ressures. In this figure, the instability line is lotted for the case where the void ratio is the same as that of the cyclic undrained triaxial test. It is observed that the erformed numerical simulation satisfactorily reroduces the exerimental observations reorted in [29]. This examle involves a fully saturated samle loaded dynamically under lane strain conditions. The geometry and boundary conditions of the roblem are dislayed in Fig. 7. As shown in this figure, the side length of the square samle is 1 m. The to boundary of the samle is loaded by a uniform vertical dislacement that varies with time as u y ¼ :75 b5tc=2þ1 t sinð5tþm; where the symbol bc denotes the greatest integer function. The side boundaries are subjected to a constant uniform confining ressure of 1,5 kpa. Undrained boundary condition is imosed on all boundaries of the samle. That is, the samle has imermeable boundaries to water. Prior to the dynamic loading, the samle is loaded monotonically under drained conditions from the isotroic confining ressure of = 1,5 kpa to the state of stress marked by oint A in Fig. 8a. The initial void ratio before the monotonic loading is e =.99. The comuted effective stress and lastic internal variables at the end of the monotonic loading serve 15 kpa u y 15 kpa 6.4 Cyclic undrained lane strain test on Toyoura sand In order to exemlify the alicability of the roosed criterion to initial boundary value roblems under dynamic loading conditions, an examle is solved emloying a fully couled dynamic finite element analysis. Fig m Geometry and boundary conditions of the lane strain test

15 Acta Geotechnica (215) 1: (a) 7 (b) CSL IL A A B Mean effective stress, (kpa) B (c) B Time (s) (d) Pore water ressure, Pw (kpa) Time (s) B det L (e) -1E (f) L2 norm of local residual Time (s) E Time (s) Iteration number Fig. 8 Finite element results of the fully saturated samle loaded dynamically under lane strain conditions

16 98 Acta Geotechnica (215) 1:83 1 Table 3 Material roerties for the finite element analysis Biot s constant a = 1 Solid hase density q s ¼ 2; kg=m 3 Water density q w ¼ 1; kg=m 3 Bulk modulus of solid hase K s ¼ : Pa Bulk modulus of water K w ¼ :2 1 1 Pa Intrinsic ermeability k ¼ m 2 Dynamic viscosity of water l w ¼ Pa s as initial values to erform the dynamic analysis under cyclic loading. Indeed, the initial conditions for the dynamic analysis are obtained from a reliminary static drained analysis to ensure the satisfaction of mass and linear momentum balance equations at time t =. The adoted model arameters are assumed to be similar to those given in Table 2 for Toyoura sand. Material roerties of the fully saturated soil considered in this examle are listed in Table 3. The domain is discretized with quadrilateral elements. The dislacement field is aroximated by biquadratic shae functions, and bilinear shae functions are used for the aroximation of the water ressure field. Figure 8a shows the effective stress ath, surrounded by an enveloe obtained from a monotonic undrained triaxial comression test for the same void ratio as in this examle. It is noted that in this examle, heterogeneity effects are negligible, and the material resonse is almost identical at all Gauss oints. Therefore, the selected Gauss oint is reresentative of the entire domain. It is observed that the samle reaches instability very close to the instability line obtained from a series of monotonic undrained triaxial comression tests erformed on samles with the same void ratio as in this cyclic test. However, neither the instability line nor the enveloe serves as redictor for. Figure 8d shows the evolution of the ore water ressure. As can be seen, during the first cycle the magnitude of the ore water ressure is not significant, but in the next cycle the ore water ressure increases suddenly. The sudden increase in the ore water ressure causes the mean effective stress to decrease remarkably, eventually resulting in the instability. Accordingly, in the last cycle, the axial strain develos much more than the revious ones, as shown in Fig. 8b. Moreover, it maintains a nearly constant deviatoric stress until instability takes lace. The develoment of the ore water ressure and the subsequent dro in the mean effective stress continues before the onset of instability. Accordingly, the determinant of the matrix obtained from Eq. (39) becomes equal to zero at the oint where the samle dislays instability, as observed from Fig. 8e. This examle clearly indicates that instability occurs when the aforementioned criterion is met, roving the caability of the roosed criterion to cature instability in boundary value roblems under dynamic loading conditions. Figure 8f shows the local convergence results of the roosed return maing algorithm at several times. It is observed that a few iterations are required to achieve the local convergence, and an asymtotic quadratic convergence rate is achieved for the given tolerance (in this examle 1-12 ). 7 Conclusions In this aer, a general criterion based on Hill s loss of uniqueness conditioned to the undrained bifurcation was resented for detecting the onset of instability in fully saturated granular media. The Manzari Dafalias lasticity model based on its recent modifications was emloyed to simulate the nonlinear behavior of granular materials. Furthermore, a fully imlicit return maing algorithm was develoed for the numerical imlementation of the Manzari Dafalias lasticity model. It was shown that the local rate of convergence of the roosed imlicit integration scheme is asymtotically quadratic. Moreover, in the finite element context when the consistent tangent oerator is used in the solution of the governing system of equations, the erformance of the imlicit method is enhanced. That is, the asymtotic quadratic convergence rate of the global Newton Rahson iterative rocess is reserved. It was observed that the resented imlicit method is more accurate and stable than the exlicit method. Relatively large increments deteriorate the numerical stability of the exlicit method, while the imlicit method roduces accurate and stable solutions with relatively large increments. Moreover, the imlicit method is able to rovide converged solution in the very low mean effective stress region, while the exlicit method fails to converge at low mean effective stress levels, and even a significant decrease in increment size does not ensure convergence. The redictive caability of the criterion resented herein was illustrated in several numerical simulations under both monotonic and cyclic loading conditions. It was shown that the roosed criterion is caable of caturing the onset of instability under static