A Simple One-Scale Constitutive Model for Static Liquefaction of Sand-Silt Mixtures

Instability of liquefaction is one of the major reasons which results in the failure of earth structure such as dam. The present study focuses on the simulation of static liquefaction behavior for granular materials such as sand and sand-silt mixtures. Based on microme-chanical analysis of inter-particle behavior, a simple one-scale model is proposed to simulate the stress-strain response of sand; then the proposed model is extended to simulate the sand-silt mixtures using the mixture theory combining the properties of sand and silt according to their proportions. Empirical expressions are introduced to fit the critical state strength and the location of the critical state line for each mixture. Parameters of the model can be divided into two categories: the first seven parameters have the same values either with pure sand or pure silt for silt-sand with any given fines content; the other three parameters are the function of fines content and three more parameters are required to estimate their values. The predicted results of triaxial test of sand and sand-silt mixtures with different fine content, which has a good agreement with the results of laboratory tests, suggest that the proposed model can simulate static liquefaction behavior of sand and sand-silt mixtures.


INTRODUCTION
Instability of liquefaction is one of the major reasons which results in the failure of earth structures such as dam."Liquefaction" is used for the first time by Hazen (1920), describing the failure of the Calaveras dam's failure.Terzaghi (1925) defined the essential processes of liquefaction and the subsequent description by Casagrande's (1936Casagrande's ( , 1965) ) together formed the basis for research of liquefaction at that time.Casagrande defined the critical void ratio (CVR) concept in his early paper.Seed and Lee (1966) took the pore pressure value as the basis for analysis of sand liquefaction, proposing the concept of "initial liquefaction".Casagrande and Castro (1975)  void ratio and proposed the concept "steady state strength", which is used to estimate sand liquefaction failure.Later several concepts such as "steady state deformation", "steady state line", "flow structure" were proposed to describe the liquefied deformation mechanism of saturated granular mixture by several researchers (Poulos,1981(Poulos, ,1985;;Castro,1992;Ishihara,1993;Bazier and Dobry,1995) .
The concept of liquefaction covers the static liquefaction caused by static loading and the cyclic liquefaction resulted from mechanical vibration such as earthquake, explosion and dynamic loading.Conceptually, when the deviatoric stress-strain curve appears obvious strain-softening phenomenon during monotonic loading and the deviatoric stress falls to zero after the peak value, the characteristics of saturated granular mixture appears like fluid, which is defined as "static liquefaction".This kind of liquefaction occurs without dynamic load, and is defined as static liquefaction distinguished from the liquefaction caused by vibration.
One important aspect of liquefaction study is to predict the stress-strain relationship of granular materials that is susceptible to liquefaction.The models for stress-strain behavior of granular materials (for example sand or sand-silt mixtures) can be generally categorized into two approaches: the conventional plasticity approach and the micromechanics based approach.The stress-strain models based on traditional plasticity is a one-scale approach, which can be found in the works of many researchers (Desai and Siriwardane, 1984;Wood, 1990;Prevost, 1985;Dafalias and Herrmann, 1982;Klisinski, 1988;Mroz, 2003).
Usually, the multi-scale model is more complex than the conventional one-scale model.The added complexity improves the realism of the model in some degree, however, this complexity also make it not easy to be used in real boundary value problem.A practical constitutive model is not only reasonable for describing the static liquefaction behavior of granular materials, but also should intend to be in a simple form with fewer parameters easy to be determined.Along this line, the present study is aimed to develop a simple but effective one-scale constitutive model based on some micromechanical analysis, to simulate the static liquefaction behavior of granular materials such as sand.
Field observations revealed that sand is often deposited in either still or running waters environment.If fine particles appear in deposits, a sand-silt mixture may form under certain deposition Latin American Journal of Solids and Structures 13 (2016) 2190-2218 environment.It is well know that fines content has a significant influence on the microstructure and behavior of soil.Terzaghi (1956) found the presence of fine particles increase the possibility of forming metastable structures.The grain-to-grain fabric of the sandy silt is responsible for their noplastic and noncohensive characteristics, which exert a considerable influence and make them susceptible to liquefaction.Consideration of this natural trend and questions regarding silt influence on engineering behavior of sandy soils has triggered the research on silty sands in recent years.Many investigators (Kuerbis et al. ,1998;Pitmanet al.,1994;Lade and Yamamuro,1997;Thevanayagam, 1998;Thevanayagam and Mohan ,2000;Salgado et al.,2000;Thevanayagam et al. ,2002;Ni et al., 2004;Murthy et al., 2007;Monkul, et al., 2011;and Dash, et al, 2011) have conducted experiment on sand with amount of silt and studied their the stress-strain behavior.
So it is important to consider the effect of fine content on the liquefaction of sand.In this study we adopt a simple but effective way to study this effect: the behavior of silt-sand mixtures is modeled using a sort of mixture theory combining the properties of the two soils according to their proportions.In particular, empirical expressions are introduced to fit the critical state strength and the location of the critical state line for each mixture.
In the following sections, we firstly introduce a simple one-scale model based on micromechanical analysis of inter-particle behavior.Then we extend the model to model sand-silt mixtures using mixture theory.Finally the predicted results from the proposed model are compared with three sets of data of sand and sand-silt mixtures to evaluate its performances in simulating stress-strain relationship of granular materials.

A SIMPLE ONE-SCALE MODEL FOR STATIC LIQUEFACTION OF GRANUALAR MATERIALS
In this section, based on micromechanical analysis of inter-particle behavior, a simple one-scale model for static liquefaction of sand is proposed.The general numerical results are presented and the response envelope and two-order work predicted from the proposed model is also discussed in simulating the liquefaction instability phenomenon of sand.

Inter-Particle Behavior of Granule
In granular assembly, particles contact each other; the orientation of a contact plane between two particles is defined by the vector perpendicular to this plane.On each contact plane, an auxiliary local coordinate can be established as shown in Fig. 1.The contact stiffness of a contact plane includes normal stiffness, n k  , and shear stiffness, r k  .The elastic stiffness tensor is defined by, Where, n , s , t are three orthogonal unit vectors that form the local coordinate system.In general, n k  and r k  is the normal and tangential elastic stiffness on contact plane, which is supposed to follow a revised Hertz-Mindlin's contact law (Chang et al., 1989): Where, n f  is the contact force in normal direction, and g G is the elastic modulus of ideal grains.
l is the branch length connecting the neighboring two particles.Three material constants and n are needed as input.
The movements of particles at contact plane often result in a dilation/contraction behavior, can be expressed as follows: Where d is the dilatancy parameter.r f  is the contact force in tangent direction.The value of 0  in Eq. 4 is assumed to be equal to the critical state friction angle (i.e.0 cs    ).
A Mohr-Coulomb type yield function, defined in a contact-force space, is assumed to be as follows, Where r f  is the resultant shear force and P r  is the resultant plastic sliding.( )   is a hardening function defined by a hyperbolic curve in p r    plane (Chang and Hicher, 2005).Two material parameters, p  and 0 p r k , are involved in the hardening function: Where, 0 p r k is related to the normal stiffness n k  by a constant p  :

A Simple One-Scale Stress-Strain Model for Granular Materials
Based on the inter-particle behavior described above, a two-scale micromechanical model may be formed as Chang (Chang, et.al 2005) through an integration process over the behavior for all contacts.In the micromechanics approach, the continuum mechanical concept "infinitesimal volume element" (IVE) is treated as a discrete mechanical concept "representative volume element' (RVE) that embodies an assembly of particle.Then the global stress-train behavior of the representative volume element can be obtained based on the behavior at local scale (i.e. the inter-particle level).However, if we assumed mechanical properties of assembly at all directions are the same as the defined plan as described in 2.1., the assembly (global) can be regarded as homogenous materials and the inter-particle behaviors is now defined for a "material point", which represents an "infinitesimal volume element" of the continuum material.Besides, a density state variable (defined as a function of the critical state void ratio) was introduced for plastic flow, which was postulated to be dependent on this variable.In this way, a simple one-scale model can be developed based on interparticle behaviors.At the inter-particle (local) level, mechanical variables are contact forces n f , r f and displace- ments n  , r  .The constitutive laws, such as yielding function, shear dilation and hardening rules, are defined at the local level using these inter-particle mechanical variables on each contact plane using Eq.(4)-Eq.( 7).At the assembly (global) level, the mechanical variables are stresses p , q and strains v  , r  , which can be obtained by integrating the corresponding inter-particle variables (forces n f , r f and displacements n  , r  ) in all orientations under isotropic condition.
Since the overall (global) physical behavior is manifested by the local physical behavior, it seems reasonable to assume that the constitutive laws at the global level takes the same form as those at the local level under the assumption that all.Thus in the one-scale model, the elastic modulus, the yielding function, the shear dilation and the hardening rules can be up scaled by replacing the local variables n f , r f , p n  and p r  by the global variables, p , q , p v  and p r  .The details are given below.

Elastic Behavior
The elastic behavior of the one-scale model can be expressed by the classical elastic theory as follows: Latin American Journal of Solids and Structures 13 (2016) 2190-2218 Where, p  is the mean stress increment, , where 2 J is the second invariant of deviator stress tensor) ; e r   is the elastic part of r   , r   is the shear strain increment ( , where 2 J  is second invariant of deviator strain tensor).G is the shear modulus and B is the bulk modulus, The bulk modulus B is considered to be pressure dependent, expressed as follows: Where 0 B and n are two material constants: 0 B is the reference bulk modulus, and n is a constant exponent.ref p is the reference pressure, which is taken to be 1 atm.

Plastic Behavior
As discussed above, the yielding condition, shear dilation and hardening rules can be described directly in the one-scale model using the macro-mechanical variables, p , q and p v  , p r  .So, accord- ing to Eq. ( 4) ~Eq.( 7), the dilatancy equation, the yielding condition and harden rules of the onescale model can be written as follows, Where, where u M is the slope of critical state line in p q  space, Where patm is the atmospheric pressure.Equation 15 involves three material constants: ref e (zero intercept),  (CSL slope), and  (CSL curvature).
The plastic stiffness p G , similar to the plastic stiffness of a contact plane, is assumed to relate to elastic stiffness of the material.It is now assumed to be related to B by a constant  : The initial slope of the hyperbolic curve (Eq.13) is We do not have the explicit form of potential function, but we have the gradients of plasticity potential function derived from dilatancy equation as follows: One can see that a non-associative flow rule is adopted in the proposed one-scale model.

Stress and Strain Relationship in p q  Space
Based on the classical plastic theory, total strain increments can be divided into two parts, i.e., elastic part and plastic part, as follows: The elastic part can be calculated by Eqs. ( 8) and ( 9) based on elastic theory, and the plastic part can be calculated based on flow rules of plastic theory, Where g is plastic potential function and d is plastic factor, which can be obtained by con- sistency condition, 0 df  .In p q  stress space the expression of d can be written as follows: Then the plastic increment strain component p p d and p q d can be calculated according to Eq.19, Latin American Journal of Solids and Structures 13 (2016) 2190-2218 Combining Eq.18, the elasto-plastic stress-strain relationship of the proposed model in p q  stress space can be obtained as follows: Thus a simple stress-strain relationship for granular materials is established and isotropic hardening incorporating one scalar internal variable is adopted in the model.The upscaled response obtained from the two-scale micromechanical model (Chang et al, 2005) is kinematic hardening, as the different plastic loading histories on contacts with different orientations are likely to impart a directional character on the macroscopic response of the grain assembly.However, one can see from the predicted response envelope and two-order work that the one-scale model is simple in from but reasonable in simulating the liquefaction instability of granular materials.At the same time, no complex integrations procedure are needed as micromechanical model does, which is a time consuming process in simulating real boundary problem.

Summary of Parameters of the Proposed Model
Ten model parameters are included in the proposed one-scale model, i.e., the global elastic constants 0   Numerical results indicate that the proposed model has ability to capture the main features of granular materials behavior.For example, different initial void ratios lead to contracting or dilating behaviors of the sand and deviatoric stress-strain curve appears obvious strain-softening phenomenon during monotonic loading and the deviatoric stress falls to zero after the peak value under undrained condition.Steady state line (SSL) and instability line (IL) defined for static liquefaction can also be identified on the undrained stress path.

Response Envelope Predicted from the Proposed Model
Response envelope is a useful tool for validating constitutive equations (Doanh, 2000;Kolymbas, 2000;Tamagnini, 2006;Sibille, 2011).The original concept of response-envelopes was presented Lewin & Burland (1970) and Gudehus (1979) in context with the development of constitutive equations.In general, to obtain a response-envelope, a soil element is subjected to a certain stress-or strain-increment.The corresponding "response" of the soil in form of either strain or stress is determined and described graphically.The direction of the applied stress-or strain increment with a constant absolute value is then varied and leads to different stress-or strain responses, endpoints of which are connected to a response envelope.The strain response envelopes predicted from the proposed model was analyzed in the present work.After an initial isotropic compression with confining pressure 200kPa, a drained triaxial loading test was simulated in axisymmetric conditions.Stress probe test is performed at 4 stress points, i.e. points A, B, C and D as shown in Fig. 3 ) is an initially isotropic stress state and the other three points are initially anisotropic stress states.(B :   et. al (2005), near the plas- tic limit condition, the strain response envelope shrinks into a straight line.This straight line indicates that, at the plastic limiting condition, the direction of the incremental strain vector is independent of the direction of the incremental stress vector.Fig. 5 shows the rose diagrams for four different levels of shear stress (q/p = 0, 0.75, 0.95, 1,2).The angles shown in Fig. 5 are the stress probing directions (see Fig. 3).Note that the direction of 210 degree is parallel to hydrostatic axis (reduction of mean stress), and the direction of 240 degree is about parallel to Coulomb friction line.Fig. 5 shows that the instabilities occur for probe directions between these two lines.Material instability is a key to understand the static liquefaction behavior of granular materials such as sand and silty sand (Lade, 1992;Yamamuro and Lade, 1998;.Nicot and Darve, 2006).Obviously, the proposed model can capture this kind of instability phenomena and be used to analyze the liquefaction of granular materials.

EXTEND THE MODEL TO SIMULATE THE BEHAVIOR OF SAND-SILT MIXTURES
Theoretically, the model described above cannot apply to sand-silt mixtures directly, but we can extend the model to model sand-silt mixtures using the mixture theory combining the properties of sand and silt according to their proportions (Thevanayagam, et al 2002;Chang and Meidani, 2013).Some empirical expressions are adopted here to fit the critical state strength and the location of the Latin American Journal of Solids and Structures 13 (2016) 2190-2218 critical state line for each mixture.In this simply way, we extend the proposed model to model liquefaction behaviors of sand-silt mixtures.

Dominant Grains Network in Sand-Silt Mixtures
It is well known that fines content has a significant influence on the microstructure and behavior of soil.Thevanayagam et al. (2002) suggested three kinds of packing structure of mixture depending on the different state of coarse grains and fine grains existed in mixtures.Dominant grains network is the grains network (either coarse grains network or fine grains network) which control the main behavior of the mixtures.Usually, a coarse grains network is formed and controlled the behavior of the mixtures if fine grains in the soil are less than 25%.On the contrary, a fine grains network is formed and controlled the behavior of the mixtures if fine grains in the soil are higher than 35%.For intermediate cases, in which the controlled grains are neither coarse nor fine grain, are not included in present study.

Void Ratio Characteristics
It is well known that fines content in soil has a considerable control on its packing structure, for that reason, a proper index to connect the pore state of soil should be supposed to take account of both void ratio and the content of fine particles.Such index as inter-granular ratio and inter-fine void ratio are proposed by some researchers (Kuerbis et al., 1989;Mitchell, 1993;Vaid , 1994;Thevanayagam ,1998Thevanayagam , ,2000Thevanayagam , ,2002) ) for sand-silt mixtures.Chang (2011) suggested a formula to calculate the initial void ratio of soil with fine content less than 25%.According to them, the void ratio of the soil mixtures has been given as:   For the case of a fine grains skeleton (for example, fc>35%), a similar relationship is proposed here to calculate the void ratio of the mixtures with fines content greater than 35%: (1 ) Where, esilt is the void ratio of the 100% silt sample.If b=0, Eq. 25 reduces to silt c e e f  , which has the same expression with that for the inter-fine void ratio.(Thevanayagam and Mohan,2000) .If e, esand, and esilt in Eqs.24 and 25 are replaced by emin, (esand)min, and (esilt)min, the above expressions can also be used to calculate the minimum void ratio of sand-silt mixtures: Two parameters "a" and "b" involved in Eqs. ( 24)-( 27), which values depend on the size and shape of particles, will also be used later to calculate the "critical void ratio" for mixtures with different fine content.can also be adopted to predict the critical state critical void ratio of the mixtures with different fine content; However, many test results indicates the critical state line is not unique for the mixtures with different fines content (Salgado et al., 2000, Murthy et al., 2007, Fourie and Papageorgiou 2001, Thevanayagam et al. 2002): For the mixtures with low fines contents, the critical state lines have the same slope and curvature but shifting downward as the amount of fines increase; On the other hand, for the mixtures with high fines contents, the critical state lines shift upward as the amount of fines increase.
We chose two sets of test data (Yang, 2004 andKonishi et al., 2007) to study the above trend in sand-silt mixture.The effect of fines content on three parameters ref e ,  , and  , were also ana- lyzed.Figure 6 is the critical state data obtained in their tests compared with the predicted results by fitting Eq. 15.
The fitting results are interesting: parameter  is found to be almost the same for the whole range of fines content; however, parameter  can be regarded as two different values, one for low fines contents (marked in green solid line) and another for high fines contents (marked in red dotted line).Parameter eref varies with the change of fines content, has the same characteristics as that for void ratio of silt-sand mixtures: The predicted results of eref from Eqs. 28 and 29 are given in Fig. 7.

Critical State Friction Angle
Many test results indicate that the critical state friction angles cs  usually increases with the amount of fine grains in mixture.Murthy et al. (2007) explained these phenomena from the point view of'flow' fabric developing at critical state.They believed that the existence of fine particles wedging between the coarse grains can contribute to the critical strength of soil and thus to the value of critical state friction angle.
From the point view of micromechanics, the critical state friction angles is origin from frictions between particles and the force chain of different packing structures.Mixtures with different fine content have different packing structure and fabric.For mixtures with very low fines contents (for example, lower than 10%), or mixtures with very high fines contents (for example, higher than 70%), the critical state friction angles are experimentally found to be almost not changed with the variation of fine content beyond the threshold content( ie.,10% or 70%).For fines contents between 10%-70%, Chang and Meidani (2013) proposed a simple model to express the effect of fines content on the critical state friction angles of mixtures as follows: where  is a constant needed to input.    Reuss (1929) suggested that the average contact stiffness of mixtures can be described as follows: Where, (kn)equ is the average contact stiffness of the mixtures, (kn)sand is the contact stiffness for pure sand and (kn)silt is the contact stiffness for pure silt.
Similarly, we assume reference bulk modulus 0 B (in Eq.10) is a function of fines content, the follow- ing equation similar to Eq. ( 31) is chosen to predict the mean bulk modulus of a mixtures comprised of sand and silt particles,

Extend the One-Scale Model to Simulate Sand-Silt Mixtures
As discussed above, based on mixture theory, the behavior of sand-silt mixtures is dominated by the sand network for low fines content, and by the silt network for high fines content.According to the different dominant grain network, the 10 parameters of the proposed one-scale model can be divided into two categories: the first seven parameters ( i.e.  ,  , v , n ,  , D , and m ) for sand- silt miture with any given fines content are assumed to be the same values of those for pure sand for sand-silt mixtures with 0% < fc < 25%; and have the same value as those for pure silt for sandsilt mixtures with 35% < fc < 100%.So for sand-silt mixtures with any fine content, these seven parameters are the same either with pure sand or pure silt.
For the other three parameters (i.e.0 B , cs  26)-( 27) and ( 30) to triaxial tests data for mixtures with different fines content.
In this simple way, we can extend the proposed one-scale model to be suitable for sand-silt mixtures.Only three more parameters (a, b and) are added to the model.

Calibration of the Model Parameters
This section explains how the model parameters are obtained for the three sets laboratory measurements (See section 4) used for comparison of the predicted results from the proposed model with sand and sand-silt mixtures test.We take the Japanese silica sand test result (Konishi et al. 2007) as an example to illustrate the calibration procedure.First, the critical state parameters are determined.Then, the elastic and plastic parameters are determined from experimental test data, and at last three parameters describing the sand-silt mixture are determined.Structures 13 (2016) 2190-2218

Critical State Parameters
Three critical state parameters, namely, ref e (intercept),  (CSL slope), and  (CSL curvature) are required to define the critical state void ratio.To do so, drained or undrained triaxial test are needed to attain the critical state line (CSL).By fitting Eq. ( 15) to the critical state data, the critical void ratio parameters ref e ,  and  are determined.Fig. 8 shows CLS of silica sand (Konishi et al. 2007), by fitting the CLS line, we get ref e = 0.68,  =0.016, and  = 0.82.(Konishi et al., 2007).
The critical state friction angle cs  (noting it also the plastic parameter) can be obtained by measuring the slope of critical state line, M, in p'-q space.The equation 6sin /(3 sin ) describes the relationship between critical state friction angle and M-line slope.For Japanese silica sand (Konishi et al. 2007) cs  = 30.96,the corresponding M is 1.243.

Elastic Parameters
The isotropic compression test is used to determine the elastic parameters B0 and n of the one-scale model.Fig. 9 shows the isotropic compression lines (ICL) of sand (Konishi et al. 2007).By fitting the ICL line, parameters B0 and n can be determined.For this sand, we obtain B0 = 6.53 and n = 0.8.The parameters for the proposed model fit well the experimental curve.Another elastic parameter for the one-scale model is Poisson's ratio  , which can be estimated from the axial and volumetric strains measured in a triaxial compression test, as discussed by many researchers.For Japanese silica sand Poisson's ratio can be taken as 0.15.

Plastic Parameters
After the 3 elastic parameters 0 B , n and  have been determined, strains can be calculated using equation ( 8) and ( 9), and the plastic strain can then be obtained by subtracting the elastic part from the measured total strain.Parameter D: Parameter D can be obtained from the initial slope of the p   curve for Japanese silica sand sample under triaxial shear (Konishi et al. 2007) with a confining pressure of 400kPa.The measured initial slope of the curve, be determined by Eq. ( 11) taking q/p=0.Using M =1.243, the value D =1.67/1.243=1.35.The average value of D = 1.2 for different confining pressures, which was later used in model prediction. can be obtained from the measured data in an undrained triaxial test.For example, from the experimental data, we obtain e =0.66, and Mp=1.60 at peak stress state for Japanese silica sand sample (Konishi et al. 2007).Using the critical state parameters, we obtain cr e =0.687 and cs  =30.96 o , then we can calculate m =3.76 using Eq. ( 14).The averaged value of m = 4.0 for different confining pressures was later used in model prediction.

Other Three Parameters for Sand-Silt Mixture
As motioned in section 3.3, these three additional parameters can be determined by fitting Eqs. ( 26)-( 27) and (30) to triaxial tests data for mixtures with different fines content.For example, these three material constants are estimated as a=-0.3,b=0.3 and =-2.5 for Japanese silica sandmarine silt mixtures (see figure 7).

COMPARISON THE PREDICTED RESULTS FROM THE PROPOSED MODEL WITH SAND AND SAND-SILT MIXTURES TEST
Three sets of data of sand and sand-silt mixtures are chosen to evaluate the performance of the proposed model.One is a typical set of drained tests on Sacramento River sands (Lee and Seed 1967) , the other two tests are a set of undrained tests on Japanese silica sand-marine silt mixtures (Konishi et al., 2007) and Hokksund sand-Chengbei silt mixtures (Yang, 2004).One can see from Fig. 12 that at the small strain range, the results predicted from the proposed model give good agreement with the test data.At large strain range, the predicted curves using the model give lower values of stress and volume dilation for a given strain level.Overall, the proposed model can capture the main characteristics of the stress-strain and dilatancy behavior of sand for different densities and confining pressures, for example, the dilation increases with the soil density and the contraction increases with the confining pressure.4.2 Undrained Tests on Japanese Silica Sand-Marine Silt Mixtures (Konishi et al., 2007) Konish (Konishi et al., 2007) conducted a series of undrained monotonic triaxial tests on mixtures of Japanese silica sand and marine silt.Three samples with the same fines content were prepared and were compressed to large axial strains after consolidation under three different confining pressures (i.e.100, 200, and 400 kPa).The stress-strain responses of the mixtures were then obtained for mixtures with different fines content.As discussed in section 3.3, the set of parameters for any given amount of fines content can be predicted from the two sets of parameters; one for pure sand and another for pure silt.Three material constants (a, b and) to determine the parameters, cs  and cr e .For Japanese silica sand- marine silt mixtures, Three material constants are estimated as a=-0.3,b=0.3 and =-2.5.The simulation results of the proposed model versus experimental tests for fine contents 5%, 15%, 50% and 100% are presented in Fig. 13.For all fine contents, one can see from the stressstrain curves that the proposed models give good predictions comparing with the test data at the small strain stage.At large strain stage, the predicted curves from the one-scale model gives lower values.The predicted results of stress paths from the proposed models are in reasonable agreement to the experimental results.
4.3 Undrained Tests on Hokksund Sand-Chengbei Silt Mixtures (Yang, 2004) A series of undrained triaxial tests on Hokksund sand mixed with different amount of Chengbei silt were conducted by Yang (2004).The consolidation confining pressures are 50, 100, and 150 kPa respectively.All samples were compressed to large axial strains to reach steady state condition if possible.Some samples collapsed in test at lower axial strains because static liquefaction occurred during monotonic loading.
Table 4 is two sets of parameters of pure sand and pure silt for Yang (2004), from which one can obtain the set of parameters for any given amount of fines content to conduct simulation.Three material constants (a, b and ) are estimated as a=-0.3,b=0.2 and =-2.5.The simulation results from the proposed model versus tests for 8 fines contents, 0%, 5%, 10%, 15%, 20%, 50%, 70%, 95% are compared in Fig. 14.    (Yang, 2004).

Latin
One can see from Fig. 14 that at the initial portion of stress-strain curve, the predicted values have a good agreement with the test data.With the development of strain, for the samples with lower fines content (i.e.0%-25%), the predicted results from the model do not match the peak stress well for confining stress of 150 kPa.However, the predicted results of stress-paths have the same trend as the stress-strain curve for the samples with lower fines content.
For the samples with fines content of 50%-95%, the predicted peak stress from the proposed one-scale model gives lower values.The predicted results of stress-paths are roughly agreed with the test results.
Experimental results show that, after peak stress, the residual shear strength reduced to almost null for fines content greater than 20%, which has significant consequences in geotechnical engineering design.This is so called instability in sand and sand-silt mixtures and this post-peak behavior can be captured by the proposed model.

SUMMARY AND CONCLUSIONS
The present study focuses on the simulation of liquefaction behavior of granular materials such as sand and sand-silt mixtures.Based on a micromechanical analysis for inter-particle behavior, a simple one-scale model for static liquefaction is proposed to simulate the stress-strain response of granular materials; the results of the analysis show the following: The proposed model has the ability to capture the main features of sand behavior.For example, different initial void ratios lead to contracting or dilating behaviors of the sand under different drained conditions.The strain response envelopes and two-order work predicted from the model shows that the proposed one-scale model can be used to analyze the instability liquefaction of granular materials.
The proposed model is extended to simulate the sand-silt mixtures using the mixture theory combining the properties of sand and silt according to their proportions.Empirical expressions are introduced to fit the critical state strength and the location of the critical state line for each mixture: the 10 parameters of the model can be divided into two categories: the first seven parameters (i.
─── model refined the definition of critical Latin American Journal of Solids and Structures 13 (2016) 2190-2218

Figure 1 :
Figure 1: Local coordinate system at an inter-particle contact.
Fig 2(a) is the predicted stress-strain and stress path curve under undrained condition and Fig. 2(b) is the predicted stress-strain and volumetric strain curve under drained condition.Latin American Journal of Solids and Structures 13 (2016) 2190-2218 (a) Undrained results (b) Drained results.

Figure 2 :
Figure 2: General numerical results of the proposed one-scale model.
stress increment dσ in all directions with the same norm ( d =10kPa σ ) was imposed, and the corresponding strain response dε was computed.

Figure 3 :
Figure 3: Different stress state of stress probe and the location of stress points on stress-strain curve.

Figure 4 :
Figure 4: The predicted strain response envelopes from the proposed model at different initially stress state.

Figure 5 :
Figure 5: Results of second-order work from the proposed model for different q/p values.

Figure 7 :
Figure 7: Comparison of the predicted and measured critical state void ratio with different fines content.

f
are the lower and up limit of fines content..Typically, as 10% and 70% respectively.


and ref e ), we assume bulk modulus 0 B , the critical state friction angle cs  and reference void ratio ref e as a function of fines content.In order to esti- mate the parameters, cs  and ref e , for sand-silt mixtures, three more parameters a, b and  are required.These three additional parameters can be determined by fitting Eqs. (

Figure 9 :
Figure 9: Simulated and measured values of Isotropic Compression Lines for silica sand tested.

Figure. 11 .
Figure.11.Determination of parameter  p Test on Sacramento River SandsThe experimental results were conducted on soil samples with two different void ratios: e = 0.87 for loosely packed sand, and e = 0.61 for densely packed sand.The model parameters are given in Table 2.The simulation results of the proposed model compared with the experimental results are shown in Fig.12.

Figure 12 :
Figure 12: Comparisons of the predictions of model and the measurement of stress-strain behavior for Sacramento Sand.

Figure 13 :
Figure 13: Stress-strain curve for Japanese silica sand-marine silt mixtures: tests and model predictions.

Table 1 :
Parameters of the proposed one-scale model.2.2.5 General Numerical Results of the One-Scale ModelNumerical simulation both in drained and undrained triaxial test is presented here to evaluate the general ability of the proposed one-scale model.Parameters used in simulation are list in table.1.
B ,  and n ; the global plastic constants  , D , m and cs  , critical state constants ref e ,  ,  .All parameters are summarized in table.1 and can be determined from stress-strain curves ob- tained from triaxial tests (see section 3.4).

Table 2 :
Parameters used in the simulation for the experimental results of Sacramento River Sand.

Table 3 :
The parameters of pure sand and pure silt involved in simulation are summarized in table 3. Parameters used in simulation for Japanese silica sand and marine silt mixtures(konish, 2007).