of daping test interfered by the noise of torsional vibration test syste was lager for low signal-to-noise ratio. Therefore, only a test ethod studyin

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1 Threshold shear strain for dynaic Raberg-Osgood odel in soils based on theroechanics Xiaoxia Guo,Xiang Sun The State Key Laboratory of Structural Analysis for Industrial Equipent, Dalian University of Technology, Dalian, P.R.China ABSTRACT Starting fro the hysteretic curves of dynaic Raberg-Osgood odel, construct sall-strain dynaic dissipation function and explain dynaic characteristics by use of the skeleton curve back stress assuption. Much use is ade of Ziegler s orthogonality principle to obtain yield function expression in the dissipative stress space. Besides, add the shift stress to the dissipative stress and deduce the for of the yield condition in the true stress space. The plotting results of yield curves indicate that there exist two threshold shear strains which represent boundaries between fundaentally different dynaic characteristics of cyclic soil behaviour. Both the two threshold shear strain does depend significantly on the axiu dynaic shear odulus coefficient and exponent. Coparison between the two threshold shear strain values and shear odulus reduction curves obtained on exactly the sae soils confirs that the soil behavior is considerably at nonlinear when dynaic strain level exceeds the second threshold shear strain, the secant shear odulus of the four soils studied is between 0.6 and 0.8 of its axiu value. UNDER KEY WORDS: theroechanics, PUBLIS Ziegler s orthogonality principle, threshold shear strain, back stress 1. INTRODUCTION Understanding and knowing the values of the cyclic threshold shear strain is often essential for solving the soil dynaics and geotechnical earthquake engineering probles that involve cyclic settleent. Such probles include earthquake induced settleents of earth fill and natural deposits and settleents caused by traffic vibration, pile driving and the vibrations of achine foundations. With the developent of econoics and the iproveent of living standards, the issue on environental vibration control has becoe ore and ore proinent. In order to control environent vibration effectively, analyzing sall strain dynaic characteristic for soils has good scientific significance and project application background. In the history of soil echanics, the knowledge and understanding of the stress-strain behavior of soils due to cyclic loading has been continuously broadened and revised by precise, innovative and well-focused laboratory testing. For exaple, Lanzo, Vucetic et al. [1], Matesic and Vucetic [2], Hsu & Vucetic [3] had a large nuber of Double Specien and Norwegian Geotechnical Institute Direct Siple Shear Tests. The results indicated that the hysteresis loop began to becoe non-sooth at cyclic shear strain aplitude c Currently, a specified accuracy of resonance colun test apparatus has been less than But at sall strain the discreteness

2 of daping test interfered by the noise of torsional vibration test syste was lager for low signal-to-noise ratio. Therefore, only a test ethod studying on sall strain dynaic characteristics had soe liitation to a certain extent. Although the concept of the threshold shear strain is relatively new and hence not widely adopted by practicing engineers, it has been given respectable coverage in the literatures on geotechnical earthquake engineering and soil dynaics by Ishihara [4], Santaarina [5], Hsu & Vucetic [3], and Duku, Stewart et al. [6] published relatively recently. Dobry [7] introduced that the cyclic shear strain aplitude is the principal paraeter governing the changes in the soil icrostructure due to cyclic loading, and confired the existence and the range of the threshold strain, and proposed that it was ore difficult to liquefy for the larger threshold strain value, and vice versa. Subsequently, Matasovic and Vucetic [8] presented the cyclic pore-water pressure and degradation odels for clay that also include the threshold shear strain as one of their critical coponents. Hsu & Vucetic [3] suggested that the threshold strain in cohesive soils was larger than in cohesionless soils and that is generally increased with the soil s plasticity index(pi). When excitation strain was below the threshold strain, soil particles were not displaced with respect to each other during shearing produced elastic vibration, and its structure has not changed(the soil icrostructure reains practically unchanged), whereas above the threshold strain, the particles were irreversibly displaced with respect to each other(the soil s icrostructure changed peranently). The early developents of therodynaics ideas were due to Ziegler [9] and Ziegler and Wehrli [10], who noted soe applications of their general theory to the classical Coulob odel. Applications of these ideas to soil echanics were pioneered by Houlsby and Collins [11-20]. Houlsby and Puzrin [14-15] generalized soe aspects of this work to non-isotheral conditions, and in a series of papers have developed a faily of sophisticated odels, based on the use of internal functions. Their work have deonstrated the generality and proising features of explicit application of the therodynaic laws to soil odeling. However, dissipation function expression suggested by Collins and Houlsby was obtained through diensional analysis ethod and having no cobination with specific test. Accordingly, the objectives of this paper are: (1) Construct sall-strain dynaic dissipation function for Raberg-Osgood odel based therodynaic approach; (2) Propose the first threshold shear strain and the second threshold shear strain respectively and explain sall-strain dynaic characteristics under different cyclic loading aplitude; (3) Discuss the agnitude of the threshold shear strain and the factors affecting it thoroughly and systeatically for any soils; (4) Indicate that the second threshold shear strain presented in this study is consistent with that of pore water increasing and volue changing in the traditional sense by coparison between

3 the threshold shear strain values and shear odulus reduction. 2. THE THERMOMECHANICAL FORMULATION OF CONSTITUTIVE LAWS The steps in constructing the ingredients of an elastic/plastic geoechanical odel in a theroechanical forulation are as follows (e.g. [16-19]): (1) Define a free energy function and a dissipation increent function. These two functions, as well as state variables and independent variables in the choice of the issues are in accordance with the experience and deterine the physical eaning. Choice of state variables due to the nature and nuber of different physical phenoena of soe relatively detailed description and precise, while soe are rougher. The Gibbs free energy function can be written for an elastic/plastic decoupled aterial: ij ij ij ij g g g (1) 1 2 (2) Deduce the elasticity law fro the elastic part of the free energy function. e ij 1 ij ij g (2) (3) Deduce the shift stresses fro the plastic part of the free energy function. ij g2 ij ij (3) (4) Deduce the dissipative stresses fro the dissipation increents function. d (4) ij ij (5) Eliinate the plastic strain increents fro the last expression and for the yield condition in dissipative stress space. g y,,, 0 (5) ij ij ij (6) Construct the flow rule for the plastic strain rates. This flow rule is always noral in the dissipative stress space. (6) g ij y ij (7) Add the shift stresses to the dissipative stresses and deduce the for of the yield condition in the true stress space. If the dissipative stress for of the yield condition does not involve the true stresses as paraeters, then the flow rule reains noral. Else, the direction of the plastic strain increent vector is not noral to the final yield loci in the true stress space. ij ij ij (7)

4 (8) If required, the increental for of the constitutive law can be generated; as usual, by differentiating the yield condition to give the consistency equation. Where, g1 ij is the elastic part of Gibbs free energy function; 2 ij energy function; e ij is the elastic coponent of the strain; the stress variable g is the pure plastic part of Gibbs free ij, which derives fro the dependence of the therodynaic potentials on plastic strain, is seen to play the role of the back, shift or drag stress in kineatic hardening odels; d is dissipation function; ij is the dissipative stress, the generalized stress or the residual stress; g ij is the rate of the internal variable; ij, ij,, ij y is the yield condition in dissipative stress space; is an arbitrary non-negative ultiplier. 3. THE CONSTRUCTION OF DISSIPATION FUNCTION FOR RAMBERG-OSGOOD MODEL 3.1 The discussion on Masing rule UNDER At present, a nuber of nonlinear visco-elastic PUBLIS odels have been developed which can be broadly classified into two types-the experiential odels and the physical odels (e.g. [21-24]). The experiential odels obtained by fitting experiental data, which ai at iproving fitting precision. Masing odel is the representation of this kind of the odel based on experiential nonlinear skeleton curves and Masing s rule. This kind of odel describes non-linearity and viscosity for soils fro different points, oreover, proposes the ethod forulating dynaic stress-dynaic strain relationship. The other is physical odel. Iwan odel is the representation of this kind of the odel, which forulates constitutive equation based on echanical eleents of elastic spring, rigid-plastic slide and viscous daper. In the Masing odel, Hardin-Drnevich odel and Raberg-Osgood odel are the ost faous. Raberg-Osgood proposed the following for of the skeleton curve describing initial loading for soils (e.g. [25]): 1 G0 C1 ax In which, α, C 1 and R are the constant paraeters which are used to deterine the shape and the position of the skeleton curve. τ ax is the axiu dynaic shear stress; G 0 is the initial shear odulus under the low strain level; τ ax and G 0 are deterined by dynaic test for soils. τ and γ are shear stress and shear strain on the skeleton R1 (8)

5 curve, respectively. For representing the stress and strain relationship during unloading and reloading, Masing s rule (e.g. [24]) is used. It is assued that the shape of different segents of the unloading/reloading curves is the sae as segents of the backbone curve stretched by a factor of two. Moreover, shear odulus when load is reversed is equal to initial-loading shear odulus. Therefore, the shear stress strain relationship during unloading or reloading is given by used of a unifor expression 1 G0 2C1 ax In the equation (8) and (9), for the backbone curve in the first quadrant and the reloading curve as the sae trend as the backbone curve in the first quadrant, the value of axiu dynaic shear stress ax is given as positive; However, for the backbone curve in the third quadrant and the unloading curve as the sae trend as the backbone curve in the third quadrant, the value of the axiu dynaic shear stress ax is given as negative. R1 (9) 3.2 Basic assuption In order to illustrate dissipative echanis of Raberg-Osgood odel, the ain assuptions for deriving the dissipation function for soil are listed below. Assuption 1, the axiu dynaic shear odulus G 0 defined through the consolidation pressure-dependent expression describes the skeleton curves under the different consolidation pressures. Assuption 2, the accuulation of the internal variable ij can be expressed through the accuulation of plastic strain on the skeleton curve. Assuption 3, the rate of the internal variable ij can be expressed by use of plastic strain rate on the delayed curve. Assuption 4, as we know, the shift stress is deduced fro the pure plastic part of the free energy function, which defines the difference between the true and the generalized stress variables. Since the pure plastic part of the free energy potential function is unknown, adopt the assuptions of the backbone curve back stress. One ay see that the above assuptions are quite coon and considered reasonable in soil constitutive odeling. With the above considerations, the construction of dissipation functions is discussed in the following sections. 3.3 The procedure of constructing the dissipative function When the back stress is assued as a backbone curve, the shift rule of the back stress can be written as

6 1 G0 C1 ax R1 (10) Forula (10) subtracts fro forula (9) and integrates along the rate of the stress, and then the dissipation function d can be expressed as R1 R1 p d d (11) G0 C1ax G0 2C1ax Non considering the coupled elastic and plastic, that the total strain increent can be divided into the elastic e p strain increent and the plastic strain increent, that is to say: d d d, therefore d d d (12) ' H G H H G H ' 0 0 Fro the forula (9), the following relationship can be obtained. R1 d 1 R 1 d G0 G0 2C1ax H (13) UNDER Cobine forula (12) and forula (13), PUBLIS we can get ' 1 H (14) R1 R G0 2C1 ax where, H is elastic-plastic odulus; H is plastic odulus. Therefore, the dissipation function is as follows. R1 R1 R1 1 p d d (15) R1 G0 C1ax G0 C1ax G0 2C1 ax R G0 2C1 ax Starting fro the dissipation function proposed above, obtain yield function expression in the dissipative stress space by use of Ziegler s orthogonality principle proposed (e.g. [9, 10]). Moreover, obtain yield function in the true stress space through adding the shift stress to the dissipative stress. By use of the forula (4), the dissipative stress can be obtained as follows R1 R1 R1 1 G0 C1ax G0 C1ax G0 2C1 ax R G0 2C1 ax R1 (16) Add the shift stress to the dissipative stress and deduce the for of the yield condition in the true stress space.

7 So the true stress can be expressed as c bc bc (17) In which bc C A C G0 1 ; RA A G 0 2 C 1 ax R1 ; C G0 C1 ax R1 ; C G0 C1 ax R1 ; G bc C p 0 1 ax R1 1 R In which, c, bc and bc are the true stress, the dissipative stress and the shift stress respectively. 4. SMALL STRAIN CHARACTERISTICS OF DYNAMIC ROMBERG-OSGOOD MODEL Cobined with a high core rock-fill da engineering, four kinds of da aterial paraeters are selected to discuss yield surface, shear threshold strain and sall strain dynaic characteristics based on the above entioned therodynaic principles. 4.1 Dynaic odel paraeters The axiu dynaic shear odulus The axiu dynaic shear odulus (e.g. [25])is given by G0 k2 p ( ) n a (18) pa Where σ is the initial average static stress of the soil; P a is the atospheric pressure; K 2 and n are the paraeters which are used to deterine the axiu dynaic shear odulus. The axiu dynaic shear stress The axiu dynaic shear stress ax can be usually obtained approxiately by Mohr-Coulob Failure Criterion, for the consolidation ratio K c 1, one has(e.g. [26]) ax cos 2c cos 1 c sin 1 3c(sin 1) ( 1 c 3c)(cos 1) 2 1 sin 2 (19) Where c and are the effective cohesion and the internal friction angle respectively, and the axial and the lateral consolidation pressure respectively. The Internal Friction Angle 1c and 3c are A large nuber of triaxial test data indicate that the effective strength index could decrease with the increase of consolidation pressure. So the internal friction angle is often expressed as a function of consolidation pressure.

8 0 3 lg p (20) where Pa is the atospheric pressure; 0 is the value of for the condition 3 p a ; and is the paraeter reflecting the decrease in internal friction angle with respect to confining pressure 3. For dynaic experiental results of a high earth-rock-fill da, dynaic test paraeters are list in table The plotting of the yield surface In the following the yield loci of Filter I are discussed, and soe interesting characteristics of the yield surfaces with respect to the accuulation of cyclic shear strain for the certain value surfaces is illustrated in Fig. 1 at hysteresis loop strain a also are explored. The yield = It can be seen fro the figure that yield surface is alost a straight line. The straight line for of yield surface indicates that the friction echanics aong the particles plays an iportant role during yielding. Since the yield surfaces of different / value approxiate the sae straight line, note that the ratio of the generalized shear stress and the volue stress is a constant with stress point oving along the hysteresis loop, which indicates that under this strain condition the friction coefficient of all the yield points is a constant. However, it can be seen fro Fig. 2 that the yield loci of different / is still straight line when the hystersis loop strain reaches to =8 10-5, but not a single one, which indicates that the stress points on the hystersis loop are still controlled by friction echanis, but when the stress points ove along the hysteresis loop, friction coefficients of different stress points are different. There inevitably exists a boundary hysteresis strain level, which can be called the first threshold shear strain t1. When the actual hysteresis loop strain is below t1, the energy dissipation echanis of the yield stress point on the hysteresis loop is governed by friction echanis, and the friction coefficient reain unchanged. However, when the actual hysteresis loop strain is above t1, even though the energy dissipation echanis of the yield stress points is still doinated by friction echanis, yet friction coefficient is different. The yield loci of Filter I at cyclic strain = are shown in Fig. 3. It is observed that the yield surfaces show curved shape. Non-linear yield surface indicates that there exist other influencing factors in addition to friction echanis. When the cyclic strain reaches a certain value, soil will appear dilatancy and negative dilatancy-related phenoenon. With the cyclic strain continuing to increase, particles crushing ay occur, which leads to the rockfill saple structure and grading change, and so on. Therefore, the bending of yield loci indicates that other echaniss such as structural changes take part in controlling plastic deforation in addition to friction echanis. So there exists another threshold shear strain which is called the second threshold strain t 2 between

9 and When the dynaic strain is between t1 and t 2, the yield of soil is doinated by friction dissipation of the variable friction coefficient. However, when the dynaic strain is larger than t 2 dilatancy-related structural variation is shown. Saples oving along the hysteresis loop, dilatancy will begin to occur on the points corresponding to the stretching branch, which will lead to the changes of soil structure. Soil saples change significantly in the structure along with the increasing hysteresis loop strain, which will induce to the pore water pressure increase during undrained shear tests and the volue change during drained shear tests. The particles will irreversibly displace with respect to each other and the soil s icrostructure will change peranently. How to deterine the first threshold shear strain and the second threshold shear strain becoes the next focus of attention., the 4.3 The definition of the two threshold shear strains and influencing factors Note that even if for the very sall hysteresis loop strain, whose shear yield loci is not strictly a single straight line by discussing the yield surfaces of the sall strain hysteresis loop for four da aterials. But fro the view UNDER of engineering application, we can consider PUBLIS it as a single straight line as long as those straight lines of shear yield are closer enough. The following definition standards are proposed by the authors. q0.8 q 0.8 q0.8 4% (21) p Where q 0.8 and q 0.8 are the shear stress values with / 0.8 and / 0.8 respectively when the ean effective stress p is fixed. The sign of the second threshold shear strain is the bending of the shear yield loci. In atheatics, adopt a radius of curvature to describe the degree of bending. The degree of bending is saller for the greater the radius of curvature and the degree of bending is greater for the saller the radius of curvature. So the iniu curvature radius of yield curves is used to easure the extent of the bending. The radius of curvature is given by y 32 '2 '' 1 / y (22) Where y and y are the first order derivative and the second order derivative of the yield curves, respectively. Fro the plotting of the yield curves of the different cyclic shear strain, one can see, it is ore accurate and ore appropriate that the yield curves could be considered as a straight line when the iniu curvature radius is less than five. Therefore, based on the iniu curvature radius of this yield function, t 2 can be obtained.

10 According to the above-proposed criteria, the first and the second threshold shear strain values of four kinds of da aterials are calculated, as listed in Table 2. Fro the above table, it can be seen that, the values of two threshold shear strains are filter I, transient aterial, ain rockfill and filter II by sort descending respectively. Fro table 1, one can see that the axiu dynaic shear odulus coefficient fro filter I to filter II is distributed by increasing sequence. Therefore, the coefficient and the exponent of the axiu dynaic shear odulus are ain factors, with two threshold shear strains decreasing with increasing coefficient and exponent. The yield loci of Filter I at the two threshold shear strain are shown in Fig. 4 and Fig.5. Fig. 4 shows that, the yield loci at the first threshold shear strain is a boundary between a single straight line and a nuber of straight lines. At the sae tie, corresponding energy dissipation echanis for soils can be regarded as the terinal controlled by constant coefficient friction, also can be seen as the starting point doinated by variable coefficient friction echanis. Prior to this point, the yield loci tends to be ore a single straight line, thereafter, the yield loci tends to be ore a nuber of straight lines. Fig. 5 shows that, if soil is subjected to cyclic shear strain aplitude, equal to the second threshold shear strain, dynaic characteristics of soil can be controlled entirely by variable coefficient friction echanis. The second threshold shear strain reflects the changes of energy dissipation echanis. When dynaic strain level exceeds the second threshold shear strain, dilatancy including negative dilatancy-induced irreversible structure changes will occur. With the increase of dynaic strain level, irreversible therodynaic process such as particle breakage will happen, which becoes a part of energy dissipation for soils. 5. RESULTS OF THE TWO SHEAR THRESHOLD STARINS AND THEIR COMPARISON WITH PUBLISHED DATA In several other investigations (e.g. [3, 27-29]), the reduction of the secant shear odulus, Gs, with was deterined for five of the seven soils tested. The five corresponding noralized shear odulus reduction curves, Gs /Gax log, are presented in Fig. 6. The data showed that the threshold strain tv = % was obtained for PI = 33.7 of UC San Diego site yellowish brown fat clay, and the corresponding GG ax was about 0.55; tv = % was obtained for PI=11.7 of Brown sandy lean clay, and the corresponding tv = % was obtained for Nevada sand with its corresponding GG ax was 0.62; GG ax =0.71; tv = % was obtained for La Cienega Los Angeles sites Brownish gray silty sand with its corresponding GG ax =0.78;

11 tv = % was obtained for Gray lean clay-fat clay, and the corresponding was about 0.8. GG ax was largest, which The results obtained in the present study listed in Table 3 are plotted together in Fig. 7 against declining dynaic shear odulus reduction. The second threshold shear strain in the Table 2 is about % for the four aterials, and the corresponding G G was about These results are consistent with s ax the data for noncohesive soils obtained indirectly by Hsu and Vucetic [29]. The newly obtained data in Fig. 7 are obviously in broad agreeent with the previously published t results by Hsu and Vucetic [29]. Coparison between the second threshold shear strain values and shear odulus reduction curves obtained confirs that the soil behavior is considerably nonlinear at t 2. Fro the echanical echanis, the second threshold shear strain presented in the present study represent that the structure of cyclic soil begin to change by shear dilatancy and shear shrinkage. The shear shrinkage causes the increase in excess static pore water pressure and the change in soil volue. Therefore, it can be thought that the second threshold shear strain in the paper is consistent with that of pore water pressure increase and volue change in the traditional sense. However, the viewpoint of the UNDER traditional threshold strain which represented PUBLIS boundaries between elastic behavior and elastic-plastic behavior had no sufficient basis. 6. CONCLUSIONS The echanical properties of granular aterials under sall deforations are of interest in probles that involve low-aplitude oscillatory loading, such as seisic ground excitation, traffic loadings and vibration of achines on soil foundations. This paper carries on dynaic constitutive odelling by use of therodynaic approach. Construct the expression of dissipation potential functions of Raberg-Osgood odel starting fro the unloading and reloading hysteretic curves. A central thee is that the constitutive behaviour is entirely deterined by the knowledge of scalar potentials. A very iportant result of this approach is that it autoatically satisfy therodynaical requireent. The following conclusions can be derived fro the plotting results of yield loci in true stress space for four kinds of aterials: (1) For dynaic characteristics of non-cohesive rockfill aterial starting fro dynaic Raberg-Osgood odel, there exist two threshold shear strains: the first threshold shear strain t1 and the second threshold shear strain t2 respectively. If the soil is subjected to cyclic shear strain aplitude, c, saller than t1, dynaic stress-strain characteristics of soil can be seen as elastic and the yield of soil is controlled by friction

12 dissipation of the constant friction coefficient. If t1 c t2, soil showed elastic-plastic characteristic gradually and the yield of soil is controlled by that of the variable friction coefficient. However, when dynaic strain aplitude is above t2, soil changed in the dilatancy-related structure. Thus the two threshold shear strains represent a boundary between fundaentally different kinds of cyclic soil behavior. (2) The agnitude of the threshold shear strain and the factors affecting it are thoroughly and systeatically discussed for any soils. It is interesting to note that in these studies the two threshold shear strains are controlled ainly by the axiu dynaic shear odulus coefficient and exponent, with two threshold shear strains decreasing with increasing coefficient and exponent. (3) The ratio of secant odulus and the axiu dynaic shear odulus corresponding to the second threshold shear strain is between 0.6 and 0.8. The second threshold shear strain presented in this study is consistent with that of pore water increasing and volue changing in the traditional sense. If dynaic strain level is saller than the second threshold shear strain level, saturated soil saples do not produce excess pore water pressure, and unsaturated soil saples do not experience a change or settle even after a large nuber of cycles. However, when cyclic shear strain aplitude is larger than the second threshold shear strain, excess pore water pressure will increase under the undrained shear condition, and the residual volue deforation will accuulate continuously under the drained shear condition. AKNOWLEDGEMENT The financial support received fro the National Natural Science Foundation of the Republic of China through grant is gratefully acknowledged. REFERENCES 1. Lanzo G, Vucetic M, et al. Reduction of shear odulus at sall strains in siple shear. Journal of Geotechnical and Geoenvironental Engineering 1997; 123: Matesic L and Vucetic M. Strain-rate effect on soil secant shear odulus at sall cyclic strains. Journal of Geotechnical and Geoenvironental Engineering 2003; 129: Hsu CC and Vucetic M. Threshold shear strain for cyclic pore-water pressure in cohesive soils. Journal of Geotechnical and Geoenvironental Engineering 2006; 132: Ishihara K. Soil behavior in earthquake geotechnics. Clarendon Press: Oxford, Santaarina JC, Klein KA, et al. Soils and waves. Wiley: England, Duku PM, Stewart JP, et al. Voluetric strains of clean sands subject to cyclic loads. Journal of Geotechnical and Geoenvironental Engineering 2008; 134:

13 7. Dobry, R. Prediction of pore water pressure buildup and liquefaction of sands during earthquakes by the cyclic strain ethod. USGPO: Washington, Matasovic N and Vucetic M. Generalized cyclic-degradation-pore-pressure generation odel for clays. Journal of Geotechnical Engineering 1995; 121(1): Ziegler H. An introduction to theroechanics. North-Holand Publishing Copany: New York, Ziegler H and Wehrli C. The derivation of constitutive relations fro the free energy and the dissipation function. Advances in Applied Mechanics 1987; 25: Houlsby GT. A study of plasticity theories and their applicability to soils. PhD dissertation, England: University of Cabridge; Houlsby GT. A derivation of the sall-strain increental theory of plasticity fro therodynaics. Proceedings of the IUTAM conference on deforation and failure of granular aterials, Delft, 1982; Houlsby GT and Puzrin AM. A theroechanical fraework for constitutive odels for rate-independent dissipative aterials. International Journal of Plasticity 2000; 16: Puzrin AM and Houlsby GT. A theroechanical fraework for rate-independent dissipative aterials with internal functions. International Journal of Plasticity 2001; 17: Puzrin AM and Houlsby GT. Strained-based plasticity odels for soils and the BRICK odel as an exaple of the hyperplasticity UNDER approach. Geotechnique 2001; 51(2): PUBLIS 16. Collins IF and Houlsby GT. Application of theroechanical principles to the odeling of geoaterials. Proceedings of the Royal Society of London(Series A), London, 1997; Collins IF and Hilder T. A theoretical fraework for constructing elastic/plastic constitutive odels of triaxial tests. International Journal for Nuerical and Analytical Methods in Geoechanics 2002; 26(11): Collins IF and Kelly PA. A Theroechnical analysis of a faily of soil odels. Geotechnique 2002; 52(7): Collins IF. A systeatic procedure for constructing critical state odels in three diensions. International Journal of Solids and Structures 2003; 40(17): Houlsby GT, Puzrin AM. Principles of hyperplasticity: an approach to plasticity theory based on therodynaic principles. Springer: London, Iwan WD. On a class of odels for the yielding behaviour of continuous and coposite syste. Journal of applied echanics 1967; 34(3): Li XJ and Liao ZP. Visco-elastic-plastic odel of soil stress-strain relationship. Earthquake Engineering and Engineering Vibration 1989; 9(3): Luan MT and Lin G. Sei-analytical and sei-discrete procedure for constructing nonlinear hysteretic constitutive odel of soils. Journal of Dalian University of Technology 1992; 32(6): Zhang KX, Li MZ and Wang ZK. Dynaic elastic-plastic odels of soils based on non-ansing s rule. Earthquake Engineering and Engineering Vibration 1997; 17(2): Xie DY. Soil Dynaics. The publishing of Xi an Jiaotong University: Xi an, 1988

14 26. Chi SC, Guo XX, et al. Sall strain characteristics and threshold strain of dynaic Hardin-Drnevich odel for soils. Chinese Journal of Geotechnical Engineering 2008; 30(2): Vucetic M, Lanzo G and Doroudian M. Effect of the cyclic loading on daping ratio at sall strains. Soils and Foundations 1998; 38(1): Vucetic M, Lanzo G and Doroudian M. Daping at sall strains in cyclic siple shear test. Journal of Geotechnical and Geoenvironental Engineering 1998; 124(7): Hsu CC and Vucetic M. Volue threshold shear strain for cyclic settleent. Journal of Geotechanical and Geoenvironental Engineering 2004; 130(1): 58-70

15 Table 1. Paraeters of four kinds of da aterials Materials Natural density (g/c 3 ) Buoyant density (g/c 3 ) Dynaic Paraeters K2 n 0 Effective strength indexes Main rockfill Transient aterial Filter I Filter II Table 2. The first and the second threshold shear strain of da aterials(10-2 %) Materials Main rockfill Transient aterial Filter I Filter II γt1 γt2 Table 3. Declining dynaic shear odulus corresponding to two threshold shear strains The first threshold The second threshold Materials shear strain γt1 shear strain γt2 Main rockfill Transient Filter I Filter II

16 List of figure captions Fig. 1. Yield curves of Filter I with γ = Fig. 2. Yield curves of Filter I with γ = Fig. 3. Yield curves of Filter I with γ = Fig. 4. Yield loci at the first threshold shear strain of fliter I Fig. 5. Yield loci at the first threshold shear strain of fliter I Fig. 6. Relation between secant shear odulus reduction and volue threshold strain [Hsu and Vucetic (2004)] Fig. 7. Relation between secant shear odulus reduction and the two shear threshold strains