An elasto-plastic model to describe the undrained cyclic behavior of saturated sand with initial static shear

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1 University of Wollongong Research Online Faculty of Engineering - Paers (Archive) Faculty of Engineering and Information Sciences 211 An elasto-lastic model to describe the undrained cyclic behavior of saturated sand with initial static shear G Chiaro University of Wollongong, gchiaro@uow.edu.au L.I.N. e Silva Univeristy of Moratuwa T Kiyota IIS, University of Toyo, iyota@iis.u-toyo.ac.j J Kosei IIS, University of Toyo, osei@iis.u-toyo.ac.j htt://ro.uow.edu.au/engaers/1363 Publication etails Chiaro, G., e Silva, L., Kiyota, T. & Kosei, J. (211). An elasto-lastic model to describe the undrained cyclic behavior of saturated sand with initial static shear. 5th International Symosium on eformation Characteristics of Geomaterials- Volume 2 ( ). Hanrimwon Co., Ltd.. Research Online is the oen access institutional reository for the University of Wollongong. For further information contact the UOW Library: research-ubs@uow.edu.au

2 International Symosium on eformation Characteristics of Geomaterials, Setember 1~3, 211, Seoul, Korea An elasto-lastic model to describe the undrained cyclic behavior of saturated sand with initial static shear Chiaro, G. School of Civil, Mining and Environmental Engineering, University of Wollongong, Australia, e Silva, L. I. N. eartment of Civil Engineering, University of Moratuwa, Sri Lana Kiyota, T. & Kosei, J. Institute of Industrial Science, University of Toyo, Jaan Keywords: constitutive model, cyclic loads, sand, static shear, stress-dilatancy behavior, stress-strain relationshi ABSTRACT: With the aim of simulating the behavior of saturated sand with initial static shear (i.e., sloed ground) undergoing undrained cyclic loading, which leads to liquefaction and large cyclic shear strain develoment, an elastolastic constitutive model which can describe both monotonic and cyclic torsional shear behaviors of saturated sand under drained and/or undrained conditions is resented. It can simulate qualitatively the stress-strain relationshi and the effective stress ath, even after the secimen enters fully liquefied state. To verify its effectiveness, the roosed model is emloyed to simulate the results of a series of hollow-cylindrical torsional shear tests on loose Toyoura sand secimens with initial static shear stress under stress-reversal and non-reversal loading conditions. 1. INTROUCTION In the last few decades, there has been the need to develo efficient and economic countermeasures against liquefaction in order to decrease seismic damage to buildings, infrastructures and lifeline facilities as well as natural and artificial sloes. In order to satisfy such needs, it is definitely required to incororate a more elaborate constitutive model into numerical codes which can describe the liquefaction behavior of sand during cyclic loading. The aim of this aer is to roose an elasto-lastic constitutive model which maes it ossible to simulate the effects of initial static shear (i.e., sloe ground conditions) on the undrained cyclic behavior of saturated sand, which leads to liquefaction-induced failure of natural and artificial sloes and consequent large shear strain develoment. The constitutive model resented herein consists of an imrovement of the one develoed at the Institute of Industrial Science (IIS), University of Toyo, by e Silva and Kosei (211). To simulate drained and undrained large cyclic behavior of sand, the roosed model requires accurate evaluation of the irreversible strain comonent. In doing so, the quasi-elastic constitutive model, named IIS model (HongNam and Kosei, 25), was used to evaluate the elastic shear strain comonent. Consequently, in analyzing the hollow cylindrical torsional shear test data, the lastic shear strain comonents were obtained by subtracting from the measured total shear strain comonent the elastic one. The resence of initial static shear stress was introduced in the model by means of a monotonic drained shear loading ath before the undrained one. The two-hase (drained followed by undrained) monotonic loading behavior which is defined in terms of shear stress vs. lastic shear strain relationshi (i.e., seleton curve) was simulated by emloying the Generalized Hyerbolic Equation (GHE) model roosed by Tatsuoa and Shibuya (1992). Based on the exerimentally obtained seleton curves, the undrained cyclic shear loading behavior was modeled by using the extended Masing s rules while considering: (a) rearrangement of articles during cyclic loading (drag effect; Tatsuoa et al., 23); (b) damage of lastic shear modulus at large level of shear stress (e Silva and Kosei, 211); and (c) hardening of the material during cyclic loading (e Silva and Kosei, 211). In addition, in modeling the undrained cyclic shear behavior, it was assumed that the total umetric strain increment during the undrained loading, which consists of dilatancy and consolidation/swelling comonents, is equal to zero. An emirical four-hase stress-dilatancy relationshi, which varies with the amount of damage to

3 the lastic shear modulus of the material (e Silva and Kosei, 211), was emloyed to model the accumulation of umetric strain increment due to dilatancy and the generation of ore water ressure during the undrained cyclic torsional shear loading. An anisotroic over-consolidation boundary surface was roosed to introduce the combined effects of overconsolidation and initial static shear into the model. Alicability of the roosed model was verified by simulating the exerimental results that were conducted by Chiaro et al. (211) in order to study the effect of initial static shear stress on the undrained cyclic behavior of saturated Toyoura sand. 2. MOELING OF CYCLIC STRESS-STRAIN BEHAVIOR OF SAN WITH STATIC SHEAR 2.1 Modeling of two-hase seleton curve by GHE Monotonic stress-strain relationshi (i.e., seleton curve) of sand subjected to torsional shear loading can be modeled by using the GHE (Tatsuoa and Shibuya, 1992), as described by Eq. (1): X Y = 1 X C1( X ) C2 ( X ) (1) where X and Y are the normalized lastic shear strain and the shear stress arameters, resectively. C 1 (X) and C 2 (X) are functions of strain: C1() C1 ( ) C1 ( X ) = 2 C () ( ) 1 C1 cos π 2 α' X C2 () C2 ( ) C2 ( X ) = 2 C () ( ) 2 C2 β ' cos π 2 X m t n t 1 1 (2) (3) All the coefficients in Eqs. (2) and (3) can be determined by fitting the exerimental data lotted in terms of Y/X vs. X relationshi. Note that, C 1 (X=) is the initial lastic shear modulus, while C 2 (X= ) reresents the normalized ea strength of the material. In this aer, X and Y were defined as follows: ref X = γ / γ (4) ( τ / ') Y = (5) ( τ / ') max ref ( τ / ') max γ = (6) ( G / ') where γ = lastic shear strain; τ = shear stress; = current effective mean rincial stress; = initial effective mean rincial stress (= 1 Pa); (τ/ ) max = ea stress in the lot τ/ vs. γ ; G = initial shear modulus (= 8 MPa). The lastic shear strain increment (dγ ) was evaluated by subtracting from the total shear strain increment (dγ t ) the elastic one (dγ e ): dγ t e = d d (7) γ γ The elastic comonent (dγ e ) was calculated by Eqs. (8) and (9), as formulated in the recently develoed quasielastic constitutive model (IIS model) roosed by HongNam and Kosei (25): dγ e = dτ / G (8) f G G = (9) f ( ( e) n / 2 ( σ z ' σ r ') n e ) σ ' where G = shear modulus; f(e) = (2.17-e) 2 /(1e), (Hardin and Richart, 1963); f(e ) = initial void ratio function at σ z = σ r = σ θ = 1 Pa; G = initial shear modulus (= 8 MPa); σ = reference isotroic shear stress (= 1 Pa); σ z and σ r = vertical and radial effective stress, resectively; and n = material arameter (=.58). In this study, for each test both drained and undrained torsional shear loadings were alied while eeing the secimen height constant. In fact, a secific amount of initial static shear was achieved by alying monotonic drained torsional shear loading; subsequently, the cyclic behavior of sand with static shear was investigated under undrained condition. As shown in Fig. 1, comarison of X-Y relationshis from drained and undrained tests revealed a similar behavior of loose sand secimens at the early stage of shearing (i.e., Y <.38). In view of this consideration and to maintain the continuity of strain develoment during the change of loading from drained to undrained, it was attemted to model the entire two-hase monotonic loading curve by emloying a single set of GHE arameters (Table 1). Tyical simulation results of two-hase seleton curve are shown in Fig. 2. Table 1. GHE arameters obtained from undrained monotonic shear tests on loose Toyoura sand secimen (e =.828). C 1 () C 1 ( ) C 2 () C 2 ( ) m t n t α β

4 International Symosium on eformation Characteristics of Geomaterials, Setember 1~3, 211, Seoul, Korea Y= (τ/')/(τ/') max Y=(τ/')/(τ/') max Test 4 (reversal) e =.833 Undrained rained (τ/') max =.65 γ ref =.81% ' =1 Pa Exeriment rained Undrained Simulation Two-hase (rained Undrained) X=γ /γ ref τ STATIC = 15 Pa Figure 2. Tyical simulation results of a two-hase seleton curve. 2.2 Modeling of subsequent cyclic loading curves Cyclic behavior of soil can be modeled by emloying the well-nown 2 nd Masing s rule. However, due to rearrangement of articles, soil behavior does not necessarily follow the original Masing s rule during cyclic loadings (Tatsuoa et al., 1997). This feature can be taen into account by dragging the corresonding seleton curve in the oosite direction by an amount β while alying the Masing s rule (Masuda et al., 1999; Tatsuoa et al., 23). In this study the following drag function roosed by HongNam (24) was used: β = X ' 1 X ' 1 Test 16 (undrained) e = Test 15 (drained) e =.81 Close-u rained Undrained X= γ /γ ref Test16 (τ/') max =.65, γ ref =.81% Test 15 (τ/') max =.72, γ ref =.9% G = 8 MPa ' = 1 Pa Figure 1. Comarison of X-Y relationshis for drained and undrained tests on loose Toyoura sand. (1) where 1 = maximum amount of drag; 2 = fitting arameter, which is equivalent to the initial gradient of the drag function; and X = Σ X, where X denotes the increment of normalized lastic shear strain. e Silva and Kosei (211) introduced two concetual factors which tae into account the damage () of lastic shear modulus at large stress level and the hardening (S) of the material during cyclic loading: { 1 ex(.8) } (1 ult = 1 ex Σ γ ).8 ult (11) where ult = minimum value of ; Σ γ = total lastic shear strain accumulated between the current and the revious turning oints; S = ( Σ X ) ( Σ X ) u to current turning oint u to current turning oint S ult 1 (12) where S ult = maximum value of S after alying an infinite number of cycles; 1 and 2 are the same arameters used in the drag function. After introducing drag, damage and hardening effects, the seleton curve during cyclic loading can be modeled as follows: ( X β ) Y = 1 X β C ( X β ) C ( X ) S 1 2 β (13) rag, damage and hardening arameters emloyed in this study for loose Toyoura sand under constant amlitude cyclic shear loading are listed in Table 2. Table 2. rag, damage and hardening arameters. 1 2 ult S ult MOELING OF UNRAINE CYCLIC BEHAVIOR OF SAN WITH STATIC SHEAR 3.1 Stress-dilatancy relationshis Volume change in drained shear, which can be considered as a mirror image of ore water ressure build-u during undrained shear, is one of the ey arameters that affect the behavior of sand under cyclic loading. Change of umetric strain in different stage of loading can be described by the stress-dilatancy relationshi, which relates the ratio of lastic strain increments (-dε /dγ ) to the stress ratio (τ/ ) (Shahnazari and Towhata, 1992; among others). It should be noted that, the theoretical stress-dilatancy relations, such as Rowe s equations (Rowe, 1962), are derived only for either triaxial (dε 2 = dε 3 or dε 2 = dε 1 ) or simle shear (dε 2 = ) loading conditions. Therefore they are not directly alicable to the case of torsional shear loading. However, the results from cyclic torsional shear tests suggest that unique relationshis between

5 -dε /dγ and τ/ exist either for loading (dγ > ) and unloading (dγ < ) conditions (Pradhan et al., 1989). e Silva (28) roosed an emirical bi-linear stressdilatancy relationshi for cyclic torsional shear loading which is lined with the damage () of lastic shear modulus as shown in Eq. (14): τ d R ε = ' dγ ± C (14) where R and C are the gradient and the intercet of the linear stress-dilatancy relationshi, resectively; and is the same as the damage factor in Eq. (11). uring cyclic loadings, the effective mean stress ( ) decreases with number of cycles and it can be associated with two otential mechanisms: (i) the soil is subjected to significant effects of over-consolidation until the stress state exceeds for the first time the hase transformation stress state (Ishihara et al., 1975) (i.e., the first time where the umetric behavior changes from contractive to dilative, d > ); and (ii) soil enters into the stage of cyclic mobility. In articular, the overconsolidation significantly alters the stress-dilatancy behavior of sand during the virgin loading and its effect evanishes with the subsequent cyclic loading. To consider the effect of over-consolidation within a certain boundary, Oa et al. (1999) roosed the following stress-dilatancy equations (after e Silva, 28): dε dγ = R τ / ' = C ln( OCR) τ τ / ' ' ln( OCR) 1.5 (15) (16) OCR = '/ ' (17) In the current study, a modification of the equations roosed by Oa et al. (1999), which consists of a rotation of over-consolidation boundary surface as schematically illustrated in Fig. 3, was made to account for the combined effects of over-consolidation and initial static shear stress on undrained cyclic torsional shear behavior of sand. To this urose, the following stress-dilatancy equations were roosed to define an anisotroic over-consolidation boundary surface: dε dγ = R τ / ' SSR = C ln( OCR) τ τ / ' SSR ' ln( OCR) 1.5 (18) (19) SSR = / ' (2) τ static where τ static = initial static shear stress. Note that, whenever SSR = (i.e., τ static = ), Eqs. (18) and (19) meet Eqs. (15) and (16), resectively. The roosed anisotroic over-consolidation boundary surface has the same features of the one resented by Oa et al. (1999) for isotroically consolidated sands, in the sense that, it defines the region within which the secimen behaves as less contractive while being affected by over-consolidation. As well, it taes into account the effects of anisotroic consolidation induced by initial static shear stress, following the same rincile of the rotation of yield surface in the stress sace due to anisotroic consolidation (e.g., Taiebat and afalias, 21). 3 Anisotroic OC bound. surface (this study) Original OC bound. surface (Oa et al, 1999) (;) (τ static /' ) ' ' (τ STATIC ;' ) τ STATIC Effective mean rincial stress, ' (Pa) Figure 3. Schematic illustration of original and anisotroic over-consolidation boundary surfaces. 3.2 Four-hase stress-dilatancy model The stress ath during undrained loading was divided into four sections namely: (A) Virgin stress ath; (B) Stress ath within the limits of hase transformation stress state; (C) Stress ath within the limits of overconsolidation boundary surface; and () Stress ath after exceeding the hase transformation stress state for the first time. The schematic illustration of the fourhase stress-dilatancy relationshi is shown in Fig. 4; the coefficients and the equations emloyed for each hase are listed in Table 3. Table 3. Four-hase stress-dilatancy arameters. Phase Eq. R C A (14) B (14) 2.2*.45* 1 C (18), (19) 2.2*.45* 1 1 (14), (11) 2.2*.36* 2 (14), (11).33*.18* *After e Silva (28)

6 International Symosium on eformation Characteristics of Geomaterials, Setember 1~3, 211, Seoul, Korea τ / ' Test 3 by simulation (A) Virgin loading (2) dγ > (1) (B) -.75 Phase transformation (B) (1) -dε / dγ dγ < (2) (C) Effects of OC for the 1 st cycle Figure 4. Tyical simulation results using the four-hase stress-dilatancy relationshi. 3.3 Modeling of ore water ressure generation In modeling the undrained cyclic shear behavior it was assumed that the total umetric strain increment (dε ) during the undrained loading, which consists of dilatancy (dε d ) and consolidation/swelling (dε c ) comonents, is equal to zero. In fact, a change of mean effective stress ( ) during undrained loading causes the re-comression/swelling of the secimen; on the other hand, a change of shear stress (τ) causes the dilatation of the secimen. Therefore the following equation is valid during undrained cyclic loading: c d dε = dε dε = (21) Exerimental evidences suggest that the bul modulus K (= d /dε c ) can be exressed as a unique function of the effective mean rincial stress ( ): d' K = dε c = K f ( e) ' f e ( ) ' m (22) where K = bul modulus at reference effective mean stress ( ); f(e) and f(e ) = void ratio function at current and reference stress state, resectively; and m = coefficient to model the stress-state deendency of K. Considering that f(e) = f(e ) in undrained tests, and that the umetric strain comonent due to consolidation/ swelling is the mirror image of the one due to dilatancy (dε c = -dε d ), the generation of ore water ressure during undrained shearing was evaluated as follows: m ( ' / ) ( d ) d d' = K ' ε (23) In this, -dε d was calculated by combining the stressstrain relationshi in Eq. (13) with the stress-dilatancy relations in Eq. (14) or Eq. (18). Emirically estimated values of K = 47 MPa and m =.5 were used for loose Toyoura sand (e =.828) at an initial confining stress state of = 1 Pa. 3.4 Modeling of initial static shear effects on the effective stress ath The resence of initial static shear widely affects the monotonic undrained behavior of sand as well as the subsequent cyclic liquefaction behavior (e.g., liquefaction resistance, failure behavior, mode of residual strain develoment, etc). As well, under the same initial condition of void ratio and effective mean rincial stress, the ea undrained strength of sand during monotonic torsional shearing increases with the initial static shear stress (Chiaro, 21). Herein, the resence of initial static shear stress was simulated by a monotonic drained loading ath before the undrained cyclic shearing, with the simlified assumtion that no change of occurs (i.e., = = constant). 4. COMPARISON BETWEEN EXPERIMENTAL AN SIMULATION RESULTS To verify the erformance of the roosed elasto-lastic constitutive model, the results from a single-element numerical simulation were comared with the exerimental data resented by Chiaro et al. (211). 4.1 Effective stress aths Tyical effective stress aths obtained exerimentally and by numerical simulations are shown in Figs. 5 through 7. One can see that, in the case of stress-reversal loading tests, the model is able to simulate the whole liquefaction behavior of sand: secimen first enters into fully liquefied state (i.e., = ) and then shows cyclic mobility. On the other hand, simulation results confirm that liquefaction does not tae lace in the case of nonreversal loading tests in a similar manner as observed in laboratory tests. Note that, stress-reversal loading tests refers to the case in which during each cycle of loading the combined shear stress value is reversed from ositive (τ max = τ static τ cyclic > ) to negative (τ min = τ static - τ cyclic < ), or viceversa. On the other hand, non-reversal loading tests refers to the case in which the combined shear stress value is always et ositive (τ min > ). 4.2 Stress-strain relationshis Tyical stress-strain relationshis obtained exerimentally and by numerical simulations are resented in Figs. 8 through 1. The model is able to simulate qualitatively the exerimental data u to a shear strain levels of about γ = 8 %, before entering a steady state (i.e., no further develoment of strain during subsequent cycles of loading).

7 4 3 Test 2 (Reversal) e=.828 τ STATIC = 5 Pa τ MAX =21 Pa τ MIN =-11 Pa 5 4 Test 7 (Non-reversal) e=.829 τ STATIC =2 Pa τ MAX =36 Pa τ MIN = 4 Pa Shear stress, τ (Pa) No liquefaction! -2 (a) Exerimental results (a) Exerimental results Effective mean rincial stress, ' (Pa) Effective mean rincial stress, ' (Pa) 4 3 Test 2 (Reversal) e=.828 τ STATIC = 5 Pa τ MAX =21 Pa τ MIN =-11 Pa 5 4 Test 7 (Non-reversal) e=.828 τ STATIC =2 Pa τ MAX =36 Pa τ MIN = 4 Pa Effective mean rincial stress, ' (Pa) Effective mean rincial stress, ' (Pa) Figure 5. Effective stress aths for Tests 2. Figure 7. Effective stress aths for Test Test 1 (Reversal) e=.828 τ MAX =3 Pa τ MIN =-1 Pa 4 3 Test 2 (Reversal) e=.828 τ MAX =21 Pa τ MIN =-11 Pa τ CYCLIC =2 Pa τ STATIC =1 Pa (a) Exerimental results (a) Exerimental results τ STATIC = 5 Pa Effective mean rincial stress, ' (Pa) Shear strain, γ (%) 4 3 Test 1 (Reversal) e=.828 τ MAX =3 Pa τ MIN =-1 Pa 4 3 Test 2 (Reversal) e=.828 τ MAX =21 Pa τ MIN =-11 Pa τ CYCLIC =2 Pa τ STATIC =1 Pa τ STATIC = 5 Pa Effective mean rincial stress, ' (Pa) Shear strain, γ (%) Figure 6. Effective stress aths for Test 1. Figure 8. Stress-strain relationshis for Test 2.

8 International Symosium on eformation Characteristics of Geomaterials, Setember 1~3, 211, Seoul, Korea Test 1 (Reversal) e=.828 (a) Exerimental results Test 1 (Reversal) e=.828 Shear strain, γ(%) τ MAX =3 Pa τ MIN =-1 Pa τ CYCLIC =2 Pa τ STATIC =1 Pa Shear strain, γ (%) τ MAX =3 Pa τ MIN =-1 Pa τ CYCLIC =2 Pa τ STATIC =1 Pa Figure 9. Stress-strain relationshis for Test Test 7 (Non-reversal) e=.829 τ STATIC =2 Pa (a) Exerimental results Test 7 (Non-reversal) e=.828 Shear strain, γ(%) τ STATIC =2 Pa τ MAX =36 Pa τ MIN = 4 Pa Shear strain, γ(%) τ MAX =36 Pa τ MIN = 4 Pa Figure 1. Stress-strain relationshis for Test Resistance against liquefaction The resistance against liquefaction (or more strictly, resistance against cyclic shear strain accumulation) was evaluated in terms of number of cycles required to develo a secific amount of single amlitude shear strain of γ SA = 7.5%, which would corresond to an axial strain ε a = 5% in case of undrained triaxial tests. Fig. 11 shows the comarison of exerimental and simulated liquefaction resistance curves, in terms of relationshis between the cyclic stress ratio CSR (= τ cyclic / ) and the number of cycles required to develo γ SA = 7.5%. Note that, the CSR is not a sufficient single arameter to describe the effects of initial static shear on the liquefaction resistance of sand (Chiaro et al., 211). Therefore, in Fig. 12, the liquefaction resistance curves are also lotted in terms of relationshis between the static stress ratio SSR (= τ static / ) and the number of cycles to develo γ SA = 7.5%. Static stress ratio, SSR = τ STATIC / ' Cyclic stress ratio,csr = τ CYCLIC / ' γ SA =7.5% CSR=.2, Exrimental results, Simulation results.2 SSR = τ STATIC / ' , Exerimental results, Simulation results CSR = τ CYCLIC / ' CSR= Number of cycles CSR=.16 ' = 1 Pa Figure 12. Liquefaction resistance curves defined in terms of static stress ratio (SSR)..5 Number of cycles SSR= γ SA =7.5% CSR=.2 ' = 1 Pa Figure 11. Liquefaction resistance curves defined in terms of cyclic stress ratio (CSR).

9 Simulation results are consistent with the exerimental data. In fact, the resistance against cyclic strain accumulation can either decrease or increase deending on the level of initial static shear and the amlitude of cyclic shear stress in a similar way as observed in laboratory tests. 5. CONCLUSIONS A cyclic elasto-lastic constitutive model is roosed to simulate the undrained cyclic torsional shear behavior of sand with initial static shear (i.e., sloing ground conditions), which leads to liquefaction and/or large cyclic shear strain develoment. The following characteristics are taen into account in the roosed model: (a) the resence of initial static shear stress is introduced by means of a monotonic drained shear loading ath before the undrained one; (b) the two-hase seleton curve (drained followed by undrained) is simulated by emloying the GHE model; (c) the subsequent cyclic stress-strain behavior is modeled by using the extended Masing s rules while considering the rearrangement of articles during cyclic loading, the damage of lastic shear modulus at large level of shear stress and the hardening of the material during cyclic loading; (d) the undrained cyclic shear behavior is modeled under the assumtion that the total umetric strain increment during the undrained loading is equal to zero; (e) an emirical four-hase stress-dilatancy relationshi, which varies with the amount of damage to the lastic shear modulus of the material, is emloyed to model the accumulation of umetric strain increment due to dilatancy and the generation of ore water ressure during the undrained cyclic shear loading; and (f) an anisotroic overconsolidation boundary surface is emloyed to account for the combined effects of over-consolidation and initial static shear stress. To verify the erformance of the roosed elasto-lastic constitutive model, the simulated results are comared with the exerimental data. The model has been found to be caable of reroducing all the features of the undrained cyclic torsional shear behavior of Toyoura sand secimens subjected to initial static shear stress, which are evaluated in terms of effective stress ath, stress-strain relationshi and liquefaction resistance curves. The authors recognize limitation of the roosed model in dealing with stress-level and density deendency due to the use of an extensive number of arameters for a single soil tye. In fact, for the same sand these arameters have to be calibrated searately if variation of initial relative density and confining ressure are concerned. REFERENCES Chiaro, G. (21). eformation roerties of sand with initial static shear in undrained cyclic torsional shear tests and their modeling, Ph Thesis, et. of Civil Eng., University of Toyo, Jaan. 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