A PROCEDURE FOR DETERMINATION OF THE ALTERNAT MODEL PARAMETERS

Transcription

1 1. Mohamme Y. FATTAH,. Omar. F. S. al DAMLUJI, 3. Yousif. J. al SHAKARCHI A PROCEDURE FOR DETERMINATION OF THE ALTERNAT MODEL PARAMETERS 1. BUILDING AND CONSTRUCTION DEPARTMENT, UNIVERSITY OF TECHNOLOGY, IRAQ -3. CIVIL ENGINEERING DEPT., COLLEGE OF ENGINEERING, UNIVERSITY OF BAGHDAD, IRAQ ABSTRACT: The ALTERNAT moel escribe in this paper is a ouble harening moel for the mechanical behaviour of san uner alternating loaing. For etermination of the parameters of the ALTERNAT moel, special types of triaxial tests are require, e.g., raine triaxial tests with monotonically increasing or ecreasing axial strain an constant isotropic stress. These tests are not easy to be conucte in soil mechanics laboratories. It is intene here to choose a simple theoretical moel to get the require stress - strain relationships for the etermination of ALTERNAT moel parameters. Of many theoretical moels available to preict the overall response of sans, the enochronic moel is chosen for this task. This moel treats the san as non-linear elastic-plastic material. Furthermore, the theory assumes inelastic changes to be cause only by the rearrangement of grains. The stress path require in the tests is applie in the computer program (ENDOCH written by the authors. The parameters are calculate for two samples for which Ng an Dobry (1994 mae experimental investigation. It is conclue that the propose proceure for the etermination of the ALTERNAT moel parameters is successful. The stress-strain relations preicte are similar to those obtaine by Molenkamp base on laboratory test results. KEYWORDS: ALTERNAT moel, enochronic, parameters, cyclic, san INTRODUCTION - The ALTERNAT Moel The moel escribe in this paper forms the major component of a ouble harening moel for the mechanical behaviour of san uner alternating loaing. The moel was evelope by Molenkamp (1987 at Delft Geotechnics. In Figure (1, the yiel surfaces of both plastic moels, namely the compressive moel an the eviatoric moel are shown in the stress space of the isotropic stress, s, an the eviatoric stress, t. t Failure Surface Kinematic Deviatoric Yiel Surface s Compressive Yiel Surface Figure 1 - The yiel surfaces of the ALTERNAT moel Kinematic Plastic Moel an Kinematic Rule Using the elastic an plastic strains an the stress together with their rates, the constitutive moels in the current state can be efine on the Cartesian co-rotational base vectors, γ i. The elastic moel will be of the form: Λ ij = function ( ij (1 The irreversible Eulerian strain rates or increments, K ij, are escribe by a plastic kinematic harening moel of the form:. copyright FACULTY of ENGINEERING HUNEDOARA, ROMANIA 97

2 G F O kl ij kl Kij = ( H Tij = ij ξ ij : pseuo stress (3 in which: F (T ij, χ = : yiel surface, G (T ij, χ 1 = : plastic potential, H (T ij, ξ kl, w mn, χ: harening, w: plastic eformation, ξ ij : tensor of anisotropy representing the effect of the anisotropic fabric, Σ χ: quantity escribing the size of the kinematic yiel surface; the so-calle harening parameter, an χ 1 : iem Σ n for plastic potential. ξ n Figure ( illustrates the above conitions. ξ new Stress Inuce Anisotropy Describe By the Kinematic Harening Σ new For simplicity, the functional form of the kinematic ξ harening H kin is chosen as follows: H Kin = H Kin ( IT, χ (4 in which: I T = invariants of the pseuo stress T ij. The kinematic harening H kin can be calculate from experimental ata using the Equation: F o F o σ kl σ Figure - Definition of the kinematic kl σ kl σ kl harening an the motion of the H Kin = = (5 K. & γ & γ kinematic yiel surfaces Kin The curve of the shear stress level (t/s c in raine triaxial compression at an isotropic pressure (s/pa = 1 (Pa is the atmospheric pressure is relate to the eviatoric strain (see Figure 3 by, (Molenkamp, Y1 1985: Y Y1 Y = (6 (t/s t 1 S c n n ( Y + Y n T 1 (t/s in which: t = shear stress, n = quantity larger than 1, cv an Y 1, Y = two approximations of the shear stress level versus plastic eviatoric strain curve of raine triaxial compression at s = Pa. Equation (6 has the properties that: s=pa S S c c Y 1 Y if Y >> Y 1 if Y 1 >> Y The two curves Y 1 an Y are efine as follows: EP Y = Eχ (7 1 an Y = + T.exp{ β ( χ χt } (8 s cv in which: E, EP = two parameters of the pre-peak fit by a power law. (t/s cv = shear stress ratio in triaxial compression at the resiual state, thus at very large strain χ. T = Y ( χ = χ t s cv β = ecay parameter. The parameters E an EP of Equation (7 can be etermine from the measure harening curve when it is expresse in terms of ln (t/s c versus ln (χ p. A fit with a straight line to the low strain range of this plot reas: c Shear Stress Level, (t/s Stress-Strain Curve χ t Deviatoric Strain, χ Figure 3 - Approximation for the stress-strain curve in raine triaxial compression test, (after Molenkamp, Tome XI (Year 13. Fascicule 3. ISSN

3 lny 1 = ln E + EP.ln χ (9 Next the parameters T an β can be etermine from a plot of ln ( (t/s c - (t/s cv versus χ. A fit with a straight line to the large strain range reas: t ln = lnt β ( χ χ t s c s (1 cv For full escription of the ALTERNAT moel, refer to Molenkamp (1987 an Fattah (1999. The Enochronic Moel - Rearrangement Measure The enochronic theory is a viscoplastic one but without introucing a yiel surface. Therefore, all complexities an ifficulties that evelop in introucing a suitable yiel criterion are avoie. The source of inelasticity in san is the irreversible rearrangement of grain configurations associate with eviatoric strains. Thus, it is convenient to characterize the accumulation of rearrangement by an appropriate variable, ξ, terme the rearrangement measure, which will be use as the inepenent variable in the stress - strain law. Assuming that the evelopment of inelastic strains is graual, ξ must be a continuous an smooth function of ε ij. It can be shown that the only expression satisfying the above conitions is (Bazant an Krizek, 1976: J 1 ξ = e e ( ε = ij ij (11 in which, J ( is the secon invariant of the eviator of the tensor in parentheses, ε = [ ε ij ] = the ~ strain tensor. ε kk e ij = ε ij δ ij (1 3 in which ε ij.= the eviator of the tensor, δ ij = Kronecker s elta. Densification Due to Deviatoric Strains The ensification of san may be characterize by the volumetric strain, λ, whose sign is chosen to be negative when the volume ecreases. If vertical accelerations are not large enough to cause any significant separation or jumping of particles (as in the case in many earthquakes, the ensification is prouce almost exclusively by interparticle slips that result in a rearrangement of grain configurations. Subject to this restriction, λ must be proportional to ξ an the epenence of the ensification increment per cycle of shear on the strain amplitue an the number of cycles may be expresse by (Bazant an Krizek, 1976: κ λ = (13 c(κ κ = C ( ε, σ ξ (14 in which: κ = the ensification measure. The function c (κ is the ensification-harening function that moels the ecrease in the ensification increments per cycle with an increase in the number of cycles, N. Expressing c (κ in a Taylor series an truncating it after the linear term yiels: c ( κ = c o (1+α κ (15 in which α an co are constants for a given san at a given relative ensity. εσ, may be regare as a function of J ( in The ensification-softening function, C(, ~ ~ aition to I1( ε an I 3 ( ε which are the first an thir invariants of ε ~. Accoringly, one suitable choice for C( ε is a function of J ( ε, an this function was ~ ~ ientifie by Bazant an Krizek (1976 as follows: 1 ( q 1 / c ( ε = q[ 4 J ( ε ] (16 in which q is a non-negative constant for a given san. Generally, the material parameters α an q epen on the relative ensity, D r. Base on the ata reporte by Silver an See (1971 an other ata, Cuellar et al. (1977 propose the following relationships: C o = 1. ε Tome XI (Year 13. Fascicule 3. ISSN

4 q =.95D r +.33Dr +.54 (17 α = 33.33D r Dr (18 in which, D r is expresse as a number (not a percentage. Intrinsic Time Since the increments of irreversible (inelastic strains are cause by interparticle slips, they must be proportional to increments of the rearrangement measure, ξ, an the proportionality coefficient may, in general, epen on the state of strain an stress. To express this fact, a new inepenent variable, η, may be expresse such that: η = F ( ε, σ. ξ (19 As in the case of ensification, the increment of inelastic strain per cycle iminishes as the number of cycles increases. To account for this effect, it is expeient to introuce a new inepenent variable, ς, terme intrinsic time, whose growth relative to η or ξ graually iminishes. This phenomenon may be escribe by the relation: η ς = ( f (η in which f (η, which may be terme a strain-harening function, is a continuously increasing positive function of η an is given by the relation: β r ( η = (1+ η (1 r in which β an r are constants. The function F is a strain-softening function an f (η is a strain-harening function in the sense of causing a ecrease or increase in the slope of the stress - strain curve. However, within the range of strains commonly encountere in seismic loaing, it appears that F may be equal to unity as an acceptable approximation, (Bazant an Krizek, Enochronic Stress-Strain Law for San Because of the limite range of ξ-values that are of practical interest, it ha been shown by Bazant an Krizek (1976 that the eviatoric stress - strain relation may be approximate by: Sij p e ij = + eij (a G e p ij Sij ξ + G Z 1 = (b in which: G = shear moulus, e ij, e p ij = the elastic an inelastic eviatoric strains, Z 1 = constant (analogous to the relaxation time in a Maxwell chain use in viscoelasticity. The volumetric stress - strain relation may be written as: 3σ ε = + λ 3 K in which: K = the bulk moulus. For a wie range of shear strains an confining pressures, Cuellar et al. (1977 foun, using simple shear tests, the following values for the parameters: Z 1 =.5 x 1-4, β =. an r =.7 Equations ( an (3 are capable of escribing experimental results quite well within a range of shear strain amplitue from about.1 to.5, but the agreement is not goo for a boarer range of amplitues, (Cuellar et al., To get better agreement, G shoul be assume to epen on I 1 ( σ, the first invariant of σ ~ ~, since that the elastic rigiity of the soli skeleton epens on the number an areas of interparticle contacts, which in turn epens on I 1( σ ~. Since the contact areas between grains increase with the confining stress, the soli skeleton becomes stiffer an grain contacts slip as the confining stress is increase. Base on extensive experimental ata for the ynamic shear moulus, G ha been shown to be essentially proportional to the square root of the confining stress. Accoringly, Bazant an Krizek (1976 propose the following expression for the shear moulus, G: ' G = M σ (4 v (3 3 Tome XI (Year 13. Fascicule 3. ISSN

5 . ANNALS OF FACULTY ENGINEERING HUNEDOARA International Journal Of Engineering in which: σ v = vertical effective stress, M = constant that is equal to.1 kn/m base on the simple shear tests conucte by Cuellar et al. (1977. It was foun by Cuellar et al. (1977 that Equation (4 i not suffice to aequately escribe the ata for large strain amplitues; generally the unloaing portions of these hysteresis loops were too steep. It is recommene here to use a new proceure base on the proceure aopte by Huang an Chen (199 for the cap plasticity moel in which the elastic bulk moulus, K, can be erive from the confine compression test as follows: P P K = = (5 e ε v Ae with Gr Ae = (6.33(1 + ei where: C r = the swelling inex, P = effective hyrostatic stress, ε e v = elastic volumetric strain. The elastic shear moulus, G, can be etermine if Poisson s ratio, ν, is evaluate: 3 (1 ν G = K (7 (1+ ν DETERMINATION OF THE ALTERNAT MODEL PARAMETERS The ouble harening kinematic moel ALTERNAT consists of three ifferent moel components, namely: (1 Non-linear elastic moel. ( Plastic compressive moel. (3 Plastic eviatoric kinematic moel. It was shown by Molenkamp (198 that the parameters A, AP, B an BP of the elastic moel can be etermine most easily from a raine isotropic loaing-unloaing test. The parameter V can be etermine from the behaviour uring initial eviatoric unloaing. If it coul be assume that the material is isotropic initially, then from one raine triaxial test at constant isotropic stress s = Pa an monotonically increasing axial eformation (see Figure 4a an b the following parameters can be etermine. This must be one after the elastic strains an the plastic compressive strains have been subtracte from the stress - strain curves. E an EP parameters can be etermine from the fit by a power law to the first part of the curve in Figure (4a. The parameters χ t, (t/s cv, T an β are etermine from the fit by an exponential function to the last part of the same curve (see Figure 4a. Dilation Shear Stress Level, (t/s (t/s cv E,EP φ o T Volumetric Strain, V χ cv s=pa Contraction. χ t Deviatoric Strain, χ Deviatoric Strain, χ a b Figure 4 - Determination of the ALTERNAT moel parameters from raine triaxial test at constant isotropic stress. In the case of only one test an no other publishe experimental ata on similar materials, only one ilatancy angle φ o can be erive from the curve of the isotropic strain versus eviatoric strain (see Figure 4b. Then it must be assume that FIMU (φ μ = FICV (φ cv = φ o. In such a case, the parameter SCV (S cv is not relevant an can be put at S cv = 1. From this curve (Figure 4b also the parameter CHICV (χ cv can be etermine if the eformation remains uniform until large eformation. Tome XI (Year 13. Fascicule 3. ISSN

6 For a proper etermination of the parameters φ μ, φ cv, S cv, LB an COH (c, at least two raine triaxial compression tests at ifferent constant isotropic stresses woul be neee. From each iniviual test, a ilatancy angle φ o can be calculate. Next the parameters φ μ, φ cv an S cv can be fitte using the values of φ o. The etermination of the parameters E, EP, EE an EEP is most simple in case of initial isotropy of the sample. The parameters E an EP are etermine from the ata of a triaxial compression test. at constant isotropic stress following irectly Equation (7. E,EP The parameters EE an EEP can be etermine from a raine triaxial extension test EE,EEP at constant isotropic effective stress as shown in Figure (5. To this en, to the measure shear stress level (t/s e in triaxial extension there exists an equivalent shear stress level (t/s c in (t/s c triaxial compression with the same measure of Triaxial Compression shear stress level f for triaxial compression an: s = Pa 1 F = 7 1 f (8 t t 1 χ (1 ( t e (1+ ( e. s s (t/s e Triaxial Extension for triaxial extension. Then the initial part of the resulting stress χ - strain curve can be fitte as (see Figure 5: Figure 5 - Determination of the parameters E, EP, EEP EE an EEP from triaxial compression an χ e = EE (9 extension tests s c For the etermination of tensile strength PTENS in isotropic loaing, either a hyraulic fracturing test or tensile test must be performe. For the etermination of viscosity VC at liquefaction, strain controlle liquefaction tests at ifferent rates must be performe. Summary of Require Tests For easy etermination of all parameters of an initially isotropic material, the following tests woul be neee: 1. 1 raine isotropic loaing-unloaing test.. 1 (or more raine triaxial compression test(s with monotonically increasing axial strain at ifferent constant isotropic stresses (or more raine triaxial extension test(s with monotonically ecreasing axial strain at ifferent constant isotropic stresses (or more raine triaxial test(s with ifferent cyclic eviatoric stress t an constant isotropic stress s hyraulic fracture test (or more strain controlle unraine triaxial compression test(s leaing to liquefaction. 7. In sintu K o test or strain controlle triaxial compression with zero lateral strain on unisturbe sample. Stress Path Molenkamp (1987 escribe a proceure for the etermination of most of the moel parameters from a raine triaxial test on apparently initially isotropic Eastern Schelt san with 5 = 15 μm an initial porosity n =.4. The stress path of the test involve: a. isotropic loaing up to a bar cell pressure. b. isotropic unloaing back to 1 bar cell pressure. c. 3 cycles of eviatoric loaing at a constant isotropic pressure of 1 bar.. isotropic loaing up to 4 bar an unloaing to 1 bar. e. another cycle of eviatoric loaing at a constant isotropic pressure of 1 bar. f. loaing towars failure at constant isotropic stress. THE PROPOSED PROCEDURE FOR THE DETERMINATION OF ALTERNAT MODEL PARAMETERS The stress path escribe in the previous section will be applie in the computer program (ENDOCH written by the authors with some equations of the ALTERNAT moel. The proceure will be illustrate through the following example. Ng an Dobry (1994 mae experimental an theoretical investigation on granular specimens compose of uniform spheres. 3 Tome XI (Year 13. Fascicule 3. ISSN

7 The iscrete element metho (DEM was use in simulating roun uniform quartz san uner monotonic raine loaing with constant mean stress an cyclic constant volume loaing (unraine. A typical -imensional specimen use by Ng an Dobry (1994 is shown in Figure (6. The porosity was calculate as the ratio between the pore area of the plane where the spheres centers lie an the box imensions. In general 3-imensional specimens, the porosity was efine as n = 1 - volume of spheres / volume of box. Two simulate granular specimens compose of spheres were use in this stuy (specimens B an C. All particles were assigne the an Dobry (1994 properties of quartz: shear moulus, G = MPa, Poisson s ratio, ν =.15, an the friction coefficient μ s = tan φ o =.5 where φ o is the interparticle friction angle. The specimens are escribe in Table (1 in which the thir an fourth columns show the total number of spherical particles N use in each specimen, as well as the ratios R 1 /R /R 3 of three ifferent particle sizes with N 1 /N /N 3 which are the numbers of particles having these sizes. Table 1 - Characteristics of specimens use in simulation, (from Ng an Dobry, Number an Sizes of Spheres in Specimens Specimen after Consoliation Total Number of (N Specimen μ 1 /N /N 3 an σ c Porosity Relative Density s Particles (R 1 /R /R 3 (kpa (n (% Cell Pressure (kn/m B C Figure 6 - Geometry of -imensional specimen use by Ng 91/79/8 (1/1.5/ 91/79/8 (1/1.5/ In all cases, N = N 1 + N + N 3 an all the specimens were isotropically consoliate to σ c = 137 kpa prior to monotonic an cyclic loaing. In the following steps, the ALTERNAT moel parameters will be calculate for specimen B. In Figure (7, the cell pressure (σ c is shown against the number of ata points (each point concerns with a loaing increment. It shoul be notice that the afore mentione stress path consists of many increments of loa, each followe by an unloaing an reloaing increment Zero-Dilatancy Line No. of Increments Figure 7 The cell pressure against number of 3,5,7,1,14 increments for the stress path followe by 1 numerical simulation In Figure (8, the effective stress path is shown in,9,11 terms of: -1 4,6,8,13 1 s = ( σ a + σ r (3 3 - t = ( σ a σ r ( where: σ a = axial stress, σ r = raial stress. Isotropic Stress (kn/m 5. Zero-Dilatancy Rule Figure 8 The effective stress path followe Dilatancy is inepenent of the increment of by the numerical simulation stress or its irection for a fixe stress point, an can be approximate by a linear function of stress ratio, (Zienkiewicz et al., 1987: Shear Stress, t (kn/m Tome XI (Year 13. Fascicule 3. ISSN

8 V& g = = ( 1+ α g ( M g η (3 & γ where: η = t/s an α g is constant. This simple rule preicts zero ilatancy whenever the line: η = M g (33 is reache. Generalization to three-imensional stress conitions can be one if a law of a Mohr-Coulomb type is assume (Zienkiewicz an Pane, 1977 for the zero ilatancy line, giving (Zienkiewicz et al., 1987: M g = 6sinφcv / (3 sinφcv sin3θ (34 where θ is Loe s angle efine by Molenkamp (1987 as: J 3 π π sin 3θ = 3 6 Tij, θ 3 (35 t 6 6 where: φ cv = a constant resiual angle of friction,j 3 = the thir invariant of the eviatoric pseuo stress tensor. In Figure (8, the zero-ilatancy lines were rawn epening on the equation of Zienkiewicz et al. (1987, i.e., Equation (34. The numbers in this figure inicate the sequence of loaing. 8.8 Spherical Stress, s (kn/m Volumetric Strain, v Figure 9 Stress strain path of the isotropic components of s an v Shear Strain, γ Shear Stress Level, t/s Shear Strain, γ Figure 1 The shear stress level against shear strain for the numerical simulation Spherical Strain, Ln(v A =.113 Ap =.37 Moel Results Best Fit Volumetric Strain, V Figure 11 The strain path in terms of shear Ln (s/pa Figure 1 Calculation of elastic parameters A an AP through logarithmic correlations strain an spherical strain In Figure (9, the stress - strain path of the isotropic components are shown in terms of s an the spherical (volumetric strain: V = 1 V o + ΔV ln 3 Vo (36 in which: V o = the initial volume of the sample. In Figure (1, the shear stress level (t/s is shown against the shear strain γ: h o + Δh vo + Δv 1 γ = 3ln + ln ho vo 6 (37 in which: h o = initial height of the sample = 76 mm. 34 Tome XI (Year 13. Fascicule 3. ISSN

9 In Figure (11, the strain path is shown in terms of the shear strain γ an the spherical strain V. It shoul be notice that mainly ue to cyclic loaing, a permanent volumetric strain an a shear strain are generate. The ratio of the rate of change of the volumetric strain an the shear strain, V & / γ&, in consecutive cycles is quite similar. The parameters A an AP can be calculate from an isotropic loaing test. In Figure (1, the relevant ata are put in the form of the logarithms of V an (s/pa, which shoul be linear. This relation is approximate by: V s A Pa AP = (38 The best fit of the relation gives: A =.113 an AP =.37. Knowing the parameters A an AP for a range of magnitues of the parameter V, the stress - strain response was calculate. Next from the calculate strains as given in Figures (9 an (11, the elastic strains, which were calculate using the calculate parameters, were subtracte. The remaining plastic components are shown in Figures (13, (14 an ( Spherical Stress, s (kn/m Plastic Volumetric Strain Figure 13 Spherical stress against plastic component of strain.1 Plastic Shear Strain Shear Stress Level, t/s Plastic Shear Strain Figure 14 Shear stress level against plastic shear strain for the numerical simulation Plastic Volumetric Strain, Ln(v -5.8 pb =.33 BP = Moel Results Best Fit Plastic Volumetric Strain Ln (s/pa Figure 15 Plastic spherical strain against plastic Figure 16 Calculation of elastic parameters B shear strain an BP through logarithm correlations Then using the remaining stress - plastic strain path of the initial loaing of Figure (13, the parameters B an BP were calculate. To this en, the initial loaing path in Figure (13 is approximate by the expression: s V = B (39 Pa In Figure (16, the relevant ata are put in the form of the logarithms of V an (s/pa which shoul be linear. The best fit of the relation gave the magnitues of B an BP as follows: B =.33 an BP =.189. The experimental ata of the shear stress level (t/s an the eviatoric plastic strain γ (Figure 14 can be use for the etermination of: 1- the loa history parameters κ an n. - the parameters of the harening E, EP, EE an EEP. 3- the initial anisotropic conition in case the material is not isotropic initially. BP Tome XI (Year 13. Fascicule 3. ISSN

10 . ANNALS OF FACULTY ENGINEERING HUNEDOARA International Journal Of Engineering Due to the straight envelopes of the alternating strain path of Figure (15, κ coul be chosen equal to 1, (Molenkamp, In Figure (17, the resulting kinematic shear strain γ kin from the stress reversal are shown as a function of the shear stress level (t/s. Cyclic Mobility It can be seen in Figure (17 that the average resulting kinematic eviatoric strain γ kin of loaing is slightly larger than the similar strain of unloaing. This ifference can be contribute to the cyclic mobility. Δ γ mob = Δγ loaing Δγ unloaing (4 It can also be notice that the shapes of the curves of loaing an unloaing are ifferent. The curve for loaing shows a continuously ecreasing tangent stiffness immeiately from the start, while for the unloaing curve the tangent stiffness remains quite large uring the first part of the unloaing path an then starts to ecrease much faster than the curve for loaing. This is also illustrate in Figure (18..8 Shear Stress Level, t/s Unloaing Loaing Shear Stress Level,t/s Loaing Unloaing Kinematic Deviatoric Plastic Strain, γ kin Figure 17 Kinematic eviation plastic strain in cyclic loaing from stress reversal against shear stress level 1. * 1. E = EP = Shear Stress Level, t/s Moel Results Best Fit Kinematic Deviatoric Strain Figure 19 Linear approximation to logarithm shear stress level against kinematic eviatoric Shear Stress Leve, t/s γ kin Figure 18 Cyclic mobility EE * = EEP =.365 Moel Results Best Fit Kinematic Deviatoric Strain Figure Linear approximation to logarithm shear stress level against kinematic eviatoric strain, unloaing strain, loaing At this stage still the harening parameters for triaxial compression E an EP an for triaxial extension E an EEP. The fit to the stress - strain curve for triaxial compression can be obtaine by using for the shear stress level below peak the following expression: { γ } EP * = E kin, c (41 s c In Figure (19, the best linear fit for shear stress levels between zero an peak of the logarithms of γ kin,c an (t/c c is shown. The fit gave the following values: E * = an EP =.4648 It is assume that the eviatoric strain rate, γ. c, in a raine triaxial compression test is relate to the eviatoric strain rate χ. at the same stress level (t/s c by, (Molenkamp, 1987: 36 Tome XI (Year 13. Fascicule 3. ISSN

11 & γ kin c in which: LB = parameter of the orer.1 up to.5. From Equation (4 it follows, if LB =.3, that: ( LB* EP Tome XI (Year 13. Fascicule 3. ISSN LB s &, χ (4 Pa * s.3*.4648 E = E = 3.881( 3 = 4.19 Pa The fit to the equivalent curve for triaxial extension is expresse by: E E * = { γ } EEP kin, e (43 s c In Figure (, the best linear fit for high shear stress levels to the logarithms of γ kin,e an (t/s c is shown. The fit gives the following values: EE * = an EEP =.365. From Equation (4, it follows, if LB =.3, that: ( LB* EP * s.3*.365 EE = EE = 1.649( 3 = 1.71 Pa In Figure (1, the average equivalent curves for triaxial compression an extension are shown together. Figure ( shows the relationship between volumetric an shear strains from which the parameter χ cv = 1. Shear Stress Level, t/s Extension Compression Loaing Unloaing Kinematic Shear Strain Figure 1 Kinematic eviatoric strain in cyclic Volumetric Strain, V χ cv = Shear Strain, γ Figure ( Shear strain against volumetric strain loaing from stress reversal against shear stress level Table ( - ALTERNAT moel parameters for the specimens use by Ng an Dobry (1994 as etermine by the propose proceure. Value Parameter Non-linear Elastic Moel: Deviatoric Plastic Moel: Harening for Triaxial compression: Harening for Triaxial Extension: Harening by Densification: Dilatancy: V A AP E EP CHIT (χ t TSINF (t/s cv TT (T BET (β CHIM (χ m NM (n m LB COH (c EE EEP POROS (n i DENSE (n KAP (κ FIMU (φ μ FICV (φ cv SCV (S cv CHICV (χ cv Tensile Strength: PTENS (σ t Viscosity in Liquefaction: VC (V c Initial State: CHII (χ I K o Specimen B (Molenkamp, (Molenkamp, (publishe ata o 6.5 o sec Specimen C (Molenkamp, (Molenkamp, (publishe ata o 6.5 o sec..5.55

12 The other parameters may be assume on the basis of publishe experimental ata of similar materials. The same proceure was followe for the etermination of the parameters for specimen C. The whole parameters are liste in Table (. CONCLUSIONS For etermination of the parameters of the ALTERNAT moel, special types of triaxial tests are require, e.g., raine triaxial tests with monotonically increasing or ecreasing axial strain an constant isotropic stress. These tests are not easy to be conucte in soil mechanics laboratories. It is intene here to choose a simple theoretical moel to get the require stress - strain relationships for the etermination of ALTERNAT moel parameters. Of many theoretical moels available to preict the overall response of sans, the enochronic moel is chosen for this task. The stress path require in the tests is applie in the computer program (ENDOCH written by the authors. The parameters are calculate for two samples for which Ng an Dobry (1994 mae experimental investigation. It is conclue that the propose proceure for the etermination of the ALTERNAT moel parameters is successful. The stress-strain relations preicte are similar to those obtaine by Molenkamp base on laboratory test results. REFERENCES [1] Bazant, Z. P. an Krizek, R. J., (1976. (Enochronic Constitutive Law for Liquefaction of San, Journal of Engineering Mechanics, ASCE, Vol. 1, EM, p. p [] Cuellar, V., Bazant, Z. P., Krizek, R. J. an Silver, M. L., (1977. (Densification an Hysteresis of San Uner Cyclic Shear, Journal of Geotechnical Engineering Division, ASCE, Vol. 13, ýgt5, p. p [3] Fattah, M. Y., (1999. (The Non-Linear Dynamic Behaviour of Soils, Ph.D issertation, University of Bagha. [4] Huang, T. K. an Chen, W. F., (199. (Simple Proceure for Determining Cap-Plasticity Moel Parameters, Journal of Geotechnical Engineering Division, ASCE, Vol. 116, No. 3, p. p [5] Molenkamp, F., (198. (Elasto-Plastic Double Harening Moel, MONOT, Report, Delft Geotechnics. [6] Molenkamp, F., (1985. (A Stress-Strain Curve for Both Harening an Softening, Proceeings of the International Conference on Numerical Methos in Engineering NUMETA85, Swansea, p.p [7] Molenkamp, F., (1987. (Kinematic Moel for Alternating Loaing, ALTERNAT, Report, Delft Geotechnics. [8] Ng, T. T. an Dobry, R. (1994. (Numerical Simulation of Monotonic an Cyclic Loaing of Granular Soil, Journal of Geotechnical Engineering Division, ASCE, Vol. 1, GT, p.p [9] Silver, M. L. an See, H. B., (1971. (Volume Changes in Sans During Cyclic Loaings, Journal of Soil Mechanics an Founations Division, ASCE, Vol. 97, SM9, p. p [1] Zienkiewicz, O. C. an Pane, G. N., (1977. (Some Useful Forms of Isotropic Yiel Sutfaces for Soil an Rock Mechanics, Chapter 5 in Finite Elements in Geomechanics, ete by G. Guehus, p.p [11] Zienkiewicz, O. C., Chan, A. H., Pastor, M. an Shiomi, T., (1987. (Computational Approach to Soil Dynamics, in Soil Dynamics an Liquefaction, eite by A. S. Cakmak, (Developments in Geotechnical Engineering-4, p.p ANNALS of Faculty Engineering Huneoara International Journal of Engineering copyright UNIVERSITY POLITEHNICA TIMISOARA, FACULTY OF ENGINEERING HUNEDOARA, 5, REVOLUTIEI, 33118, HUNEDOARA, ROMANIA 38 Tome XI (Year 13. Fascicule 3. ISSN