A Critical State Sand Model with Elastic-Plastic Coupling

Transcription

1 A Critical State Sand Model with Elastic-Plastic Couling Ali Lashkari * and Ali Golchin Deartment of Ciil & Enironmental Engineering Shiraz Uniersity of Technology, Shiraz, Iran lashkari@sutech.ac.ir, lashkari_ali@hamyar.net Summary. Exerimental studies hae reealed that lastic strains change the elastic roerties of granular soils. This obseration is an elastic-lastic couling henomenon which is considered here through the deendence of a free energy function to a roer hardening arameter. It is shown that imlementation of the resulted hyerelastic model in a critical state bounding surface lasticity latform leads to more reasonable simulations. Keywords: sand, bounding surface, elastic-lastic couling, hyerelasticity. 1 Introduction In liuefaction henomena, the restrain on olume change necessitates a balance between the elastic and lastic strains. Hence, the elastic ortion of behaior has a rofound influence in liuefaction analysis. The elastic behaior of granular soils deends on soil density as well as mean rincial effectie stress. Moreoer, the eolution of fabric associated with shear stress affects the elastic resonse of granular soils (e.g., [1,2]). Seeral noel constitutie models in the bounding surface lasticity framework hae been suggested accounting for the effects of induced anisotroy through modification of elements dealing with the lastic art of behaior such as lastic hardening modulus and dilatancy [3,4]. In these models, howeer, the elastic ortion always remains isotroic een when the mentioned lastic ingredients reach the so-called fully saturated state of anisotroy. It is clear that the assumtion of the isotroy of elasticity has been made mainly due to the simlicity. Recently, Lashkari [5] suggested a simle elastic-lastic couled hyoelastic model and showed that induced anisotroy of elasticity articiates actiely in sudden loss of strength in liuefaction of dense sands subjected to cyclic loading. The so-called small strain shear modulus of granular soils (measured at shear strains less than 1-5 ) is usually exressed by the following relationshi: G = G ref F( e) (1) ref * Corresonding author. Q. Yang et al. (Eds.): Constitutie Modeling of Geomaterials, SSGG, sringerlink.com Sringer-Verlag Berlin Heidelberg 213

2 2 A. Lashkari and A. Golchin where G is a dimensionless constant and ref is a reference ressure ( 11 kpa). F(e) imoses the influence of oid ratio, e, on G. The best known suggestions are F(e)=(2.97-e) 2 /(1+e), and F(e)=(2.17-e) 2 /(1+e), resectiely, for soils with angular and round grains [6]. Mean rincial effectie stress,, has a great influence on the elastic shear moduli of soils through the term (/ ref ). Hardin & Black [6] recommended.5 for sands under extremely low strain leels ( 1-5 ) when soil behaior is urely elastic. Surrisingly, when granular soils are subjected to moderate-large shear strains beyond the retty small elastic domain, it has been reealed that is no more constant and increases with the irrecoerable shear strains (e.g., [1,7-9]). Using the aailable exerimental data in the literature, the measured -shear strain relationshi is illustrated in Fig. 1. Eolution of with irrecoerable shear strains is an eidence of elastic-lastic couling which is not addressed in the currently aailable soil elastic models (e.g., [5,1,11]). 2 Hyerlasticity with Elastic-Plastic Coulng In most soil constitutie models, the elastic art of behaior is simulated using Hyoelastic models which do not consere energy (e.g., [3-5,12]). The remedy for this deficiency is in Hyeleasticity theory in which elastic constitutie euations are deried from free energy functions (e.g., [1,11]). Here, it is assumed that the hyerelastic resonse of granular media is uniuely defined by a Gibbs free energy function exressed by Γ=Γ(,,). Variables (=(σ 1 +2σ 3 )/3) and (=σ 1 -σ 3 ) are, resectiely, mean rincial effectie stress and deiator stress defined in triaxial sace. Now, elastic strain rates are calculated by: e =Γ, +Γ, + Γ, ; e =Γ, +Γ, + Γ, re co erable irre coerable r e co erable irre co erable ε ε (2) e e where ε ( = ε 1 + 2ε 3) and ε = 2 / 3( ε 1 ε 3) are, resectiely, the elastic olumetric and shear strains. X, Y is the comact form for oeration X/ Y. In E. (2), aries with a roer hardening arameter. Hence, for ure elastic resonse, one has =. The first two terms in each elastic strain rates are recoerable, but the last term (in each) is irrecoerable uon unloading due to fabric eolution associated with loading. Rearrangement of terms yields the following relationshi to relate the rates of the stress ariables to the rates of the recoerable elastic strains: = Γ, e er 1 Γ, Γ, ε Γ, K J ε = e er (3) 2 Γ, Γ Γ Γ,,, ε Γ, J 3G ε

3 A Critical State Sand Model with Elastic-Plastic Couling 21 1 Silica sand No. 2 (e=.71, Dr=8%): Siler and Seed (1971) Silica sand No. 2 (e=.83, Dr=45%): Siler and Seed (1971) Ottawa sand No. 2-3 (e=.46): Drneich et al. (1967) Ottawa sand No. 2-3 (e=.62): Drneich et al. (1967) Ottawa sand No. 3-5 (e=.55): Drneich et al. (1966) Dry Ottawa sand No. 2-3 (dense): Hardin and Richart (1963) Crushed Quartz sand No. 2-3 (dense): Hardin & Richart (1963) Aerage of 4 sands (e=.6-.88): Kuribayashi et al. (1975) Toyoura sand (e=.64): Kokusho (198) Toyoura sand: Iwasaki et al. (1976) Toyoura sand (e=.62-.7): Iwasaki et al. (1978).9.8 Exonent, (-) Shear Strain, γ =3/2 ε (%) Fig. 1. Effect of shear strain on eolution of (exerimental data after [11]) er er where ε and ε are recoerable olumetric and deiator elastic strain rates (see E. 2). K, J, and G are, resectiely, the elastic bulk, couling, and shear moduli which are calculated by artial differentiation of the Gibbs free energy function. The yield function, domain of ure elasticity, in - lane is exressed by: f = η α m = (4) where η=/ is stress ratio, α is a hardening arameter, and m reresents the yield function size [3-5]. In the bounding surface latform suggested by Dafalias & Manzari [4], the following constitutie euations control the hardening: α = η = K ε K s L. (5) = In E. (5), ε is deator lastic strain rate, and L is the lastic multilier. One has s = +1 if η α = m, and s = -1 if α η = m. Finally, < > are Macauley brackets. In this study, lastic modulus, K, is defined by:

4 22 A. Lashkari and A. Golchin b α s α ref K = ( ) α α h e (6) in where h(e) is a function of oid ratio, α in is the initial alue of α in the most recent loading, and α b is the bounding surface back stress ratio. Detailed definition of these terms can be found in [4,5]. It is suggested that can eole with α through the following relationshi: α = ( α) = + ( max ) (7) b α + B ( α α ) where Β is a arameter controlling the ace of with change in α. In aboe euation, and max are the boundaries of associated with ery small and large shear strains, resectiely. Each strain rate can be arted into the elastic and lastic arts. Now by referring to decomosition resented in E. (2), and recalling that lastic strains are irrecoerable uon stress unloading, one has: 1 irrecoerable = e + = er ε ε ε ε + ei + ε ε = er + i ε ε e er ei er i ε = ε + ε = ε + ε + ε = ε + ε irreco erable (8) i where ε and ε i are the olumetric and deiatoric irrecoerable strain rates. Considering Es. (5), the irrecoerable strain rates can be written by: i i ε = Γ,, K ε + ε ; ε = (1 + Γ,, K ) ε. (9) α None of the existing conentionally aailable exerimental deices and their corresonding testing techniues is caable of direct searating the irrecoerable elastic strain rate from the lastic strain rate in olumetric and shear irrecoerable strain rates. Based on this conclusion, the following definition is made for dilatancy in elastic-lastic couled geomaterials: where d is dilatancy function [3-5]. i ε Γ,, α K ε + ε d = = (1) i s ε s ( Γ,, α K ε + ε ) α 3 Ealuation Simulation of three loading/unloading triaxial tests on dense samles of Toyoura sand is studied here. Isotroically consolidated samles ( in =1 kpa) were

5 A Critical State Sand Model with Elastic-Plastic Couling 23 firstly subjected to shear stress u to 25% axial strain. Then, they were unloaded to isotroic stress condition. To inestigate the influence of different elasticity theories on the redictie caacity of constitutie models, seeral elasticity theories are imlemented within the model of Dafalais & Manzari [4]. In Fig. 2(a), redicted stress aths using the model of Manzari & Dafalias [12] are shown. In the first shear loading, fabric is nearly isotroic. Therefore, the model redictions are accetable. Howeer, rior to unloading, soil fabric reaches a saturate or nearly saturate state of anisotroy. As a result, this model which ignores the influence of anisotroy on both elastic and lastic ingredients is incaable of roiding reasonable redictions in the unloading hase. In Fig. 2(b), it can be obsered that the modification of lastic modulus by fabric-dilatancy effects in the Dafalias & Manzari [4] model leads to imroement of redictions. Howeer, one may find shar return of stress aths following the unloading oint in measured data which can not be simulated by this model due to the assumtion of isotroy in elasticity. As a conseuence, the simulated half butterfly loos are not similar to those of data. Predictions by the model of Lashkari [5] with anisotroic hyoelasticity are shown in Fig. 2(c). A reasonable agreement between exeriments and simulations indicates the imortance of considering the influence of induced anisotroic elasticity on the redictie caacity; howeer this aroach does not consere energy. Predictions by the imlementation of the hyerelastic models of Eina & Puzrin [1] and Houlsby et al. [11] are shown in Fig. 2(d). The shar return of stress aths following the unloading oints indicates that the hyerelastic aroaches are caable of considering the effect of induced anisotroy; howeer it is simly becomes clear that the magnitude of generated anisotroy in these models is inadeuate. Finally, redictions by the alication of the hyerelastic formulation of this study with the ossibility of elastic-lastic couling are illustrated ersus exeriments in Fig. 3(a). In the later aroach, a modified Eina & Puzrin [1] Gibbs free energy function in the following form is adoted: ( (2 ) (2 Γ,, ) = (2 )(1 ) K G G ref ref ref ). (11) where and are, resectiely, alues of mean rincial effectie and deiator stresses at zero elastic strains. It must be noted that desite the original definition of [1], in this study eoles in range though E. (7). The eolution of with shear strain is shown together with exerimental data in Fig. 3(b). Considering that the samles initial state arameter alues are relatiely the same, nearly identical eolution cures for are obtained. Howeer, lower cures for samles in loose state are exected which are not resented here. Comaring Fig. 3(a) to Fig. 2(d) indicates that considering the elastic-lastic couling leads to further generation of anisotroy in elasticity and imroement of simulations.

6 24 A. Lashkari and A. Golchin Shear Stress, [kpa] Shear Stress, [kpa] Shear Stress, [kpa] Shear Stress, [kpa] (a) Isotroic Hyoelasticity (Manzari & Dafalias, 1997) e= e= e= Mean Princial Effectie Stress, [kpa] (b) Isotroic Hyoelasticity + Fabric-Dilatancy effects (Dafalias & Manzari, 24) e= e= e= Mean Princial Effectie Stress, [kpa] (c) Anisotroic Hyoelasticity (Lashkari, 21) e= e= e= Mean Princial Effectie Stress, [kpa] (d) Hyerelasticity (Eina & Puzrin, 24) and (Houlsby et al., 25) e= e= e= Mean Princial Effectie Stress, [kpa] Fig. 2. Ealuation of elasto-lastic models with different elasticity theories: (a) isotroic hyoelasticity [12]; (b) isotroic hyoelasticity with fabric-dilatancy effects [4]; (c) anisotroic hyoelasticity of Lashkari [5]; (d) hyerelastic theories of Eina & Puzrin[1] and Houlsby et al. [11] (data taken from [13])

7 A Critical State Sand Model with Elastic-Plastic Couling 25 Shear Stress, [kpa] (a) This Study e=.734 e=.764 e= Mean Princial Effectie Stress, [kpa] Exonent, [-] (b) Toyoura sand Other sands Simulation Shear Strain, γ = 3/2 ε [%] Fig. 3. Simulations using the roosed aroach of this study: (a) stress aths (data taken from [13]); (b) eolution of with shear strain (data after [9]) References [1] Koseki, J., Kawakami, S., Nagayama, H., Sato, T.: Change of small strain uasielastic deformation roerties during undrained cyclic torsional shear and triaxial tests of Toyoura sand. Soils and Foundations 4(3), (2) [2] Kuwano, R., Jardine, R.J.: On the alicability of cross-anisotroic elasticity to granular materials at ery small strains. Géotechniue 52(1), (22) [3] Paadimitriou, A.G., Bouckoalas, G.D., Dafalias, Y.F.: Plasticity model for sand under small and large cyclic strains. ASCE Journal of Geotechnical and Geoenironmental Engineering 127(11), (21) [4] Dafalias, Y.F., Manzari, M.T.: Simle lasticity sand model accounting for fabric change effects. ASCE Journal of Engineering Mechanics 13(6), (24) [5] Lashkari, A.: A SANISAND model with anisotroic elasticity. Soil Dynamics and Earthuake Engineering 3, (21)

8 26 A. Lashkari and A. Golchin [6] Hardin, B.O., Black, W.L.: Sand stiffness under arious triaxial stresses. ASCE Journal of the Soil Mechanics and Foundation Diision 92(SM2), (1966) [7] Kokusho, T.: Cyclic triaxial test of dynamic soil roerties for wide strain range. Soils and Foundations 2(2), 45 6 (198) [8] Siler, M.L., Seed, H.B.: Deformation characteristics of sands under cyclic loading. ASCE Journal of the Soil Mechanics and Foundation Diision 97(SM8), (1971) [9] Ishihara, K.: Soil behaior in earthuake Geotechnics. Oxford Science Publications (1996) [1] Eina, I., Puzrin, A.M.: Pressure-deendent elasticity and energy conseration in elastolastic models for soils. ASCE Journal of Geotechnical and Geoenironmental Engineering 13(1), (24) [11] Houlsby, G.T., Amorosi, A., Rojas, E.: Elastic moduli of soils deendent on ressure: a hyerelastic formulation. Géotechniue 55(5), (25) [12] Manzari, M.T., Dafalias, Y.F.: A critical state two-surface lasticity model for sands. Géotechniue 47(2), (1997) [13] Verdugo, R., Ishihara, K.: Steady state of sandy soils. Soils and Foundations 36(2), (1996)