E3S Web of Conferences 9, 17001 (2016) DOI: 10.1051/e3sconf/20160917001
E-UNSAT 2016
A finite strain elastoplastic constitutive model for unsaturated soils incorporating mechanisms of compaction and hydraulic collapse
Keita Nakamura1 and Mamoru Kikumoto1,a
1Yokohama National University, Tokiwadai 79-5, Hodogaya-ku, Yokohama 240-8501, Japan
Abstract. Although the deformation of unsaturated soils has usually been described based on simple infinitesimal theory, simulation methods based on the rational framework of finite strain theory are attracting attention especially when solving geotechnical problems such as slope failure induced by heavy rain in which large a deformation is expected. The purpose of this study is to reformulate an existing constitutive model for unsaturated soils (Kikumoto et al., 2010) on the basis of finite strain theory. The proposed model is based on a critical state soil model, modified Cam-clay, implementing a hyperelastic model and a bilogarithmic lnv-lnP’ (v, specific volume; P’, effective mean Kirchhoff stress) relation for a finite strain. The model is incorporated with a soil water characteristic curve based on the van Genuchten model (1990) modified to be able to consider the effect of deformation of solid matrices. The key points of this model in describing the characteristics of unsaturated soils are as follows: (1) the movement of the normal consolidation line in lnv-lnP’ resulted from the degree of saturation (Q, deviatoric Kirchhoff stress), and (2) the effect of specific volume on a water retention curve. Applicability of the model is shown through element simulations of compaction and successive soaking behavior.
1 Introduction In this study, we aim to develop a method that can suitably predict the entire life of an embankment from its Although the stress-strain relationship of unsaturated construction process to the failure stage owing to heavy soils has usually been described based on simple rain, for instance. In order to achieve this, an infinitesimal infinitesimal theory, application of the framework of constitutive model for unsaturated soils incorporating finite strain theory to unsaturated soil mechanics is compaction and collapse mechanisms (Kikumoto et al. attracting attention (e.g. [1, 2]), especially when [16]) is extended to a finite-strain model that can capture predicting geotechnical issues in which large deformation the behavior of unsaturated soils at a large strain level. In of the ground (such as a failure of slope or embankment the model, the shear strength of unsaturated soil is owing to heavy rain or earthquake) is expected. controlled by the movement of the normal consolidation The constitutive framework of finite strain line in the direction of the specific volume (v) axis with a elastoplasticity has been developed [3–6] based on the variation in the degree of saturation (Sr), and a soil water multiplicative decomposition of the deformation gradient characteristic curve incorporating the effect of changes in [7]. For geomaterials, several researchers [8–10] have the void ratio (e) is adopted. In this paper, we present an proposed stress-update algorithms based on return outline of a constitutive model for unsaturated soils based mapping in principal space [3, 11, 12]. Borja and on finite strain theory. The applicability of the model is Tamagnini [8] developed the infinitesimal version of the finally discussed through element simulations of modified Cam-Clay model for finite strain theory based compaction and the soaking behavior of unsaturated soils. on a multiplicative decomposition of the deformation ep gradient ( FFF). In their formulation, a yield 2 Outline of a constitutive model for function is defined as a function of the mean Kirchhoff stress (P) and deviatoric Kirchhoff stress (Q) instead of unsaturated soils mean Cauchy stress (p) and deviatoric Cauchy stress (q), In this section, a model for unsaturated soils respectively. In addition, for modeling the behavior of incorporating compaction and collapse mechanisms isotropic consolidation of soil, a bilogarithmic lnv-lnP (v, under a finite strain range is derived. This model consists specific volume) relation [13] is applied for some of two parts, namely, an elastoplastic stress-strain significant advantages [14, 15]. For unsaturated soil, relationship taking account of the effect of the degree of Song and Borja [1, 2] developed a mathematical saturation, and an advanced soil water retention curve framework for coupled solid-deformation/fluid-diffusion model considering effect of volumetric deformation. in a finite strain range. a Corresponding author: [email protected]
© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/).
E3S Web of Conferences 9, 17001 (2016) DOI: 10.1051/e3sconf/20160917001
E-UNSAT 2016
v v n n l l
e N e lnvref+lnx CL m f m u or u l dr l lnc0 o N ied o P′ v lnv +lny C v lln s ref L oi lnv0 P′ c f l c 0 i o (l i f r d ny f i ri = i N c NC ed x c CL e lnvref L so ) e -v(c) (y p fo il p y = s r (l s d ny -ln y f rie = f y0 0 ) o d x o s N oil (1- C m S m lnv lnc L (ln r) ( h y ) h y t = t = i l i y r 0) r ) a 1 a g g o o L L ln(-P′ref) ln(-P0′) ln(-P′) Logarithm of mean effective Kirchhoff stress ln(-P′) Logarithm of mean effective Kirchhoff stress ln(-P′) Figure 1. Assumed normal consolidation line for unsaturated Figure 2. Volumetric behavior of unsaturated soil considering soils: dependence of the position of NCL on variation of Sr. the effects of degree of saturation Sr and specific volume v.
W(,ee) 3 2.1 Elastoplastic constitutive model PP vsexp 1 e2 (5) e ref s v 2ˆ We first derive a stress-strain relationship for unsaturated W(,ee) soils based on the effective Kirchhoff stress: QP vs3( ex p ) e (6) e 0 ref s τ τ {Sr w(1S r ) a }I (1) s where τ Jσ (J, Jacobian; σ , Cauchy stress tensor) is the Kirchhoff stress tensor, Sr is degree of saturation, and 2.1.2 Yield function
Juf ( uf , pore fluid pressure) is the Kirchhoff pore fluid pressure. I is the second-order identity tensor. Since unsaturated soils having a lower degree of Subscripts w and a denote water and air, respectively. saturation usually exhibit stiffer behavior, the normal consolidation line (NCL) defined in the ln P - ln v plane is assumed to move in the direction of the specific 2.1.1 Hyperelastic model volume v axis owing to the variation of degree of saturation S (Figure 1) in the proposed model. In infinitesimal strain theory, a hypoelastic formulation is r usually employed. This approach, however, causes an ˆ P lnvvNCL ln ref ly ln ln (7) irrational dissipation of stored energy especially under Pref conditions of cyclic loading [17]. On the other hand, the vref is the specific volume v on the normal consolidation hyperelastic model guarantees the conservation of stored line at PP . lˆ is the compression index representing energy. Therefore, we apply a hyperelastic model [8] ref the slope of the compression line in the - with pressure-dependent bulk and shear moduli to ln P ln v describe the elastic response, which was originally relation, which is linked to the compression index l that proposed by Houlsby [18] in an infinitesimal strain range defines the slope of the compression line in ln p - and extended to finite strain theory by Borja and ln v [8]. Tamagnini [8]. l The hyperelastic potential function is defined as a lˆ (8) function of the elastic volumetric strain e and elastic 1 l v lny in Equation (7) is the separation between the NCL deviator strain e as s for current, the unsaturated state, and that for the e e3 e e2 saturated state in logarithmic scale. works as a state W(v,) s P ref ˆ exp s (2) 2 variable considering the effect of the degree of saturation where and e are given as Sr on the strength of the unsaturated soils. is ee assumed to increase as S decreases: v v,ref (3) r ˆ lnyx (1Sr ) (9) e ref P ref exp (4) x is a material parameter representing the vertical e distance of the state boundary surface for dried and Here, v,ref is the elastic volumetric strain at a reference saturated samples in the compression plane. The pressure P , and ˆ is the elastic compressibility index ref volumetric movements of NCL and CSL are also associated with the Kirchhoff pressure [8]. The elastic considered in a similar way in the previous work [19]. shear modulus contains a constant term ref and In order to take the effect of density or the variable term with a parameter for pressure overconsolidation ratio into account in the proposed dependency. From Equation (2), the mean and deviatoric model, we also define a logarithmic volumetric distance Kirchhoff stresses are given as follows: ln c ( c 1) between the current specific volume v and
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E3S Web of Conferences 9, 17001 (2016) DOI: 10.1051/e3sconf/20160917001
E-UNSAT 2016 specific volume vNCL on the normal consolidation line at the current mean effective stress as a state variable
lnv
NC L (Un sat)
lnvref
N lny CS CL L ( (S lnvref-(l-)ln2 Un at) sat)
ln(-P′ref) CS ln(-P′) L (Sa 2 t) Q ln 1 22 M P
Figure 3. State boundary surface. Figure 4. Soil water characteristic curve for three kinds of specific volumes. (Figure 2). Thus, the specific volume v can be written as follows: Table 1. Material parameters for water retention curve P max lnvv lnv lnc ln lˆ ln y ln c (10) Maximum saturation Sr 1.0 NCL ref min Pref Minimum saturation Sr 0.0 The state variable c is assumed to decrease with the Fitting parameter [1/kPa] 0.028 development of the plastic deformation and finally Fitting parameter n 1.5 converges to 1 as follows: Fitting parameter m 0.33 Reference specific volume v 1.9 c aln c (11) ref Effect of volumetric deformation xv 9.4 a is a material parameter controlling the effect of the density or overconsolidation ratio, where is the 2 increment of plastic multiplier. p ˆ PQ f (PQ,,,,)(y c v l ˆ )ln ln1 From Equation (10), the volumetric strain for P MP22 0 (16) isotropic consolidation can be written as yc ln ln ep 0 vPˆ yc v0 v l ln ln ln ln (12) yc00 v(c) vP yc 0 0 0 0 where p p p . where subscript 0 denotes the values at initial state. The v v(c) v(d) elastic component of the volumetric strain for pure For the plastic flow rule, we employ a formulation of isotropic loading is derived from Equation (5) as follows: the exponential approximation [3, 12]: f ee P e e,tr n1 ˆ ln (13) A, n 1 A , n 1 , A 1,2,3 (17) v(c) v0 P0 An,1 e,tr p From Equations (12) and (13), we obtain the plastic where A, n 1 A , n 1 A , n is the trial elastic principal volumetric strain for isotropic consolidation as Equation logarithmic strain, is the principal Kirchhoff effective (14): A pe stress, and t ( t tnn1 t ). v(c) v(c) v(c) (14) ˆ P yce (l ˆ)ln ln ln v0 2.2 Water retention curve model P0 yc 0 0 Selecting an elliptic-shaped yield surface as the modified In this study, an extended version of a water retention Cam-Clay model [20], the plastic volumetric strain owing curve model proposed by van Genuchten [21] is applied. to shearing, namely dilatancy, is defined as This model is capable of considering the effect of volume 2 change on retention characteristics. The model proposed p ˆ Q v(d) ( l ˆ)ln 1 (15) by van Genuchten is first given as a function of the MP22 Kirchhoff suction as: where M is a material constant and a critical state stress min max min n m ratio, which controls the aspect ratio of the ellipsoidal SSSSr r ( r r ) 1 (S) (18) yield surface. From Equations (14) and (15), the yield min max where Sr and Sr are the minimum and maximum function can be finally written as degrees of saturation, respectively. , n, m are material
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E3S Web of Conferences 9, 17001 (2016) DOI: 10.1051/e3sconf/20160917001
E-UNSAT 2016 parameters, and S Js (s, suction) is the Kirchhoff A differentiation of the plastic flow rule (17) results in suction. In order to consider the effect of the density of soils, we propose to replace S with a modified Table 2. Material parameters for elastoplastic model Kirchhoff suction S * as Equations (19) and (20). min max min * n m Elastic compressibility index ˆ 0.012 SSSSr r ( r r ) 1 (S ) (19) Elastic shear modulus 0 [MPa] 20.0 xv * v SS (20) Elastic parameter 100.0 v ref Elastoplastic compressibility index lˆ 0.108
Here, vref is a reference specific volume, and xv is a Critical stress ratio M 1.2 material parameter for controlling the effect of the Reference pressure P’ref [kPa] -98.0 volumetric deformation of soil. Figure 4 shows the water Reference specific volume vref 1.9 retention curves for three kinds of specific volume (v = 0.28 1.90, 1.75, and 1.60). It is properly simulated by the Effect of movement of NCL x model that soil with a higher density has a higher degree Effect of density and confining pressure a 0.03 of saturation under the same value of suction, which is reported by experimental results [22]. Table 3. Initial state of soil for simulations
Suction s [kPa] 0.0 e 2.3 Return mapping Elastic volumetric strain v0 0.0 Mean Kirchhoff stress P’ [kPa] -98.0 Next, we derive the return mapping procedure based on 0 Specific volume v0 1.85 the elastic principal logarithmic strains [23]. In the return e mapping of proposed model, An,1 (A = 1, 2, 3), cn1 , e2 dεn1 dε n 1ββ f n 1 dβ n 1 β f n 1 d n 1 . (23) and are unknown variables evaluated for given From Equations (2) and (23), the stress-strain relation values An,1 and Sn1 . Thus, the unknown variable with the Hessian matrix is derived as [4] vector x and residual vector rx() take form, dβn1Ξε n 1 d n 1 β f n 1 d n 1 (24) respectively, as 1 e-1 2 Ξcn1 n 1 ββ f n 1 (25) e e,tr fn1 ε ε e 2 e e e nn11 where c W(,), and Ξ is the Hessian matrix. εeeε vs εn1 βn1 , (21) dcn+1 is obtained by differentiating the backward-Euler x cn1 rx( )cn11 c n a ln c n p approximation of Equation (11) as follows: f(,,, Pn1 Qn 1y n 1 c n 1, v, n 1 ) accln ddc nn11 (26) n1 c a where β is the principal effective Kirchhoff stress n1 Inserting Equations (23), (24), and (26) into the vector. The backward-Euler approximation is applied to consistency condition, we obtain Equation (11). y n1 can be calculated from Equation (9) and (19). Equation (21) is nonlinear, so the solution x dfn1 β ff n 1 dβ n 1 y n 1 dy n 1 needs to be evaluated iteratively. To this end, we use p (27) c ffn1ddc n 1 p n 1 v 0 Newton’s method with a local Jacobian matrix r()/ x x v until rx()< TOL is satisfied. Solving for d gives
In the elastic regime (i.e., unloading), the above dxn1N n 1 Ξε n 1 d n 1 dS r / D n 1 (28) procedure is not implemented. This can be judged by the where value of the yield function with trial values Ptr , Qtr , 2 n1 n1 Nn1βff n 1δ β β n 1 , (29) tr p,tr p y n1 , ccnn1() , and v,nn 1() v, as follows: aln c D N Ξδ ff n1 . (30) tr tr tr tr p,tr n1 n 1 n 1β n 1β n 1 fn1 f(,,,,)P n 1 Q n 1y n 1 c n 1 v, n 1 (22) cn1 a tr δ : (1,1,1)T is the vector in the principal space. The The trial values except for cn1 hold as updated values if tr algorithmic tangent moduli β / ε and fn1 0 (elastic state), and cn1 is obtained by solving n1 n1
βnn1/ S r, 1 are obtained by inserting Equation (29) fn1 0. into Equation (24) as follows:
βn1 2.4 Algorithmic tangent moduli ΞΞNΞn1 n 1β f n 1 n 1 n 1 / D n 1 (31) εn1 An algorithmic tangent moduli that is consistent with a βn1 formulation based on the principal space is presented xΞn1 β f n 1 / D n 1 (32) S here. The use of the algorithmic tangent moduli is r,n 1 necessary to ensure the convergence of iterations.
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E3S Web of Conferences 9, 17001 (2016) DOI: 10.1051/e3sconf/20160917001
E-UNSAT 2016
These algorithmic tangent moduli correspond with same set of initial parameters (such as density and the conventional elastoplastic tangent moduli when 0 . stress state) is also used in the simulations, which are summarized in Table 3. As shown in Table 3, the initial
(a)
Figure 7. Compaction curves for several kinds of applied (b) maximum stresses (final states of compaction simulations)
(a)
Figure 5. Simulations of isotropic compression of unsaturated soils under constant suction.
(a)
(b)
(b)
Figure 8. Simulations of triaxial test under constant suction and confining pressure.
state of the soil is assumed to be saturated. Figure 6. Simulations of soaking soils. The compression behavior of unsaturated soils having different values of degree of saturation is shown. Figure 5 3 Simulations shows the isotropic compression curves of unsaturated soil under constant suction. In this simulation, the suction Soaking and compaction behaviors of unsaturated soils was first increased to prescribed values (50, 100, and 150 are simulated by the proposed model, and the kPa) under constant net stress ( net 98kPa ) and three applicability of the model is discussed here. In the different values of the degree of saturation are obtained simulations presented here, the same sets of material for each case. It is indicated that the proposed model can parameters for the water retention curve and elastoplastic described the typical behaviors of unsaturated soils model summarized in Table 1 and 2 are adopted. The where: unsaturated soil is able to stay over the NCL for
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E3S Web of Conferences 9, 17001 (2016) DOI: 10.1051/e3sconf/20160917001
E-UNSAT 2016 saturated soil in the beginning stage of compression; the References degree of saturation increases owing to volumetric compression (even though suction is kept constant); there 1. X. Song, R.I. Borja, Vadose Zone J. 13, 5 (2014) is rapid compression behavior to the NCL for saturated 2. X. Song, R.I. Borja, Int. J. Numer. Meth. Eng. 97, soil with an asymptotic increase in the degree of 658-682 (2014) saturation to 100%. Such a tendency has been reported in 3. J.C. Simo, Handbook of Numerical Analysis, 183– past experimental works by authors such as Wheeler and 499 (1998) Sivakumar [24] and Kayadelen [25]. 4. J.C. Simo, T.J.R. Hughes, Computational Inelasticity We show a response of the proposed model from 7 (1998) soaking. In Figure 6, the soil is soaked under constant net 5. F. Dunne, K. Runesson, Introduction to stress (-200, -300, -500, and -800 kPa) after the isotropic Computational Plasticity (2005) compression shown in Figure 5 (case: s = 100 [kPa]). In 6. E.A. de Souza Neto, D. Peric, D.R.J. Owen, simulations, the soaking collapse behavior of soils can be Computational Methods for Plasticity: Theory and seen by decreasing suction s to zero. Applications (2008) The compaction behavior of soils is simulated in 7. E.H. Lee, Elastic-plastic decomposition at finite Figure 7. Suction s is first increased under a constant net strains, J. Appl. Mech. 36, 1–6 (1969) 8. R. Borja, C. Tamagnini, Comput. Method. Appl. M. stress ( net 98kPa ) until the prescribed water content w is achieved. Compaction behavior can be regarded as 155, 73–95 (1998) the exclusion of entrapped air without significant 9. G. Meschke, W.N., Liu, Comput. Method. Appl. drainage of void water. Thus, in this simulation, the total Mech. Eng. 173, 167–187 (1999) Cauchy stress is increased from -98.0 kPa to a 10. Y. Yamakawa, K. Hashiguchi, K.Ikeda, Int. J. predetermined maximum value (-200, -400, -600, -800, - Plasticity 26, 634–658 (2010) 1000, and -1200 kPa) with constant water content, and 11. J.C. Simo, Comput. Method. Appl. Mech. Eng. 85, returns to -98.0 kPa. Figure 7 shows that the proposed 273–310 (1991) model can simulate the typical compaction behavior of 12. J.C. Simo, Comput. Method. Appl. Mech. Eng. 99, soils. 61–112 (1992) We validate the proposed model in terms of the shear 13. K. Hashiguchi, M. Ueno, Constitutive equations of strength of unsaturated soils. In Figure 8, we simulate soils, Proc., Ninth Int. Conf. Soil Mech. Found. triaxial tests under constant suction (drained water and Engrg., 73–82 (1977) drained air). Suction was first increased to prescribed 14. R. Butterfield, Geotechnique 29, 469–480 (1979) values (100, 200, and 300 kPa) before shearing. Shearing 15. K. Hashiguchi, Int. J. Numer. Anal. Met. 19, 367– is then simulated under a constant confining pressure. 376 (1995) Figure 7 shows that the proposed model can simulate the 16. M. Kikumoto, H. Kyokawa, T. Nakai, H.M. Shahin, difference in the strength against shearing, i.e., Unsaturated Soils, 849–855 (2010) unsaturated soil is stiffer than saturated soil. 17. M. Zytynski, M.K. Randolph, R. Nova, C.P. Wroth, Int. J. Numer. Anal. Methods Geomech. 2, 87–93 (1978) 4 Conclusion 18. G.T. Houlsby, Comput. Geotech 1, 3–13 (1985) 19. D. Gallipoli, A. Gens, R. Sharma, J. Vaunat, A model for unsaturated soils that can describe Geotechnique 53, 123–135. compaction and collapse mechanisms under a finite strain 20. K.H. Roscoe, J.B. Burland, Engineering Plasticity, range is presented here. The model is an extended version 535–609 (1968) of an infinitesimal model proposed by Kikumoto et al. 21. M.T. van Genuchten, Soil Sci. Soc. Am. J. 44, 892– [16]. Essential concepts of the model in describing the 898 (1980) behavior of unsaturated soils are as follows: shifting the 22. A. Tarantino, S. Tombolato, Geotechnique 55, 307– NCL in the direction of the specific volume with a 317 (2005) change in the degree of saturation, and a dependency of 23. R.I. Borja, K.M. Sama, P.F. Sanz, Comput. Methods the degree of saturation on specific volume or density. Appl. Mech. Eng. 192, 1227–1258 (2003) The proposed model is formulated based on Kirchhoff 24. S.J. Wheeler, V. Sivakumar, Geotechnique 45, 35–53 stress invariants in order to predict the large deformation (1995) behavior of unsaturated soil based on finite strain theory. 25. C. Kayadelen, Environ. Geol. (54), 325–334 (2008) It is indicated through the simulations presented in this paper that the proposed model can describe the typical behavior of unsaturated soils such as soaking collapse phenomena, compaction of soils and shearing behavior.
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