302 Three-dimensional Analysis of Unsaturated Soil using an Elasto by nak14542


									302                             Three-dimensional Analysis of Unsaturated Soil using an Elasto-viscoplastic Model

Key word:Unsaturated soils                                                     ○Kyoto University        Student member                 Huaiping      Feng
                 Elasto-viscoplastic model                                      Kyoto University        International member            Sayuri     Kimoto
                 Multiphase media                                                Kyoto University        International member            Fusao     Oka
                                                                                 Meijo University        International member            Takeshi Kodaka
1.   Introduction
                                                                                                                               &      ( ρ G V i G ) ,i Q M
    To analyze the unsaturated soil, a great number of                                    (1 − s w )n − s w n + (1 − s w )n
                                                                                                    & &                            =−                 + G (5)
constitutive models have been proposed (e.g., Alonso et                                                                        ρ G
                                                                                                                                          ρ  G
al.1990), most of those models are elasto-plastic models. By
                                                                                   where, n is the volume fraction of void, and sw is degree of
adopting the average skeleton stress from the viewpoint of
mixture theory and introducing the suction effect into model                       saturation. Here dencity of gas is assumed to change with gas
into an elasto-viscoplastic constitutive model for unsaturated                     pressure, while dencity of water remain             constant.
soil considering structure degradation (Kimoto and Oka 2005)                            The rate type of equlibrium equation is expressed as
for saturated soil, an elasto-viscoplastic model for unsaturated                   follows:
soil has been constructed to analyze the unsaturated soil.
Although the collapse behavior, which is brought about by a                                 &             v∫&
                                                                                                            S ji , j dV = 0
                                                                                   where S ji is the total nominal stress rate tensor.
decrease of suction, can be reproduced, the developed model
                                                                                       The van Genuchten type of equation is extended as
is still based on two phases, which can not reflect the pore
                                                                                   equation (7), which works also in the cases where suction is
water pressure and the pore air pressure separately. For this
                                                                                   less than zero.
reason, a van Genuchten type of equation is employed as the
constitutive equation between the saturation and the suction.                                             {                }
                                                                                        ⎧ (S max − S min ) (1 + (αP C ) n −m + S min
                                                                                                                                                 PC ≥ 0
Based on this, an air-water-soil three-phase coupled model                         sw = ⎨                                                                   (7)
has been proposed (Oka et al.2006), In the present study, a
three-dimensional multiphase finite element method using an                             ⎩ max     {
                                                                                        ⎪2S − (S − S ) (1 + (−αP C ) n −m + S
                                                                                                   max       min   {              }  min    }   PC < 0
elasto-viscoplastic constitutive equation is developed to                          4.   Material parameter and boundary condition
simulate the triaxial behaviour of unsaturated cylindrical
                                                                                       A twenty-node isoparametric element with a reduced
                                                                                   Gaussian(2×2×2) integration for the soil skeleton and an
                                                                                   eight-node isoparametric element with a full(2×2×2)
2. Elasto-viscoplastic constitutive model considering the
                                                                                   integration for pore water and pore air are used. Material
effect of suction
                                                                                   parameters used in this analysis are listed in table 1. Fig. 1
    As the basic stress variable in the model, the average
                                                                                   shows the finite element mesh together with boundary
skeleton stress tensor is given as follows.
                                                                                   conditions. A constant axial (z-direction) displacement of
       σ ij = σ ij − P F δ ij ; P F = s w P w + (1 − s w ) P G
                                                                         (1)       0.5%/min is applied to the nodes on the top surface. All
                                                                                                      Table 1 Material parameters
    where σ ij is the average skeleton stress tensor, P w and

 P are the partial stress values for the pore water pressure,
                                                                                        Initial suction (kPa)                              50             100
and the pore gas pressure, respectively. s w is the degree of                           Initial void ratio    e0                           1.07           1.05
saturation, and P F is the averaged pore pressure.                                      Initial saturation    sw                           33.61          21.50
    The viscoplastic stretching tensor is given in the following                        Elastic shear modulus G0(kPa)                      45100          46800
                                                                                        Consolidation yield stress σ mbi
                                                                                                                     '                     217            222
equation as
                                                                                        Initial air pressure Pg(kPa)
                                                                                                                                           250            250
                   ⎪ ⎛           ~      σ       '     ⎫
                                                    ⎞ ⎪ ∂f p                            Initial water pressure Pw(kPa)                     200            150
     D = Cijkl exp ⎨m ' ⎜η(* ) + M * ln
       vp                                      m
                                                    ⎟⎬ '         f >0    (2)
                   ⎪ ⎜                  σ           ⎟ ⎪ ∂σ
      ij                   0                  '
                   ⎩ ⎝                        mb    ⎠ ⎭ kl                              Swelling index            κ                        0.0102
                                                                                        Compression index λ                                0.114
where, σ mb is the strain hardening parameter, which control
                                                                                        Viscoplastic parameter m                           52
the size of boundary surface. The suction effect is introduced                          Viscoplastic parameter C1(1/s)                     1.0×10-11
                                                                                        Viscoplastic parameter C2(1/s)                     1.5×10-11
into the value of σ mb by

                                                                                        Stress ratio at critical state M*m                 1.0
                           1 + e vp ⎡               ⎧       Pc      ⎫⎤                  Suction parameter SI                               0.5
        σ mb = σ ma exp(
          '      '
                                ε kk ) ⎢1 + S I exp ⎨− S d ( i c − 1⎬⎥   (3)            Suction parameter Sd                               0.25
                           λ −κ        ⎢
                                       ⎣            ⎩       P       ⎭⎥
                                                                     ⎦                  Van Genuchten papameter α(1/kPa)                   0.065
                                                                                        Van Genuchten parameter n                          1.6
3.     Finite element method formulation for unsaturated soil                           Permeability of water at sw=1 kW(m/s)              1.0×10-6
      The conservation laws of mass for the water phase(W)                              Permeability of gas at sw=0 kG(m/s)                1.0×10-5
                                                                                        Shape parameter a                                  3.0
and the gas phase(G) are given in following equations:
                                                                                        Shape parameter b                                  2.3
                                                        QM                              Maximum saturation smax                            0.7
                            s w n + s w n = −ViW +
                                & &                                      (4)
                                                                                        Minimum saturation smin                            0

Three-dimensional Analysis of Unsaturated Soil using an Elasto-viscoplastic Model, Kyoto University, Huaiping Feng, Sayuri
Kimoto, Fusao Oka, Takeshi Kodaka
boundaries are assumed to be impermeable. Furthermore,                                                                                          agreement with the experimental data. Fig. 5 shows the
horizontal deformation is constrained at both top and bottom                                                                                    distribution of viscoplastic deviatoric strain with the
boundaries.                                                                                                                                     development of the axial strain, and it shows that the
                                                                                                                                                maximum value of γ                             p
                                                                                                                                                                                                       concentrates around the center of the
                                                                                                    Displacement with
                                                                                                                                                                              5 00
                                                                                                        Fixed                                                                             4 50 kP a(exp )
                                                                                                                                                                                          3 50 kP a(exp )

                                                                                                       Free only at z direction                                                           4 50 kP a

                                                                                                                                                      Deviator stress (kPa)
                                                                                                                                                                              4 00
                                                                                                                                                                                          3 50 kP a

                                                                                                                                                                              3 00

                                                                                                            Impermeable                                                       2 00

                                                                Y                                                                                                             1 00                     Pc =50kPa
                                                                                                                                                                                                       Strain rate: 0.5%/min
                                                                       X                                                                                                        0
                                                                        2.5cm                                                                                                        0    2        4       6       8    10    12   14    16
                                                            Fig.1. Finite element mesh and boundary                                                                                                     A xia l strain (% )
                                                                                                                                                                              Fig.4. Deviator stress vs. axial strain under
5.   Simulation results                                                                                                                                                                 different cell pressures
       Under undrained conditions for both air and water,
several cases were numerically analyzed with different initial
suctions, strain rates and confining pressures. Predicted
volumetric strain during compression with initial suction
(Pc=50kPa) was compared with experimental data for silty
clay as shown in Fig. 2. Good agreement is observed between
the model predictions and the experimental data up to axial
strain 8%. The relationship between degree of saturation and                                                                                                                         5%            10                15             20
axial strain with different initial suctions are shown in Fig. 3.                                                                                                                    Fig.5. Distribution of the γ p in
Using the extended soil water characteristic curve, predicted                                                                                                                        undrained-for-air-and-water test
results well agree with the experimental                                                                                  ones. The effect of
                                                                                                                                                6.   Conclusion
the confining pressure is also verified and results are shown in                                                                                      An air-water-soil three-phase coupled finite element
Fig. 4. From the curve, it can be observed that higher                                                                                          method based on an elasto-viscoplastic model has been
confining pressure leads to higher strength, which is in good                                                                                   introduced. Three-dimensional numerical simulations show
                                                                                                                                                that with this analysis method it is possible to reproduce the
                                                                                                                                                behaviors of unsaturated soil during triaxial compression,
                                                                                                                                                such as changing of pore-air pressure, pore-water pressure,
                                Volumetric strain (%)

                                                        1                                                                                       degree of saturation and volumetric strain. With the proposed
                                                                                                          P c =50kPa(exp)
                                                                                                          P c =50kPa                            method, it is possible to analyze the effect of initial suction,
                                                        2                                                                                       confining stress and strain rate on strength and pore pressures
                                                                                                                                                of unsaturated soil.
                                                                                                                                                7. Reference
                                                        4           σ 3 = 450kPa                                                                1) Alonso, E.E., Gens, A. and Josas, A., “A constitutive
                                                                0          2       4     6      8    10             12     14   16              model for partially saturated soils”, Geotechnique, 40(3), pp.
                                                                                       Axial strain (%)                                         405-430, 1990.
                                       Fig.2.Comparison of the volumetric strain vs.                                                            2) Kimoto, S. and F. Oka, An elasto-viscoplastic model for
                                                                                                                                                clay considering destructuralization and consolidation
                                                                                                                                                analysis of unstable behavior, Soils and Foundations, 45, 2,
                                                                       σ 3 = 450kPa
     Degree of saturation (%)

                                               40                                                                                               pp.29-42, 2005.
                                                                                                                                                3) Oka, F., Kodaka, T., Kimoto, S., Kim, Y.-S and Yamasaki,
                                                                                                                                                N., An Elasto-viscoplastic Model and Multiphase Coupled FE
                                               20                                                                                               Analysis for Unsaturated Soil, Unsaturated Soils Conference
                                                                                             P c = 5 0 k P a (e x p )                           April 2-6 2006, ASCE, Carefree Arizona, Geotechnical
                                                                                             P c = 1 0 0 k P a (e x p )
                                                                                             P c= 5 0 k P a                                     Special Publication, No.147, ASCE, 2006,
                                                                                             P c= 1 0 0 k P a
                                                            0          2       4        6         8      10         12     14   16
                                                                                       A x ia l s tra in (% )
                                   Fig.3. The degree of saturation vs. axial strain
                                           under different initial suction

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