EXPERIMENTAL AND COMPUTATIONAL MODELING OF UNSATURATED SOIL RESPONSE UNDER TRUE TRIAXIAL STRESS STATES Laureano R. Hoyos The University of Texas at Arlington lhoyos@uta.edu Workshop on Nonlinear Modeling of Geotechnical Problems Johns Hopkins University, Maryland, November 3, 005
OUTLINE PART I CONSTITUTIVE MODELING OVERVIEW PREVIOUS EXPERIMENTAL WORK MODEL PREDICTIONS PART II CURRENT RESEARCH WORK FUTURE WORK (FROM THEORY TO PRACTICE)
Background and Importance Traffic load Foundation load Pavement (σ 1 ) (u a u w ) (σ 1 ) (u a u w ) (u a u w ) (u a u w ) (σ ) (σ ) (σ 3 ) (u a u w ) (σ 3 ) (u a u w ) In nature, soils well above the ground-water table are often subject to three-dimensional stress gradients due to continuous changes in the stress state variables (σ ij δ ij ) and (u a u w )δ ij It is in this context where a true triaxial testing apparatus, capable of testing soil specimens along a wide range of simple-to-complex stress paths under controlled suction states, plays a fundamental role in a complete stressstrain-strength characterization of this type of materials.
Previous Work An existing implicit integration algorithm was further refined with the aim of predicting unsaturated soil response along suction-controlled multi-axial stress paths that are not achievable in a conventional cylindrical cell. The developed algorithm is based on a few modifications made to the constitutive framework postulated by Alonso et al. (1990) for unsaturated soils, referred to as Barcelona model in this work. The refined version of the algorithm supports numerical analyses in deviatoric plane (π-plane) using a mixed control constitutive driver, along with the Generalized Cam-Clay model, that also accounts for the influence of a third stress invariant Lode-angle θ within a constant-suction scheme. True triaxial data (σ 1 > σ > σ 3 ) from a series of constant-suction triaxial compression (TC), triaxial extension (TE), and simple shear (SS) tests, conducted on 10-cm (4-in) side, cubical specimens of silty sand, were used for the tuning and validation of the refined algorithm.
Barcelona Model (Alonso et al. 1990) q CSL(s) M 1 M 1 q CSL (s = M { p + p }{ p (s) p} = 0 s 0) o p dε q p dε p p s s s = 0 s p o (0) p o (s) p ( LC) LC s = s o SI s k Elastic region 1 po(s) po(0) c = c p p s = 0 c p p o (0) p o (s) p s λ(0) k λ(s) k p
Implicit CPPM-Based Integration σ e, s e ( n+1 σ e, n+1 s e ) ( n σ, n s ) ( n σ, n s ) ( n+1 σ, n+1 s ) o n F(σ, s, p o, s o ) = 0 n+1 F(σ, s, p o, s o ) = 0 The computational implicit integration procedure was developed as a Backward Euler (BE) return rule-based scheme for integrating the constitutive relations originally postulated by the Barcelona model. The solution of the unsaturated soil implicit integration problem can be devised as the projection (CPPM-based) of a trial stress state (σ, s) onto an updated yield surface n+1 F.
Adaptation of Willam-Warnke Function c = 1.0 c = 0.7 c = 0.53 g( θ, c) = (1 c )cos( θ π / 3) (1 c) 4(1 c 4(1 c )cos ( θ π / 3) + (1 c) )cos ( θ π / 3) + 5c 4c The Willam-Warnke function g(θ,c) for characterization of concrete behavior under general stress states was adopted herein. The function has been successfully used to capture constitutive response of soils. With the developed algorithm, the influence of Lode-angle θ on unsaturated soil response in the (p : q : θ) space is verified against a full set of results from a series of suction-controlled true triaxial tests on compacted silty sand.
Axis-Translation Technique Pore-air (u a raised to 0 kpa) Matrix suction, s = (u a u w ) = 101 kpa Soil solids Pore water u w = 101 kpa Water Translation of u w to 0 kpa absolute Air bubble (by diffusion) -bar ceramic disk (air-entry value = 0 kpa)
True Triaxial Testing Scheme Z u a X Y 9 Bolt 5-bar HAE disk u a u a Flush-in Flush-out u w (σ 1 ) (Hoyos 1998) SS (b = 0.5, θ = 30 o ) TC (b = 0, θ = 0 o ) b = σ σ 3 σ 1 σ 3 TE (b = 1, θ = 60 o ) θ A σ oct = 50, 100, or 00 kpa s = (u a u w ) = 50, 100, or 00 kpa (σ ) (σ 3 )
Silty Sand Response: Isotropic Loading O (after tamping compaction) Matric suction, s : kpa 400 00 Equalization ( I ) A Ramped consolidation ( II ) B 100 50 0 0 50 100 150 00 50 300 350 400 Net mean stress, p : kpa
Silty Sand Response: Isotropic Loading. A.18 B C D Barcelona model Experimental (test cell).14 s= 400 kpa Specific volume, v = 1 + e.10.06.0 1.98 A : p o ( 50 ) = 50 kpa B : p o (100) = 56 kpa C : p o (00) = 7 kpa s= 00 kpa s= 100 kpa s= 50 kpa 1.94 D : p o (400) = 78 kpa 1.90 0 50 100 150 00 50 300 350 400 450 Net mean stress, p : kpa
Silty Sand Response: Axi-symmetric Loading q : kpa s : kpa Soil state after tamping compaction T-S5-TC T-S6-CTC 00 T1-S5-TC T1-S6-CTC 150 T-S3-TC T-S4-CTC 100 T1-S3-TC T1-S4-CTC T-S1-TC T-S-CTC 50 T1-S1-TC T1-S-CTC 0 0 50 100 150 00 50 300 350 400 450 500 p : kpa
Silty Sand Response: Axi-symmetric Loading 100 1000 Deviatoric stress, q : kpa 800 600 400 M (s) 1 CSL (s = 00 kpa) CSL (s = 100 kpa) 00 CSL (s = 50 kpa) µ(s) 0 0 00 400 600 800 Net mean stress, p : kpa
Barcelona Model Parameters Parameter Value Units λ(0) 0. - κ 0.011 - β 17.89 (MPa) -1 r 0.1 - p c 0.036 MPa G 8.8 MPa M 0.98 - k 1.34 - p o (0) 0.041 MPa s o - MPa Model parameters were obtained from a comprehensive series of suctioncontrolled isotropic and axisymmetric loading tests conducted on compacted silty sand (Hoyos, 001).
Silty Sand Response: True Triaxial Loading TC: 0.4 (a) s = 50 kpa TE: 0.3 (a) s = 50 kpa SS: 0.3 (a) s = 50 kpa 0.3 Deviatoric stress, q : MPa 0.3 0. 0.1 Experim. Numerical Deviatoric stress, q : MPa 0. 0.1 Experim. Numerical Deviatoric stress, q : MPa 0. 0.1 Experim. Numerical 0. 0.1 0.0-0.10-0.05 0.00 0.05 0.10 0.0-0.10-0.05 0.00 0.05 0.10 0.0-0.10-0.05 0.00 0.05 0.10 0.0-0.05 0.00 0.05 0.4 (b) s = 100 kpa 0.3 (b) s = 100 kpa 0.3 (b) s = 100 kpa 0.3 Deviatoric stress, q : MPa 0.3 0. 0.1 Deviatoric stress, q : MPa 0. 0.1 Deviatoric stress, q : MPa 0. 0.1 0. 0.1 0.0-0.10-0.05 0.00 0.05 0.10 0.0-0.10-0.05 0.00 0.05 0.10 0.4 (c) s = 00 kpa 0.3 (c) s = 00 kpa 0.0-0.10-0.05 0.00 0.05 0.10 Major/minor principal strain : cm/cm 0.0-0.05 0.00 0.05 Intermediate strain : cm/cm Deviatoric stress, q : MPa 0.3 0. 0.1 0.0-0.10-0.05 0.00 0.05 0.10 Principal strain : cm/cm Deviatoric stress, q : MPa 0. 0.1 0.0-0.10-0.05 0.00 0.05 0.10 Principal strain : cm/cm τ 1 3 σ oct σ = 1 + σ 3 + σ 3 u oct = ( σ1 σ) + ( σ σ3) + ( σ1 σ3) q = 3 τ oct a
θ = 0 o θ = 30 o Response in Octahedral (π) Plane (a) (b) (c) (σ 1 ) : MPa θ = 60 o (σ 3 ) : MPa (σ ) : MPa (σ 1 ) : MPa θ = 30 o θ = 60 o θ = 30 o θ = 60 o θ = 0 o θ = 0 o s = 00 kpa s = 100 kpa s = 50 kpa s = 00 kpa s = 100 kpa s = 50 kpa (σ 3 ) : MPa (σ ) : MPa (σ 1 ) : MPa s = 00 kpa s = 100 kpa s = 50 kpa σ oct = 50 kpa σ oct = 100 kpa σ oct = 00 kpa (σ 3 ) : MPa (σ ) : MPa
True Triaxial Systems: Recent Developments Matsuoka et al. (00) developed a true triaxial apparatus with three pairs of rigid loading plates in three orthogonal directions. A 10-cm per side, silty soil specimen seats between upper and lower loading plates with the remaining four lateral surfaces covered by membranes. Upper and lower loading plates house HAEV ceramic disk (300-kPa air-entry value) and porous stones (5-mm diameter and covered with polyfluorotetraethylene filters). Suction states in the specimen are attained by inducing negative pore-water pressures via an external vacuum-based system.
True Triaxial Systems: Recent Developments (Matsuoka et al. 00)
True Triaxial Systems: Recent Developments Stress increment applied when all axial strain rates reached less than 10-5 /min. Rigid steel plates limit stress paths to θ = 0 30 o. Responses from controlled suction triaxial and true triaxial show good agreement. (Matsuoka et al. 00)
True Triaxial Systems: Recent Developments Hoyos et al. (005) have recently begun implementation of a novel true triaxial apparatus aimed at yielding a considerably enhanced performance. Both u a and u w are applied at the bottom face of a 3-in cubical soil specimen, while distilled de-aired water is used as pressurizing fluid against latex membranes. Aluminum cubical base piece Y Sintered stainless steel 0.15 0.050 5-bar disk Coarse stone 5-bar disk 0.750 0.050 0.100 3.0 Coarse stone 5-bar disk Coarse stone X 5-bar disk Coarse stone 5-bar disk 3.0
True Triaxial Systems: Recent Developments Air/water pressure are both supplied via nylon tubing from a PCP-5000-UNSAT pressure control panel. PCP-5000-UNSAT Pressure Panel External Pressure Control Panel Assembled Test Cell On-going testing involves a wide range of stress paths that are not achievable in a cylindrical apparatus. Further refinement contemplated in the near future includes temperature control in pore fluids, tip tensiometers, BE testing, and digital imaging processing. The device is being developed under a Major Research Instrumentation project (Award # 016545) sponsored by the U.S. National Science Foundation. D.A.S. This support is gratefully acknowledged.
FUTURE WORK: FROM THEORY TO PRACTICE Constitutive Model Parameters via SWCC 100 Degree of saturation, S (%) 80 60 40 0 Sand Botking Silt Regina Clay Indian Head Till 0 1 100 10000 1000000 Soil suction, (kpa)
FUTURE WORK: FROM THEORY TO PRACTICE q Uncontrolled-Suction Testing s p s Strain/deformation
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