Abstract
This paper investigates three-dimensional elastic–viscoplastic consolidation behaviors of transversely isotropic saturated soils. The Drucker–Prager yield criterion for isotropic materials is extended for modeling transversely isotropic medium. By coupling Perzyna’s viscoplastic theory with a transversely isotropic soil model, the incremental one-dimensional (1D) Nishihara’s constitutive model is established and then extended to formulate a three-dimensional (3D) material model. The method to obtaining the material parameters for the proposed model is also provided. The return mapping algorithm and algorithmic tangent matrix are presented to numerically implement the proposed theory into the finite element package—ABAQUS. We have validated the proposed theory by comparing numerical results with the uniaxial compression testing data of Shanghai soft clay and the field observations of soft soil embankments in the Dongting Lake area in China. Then, several numerical examples are conducted to study the influences of elastic–viscoplastic and transversely isotropic parameters on the time-dependent behavior of saturated soils.
Similar content being viewed by others
References
Abousleiman YN, Cheng HD, Jiang C, Roegiers JC (1996) Poroviscoelastic analysis of borehole and cylinder problems. Acta Mech 119(1):199–219
Adachi T, Oka F (1982) Constitutive equations for normally consolidated clay based on elasto-viscoplasticity. Soils Found 22(4):57–70
Ai ZY, Cheng YC (2014) Extended precise integration method for consolidation of transversely isotropic poroelastic layered media. Comput Math Appl 68(12):1806–1818
Ai ZY, Dai YC, Cheng YC (2019) Time-dependent analysis of axially loaded piles in transversely isotropic saturated viscoelastic soils. Eng Anal Bound Elem 101:173–187
Ai ZY, Jiang XB, Hu YD (2014) Analytical layer-element solution for 3D transversely isotropic multilayered foundation. Soils Found 54(5):967–973
Ai ZY, Ye Z, Zhao YZ (2020) Consolidation analysis for layered transversely isotropic viscoelastic media with compressible constituents due to tangential circular loads. Comput Geotech 117: 103257
Ai ZY, Zhao YZ, Liu WJ (2020) Fractional derivative modeling for axisymmetric consolidation of multilayered cross-anisotropic viscoelastic porous media. Comput Math Appl 79(5):1321–1334
Ai ZY, Zhao YZ, Song XY, Mu JJ (2019) Multi-dimensional consolidation analysis of transversely isotropic viscoelastic saturated soils. Eng Geol 253:1–13
Biot MA (1941) General theory of three-dimensional consolidation. J Appl Phys 12(2):155–164
Biot MA (1955) Theory of elasticity and consolidation for a porous anisotropic solid. J Appl Phys 26(2):182–185
Biot MA (1956) Theory of deformation of a porous viscoelastic anisotropic solid. J Appl Phys 27(5):459–467
Booker JR, Randolph MF (1984) Consolidation of a cross-anisotropic soil medium. Q J Mech Appl Math 37(3):479–495
Booker JR, Small JC (1977) Finite element analysis of primary and secondary consolidation. Int J Solids Struct 13(2):137–149
Borja RI, Yin Q, Zhao Y (2020) Cam-Clay plasticity. Part IX: On the anisotropy, heterogeneity, and viscoplasticity of shale. Comput Meth Appl Mech Eng 360:112695
Callisto L, Calabresi G (1998) Mechanical behaviour of a natural soft clay. Géotechnique 48(4):495–513
Cao ZG, Chen JY, Cai YQ, Zhao L, Gu C, Wang J (2018) Long-term behavior of clay-fouled unbound granular materials subjected to cyclic loadings with different frequencies. Eng Geol 243:118–127
Carrier GF (1946) Propagation of waves in orthotropic media. Q Appl Math 4:160–165
Chen JJ, Lei H, Wang JH (2014) Numerical analysis of the installation effect of diaphragm walls in saturated soft clay. Acta Geotech 9(6):981–991
Collins IF (2005) Elastic/plastic models for soils and sands. Int J Mech Sci 47(4–5):493–508
Cryer CW (1963) A comparison of the three-dimensional consolidation theories of Biot and Terzaghi. Q J Mech Appl Math 16(4):401–412
Deng Z, Tang J, Zhu Z, Fu G, Nie R (2014) Analytical solution for rheological one-dimensional consolidation of soft soil based on improved Nishihara model. J Hunan Univ (Nat Sci) 41(6):85–91
Drucker DC, Prager W (1952) Soil mechanics and plastic analysis or limit design. Q Appl Math 10(2):157–165
Duvaut G, Lions JL (1976) Inequalities in Mechanics and Physics. Springer-Verlag, Berlin-Heidelberg
Hill R (1950) The mathematical theory of plasticity. Oxford University Press, Oxford
Kabbaj M, Tavenas F, Leroueil S (1988) In situ and laboratory stress-strain relationship. Géotechnique 38:83–100
Kutter BL, Sathialingam N (1992) Elastic-viscoplastic modelling of the rate-dependent behaviour of clays. Géotechnique 42(3):427–441
Leoni M, Karstunen M, Vermeer PA (2008) Anisotropic creep model for soft soils. Géotechnique 58(3):215–226
Li XB, Jia XL, Xie KH (2006) Analytical solution of 1-D viscoelastic consolidation of soft soils under time-dependent loadings. Rock Soil Mech 27(suppl):140–146
Liao JJ, Wang CD (2015) Elastic solutions for a transversely isotropic half-space subjected to a point load. Int J Numer Anal Methods Geomech 22(6):425–447
Liu JC, Lei GH, Wang XD (2015) One-dimensional consolidation of visco-elastic marine clay under depth-varying and time-dependent load. Mar Geores Geotechnol 33(4):342–352
Lu DC, Miao JB, Du XL, Tian Y, Yao YP (2020) A 3d elastic-plastic-viscous constitutive model for soils considering the stress path dependency. Sci China-Technol Sci 63(5):91–108
Madaschi A, Gajo A (2017) A one-dimensional viscoelastic and viscoplastic constitutive approach to modeling the delayed behavior of clay and organic soils. Acta Geotech 12(4):827–847
Mesri G, Choi YK (1985) Settlement analysis of embankments on soft clays. J Geotech Eng ASCE 111(4):441–464
Nayak GC, Zienkiewicz OC (2010) Elasto-plastic stress analysis. a generalization for various constitutive relations including strain softening. Int J Numer Anal Methods. Geomech 5(1):113–135
Nishihara M (1952) Creep of shale and sandy-shale. J Geol Soc Japan 58:373–377
Pan E (1989) Static response of a transversely isotropic and layered half-space to general surface loads. Phys Earth Planet Inter 58(2):103–117
Pan E (1994) An exact solution for transversely isotropic, simply supported and layered rectangular plates. J Elast 25(2):101–116
Perzyna P (1963) The constitutive equations for rate sensitive plastic materials. Q Appl Math 20(4):321–332
Qian JG, Du ZB, Yin ZY (2018) Cyclic degradation and non-coaxiality of soft clay subjected to pure rotation of principal stress directions. Acta Geotech 13(4):943–959
Rezania M, Taiebat M, Poletti E (2016) A viscoplastic SANICLAY model for natural soft soils. Comput Geotech 73:128–141
Roscoe KH, Burland JB (1968) On the generalized stress-strain behaviour of wet clay. Engineering Plasticity. Cambridge University Press, London, pp 535–609
Roscoe KH, Thurairajah A, Schofield A (1963) Yielding of clays in states wetter than critical. Géotechnique 13(3):211–240
Rowe RK, Hinchberger SD (1998) Significance of rate effects in modelling the Sackville test embankment. Can Geotech J 35(3):500–516
Schofield AN, Wroth CP (1968) Critical State Soil Mechanics. McGraw Hill, London
Sergio AM, Miguel PR (2018) Assessment of an alternative to deep foundations in compressible clays: the structural cell foundation. Front Struct Civ Eng 12(1):67–80
Shahbodagh B, Mac TN, Esgandani GA, Khalili N (2020) A bounding surface viscoplasticity model for time-dependent behavior of soils including primary and tertiary Creep. Int J Geomech 20(9):04020143
Shahrour I, Meimon Y (1995) Calculation of marine foundations subjected to repeated loads by means of the homogenization method. Comput Geotech 17(1):93–106
Shi Q (1998) Experimental investigation of creep behaviour of saturated soft clay. Soil Eng Found 12(3):40–44
Shi X, Wu J, Ye S, Zhang Y, Xue Y, Wei Z, Li Q, Yu J (2008) Regional land subsidence simulation in Su-Xi-Chang area and Shanghai City. China Eng Geol 100(1–2):27–42
Simo JC, Hughes TJR (1998) Computational Inelasticity. Springer, New York
Singh SJ, Rosenman M (1974) Quasi-static deformation of viscoelastic half-space by a displacement dislocation. Phys Earth Planet Inter 8(1):87–101
Taylor DW, Merchant W (1940) A theory of clay consolidation accounting for secondary compressions. J Math Phys 19(3):167–185
Vermeer PA, Borst R (1984) Non-associated plasticity for soils, concrete and rock. Heron 29(3):1–64
Vermeer PA, Neher HP (2000) A soft soil model that accounts for creep. Proc Plaxis Symp Beyond Comput Geotech Amsterdam 1999:249–262
Wang CD, Liao JJ (2002) Elastic solutions for stresses in a transversely isotropic half-space subjected to three-dimensional buried parabolic rectangular loads. Int J Numer Anal Methods Geomech 39(18):4805–4824
Xie KH, Xie XY, Li XB (2008) Analytical theory for one-dimensional consolidation of clayey soils exhibiting rheological characteristic under time-dependent loading. Int J Numer Anal Methods Geomech 32(14):1833–1855
Xu GL, Zhang JW, Liu H, Ren CQ (2018) Shanghai center project excavation induced ground surface movements and deformations. Front Struct Civ Eng 12(1):26–43
Yimsiri S, Soga K (2011) Cross-anisotropic elastic parameters of two natural stiff clays. Geotechnique 61(9):809–814
Yin ZY, Chang CS, Hicher PY, Karstunen M (2009) Micromechanical analysis of kinematic hardening in natural clay. Int J Plast 25(8):1413–1435
Yin JH, Feng WQ (2017) A new simplified method and its verification for calculation of consolidation settlement of a clayey soil with creep. Can Geotech J 54(3):333–347
Yin JH, Graham J (1999) Elastic viscoplastic modelling of the time-dependent stress-strain behaviour of soils. Can Geotech J 36(4):736–745
Yin JH, Zhu JG, Graham J (2002) A new elastic-viscoplastic model for time-dependent behaviour of normally and overconsolidated clays: theory and verification. Can Geotech J 39(1):157–173
Yue ZQ (1995) Elastic fields in two joined transversely isotropic solids due to concentrated forces. Int J Eng Sci 33(3):351–369
Yue ZQ, Xiao HT, Tham LG, Lee CF, Yin JH (2005) Stresses and displacements of a transversely isotropic elastic halfspace due to rectangular loadings. Eng Anal Bound Elem 29(6):647–671
Zhou C, Yin JH, Zhu JG, Cheng CM (2005) Elastic anisotropic viscoplastic modeling of the strain-rate-dependent stress-strain behaviour of K0-consolidated natural marine clays in triaxial shear tests. Int J Geomech ASCE 5(3):218–232
Zhu QY, Yin ZY, Hicher PY, Shen SL (2016) Nonlinearity of one-dimensional creep characteristics of soft clays. Acta Geotech 11(4):887–900
Zhu HH, Zhang CC, Mei GX, Shi B, Gao L (2017) Prediction of one-dimensional compression behavior of Nansha clay using fractional derivatives. Mar Geores Geotechnol 35(5):688–697
Acknowledgements
This study was supported by the National Natural Science Foundation of China (Grant No. 41672275). We would like to express our sincere appreciation to Professor Xiaoyu Song in University of Florida for his careful and constructive comments on this paper.
Funding
National Natural Science Foundation of China, 41672275, Zhi Yong Ai.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix
Appendix
The kernel of the subroutine in ABAQUS is presented as follows:
"DO I = 1,NTENS.
DO J = 1,NTENS.
VERATE(I) = VERATE(I) + VMA(I,J)*STRESS(J)*(1.0 + E1/E0)/eta1.
ENDDO.
VERATE(I) = VERATE(I)-E1*STRAN(I)/eta1.
ENDDO.
DSTRIAL = 0.0
DO I = 1,NTENS.
DO J = 1,NTENS.
DSTRIAL(I) = DSTRIAL(I) + VEJM(I,J)*(DSTRAN(J)-Bn*VERATE(J)).
ENDDO.
STRIAL(I) = DSTRIAL(I) + STRESS(I).
ENDDO.
I1 = STRIAL(1) + STRIAL(2) + STRIAL(3)
J2 = 0.5*((STRIAL(2)-STRIAL(3))**2.0 + (STRIAL(3)-STRIAL(1))**2.0 + lambda*(STRIAL(1)-STRIAL(2))**2.0 + 12.0*lambda_tao*(STRIAL(5)**2.0 + STRIAL(6)**2.0) + (4.0*lambda + 2.0)*STRIAL(4)**2.0)/(2.0 + lambda)
q_tri = SQRT(3.0*J2).
p_tri = I1/3.0
F = q_tri + p_tri*A_phi.
c = c0 + H*EVPSTRAN.
F0 = K*c
Dgamma = 0.0
IF (F.LE.F0) THEN.
STRESS = STRIAL.
DDSDDE = VEJM.
VESTRAN = VESTRAN + DSTRAN.
ELSE.
EVPSTRAN_G = EVPSTRAN.
EVPSTRAN_E = EVPSTRAN.
c_G = c0.
c_E = c0.
d = -SQRT(3.0)*VG-VK*A_phi*A_psi-K*K*H.
DO I = 1,100.
Dgamma = Dgamma-(F-F0)/d.
EVPSTRAN_G = EVPSTRAN_G + K*Dgamma.
c_G = c0 + H*EVPSTRAN_G.
F = q_tri-SQRT(3.0)*VG*Dgamma + (p_tri-VK*A_psi*Dgamma)*A_phi.
F0 = K*c_G
F (F-F0.LE.TOL) THEN.
I1_New = I1-3.0*VK*A_psi*Dgamma
DO J = 1,NDI.
STRESS(J) = (1.0-SQRT(3.0)*VG*Dgamma/q_tri)*(STRIAL(J)-I1/3.0) + I1_New/3.0
STRESS(J + NDI) = (1.0-SQRT(3.0)*VG*Dgamma/q_tri)*STRIAL(J + NDI).
ENDDO.
goto 10.
ENDIF.
if(I = = 100)THEN.
call xit().
endif.
ENDDO.
10 continue.
IF ((SQRT(J2)-VG*Dgamma).LT.0.0) THEN.
DVPSTRAN = 0.0
d = K*K*H/(A_psi*A_phi) + VK.
DO I = 1,100.
R = c_E*K/A_psi-p_tri + VK*DVPSTRAN.
DVPSTRAN = DVPSTRAN-r/d.
EVPSTRAN_E = EVPSTRAN_E + DVPSTRAN*K/A_phi.
c_E = c0 + H*EVPSTRAN_E.
IF (ABS(r).LE.TOL) THEN.
EVPSTRAN = EVPSTRAN_E.
c = c_E.
FORALL(J = 1:NDI) STRESS(J) = p_tri-VK*DVPSTRAN.
FORALL(J = 1:NDI) STRESS(J + NDI) = 0.0
goto 20.
ENDIF.
if(I = = 100)THEN.
call xit().
endif.
ENDDO.
ELSE.
EVPSTRAN = EVPSTRAN_G.
c = c_G.
ENDIF.
20 continue.
ENDIF.
STATEV(1) = EVPSTRAN.
RETURN.
END".
Rights and permissions
About this article
Cite this article
Ai, Z.Y., Zhao, Y.Z., Dai, Y.C. et al. Three-dimensional elastic–viscoplastic consolidation behavior of transversely isotropic saturated soils. Acta Geotech. 17, 3959–3976 (2022). https://doi.org/10.1007/s11440-021-01423-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11440-021-01423-2