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Influences of loading direction and intermediate principal stress ratio on the initiation of strain localization in cross-anisotropic sand

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Abstract

Since cross-anisotropic sand behaves differently when the loading direction or the stress state changes, the influences of the loading direction and the intermediate principal stress ratio (b = (σ 2 − σ 3)/(σ 1 − σ 3)) on the initiation of strain localization need study. According to the loading angle (angle between the major principal stress direction and the normal of bedding plane), a 3D non-coaxial non-associated elasto-plasticity hardening model was proposed by modifying Lode angle formulation of the Mohr–Coulomb yield function and the stress–dilatancy function. By using bifurcation analysis, the model was used to predict the initiation of strain localization under plane strain and true triaxial conditions. The predictions of the plane strain tests show that the major principal strain at the bifurcation points increases with the loading angle, while the stress ratio decreases with the loading angle. According to the loading angle and the intermediate principal stress ratio, the true triaxial tests were analyzed in three sectors. The stress–strain behavior and the volumetric strain in each sector can be well captured by the proposed model. Strain localization occurs in most b value conditions in all three sectors except for those which are close to triaxial compression condition (b = 0). The difference between the peak shear strength corresponding to the strain localization and the ultimate shear strength corresponding to plastic limit becomes obvious when the b value is near 0.4. The influence of bifurcation on the shear strength becomes weak when the loading direction changes from perpendicular to the bedding plane to parallel. The bifurcation analysis based on the proposed model gives out major principal strain and peak shear strength at the initiation of strain localization; the given results are consistent with experiments.

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Abbreviations

A :

The fitting parameter for stress–strain relationship

A d :

Parameter of stress–dilatancy

b :

Intermediate principal stress ratio

C 1, C 2, C 3 :

Parameters of the criterion of strain localization

\(C_{ijkl}^{np}\) :

Non-coaxial compliance tensor

\(d{\tilde{\mathbf{\sigma }}}\) :

Vectorial notations of the stress increment tensor

\(d{\tilde{\mathbf{\varepsilon }}}\) :

Vectorial notations of the strain increment tensor

D :

Dilatancy function

\(D_{ijkl}^{e}\), D e :

Elasto-plastic modulus tensor

\(D_{ijkl}^{p}\), D p :

Plastic modulus tensor

\(D_{ijkl}^{\text{ep}}\), D ep :

Elasto-plastic modulus tensor

(D ep)sys :

Symmetric part of elasto-plastic modulus tensor

E :

Elastic modulus

e :

Void ratio

e ij :

Deviatoric strain

F :

Yield function

\(g(\theta_{\sigma } )\) :

Shape function in deviatoric plane

G :

Pressure-dependent shear modulus

G 0, G 01, G 02 :

Regression constant of elastic shear modulus

H p :

The hardening modulus

H t :

Non-coaxial hardening modulus

J 2 :

Second stress invariant

K :

Bulk elastic modulus

L i :

The loading direction

l i :

The unit vector specifying the loading direction

M :

Stress ratio

M f , M f0, M f1 , M f2 , M f3 :

Peak stress ratio

M d :

Dilatancy–stress ratio

n :

The unit vector which is normal to the shear band

p :

Mean stress

p at :

Atmospheric pressure

q :

Equivalent shear stress

Q :

Plastic potential

s ij :

Effective deviatoric stress

S ij :

Stress tensor independent of δ ij and s ij

W 2 :

Second-order work

β :

The shear strength difference between triaxial tension compression conditions

δ ij :

Kronecker delta

δ :

Angle between the major principal stress and the normal of bedding plane

ε ij :

Strain tensor

\(\varepsilon_{s}^{p}\) :

Equivalent plastic shear strain

ε v :

Volumetric strain

φ, φ c, φ E :

Friction angle

Ω1 :

Cross-anisotropic parameter

ξ, η :

Shear band angle

\(\dot{\lambda }\) :

Plastic multiplier

Ν :

Poisson ratio

σ ij :

The stress tensor

θ :

Equals to \({\pi \mathord{\left/ {\vphantom {\pi 6}} \right. \kern-0pt} 6} + \theta_{\sigma }\)

\(\theta_{\sigma }\) :

Lode angle

σ ij :

The stress tensor

\(\zeta\) :

Angle between the intermediate principal stress and fabric tensor

References

  1. Abelev AV, Lade PV (2003) Effects of cross anisotropy on three-dimensional behavior of sand. I: stress–strain behavior and shear banding. J Eng Mech ASCE 129(2):160–166

    Article  Google Scholar 

  2. Alshibli KA, Sture LS (2000) Shear band formation in plane strain experiments of sand. J Geotech Geoenviron Eng ASCE 126(6):495–503

    Article  Google Scholar 

  3. Alshibli KA, Batiste SN, Sture S (2003) Strain localization in sand: plane strain versus triaxial compression. J Geotech Geoenviron Eng ASCE 129(6):483–494

    Article  Google Scholar 

  4. Andrade JE, Baker JW, Ellison KC (2008) Random porosity fields and their influence on the stability of granular media. Int J Numer Anal Methods Geomech 32(10):1147–1172

    Article  MATH  Google Scholar 

  5. Bardet JP (1990) A comprehensive review of strain localization in elastoplastic soils. Comput Geotech 10:163–188

    Article  Google Scholar 

  6. Bardet JP (1990) Lode dependences for pressure-sensitive isotropic elastoplastic materials. J Appl Mech ASME 57(3):498–506

    Article  Google Scholar 

  7. Bauer E (1996) Calibration of a comprehensive hypoplastic model for granular materials. Soils Found 36(1):13–26

    Article  Google Scholar 

  8. Bigoni D, Hueckel T (1991) Uniqueness and localization-I. Associative and nonassociative elastoplasticity. Int J Solid Struct 28(2):197–213

    Article  MathSciNet  MATH  Google Scholar 

  9. Bolton MD (1986) The strength and dilatancy of sands. Geotechnique 36(1):65–78

    Article  Google Scholar 

  10. Borja RI, Song X, Rechenmacher AL, Abedi S, Wu W (2013) Shear band in sand with spatially varying density. J Mech Phys Solids 61(1):219–234

    Article  Google Scholar 

  11. Chang CS, Bennett K (2015) Micromechanical modeling for the deformation of sand with noncoaxiality between the stress and material axes. J Eng Mech ASCE. doi:10.1061/(ASCE)EM.1943-7889.0000966

    Google Scholar 

  12. Chen Q, Andrade JE, Samaniego E (2011) AES for multiscale localization modeling in granular media. Comput Methods Appl Mech Eng 200(33–36):2473–2482

    Article  MATH  Google Scholar 

  13. Gao Z, Zhao J, Li X, Dafalias YF (2014) A critical state sand plasticity model accounting for fabric evolution. Int J Numer Anal Methods Geomech 38(4):370–390

    Article  Google Scholar 

  14. Gudehus G (1996) A comprehensive constitutive equation for granular materials. Soils Found 36(1):1–12

    Article  Google Scholar 

  15. Gutierrez M, Ishilhara K, Towhata I (1991) Flow theory for sand during rotation of principle stress direction. Soils Found 31(4):121–132

    Article  Google Scholar 

  16. Hill R (1958) A general theory of uniqueness and stability in elastic-plastic solids. J Mech Phys Solids 6(3):236–249

    Article  MATH  Google Scholar 

  17. Huang MS, Lu XL, Qian JG, Wang WD (2009) Prediction of the onset of strain localization in non-coaxial plasticity. In: Proceedings of the 17th international conference on soil mechanics and geotechnical engineering, IOS, Alexandria, Egypt

  18. Huang MS, Lu XL, Qian JG (2010) Non-coaxial elasto-plasticity model and bifurcation prediction of shear banding in sands. Int J Numer Anal Methods Geomech 34(9):906–919

    MATH  Google Scholar 

  19. Kolymbas D (1991) An outline of hypoplasticity. Arch Appl Mech 61:143–151

    MATH  Google Scholar 

  20. Kolymbas D (2009) Kinematics of shear bands. Acta Geotech 4(4):315–318

    Article  Google Scholar 

  21. Lade PV (2008) Failure criterion for cross-anisotropic soils. J Geotech Geoenviron Eng ASCE 134(1):117–124

    Article  Google Scholar 

  22. Lade PV, Abelev AV (2003) Effects of cross anisotropy on three-dimensional behavior of sand. II: volume change behavior and failure. J Eng Mech ASCE 129(2):167–174

    Article  Google Scholar 

  23. Lade PV, Wang Q (2001) Analysis of shear banding in true triaxial tests on sand. J Eng Mech ASCE 127(8):762–768

    Article  Google Scholar 

  24. Lade PV, Nam J, Hong WP (2008) Shear banding and cross-anisotropic behaviour observed in laboratory sand tests with stress rotation. Can Geotech J 45:74–84

    Article  Google Scholar 

  25. Lashkaria A, Latifi M (2007) A simple plasticity model for prediction of non-coaxial flow of sand. Mech Res Commun 34(2):191–200

    Article  MATH  Google Scholar 

  26. Li XS, Dafalias YF (2002) Constitutive modeling of inherently anisotropic sand behavior. J Geotech Geoenviron Eng ASCE 128(10):868–880

    Article  Google Scholar 

  27. Li XS, Dafalias YF (2004) A constitutive framework for anisotropic sand including non-proportional loading. Géotechnique 54(1):41–55

    Article  Google Scholar 

  28. Li XS, Li X (2009) Micro-macro quantification of the internal structure of granular materials. J Eng Mech ASCE 135(7):641–656

    Article  Google Scholar 

  29. Li X-S, Dafalias YF, Wang Z-L (1999) State-dependant dilatancy in critical-state constitutive modelling of sand. Can Geotech J 36(4):599–611

    Article  Google Scholar 

  30. Lu XL, Huang MS, Qian JG (2011) The onset of strain localization in cross-anisotropic soils under true triaxial condition. Soils Found 51(4):693–700

    Article  Google Scholar 

  31. Lü X, Huang M, Andrade JE (2016) Strength criterion for cross-anisotropic sand under general stress conditions. Acta Geotech 11(6):1339–1350

    Article  Google Scholar 

  32. Lü X, Huang M, Andrade JE (2017) Predicting the initiation of static liquefaction of cross-anisotropic sands under multi-axial stress conditions. Int J Numer Anal Methods Geomech. doi:10.1002/nag.2697

    Google Scholar 

  33. Mortara G (2010) A yield criterion for isotropic and cross-anisotropic cohesive-frictional materials. Int J Numer Anal Methods Geomech 34(9):953–977

    MATH  Google Scholar 

  34. Oda M, Koishikawa I, Higuchi T (1978) Experimental study of anisotropic shear strength of sand by plane strain test. Soils Found 18(1):25–38

    Article  Google Scholar 

  35. Papamichos E, Vardoulakis I (1995) Shear band formation in sand according to non-coaxial plasticity model. Géotechnique 45(4):649–661

    Article  Google Scholar 

  36. Pietruszczak S, Mroz Z (2000) Formulation of anisotropic failure criteria incorporating a microstructure tensor. Comput Geotech 26(2):105–112

    Article  Google Scholar 

  37. Pietruszczak S, Stolle FE (1987) Deformation of strain softening material, part II: modelling of strain softening response. Comput Geotech 4(2):109–123

    Article  Google Scholar 

  38. Qian JG, Yang J, Huang MS (2008) Three-dimensional noncoaxial plasticity modeling of shear band formation in geomaterials. J Eng Mech ASCE 134(4):322–329

    Article  Google Scholar 

  39. Rice JR (1976) The localization of plastic deformation. In: Proceedings of the 14th international congress on theoretical and applied mechanics, North-Holland Publishing, Delft

  40. Rodriguez NM, Lade PV (2013) True triaxial tests on cross-anisotropic deposits of fine Nevada sand. Int J Geomech ASCE 13(6):779–793

    Article  Google Scholar 

  41. Rodriguez NM, Lade PV (2013) Effects of principal stress directions and mean normal stress on failure criterion for cross-anisotropic sand. J Eng Mech ASCE 139(11):1592–1601

    Article  Google Scholar 

  42. Rudnicki JW, Rice JR (1975) Conditions for the localization of the deformation in pressure sensitive dilatant materials. J Mech Phys Solids 23(6):371–394

    Article  Google Scholar 

  43. Schanz T, Vermeer PA (1996) Angles of friction and dilatancy of sand. Geotechnique 46(1):145–151

    Article  Google Scholar 

  44. Tatsuoka F, Nakamura T, Huang CC, Tani K (1990) Strength of anisotropy and shear band direction in plane strain test of sand. Soils Found 26(1):65–84

    Article  Google Scholar 

  45. Tejchman J, Niemunis A (2006) FE-studies on shear localization in an anisotropic micro-polar hypoplastic granular material. Granul Matter 8(3–4):205–220

    Article  MATH  Google Scholar 

  46. Tejchman J, Wu W (2007) Modeling of textural anisotropy in granular materials with stochastic micro-polar hypoplasticity. Int J Non-Linear Mech 42(6):882–894

    Article  Google Scholar 

  47. Tejchman J, Herle I, Wehr J (1999) FE-Studies on the influence of initial void ratio, pressure level and mean grain diameter on shear localization. Int J Numer Anal Methods Geomech 23:2045–2074

    Article  MATH  Google Scholar 

  48. Tejchman J, Bauer E, Wu W (2007) Effect of fabric anisotropy on shear localization in sand during plane strain compression. Acta Mech 189(1):23–51

    Article  MATH  Google Scholar 

  49. Tvergaard V (1981) Influence of voids on shear band instabilities under plane strain conditions. Int J Fract 17(4):389–407

    Article  Google Scholar 

  50. Vardoulakis I (1979) Bifurcation analysis of the triaxial test on sand samples. Acta Mech 32(1–3):35–54

    Article  MATH  Google Scholar 

  51. Wan RG, Guo PJ (2004) Stress dilatancy and fabric dependencies on sand behavior. J Eng Mech ASCE 130(6):635–645

    Article  Google Scholar 

  52. Willam KJ, Warnke EP (1975) Constitutive model for the triaxial behavior of concrete. In: International association for bridge and structure engineering proceedings, Bergamo, Italy

  53. Wu W (1998) Rational approach to anisotropy of sand. Int J Numer Anal Methods Geomech 22(11):921–940

    Article  MATH  Google Scholar 

  54. Xiao Y, Liu H, Yang G (2012) Formulation of cross-anisotropic failure criterion for granular material. Int J Geomech ASCE 12(2):182–188

    Article  Google Scholar 

  55. Yoshimine M, Ishihara K, Vargas W (1998) Influence of principal stress direction and intermediate principal stress on undrained shear behavior of sand. Soils Found 38(3):179–188

    Article  Google Scholar 

  56. Zhao J, Gao Z (2016) Unified anisotropic elastoplastic model for sand. J Eng Mech ASCE 142(1):04015056–04015061

    Article  Google Scholar 

  57. Zhao J, Guo N (2015) The interplay between anisotropy and strain localisation in granular soils: a multiscale insight. Géotechnique 65(8):642–656

    Article  Google Scholar 

Download references

Acknowledgements

The financial supports by National Science Foundation of China (NSFC through Grant Nos. 11372228 and 41672270) and National Key Research and Development Program (through Grant No. 2016YFC0800202) are gratefully acknowledged.

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Authors and Affiliations

  1. Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Tongji University, Shanghai, 200092, China

    Xilin Lü, Maosong Huang & Jiangu Qian

  2. Department of Geotechnical Engineering, Tongji University, Shanghai, 200092, China

    Xilin Lü, Maosong Huang & Jiangu Qian

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  1. Xilin Lü
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Correspondence to Maosong Huang.

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Lü, X., Huang, M. & Qian, J. Influences of loading direction and intermediate principal stress ratio on the initiation of strain localization in cross-anisotropic sand. Acta Geotech. 13, 619–633 (2018). https://doi.org/10.1007/s11440-017-0582-9

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