Non-coaxial behavior modeling of sands subjected to principal stress rotation

  • Zhouquan Chen
  • Maosong HuangEmail author
Research Paper


This paper attempts to clarify the necessary elements for describing the non-coaxial behavior of sand in the plasticity framework based on the critical-state soil mechanics. Such sand response to two representative loading types, the monotonic loading with fixed principal stress direction and rotation of principal stress direction, is the main focus. The present constitutive model can be regarded as an extension of the existing platform model for sand considering the inherent anisotropy. A novel non-coaxial flow direction involved in the plastic flow rule is the crucial feature, which is defined as the part of stress increment perpendicular to a reference direction related to the current stress direction through the Gram–Schmidt orthogonalization process. Such incorporation of the effects of the stress increment on the plastic deformation makes the model capture the non-coaxiality for the monotonic loading with fixed principal stress direction. In order to describe the plastic response induced by the pure rotation of principal stress direction, a set of mapping rules included in the bounding surface plasticity are formulated to develop plastic flow mechanism in the general multi-axial stress space. Besides, the effects of fabric anisotropy on simulating the non-coaxial plastic behavior are discussed. The results show that the above-mentioned modifications on these model ingredients make the theoretical simulation fit well with test data for Toyoura sand under the foregoing two loading conditions.


Anisotropy Bounding surface plasticity Non-coaxiality Principal stress axes rotation Sand critical-state model Tangential effect 

List of symbols


Anisotropic variable


Intermediate principal stress ratio

e, ec

Void ratio and critical-state void ratio

f, F

Yield surface and bounding surface


Elastic shear modulus


Interpolation function for the failure stress ratio


Elastic bulk modulus


Loading plastic modulus and unloading plastic modulus


Internal state variable related to stress history

Mc, Me

Critical-state stress ratio in triaxial compression and triaxial extension

Md, Mp

Phase transformation stress ratio and peak stress ratio


Loading direction defined in the yield surface


Loading direction defined in the bounding surface


Integrated plastic flow direction

\(n_{ij}^{gc}\), \(n_{ij}^{gn}\)

Coaxial part and non-coaxial part of the integrated plastic flow direction

\(n_{p}^{g}\), \(n_{q}^{g}\), \(n_{\theta }^{g}\)

Three components related to the first derivatives of p, q and θ in \(n_{ij}^{gc}\)

\(n_{ij}^{{{\text{d}}\sigma }}\)

Normalized unit tensor of stress increment tensor


Non-coaxial flow direction


Stress ratio related to a modified stress

R1, R2

Similarity ratio related to radial mapping and stress rate direction mapping

p, \(\bar{p}\)

Mean pressure related to current stress and image stress

q, \(\bar{q}\)

Equivalent stress related to current stress and image stress


Deviatoric stress tensor and image deviatoric stress tensor


Deviatoric part of stress increment \({\text{d}}\sigma_{ij}\)


Component of dsij along the direction orthogonal to sij


Principal stress tensor of sij


Combined stress tensor related to \(s_{ij}^{p}\)


Modified stress tensor


Angle of the major principal stress in relation to orientation of deposition


Angle of the major principal plastic strain increment to the vertical axis


Kronecker delta


Vector magnitude characterizing the fabric anisotropy

\(\varepsilon_{ij}\), \(\varepsilon_{ij}^{e}\), \(\varepsilon_{ij}^{p}\)

Strain tensor, elastic strain tensor and plastic strain tensor

\(\varepsilon_{ij}^{pc}\), \(\varepsilon_{ij}^{pn}\)

Coaxial plastic strain tensor and non-coaxial plastic strain tensor

\(\varepsilon_{z}\), \(\varepsilon_{r}\), \(\varepsilon_{\vartheta }\), \(\varepsilon_{z\vartheta }\)

Axial, radial, circumferential and torsional strains

\(\theta\), \(\bar{\theta }\), \(\hat{\theta }\)

Lode angle related to current stress, image stress and modified stress

\(\sigma_{ij}\), \(\bar{\sigma }_{ij}\)

Current stress tensor, image stress tensor

\(\bar{\sigma }_{ij}^{1}\), \(\bar{\sigma }_{ij}^{2}\)

Radial mapping and stress rate direction mapping

\(\sigma_{1}\), \(\sigma_{2}\), \(\sigma_{3}\)

Major, intermediate and minor principal stresses

\(\sigma_{z}\), \(\sigma_{r}\), \(\sigma_{\vartheta }\), \(\tau_{z\vartheta }\)

Axial, radial, circumferential and torsional stresses


State parameter


Projection center related to radial mapping

\(\sigma_{ij} = \left[ {\sigma_{11} \sigma_{33} \sigma_{13} } \right]\left( {i,j = 1,2} \right)\)

Mapping from tensor to vector in stress space (\(\sigma_{11}\), \(\sigma_{33}\), \(\sigma_{13}\))

\(t_{ij}^{\text{non}} \left( {i,j = 1,2} \right)\)

Direction of current stress in stress space (\(\sigma_{11}\), \(\sigma_{33}\), \(\sigma_{13}\))

\(m_{ij} \left( {{\text{i}},{\text{j}} = 1,2} \right)\)

Direction orthogonal to current stress in stress space (\(\sigma_{11}\), \(\sigma_{33}\), \(\sigma_{13}\))

\(m_{ij}^{\text{non}} \left( {i,j = 1,2} \right)\)

Non-coaxial flow related to \(m_{ij}\)



This work was financially supported by the National Natural Science Foundation of China (Grant No. 51738010) and the National Key R&D Program of China (Grant No. 2016YFC0800200).


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Geotechnical EngineeringTongji UniversityShanghaiChina
  2. 2.Key Laboratory of Geotechnical and Underground Engineering of Ministry of EducationTongji UniversityShanghaiChina

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