Constitutive modeling of principal stress rotation by considering inherent and induced anisotropy of soils

Abstract

In order to simulate the soil response during principal stress rotation, anisotropic unified hardening (UH) model is developed within the framework of elastoplastic theory. Without introducing any additional mechanism to display the role of stress rotation specifically, this model achieves the simulation by considering the material anisotropy. The effect of inherent anisotropy is reflected using the anisotropic transformed stress method, but a new formula for the stress map** is adopted to keep the mean stress unchanged. Analysis indicates that from the view of the transformed stress tensor, the anisotropic soil is subjected to loading during pure rotation of principal stress axes, so that plastic strains can be calculated. To represent the induced anisotropy, a fabric evolution law is proposed based on laboratory and numerical test results. At the critical state, the fabric tensor reaches a stable value determined by the stress state, while the critical state line is unique in the plane of void ratio versus mean stress. The anisotropic UH model has concise formulation and explicit elastoplastic flexibility matrix and can provide reasonable predictions for the deformation of anisotropic soils when principal stresses rotate.

Appendix: Explicit expressions of the partial differentials in the plastic flexibility matrix

• 1.
  $$\frac{\partial f}{{\partial \tilde{\sigma }_{ij} }}$$

According to the chain rule,

$$\frac{\partial f}{{\partial \tilde{\sigma }_{ij} }} = \frac{\partial f}{{\partial \tilde{p}}}\frac{{\partial \tilde{p}}}{{\partial \tilde{\sigma }_{ij} }} + \frac{\partial f}{{\partial \tilde{q}}}\frac{{\partial \tilde{q}}}{{\partial \tilde{\sigma }_{ij} }}$$

(26)

where

$$\frac{{\partial \tilde{p}}}{{\partial \tilde{\sigma }_{ij} }} = \frac{{\delta_{ij} }}{3},\;\;\frac{{\partial \tilde{q}}}{{\partial \tilde{\sigma }_{ij} }} = \frac{{3\left( {\tilde{\sigma }_{ij} - \tilde{p}\delta_{ij} } \right)}}{{2\tilde{q}}} ,$$

(27)

Besides, $\frac{\partial f}{{\partial \tilde{p}}}$ and $\frac{\partial f}{{\partial \tilde{q}}}$ can be derived from the yield function (Eq. 13) as follows

$$\frac{\partial f}{{\partial \tilde{p}}} = \frac{1}{{\tilde{p}}}\frac{{\tilde{M}^{2} - \tilde{\eta }^{2} }}{{\tilde{M}^{2} + \tilde{\eta }^{2} }} , \frac{\partial f}{{\partial \tilde{q}}} = \frac{1}{{\tilde{p}}}\frac{2{\tilde{\eta } }}{{\tilde{M}^{2} + \tilde{\eta }^{2} }}$$

(28)

• 2.
  $$\frac{\partial f}{{\partial \sigma_{ij} }}$$

First, the partial differential with respect to $\bar{\sigma }_{ij}$ is calculated by

$$\frac{\partial f}{{\partial \bar{\sigma }_{ij} }} = \frac{\partial f}{{\partial \tilde{p}}}\frac{{\partial \tilde{p}}}{{\partial \bar{\sigma }_{ij} }} + \frac{\partial f}{{\partial \tilde{q}}}\frac{{\partial \tilde{q}}}{{\partial \bar{\sigma }_{ij} }}$$

(29)

Note that one can always get $\tilde{p} = \bar{p}$ and $\tilde{q} = \bar{q}_{\text{c}}$ from the transformed stress Eq. (4), so that

$$\frac{{\partial \tilde{p}}}{{\partial \bar{\sigma }_{ij} }} = \frac{{\partial \bar{p}}}{{\partial \bar{\sigma }_{ij} }} = \frac{{\delta_{ij} }}{3},\;\;\frac{{\partial \tilde{q}}}{{\partial \bar{\sigma }_{ij} }} = \frac{{\partial \bar{q}_{\text{c}} }}{{\partial \bar{\sigma }_{ij} }} = \frac{{\left( {\bar{q}_{\text{c}}^{2} - \bar{I}_{1}^{2} } \right)\frac{{\partial \bar{I}_{1} }}{{\partial \bar{\sigma }_{ij} }} + 9\frac{{\partial \bar{I}_{3} }}{{\partial \bar{\sigma }_{ij} }}}}{{2\bar{q}_{\text{c}} \left( {\bar{q}_{\text{c}} - \bar{I}_{1} } \right)}}$$

(30)

Then according to the chain rule, the partial differential with respect to $\sigma_{ij}$ is equal to

$$\frac{\partial f}{{\partial \sigma_{ij} }} = \frac{\partial f}{{\partial \bar{\sigma }_{kl} }}\frac{{\partial \bar{\sigma }_{kl} }}{{\partial \sigma_{ij} }}$$

(31)

where based on the modified stress Eq. (3),

$$\frac{{\partial \bar{\sigma }_{kl} }}{{\partial \sigma_{ij} }} = \frac{3}{2}\left( {F_{jl} \delta_{ki} + F_{ki} \delta_{jl} } \right) - \left( {F_{ji} - \frac{{\delta_{ji} }}{3}} \right)\delta_{kl}$$

(32)

• 3.
  $$\frac{\partial f}{{\partial F_{ij} }}$$

According to the chain rule,

$$\frac{\partial f}{{\partial F_{ij} }} = \frac{\partial f}{{\partial \bar{\sigma }_{kl} }}\frac{{\partial \bar{\sigma }_{kl} }}{{\partial F_{ij} }} = \frac{\partial f}{{\partial \bar{\sigma }_{kl} }}\left[ {\frac{3}{2}\left( {\sigma_{ki} \delta_{jl} + \sigma_{jl} \delta_{ki} } \right) - s_{ji} \delta_{kl} } \right]$$

(33)