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

Abstract

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 map** 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.

Appendices

Appendix 1

Due to ${\text{d}}s_{ij} /s_{ij} = {\text{const}}$, substituting ${\text{d}}s_{ij} = {\text{const}} \cdot s_{ij}$ into Eq. (5) yields

$${\text{d}}s_{ij}^{\text{non}} = {\text{const}} \cdot s_{ij} - {\text{const}} \cdot \frac{{s_{kl} s_{kl} }}{{s_{mn} s_{mn} }}s_{ij} - {\text{const}} \cdot \frac{{s_{kl} S_{kl} }}{{S_{mn} S_{mn} }}S_{ij} = - {\text{const}} \cdot \frac{{s_{kl} S_{kl} }}{{S_{mn} S_{mn} }}S_{ij}$$

(43)

Due to $S_{ij} = s_{ik} s_{kj} - \frac{2}{3}J_{2} \delta_{ij} - \frac{{3J_{3} }}{{2J_{2} }}s_{ij}$,

$$s_{ij} S_{ij} = s_{ik} s_{kj} s_{ji} - \frac{2}{3}J_{2} \delta_{ij} s_{ij} - \frac{{3J_{3} }}{{2J_{2} }}s_{ij} s_{ij}$$

(44)

Considering that $\delta_{ij} s_{ij} = 0$, $J_{2} = \frac{1}{2}s_{ij} s_{ij}$ and $J_{3} = \frac{1}{3}s_{ij} s_{jk} s_{ki}$, Eq. (44) can be reduced to

$$s_{ij} S_{ij} = 3J_{3} - 0 - \frac{{3J_{3} }}{{2J_{2} }}2J_{2} = 0$$

(45)

As a result, Eq. (7) is proved due to $s_{ij} S_{ij} = s_{kl} S_{kl} = 0$.

Appendix 2

The corresponding partial derivatives are given as follows

$$\frac{\partial p}{{\partial \sigma_{ij} }} = \frac{1}{3}\delta_{ij} ,\quad \frac{\partial q}{{\partial \sigma_{ij} }} = \frac{{3s_{ij} }}{2q}\quad \frac{\partial \theta }{{\partial \sigma_{ij} }} = - \frac{\sqrt 3 }{2\cos 3\theta }\left( {J_{2}^{{{{ - 3} \mathord{\left/ {\vphantom {{ - 3} 2}} \right. \kern-0pt} 2}}} \frac{{\partial J_{3} }}{{\partial \sigma_{ij} }} - \frac{3}{2}J_{3} J_{2}^{{{{ - 5} \mathord{\left/ {\vphantom {{ - 5} 2}} \right. \kern-0pt} 2}}} \frac{{\partial J_{2} }}{{\partial \sigma_{ij} }}} \right)$$

(46)

$$\frac{{\partial J_{2} }}{{\partial \sigma_{ij} }} = s_{ij} ,\quad \frac{{\partial J_{3} }}{{\partial \sigma_{ij} }} = s_{ik} s_{kj} - \frac{1}{3}\delta_{ij} s_{kl} s_{kl}$$

(47)

$$\frac{\partial g(\theta )}{\partial \theta } = \frac{12c\cos (3\theta )(c - 1)}{{[c - \sin (3\theta )(c - 1) + 1]^{2} }} - \frac{3\cos (3\theta )(c - 1)}{2}$$

(48)

Appendix 3

Given that the integrated plastic flow rule is written as

$${\text{d}}\varepsilon_{ij}^{p} = {\text{d}}\lambda n_{ij}^{g} = {\text{d}}\lambda \left( {n_{ij}^{gc} + n_{ij}^{gn} } \right)$$

(49)

For the stress-controlled loading condition that ${\text{d}}\sigma_{ij}$ is known, Eq. (25) can be used directly to calculate the plastic loading index. For the strain-controlled loading condition that ${\text{d}}\varepsilon_{ij}$ is given, substituting Eq. (2) and Eq. (49) into Eq. (25) gives

$${\text{d}}\lambda = \left\{ {\begin{array}{*{20}c} {\frac{{n_{ij} D_{ijkl}^{e} {\text{d}}\varepsilon_{kl} }}{{H_{\text{L}} + n_{ij} D_{ijkl}^{e} n_{kl}^{g} }}} & {n_{ij} D_{ijkl}^{e} {\text{d}}\varepsilon_{kl} > 0} \\ {\frac{{\bar{n}_{ij} D_{ijkl}^{e} {\text{d}}\varepsilon_{kl} }}{{H_{\text{U}} + \bar{n}_{ij} D_{ijkl}^{e} n_{kl}^{g} }}} & {n_{ij} D_{ijkl}^{e} {\text{d}}\varepsilon_{kl} \le 0} \\ \end{array} } \right.$$

(50)

where $D_{ijkl}^{e}$ is the elastic stiffness tensor, expressed as

$$D_{ijkl}^{e} = (K - 2G/3)\delta_{ij} \delta_{kl} + G(\delta_{ki} \delta_{lj} + \delta_{li} \delta_{kj} )$$

(51)

Combination of Eqs. (50) and (2) yields

$${\text{d}}\sigma_{ij} = D_{ijkl}^{ep} {\text{d}}\varepsilon_{kl}$$

(52)

where

$$D_{ijkl}^{ep} = \left\{ {\begin{array}{*{20}c} {D_{ijkl}^{e} - \frac{{D_{ijpq}^{e} n_{pq} n_{mn}^{g} D_{mnkl}^{e} }}{{H_{\text{L}} + n_{ij} D_{ijkl}^{e} n_{kl}^{g} }}} & {n_{ij} D_{ijkl}^{e} {\text{d}}\varepsilon_{kl} > 0} \\ {D_{ijkl}^{e} - \frac{{D_{ijpq}^{e} \bar{n}_{pq} n_{mn}^{g} D_{mnkl}^{e} }}{{H_{\text{U}} + \bar{n}_{ij} D_{ijkl}^{e} n_{kl}^{g} }}} & {n_{ij} D_{ijkl}^{e} {\text{d}}\varepsilon_{kl} \le 0} \\ \end{array} } \right..$$

(53)