Non-coaxiality of soft clay generated by principal stress rotation under high-speed train loading

Abstract

Plenty of geomechanics tests and theories have confirmed the existence of non-coaxiality while soil is subjected to principal stress rotation. This paper investigated the influence of one particular principal stress path, which is a ‘heart-shape’ stress path that is normally induced by high-speed train loading, on the non-coaxiality of reconstituted soft clay. Hollow cylinder apparatus was employed to carry out series of undrained dynamic tests. The goals of this study were to (1) reveal the essential factors of complex cyclic loading paths that influence non-coaxiality in clayey soil and (2) quantify the influence of the factors on variation in non-coaxiality under the high-speed training loading. To analyze the non-coaxiality under high-speed train loading, (a) the pure rotation stress path was utilized as comparison for underling the different influence that ‘heart-shape’ stress path has from other conventional cyclic stress paths. (b) Two variables, dynamic stress ratio and tension–compression amplitude ratio, were introduced in analyzing the evolution of the non-coaxial angle. (c) Based on the test results, equations for describing the revolution of non-coaxiality were proposed which can help to describe the variation in non-coaxial angle under complex loadings quantitatively and understand the influence of the major factors of the stress path intensively.

Appendices

Appendix 1: Nomenclature

See Table 4

Table 4 Nomenclature

Appendix 2: Theoretical calculation of principal stress direction angle

The theoretical formula for calculating principal stress direction angle is derived as follows.

Principal stress angle in one cycle period is calculated as:

$$ \alpha_{\sigma } = \left\{ {\begin{array}{*{20}l} {\frac{1}{2}\arctan \frac{{2\tau_{z\theta } }}{{\sigma_{z} - \sigma_{\theta } }},} \hfill & {\frac{{\sigma_{z} - \sigma_{\theta } }}{2} \ge 0} \hfill \\ {\frac{1}{2}\arctan \frac{{2\tau_{z\theta } }}{{\sigma_{z} - \sigma_{\theta } }} + \frac{\pi }{2},} \hfill & {\frac{{\sigma_{z} - \sigma_{\theta } }}{2} < 0 \wedge \tau_{z\theta } \ge 0} \hfill \\ {\frac{1}{2}\arctan \frac{{2\tau_{z\theta } }}{{\sigma_{z} - \sigma_{\theta } }} - \frac{\pi }{2},} \hfill & {\frac{{\sigma_{z} - \sigma_{\theta } }}{2} < 0 \wedge \tau_{z\theta } < 0} \hfill \\ \end{array} } \right. $$

(8)

where $\frac{{\sigma_{z} - \sigma_{\theta } }}{2}$ and $\tau_{z\theta }$ are computed as follows:

$$ \frac{{\sigma_{z} - \sigma_{\theta } }}{2} = \frac{1}{2} \times q_{0} \times \left( {1 + \cos \alpha_{q} } \right) \times \cos \alpha_{q} + \frac{1}{8} \times q_{0} \times \left( {\frac{9}{a + 1} - 8} \right) $$

(9)

$$ \tau_{z\theta } = \frac{1}{2} \times q_{0} \times \left( {1 + \cos \alpha_{q} } \right) \times \sin \alpha_{q} $$

(10)

where $\alpha_{{{q}}} = \left( {2 \times t \times f - 1} \right) \times \pi$, t represents time and f represents frequency.

Considering:

$$ \Gamma \left( {t,f} \right) = \arctan \frac{{2\tau_{z\theta } }}{{\sigma_{z} - \sigma_{\theta } }} = \arctan \frac{{\left( {1 + \cos \alpha_{q} } \right) \times \sin \alpha_{q} }}{{\left( {1 + \cos \alpha_{q} } \right) \times \cos \alpha_{q} + \frac{9}{4a + 4} - 2}}. $$

(11)

Principal stress direction angle is expressed as:

$$ \alpha_{\sigma } = \frac{1}{2} \times \Gamma \left( {t,f} \right) + \left\lfloor { \pm \frac{\pi }{2}} \right\rfloor $$

which is Eq. (5).

Appendix 3: Stress and strain interpretation

See Fig. 12 and Table 5.

Fig. 12

Stress and strain state in the hollow cylinder specimen

Table 5 Parameters calculation