Dynamic Response and Failure Mechanism of Brittle Rocks Under Combined Compression-Shear Loading Experiments

Abstract

A novel method is developed for characterizing the mechanical response and failure mechanism of brittle rocks under dynamic compression-shear loading: an inclined cylinder specimen using a modified split Hopkinson pressure bar (SHPB) system. With the specimen axis inclining to the loading direction of SHPB, a shear component can be introduced into the specimen. Both static and dynamic experiments are conducted on sandstone specimens. Given carefully pulse shaping, the dynamic equilibrium of the inclined specimens can be satisfied, and thus the quasi-static data reduction is employed. The normal and shear stress–strain relationships of specimens are subsequently established. The progressive failure process of the specimen illustrated via high-speed photographs manifests a mixed failure mode accommodating both the shear-dominated failure and the localized tensile damage. The elastic and shear moduli exhibit certain loading-path dependence under quasi-static loading but loading-path insensitivity under high loading rates. Loading rate dependence is evidently demonstrated through the failure characteristics involving fragmentation, compression and shear strength and failure surfaces based on Drucker–Prager criterion. Our proposed method is convenient and reliable to study the dynamic response and failure mechanism of rocks under combined compression-shear loading.

Appendix

In Cartesian coordinates, a stress state can be expressed as a set of stress components σ _x , σ _y , τ _xy and τ _yx . Assuming a plane (Φ) making angle α with the plane σ _x , the normal stress σ _α and shear stress τ _α on plane Φ can be calculated as Eq. (10) (Timoshenko and Goodier 1934). This stress transformation can be depicted by a Mohr’s stress circle (Fig. 4e, where we treat σ ₀ and τ ₀ as σ _α and τ _α , respectively).

$$ \left\{ {\begin{array}{*{20}l} {\sigma_{\alpha } = \sigma_{x} \cos^{2} \alpha + \sigma_{y} \sin^{2} \alpha + 2\tau_{xy} \sin \alpha \cos \alpha } \hfill \\ {\tau_{\alpha } = (\sigma_{y} - \sigma_{x} )\sin \alpha \cos \alpha + \tau_{xy} (\cos^{2} \alpha - \sin^{2} \alpha )} \hfill \\ \end{array} } \right. $$

(10)

Similarly, given the strain state by ε _x , ε _y , γ _xy and γ _yx , the normal strain ε _α and shear strain γ _α on plane Φ can be calculated as Eq. (11) (Timoshenko and Goodier 1934). This strain transformation can also be depicted by a Mohr’s strain circle (Fig. 4f, in which we treat ε ₀ and γ ₀ as ε _α and γ _α , respectively).

$$ \left\{ {\begin{array}{*{20}l} {\varepsilon_{\alpha } = \varepsilon_{x} \cos^{2} \alpha + \varepsilon_{y} \sin^{2} \alpha + 2(\frac{{\gamma_{xy} }}{2})\sin \alpha \cos \alpha } \hfill \\ {\frac{{\gamma_{\alpha } }}{2} = (\varepsilon_{y} - \varepsilon_{x} )\sin \alpha \cos \alpha + \frac{{\gamma_{xy} }}{2}(\cos^{2} \alpha - \sin^{2} \alpha )} \hfill \\ \end{array} } \right. $$

(11)

For an elastic body under a plane strain condition, the relationship of the stress and strain follows the generalized Hook’s law:

$$ \left\{ {\begin{array}{*{20}l} {\varepsilon_{x} = \frac{1}{E}(\sigma_{x} - \mu \sigma_{y} )} \hfill \\ {\varepsilon_{y} = \frac{1}{E}(\sigma_{y} - \mu \sigma_{x} )} \hfill \\ {\gamma_{xy} = \frac{2(1 + \mu )}{E}\tau_{xy} } \hfill \\ \end{array} } \right. $$

(12)

Substitute Eq. (12) into Eq. (11), Eq. (13) can be obtained:

$$ \left\{ {\begin{array}{*{20}l} {\varepsilon_{\alpha } = \frac{1 + \mu }{E}\left[ {(\sigma_{x} \cos^{2} \alpha + \sigma_{y} \sin^{2} \alpha ) - \frac{\mu }{1 + \mu }(\sigma_{x} + \sigma_{y} ) + 2\tau_{xy} \sin \alpha \cos \alpha } \right]} \hfill \\ {\frac{{\gamma_{\alpha } }}{2} = \frac{1 + \mu }{E}\left[ {(\sigma_{y} - \sigma_{x} )\sin \alpha \cos \alpha + \tau_{xy} (\cos^{2} \alpha - \sin^{2} \alpha )} \right]} \hfill \\ \end{array} } \right. $$

(13)

Compare Eqs. (13) and (10), we can obtain Eq. (14):

$$ \left\{ {\begin{array}{*{20}l} {\varepsilon_{\alpha } = \frac{1 + \mu }{E}\left[ {\sigma_{\alpha } - \frac{\mu }{1 + \mu }(\sigma_{x} + \sigma_{y} )} \right]} \hfill \\ {\frac{{\gamma_{\alpha } }}{2} = \frac{1 + \mu }{E}\tau_{\alpha } } \hfill \\ \end{array} } \right. $$

(14)

Equation (14) indicates that for a given stress state, the strain circle can be acquired from the corresponding stress circle by offsetting the ordinate to the right and then scaling both the abscissa and ordinate. The offset amount in the σ–τ coordinates is $\frac{\mu }{1 + \mu }(\sigma_{x} + \sigma_{y} )$ and the scaling coefficient is $\frac{1 + \mu }{E}$.

In particular, for the given stress state in Fig. 4e, the offset amount Δε in the ε-γ/2 coordinates (see strain circle in Fig. 4e) can be calculated as:

$$ \Delta \varepsilon = \frac{\mu }{1 + \mu }(\sigma_{0} + \sigma_{0} { \tan }^{2} \theta ) \cdot \frac{1 + \mu }{E} = \frac{{\mu \sigma_{0} }}{{E\cos^{2} \theta }} $$

(15)