The Micromechanics of Westerley Granite at Large Compressive Loads

Abstract

The micromechanical damage mechanics formulated by Ashby and Sammis (Pure Appl Geophys 133(3) 489–521, 1990) has been shown to give an adequate description of the triaxial failure surface for a wide variety of rocks at low confining pressure. However, it does not produce the large negative curvature in the failure surface observed in Westerly granite at high confining pressure. We show that this discrepancy between theory and data is not caused by the two most basic simplifying assumptions in the damage model: (1) that all the initial flaws are the same size or (2) that they all have the same orientation relative to the largest compressive stress. We also show that the stress–strain curve calculated from the strain energy density significantly underestimates the nonlinear strain near failure in Westerly granite. Both the observed curvature in the failure surface and the nonlinear strain at failure observed in Westerly granite can be quantitatively fit using a simple bi-mineral model in which the feldspar grains have a lower flow stress than do the quartz grains. The conclusion is that nonlinearity in the failure surface and stress–strain curves observed in triaxial experiments on Westerly granite at low loading rates is probably due to low-temperature dislocation flow and not simplifying assumptions in the damage mechanics. The important implication is that discrepancies between experiment and theory should decrease with increased loading rates, and therefore, the micromechanical damage mechanics, as formulated, can be expected to give an adequate description of high strain-rate phenomena like earthquake rupture, underground explosions, and meteorite impact.

Appendices

Appendix A: Calculating the Strain Energy and Axial Strain Due to Sliding on the Initial Inclined Cracks

The average value of the stress intensity factor, K around the edge of the crack in Eq. 6 is

$$ \begin{aligned} \langle K_{II}^{2}+K_{III}^{2} \rangle &= \frac{32a\sigma_{s}^{2}}{\pi^{2}(2-\nu)^{2}} \times\\ &\int\limits_{0}^{\pi/2}{\left[\cos^{2}{\phi}+(1-\nu)^{2}\sin^{2}{\phi}\right]}d\phi \nonumber \\ &=\frac{8a\sigma_{s}^{2}}{\pi}\left[\frac{1+\left(1-\nu \right)^{2}}{\left(2-\nu \right)^{2}}\right] \end{aligned} $$

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and Eq. 6 becomes

$$ \begin{aligned} \Updelta W_{1} &= \frac{16\sigma_{s}^{2}}{E_{o}}\left[\frac{1+\left(1-\nu \right)^{2}}{\left(2-\nu \right)^{2}}\right] \int\limits_{0}^{a}r^{2}dr \\ &= \frac{16a^{3}\sigma_{s}^{2}}{3E_{o}}\left[\frac{1+\left(1-\nu \right)^{2}}{\left(2-\nu \right)^{2}}\right] \end{aligned} $$

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The uniaxial strain $\epsilon_{1}$ can be calculated as (recall that $\sigma_{s} = \tau + \mu \sigma_{n}$)

$$ \begin{aligned} \epsilon_{1} &= \frac{\partial W}{\partial\sigma_{1}} = \frac{\sigma_{1}}{E_{o}} +\\ &N_{V}\left(\frac{\partial\Updelta W_{1}}{\partial\sigma_{s}}\right) \left(\frac{\partial\sigma_{s}}{\sigma_{1}}\right)\\ &= \frac{\sigma_{1}}{E_{o}} + \\ & N_{V}\left(\frac{\partial\Updelta W_{1}}{\partial\sigma_{s}}\right) \left(\frac{\partial\tau}{\partial\sigma_{1}} + \mu \frac{\partial\sigma_{n}}{\partial\sigma_{1}}\right)\\ &= \frac{\sigma_{1}}{E_{o}} + N_{V}\left(\frac{\partial\Updelta W_{1}}{\partial\sigma_{s}}\right) \left( \frac{1+\mu}{2} \right)\\ \Rightarrow \epsilon_{1} &= \frac{\sigma_{1}}{E_{o}} - \frac{32N_{V}a^{3}\sigma_{s}}{6E_{o}}(1+\mu) \times\\ &\left[\frac{1+\left(1-\nu \right)^{2}}{\left(2-\nu \right)^{2}}\right] \end{aligned} $$

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In terms of the principal stresses

$$ \begin{aligned} \epsilon_{1} &= \frac{\sigma_{1}}{E_{o}} + \frac{2D_{o}}{\pi \alpha^{3}E_{o}}(1-\mu)^{2} \times\\ & \left[\frac{1+\left(1-\nu \right)^{2}}{\left(2-\nu \right)^{2}}\right] \times \nonumber \\ & \left[ \sigma_{1} - \frac{(1+\mu)}{(1-\mu)}\sigma_{3}\right] \end{aligned} $$

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Appendix B: Estimation of Nonlinear Strain From the Moment Tensors

A second estimate of the strain associated with the damage can be obtained by summing the seismic moments of the individual flaws. Kostrov (1974) gives the following expression for the macroscopic strain in a volume V containing n flaws

$$ \epsilon_{ij} = \frac{1}{\mu V}\sum_{n=1}^{N}M_{ij} $$

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Johnson and Sammis (2001) express the scalar moment density associated with an individual angle crack and its wing cracks as a shear moment associated with sliding on the angle crack and a tensile moment associated with the opening of the wing cracks.

$$ m = m_{s}+m_{t} $$

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where

$$m_{s}=\frac{9}{2}\left(\frac{\lambda+2G}{\lambda+G}\right)\frac{D_{o}K_{IC}}{\sqrt{\pi a}\cos^{5/2}{\Uppsi}\sin{\Uppsi}} \times \left[\left(\frac{D}{D_{o}}\right)^{1/3}-1\right]^{1/2}$$

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$$m_{t}=\frac{3}{2}\frac{(\lambda+2G)^{2}}{G(\lambda+G)}\frac{D_{o}K_{IC}}{\sqrt{\pi a \cos{\Uppsi}}} \times\left[\left(\frac{D}{D_{o}}\right)^{1/3}-1\right]^{5/2}$$

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$$\frac{m_{t}}{m_{s}} = \frac{\lambda+2G}{3G}\sin{\Uppsi}\cos^{2}{\Uppsi} \times\left[\left(\frac{D}{D_{o}}\right)^{1/3}-1\right]^{2}$$

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In these expressions Johnson and Sammis (2001) ignore the β factor in Ashby and Sammis (1990) since it gives a contribution to the moment at zero stress. Note that the tensile opening of the wing cracks makes an increasingly dominant contribution as the damage increases.

For a crack opening in the x ₂ direction, the moment density tensor is

$$\mathbf{m} = m_{s}\left[\begin{array}{lll} -1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 0 \end{array}\right] + m_{t} \left[\begin{array}{lll} \zeta & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & \zeta \end{array}\right]$$

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where $\zeta = \frac{\lambda}{\lambda + \mu} = 1-\frac{c_{s}^{2}}{c_{p}^{2}}$. Here c _s and c _p are the S- and the P-wave speeds, respectively.

Kostrov’s expression for the axial strain is

$$\epsilon_{ij} = \frac{1}{\mu V}\sum_{n=1}^{N}M_{ij} = \frac{1}{\mu}(m_{s}+\zeta m_{t})$$

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For $\Uppsi = 45\deg$ we have

$$\begin{aligned} \epsilon_{11}&=\frac{3}{2^{3/4}}\left(\frac{\lambda+2G}{\lambda+G}\right) \times\\ &\frac{K_{IC}D_{o}}{G\sqrt{\pi a}} \left[\left(\frac{D}{D_{o}}\right)^{2/3}-1\right]^{1/2} \times\\ &\left \{\frac{12}{\sqrt{2}} + \frac{\lambda}{G} \left[\left(\frac{D}{D_{o}}\right)^{2/3}-1\right]^{2} \right \} \end{aligned}$$

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The first term in the curly brackets is associated with sliding on the initial angle cracks while the second term is associated with tensile opening of the wing cracks.