structural geology, propagation mechanics and - Stanford School of ...

plastic compaction, as suggested by the elliptical band profiles, then the assumption of at
least quasi-elastic behavior inside the bands on the time scale of propagation might be
justified (Sternlof et al., 2005).
Despite these uncertainties, the distribution of Dn measured for the 25-m-long CB
profile featured by Sternlof et al. (2005) can be used to estimate an approximate effective
elastic stiffness (Eb) for the anticrack elements (Figure 4.16). For Eb = 0 (i.e. a perfect,
traction-free anticrack), the model calculates a distribution of Dn two orders of magnitude
too high (~10 cm versus ~1 mm in the middle and ~1 cm versus ~.1 mm at the tips). The
addition of even a small amount of element stiffness, however, drastically reduces Dn,
and we found that setting Eb = 15 (3% of Es) produced a slight over estimate. Of course,
the measured distribution of Dn could be matched exactly by independently varying Eb
for every element, but the benefits of such specificity do not justify the effort, particularly
as the assumption of elastic behavior is dubious at best. For our immediate purposes, it is
enough that the near-tip magnitudes of Dn calculated by the model generally coincide
with those measured for natural bands, since these displacement discontinuities, in
concert with any Ds accommodated, dictate the stress perturbations produced outside the
band.
The next aspect of essential model calibration involves choosing the characteristic
distance r at which σθθ max will be calculated. As demonstrated by Sternlof et al. (2005),
because the model represents CBs as trains of constant displacement dislocations, stress
magnitudes calculated in the very near-tip field (r < 0.1 mm) vary as 1/r (Figure 4.17).
For r ranging from about 0.5 mm to 5 mm, the calculated stresses vary along a more a
crack-like 1/√r trend. Beyond 5mm, the BEM model-generated stresses basically
coincide with those calculated analytically using the Eshelby inclusion method (Eshelby,
1959; Mura, 1987), which we consider the most accurate representation of the ideal,
isolated CB based on the field data, and which predicts a normalized σθθ max of about 8.75
as r → 0. The stress singularity generated by the model at r = 0 has no significance in
physical reality, both because the nature of deformation at the tip is actually plastic, and
because the assumption of homogeneous, isotropic elasticity cannot be presumed to hold
as r drops below the grain size of the sandstone. In fact, with an average grain size of
~0.2 mm, any r less than about 5 mm becomes suspect (Amadei and Stephansson, 1997).
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