structural geology, propagation mechanics and - Stanford School of ...
structural geology, propagation mechanics and - Stanford School of ...
structural geology, propagation mechanics and - Stanford School of ...
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⎡ ∆ ⎤ ⎧ p 1 ∆ ⎛ M ⎞⎫<br />
Eb<strong>and</strong> = M ξh ε<br />
1<br />
h 2 h Mb<br />
⎪ ⎛ ⎞ ⎛ ⎞ ⎪<br />
⎢ ⎜ ⎟⎥ ⎨ + ⎜ ⎟⎜<br />
− ⎟⎬.<br />
(10)<br />
⎣ ⎝ ⎠⎦ ⎪⎩ ⎝ ⎠⎝<br />
⎠⎪⎭<br />
Equation (10) suggests that the ratio <strong>of</strong> moduli is probably not significant unless it<br />
p<br />
exceeds two <strong>and</strong> ∆/ h is <strong>of</strong> the same order <strong>of</strong> magnitude as ε . This result is consistent<br />
with the finding <strong>of</strong> Sternl<strong>of</strong> <strong>of</strong> al. (2005) that the internal stiffness does not appreciably<br />
p<br />
affect the state <strong>of</strong> stress around a highly eccentric ellipsoidal CB when ε is on the order<br />
<strong>of</strong> 10 %. If the CB is modeled as a rigid inclusion, then Mb→∞<br />
<strong>and</strong> the last parenthesis in<br />
(10) becomes −1.<br />
If the inelastic compactive strain is much greater than the nominal<br />
p<br />
strain (i.e. ε >> ∆/ h ), then this reduces to the same expression as (9). Consequently,<br />
unless the stiffness <strong>of</strong> the CB material is much less than that <strong>of</strong> the surrounding material,<br />
(9) gives a good approximation to the energy release rate.<br />
5. Discussion<br />
The simple model presented here suggests that CB <strong>propagation</strong> can reasonably be<br />
considered to occur when the energy released per unit advance <strong>of</strong> the b<strong>and</strong> Eb<strong>and</strong> is equal<br />
to some critical value Ecrit that reflects the resistance <strong>of</strong> the material to compaction.<br />
Expressions (6), (7), (9), <strong>and</strong> (10) for Eb<strong>and</strong> give the energy released in terms <strong>of</strong> the<br />
compactive strain <strong>of</strong> the b<strong>and</strong>, the imposed strain, the elastic properties <strong>of</strong> the b<strong>and</strong> <strong>and</strong><br />
the surrounding material, <strong>and</strong> the thickness <strong>of</strong> the b<strong>and</strong> <strong>and</strong> the surrounding material<br />
(spacing between b<strong>and</strong>s). Assuming that b<strong>and</strong> <strong>propagation</strong> does occur when Eb<strong>and</strong> = Ecrit ,<br />
it is possible to estimate the minimum value <strong>of</strong> the material parameter Ecrit from<br />
representative values <strong>of</strong> the parameters derived from the Aztec s<strong>and</strong>stone (Sternl<strong>of</strong> et al.,<br />
2005). Taking σ + = 40 MPa, ξ h = 001 . m, corresponding to 1 cm thick CBs spaced 1<br />
p<br />
meter apart, <strong>and</strong> ε = 01 . , corresponding to a porosity reduction <strong>of</strong> 10 % , yields Eb<strong>and</strong><br />
=<br />
2<br />
40 kJ/m . Ecrit<br />
must be at least this large when <strong>propagation</strong> <strong>of</strong> the b<strong>and</strong> ceased, otherwise<br />
it would have continued advancing. Using the smallest <strong>and</strong> largest values for σ +<br />
estimated by Sternl<strong>of</strong> et al. (2005) <strong>and</strong> keeping the same values <strong>of</strong> ξ h <strong>and</strong><br />
2<br />
range <strong>of</strong> Ecrit from 10 to 60 kJ/m . For<br />
ν = 02 . , the value <strong>of</strong> σ + implies<br />
h<br />
p<br />
ε , yields a<br />
M = 22 GPa, corresponding to G = 8.3 GPa <strong>and</strong><br />
18 10<br />
−3<br />
∆/ = . × . However, ∆ is difficult to estimate <strong>and</strong><br />
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