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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Equilibrium requires that σ b<strong>and</strong> = σ out . This equation can be combined with (3) to<br />
determine the strains in terms <strong>of</strong> ∆/ h , the moduli, the geometry <strong>and</strong><br />
strain energy per unit area <strong>of</strong> a vertical slice far behind the edge <strong>of</strong> the CB is<br />
p<br />
ε . Therefore, the<br />
1<br />
p<br />
W h b<strong>and</strong> b<strong>and</strong> (1 ) h out out<br />
2 2<br />
1<br />
− ⎛ ⎞<br />
= ξ σ ⎜ε − ε ⎟+<br />
− ξ σ ε . (5)<br />
⎝ ⎠<br />
+ −<br />
Taking the difference W − W yields the energy release per unit area created <strong>of</strong> CB<br />
with thickness ξ h :<br />
⎧ 2<br />
⎫<br />
⎪ ⎛ ⎞<br />
2 ⎪<br />
⎨ ⎜ ⎟ ⎬<br />
⎪ ⎝ ⎠ ⎪<br />
⎩⎪ ⎭⎪<br />
1 Mh ⎛∆⎞ ⎛ M ⎞ p⎛∆⎞<br />
p<br />
Eb<strong>and</strong> =<br />
⎜ ⎟ξ⎜ − 1⎟+ 2ξε<br />
⎜ ⎟−<br />
ξε<br />
2( MM / b) ξ + (1 −ξ) ⎝ h⎠ ⎝Mb ⎠ ⎝ h⎠<br />
4. Special cases<br />
If the b<strong>and</strong> modulus vanishes (i.e. M b = 0 ), then (6) reduces to<br />
. (6)<br />
Eb<strong>and</strong> = 1<br />
2 M ⎡ ⎛∆⎞⎤ ⎢ ⎜ ⎟ ∆<br />
⎣ ⎝ h ⎠⎦<br />
⎥ , (7)<br />
where σ + = M ( ∆/ h)<br />
is the uniform stress ahead <strong>of</strong> the b<strong>and</strong>. This is equivalent to the<br />
result for a crack with zero tractions on the surfaces <strong>and</strong>, notably, does not depend on the<br />
inelastic compactive strain. Equating (7) <strong>and</strong> (1) yields the following expression for the<br />
stress intensity factor<br />
K<br />
b<strong>and</strong><br />
=∆<br />
MG<br />
, (8)<br />
(1 −ν<br />
) h<br />
which is independent <strong>of</strong> the b<strong>and</strong> length.<br />
If the elastic modulus <strong>of</strong> the CB remains the same as the material outside (i.e.<br />
M b = M ) <strong>and</strong> we neglect the term (ξε p ) 2 as inconsequentially small, then (6) reduces to<br />
p<br />
Eb<strong>and</strong> = σ + ξε h . (9)<br />
Thus, (9) has the simple interpretation <strong>of</strong> the stress multiplied by the compactive<br />
displacement accommodated within the CB.<br />
An indication <strong>of</strong> the effect <strong>of</strong> the difference in moduli can be obtained from Eb<strong>and</strong> by<br />
simplifying the expression (6) for ξ