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

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