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This Appendix will clarify the derivation of the product formulas relating
the bulk modulus K = K_{R} and G_{eff}^{(2)}. The product formula
for K_{V} and G_{eff}^{(1)} is just the dual, and is obtained
in a very similar fashion.
Since the relevant excitation for G_{eff}^{(2)} has been shown to be
a shear strain proportional to (1, 1, 2)^{T}, consider
 
(56) 
Then, by applying the inverse of the matrix in (62)
to the left side of the equation, we get the useful formula:
 
(57) 
which supplies two independent identities among the
elastic coefficients. These are
1 = (<I>S_{11}I>+<I>S_{12}I>)(<I>C_{11}I>+<I>C_{12}I>2<I>C_{13}I>)  2<I>S_{13}I>(<I>C_{33}I><I>C_{13}I>) 


(58) 
and
1 = <I>S_{13}I>(<I>C_{11}I>+<I>C_{12}I>2<I>C_{13}I>)  <I>S_{33}I>(<I>C_{33}I><I>C_{13}I>). 


(59) 
Adding these together and switching to the a,b,c notation for stiffnesses, we find
 
(60) 
Recalling that
 
(61) 
and then substituting (66), we find
 
(62) 
Then, since
 
(63) 
we find immediately that
 
(64) 
because G_{eff}^{(2)} = [(amf)+(cf)]/3.
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Stanford Exploration Project
7/8/2003