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REVIEW OF KNOWN INEQUALITIES FOR ELASTIC CONSTANTS

Since the stress-strain relation (TIss) is derivable from an energy functional, it is not hard to show that the matrix must be nonnegative or the material will be mechanically unstable. Nonnegativity of the matrix implies that all its principal minors must be nonnegative, which in turn implies the following inequalities:

a = b + 2m 0,     c 0,  

l 0,     m 0,   and

(a^2 - b^2)/4m = b+m 0,     ac - f^2 0,   and

[a(ac-2f^2) - b(bc-2f^2)]/4m = (b+m)c-f^2 0.   The second inequality in (2by2det) follows from (3by3det), (diagshr), and the second inequality in (diagonal) and is therefore often omitted from such listings. Similarly, the inequality for a follows from those for m and b+m. All of these inequalities must be satisfied regardless of the source of the anisotropy.

The formulas (avea)-(avem) can be used to derive some very simple relations among the constants. For example,

c f   follows directly from (avec) and (avef), simply noting that $\lambda/(\lambda+2\mu) \le 1$ in every layer. The inequality

c 43l   is derived directly from the fact that

<1+2> <14/3> = 34<1>,   which follows from the fact that the bulk modulus must be nonnegative in each layer so that $\lambda+ 2\mu/3 \ge 0$ everywhere. Then, the two shear moduli must satisfy

l m   since

1 <><1>.   follows easily from the well-known Cauchy-Schwartz inequality $\left<\alpha\beta\right\gt^2 \le \left<\alpha^2\right\gt\left<\beta^2\right\gt$by setting $\alpha= \mu^{1/2}$ and $\beta= 1/\mu^{1/2}$.Equality applies in the Cauchy-Schwartz inequalities only when $\alpha= const\times\beta$, which implies in the present circumstances that $\mu$ must be constant for l = m. But this is precisely the condition mentioned earlier for the layer equations to be isotropic, so I generally exclude this case from consideration.

Another inequality can be derived from the formulas obtained for finely layered media. I showed earlier that the anellipticity parameter given by

A (a-l)(c-l) - (f+l)^2   has the property that the dispersion relations for both qP- and qSV-waves are simple ellipses when A = 0 and are anelliptic otherwise. Using the results (avea)-(avem), A is shown to satisfy A > 0 for any fine layered transversely isotropic medium by noting that

A = cl[<(++2)> <1(++2)> -<++2>^2] 0.   The inequality follows again from the Cauchy-Schwartz inequality in another form somewhat more transparent for the present application

<>^2 = <()^1/2(/)^1/2> <></>.   Equality again applies only when $\mu$ is identically constant. But, it was mentioned earlier that finely layered media are isotropic if $\mu$ does not vary, so A > 0 holds for all such finely layered media if their overall constants are anisotropic. The result (Ainequality) was first obtained by Postma (1955) for two-component layered materials and later by Berryman (1979) for the multicomponent case considered here.


previous up next print clean
Next: RANGE OF THE ANISOTROPY Up: Berryman: Range of the Previous: AVERAGING THIN LAYERS FOR
Stanford Exploration Project
11/11/1997