KNOWN INEQUALITIES AND THOMSEN PARAMETER $\gamma$

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. The two shear moduli must satisfy

1 <><1> = ml,   because the well-known Cauchy-Schwartz inequality $\left<\alpha\beta\right\gt^2 \le \left<\alpha^2\right\gt\left<\beta^2\right\gt$gives this result when $\alpha= \mu^{1/2}$ 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 we generally exclude this case from consideration. The implication of (CSineqform) for Thomsen's $\gamma$ is that

= m-l2l 0.