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Consider the 2-dimensional (plane strain) case of a linear elastic general anisotropic material. Its elasticity tensor in engineering (Voigt) notation is positive definite, and looks like: \begin{equation} C_{ij} = \begin{bmatrix} c_{11} & c_{12} & c_{13}\\ c_{21} & c_{22} & c_{23}\\ c_{31} & c_{32} & c_{33} \end{bmatrix} \end{equation} I was wondering if the condition $c_{11}c_{33}-c_{13}^2>0$ a consequence of the positive definiteness of $C_{ij}$ or a consequence of its ellipticity? I am not even sure there exist necessary conditions for ellipticity in the general anisotropic case.

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In order for the strain energy density to be positive, the elasticity tensor has to be positive definite. According to Sylvester's criterion, a necessary condition for positive definiteness is that all of the principal minors are positive. The condition you cite follows directly from the second principal minor being positive (under an appropriate symmetry operation).

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