Ting, T. C. T. Positive definiteness of anisotropic elastic constants. (English) Zbl 1001.74511 Math. Mech. Solids 1, No. 3, 301-314 (1996). Summary: When the elastic constants of an anisotropic material are written as a \(6\times 6\) symmetric matrix C, the elastic energy of the material is positive if the matrix C is positive definite. There are two criteria that one can use to see whether C is positive definite. We present each criterion and discuss its merits and drawbacks. For a two-dimensional deformation, it suffices to consider a \(5\times 5\) symmetric matrix \(C^0\). In the Stroh formalism for two-dimensional deformations, the matrix \(C^0\) is replaced by three \(3\times 3\) matrices \(N_1,N_2,N_3\). Deleting the elements that are either zero or unity reduces \(N_3\) to a \(2\times 2\) matrix \(\hat N{}_3\), and \(N_1\) to a \(3\times 2\) matrix \(\hat N{}_1\). We show that \(C^0\) is positive definite if and only if \(N_2\) and \(-\hat N{}_3\) are positive as well. The matrix \(\hat N{}_1\) can be arbitrary. In particular, a new relation \(|C^0|=|-\hat N{}_3|\cdot|N_2|^{-1}\) is obtained. Generalizing to three-dimensional deformations, we show that positive definiteness of the \(6\times 6\) matrix C is equivalent to positive definiteness of two \(3\times 3\) matrices. In the special case of monoclinic materials with the symmetry plane at \(x_1=0, x_2=0\), or \(x_3=0\), positive definiteness of C is equivalent to positive definiteness of three \(2\times 2\) matrices. Cited in 24 Documents MSC: 74B05 Classical linear elasticity 74E10 Anisotropy in solid mechanics PDFBibTeX XMLCite \textit{T. C. T. Ting}, Math. Mech. Solids 1, No. 3, 301--314 (1996; Zbl 1001.74511) Full Text: DOI References: [1] Lekhnitskii, S. G., Theory of Elasticity of an Anisotropic Elastic Body (1950) [2] Hohn, F. E., Elementary Matrix Algebra (1965) · Zbl 0132.25303 [3] Mehrabadi, M. M., Q. J. Mech. Appl. Math. 43 pp 15– (1990) · Zbl 0698.73002 [4] Mehrabadi, M. M., J. Elasticity 30 pp 191– (1993) · Zbl 0773.73013 [5] Ting, T. C. T., Q. Appl. Math. 52 pp 363– (1994) · Zbl 0812.73010 [6] Eshelby, J. D., Acta Metall. 1 pp 251– (1953) [7] Stroh, A. N., Phil. Mag. 3 pp 625– (1958) · Zbl 0080.23505 [8] Ingebrigtsen, K. A., Phys. Rev. 184 pp 942– (1969) [9] Barnett, D. M., Phys. Norv. 7 pp 13– (1973) [10] Chadwick, P., Adv. Appl. Mech. 17 pp 303– (1977) · Zbl 0475.73019 [11] Ting, T. C. T., Q. Appl. Math. 46 pp 109– (1988) · Zbl 0644.73016 [12] Barnett, D. M., SIAM Proceedings Series, in: Modern Theory of Anisotropic Elasticity and Applications pp 199– (1991) [13] Ting, T. C. T., J. Elasticity 27 pp 143– (1992) · Zbl 0755.73031 [14] Ting, T. C. T., Anisotropic Elasticity: Theory and Applications (1996) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.