This Article Full Text (PDF) References Alert me when this article is cited Alert me if a correction is posted Services Email this article to a friend Similar articles in this journal Alert me to new issues of the journal Add to Saved Citations Download to citation manager Request Permissions Request Reprints Add to My Marked Citations Citing Articles Citing Articles via HighWire Citing Articles via Google Scholar Citing Articles via Scopus Google Scholar Articles by Ting, T. C. T. Search for Related Content Social Bookmarking                   What's this?

Positive Definiteness of Anisotropic Elastic Constants

Department of Civil and Materials Engineering, University of Illinois at Chicago, Chicago, Illinois 60607-7023

When the elastic constants of an anisotropic material are written as a 6 x 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 if 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 x 5 symmetric matrix C'. In the Stroh formalism for two-dimensional deformations, the matrix C⁰ is replaced by three 3 x 3 matrices N₁, N₂, N₃. Deleting the elements that are either zero or unity, the N₃ is reduced to a 2 x 2 matrix &3, and the N₁ is reduced to a 3 x 2 matrix N1. We show that C' is positive definite if and only if N₂ and - N₃ are positive as well. The matrix N₁ can be arbitrary. In particular, a new relation 1C⁰ I = I -N₃ j IN2 I - is obtained. Generalized to three-dimensional deformations, it is shown that positive definiteness of the 6 x 6 matrix C is equivalent to positive definiteness of two 3 x 3 matrices. In the special case of monoclinic materials with the symmetry plane at xl = O,x2 = 0, or x₃ = 0, positive definiteness of C is equivalent to positive definiteness of three 2 x 2 matrices.

Mathematics and Mechanics of Solids, Vol. 1, No. 3, 301-314 (1996)
DOI: 10.1177/108128659600100302


CiteULike    Connotea    Del.icio.us    Digg    Reddit    Technorati    What's this?

This article has been cited by other articles:

				T. C.T. Ting Can a Linear Anisotropic Elastic Material Have a Uniform Contraction under a Uniform Pressure? Mathematics and Mechanics of Solids, June 1, 2001; 6(3): 235 - 243. [Abstract] [PDF]