Extrema of Young's modulus in hexagonal materials

The optimum values of Young's modulus for hexagonal materials are obtained by using the representation of the elasticity tensor by Ahmad and Khan [F. Ahmad, R. A. Khan, Eigenvectors of a rotation matrix, Q. J. Mech. Appl. Math. 62 (2009) 297-310]. The expression of Young's modulus E(n) for a hexagonal material is written in terms of n(3) only such that it reveals an axis of rotational symmetry in the direction x(3), which is perpendicular to the transverse isotropy plane, that is x(1)x(2)-plane: indeed the components n(1), n(2) of the unit vector n have no influence on the value of Young's modulus. Moreover Young's modulus is expressed in terms of invariant quantities, i.e. eigenvalues rather than components of the compliance tensor. The problem is solved in a simple manner and is applied to some real materials. (C) 2012 Elsevier Inc. All rights reserved.