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)E(n) for a hexagonal material is written in terms of n3n3 only such that it reveals an axis of rotational symmetry in the direction x3x3, which is perpendicular to the transverse isotropy plane, that is x1x2x1x2-plane: indeed the components n1,n2n1,n2 of the unit vector nn 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.