Extrema of Young’s modulus for elastic solids with tetragonal symmetry

For a homogeneous and linearly elastic solid the general expression of Young’s modulus E(n) is given, and a constrained extremum problem is formulated for the evaluation of the directions n corresponding to stationary values of the modulus. The formulation follows that presented in [International Journal of Solids and Structures 40 (2003) 1713–1744] for the cubic and transversely isotropic elastic symmetries. In this paper the tetragonal elastic symmetry class is considered, and explicit solutions for the directions n associated to critical points of E(n) are analytically evaluated. Properties of these directions and of the corresponding values of the modulus are discussed in detail. The results are presented in terms of three material parameters, which are responsible of the degree of anisotropy. For the tetragonal system, the complete description of the directional dependence of Young’s modulus leads to the identification of 12 classes of behavior. For each of these classes several examples of real materials are shown and suitable graphical representations of the function E(n) are given as well.