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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.

The existence of the counterintuitive property for which a material laterally shrinks when compressed and expands when stretched, is shown for crystalline solids of the monoclinic symmetry class. The domains where the auxeticity property appears are shown and it is found, in some cases, the existence of domains where the Poisson’s ratio is always negative.

Homogeneous, anisotropic and linearly elastic solids, subjected to a given state of strain (or stress), are considered. The problem dealt with consists in finding the mutual orientations of the principal directions of strain to the material symmetry axes in order to make the strain energy density stationary. Such relative orientations are described through three Eulers angles. When the stationarity problem is formulated for the generally anisotropic solid, it is shown that the necessary condition for stationarity demands for coaxiality of the stress and the strain tensors. From this feature, a procedure which leads to closed form solutions is proposed. To this end, tetragonal and cubic symmetry classes, together with transverse isotropy, are carefully dealt with, and for each case all the sets of Eulers angles corresponding to critical points of the energy density are found and discussed. For these symmetries, three material parameters are then defined, which play a crucial role in ordering the energy values corresponding to each solution. 2003 Elsevier Ltd. All rights reserved.

For a homogeneous anisotropic and linearly elastic solid, the general expression of Youngs modulus EðnÞ, embracing all classes that characterize the anisotropy, is given. A constrained extremum problem is then formulated for the evaluation of those directions n at which EðnÞ attains stationary values. Cubic and transversely isotropic symmetry classes are dealt with, and explicit solutions for such directions n are provided. For each case, relevant properties of these directions and corresponding values of the modulus are discussed as well. Results are shown in terms of suitable combinations of elements of the elastic tensor that embody the discrepancy from isotropy. On the basis of such material parameters, for cubic symmetry two classes of behavior can be distinguished and, in the case of transversely isotropic solids, the classes are found to be four. For both symmetries and for each class of behavior, some examples for real materials are shown and graphical representations of the dependence of Youngs modulus on direction n are given as well.

The existence of the counterintuitive property for which a material laterally expands when stretched, is described with reference to the orthorhombic CuAlNi alloy. In particular, it is shown that there is a set of planes for which Poisson’s ratio is always negative.

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