Fourth-rank tensors of [[V2]2]-type and elastic material constants for 2D crystals

Fourth-rank symmetric tensors, that embody the elastic properties of crystalline anisotropic substances, were constructed for all 2D crystal systems. Using them we obtained explicit expressions for inverse of Young's modulus E(n), inverse of shear modulus G(m,n) and Poisson's ratio v(m, n), which depend on components of the elastic compliances tensor S, on direction cosines of vectors n of uniaxial load and the vector m of lateral strain with crystalline symmetry axes. All 2D crystal systems are considered. Such representation yields decomposition of the above elastic characteristics to isotropic and anisotropic parts. Expressions for Poisson's coefficient are well suited for studying the property of auxeticity and anisotropy of 2D crystals. Christoffel's tensor is calculated for all 2D crystal classes. (c) 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.