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

Abstract

Fourth-rank tensors [[V 2 ] 2 ] (Voigt's) type, 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 ν(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

Auxetic properties of polycrystals

“…This yields the familiar inequalities (cf. 1, 9), namely: and where d depends on dimensionality D : $d = 2$ for 2D and $d = 3$ for 3D. We shall underline that the above inequalities hold only for initially unstrained crystals 9.…”

“…They are related to the eigenvalues of the stiffness tensor C , namely $s_u = c_u^{ - 1} $ ($u = J,L,M$ ). For quadratic materials 1: …”

“…In our recent papers we considered mechanical characteristics of all two‐dimensional (2D) symmetry systems 1 and of three‐dimensional (3D) crystalline structures of high and middle symmetry 2. In particular we derived explicit expressions for Young's E and shear G moduli as well as for Poisson's ratio ν depending on directional cosines of angles between directions n of the load and the direction m of the lateral strain, respectively, and directions of the crystalline symmetry axes.…”

“…Results presented in Refs. 1, 2 allow us to calculate their mean values in a rather simple way described in Section 4. Of course one may calculate Y , G , and ν using effective values of components of the stiffness tensor C .…”

Another Look at Auxeticity of 2D Square Media

“…Expressions for the PR of 2D materials are known, and the exact general formulas in terms of the compliances tensor S have been derived by Jasiukiewicz et al [25,26] with the fourth-rank symmetric tensorial bases, which are the counterparts of the tensor bases introduced by Walpole. For the square media from their approach, one can obtain the following expression (2)…”