Elastic properties of a Poisson–Shear material

Abstract

A Poisson–Shear material is one which displays significant in-plane shear strain when loaded in the outof-plane direction. Hence for a prescribed out-of-plane strain (e3), we have e1 ≈ −e2. To attain this behavior a Poisson–Shear material should, for example, possess a positive Poisson’s ratio in the 1–3 plane but a negative Poisson’s ratio in the 2–3 plane, or vice versa, such that the magnitudes are almost equal. A Poisson–Shear material, therefore, can be used as a micro-capsule for squeezing into veins or any small ducts such that axis-1 is parallel to the micro-duct. Hence when squeezed along axis-3 the micro-capsule contracts along axis-2, or vice versa, such that the transverse cross-section of the micro-capsule contracts to ease entry. Allowance for expansion along axis-1, on the other hand, prevents excessive densification, hence enabling the micro-capsule to perform as drugdelivery media or for any storage purposes during transport (see Fig. 1 for illustration). Based on strain energy formulation, the Poisson’s ratio is in the range −1 0 in 2–3 plane, we let 0 < θ2 < (π/2) < θ1 < π . Recently, a unified study on the elastic stiffness of a generalized honeycomb structure, applicable for Poisson’s ratio of either signs, has been performed [30], whereby the mode of deformation is confined to be two-dimensional. In this paper, we extend a similar approach [30] for the case of Poisson–Shear material [29], in which deformation is three-dimensional. For brevity, we employ the notation θi for i = 1, 2. By virtue of symmetry, one quarter of the RVE is isolated for analysis, as shown in Fig. 5. A kinematics proof for Poisson–Shearing is given in the Appendix. In the following analysis, we consider i, j = 1, 2 = 3 unless specified otherwise. Since (OC) = [bi − (li − bi ) cos θi ]/2 and (OA) = [(li − bi ) sin θi ]/2, we have the in-plane extensions