Numerical calculation and finite element analysis for anisotropic elastic properties of carbon fibers: dependence of integration subinterval and mesh size on indentation-derived elastic modulus

Abstract

The elastic modulus measured by indentation of carbon fibers with various anisotropic elasticity is calculated by two numerical approaches, the Vlassak–Nix model and finite element analysis, to reveal the acceptable calculation condition for highly anisotropic materials. Five commercially available carbon fibers that varied in anisotropy index in the range of 0.5–7.9 are used (either polyacrylonitrile- or pitch-based). The numerical error in the calculated modulus increases with the decrease in fiber angle and with the increase in the anisotropy index under the same mesh condition, indicating finer mesh is required for a highly anisotropic material. The acceptable mesh size linearly increases with anisotropic index. The Vlassak–Nix model overestimates the elastic modulus at a small tilt angle if few integration subintervals are used. Conversely, finite element analysis of the Hertz contact problem with coarse mesh underestimates the modulus at a small tilt angle, and a maximum modulus is observed when the fiber is tilted a few degrees against the indentation axis. These findings are expected to assist the future determination of ideal calculation conditions for materials with large anisotropic elasticity including fibers and composites.