Dynamical analysis of homogenized second gradient anisotropic media for textile composite structures and analysis of size effects

In order to predict the dispersion relation of 3D composite structures in the low frequency range, we construct effective first and second order grade continuum models. The effective properties of textile composites are obtained computationally by an equivalent strain energy method based on the response of the representative volume unit cell (RUC) under prescribed boundary conditions as described in Goda and Ganghoffer (2016). The expressions of the phase velocities for the three modes of wave propagation in a 3D context (longitudinal, horizontal shear and vertical shear) reveal that the second order continuum is dispersive, due to the presence of the second order elasticity constants. The shape change of the phase velocity when increasing the wave number shows the dispersive behavior of the second gradient medium, whereas Cauchy medium is non dispersive. Plots of the iso-frequency contour for the two investigated composites in the case of second gradient and Cauchy effective medium show that the second gradient contributions does not modify the anisotropic behavior of the considered composites. Important size effects on the dynamical behavior are shown, especially reflected by the dispersive behavior and the anisotropic dynamic responses, due to the significant overall increase of the second order rigidity matrix when increasing the RUC size.