Microstructural model for a three-dimensional fiber network

A three-dimensional (3-D) anisotropic angular-distribution function (ADF) was developed to quantitatively characterize fiber orientation in paper. The ADF uses five independent coefficients to characterize 3-D fiber orientation in machine-made papers. By applying the ADF to a 3-D microstructural model of paper, it is possible to predict the nine elastic constants that describe paper in three dimensions. Modelings for both machine-made paper and handsheets are presented. Young's modulus and Poisson ratio were obtained in the machine, cross-machine, and thickness directions for six machine-made papers and six handsheets over a range of z-directional orientation. The ratio E(11)/E(33) decreased with increasing fiber orientation in the z-direction, with the ratio dropping dramatically at small angles of z-directional orientation. Also, one of the ADF coefficients serves as a global ordering parameter to determine the degree of z-directional orientation.