Papermaking

An aggregate (mosaic) model is proposed to represent the structure of paper and model the mechanical properties. The model treats paper as an aggregate of three subregions of characteristic materials, viz. bonded regions, unbonded regions (free fiber segments) and voids. A computer simulation based on the Monte Carlo method is performed to generate random and oriented paper sheets and input parameters for the aggregate model. The number of fiber crossings, total bonded area, average free fiber segment length and volume fractions of bonded material and free fiber segments and apparent sheet density are obtained from the statistical geometry description of the paper structure. The upper and lower bounds on the elastic moduli and moisture swelling coefficients of void-free paper are derived based on anisotropic elasticity theory and a fiber orientation distribution parameter. The finite element method is applied to generate effective elastic moduli and moisture swelling coefficients of the aggregate model consisting of fiber crossings and segments, but no voids. The elastic moduli of paper so obtained are corrected for the voids present in paper. The predictions are compared with previously published experimental results, and it is demonstrated that the results generally fall within the theoretical bounds. The mosaic model was shown to approximate the mechanical properties of paper.

The influence of voids on the hygroelastic properties of paper has been investigated using analytical and numerical methods. Paper was modeled as a laminate made of cell-wall layers. A continuous fiber orientation distribution was introduced into the laminate model to derive the baseline properties of the papersheet. The voids in the papersheet were modeled as reinforcements with zero elastic properties. The reduction of elastic stiffnesses of isotropic materials containing different shapes and volume fractions of pores were analyzed using Voigt, Reuss, foam and combination models. Hashin's two-phase bounding model and Christensen's three-phase self-consistent models were also used to predict the elastic stiffnesses of isotropic porous materials. The influence of voids on the engineering constants of orthotropic materials was analyzed using 2-D and 3-D finite element models. The invariance of hygroexpansion in the presence of voids was demonstrated using analytical and numerical methods. The theoretical model predictions were correlated with previously published experimental results.

A computational approach for characterization of curl of paper under humidity changes is presented. Asymmetric papers with nonuniform through-thickness fiber orientation distribution are considered. Testing of the constituent layers of the papers considered was conducted at various constant relative humidities to obtain the mechanical properties, moisture content, moisture expansion coefficients and stress relaxation curves. Experiments were performed on asymmetric two-ply laboratory made papers to determine the curl response under moisture loading. The influence of viscoelastic stress relaxation on the curl response was first investigated. Geometrically nonlinear finite element analysis was conducted. It was found that the curvatures relax at an increasing rate with increasing humidities because of moisture enhanced viscoelastic dominance. Computed time-dependent curvatures were compared to experimental measurements which verified the mode shape and time-dependent relaxation response. Geometrically nonlinear finite element analysis revealed that initial deflections may strongly influence the subsequent curl behavior. A sheet with initial curvatures may undergo a bifurcation transition (buckling curl response) if the curvatures strongly interact. After the bifurcation transition, the sheet may or may not assume an unexpected shape. Experiments showed sensitivity of the response to the directions of the initial curvatures, and there are indications of a bifurcation as a result of curvature interaction. A two-ply laminate model was used to analyze curvatures of various asymmetric papers. Differences in fiber orientation distribution and principal fiber orientation angle between the two plies were considered. The analysis showed that the sheet typically bifurcated into a cylindrical and/or twisted shape. A sheet with known through-thickness fiber orientation demonstrated a complex curl response that could be simulated using the approach presented, given that the initial curl shape is known.