Random fiber networks and special orthotropic elasticity of paper

The objective of this study is to develop an improved understanding of the microscopic (fiber scale) interactions in cellulosic fiber networks, and of their influence on the macroscopic in-plane elastic properties of paper. In particular, the study is motivated by a particular in-plane elastic orthotropy of paper observed experimentally for various types of paper dating back to Horio and Onogi (1951) and Campbell (1961), namely: S₁₁₁₁+S₂₂₂₂-2S₁₁₂₂-S₁₂₁₂ = 0. Here, S_ijkl are the components of an in-plane compliance tensor. This is a statement of the invariance of in-plane shear compliance S₁₂₁₂, which has been observed in some studies but contradicted in others. We present a possible explanation of this `special orthotropic elasticity' of paper, using an analysis in which paper is modeled as a quasi-planar random microstructure of interacting fiber-beams. This model is especially well suited for low basis weight papers. It is shown that geometric disorder in a fiber network is necessary to explain this orthotropy; it is shown analytically that without disorder a periodic fiber network fails this relationship. On the other hand, it is disordered networks with low flocculation that best satisfy the relationship. It also follows from the micromechanical analysis that no special angular distribution function of fibers is required (such as proposed by Schulgasser and Page 1988), and that the uniform strain assumption should not be used.