Random fiber networks and special elastic orthotropy of paper

We consider a particular in-plane elastic orthotropy observed experimentally for various types of paper, namely: S₁₁₁₁ + S₂₂₂₂ - S₁₁₂₂ = S₁₂₁₂, where S_ijkm are components of the inplane compliance tensor. This is a statement of the invariance of in-plane shear compliance S₁₂₁₂, which has been observed in some studies but questioned in others. We present a possible explanation of this "special orthotropy" of paper, using an analysis in which paper is modeled as a quasi-planar random microstructure of interacting fiber-beams - a model especially well suited for low basis weight papers. First, it is shown analytically that without disorder a periodic fiber network fails the special orthotropy. Next, using a computational mechanics model, we demonstrate that two-scale geometric disorder in a fiber network is necessary to explain this orthotropy. Indeed, disordered networks with weak flocculation best satisfy this relationship. It is shown that no special angular distribution function of fibers is required, and that the uniform strain assumption should not be used. Finally, it follows from an analogy to the thermal conductivity problem that the kinematic boundary conditions, rather than the traction ones, lead quite rapidly to relatively scale-independent effective constitutive responses.