Random fiber networks and special elastic orthotropy of paper

We consider a particular in-plane elastic orthotropy observed experimentall y for various types of paper, namely: S-1111+S-2222-2S(1122)=S-1212, where S-ijkm are components of the in-plane compliance tensor. This is a statemen t of the invariance of in-plane shear compliance S-1212, which has been obs erved in some studies but questioned in others. We present a possible expla nation of this "special orthotropy" of paper, using an analysis in which pa per 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 fa ils the special orthotropy. Next, using a computational mechanics model, we demonstrate that two-scale geometric disorder in a fiber network is necess ary to explain this orthotropy. Indeed, disordered networks with weak flocc ulation best satisfy this relationship. It is shown that no special angular distribution function of fibers is required, and that the uniform strain a ssumption should not be used. Finally, it follows from an analogy to the th ermal conductivity problem that the kinematic boundary conditions, rather t han the traction ones, lead quite rapidly to relatively scale-independent e ffective constitutive responses.