Random formation, inelastic response and scale effects in paper

We consider the constitutive law of paper and its relation to a spatially inhomogeneous, random paper structure, the so-called formation. Under biaxial tensile loading, paper may be taken as an elastic-plastic hardening material, modelled here by a hyperbolic tangent law. The fact that paper is a quasi-two-dimensional solid offers a chance to inspect its formation via a basis-weight distribution (mass per unit area), and then carry out cross-correlations of two planar fields: the formation versus the strain (or stress) field resulting from a fine-mesh finite-element simulation of a sheet of paper subjected to a specified in-plane loading. The simulation requires an assumption of constitutive coefficients as a function of formation, and the higher the resultant cross-correlation, the better the assumed constitutive law. With this methodology we conclude that the higher the basis-weight, the higher the local stiffness and strength of the paper (on millimetre scales) in two very different cases: a Crayola sketch-paper and a paperboard. We next set up a Boolean model of paper formation, where each grain plays the role of a floc of cellulose fibres. First, we employ a computational-mechanics fibre-network model to demonstrate the effects of formation on non-uniform strain fields in the elastic regime. Moving on to the inelastic regime, we treat flocs as continuum ellipses, and, employing the resulting relation between formation and constitutive coefficients, we study scale-dependent homogenization of paper and establish that the representative volume element in the sense of Hill (1963 J. Mech. Phys. Solids 11, 357-372) is approximately reached on scales about 10 times larger than the floc size.