8 - Hygro-mechanics of fibrous networks: A comparison between micro-scale modelling approaches

Abstract

This chapter presents a multiscale approach to investigate the dependence of the effective hygro-mechanical behaviour of paper sheets on the properties of the underlying fibrous network. Despite the vast amount of literature on the hygro-expansion of paper, the relation between the effective material properties and the underlying micro-structural features is not fully understood. The point of departure of this work is a highly idealised micro-structural model of paper that is based on two main assumptions: (i) the underlying fibrous structure is described as a lattice, consisting of two orthogonal fibre families and (ii) paper is simplified to a two-dimensional (2D) material. Despite its simplicity, this model reproduces representative features, such as the network-level hygro-elastic properties, the areal coverage, etc. The model can be solved analytically, providing closed-form expressions that explicitly reveal the influence of the individual micro-scale parameters on the effective hygro-mechanical response. Taking this idealised model as a reference, the effect of the two main underlying modelling assumptions on the predicted effective hygro-elastic response is explored, by relaxing them one at a time. First, instead of a lattice description, a planar fibrous network is considered. The network model is generated by random deposition of the fibres within a planar region according to a uniform probability density function. Asymptotic homogenisation is used to determine its effective properties numerically. Second, instead of a 2D representation, a three-dimensional (3D) model is considered, which extends the reference lattice description in the out-of-plane direction. In the 3D representation, the fibres may bend and be wrapped around each other, which may affect the effective hygro-elastic response. The effective properties of the 3D model are computed by numerical homogenisation. The properties predicted by the reference 2D lattice model are compared with those obtained from the network model and the 3D lattice model. Assuming a 2D lattice description has in general a strong influence on the predicted mechanical behaviour of the material. For realistic values of the average thickness of the network, the elastic properties calculated with the 2D lattice model overestimate the response of the planar random network model by about 30%. Moreover, the elastic properties of the 3D lattice are overestimated by less than 5%, up to a factor of approximatively 2 for high degrees of fibre waviness. As for the hygroscopic behaviour, the 2D lattice idealisation appears to have a smaller influence on the predicted response. The hygroscopic properties obtained from the 2D lattice model match those of the random network well, to within a range of 10% for realistic coverages, while the difference with those of the 3D unit cells is within approximately 30% for realistic average network thicknesses.