Compression of Fibrous Assemblies: Revisiting the Stress–Density Relation

Many engineering materials are made from fibers, and fibrous assemblies are often compacted during the fabrication process. Compression leads to the formation of contacts between fibers, and this causes stiffening. The relation between the uniaxial stress, S, and the volume fraction of fibers, φ, is of power law form. The derivation of this relation based on micromechanics considerations takes as input the structural evolution represented by the dependence of the mean segment length of the network, l_c, on the current density, ρ (ρ is defined as the total length of fiber per unit volume of the network). In this work, we revisit this problem while considering that the mean segment length should be defined exclusively by fiber contacts that transmit load. We use numerical simulations of the compression of crimped fiber assemblies to show that, when using this definition, $ρ∼1/lc2$ at large enough strains. Purely geometric considerations require that $ρ∼1/lc$⁠, and we observe that this applies in the early stages of compaction. In pre-stressed networks, the density–mean segment length scaling is of the form $ρ∼1/lc2$ at all strains. This has implications for the relation between stress and the fiber volume fraction. For both ρ versus l_c scalings, $S∼(φn−φ0n)$⁠, where φ₀ is the initial or reference fiber volume fraction; however, n = 3 when $ρ∼1/lc$ and n = 2 for $ρ∼1/lc2$⁠. These predictions are compared with experimental data from the literature.