Energy absorption in compression of low-density fibre material

Lightweight cellulose fibre materials provide a potential alternative for fossil-based polymer foam cushions in packaging applications. By relying on foam or air-laid forming technology, it is possible to achieve comparable material density to expanded polystyrene or expandable polyethylene foams. However, the strength of fibre materials is typically lower than for their polymer-based equally light counterparts. At low density, the fibre network structure deviates significantly from the one in typical paperboard. In particular, the free segments between neighboring inter-fibre joints can be rather long, which gives room for new deformation modes. On the other hand, the exponential segment-length statistics of a random fibre network are preserved, which provides a basis for making quantitative predictions on how stress σ and energy absorption E develop during material compression.

By assuming that inelastic buckling proceeds from the longest fibre segments to shorter ones [1], we predict how the so-called cushion factor, σ/E, should behave with increasing compression strain ϵ [2]. In terms of network deformations, dimensionless absorption energy is described by a simple formula

E/σ_0 = ∫_s(ϵ)^∞ exp(-s)/s ds (1)

where 1/s corresponds to the average dimensionless work required to buckle a segment of length s, and exp(-s) is the probability density of these segments. Other parameters such as material density, fibre stiffness etc. affect only the absolute level described by σ_0. In Eq. (1), s(ϵ) is a universal function, which can be determined by a geometric argument [1].

On average, the above prediction is followed well by a vast set of tested fibre materials despite their inherent heterogeneity. The observed deviations for some samples seem to arise from fibre bundles or micro membranes formed by fines at inter-fibre joints. By taking advantage of such special features of the microstructure, the cushioning behavior of fibre materials could be further improved from the already promising level found for random fibre networks.


References

[1] J. A. Ketoja, S. Paunonen, P. Jetsu, E. Pääkkönen, Compression strength mechanisms of low-density fibrous materials. Materials, 12: 384, 2019. https://doi.org/10.3390/ma12030384

[2] E. Pääkkönen, J. A. Ketoja, J. Paltakari, Energy absorption and resilience in quasi-static loading of foam-formed cellulose fibre materials. Cellulose, 31:7137-7152, 2024. https://doi.org/10.1007/s10570-024-06030-4