# Fibre network in compression

Compression strength is usually described at the level of observations. The theoretical background is not very well understood, so there are various theories of compressive failure. Development of the theories is also made more difficult because there are no experimental results for the compression strength of fibres ^{1}. The main elements in the theories are fibre-level buckling (structural failure) and shear failure of bonds or fibres (material failure), but various theories give them opposite roles.

The observable effect of a compressive failure is seen in Figure 1. In addition to bending of the layers caused by buckling, one can see areas where the sheet has delaminated. Delamination reduces the compressive load-carrying capacity much more than mere buckling, because the replacement of a thick solid structure with several thinner and thus less stiff elements that can move relative to each other reduces bending stiffness dramatically. The location of the delamination has been observed to depend on the z-directional profiles. With chemical strength additives on the surfaces, the location is more commonly near the middle of the sheet, and with a two-sided sheet, the location is more commonly at the fines-rich felt side ^{2}.

*Figure 1. Paperboard specimen after compressive failure illustrating various local failure modes ^{3,4}.*

The buckling theory has been developed by Fellers ^{3}. It describes compressive failure as a structural failure of the fibrous network, whereas shear failures of bonds have a minor role ^{4}. The insignificance of shear failure is in contrast with several other failure modes of paper where it is often important. For example, in tension, the failure of paper and board arises from either the shear failure of bonds or the tensile failure of fibres.

Buckling refers generally to a situation where axial compression changes abruptly into bending and the load-carrying capacity of the object decreases. How much it decreases depends on the situation. In buckling, the material itself does not fail. If the span is long, it even remains in the elastic region.

Buckling takes place when the work done in compression exceeds the work that would be needed to bend the beam instead. For an elastic beam, the critical stress of buckling *σ _{c,b}* is given by

(1)

where *S _{b}* is the bending stiffness per unit width,

*l*is the length and d the thickness of the beam, and

*C*is a geometric constant

^{3}. For a beam with fully fixed end

*C*= 4 and for rotating ends

*C*= 1. Fibre bonds can be considered fixed, so

*C*= 4 is applicable. In a fibre network, the beams would be the fibre segments between bonds.

The theory explains some observed aspects of compressive strength, for example that it is relatively independent of pulp yield, and thus independent of bond strength, and why sometimes non-linear elasticity is observed. However, some other observations are not easy to explain with it, like the orientation of compressive strength. Buckling load according to Eq. 1 is proportional to ≈ l^{-2}. Because the fibre segments are longer in MD in proportion to the geometric fibre orientation, the buckling load would be lower in MD, and the correct value would be obtained with some complex dependence of fibre stiffness and orientation. Also, the buckling load of individual fibre segments calculated from Eq. 1 would be in the Giga-Pascal range, which is clearly above what one would expect with compression strengths in the tens of Mega-Pascals range.

In many other planar materials, compressive failure can be linked to shear failure of the material, and the shear strength has been correlated with a slide modulus, which is a derivation of the shear modulus. There have been attempts to generalise this theory to paper but without much success ^{5}. Habeger created his theory from this starting point ^{6}. He observed the important difference that paper is made of particles that have a size similar to the wavelengths of the typical failure modes, whereas with the other materials the building blocks, typically crystalline grains, are significantly smaller. Therefore, the theories based on a homogeneous material are not directly applicable, because paper is inhomogeneous in the relevant scales. The failure of the general theory is seen also in that compressive strength depends on the product of in-plane elastic modulus and out-of-plane shear modulus (Figure 2), instead of shear modulus only. Because the above parameters are not very well correlated, if compressive strength were plotted against any individual parameter, like elastic modulus, the result would be a breakdown into different lines depending on the furnish. The relationship is not very practical for paper testing, because out-of-plane shear modulus is not easy to measure, though it offers a good starting point for modelling ^{7,8}.

*Figure 2. Observed relationship between compressive strength of handsheets and the product of in-plane elastic modulus and out-of-plane shear modulus ^{6}.*

The observation in Figure 2 can be explained with a theory where a layered structure is assumed, with the thickness of each layer being similar to fibre thickness. Some of the layers are initially curved, some straight. In compression, the curvature of the initially curved layers increases. This is called prebuckling deformation, and it is the critical phenomenon initiating the failure. During compression, shear stresses arise between the curved and the straight layers, and compressive failure takes place when the shear strength is exceeded locally.

After shear failure, buckling of critical laminae will follow, because with lost support the buckling load drops dramatically.

An important assumption is that the critical laminae are not straight, but curved. This means that the observed compressive failure phenomena depend on the imperfections of the fibre network. The theory explains compressive failure mainly qualitatively, with the general factors that decrease compressive strength being:

- larger initial curvature of the layers (imperfections of the fibrous network)
- large shear modulus (stiff structure)
- low shear failure stress (probably relates to weak fibres or weak fibre bonds)
- low layer thickness (relates to low fibre wall thickness).

Experimental evidence of the role of shear failure is given in a study in which deformations were observed with a video camera during compression ^{2}. It was noticed that shear failure between the S1 and S2 layers of the fibre wall takes place in compressive failure. Extensive deformation in the z-direction was also noticed; with a 1% compression the sheet became on average 13% thicker than before the failure, also supporting the assumption of increasing prebuckling, or increasing curvature of fibre layers caused by buckling.

Neither theory is very practical, because they explain compressive failure only qualitatively, but do not link fibre properties to failure. There have been some attempts to make other theories. Shallhorn ^{1} used a generalisation analogously to Page’s tensile strength theory, but the resulting theory is valid only for the low-density range (<300 kg/m³), and is therefore not very practical either.