# Microscopic strength mechanisms

When a piece of paper is stretched to failure, inter-fibre bonds gradually fail, as we discussed above in relation to the load-elongation behaviour of paper. Microscopic examinations of resulting fracture surfaces readily reveal that especially in well-bonded papers, also fibres break. The two microscopic failure models have been used in theories on the tensile strength of paper. Although such models cannot describe in a sensible manner the actual tensile strength of paper, they are still instructive.

The simplest estimate for tensile strength uses the assumption that fibres are solely responsible for the failure of paper. The tensile strength, *T*, of paper is then simply the elastic modulus of paper times the breaking strain of the fibres. Empirically, this argument has some validity, since the ratio of tensile strength to elastic modulus (=elastic breaking strain) is similar in various papers even with different bonding degrees. The elastic breaking strain of paper might therefore relate to a generic or universal breaking threshold of fibres or bonds.

The *zero-span strength* makes an even stronger connection between paper strength and fibre strength. In the zero-span measurement, the nominal gauge length is zero, in practice a fraction of a millimetre. The macroscopic failure should therefore always be triggered by fibre rupture. The fibres that fail first are parallel to the elongation. If the fibres are linearly elastic and their failure strain is *ε _{f}*, the zero-span strength in GPa is

*Z = E*, where

_{s}ε_{f}*E*is the effective elastic modulus for the short test span. Fibre strength,

_{s}*F*, is equal to

_{f}*A*(

_{f}E_{f}ε_{f}*A*is the cross-sectional area of a fibre). If one further assumes that , the following connection between zero-span strength and fibre strength results in

_{f}^{1}:

(1)

The prefactor 9/8 comes from the uniaxial strain condition assuming that the Poisson factor is 1/3. Since the true test span is not zero, it is not obvious that the assumed modulus relationship, , is always valid. The interpretation of the test is further complicated by the observations that the results follow a complicated distribution (normal or Weibull), and that the zero-span strength seems to have a rather non-trivial dependence on grammage ^{2}.

Bond failures result because the external load transmits across the fibre network from fibre to fibre through shear forces at the inter-fibre bonds. When the external load increases, a bond fails if the shear force on the bond exceeds its shear strength. Figure 1 illustrates the stress-transfer mechanism schematically. Stress transfers to and from every fibre through many bonds. The shear forces on the bonds vary stochastically, depending on the local network structure around each bond.

*Figure 1. Schematic illustration of stress transfer across paper along a chain of fibres. The external load transmits through shear forces at inter-fibre bonds.*

Microscopic models of paper strength have studied the competition of fibre and bond failures ^{3}. These give predictions on the dependence of strength on average fibre properties. Similarly, to models for fibre-reinforced composite materials, they assume that fibres are pulled out as bonds break. The predicted tensile strength is then proportional to the force needed to pull a typical fibre out of the sheet, i.e. the sum of bond strengths calculated over one half of a typical fibre length.

Although these microscopic models are consistent with many experimental observations (such as the increase in tensile strength with bonding degree or fibre length), there are many fundamental problems that make these models unphysical. Most notably, when a piece of paper breaks, the whole network structure disintegrates in the fracture process zone. This is therefore very different from the failure of a composite material, where fibres pull out from a continuous and by-and-large intact matrix. Also, there are no methods to directly measure the microscopic quantities that are needed. They can only be inferred from the same tensile strength data that they should account for.

Finally, we recall the discussion of the load-elongation behaviour of paper in the preceding section. The dependence of the elastic strain vs. total strain is largely independent of sheet structure, and only the location of the breaking point on the master curve varies. Our experience is that in machine-made papers, the ratio of tensile index over tensile stiffness is often equal to 1.0 ± 0.1%. Even in handsheets, the ratio is almost exclusively between 0.5% and 1.5%, as we will demonstrate at the end of this chapter. These observations strongly suggest that much of the variation observed in tensile strength actually arises from changes in elastic modulus. This fact was ignored in the traditional microscopic tensile strength theories.

Instead of trying to relate the tensile strength of paper directly to microscopic strength properties (such as fibre and bond strength), one should use the continuum fracture mechanics discussed above. In particular, when we combine Eqs. 2 and 4 in Formation effects on tensile strength, we get the following approximation for tensile strength T in the absence of defects (i.e., *a* = 0):

(2)

where the fracture process scale ξ has been replaced with a “*damage width*” *w _{d}* and the geometry factor β is approximated by

^{4}

*2π*. The damage width can be measured using the same visualisation technique as in

^{5}Figure 2 in Formation effects on tensile strength, and several methods are available to measure fracture energy

*G*, as we will explain shortly. Figure 2 shows an example of the agreement between Eq. 2 and experiments.

_{c}*Figure 2. Tensile index of handsheets formed from various chemical pulps and also by mixing in mechanical pulp vs. the prediction of ^{4} Eq. 2.*

Even if Eq. 2 is only a simplified approximation, it has the important virtue that all the quantities on the right-hand side, needed to calculate the tensile strength estimate, have a macroscopic average value. While this is naturally true for elastic modulus, it also holds for damage width and fracture energy whose measurements use paper specimens where fracture starts from a cut, and not from a “weak spot” that governs the tensile strength. By the same token, Eq. 2 also fails to account for the effects that formation and other non-uniformities can have on tensile strength. See Ref. 5 for further discussion.

Damage width appears to be linearly proportional to mean fibre length, unless fibres are weak relative to inter-fibre bonding ^{6}. In the latter case, fibres instead of bonds break when paper is elongated, and this shows up as a reduction in damage width. Also, the fracture energy decreases, and Eq. 2 reproduces the combined effect in tensile strength. The fracture energy divided by damage width is, at least in some cases, linearly related and close in magnitude to various measures of out-of-plane fracture energy and out-of-plane tensile energy absorption (i.e., Scott bond) ^{7}. This suggests that the fracture energy *G _{c}* is a related to inter-fibre bonding energy times damage width.

With some more elaboration one can even transform Eq. 2 into one where the in-plane fracture energy *G _{c}* and damage width

*w*can be replaced with different out- of-plane fracture energy or tensile energy absorption values, with reasonable agreement with measured values of tensile strength

_{d}^{6}.