Formation effects on tensile strength
The local minimum value of strength within a test strip determines the tensile strength of that strip. Tensile strength therefore would be expected to decrease if formation becomes more non-uniform. Norman demonstrated this 1 by varying the forming stock concentration in a handsheet mould, see Figure 1. The average of local strength values measured using strips with a narrow neck was independent of the forming concentration. The example provided by Norman is rather atypical. Usually, the connection between formation and tensile strength is much smaller, or completely absent 2. Other factors in the structure of paper may also enhance or reduce the effect of grammage variations. These include the spatial variations in fibre orientation, bonding degree, and drying shrinkage or wet strain.
Figure 1. Tensile index vs. forming stock concentration for standard specimens using a strip width of 15 mm and small specimens with a narrow 3-mm waist 1.
The mechanism how formation influences the tensile strength of paper can be appreciated if one observes how paper deforms under loading. Figure 2 shows a typical example. With increasing elongation, localised areas of “damage” or inter-fibre bond failures appear in the paper specimen. These damage areas act as crack precursors. When the elongation is increased sufficiently, one of them gives rise to a macroscopic crack that propagates across the specimen. This is how the formation of paper controls the location of failure in the paper.
Figure 2. Localized “damage” areas (shown as white) in a silicone-impregnated newsprint specimen that was viewed in reflected light against a black background. The specimen was strained in the horizontal direction. The “damage” corresponds to the opening of inter-fibre bonds that are then seen to reflect light 3.
From Figure 2 it is also obvious that the tensile strength of the specimen is not equal to the minimum of local strength values. The critical factor for failure is the ratio of local strength or rather local strain over the local breaking strain. In papers with poor formation, local stresses vary more than in papers with good formation 4-6. Figure 3 illustrates the variation of local strains in high and low-grammage areas. In a sample of good formation, such as in Figure 3, the strain difference between low- and high-grammage areas is smaller than in a sample of relatively bad formation. The highest local strains are measured in the zone that finally fails. The more uniform paper tolerates higher local strain in the failure zone than the less uniform paper.
Figure 3. Local strain vs. external strain in copy paper specimens of good (a) and bad (b) formation, COV(b) = 6% and 10%, respectively. Low and high-grammage areas and the final fracture area are followed separately 6.
The topology of structural variations has a significant effect on the local stress distribution and probably on the local strength distribution. The first fact explains why tensile strength decreased when Norman 1 deposited stock spots on wet handsheets. Next to a high-grammage spot in the longitudinal direction, the local strain is higher than the average. Paper fails there more readily than it would in the absence of the spot. The non-uniformity of paper also leads to size-dependence in tensile strength. Tensile strength is lower in large specimens than in small specimens.
In general fracture mechanics, the size-effect of strength without flaws is studied as a stochastic phenomenon. The important idea is that the weakest-spot phenomenon also dictates the distribution of the strength values. The classical expression is the Weibull formula that describes the strength distribution via the Weibull exponent β: the smaller is β, the more heterogeneous and “weak” is the material. The Weibull distribution also seems to be the best one for the tensile strength values and web breaking tension values of paper 7. Note that there are no studies of the change of the strength distribution with the rate of elongation.
For paper, the Weibull exponent β is of the order of 15–20. The values of β in paper are higher than those found for single pulp fibres, and consistent with the fact that differences in the formation of paper usually have relatively little influence on the tensile strength values of paper. We also note that the elastic modulus of paper is only weakly dependent on formation 8 since the modulus is proportional to the average elastic energy in the specimen.