Proceeding Articles

An examination has been made of the size and shape of fibre-to fibre bonds and their frequency of occurrence in paper sheets. Definitions have been proposed for parameters associated with these bond properties. The effects of beating and drying tension on these parameters have been investigated and the likely effect of other papermaking variables has been considered. The significance of the parameters in controlling the physical properties of paper is discussed.

In earlier papers,^{(1, 2)} the possibility of the use of light microscopic methods for the observation of optical contact regions between fibres in paper sheets was described and arguments were put forward that these areas of contact define the regions within which strong adhesive forces operate. While sufficient confidence in this view was held to use the technique for studies relating to geometry of fibre-to-fibre bonding in paper, it was considered that the question of the fine structure within these contact regions was sufficiently important to warrant a separate investigation, the initial results of which are given here.

Mathematical theories for some of the mechanical properties of a well-consolidated anisotropic fibrous web are developed in two major divisions : the elastic and the plastic regimes of stress/strain relationships . For each of the principal directions in the plane of the sheet, theories of the elastic regime are developed for external load, Poisson’s ratio, Young’s modulus and the modulus of rigidity. Applications of the theory to improve pulp evaluation and to studies of important sheet properties (in the elastic regime) such as stiffness and sheet rigidity are discussed. A complex of phenomena of the plastic regime is inferred from theory . Stresses tending to cause rupture of fibre-to-fibre bonds are found in two important groups: those associated with torque on bonds resulting from shearing force in fibre segments (and combined with stress caused by anisotropic shrinkage of the fibres) and those associated with tension in fibres (and combined with the anisotropic shrinkage stress) . The incidence of fibre-to-fibre bond rupture as the sheet strain is increased from the elastic range into the plastic regime is governed by equations developed for torque and tension bond failures. A brief discussion of the theory of the zero-span tensile test is included.

The purpose of this contribution is to describe how statistical-geometrical analysis [such as that developed in the paper by Corte and Kallmes, this vol., pp. 13-46] can be extended and applied to the determination of the joint probability density function p(s, e), which arises so naturally and importantly in Van den Akker’s very interesting theoretical analysis of some of the fundamental mechanical properties of paper exhibited in the elastic regime (this vol., pp. 205-241).

Using the techniques described in a previous paper, a study has been made of the breakage of fibre-to-fibre bonds in paper sheets under tensile strain . The frequent occurrence of partial separation in the areas of bonding and the relative rarity of total breakages are of particular interest. The way in which the bonded area loss is distributed between the bonds has been examined with particular reference to the effect of drying tension and beating . Both partial and total breakages are more common in freely dried sheets than in sheets dried under tension. Possible reasons for the occurrence of partial bond breakages are discussed.

Microscopic examination of the nature of paper failure, when tensile stresses are imposed upon the sheet, has shown that two important factors governing paper strength are the strength of individual fibres and the strength of bonding between fibres . A satisfactory method of determining individual fibre tensile strength has been published by this laboratory.⁽¹⁾ Using this method, the changes in strength of individual fibres of Loblolly pine springwood and summerwood holocellulose were followed during extraction with alkali of increasing concentration . The results of this work are reproduced in part in Table 1.⁽¹⁾

Following considerations of previous work, the authors’ investigations indicate that the first interfibre bonds to rupture under tensile strain do so by a peeling action with shear deformation of the paper in locally weakened regions. Progressive breakdown occurs along specific narrow bands within which strain is largely concentrated. The direction of these `strain lines’ has been predicted. Only for weak papers does final fracture occur along a single strain line, when considerable shear also results. For strong papers, only short lengths of strain lines form in a widely scattered cross-cross formation. In all cases, ultimate failure depends on the relationship between the tensile strength of fibres and the shear strength of bonds.

The mechanism of strain line formation results in paper thickness increases. Permanent deformation, frozen-in stresses and the effects of drying stresses are largely associated with strain line formation and frictional forces between fibre ‘mats’. Changes in moisture content modify the frictional effects and stress distribution and some aspects of dimensional stability can be explained. The finer details of fibre orientation distribution in machine-made papers can be attributed to shear in strain lines.

A stress/strain law is derived for a hydrogen-bonded network that is isotropic for all strains. The calculus of variations is used in conjunction with the Principle of Least Work to obtain the distribution of strain among the hydrogen bonds of the system as a function of their orientation . In addition to unidirectional applied stress, the theory can be used to predict the effects of two- and three-dimensional tension. The solution for unidirectional strain is compared with a modified form of the original treatment of the theory by Nissan and experimental evidence is used to corroborate the postulate that these two solutions `bracket’ the stress/strain behaviour for anisotropic (oriented) systems .

The physical properties of paper have been interpreted on the basis of the theory of the statistical geometry of fibrous networks. One example of each of the three main groups of properties is discussed: The maximum pore size and the optically bonded area on the basis of the theory of multiplanar structures and of the number of bond failures in the fracture of random two-dimensional networks . The experimental results support the statistical geometric approach to structure and physical properties.

A theory of the elasticity of handsheets has been formulated, based on the hypothesis that elastic deformation is caused by bending and, to a small extent, shearing and stretching of the unbonded parts of the fibres within the plane of the sheet. Experiments have shown that the theory holds for dense, well-bonded sheets, but deviates for light, poorly bonded sheets by a factor directly related to a geometric property the mean free fibre length . It is concluded that the arrangement of the fibres in handsheets is the connecting link between the fibre properties and sheet elasticity.