Peer Reviewed Applications of an analytical method to calculate the load distribution along a fibre in a loaded network

A new analytical solution for the load distribution along a fibre in a network has been used to investigate some aspects of paper tensile strength and elastic modulus. The method uses a similar approximation to the shear-lag formulation but allows stress transfer at individual contacts, rather than specifying a single stress transfer function applying along the entire length of the fibre. Measured elastic modulus data, where the fibres only varied in length and not cross-section showed only a small effect of fibre length on modulus. This is consistent with a high overall stress transfer coefficient for each fibre-fibre contact, resulting in the contacts at the ends of the fibres being heavily loaded. The maximum force at the middle of the fibre was calculated as a function of the fibre-fibre shear bond strength. The data showed that most literature values are too low to allow the fibre to break during paper fracture. The simulation method was able to explain the reduction in sheet tensile strength with a reduction in density, but was unable to explain the reduction in sheet strength with reduced fibre length. The assumption that a fibre-fibre bond fails completely once its breaking load is exceeded is believed to cause the discrepancy. INTRODUCTION Paper mechanical properties include strength, elastic modulus and stretch at break. Strength and stiffness are very important for paper performance in many converting and end-use applications. Paper itself is a network of ligno-cellulosic fibres bonded together through hydrogen bonds. The fibres are positioned stochastically, that is with their centres randomly located, and with orientation determined according to an orientation distribution. Additional perturbing influences on fibre location are flocculation, where fibre clumping produces local areas of high grammage, with corresponding areas of low grammage elsewhere in the sheet. Acting in opposition to this tendency, fibres may also be more uniform than would be predicted from randomly located positions, since areas of low grammage will experience greater drainage during sheet formation, concentrating fibres at that point, a process known as self-healing. The orientation and distribution of fibres in contact with any given fibre of interest are determined by the overall stochastic distribution as well as influences from flocculation and self-healing. For a stochastic three dimensional network, distances between contacting fibres have been shown to be given by a Γ function (1), while experimental data (2) have been fitted by a two-parameter Weibull probability density function yielding similarly shaped distributions