Stress-strain curve of paper revisited - Innventia.com

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PAPER PHYSICS a) b) Fig 23. (a) Stress-strain curves for different number of fiber-fiber bonds in the simulated tensile test on 10 x 4 mm 2 , 27 g/m 2 network: (A) control network, 100% bonds; (B) 25% of removed bonds; (C) 50% of removed bonds. (b) Transposed curves. a respective curve. Stress-strain curves are transposed by diving them by the efficiency factor Ф, which has a maximum value of 1.0 for a network with all bonds. The results of reducing the number of bonds by 25 and 50 percent are presented in Fig 23a and the transposed curves in Fig 23b. The curves coincide in the linear region and up to 1.4% strain. Seth and Page 1983 reported corresponding experimental findings by modifying the number of bonds by wet-pressing and beating. Since the wet-pressing affects the thickness and we removed the bonds at given thickness, we referred to the experiment with beating for comparison. The results agree rather well for the range of efficiency factors considered in the study. It should be noted, however, that even with 50% removed bonds we could not reach as low values of the efficiency factor as reported by Seth (Ф=0.686). Even with 50% bonds removed, the efficiency factor dropped to 0.9 only. This can indicate that fiber properties were modified by beating in the physical experiments but remained unchanged in our numerical simulations. Decreasing the number of bonds by 50% resulted in nearly Fig 24. Fiber axial stress distribution for different number of contacts at specific stress level of 13 kN·m/kg: (a) control network, 100% bonds; (b) 75% of bonds; (c) 50% of bonds. 30% lower strength. The stiffness of the network was not significantly affected. The reduction in strength can be explained by a simple fact that a lower number of contacts would have a higher average stress at a given global strain level, since the elastic stiffness of the network was not affected, and the global stress at a given strain level is the same. Lower number of contacts also imposes a greater stress variation as well as higher mean stress along the fibers. Fig 24 shows that distribution of axial stress in the beam element expressed through probability density function. This data was collected along the linear slope of the curve at a specific stress of 13 kN·m/kg. It shows that the distribution is bimodal. The left mode represents unstressed fiber segments and the right mode corresponds to the fiber segments bearing the load. With 50% removed bonds, the average stress in the right-side mode is greater. It explains the fact the network with fewer bonds deviates from the straight line sooner. Conclusions We used a three-dimensional fiber network model that encompasses fiber nonlinearities and bond failures to examine the phenomena, which take place in a network of fibers along the stress-strain curve. The main outcome is that the original strain inhomogeneities due to the structure are transferred to the local bond failure dynamics. The results show that failed bonds are located in the places with high local strain. By comparing networks with weak and unbreakable bonds, we concluded that strain concentrations are the precursors of bond failures and not the other way around. The width of strain concentrations regions have a size on a millimeter scale and obviously depend on the initial details of the network structure, such as local fiber orientations and bond density. The network with elastic fibers showed no sign of softening up to the point close to failure. It confirmed again that non-linear response of the network has its origin in the Nordic Pulp and Paper Research Journal Vol 27 no.2/2012 327
PAPER PHYSICS fibers. Plasticity on the fiber level reduced the strain variations since the energy dissipated through plastic deformations in the areas of high strain. The influence of the bond strength was significant. A factor of 2-3 in bond strength, which is relatively low in the view of the scatter encountered throughout the literature, changed the strength of the network dramatically. At the same time, a "non-traditional" bonding parameter, namely, the compliance of the bond regions showed a comparably strong effect on the strength as the bond strength. More compliant bond regions increase the strength by accumulating more energy prior to failure. This suggests that the critical fracture energy, which account for both the strength and compliance of the bonds, is a better measure for relating bond and network strength. The work of separation (another unconventional bonding property) showed a relatively small impact on the stress-strain curve and can be discarded in practical applications. Decreasing the number of bonds in the network by 50%, did not change the elastic stiffness significantly but decreased the strength. It also increased stress variation inside the network as well as the mean axial stress in the load-bearing fiber segments. Acknowledgement The authors appreciate WoodWisdom-NET and BiMaC Innovation together with their industrial partners for the financial support. Literature Alava, M. and Niskanen, K. (2006): The physics of paper, Reports on Progress in Physics 69(3), 669-723. Aulin, C., Gällstedt, M. and Lindström, T. (2010): Oxygen and oil barrier properties of microfibrillated cellulose films and coatings, Cellulose 17(3), 559-574. Batchelor, W. (2010): Application of an analitical method to calculate the load distribution along a fibre in a loaded fibre network, Appita Journal 63(4), 287-293. Gradin, P., Graham, D., Nygård, P. and Vallen, H. (2008): The Use of Acoustic Emission Monitoring to Rank Paper Materials with Respect to Their Fracture Toughness, Experimental Mechanics 48(1), 133-137. Groom, L., Mott, L. and Shaler, S. (2002): Mechanical properties of individual southern pine fibres. Part I. Determination and variability of stress-strain curves with respect to tree height and juvenility, Wood and Fibre Science 34(1), 14-27. Henriksson, M., Berglund, L. A., Isaksson, P., Lindström, T. and Nishino, T. (2008): Cellulose Nanopaper Structures of High Toughness, Biomacromolecules 9(6), 1579-1585. Heyden, S. (2000): Network Modelling for the Evaluation of Mechanical Properties of Cellulose Fluff, Lund University, Lund. Hägglund, R. and Isaksson, P. (2008): On the coupling between macroscopic material degradation and interfiber bond fracture in an idealized fiber network, International journal of solids and structures 45, 868-878. Joshi, K., Batchelor, W., Parker, I., Nguyen, L. (2007): A new method for shear bond strength measurement, International Paper Physics conference, Queensland, Australia, pp. 7-13. Kulachenko, A. and Uesaka, T. (2010): Simulation of Wet Fiber Network Deformation, Progress in Paper Physics, Montreal, Canada. Lindström, T., Wågberg, L., Larsson, T. (2005): On the nature Advances in Paper Science and Technology, Cambridge, The Pulp and Paper Fundamental Research Society, pp. 457-562. Niskanen, K. (1998): Paper Physics, Fapet Oy. Niskanen, K. J., Alava, M.J., Seppäla, E.T., Åström, J. (1999): Fracture Energy in Fibre and Bond Failure, Journal of Pulp and Paper Scince 25(5), 167-169. Page, D. H. and El-Hosseiny, F. (1983): The mechanical properties of single wood pulp fibers. Part IV. Fibril angle and the shape of the stress-strain curve, Journal of Pulp and Paper 9(4), 99-100. Retulainen, E., Ebeling, K. (1993): Fibre-fibre bonding and ways of characterizing bond strength, Appita Journal 46(4), 282-288. Räisänen, V. I., Alava, M. J., Nieminen, R. M. and Niskanen, K. J. (1996): Elastic-plastic behaviour in fibre networks, Nordic Pulp and Paper Research Journal 11(4), 243-248. Seth, R. S. and Page, D. H. (1983): The Stress Strain Curve of Paper, The Role of Fundamental Research in Paper Making, Mechanical Engineering Publication, London, pp. 421-452. Zavarise, G. and Wriggers, P. (2000): Contact with friction between beams in 3-D space, International Journal for Numerical Methods in Engineering 49, 977-1006. 328 Nordic Pulp and Paper Research Journal Vol 27 no.2/2012
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