Stress-strain curve of paper revisited - Innventia.com

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$Engineering fracture mechanics analysis of paper ... - Innventia.com$

PAPER PHYSICS energy cannot dissipate in highly strained areas by other contribute to the response of a dry network. a) Fig 17. Stress-strain curves for different fiber material models utilized on fiber level in 10 x 4 mm 2 , 27 g/m 2 network: (a) bilinear isotropic hardening plasticity; (b) linear elastic fiber material model. b) Fig 19.Normalized strain in the network with: (a) linear elastic; (b) bilinear isotropic hardening plasticity fiber material model at network fracture. a) Fig 18. Total amount of fractured (solid line) and separated (dashed line) bonds for networks with different fiber material models: (a) bilinear isotropic hardening plasticity; (b) linear elastic fiber material model. means than bond failure. The strain variations can be observed in Fig 19a, in which we plot the strain fields at the moment of failure normalized with the global strain to failure recorded at this point. It is also reflected in the map of the failed bonds, Fig 20, which shows more clustered concentrations of bond failures in the elastic fiber networks. Note on the effect of friction Numerical experiments showed that friction coefficient varied in a reasonable range had virtually no effect on either stress-strain curve or strength. The observed frictional forces were two orders of magnitude lower than the forces developed in the bonds and thus could not significantly b) Fig 20. Total number of contacts at different levels of fracture, represented by the contour legend, where 0 and 1 corresponds to reaching fracture and separation distances respectively: (a) linear elastic; (b) bilinear isotropic hardening plasticity fiber material model. The role of unconventional contact parameters As mentioned earlier, the bond strength is one of three parameters characterizing a bond failure under the adopted bilinear cohesive zone model. Measuring the two remaining properties, namely, the compliance of bond regions and the work of separation have not been possible with reported techniques. During experiments, the fiber-fiber bonds were not isolated, and the measured force encompassed the response from both the bonds and from global deformation of the fibers. We look at the effect of the bond compliance and the work of separation on the stress-strain curve with the help of modeling on a single realization of the network. Nordic Pulp and Paper Research Journal Vol 27 no.2/2012 325
PAPER PHYSICS Effect of compliance of contact regions Beam elements that we use to represent the fibers have no through-thickness normal strains, which makes their cross-section rigid against the load in the normal direction. In reality, the fibers deform locally due to reciprocal forces in the bond regions. We express the compliance of the bond regions through the contact stiffness, which is usually a numerical parameter in a penalty-based contact algorithm. In our case, when the cross-sections of the fibers are rigid against point-wise loads, the contact stiffness alone is appropriate enough to represent the compliance of the bonding regions. Considering its influence in the frames of the cohesive zone model, it is nothing but the slope of the elastic region in Fig 3. Consequently, with a given bond strength, the bond with a lower stiffness can accumulate greater critical fracture energy (the area under the curve up to the fracture distance). We varied the contact stiffness in the normal and tangent directions according to Table 4. Stress-strain curves plotted for these cases (Fig 21) show that the bond compliance has a very limited effect on the elastic part of the curve. Decreasing the bond stiffness in both directions by a factor of two did not change the initial slope of the curve simply because the amount of elastic energy stored in the bonds is relatively small. Having softer bond regions delays, however, the point of failure. On the contrary, the stiffer the bond regions the lower the strength, which together with the results on the impact of bond strength means that the critical fraction energy accounting for both the strength and compliance is a better measure for relating the bonds and network strength properties. Effect of work of separation The work of separation is the amount of energy needed to separate completely the bond from the point of fracture. In practice this means creating a delay between failure and full separation in a displacement-controlled test of individual bonds. We varied the separation distance, Fig 22, according to Table 5 leaving bond strength and contact stiffness intact. The work of separation was set to be 15 percent of the fracture energy in the reference and was changed to 5 and 25 percents in two selected cases. Fig 22 shows that the work of separation varied in a reasonable range has an expected but rather limited effect on the strength of the fiber network. Increasing the work of separation from 5 to 25 percents of the critical fracture energy increased the strength by about 12 percent. Effect of the number of bonds The number of bonds in the network can be related to a commonly used term “the level of interfiber bonding”. The latter is usually changed by wet pressing or beating. In numerical simulations, we reduced the number of bonds by randomly removing contacts all over the network. Along with the stress-strain curves, we computed the efficiency factors introduced by Seth and Page 1983. The efficiency factor Ф is the ratio of the initial Young’s modulus (for all bonds) to the current Young’s modulus for Fig 21. Stress-strain curves for different bonding compliance in 10 x 4 mm 2 , 27 g/m 2 network: (a) control compliance; (b) softer bond regions; (c) stiffer bond regions. Fig 22. Stress-strain curves for different separation distance the simulated tensile test on 10 x 4 mm 2 , 27 g/m 2 network at normal 25.5 mN and tangent 4.2 mN bond strength kept constant: (a) reference case; (b) minimized; (c) maximized separation distance (see Table 5). Table 4. Varying contact stiffness (CS) in the numerical tests. Contact stiffness Normal CS Tangent CS (CS) [GPa] [GPa] Soft bonds 20 4 Control 40 8 Stiff bonds 80 16 Table 5 Effect of varying separation distances (SD) keeping bond strength constant (25.5 mN in the normal and 4.2 mN in the tangent directions). Separation distance (SD) Normal SD Tangent SD [μm] [μm] a (reference) 0.73 0.60 b 0.67 0.55 c 0.80 0.66 326 Nordic Pulp and Paper Research Journal Vol 27 no.2/2012
Page 1 and 2: PAPER PHYSICS Stress-strain curve o
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