Stress-strain curve of paper revisited - Innventia.com

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

PAPER PHYSICS Fig 12. Comparison of the experimental and simulated results from the tensile test on 10 x 4 mm 2 , 27 g/m 2 sample. fibers and agrees with the prediction by Alava and Niskanen 2006. Fibers not connected to the network were removed during computations. We performed numerical simulations until the stress-strain curves showed the persistent signs of softening. Capturing the entire softening part of the curve would require a fine time-step and consume a significant amount of computation time without providing much additional information. Fig 12 shows the comparison of the computed stress-strain curve with the experimental ones. Although the initial part of the curve was captured, the strength and the hardening regions were not. The fact that the strength was considerably overestimated can be attributed to an overestimated number of contacts or other contact parameters. We have investigated these factors on the very same network and mesh in order to exclude the influence of the variations due to disordered network structure. Effect of the bond strength We considered four cases: (a) bond failure with the reference bond strength “Control”; (b) with the reference bond strength divided by 3 “Weak”; (c) with the reference bond strength multiplied by a factor of 5/3 “Strong”, as defined in Table 3; (d) no bond failure “No debonding”. The separations distance was 15% longer than the fracture distance in all the cases. Fig 13 shows the stress-strain curves for different bond strengths. The appearance of the curves is similar to the results of Seth and Page 1983, who varied the bonding strength by addition of bonding agents – the networks show identical response until the point close to failure. The higher the bond strength is, the longer the curve follows the case Table 3 Different bond strengths used in parametric study. Bond strength (BS) Normal BS Tangent [mN] [mN] Strong 42.5 7 Control 25.5 4.2 Weak 8.5 1.4 BS Fig 13. Stress-strain curves for different bond strengths from the simulated tensile test on 10 x 4 mm 2 , 27 g/m 2 network. with no bond failure. Remarkably, a factor of 3 in bond strength, which is lower than the variation found in the literature (partially with data reported in stress units), changed the strength of the network dramatically with a given bond compliance and relation between fracture and separation distances. The intensity of damage can be related to the number of broken bonds. Using acoustic emission, Gradin et al. 2008 showed that during the elastic deformation of paper only a very limited number of acoustic events was registered. It remained unclear, however, whether all the fiber bond breakage could be detected with the utilized equipment, since some of the broken bonds could not emit sufficient amount of energy at failure to be registered. For example, similar acoustic emission testing applied to cellulose nanopaper by Henriksson et al. 2008 showed only very few events registered until the rupture. By observing the number of bonds failed in the network, Fig 14, we can conclude that the growth rate in the number of contacts follows an exponential law. At the same time, relatively few bonds, namely, 3% of the total number, failed completely in the network with strong bonds before the catastrophic failure. Although more contacts failed in the network with weaker bonds, the total number of them stayed below 5%. By observing the difference between the number of fractured and fully separated bonds at a given substep, Fig 14, we can conclude that there is no observable delay between the damage and complete separation of the bonds (dashed and solid lines almost coincide). Let us now examine the processes on the network level which preceded the failure and compare to what was measured in the experiments. We looked first how the failed bonds were distributed in the network. Fig 15a shows initial contact positions in the network while Fig 15b and Fig 15c show the fractured bonds at the time of onset of softening. The color scale indicates the state of damage where 0 corresponds to undamaged state and 1 to separated state according to Fig 3. Only the bonds having some degree of Nordic Pulp and Paper Research Journal Vol 27 no.2/2012 323
PAPER PHYSICS damage were plotted. towards the end. It demonstrates again that the hardening a) b) Fig 14. Total amount of fractured (solid line) and separated (dashed line) bonds for networks with different bond strengths: (a) Control, (b) Weak and (c) Strong bonds. Although the stronger network has fewer failed bonds, the failure pattern remains similar. There is a clear localization zone which indicates the failure path. The fibers which were pulled out were clearly indicated by the continual chain of separated contacts. Fig 16 shows the strain fields in the direction of loading. It was calculated as a continuous field after mapping the displacement field onto a 2D mesh. The calculated strains fields agree well with the measured strain fields, Fig 11, in terms of size and magnitude of variations. The strain field in the Fig 16a was output when the network with weak bonds reached the maximum stress, that is, around 0.7% strain. Fig 16b shows the strain field in the network with no debonding also at 0.7% strain. Apart from greater strain variations in the network with weaker bonds, both strain fields have similar features. The locations of maximum strains were the same for both cases. At the same time, the locations of the separated bonds, Fig 15b, correlated with the areas where the local strains were larger, Fig 16a. This means that the bond failures are largely affected by the local strains and not the other way around. In other words, large local strains are the precursors of bond failures. The local strain variations depend, in their turn, on the initial details of the network structure, such as local fiber orientations, number of bonds and density. This is different from wet networks, where strain variations are largely affected by the stick-slip behavior of the bonds. Effect of plasticity We investigated how plasticity on the fiber level affects the stress strain curve and micromechanical processes during straining. We compared the simulation with and without invoking bilinear plasticity for the fibers. Fig 17 shows that the network with elastic fibers follows a straight line up until the failure showing some deviation 324 Nordic Pulp and Paper Research Journal Vol 27 no.2/2012 c) Fig 15. (a) The initial contact positions prior to loading. 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: (b) weak bonds; (c) strong bonds. a) b) Fig 16. Strain field in the network with (a) weak bonds prior to failure at a global strain of 0.7%, (b) no debonding at a global strain of 0.7%. behavior of the network is mainly controlled by the fibers and not by the bonds. Interestingly, the strength of these two networks was almost identical. Assigning a linear elastic material model for fibers resulted in almost 20% fewer fiber-fiber bonds completely separated prior to network rupture compared to the case when a plastic behavior of fibers was assumed, Fig 18. This can be explained by the fact that a network consisting of elastic fibers sustains sharper strain variations as the
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Stress-strain curve of paper revisited - Innventia.com

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