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TAPPI test method T-494, [45] and other tensile properties, such as elastic modulus, where needed. Test method T-494 de fi nes tensile strength as the maximum load overall and tensile stiffness as the maximum load within the linear part of the load-extension curve, both expressed per unit width of the specimen. [45] It was found (through visual observation) that the load-extension curve was linear for all handsheets at least below 0.1% strain. 3. Results and Discussion Thickness change with extension is plotted in Figure 3 . Thickness strain with axial strain is plotted in Figure 4 . Table 1 shows the cumulative Poisson ’ s ratio values obtained at different strain levels. Barring two heavily re fi ned hardwood
handsheets (HW1R þ and HW3R þ ), all samples showed a net increase in thickness before failure. Figure 3 reveals that larger thickness increases occurred in softwood (compared to hard- wood) and unre fi ned (compared to re fi ned) handsheets. Interestingly, a majority of all of the 18 types showed a measur- able decrease in thickness fi rst, before the thickness began to increase (see Figure 4). We call this a “ dip ” and quantify it as the cumulative Poisson ’ s ratio calculated at the lowest thickness measured (Table 1c). Thickness of unre fi ned softwood handsheets (SW 1,3,5) increased consistently throughout the extension regime. In contrast, the lightest/thinnest heavily re fi ned hardwood handsheet (HW1R þ ) showed a consistent decrease in thickness. For all handsheets, the general shape of thickness-extension curve was concave up (increasing slope), reminiscent of previous observations. [1,2,5,9] Post failure, thickness of all samples was found to have decreased sharply from the one observed right before failure (see Figure 3), hinting that a signi fi cant number of structural changes responsible for causing thickness increase in paper were largely reversible and not destroyed during deformation. In some specimens, a full return to the original thickness was observed. If the thickness increase upon straining was due to irreversible changes alone, the thickness would have stayed the same before and after failure. Our previous study delineating various revers- ible and irreversible changes during tensile deformation of a nonwoven network can help explain this behavior in more detail: straightening of bent fi bers was found to be reversible, while slip- ping and breaking of fi bers were found to be irreversible changes contributing to auxeticity. [28] Before we interpret these observations further, a brief discus- sion of the network structure and the mechanistic picture of deformation is presented in the following section.
HW5 HW3 HW1 SW5 SW3 SW1
480
440
HW5R HW3R HW1R SW5R SW3R SW1R
400
360
320
HW5R+ HW3R+ HW1R+ SW5R+ SW3R+ SW1R+
3.1. Structure and Mechanism
280
The mathematical model we previously developed that governs the thickness increase in paper is shown in Figure 5 a. [5,9] Adapted from Öhrn, [1] it assumes that although the real structure of paper is a random fi ber network, local fi ber arrangements that look similar to Figure 5a, are widespread. It shows a fi ber oriented along x -axis, but not fully extended, meandering above and below other fi bers. When stretched, this fi ber gets extended/ straightened and pushes the other in-plane fi bers that it is in con- tact with, out of the plane of paper, causing an overall increase in thickness (Figure 5b). If d represents the thickness/height of fi bers (not diameter, since they are not perfectly round); l , the length of the fi ber segment between two contacts; x 0 , the spacing between two contacts along x -axis; and θ , the out-of-plane angle a fi ber segment makes with the xy -plane; then, the change in thick- ness ( dz ) corresponding to an extension ( dx ) was shown to be equal to [5,9] dz ¼ d ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d 2 2 x 0 dx dx 2 q (1) Thus, thickness strain ( dz /2 d ) was found to be dependent on the fi ber thickness ( d ) as well as the contact spacing ( x 0 ). The length l was assumed to be constant; a fair assumption since the fi ber segment is not expected to be strained before it has
240
200
160
120
80
0.0 1.0 2.0 3.0 4.0 Extension (mm)
Figure3. Thickness versus extension plots for handsheets. Each datapoint represents the average thickness of fi ve specimens. Last datapoint in each curve corresponds to the thickness measured after the specimen had failed and had been removed from the clamps. Softwood samples are rep- resented by yellow and hardwoods by blue. Larger squares indicate heavier samples. Open squares indicate unre fi ned handsheets, while lightly shaded and solid squares indicate re fi ned (R) and heavily-re fi ned (R þ ) handsheets respectively.
2400589 (5 of 10)
Phys. Status Solidi B 2025 , 2400589
© 2025 The Author(s). physica status solidi (b) basic solid state physics published by Wiley-VCH GmbH
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