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with strain 2) thickness fi rst decreased and then increased with strain and 3) thickness decreased consistently with strain. In ret- rospect, we had obtained identical patterns previously when examining commercial paper samples: pattern-1 was observed for copy paper, paperboard, and cotton paper; pattern-2 was observed for bamboo paper; and pattern-3 was observed for glass- ine paper. [5,9] Based on the mechanistic picture just described, we believe that these patterns arise from the starting fi ber arrangement and how closely it resembles the simpli fi ed auxetic unit depicted in the model. Predicting auxetic response stem- ming from the structure and processing history of paper types was the focal point of this research.
Table 1. Cumulative Poisson ’ s ratio values calculated from average thickness strains at axial strains a) just before break, b) just below 0.8% strain corresponding to the inset shown in Figure 4, and c) corresponding to lowest thickness value (dip).
Sample
Poisson ’ s ratio (cumulative)
Just before break (a)
Just below 0.8% strain (b)
At lowest thickness value (c)
HW5
1.92 0.12 2.03 0.17 1.68 0.14 3.81 0.19 2.44 0.09 2.18 0.05 0.89 0.09 0.65 0.13 0.52 0.17 1.40 0.07 1.17 0.13 1.39 0.21
0.06 0.14 0.25 0.34 0.23 0.52 0.32 0.70 0.42 0.58 1.18 0.03 0.61 0.78 0.67 0.67 0.88 0.79 0.55 0.38 0.84 0.91 1.68 0.67 0.89 0.58 1.07 0.91 1.56 0.03 0.76 0.52 1.16 0.79 2.04 0.74
0.28 0.25 0.30 0.41 0.23 0.52
HW3
HW1
SW5
NA
SW3
NA
3.2.1. Softwood versus Hardwood
SW1
NA
HW5R
0.65 0.39 0.60 0.42 0.88 0.79 1.09 0.36 0.84 0.91 1.47 0.58 0.67 0.34 0.57 0.43 0.85 0.01 0.76 0.52 1.06 0.36 1.38 0.66
Longer fi ber lengths in softwoods keep more auxetic units connected, allowing for better load propagation as well as better transfer of local auxetic response. Thus, SW, SWR, and SWR þ handsheets exhibited a larger net auxetic response than corre- sponding HW, HWR, and HWR þ handsheets respectively (see Figure 4 and Table 1a). Additionally, light re fi ned softwood handsheets, SW1R and SW1R þ , exhibited a markedly larger dip in thickness when compared to their hardwood counterparts (see Figure 4). Strong dips may partly arise from fi bers not being fully extended in their initial arrangement (see Figure 5(d6)), which is more likely a possibility with longer softwood fi bers.
HW3R
HW1R
SW5R
SW3R
SW1R
HW5R þ 0.35 0.12
HW3R þ HW1R þ
0.20 0.30 0.67 0.01
SW5R þ 0.71 0.11 SW3R þ 0.36 0.08 SW1R þ 0.23 0.26
3.2.2. Effect of Re fi ning
One of the most striking observations during this study was the suppression of the auxetic response due to re fi ning. Thickness strains for unre fi ned softwood and hardwood handsheets (of all thicknesses) remained larger than their re fi ned counterparts across almost the entire strain regime (compare open squares with shaded/solid squares in Figure 4), simultaneously resulting
3.2. Effect of Structural Parameters
In Figure 3 and 4, there were at least three distinct thickness change patterns observed — 1) thickness increased consistently
Figure5. Mechanistic ideas for out-of-plane auxetic response in paper; a) model with key network parameters like fi ber diameter ( d ), fi ber length between contacts ( l ), contact-spacing ( x = x 0 ) and out-of-plane fi ber angle ( θ = θ 0 ) in its initial state; b) model in its fully extended state (with no straining of individual fi bers); c) theoretical plots of thickness versus axial strain for the model showing variations with respect to changing fi ber diameter ( d ) and initial contact spacing ( x 0 ); d) cartoons of various possible fi ber arrangements and mechanistic factors at play — d1) model auxetic unit, d2) a possibility where θ > θ 0 , d3) fi ber ends or disjointed fi bers, d4) possibility where neighboring auxetic units do not touch each other, d5) smaller diameter often resulting in smaller spacing, and d6) possibility of not fully extended fi bers between contact points.
2400589 (7 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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