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of two heavily re fi ned hardwood handsheets (HW1R þ and HW3R þ ), all other softwood and hardwood handsheets exhibited a net increase in thickness before failure. Considering the values of Poisson ’ s ratio at lower strain levels, handsheets containing unre fi ned softwood fi bers showed a consistent auxetic response, whereas other handsheets showed more variability. This result suggested that fi ber networks having longer fi bers (softwood) with fewer network attachments per fi ber (unre fi ned), led to a larger auxetic response, providing a fundamental understanding that could be used to design other fi ber network mechanical metamaterials. Additionally, many samples exhibited a decrease in thickness initially before the thickness started to increase, which has not been observed commonly for commercially pro- duced papers but has been seen in other studies with handsheets. The difference in the nature of the thickness-extension curves and the values of Poisson ’ s ratio for different types of paper sug- gests a strong correlation with the fi ber-network structure and the processing conditions employed during papermaking. Strong hydrogen bonds between fi bers at junction points and the locally interwoven organization of fi bers, albeit irregular, are thought to be critical elements in the tensile deformation mechanism leading to auxetic response. Overall, the results of this study show clearly that pulp type ( fi ber length and thickness, fi ber morphology) and process conditions (such as web forming and re fi ning) affect various structural parameters such as paper density, spacing between fi ber contacts, fi ber orientation, and fi ber morphology that consequently affect the nature and extent of auxetic response. Acknowledgements The authors thank the Renewable Bioproducts Institute at Georgia Tech for a Paper Science and Engineering (PSE) Fellowship to P.V. The authors also thank Dr. Rallming Yang at PSE for his help with the procurement of pulp and the use of the papermaking labs. A.C.G. wishes to thank Georgia Tech for an ACE/SLOAN Retirement Transition Legacy Award.
3.2.5. Machine-Made Paper versus Handsheets
The factors enhancing the auxetic response in paper — long fi ber lengths between contact points, the initial extension or tautness of fi bers between contact points, and improved connectivity between model auxetic layers in the z -direction, are enhanced in many commercially made papers. Commercial papermaking processes such as calendering, drying under tension, application of pressure in the thickness direction during both wet and dry processes, and the preferential orientation of fi bers in the machine ’ s movement direction amplify these factors. Additionally, the use of certain fi llers, like adhesives, might con- tribute to higher magnitudes of Poisson ’ s ratio in commercially produced paper. In contrast, our samples, which lacked these processing steps and fi llers, exhibited lower Poisson ’ s ratios mag- nitudes, when compared to Öhrn ’ s MG kraft paper ’ s Poisson ’ s ratio of 6.3. [1]
3.2.6. A Word on Reversibility
The decrease in thickness of auxetic specimens post failure indicated that the thickness increase was a partially reversible phenomenon. Close observation of the last two datapoints for each series in Figure 3 reveals that all samples exhibited this behavior (with the exception of HW1R þ , which was not auxetic at all) with no noticeable trend with either the pulp type, thick- ness of handsheet, or the degree of re fi nement. Given the simi- larity in the shape of the experimental curves (Figure 4) to the simulated curves (Figure 5c), it can be argued that the network con fi guration in the handsheets resembled that of the model. If auxeticity predominantly stems from straightening of bent fi bers, the post failure thickness measurements indicate that this straightening was a partially reversible process, no matter the kind of sample. Hydrogen-bonding at contact points, along with elasticity of other network deformations, must have aided in the reversibility of straightening of bent fi bers upon failure (break- ing) of the handsheet. 4. Conclusions Laboratory handsheets were produced out of bleached hardwood and softwood kraft pulp. This process allowed the production of themost “ clean ” form of a paper structure due to the lack of use of any fi llers, additives, or coatings and no in-plane preference of fi ber orientation due to the absence of an industrial-type contin- uous web-forming process. Examining this type of paper sample allowed us to study the effects of fi ber type and re fi ning explicitly and provided a basis for separating the effects of additives and commercial processing on the auxetic response of paper. Additionally, producing papers with a largely isotropic network structure in a plane provided direct insight into the role of fi ber alignment on the magnitude of the auxetic response, building upon previous results that have made inferences based on com- parisons between commercial papers produced by different methods but not necessarily composed of the same starting materials. In this work, out-of-plane auxetic behavior was extensively observed in our laboratory-made handsheets. With the exception
Con fl ict of Interest The authors declare no con fl ict of interest.
Data Availability Statement The data that support the fi ndings of this study are available from the corresponding author upon reasonable request. Keywords auxetic, cellulose fi bers, fi ber networks, hydrogen bonding, nonwovens, paper handsheets, structural models
Received: November 1, 2024 Revised: February 4, 2025 Published online:
[1] O. E. Öhrn, Sven. Papperstidn. 1965 , 68 , 141. [2] L. Gottsching, H. L. Baumgarten, in Fundamental Properties of Paper Related to Its Uses (Ed: F. Bolaim) British Paper and Board Ind. Federation, Manchester 1976 , pp 227 – 252.
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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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