RESEARCH ARTICLE
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On the Origins of Out-of-Plane Auxetic Response in Paper
Prateek Verma,* Anselm C. Grif fi n, and Meisha L. Shofner*
earliest reports of thickness measurements during extension, Öhrn (in 1965) reported out-of-plane Poisson ’ s ratios such as 6.3 for MG kraft paper and þ 0.7 for tracing paper, among a few different kinds of papers he examined. [1,10] Öhrn also pro- posed a mechanistic hypothesis for this response, based on the straightening of in-plane fi bers upon strain, which has largely been accepted by researchers since. Later, Baumgarten (in 1976) examined a few commercial and handsheet paper sam- ples for their in-plane and out-of-plane Poisson ’ s ratios. In addition to observing a negative out-of-plane Poisson ’ s ratio for some commercial papers, Baumgarten also observed the thickness of spruce ground- wood handsheets to fi rst decrease and then increase with axial strain. [2] Baum
Previously, several commercially available papers were examined and an increase in thickness upon stretching in certain types was found. A mechanistic expla- nation for the origin of this auxetic response was also devised, utilizing earlier reports. In this article, the understanding of this mechanism is applied to examine auxeticity in lab-produced paper handsheets. Only bleached kraft wood fi ber pulp is employed, with no fi llers, additives, or coatings incorporated. In this way, the contribution of a small number of structural factors to any auxetic response in the handsheets can be directly examined. Key structural parameters of a nonwoven cellulose fi ber network in paper — staple fi ber length, bulk density, and fi ber contact spacing — which are predicted to affect the magnitude and sign of Poisson ’ s ratio previously, are altered by using softwood and hardwood pulps with and without re fi ning to produce a range of handsheets. It is found that longer fi bers favor auxeticity while re fi ning hampers it.
1. Introduction Several kinds of papers are auxetic. The thickness of these papers can increase when they are uniaxially stretched. [1 – 5] They are partially auxetic, [6] which means that the increase is observed only in the thickness and not in the width. Because this increase is observed in a direction that is normal to the sheet of paper, it is called an out-of-plane auxetic response. Additionally, the width of many kinds of paper has been reported to decrease upon stretch- ing, that is, the in-plane Poisson ’ s ratio is positive. [2,7,8] For the sake of simplicity, in the remainder of this manuscript, we use the term auxetic to mean out-of-plane auxetic, when discussing paper. In one of our previous works, we determined that certain kinds of commercially available paper (copy paper, paperboard, and cot- ton paper) were auxetic while certain other kinds (glassine paper) were not. [5,9] For instance, the out-of-plane Poisson ’ s ratios for copy paper and glassine paper were found to be around 3 and þ 0.6, respectively at 0.7% in-plane strain. [5] In one of the E-mail: prateekaverma@gmail.com; meisha.shofner@mse.gatech.edu The ORCID identi fi cation number(s) for the author(s) of this article can be found under https://doi.org/10.1002/pssb.202400589. © 2025 The Author(s). physica status solidi (b) basic solid state physics published by Wiley-VCH GmbH. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited and is not used for commercial purposes. DOI: 10.1002/pssb.202400589 P. Verma, A. C. Grif fi n, M. L. Shofner School of Materials Science and Engineering Georgia Institute of Technology Atlanta, GA 30332, USA
et al. (in 1984) found that the thickness of wet paper, too, increased upon stretching. Poisson ’ s ratios for Baum ’ s wet blotting paper ranged approximately between 1 and 4. [3,10] Baum adapted Öhrn ’ s mechanistic picture to develop a mathematical model for the behavior of wet paper and emphasized the role of fi ber bonding, segment length, and initial fi ber arrangement in causing the thickness increase. [3] Similarly, Stenberg and Fellers (in 2002) found that a variety of paper types, as would now be expected, exhibited a negative out-of-plane Poisson ’ s ratio: 4.4 for calen- dered sackpaper and 0.24 for copy paper, for instance, as deter- mined from a linear fi t of strain data. [4] Johansson et al. have shown that fast synchrotron X-ray tomographic imaging of paper- board in tension enabled the study of deformation and failure mechanisms related to observed Poisson ’ s ratios. [11] In a study of particular relevance to our report here, Golkhosh et al. observed auxetic behavior in softwood paper handsheets and attributed it to fi ber straightening followed by breakage of inter fi ber bonds lead- ingto fi ber pull-out and an out-of-plane negative Poisson ’ sratio. [12] In summary, we believe that the size and shape of cellulose fi bers, spacing between fi ber- fi ber contacts, frequency of these contacts, mechanical properties of fi bers themselves, and fi ber arrangement, which together de fi ne the nonwoven network structure of paper, affect the sign and magnitude of the Poisson ’ s ratio of paper. [5,9] Structurally, a sheet of paper is an anisotropic network of cellulose fi bers lying predominantly in the plane of the paper. Paper is typically 50 – 250 μ m in thickness, and the fi bers are about a millimeter to a few millimeters long and 10 – 30 microns in diameter. [13] Thus, only a few in-plane fi bers are present if observed through the thickness direction of paper. The cellulose fi bers attach to each other through hydro- gen bonding forming an interconnected nonwoven network. Like paper, other nonwoven fi ber networks too, can exhibit auxetic behavior. Delannay et al. produced, [14] and later
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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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