In the Classroom edited by
Overhead Projector Demonstrations
Doris K. Kolb Bradley University Peoria, IL 61625
Illustrating Poisson’s Ratios with Paper Cutouts
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Dean J. Campbell* and Moira K. Querns Department of Chemistry, Bradley University, Peoria, IL 61625-0208; *
[email protected] Flat, flexible lattices can be used to illustrate Poisson’s ratios of materials. These lattices can be produced from ordinary sheets of paper. Poisson’s ratio can be described as the percentage of crosswise contraction divided by the percentage of fractional elongation when a material is stretched (1–3). An equation for the Poisson’s ratio ν is given as
ν = % decrease in width % increase in length
(1)
Most materials have a positive Poisson’s ratio, which may be easily demonstrated with a wide rubber band (2). As the rubber band is stretched, its width decreases. The flat, flexible lattices described in this paper can illustrate both positive and negative Poisson’s ratios. The idea of the positive Poisson’s ratio is so prevalent in popular culture that many cartoon characters are drawn as becoming thinner as they are stretched and fatter as they are squashed. Materials with negative Poisson’s ratios, on the other hand, tend to be counterintuitive. Note that a positive Poisson’s ratio involves a decrease in width, whereas a negative ratio involves an increase in width upon stretching. Some materials with negative Poisson’s ratio structures are metal and polymer foams (1, 4, 5), certain microfabricated metal structures (6 ), plastic composite materials (1), and the plastic Hoberman spheres sold in toy stores. A feature associated with many of these open porous structures is a set of units or pores that are “crushed” or contain concave boundaries, as in Figure 1. When these units or pores are stretched, the units expand, expanding the entire structure (2). The flat, flexible lattices described in this paper are simplifications of aforementioned structures. They are generally easier to create (and less expensive) than the three-dimensional structures. Some nonporous materials have structure at the molecular level that results in a negative Poisson’s ratio. These include some minerals, such as α-cristobalite (7), and some metals, such as zinc (when stretched in particular directions). Ni3Al, an alloy used in aircraft gas turbines, also has a negative Poisson’s ratio (8). The flat lattices described here do not necessarily reflect how the atoms move in these materials to produce such a behavior; but the lattices do illustrate the overall phenomenon. Applications of materials and structures with negative Poisson’s ratios are not yet widely utilized. One potential use of these materials is to compensate structurally for materials that have positive Poisson’s ratios. Some other proposed uses include snap-like fasteners that are easy to connect but difficult to disconnect, climbing ropes that expand and grip in crevices when the ropes stretch, foam pads such as wrestling
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mats that reduce impact forces, sound baffling panels, and piezoelectric amplifiers (1, 3). It should be noted that to make materials with a negative Poisson’s ratio and high stiffness, the structure that causes this negative Poisson’s behavior must exist at the molecular level in the material; if the structure exists at some larger level, the material will not be as stiff (1). Design of these materials at the molecular level will require an intimate knowledge of molecular structures. Examples of flat lattices that can demonstrate positive and negative Poisson’s ratios are shown in Figure 1.W These lattices have been successfully used on an overhead projector. To stretch the lattices, it is best to lay them on a smooth, flat surface (like the stage of an overhead projector, or a desktop) and hold one long, straight edge of the structure while pulling on the other side. Gently stretch the lattices lengthwise by pulling the long, straight edges of the cutouts away from each other. (It is easier if supporting strips of cardboard are glued, taped, or stapled to these long straight edges.) These demonstrations are likely most suited to courses that have an emphasis on materials, such as a Chemistry for Engineers course. The paper lattices in Figure 1 exhibit ratios of about 2 (top) and ᎑0.4 (bottom). An additional classroom challenge would be to find the unit cell for each of the patterns.
Figure 1. Positive Poisson’s lattice: (A) unstretched and (B) stretched. Negative Poisson’s lattice: (C) unstretched and (D) stretched. In each case, the rectangle under the stretched lattice has the same overall dimensions as the unstretched lattice.
Journal of Chemical Education • Vol. 79 No. 1 January 2002 • JChemEd.chem.wisc.edu
In the Classroom
Acknowledgments
Literature Cited
We would like to thank Bradley University, the National Science Foundation through the Materials Research Science and Engineering Center for Nanostructured Materials and Interfaces (DMR-96325227), and the LEGO Corporation for generous support. We would also like to acknowledge the assistance of general chemistry students at BU and participants at the NSF-sponsored ICE Materials Science Workshop at the University of Wisconsin–Madison. Finally, we would like to thank Doris Kolb at Bradley University for very helpful editing. Supplemental Material Templates for the flat paper lattices and building instructions for lattices constructed with LEGO bricks are available in this issue of JCE Online. LEGO® is a trademark of the LEGO Group. Demonstrations and experiments using LEGO bricks can be found at http://mrsec.wisc.edu/edetc/LEGO/index.html. W
1. Lakes, R. Adv. Mater. 1993, 5, 293–296. 2. Ellis, A. B.; Geselbracht, M. J.; Johnson, B. J.; Lisensky, G. C.; Robinson, W. R. Teaching General Chemistry: A Materials Science Companion. American Chemical Society: Washington, DC, 1996; pp 2–5. 3. Lakes, R. Nature 1992, 358, 713–714. 4. Lakes, R. Science 1987, 235, 1038–1040. 5. Lakes, R. Foam Structures with a Negative Poisson’s Ratio; http://silver.neep.wisc.edu/~lakes/sci87.html (accessed Oct 2001). 6. Jackman, R. J.; Brittain, S. T.; Adams, A.; Prentiss, M. G.; Whitesides, G. M. Science 1998, 280, 2089–2091. 7. Keskar, N. R.; Chelikowski, J. R. Nature 1992, 358, 222– 224. 8. Baughman, R. H.; Shacklette, J. M.; Zakhidov, A. A.; Stafstrom, S. Nature 1998, 392, 362–365.
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