The Miura Ori tessellation is recognized for having unique properties, one of which is its negative Poisson’s ratio. The Poisson’s ratio is the ratio of how much a material contracts or extends in the transversal and longitudinal direction, when subject to a tensile force. Most materials would decrease in width when stretched, so they would have a positive Poisson’s Ratio. However, some materials, like the Miura Ori, increases in width, indicating a negative Poisson’s ratio.
Calculating the Poisson’s Ratio
The Poisson’s Ratio (usually denoted by the greek letter nu) is calculated by
n = (-etransversal) / (elongitudinal)
e = (change in length) / (original length)
where the transversal strain is perpendicular to the tensile force and the longitudinal strain is parallel to the tensile force.
Therefore, we can calculate this Poisson’s Ratio of our Miura Ori, as shown below.
Let # of columns = x
Let # of rows = y
*Note that in this context, each column or row is the distance between two vertexes adjacent in the transversal or longitudinal direction respectively.
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Calculating etransversal:
c = (2a2 – 4aCos C)1/2
original length = x(2a2 – 4aCos C)1/2
etransversal = (b – (2a2 – 4aCos C)1/2) / ((2a2 – 4aCos C)1/2)
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Calculating elongitudinal:
original height = d cos(tan-1(b / d))
original length = y(d2 – (d cos(tan-1 (b / d)))2)1/2
elongitudinal = y(d – (d2 – (d cos(tan-1 (b / d)))2)1/2)
Explanation: For the transversal strain, I used the Law of Cosines to find the distance c. Multiplying c by the number of columns gives the original transversal length. Next, using the strain formula, I was able to plug in and simplify the transversal strain. For the longitudinal strain, I used the height formula, from On the out-of-plane compression of a Miura-ori patterned sheet by Yang Lv, Ying Zhang, Neng Gong, Zhong-xian Li, Guoxing Lu, and Xinmei Xiang. Knowing the slope of the diagonal lines in the Miura Ori crease pattern is b/d, I was able to convert it into degrees by using its inverse tangent. Using the pythagorean theorem and multiplying the result by the number of rows, I determined the original length. Finally, using the strain formula once again gave me the longitudinal strain. Therefore:
| Poisson’s Ratio for Miura Ori: n = ((-x(b – (2a2 – 4aCos C)1/2))(d2 – (d cos(tan-1 (b / d)))2)1/2)) / ((y(d – (d2 – (d cos(tan-1 (b / d)))2)1/2))(2a2 – 4aCos C)1/2)) |
DISCLAIMER: this formula only works in very specific scenarios