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A Review of Poisson’s Ratio of Paper
P R A T E E K V E R M A
p v e r m a @ g a t e c h . e d u
Material Science and Engineering
G e o r g i a I n s t i t u t e o f Te c h n o l o g y
Auxetic Materials
Positive Poisson’s ratio – Material contracts when stretched Conventional Materials
Negative Poisson’s ratio – Material expands when stretched ! These are called ‘Auxetic Materials’ Paper has been reported to show negative Poisson’s ratio values in
thickness direction
Image: published by Workshop on “Auxetic and Related Systems”, August 2005 (Poznan Supercomputing Center)
A Unique Property
Assuming isotropic material, Poisson’s ratio for Metals, Plastics etc. = 0.3 Rubber, Elastomers = 0.5 Cork is close to Zero
For isotropic and linear elastic materials
ν = -1.0 to 0.5
Anisotropic Materials can have Higher magnitudes of ν Multiple values of ν
Auxetic behavior is rare in nature
Animation by: Prof. Katia Bertoldi, Harvard University, on iMechanica
A Unique Property
Naturally Auxetic Bones Skin Arteries Certain Rocks
Synthetic Auxetic Lakes’ Foams Expanded PTFE Ultra high mol. wt. PE
Paper
? ?
Not all bones/skins are auxetic. Not all rocks are auxetic. Auxetic behavior (natural or synthetic) achieves specific properties
and applications
Properties and Applications
Fiber Composites Made of Auxetic fibers Fibers increase in diameter when stretched Do not detach from the matrix Can we think of making paper composites too!
High Impact Ballistics protection
Auxetic materials become dense when compressed Thus impact strength is very high Self reinforcing on impact, used to make bulletproofs
Double curvature
Auxetic materials can bulge without thinning Make comfortable mattresses and strong dome shapes
Image by – (top) Auxetic Materials Network, U of Bolton (UK); (bottom) Gilbert Auxetic Products
Poisson’s Ratio of Paper
Recently in 2002, N Stenberg1 has reported negative values of Poisson’s ratio in the thickness direction of paper o Thickness ~ 0.07 to 0.458 mm o Grammage ~ 45 to 282 g/m2
Material, t vxz vyz
Paperboard (uncoated), 458
-2.0 -1.2
Paperboard (coated), 387
-3.2 -1.8
Liner, 286 -2.3 -0.8
Copy paper, 99 -0.2 +0.5
Sackpaper (uncalandered), 89
-0.2 +0.6
Sackpaper (calandered), 72
-3.0 -0.9
Sackpaper (calandered), 70
-4.4 -1.2
Newsprint, 62 -3.8 -0.4
[1] Out-of-plane Poisson’s ratios of Paper and Paperboard, Nordic Pulp and Paper Research Journal (2002)
Continuous thickness measurement using displacement transducers
Stress was applied in the plane of paper in both MD and CD
Average δ ~ 1.5
Poisson’s Ratio of Paper
[1] Out-of-plane Poisson’s ratios of Paper and Paperboard, Nordic Pulp and Paper Research Journal (2002)
Machine Direction Cross Direction
Sample: uncoated paperboard t = 458 ± 10 μm
Poisson’s Ratio of Paper
[1] Thickness Variations of Paper on Stretching, Olov E. Ohrn, Svensk papperstidning, 1965, 68(5), 141-149
Ohrn1, 1965
Method of Measurement o Mechanical thickness gauge – pressure by a roller o Mercury dilatometer – f(volumer change of paper) o 2D – bulge stretching
Poisson’s Ratio of Paper
[1] Thickness Variations of Paper on Stretching, Olov E. Ohrn, Svensk papperstidning, 1965, 68(5), 141-149
Ohrn1, 1965
Type Basis
Weight g/m2
Thickness μm
Increase of thickness*
(at rupture)
μ %
Kraft paper, Billerud 75 130 12 9%
Semi chemical lab handsheet 75 130 7 5%
Tracing paper (grease-proof) 57 50 2 4%
MG kraft paper, Billerud 80 100 12 12%
*MD values, CD values lower but similar
Poisson’s Ratio of Paper
[1] Determination of all nine orthotropic elastic constants of paper, Baum, Mann, Habeger, IPC Tech. Paper Series 84
Property Value Error %
v13 0.59-2.45 60
v23 1.32-2.36 30
v in-plane 0.01-0.30 10
Baum et al1, 1979
Paper Sample o Heavy bleached Kraft Milk Carton o t = 0.68 mm, 14x28 inch o Density = 0.78 g/cc
Method of Measurement o Ultrasonic –> high frequency, low strain o Wave velocity = f (elastic constants)
Conditions
o Preconditioned at low humidity – 48 hrs o 23 °C, 50% RH