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

Poisson’s Ratio

Poisson’s Ratio

Index Notation

and

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

Poisson’s Ratio of Paper

Thank You! Questions?

Poisson’s Ratio of Paper

What we want to do? Electron Microscopy / Tomography Study microstructure of paper Confirm the reported measurements Find length scales where auxetic behavior Operates and originates

We need thick paper samples….. Of same structure as normal paper