Download the book, free !! contact roman@gatech.edu, also available - - PowerPoint PPT Presentation

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Download the book, free !! contact roman@gatech.edu, also available in paperback from Amazon

Renewable Bioproducts Institute (formerly IPST) Georgia Institute of Technology 50 10th St. N.W., Atlanta, Georgia, U.S.A.

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In this talk, we’ll cover :

• Basic paper physics, all you need to know
• A quick look at a few basic tests
• How the tests all tie together
• A few useful models

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Paper structure

Paper can be thought of a conglomeration of pine straw clumps on a millimeter scale , paper is thus inherently non-uniform . In cross section, paper is a stack of tubes, many are collapsed.

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Paper Structure

Laser confocal shot, torn edge 40 X Same as left but lighting up a single layer Paper as a mass of fettucine ! Same paper but a different spot

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3 Principle Directions, Paper and Board

Introducing Stiffness or Elastic Modulus E: EMD > ECD >> EZD

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Moisture effects: hysteresis

• 1. Samples for testing should always be preconditioned

at 20% RH

• 2. Then, conditioned at 50% RH and tested at 50% RH

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Moisture effects: strength properties

𝑻𝒖𝒔𝒇𝒐𝒉𝒖𝒊𝟔𝟏% 𝑺𝑰 = 𝑻𝒖𝒔𝒇𝒐𝒉𝒖𝒊 (%𝑵) × %𝑵 × . 𝟏𝟖 + 𝟏. 𝟓𝟖 When conditioning samples takes too long , for intra-lab comparison of specs and reproducibility , use the correction:

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Paper variability and the need for testing

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Variability increases when testing pieces on a smaller scale e.g. SCT vs RCT, or 15 mm wide strips vs 25 mm wide strips. Samples taken from Front or Back positions show greater CD variation than those near the Center

The Tensile Test Tells All

l

w

Load cell measures F

Cross head moves down distance ∆l

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Tensile test tells all

Define tensile stiffness Sb , using values from the tensile test:

𝑇𝑐 = ∆𝐺 × 𝑚 𝛦𝑚 × 𝑥

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F

l

Δ

𝐹𝑁𝐸,𝐷𝐸 × 𝑢 = 𝑇𝑐𝑁𝐸,𝐷𝐸 𝜁𝑁𝐸,𝐷𝐸 = ∆𝑚𝑁𝐸,𝐷𝐸 𝑚𝑁𝐸,𝐷𝐸 𝜏𝑁𝐸,𝐷𝐸 = ∆𝐺

𝑁𝐸,𝐷𝐸

𝑢 × 𝑥

𝐹𝑁𝐸,𝐷𝐸 = 𝑇𝑐𝑁𝐸,𝐷𝐸 𝑢 = 𝜏𝑁𝐸,𝐷𝐸 𝜗𝑁𝐸.𝐷𝐸

Elastic Modulus E, from Tensile Testing

1) For both MD and CD orientation, where t is the sheet “caliper”*, tensile stiffness Sb: 2) Defining strain ε : 3) Stress σ, is Force/Area 4) Hooke’s law

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Elastic Modulus of Paper

• E is principally governed by the fiber modulus which is

determined by species, fiber length

• E appears as the dominant factor in the theory for tensile

strength T (Page equation) and compression stress σ (Shallhorn Gurnagul)

1 𝜏𝑑 = 𝟐 𝝉𝒑 + 𝐷𝑥 2𝛽𝐹

𝑔𝑢3

𝜍𝑔 𝜍 − 1

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Shallhorn Gurnagul SCT model

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Compression and Tension Relation to Modulus

Modulus E is the same in Tension and Compression, and EMD ~ 1.5 - 2 x ECD

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Modulus and Tensile Strength Unaffected by Bonding

~ ± 12 % about Control Page and Seth experiments 1981

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Compression Strength Follows Modulus

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Example: Modulus and Corrugated Board Pin Adhesion

Pin adhesion test

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Tensile Stiffness in Bending: Paper

𝐸 = 𝐹𝑢 × 𝑢2 12 Bending stiffness D :

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So, D can be checked using Sb and t

Tensile Stiffness in Bending: Board

𝐸 = 𝐹 ∙ 𝑢 ∙ ℎ2 2

t – liner caliper h – board caliper E – liner modulus

“Sandwich” beam approximation:

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Tensile Stiffness in Bending: Boxes

McKee equation for BCT failure load:

BCT = C(ECT)3/4{(DMD x DCD)1/2}1/4 x Z1/2

= C(ECT)3/4 {(EMDth2/2 x ECDth2/2)1/2}1/4 x Z1/2 = C’(ECT)3/4 (E x t) ¼ (h x Z)1/2

= 5.87 ECT x √(h x Z) E x t = Sb of liners is assumed to be proportional to board ECT

Usual simplified McKee equation

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Vertical load

Tensile Stiffness in Ultrasonics

𝜍𝑾𝟑

𝑇𝑐 = 𝐹𝑢 = 𝜍𝑊2 𝑢 = 𝛾 𝑢 𝑊2 𝑢 = 𝛾𝑊2

ρ – density = Basis weight/caliper = β/t V2 – sound velocity squared aka “specific stiffness” Tensile stiffness is related to specific stiffness:

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Ultrasonic Tensile Stiffness in Ring Crush

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SCT RCT

V2

A proposed “McKee” equation for RCT: Buckling load for a cylinder: Model for RCT:

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The data fit:

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Example: Ultrasonic tensile stiffness correlates with CMT and SCT

Ultrasonic Tensile Stiffness βV2 (N/mm) x 103

CMT

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Models for End-use: Writability

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=

Bristow wheel emulating handwriting Applying a swath of aqueous ink Top printed side Bottom unprinted side

Challenge: Quantify and Relate Show- through to Physical Properties

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=

∆𝑭 = 𝑴𝒗

∗ − 𝑴𝒒 ∗ 𝟑 + 𝒃𝒗 ∗ − 𝒃𝒒 ∗ 𝟑 + 𝒄𝒗 ∗ − 𝒄𝒒 ∗ 𝟑

ΔE = a1x1 + a2x2 + ...+ anxn + b

Underside show-through Tappi method colorimeter CIE L*a*b* values Show-though quantified as difference in unprinted and printed: Seek a regression predictive model of the form:

Physical Properties and Show-through

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Contact Angle Cobb water absorption Air permeability Surface roughness (stylus)

+ ?..

Hercules size (HST)

Show-through Analysis

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ΔE = - 0.0312 x (HST) + 0.170 x (contact angle) + 2.60 x (angle rate) + 7.86 x (Cobb30/β) -10.3

Using Excel LINEST multiple regression analysis, retain significant terms, optimize r2:

This model is used to evaluate candidate commercial papers for notebooks

Summary

• Elastic Modulus, “E” is key to understanding the

interrelationships between:

– Compression and Tension strengths – Burst – Bending stiffness

• “E” can be measured from tensile testing and

sonic (speed of sound) testing

• Physical testing of paper is necessary for wood

fiber based products – and gives me (and a few

• thers) a job !!

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E is key…

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E is key It’s the modulus, you see It is there, for you and me It’s good for tension Nice for compression, And deserves, your attention !

Thank you for your Attention !! Questions, Comments, to:  Roman@gatech.edu