x = 0 x = 0 δ 0 F = βδ A B

Transcription

1 Modelling Paper Tensile Strength from the Stress Distribution along Fibres in a Loaded Network Warren Batchelor Australian Pulp and Paper Institute

2 Tensile strength of paper Important to predict Approaches Analytical models Finite it Element simulations of low density sheets

3 Problems with strength modelling Qualitative not quantitative predictions Many variables difficult to measure Unverified assumptions about fracture process Simplified models with average age fibre properties

4 New approach presented here Model paper behaviour with single fibre stress development Usually fibres in stress direction Micro-Macro Macro approach Single fibre of interest Discrete contacts transferring stress from solid Matrix Apply strain to matrix Cox type shear lag assumption of stress transfer at each contact

5 x = 0 x = 0 δ F βδ = A B

6 The Problem! Consider steps: ONE contact with F δ Strain Matrix Displaces contact Produces Force in fibre Force reduces contact displacement Reduces force etc etc.. Equilibrium: force produces displacement required to generate force How do find displacements for multiple contacts??

7 Seg. 1 Fibre mid-point: x=0 Solution x 1 x 2 x i-1 x i Express δ at each contact in θ i terms of displacement at last contact Fibre end: x=l ( x x ) = th : Elastic Modulus of n segment j= i n n 1 δn 1 = δn βn δn ε EA n n j= n E n th A: Cross sectional area of n segment ε: Matrix strain Solve to obtain all displacements

8 Equation to solve for three contacts Complexity increases as power to the number of contacts, i. More than four contacts- very tedious to write Instead solve symbolically using Matlab 0 = x 1 1 ( x x )( ) δ ε + β3δ3 + β ( x )( ( ( ( )( )))) 2 x1 ε + βδ 3 3+ β 2 δ3 + x3 x2 ε+ βδ 3 3 ( )( ) ( ) + x β δ + x x ε + β δ + x β δ ε x δ β = β / EA

9 An example: fixed contact positions, randomly varying β / (0-10, fibres 1-1 3, fibres 4-6) Lo ocal fibre strain/netw work strai in fibre #1 fibre #2 fibre #3 fibre #4 fibre #5 fibre # Contact position

10 Example: stress development in a fibre. Randomly placed contacts. t Contacts t removed when load exceeds bond strength of bond ε= ε=0.012 ε=0.018 ε=0.024 ε= ε=0.036 d (N) Loa Position along fibre (μm)

11 Advantages of method Random contact positions β can vary contact to contact Elastic modulus, X-section area can vary segment to segment Disadvantages of method Linear elastic only Data needed: β,, E, A Bond strength Fibre contact positions

12 Comparison Experimental Data Radiata pine Never dried Unbleached Kappa number 30 Fibre dimension variation: cutting and fractionation

13 Samples Label Length weighted fibre length (mm) Fibre wall area (µm 2 ) Pressing Levels (Contacts) L Middle (P3) L Middle (P3) L Middle (P3) L Middle (P3) Accepts P1, P3, P5 Rejects P1, P3, P5 Hydrocyclone fractionation

14 Measurements Sheet density, elastic modulus, tensile strength Fibre shape (fill factor), cross-sectional sectional area, length Fibre contacts: distances between contacts, Weibull distributions of contacts Full / partial contacts

15 Contact measurements A B Fibre of interest 1 Cross-section image before (A) and after (B) thresholding h and binarisation. Fibre 2 and 3 in (B) make two full contacts, fibre 1 makes a partial contact, and fibre 4 is not in contact with the fibre of interest.

16 Distance between contact frequency distribution 35 Fr requency (% %) AccP0-measured AccPM-measured AccPH-measured AccP0-fit b = 82.1, c = 1.62 AccPM-fit b =48.6, c =1.55 AccPH-fit b =42.2, c = 1.53 c f g g b g b c 1 c ( ) = ( / ) exp( ( / ) ) b b is the scale parameter c is the shape parameter Free fibre length (μm)

17 Average elastic modulus along fibre Average: 30 simulations per point Fibre elastic mod: assumed 30 Gpa Effective elastic modulus: average load along fibre/x-section area/matrix strain Effective e elastic mod=30 Gpa Perfect stress transfer

18 Average elastic modulus along 3.00E+10 fibre dulus (P Pa) Effective fibre mo 2.50E E E E E E+00 L0 P3 L1 P3 L2-P Stress transfer coefficient (β / )

19 Elastic Stiffness Index for β / 9 8 ex (MNm /kg) Te ensile Stif ffness Ind ~5% reduction in stiffness with fibre length Implies β / ~ 2500 L0 L1 L2 L Sheet Density (kg/m 3 )

20 Average elastic modulus along 3.00E+10 fibre dulus (P Pa) Effective fibre mo 2.50E E E E E E+00 L0 P3 L1 P3 L2-P Stress transfer coefficient (β / )

21 β / = 2500 Strength modelling Bond strength= 25 MPa (IPPC 2007) Average 30 simulations with measured fibre contact statistics Fibre elastic modulus= 30 GPa Contacts Assumed both full, partial contacts contribute But bond breaking load scaled to contact area Fibre fracture not considered Model fibres in stress direction only

22 Six simulations for L0 P3 Av verage Lo oad (N) Series1 Series2 Series3 Series4 Series Matrix strain

23 Avera age fibr re load (N) Six simulations for RR P Axis Title Series1 Series2 Series3 Series4 Series5 Series6

24 30 Simulations Averages 0.18 Average e force along fib bre (N) L0 P3 L1 P3 L2 P3 L3 P3 RR P1 RR P3 RR P5 AA P1 AA P3 AA P5 Network strain

25 Tensile Index data Tens sile Index (knm/k kg) L0 L1 L2 L3 Accepts Rejects Sheet density (kg/m 3 )

26 Max. Calculated force v measured tensile index Maximu um avera age force Tensile Index knm/kg

27 Where to next? Method to treat partial contacts Relation to segment activation? Relationship between stress transfer and crossing angle? Inclusion of fibre fracture

28 Acknowledgements The Australian Research Council through the Discovery Grants program. Dr Jihong He, for the experimental data from his PhD thesis