1. 4th Semester Seminar
Finite Difference Element Method
for Modeling Paper Response
Institute of Paper Science and
Technology
Atlanta, Georgia
by Jaime Castro, Ph.D. Candidate
Professor Martin Ostoja-Starzewski, Advisor

2. Finite Difference Element Method for
Modeling Paper Response
• Local variability of paper properties
• Computer model
• Modeling examples
• Proposed verification

3. Paper Strength
• Non uniformity of basis weight
• Spatial variation of fiber orientation
• Spatial variation of bonding degree
• Spatial variation of drying shrinkage
• Furnish or fiber properties
Local variability of paper properties

4. FLAC
Fast Lagrangian Analysis of Continua
FLAC is a two-dimensional explicit finite difference
program for engineering mechanics computation.
Traditional finite element method is an implicit method
This method has already been extensively used in
geomechanics

5. Finite difference form of Newton’s second law:
m
t
Fuu t
i
tt
i
tt
i

  )2/()2/(

New coordinate:  
tuuu tt
i
t
i
tt
i   2/)()(

Motion and Equilibrium







dt
ud
mF


6. Elements and Grid

7. )( ijij ef 














i
j
j
i
x
u
x
u
ij
e


2
1
i
j
iji
g
xt
u


 




 
  snuu
Ax
u
j
b
i
a
i
j
i


 )()(
2
1


Explicit Calculation Cycle

8. Comparison of Explicit and Implicit
Solution Methods
Explicit (FLAC)
Timestep must be smaller than a
critical value for stability
Small amount of computational
effort per timestep
No significant numerical damping
introduced for dynamic solution
No iterations necessary to follow
nonlinear constitutive law
Implicit (FE)
Timestep can be arbitrarily large,
with unconditionally stable
schemes
Large amount of computational
effort per timestep
Numerical damping dependent on
timestep present
Iterative procedure necessary to
follow nonlinear constitutive law

9. …Comparison of Explicit and Implicit
Solution Methods
Explicit (FLAC)
Provided that the timestep criterion
is always satisfied, nonlinear laws
are always followed in a valid
physical way.
Matrices are never formed. Memory
requirements are always at a
minimum.
Since matrices are never formed,
large displacements and strains are
accommodated without additional
computing effort.
Implicit (FE)
Always necessary to demonstrate that
the above - mentioned procedure is:
(a) stable; and (b) follows the
physically correct path.
Stiffness matrices must be stored.
Memory requirements tend to be
large.
Additional computing effort needed
to follow large
displacements and strains.

10. Previous Paper FE-models
L. Wong, M. T. Kortschot, and C. T. J. Dodson, Finite element analysis and experimental
measurement of strain fields cand failure in paper, International Paper Physics
Conference (CPPA and TAPPI): p. 131-135 (September 11, 1995).
ThicknessFront View

11. M. J. Korteoja, A. Lukkarinen, K. Kaski, D. Gunderson, J. Dahlke, and K. J. Niskanen,
Local strain fields in paper, Tappi Journal 79, No. 4: p. 217-223 (April 1996).
Previous Paper FE-models

e

e

e

12. Basis Weight (g/m²)
315-325
305-315
295-305
285-295
275-285
Beta-ray radiography. Paperboard. M. Bliss

14. Assigning Material Properties and
Constitutive Law to Each Element

15. Assigning Material Properties and
Constitutive Law to Each Element
BWE  5
10
Elastic model – Plane Stress
3/1
)21(3 

E
K
)1(2 

E
G
MPaEAvg 4.30

17. Average= 30.4 MPa

24. 15.50
15.60
15.70
15.80
15.90
16.00
16.10
16.20
0 20 40 60 80 100
Number of side elements
N-m Mesh Independence
Work Done on the
System (N-m)

26. Homogeneous case Syy=6.19E+05c

27. Homogeneous case:
Maximum Sxx=1.75E+02

28. Homogeneous case:
Minimum Sxy=-1.5E+01
Maximum Sxy=+1.5E+01

29. Model Verification
• Model the inelastic regime and calculate
energy dissipation in each zone.
• Measure energy dissipation with an infrared
camera with less than 1mm resolution.
• Compare basis weight map with the
evolution of inelastic zones.
• Influence of basis weight on energy
dissipation.

30. 28.5°C
30.8°C
29
30
Infrared Measurements

31. Obtain Mech. Properties From a
Fiber Network Model
c

e
q
BW
Bond