4th Semester Seminar
Finite Difference Element Method Finite Difference Element Method for Modeling Paper Responsefor Modeling Paper Response
Institute of Paper Science and Institute of Paper Science and TechnologyTechnology
Atlanta, GeorgiaAtlanta, Georgia
by Jaime Castro, Ph.D. CandidateProfessor Martin Ostoja-Starzewski, Advisor
Finite Difference Element Method for Modeling Paper Response
• Local variability of paper properties• Computer model• Modeling examples• Proposed verification
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
FLACFast 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
Finite difference form of Newton’s second law:
mtFuu t
itt
itt
i
)2/()2/(
New coordinate: tuuu tti
ti
tti 2/)()(
Motion and Equilibrium
dtudmF
Elements and Grid
)( ijij ef
i
j
j
i
xu
xu
ije
21
ij
iji gxt
u
snuu
Axu
jb
ia
ij
i )()(
21
Explicit Calculation Cycle
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
…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 largedisplacements and strains.
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
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
Basis Weight (g/m²)
315-325305-315295-305285-295275-285
Beta-ray radiography. Paperboard. M. Bliss
Assigning Material Properties and Constitutive Law to Each Element
Assigning Material Properties and Constitutive Law to Each Element
BWE 510
Elastic model – Plane Stress
3/1
)21(3
EK)1(2
EG
MPaEAvg 4.30
Average= 30.4 MPa
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)
Homogeneous case Syy=6.19E+05c
Homogeneous case:Maximum Sxx=1.75E+02
Homogeneous case:Minimum Sxy=-1.5E+01Maximum Sxy=+1.5E+01
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.
28.5°C
30.8°C
29
30
Infrared Measurements
Obtain Mech. Properties From a Fiber Network Model
c
BW
Bond