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