lecture 16: design of paper and board packaging/menu/general/column... · analysis of output,...

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Lecture 16:Design of paper and board

packaging

Advanced concepts:FEM, Fracture Mechanics

After lecture 16 you should be able to

• describe the finite element method and its use for paper-based industrybased industry

• illustrate how a finite element model is created• discuss the results from finite element analysis of:

– Compression test of a package– Creasing and folding

• discuss concepts such as the J-integral and the fracture toughnesstoughness

• account for the procedures of failure predictions in paper materials

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Literature• Pulp and Paper Chemistry and Technology - Volume 4, Paper

Products Physics and Technology, Chapters 2 & 12• Östlund, S. and Mäkelä, P., Fracture properties, KTH, 2011

Three-dimensional modelling and analysis of paper and board, FEM

Adapted fromMikael NygårdsMikael Nygårds

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Why use numerical methods?• Different types of loadings can be

investigated.Th ff t f diff t t i l

FEM

MaterialPerformance

Process

• The effect of different material properties can be investigated.

• More information about damage and deformation mechanisms can be gained.

• Material behaviour and structural behaviour can be predicted P ti th t i t t f

Solutions• Properties that are important for

the manufacturer can be linked to properties that are important for the user.

• Saves costs and enables more efficient product development

Material performanceEstablish the important/critical material properties

Converting

p p

DelaminationPackaging Design

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The Finite Element-Method (FEM) is a…

• … method to solve partial differential equations• procedure to solve field problems with engineering• … procedure to solve field problems with engineering

accuracy– In field problems the parameters of interest are described by

continuous or discrete field variables, e.g.– Continuum mechanics

• Solid mechanics – stress, strain• Moisture transport• Fluid mechanics• Heat transfer, temperature• Electro magnetic field theory• ...

Finite element methodAssume a deformation (displacement field)

Nu

NN uNu =

Calculate displacements

NN uβε =

Calculate strain

Calculate stress (in general non linear relations)

( )εσ F=(in general non-linear relations)

∫∫∫ ⋅+⋅=V

T

S

T

V

udVfudStdV0:0

εσ

Principle of virtual work

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Way of working1. Understand the problem (boundary conditions,

symmetries, material models, approximations)symmetries, material models, approximations)2. Modelling, discretisation, input data, selection of

elements, Verification (PRE-PROCESSING)3. Solving of systems of equations and calculation of

relevant properties e.g. stress and strain4. Analysis of output, graphical presentation (POST-

PROCESSING)PROCESSING)5. Consequences of analysis

Modelling, discretisation, input data, selection of elements, Verification (PRE-PROCESSING)

Geometry(discretisation)(discretisation)

Material modelσ = f(ε)Boundary conditions

6

Challenges within paperboard research• Constitutive models for paper that involve

the through thickness direction as well as– the through thickness direction, as well as

– moisture and temperature dependence is still active research

• Incomplete knowledge of material data for many materials

• Simulation tools for many applications are still missing

L k f b t d bl• Lack of robustness and convergence problems

Compression of boxesExperiments

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Compression of cylindersExperiments

What is the difference compared to rectangular boxes?

FE-analysis of compression of boxMATERIALPROPERTIESANALYSISUSE OF

PRO-

CRITICAL

DETAILMODEL(FEM)

LOADS ON DETAILS

BOX-MODEL (FEM)

LOADS

DUCT

CRITICALLOADS

(FEM)

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Static compressive load on boxFEM and experiment

Reference: L. Beldie, Lund University, 2001

Top and bottom segments


9

Middle segment


Experimental characterization of crease


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Static compressive load on boxFEM with crease elements and experiment

Test of whole package


Development of a three dimensional paperboard model

Real process/object

Laws of mechanics

p j

Experimentalverification

Modelformulation

Mathematical modelSolution of

mathematical problems Numerical

method

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Mechanical behaviour of paperboard

Top layer

Bottom layer

Middle layer/s

otto aye

In-plane behaviourCD

[MP

a]

30

40MD

CD

MD

Strain [%]

Stre

ss

2 40-1

-10

0

10

20CD

• Anisotropic elasticityThe elastic modulus in MD are 2-3 times larger than the elastic modulus in CD

Reference: Q. Xia, MIT, Boston, 2002.

larger than the elastic modulus in CD• Anisotropic initial yield stress

The initial yield stress in MD are 2-3 times greater then the initial yield stress in CD

• Anisotropic plastic strain hardeningPaper hardens more in MD than in the CD

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Out-of-plane behaviourrm

al S

tress

[MP

a]

0 1

0.2

0.3

0.4

ear S

tress

[MP

a]

0.

0.6

0.9

1.2

Normal displacement [mm]

Nor

00

0.1

0.5 1.51.0

Shear displacement [mm]

She

00

0.3

0.5 1.51.0 2.0

Reference: N. Stenberg, STFI/KTH, Stockholm, 2000.

Out-of-plane behaviourSummary

• Small amount of non-linearity before pre-peak loadbefore pre peak load

• Dominating softening behavior after the peak load

• Shear strength is pressure dependent

• Shearing causes normal dilatationdilatation

• Residual shear-load remains under normal compressive loading.

Reference: N. Stenberg, STFI, Stockholm, 2000. Q. Xia et. al., MIT, Boston, 2002.

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ModelIn-plane: elastic-plastic continuum • Anisotropic elasticity

Constitutive model:Anisotropic elasticity

• Anisotropic initial yield• Anisotropic plastic strain hardeningOut-of-plane: interfacial model• Post-peak softening tensile and

shear behavior.• Pressure dependent shear

resistancex

zy

Two‐dimensional view of tractions and displacements at interface

x

z

interface

xδzδ

xT

xT

zT

zT

damaged region

resistance• Normal dilation under shearing • Existence of shear friction• History dependent

Reference: Q. Xia et. al., MIT, Boston, 2002.

Continuum model Stresses:

Strains:

Interface modelTractions:

Displacements:, , , , ,x y z xy zx yzσ σ σ τ τ τ

, , , , ,x y z xy zx yzε ε ε γ γ γ

, ,x y zT T T

, ,x y zδ δ δ

Verification – Uniaxial tensionF F

References: N. Stenberg, STFI, Stockholm, 2002Q. Xia, MIT, Boston, 2002.

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Verification – Biaxial, compression

Out-of-plane compression

Reference: N. Stenberg, STFI, Stockholm, 2002Q. Xia, MIT, Boston, 2002.In-plane compression

Delamination modelFailure surface

Reference: N. Stenberg, STFI, Stockholm, 2002

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Delamination modelZD tension

Reference: N. Stenberg, STFI, Stockholm, 2002Q. Xia, MIT, Boston, 2002.

Delamination modelShear

Thickness increase under shear

Reference: N. Stenberg, STFI, Stockholm, 2002Q. Xia, MIT, Boston, 2002.

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Creasing and foldingModel setup

Top ply

Middle ply

Bottom ply

Creasing of Paperboard

Reference: H. Dunn, MIT, 2000. Reference: STFI-Packforsk/Tetra Pak 2004

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Verification – CreasingMeasurements

• Reaction force, F

• Displacement, u

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Verification –FoldingMeasurements

• Reaction force, F

R t ti l θ

Centre of rotation

• Rotation angle, θ

SpecimenSpecimen

Load cell

Clamps

Creasing and folding

Mikael Nygårds, STFI-Packforsk

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Creasing and foldingComparison with experiments

MD folding at two different creasing depths

Fracture Mechanics

Adapted fromPetri MäkeläInnventia AB

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The A4-exampleLoad: 1450 NElongation: 5.2 mm

Load: 750 NElongation: 1.4 mm

10 mmedge crack

48 % reduction of load carrying capacity and73 % reduction of strain at break

Reduced effective width of a paper web

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Corresponding stress distribution

Reduced strength

Fracture mechanics

Geometry

Fracturetoughness

Loading

toughness

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Paperboard applicationsCut-outs and cracks in corrugated board Failure of sacks

Crack

Web breaks K-cracks

Perforation

Nicks

Crack initiation spots

K cracks

Web breaksnomσ̂

nomσ

σedge crack

interior crackσ

σ

a) b)

RR

p

p

t

2a

σ

σ

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Delamination in printing nipsa) b) c)

Crack tip modellingFPZ appearance and modes of crack opening

Crack Tip3‐4 mm

Fracture Process Zone of considerable size

Damage

Modes I and II are predominant under in-plane loading and mode I is considered most severe.

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Linear elastic fracture mechanicsLEFM

σ

( )Iij ij

K kσ ϕ=

Stress state in the crack tip regionFailureprocess zone

Log (σyy)

εE

( )ij ijrϕ

(material, geometry,loading)

IK f=

Singularity‐dominatedzone controlled by K inEquations (6.1)

Log (r/rp)

12

−

Fracture criterion, LEFM

Tensile strength

Stress

Strain

Tensile strength

KI

Critical K

Strain

Critical KI

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1 2 1 2

MD CD

Linear elastic fracture mechanicsσ n

om/σ

b, ε n

om/ ε

b

0.2

0.4

0.6

0.8

1.0

1.2

Stress at break, num.Strain at break num

Stress at break, exp.Strain at break exp

σ nom

/σb,

ε nom

/ εb

0.2

0.4

0.6

0.8

1.0

1.2

Stress at break, num.Strain at break, num.

Stress at break, exp.Strain at break, exp.

Crack size [mm]0 5 10 15 20

0.0Strain at break, num.Strain at break, exp.


0.0Strain at break, num.Strain at break, exp.

Conclusions, LEFM

• Does not apply to paper materials in general.

• Large cracks in very large structures are required in order for LEFM to be applicable to paperboard.

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Non-linear fracture mechanics, NLFMThe HRR crack tip fields

σ

Stress state in the crack tip region

εNEE ,, 0

1

Failure process zone

Log (σyy)

J‐dominated zone

(material,geometry,loading)J f=

( )1

1,

n

ij ijJa f nr

σ ϕ+⎛ ⎞= ⎜ ⎟

⎝ ⎠

Hutchinson 1968, Rice, Rosengren 1968

K‐dominated zone

Log (r/rp)

12

−

11n

−+

J is the energy release-ratein a non-linear “elastic” material

• In the special case of a linear elastic t i l J i ti l t K 2material, J is proportional to KI

2

• This means that LEFM is a special case of NLFM.

• J is defined as a path-independent line-integral around the crack-tip involvingintegral around the crack-tip involving expressions containing stress, strain and displacement.

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Fracture criterion, NLFM

Tensile strength

Stress

Strain

Tensile strength

J

C iti l J

Strain

Critical J

1 2 1 2

MD CD

NLFMThe J-integral method

σ nom

/σb,

ε nom

/ εb

0.2

0.4

0.6

0.8

1.0

1.2

σ nom

/σb,

ε nom

/ εb

0.2

0.4

0.6

0.8

1.0

1.2






0.0


0.0Strain at break, num.Strain at break, exp.Strain at break, num.Strain at break, exp.

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Conclusions, NLFM(J-integral method)

• Does predict failure quantitatively for large k d lit ti l f h t kcracks and qualitatively for short cracks

• Numerically “cheap”• Easy calibration

What information does J carry?

• J is a loading parameter

• J expresses the severity of the stresses at the crack-tip

• When J reaches a critical value, the crack starts to growg

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Critical value of J (or K) is called fracture toughness1. In order to formulate a fracture criterion, we

need to know how severe stress states theneed to know how severe stress states the material is able to withstand.

2. We need to know the critical value of J for the material, i.e. the fracture toughness (Jc) of the material.

3. The information from material testing on a test piece with a man-made crack is required to evaluate the fracture toughness.

Predictions of failureGenerally requires numerical methods1. Material

behaviour 3.

Fracturetoughness

2.

behaviour 3.Full-scale

predictions of failure

FE-analysisFE-analysis

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Verification of transferabilityJ-integral method for 1 m wide paper webs

ion

/ mm

8

10

12

e / k

N

4

5

6

PredictionsExperiments

on /

mm

15

20

e / k

N4

5PredictionsExperiments

Fluting Sackpaper

Crit

ical

elo

ngat

ion

/ mm

2

4

6

8

10

12

14

Crit

ical

forc

e / k

N

1

2

3

4PredictionsExperiments

Crack length / mm0 5 10 15 20 25 30

Crit

ical

elo

ngat

i

0

2

4

6

Crit

ical

forc

e

0

1

2

3

Crack length / mm

0 5 10 15 20 25 30

Crit

ical

elo

ngat

io

0

5

10

Crit

ical

forc

e

0

1

2

3

Crit

ical

elo

ngat

ion

/ mm

2

4

6

8

10

12

Crit

ical

forc

e / k

N

0.5

1.0

1.5

2.0

2.5

PredictionsExperimentsNewsprint Testliner

Crack length / mm0 5 10 15 20 25 30

0

2

0

Crack length / mm0 10 20 30 40

0

2

0.0

Crack length / mm

0 5 10 15 20 25 30

Crit

ical

elo

ngat

ion

/ mm

0

2

4

6

8

10

12

14

16

Cro

tical

forc

e / k

N

0

1

2

3

4PredictionsExperimentsMWC

Crack length / mm0 10 20 30 40

Crit

ical

elo

ngat

ion

/ mm

0

2

4

6

8

10

12

Crit

ical

forc

e / k

N

0.0

0.5

1.0

1.5

2.0PredictionsExperimentsSC

Model Real material

Stress state in vicinity of crack tip

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Damage behaviourElastic unloadingsupports energy

Pa

50

60

Tensile testing

Elastic unloadingsupports energy

Damage evolutionconsumes energy

Strain / %0.0 0.5 1.0 1.5

Tens

ile S

tress

/ M

0

10

20

30

40

50

Instability when rate of supported energy from elastic unloading exceeds consumed energy during damage evolution

Strain / %

Tensile test resultsLong and short test pieces

ensi

le S

tress

[MPa

]

20

30

40

50

60

70

Short stripLong strip

Ordinary tensile test piece

Apparant strain [%]0 2 4 6 8 10

Te

0

10

“Short” tensile test piece

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Modelling of damage – cohesive zonew

σ 0s (

σ )

σ (w)σ (w)

L+δ Elongation (δ )δ0

Stre

ss

LE

w σσδδ −+−= 0

0 Damage zone

r

Widening (w)

Stre

ss (

σ )

( ) βασσ ww −= e0

avw(r)

σy(r)y

x

Elastic-plastic material + cohesive zone

1 2 1 2

MD CD

σ nom

/σb,

ε nom

/ εb

0.2

0.4

0.6

0.8

1.0

1.2

σ nom

/σb,

ε nom

/ εb

0.2

0.4

0.6

0.8

1.0

1.2





66


0.0


0.0Strain at break, num.Strain at break, exp.Strain at break, num.Strain at break, exp.

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Conclusions, Cohesive crack model

• Accurate predictions of failure for all crack sizes

• Numerically expensive

• Expensive, cumbersome and

• Time-consuming “calibrationTime consuming calibration

• No explicit fracture criterion needed

Final remarks• FPZ size important for the choice of fracture mechanics

model.model.• If FPZ is small crack tip fields are singular• J-integral method predicts mode I in-plane failure of

notched paper structures well.• Cohesive stress models excellently predicts mode I in-

plane failure.• Such models are also applicable for out-of-plane failure

where the crack tip singularity concept is not unambiguously applicable.

• Fracture mechanics can be used for damage tolerance analysis of structures containing assumed defects.

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After lecture 16 you should be able to• Illustrate the finite element method and its use for paper-

based industry• Illustrate how a finite element model is created• Discuss the results from finite element analysis of:

– Compression test of a package– Creasing and folding

• Discuss concepts such as the J-integral and the fracture toughness

• Account for the procedures of failure predictions in paper materials

lecture 16: design of paper and board packaging/menu/general/column... · analysis of output,...

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