Numerical Investigation of Paperboard Forming - · PDF fileNumerical Investigation of...

Numerical Investigation of Paperboard Forming Hui Huang and Mikael Nygårds

KEYWORDS: Paperboard, Forming, Numerical

investigation, Mechanical properties

SUMMARY: A three dimensional numerical investiga-

tion of a commercial four-ply paperboard formed into a

pear-shaped mould was presented. The numerical

investigation included the effect of pressure, boundary

conditions, material properties and different deformation

and damage mechanisms such as delamination and

plasticity. Simulations were done in both the MD and CD

using different pressures. A paperboard model with a

combination of anisotropic continuum model and a

softening interface model had good deformation behavior

during the forming simulations. Forming experiment that

mimicked the simulations was performed. Numerical and

experiment results were compared with good agreement.

ADDRESSES OF THE AUTHORS: Hui Huang

([email protected]), Mikeal Nygårds ([email protected]),

KTH, Department of Solid Mechanics, SE-100 44,

Stockholm.

Corresponding author: Mikael Nygårds

Forming of paperboard into double-curved surfaces has

the potential to expand the utilization space for paper

products. This since packages can be made more flexible

and individualized; hence package design can bring

significant advantages for producers, customers and also

the society. However, many deformation and damage

mechanisms within the paperboard during forming

processes are still unknown; therefore parameters need to

be identified and analyzed. Compared to e.g. sheet metals

that often are formed, there are several complicating

factors for paper materials, such as anisotropy, gradients

in the thickness direction, moisture and temperature

dependency etc, which need to be understood in order to

enable forming of paperboard. Since paperboard is highly

anisotropic and has complex mechanical properties it will

crack and wrinkle differently than e.g. metals and

plastics, that normally is formed into double-curved

structures. Moreover, commercial paperboard is usually

composed of three to five plies. Pulp fibers are generally

lying along one direction on each ply, which direction is

called machine direction (MD). The direction

perpendicular to machine direction is called cross

machine direction (CD). The third direction is out-of-

plane direction (ZD), as Figure 1 shows.

Nowadays, two main strategies can be used to form

double-curved paper surfaces. First, a fiber suspension

can be sprayed onto a mould with desired shape. This is

how egg box like packages are made. Second, a paper

sheet can be formed into a mould with desired shape by

applying load on an initial flat surface. This can be done

by having hard male and female mould, which is

typically used to form plastic containers. A strategy of

this kind has been presented by Morris and Siegel (1997)

for deep drawing of a food container. Alternatively, the

load can be applied by a membrane using pressurized air

or a liquid on the concave side of the package, or a

vacuum on the convex side of the package (Östlund et al.

2011). To maintain high stiffness and strength that

paperboard sheets have, it is advantageous if any of the

latter methods can be used, rather than spraying fiber

suspensions.

The finite element method is advantageous since it can

complement experimental studies in order to study

deformation and damage mechanisms in paper material in

more detail. The method has been used to study the

mechanical response of paperboard at relatively large

scale deformation, such as analyzing paperboard in-tack

delamination (Hallbäck et al. 2006), compression loading

of paperboard package (Beldie et al. 2001), creasing of

paperboard (Nygårds et al. 2008, Beex et al. 2008),

folding of paperboard (Huang, Nygårds, 2010), paper

webs during printing (Kulachenko et al, 2005).

This paper is aiming to use the finite element method

to investigate the behavior of paperboard during forming

of complex shaped surfaces. First, the choice of element

type and interfaces in the finite element model was

investigated. Second, the forming performance due to

pressure, boundary conditions and material properties

was investigated. Lastly, forming experiments were done

and its results were compared with numerical results.

Material and material model

The paperboard that was used in this work was a multiply

paperboard, with a top ply, two middle plies and a bottom

ply. The elastic-plastic material properties of the different

plies in this paperboard have been characterized by

Nygårds (2008). In addition, the shear strength profiles

have been characterized by Huang and Nygårds (2011).

This paperboard has relatively strong bottom ply and

relatively weak middle plies.

The creasing and folding behaviour of paperboard has

previously been shown to be well predicted by a

combination of continuum and delamination models, as

shown in Fig 1 (Huang, Nygårds, 2010, 2011). In the

finite element simulation, the plies in the paperboard

were represented by an elastic-plastic continuum model,

and the interfaces were represented by an elastic-plastic

interface model. In the forming operation there are no

excessive opening of interfaces, therefore each individual

ply was assumed to have uniform properties in the

thickness direction, i.e. the material mapping suggested

by Huang and Nygårds (2011) was not utilized. Another

purpose for this study was to study the strain distribution

during double-curve forming so that the results can be

used to better understand how the paperboard properties

influence the forming behaviour. Since damage is not

desired during forming, in-plane damage was not

considered in the modelling. The finite element

simulations were performed by using Abaqus/Explicit

(ABAQUS, 2010). Alternatively, Abaqus/Standard or

Abaqus/Dynamic implicit could have been used. Initial

forming simulations did however show that

Abaqus/Explicit was the fastest and most reliable solver.

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Nordic Pulp and Paper Research Journal Vol 27 no.2/2012 211

Fig 1. Illustration of the paperboard structure and coordination system.

Fig 2. 3D view of the forming mould with dimensions.

[1]

The forming mould used in the simulations is seen in

Fig 2. The mould was pear-shaped and has three different

curvatures: r1, r2, and r3, the largest depth in the mould

was 25 mm. Simulations were done with both the

machine direction and cross machine direction along the

centreline. MD forming indicates that the centreline of

the mould was parallel with the paperboard MD, and CD

forming indicates the centreline of the mould was parallel

with paperboard CD.

Continuum model

The continuum model was used to represent the plies of

the paperboard, which was an elastic-plastic model with

orthotropic linear elasticity, Hill’s yield criterion and

isotropic hardening, available in ABAQUS (2010). Hill’s

yield criterion is defined in Eq 1 (see above) where

represent the initial yield stresses in the different material

directions, and the constants F, G, and M are defined as

,

, [2]

.

In the model, the hardening was assumed to be linear

with hardening modulus H.

Delamination model

The delamination model was used to represent the

interfaces in the paperboard such that different plies

could be combined together to represent the paperboard.

The delamination model was an orthotropic elastic-plastic

cohesive law which relates the interface tractions to the

opening and sliding of the interface. Eq 3 was defined to

govern the change in the traction vector t across the

interface due to incremental relative displacements.

,

, [3]

, where is the stiffness in the interface normal

direction, i.e. ZD, and are the shear stiffness

components in MD and CD respectively. Damage

initiation was governed by

, [4]

where ,

, are the maximum stresses needed to

initiate damage in the respective directions. <> indicated

that purely compressive stress (i.e., a contact penetration)

does not initiate damage. Softening behaviour was

defined by exponential damage evolution, which means

when the initiation criterion was reached, the cohesive

strength decrease with exponential behaviour. The

damage evolution D was defined as:

Mould

r2

r3 r1

L

L=94 mm r1=35 mm r2=95 mm r3=15 mm

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212 Nordic Pulp and Paper Research Journal Vol 27 no.2/2012

, [5]

where is a non-dimensional parameter that defines the

rate of damage evolution, is the maximum value of

effective displacement at during loading history, and

is the effective separation at complete failure. The

effective displacement is defined as

, [6]

which describes the evolution of damage under a

combination of normal and shear separations across the

interface.

When damage evolution is activated, the stiffness and

traction components are degraded as:

,

, [7]

,

,

, [8]

.

Fig 3 Forming equipment and sketch of forming process. The mould is located within the support and balloon is mounted inside of the rings. (Östlund et al. 2011).

Fig 4. Illustration of numerical model mesh picture with loading definition. Pressure is perpendicularly applied on the paperboard surface with linearly increasing pressure as function of time.

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Finite element model

Simulations were done to mimic the experiment forming

apparatus as show in Fig 3. During forming experiments,

an adjustable pressure was applied on a rubber membrane

above the paper specimen. This forms the paper sample

into the desired mould shape. In the finite element model,

the loading was simulated by applying a uniform pressure

on the paperboard top surface using a linearly increasing

load as function of time, as displayed in Fig 4. Since

Abaqus/Explicit (ABAQUS, 2010) was used, mass

scaling was utilized to shorten the simulation time. By

performing simulations with different mass scaling

coefficients, it was observed that results became stable

and reasonable when the mass scaling coefficient was

decreased to 5×10-6

. To ensure stable results, the mass

scaling coefficient was chosen as 5×10-7

in the whole

model. The forming mould was in the analysis

represented by an analytical rigid body (ABAQUS,

2010). Surface contact between the paperboard and

forming mould was defined as exponential overclosure

behavior (ABAQUS, 2010) with 0.5 MPa contact

pressure in the normal direction and 10-3

mm clearance,

as done previously by Nygårds et al. (2009). In the

tangential direction, a Coulomb friction model in

ABAQUS (2010), with friction coefficient 0.5, between

the paperboard surface and forming mould top surface

was used.

Results and discussion The aim was to investigate if it was possible to perform

simulation of forming, and to see limitations. Therefore, a

fairly deep mould was used, which made the simulations

more challenging. In the simulations the determined

material properties of a commercial paperboard was used,

which are found in Appendix 1. To evaluate the

simulation results contour plots of in-plane strain

components, as well as the strain along the centre line

and cross line were plotted versus the deformation down

into the mould.

The effect of interface model, shell element and continuum elements in the model

The modeling concept, where a combination of

continuum and delamination models, is used have been

verified to predict paperboard converting operations well

(Nygårds et al 2009, Huang and Nygårds 2010, 2011). In

the forming of double curves surfaces there is only a

small likelihood that the paperboard will delaminate,

since it is mainly loaded in the in-plane directions and

compressed in the out-of-plane direction. Therefore, the

out-of-plane properties should be of less importance in

forming than increasing and folding operations, and a

simpler modelling concept could be used.

In numerical simulations of material behaviour it is

important to consider the deformation and damage

mechanisms that are important in different operations.

Correct model representation will shorten simulation time

and improve the numerical convergence of the Newton-

Raphson loops to solve the constitutive equations, such

that the problem is solvable. If too few details are

included in the models adequate mechanisms will not be

accounted for, but if too many details are accounted for

convergence problems and long simulation times will

occur. Especially for paper material that is anisotropic

convergence problems are problematic. The effect of

modelling concept for forming was investigated, where

three different models were used to represent the

paperboard during forming, namely:

1. Paperboard model where the plies were represented

by continuum elements (C3D4), and delamination

was accounted for by a surface representation, named

Cohesive behaviour in Abaqus (2010). The model

was composed by four continuum plies and three

interface layers. The four continuum plies were

assigned material parameters from the

characterization (Nygårds, 2008), (Huang and

Nygårds, 2011). The three interface layers connected

the four continuum plies. The interfaces were

assigned material parameters from the

characterization. In total 67984 elements were used in

the model, where the thickness of each ply was

represented by one element, hence totally four

elements were used in the thickness direction.

2. Continuum model where the plies were represented

by continuum elements (C3D8), and no delamination

was considered. The model consisted of four

continuum plies that were tied. Each continuum ply

was assigned material parameters from the

characterization. Hence, the same continuum

parameters as for paperboard model were used. In

total 35768 elements were used in the model, where

the thickness of each ply was represented by one

element, hence totally four elements were used in the

thickness direction.

3. Shell model where the plies only were represented by

composite shell element (SC8R). Hence, out of plane

properties were not accounted for. Apart from that,

the same material parameter as in continuum model

was used. Therefore no delamination was considered.

In total 35488 elements were used in the model,

where the thickness of each ply was represented by

one element, hence totally four elements were used in

the thickness direction.

Thus, the paperboard model accounts for all

mechanisms that have been characterized. The continuum

model disregards delamination, and will as a consequence

have different ZD tension behaviour than the real

material. The use of shell element will shorten the

simulation time, but no delamination, or out-of-plane

deformation was accounted for. In the simulations a

forming pressure of 10 bar was used and the boundaries

were free.

In Fig 5, the deformations along the centre line and

cross line are plotted for the three modeling concepts.

Although the pressure was the same the models deform

differently, due to the different assumptions. The shell

model was pushed further into the mould compared to the

continuum model and paperboard model. When more

mechanisms were accounted for in the model

formulation, the less the model was pushed into the

mould, which was due to the fact that energy was

consumed performing the included deformations.

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In Fig 6, strain components along the centre and cross

lines have been plotted, and some interesting

observations was done. First, the strain in MD for the

paperboard model and continuum model was similar,

apart from the fact that the paperboard model

delaminates, and hence gives larger scatter and locally

high strains. Second, the strain in the shell model is less

than the other models, although it deformed deeper into

the mould, which was due to the fact that more of the

model was pulled into the mould. Third, the strain in CD

was smaller in the continuum model than in paperboard

model.

The contours of in-plane strain components for three

different modeling concepts are found in Fig 7

(Continuum model), Fig 8 (Shell model) and Fig 9

(Paperboard model). The deformation behaviour was

different between the different concepts. In the shell

model, several wrinkles appeared in the model. For both

the shell and continuum models, the model edge was

pulled into the mould during MD forming. For the

paperboard model, local delamination was observed at

the boundary and also the edges of the mould. This

delamination phenomenon could be used to explain that

paperboard model displayed highest strain profile among

the three models.

The relation between maximum displacement U2 and

maximum strain ɛ within each model was plotted in

Figure 10. As we can see this relation deviate more for

three models for CD forming than MD forming.

Although, in Figure 5 continuum model deformed more

than the paperboard model under 5 bar, when the pressure

in the end reached 10 bar paperboard model deformed

more than the continuum model in CD forming. The only

difference between the continuum model and the

paperboard model was delamination. Therefore, we can

conclude that delamination had large impact on forming

performance, especially on CD forming.

The paperboard model has been verified previously for

creasing and folding (Huang and Nygårds 2010, 2011), it

will be used to investigate the effect of other model

parameters. Besides, according to the results, the

paperboard model captured more information than the

continuum and shell models, and the paperboard model

had good convergence, which the shell model did not

have. One drawback with the paperboard model was

however the long computational time. One simulation

took roughly 200 hours, while the same simulation it took

85 hours for continuum model and only 48 hours for the

shell model, by using a 3 Ghz CPU with 64-bit operating

system.

Fig 5. U2 Displacement comparisons for Paper, Shell and Paperboard model simulations with free boundary condition under same loading pressure 5 bar. Left column is CD forming and right column is MD forming. Two black dashed-lines indicate the position of mould shape edges. Displacement values were taken from the paperboard top surface.

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Fig 6. Strain comparison for Paper, Shell and Paperboard model simulations with free boundary condition. Left column is MD forming and right column is CD forming. The respective deformations are U2max= 24.2 mm at MD forming and U2max= 18.54 mm at CD forming. Two black dashed-lines indicate the position of mould shape edges. Strain values were taken from paperboard top surface.

MD Forming CD Forming

Fig 7. Strain contour plots strain in MD (LE11) and in CD (LE22) of continuum model simulations with free boundary condition under same U2 displacement as in Figure 5. Left column is MD forming and right column is CD forming. The respective displacements are U2max= 24.2 mm at MD forming and U2max= 18.54 mm at CD forming.

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Fig 8. Strain contour plots strain in MD (LE11) and in CD (LE22) of shell model simulations with free boundary condition. Left column is MD forming and right column is CD forming. The respective displacements are U2max= 24.2 mm at MD forming and U2max= 18.54 mm at CD forming


Fig 9. Strain contour plots strain in MD (LE11) and in CD (LE22) of paperboard model simulations with free boundary condition. Left column is MD forming and right column is CD forming. The respective displacements are U2max= 24.2 mm at MD forming and U2max= 18.54 mm at CD forming.

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Fig 10. Comparison among different element models results by maximum deformation U2 versus maximum logarithmic strain under pressure from 0bar till 10bar (Shell-Forming crashed at 5bar).

Fig 11. Contour plots of strain in MD (LE11) and in CD (LE22) for MD forming (left Column) and CD forming (right column) under 10 bar. The respective displacements are U2max= 18.1 mm at MD forming and U2max= 14.14 mm at CD forming.

The effect of pressure

Since the paperboard was anisotropic it will behave

differently depending on how it is placed in the mould.

This also generates an inhomogeneous strain field within

the paperboard. The paperboard model was pressed into

mould using fixed boundaries and simulating both MD

and CD forming by adding a 10 bar pressure gradually. In

Fig 11, it can be observed that the in-plane strain fields in

MD and CD depends on how the paperboard was placed

in the mould. For both MD and CD forming high strains

were found in the deepest part of the mould. In Figure 11,

it was observed that for both MD and CD forming, a

more uniform strain field develops in MD compared for

CD, which was due to the higher stiffness in MD.

The gradual deformations of the paperboard into the

mould were evaluated along the centre and cross line by

evaluation of the deformation at different pressures,

ranging from 2 to 10 bar. As the pressure increased the

paperboard model deformed more and with gradual

increasing strain, as seen in Fig 12. A comparison of MD

and CD forming shows that the paperboard deforms

differently in the two cases. For equal pressures the MD

forming goes deeper into the mould then CD forming. In

Fig 12, it should be noted that neither MD forming nor

CD forming touched the mould surface completely.

Noteworthy in Fig 12, was also that, although the

paperboard did not touch any part of mould bottom, the

paperboard deformation followed the mould shape

instead of a symmetrical arc shape, which was because of

the shape of the mould edge. For MD loading it should

also be noted that the paperboard comes partly into

contact with the mould at a pressure of 6 bar. This can be

observed at the right part of the centreline. As a

consequence this affects the strain field in the

paperboard, since the friction between the paperboard and

mould obstructs straining of the paperboard. Therefore,

the strain increased less in the areas in contact with the

mould during continued loading, as seen in Fig 12.


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Fig 12. Displacement and strain vs. normal distance (cross line and centreline) plot comparison under increasing loading pressure for CD forming (right column) and MD forming (left column). Two black dashed-lines indicate the position of mould shape edges. Displacement and strain values were taken from the paperboard top surface.

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Fig 13. Displacement comparison between the forming simulations with fixed and free boundary condition under same loading pressure 10 bar on mould. Left column is MD forming and right column is CD forming. Two black dashed-lines indicate the position of mould shape edges.

Fig 14. Strain comparison between the forming simulations with fixed and free boundary condition under same U2 deformation. Left column is MD forming and right column is CD forming. The respective deformations are U2max= 18.1 mm at MD forming and U2max= 14.14 mm at CD forming, same as in Fig 10. Two black dashed-lines indicate the position of mould shape edges. Strain values were taken from the paperboard top surface.

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Fig 15. Contour plots of strain in MD (LE11) and in CD (LE22) with free boundary condition for CD forming (right column) and MD forming (left Column) on Mould. The respective displacements are U2max= 18.1 mm at MD forming and U2max= 14.14 mm at CD forming, same as in Fig 11.

Fig 16. Sketch explanation for different material combination of paperboard plies. In the colour bar, the darker colour represents stiffer material property.

For both MD and CD forming it was observed that the

highest strains occurs in the CD direction, 26% and 18%

respectively, these appeared in the deepest part of the

mould. The maximum strain in the MD direction was 8%

and 11% respectively, which appeared close the edges,

where initial contacts between the paperboard model and

the mould were established. Interesting was also to note

that compressive strains in CD appeared along the

centreline in the shallow part of the mould. The negative

strains would in an experimental setup contribute to

wrinkling of the paperboard.

The effect of boundary conditions In the previous section forming with fixed paperboard

boundaries were simulated. A possible route to reduce

the straining of the paperboard during forming is to alter

the boundary conditions. The two extreme cases are fixed

and free boundaries. The free boundaries will allow the

paperboard to move into the mould, which will reduce

the stretching of the paperboard, but instead it can cause

wrinkling.

In this section, simulations with fixed and free

boundaries were done, which was done using two

different pressures. First, both models were loaded with

the same pressure p=10 bar, displacement was compared

for the two boundary conditions. Second, the model with

free boundaries was compared with the model with fixed

boundaries, when the deformation into the mould was

similar, which occurred when p=6 bar using free

boundary conditions and p=10 bar using fixed boundary

conditions. The deformation of all models along the

centreline and cross line can be seen in Fig 13, and the

corresponding strains can be seen in Fig 14.

In Fig 13 it is obvious that the model with free

boundaries deforms longer into the mould when the same

pressure was used. The model shrinks more along MD

than CD with free boundary condition. In Fig 14 it is

observed that the straining in MD was very sensitive to

the boundary conditions, ɛMD decreased from 7% to 3.5%

for MD forming along the centreline, and from 11% to

4% for CD forming along the cross line. On the contrary,

the strain in CD was not as sensitive to the boundary

conditions. Especially for MD folding, the CD strain ɛCD

profile showed similar tendency between two different

boundary conditions. Moreover, ɛMD had a flatter profile

inside the mould, with maximum values at mould shape

edges, while ɛCD had maximum values at the deepest

position of the mould. This shows that the paperboard

was strained more in CD than in MD inside the mould.

When looking into the ɛCD profile along the centre line

Type 1

Type 2

Type 3

Type 4

Type 5 Type 6

Stiff

Soft


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for CD forming in Fig 14, it was observed that ɛCD using

fixed boundary conditions resulted in negative values,

which indicated that it could be higher possibility for

wrinkling for the model with fixed boundary conditions

than free boundary conditions.

Fig 15 shows contour plots for simulations with free

boundary condition, which had the same displacement as

the contour plots in Fig 11.

The effect of material properties during forming

The characterization of material properties showed that

the paperboard has an I-beam like structure in the

thickness direction; compare the stiffness and strength

values in Appendix 1. However, it was also observed that

the bottom ply was stiffer than the top ply. Hence, the

characterized plies have tree different stiffness and

strength values, according to Type 1 in Fig 16. In order

to investigate the influence of paperboard material

properties on forming performance, five artificial

paperboard combinations were tested, illustrated as

Types 2-6 in Figure 16. In Type 2 and 3, the middle ply

was taken out to one side, to generate gradual increasing

and decreasing stiffness and strength properties in the

thickness direction respectively. Types 4 to 6 had

uniform material properties based on the middle, top and

bottom plies. Therefore, Type 4 is the most compliant

material while Type 6 is the stiffest material.

Simulations for all six type of paperboard material

were done using a forming pressure of 10 bar. However,

Type 4 and Type 6 had convergence problems when the

pressure reached 6 bar and 5 bar respectively. Hence,

displacement comparisons for the six material types were

done using 5 bar in Figure 17. Since Type 4 is the most

compliant model, it had much larger deformation than

the other material types. Besides, the top ply was more

isotropic, and the bottom ply was most anisotropic, as

showed in Appendix 1.

The effect of this could be observed in Fig 17, by

comparing Type 5 and 6. Due to the anisotropy effect the

deepest deformation of Type 5 and 6 differs; Type 5 has

its deepest deformation to the left of Type 6 in the figure.

This phenomenon also occurred for CD forming, but not

as obvious as for MD forming. Moreover, Fig 17 also

showed that when the properties were varied from Type

1 to Type 3, it had similar displacement performance.

Fig 18 illustrated strain profile for the six types of

material under similar deformation. It was interesting to

see that, the modified material properties did not alter the

strain profile in MD in Figure 18 much. However, the

strain in CD was more sensitive to changes in material

properties. All six material types showed different ɛCD

properties in Fig 18. Type 1 and 5 had the highest strain

in CD, while Type 2 and 4 had the lowest.

Experiment verification

Paperboard forming experiments were done by using the

forming apparatus that has been developed by Östlund et

al. (2011). Both MD and CD forming with free and fixed

boundary condition were tested. The apparatus was

capable of performing forming on many papers qualities.

However, the experimental procedures are still under

development, and there are currently no methods to

evaluate strain/stress profiles or deformation profiles.

Therefore, only the formed shape and fracture pattern

will be illustrated here.

In the experiments, MD forming penetrated deeper

into the mould than CD forming, when fixed boundary

conditions was used, which was in accordance with the

numerical results. In general, forming with free boundary

condition was better, although it resulted in more

wrinkles compared to forming with fixed boundary

condition. For MD forming with fixed boundaries, the

paperboard failed when the pressure reached 3 bar.

Interpolation of the strain in Fig 12 indicates that this

corresponds to local strain levels of MD=4% and

CD=6%. A crack was observed close to the mould edges

on the failed specimen in Figure 19. Standard in-plane

tension tests of this paperboard showed that the failure

strain in MD was 2.0% and in CD 5.5%. Probably the

crack was initiated by the high local strain in MD, MD,

which was peaking along the crack path when evaluated

along the centerline, compare the line plot in Fig 12 and

contour plot in Fig 11.

With free boundary condition, the paperboard failed

when the pressure reached roughly 5 bar both for MD

and CD forming. In Fig 20, the failed specimens are

shown. More wrinkles appeared when free boundary

conditions were used, compared to the fixed boundary

conditions that was shown in Fig 19. The evaluated

strains in MD and CD, as shown in Figure 14, for MD

and CD forming can be used to interpret the failures.

Note the data in Fig 14 was plotted for 10 bar pressure,

but the data in Fig 12 showed that the strain components

increase linearly with pressure. For MD forming, Fig 14

showed that the highest strain in MD was 3%

(presumably MD=1.5% for 5 bar), and the highest strain

in CD was 20% (presumably CD=10% for 5 bar). Hence,

the specimen should fail due to the high CD strain that

was located in the middle of the cross line, which was

also supported by the centered crack path in Fig 20. For

CD forming, Fig 14 showed that the highest strain in MD

was 4% (presumably MD =2 % for 5 bar), and the highest

strain in CD was 13 % (presumably CD =6.5% for 5 bar).

Both these strain values were close to the experimental

failure strains. In Fig 20, it was however interpreted as if

the crack was initiated by the high strain in MD close to

the mould boundary to the left in the picture.

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Fig 17. Displacement comparison among six material type combinations (as Fig 15 shows) with fixed boundary condition under same loading pressure 5 bar on mould. Left column is MD forming and right column is CD forming. Two black dashed-lines indicate the position of mould shape edges. Displacement values were taken from the paperboard top surface.

Fig 18. Strain comparison among six material type combinations (as Fig 15 shows) with fixed boundary. Left column is MD forming and right column is CD forming. The respective displacements are U2max= 9.3 mm at MD forming and U2max= 6.4 mm at CD forming. Strain values were taken from the paperboard top surface.

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Fig 19. Experimental MD forming using 3 bar pressure and fixed boundaries.

Fig 20. Experimental forming using 5 bar pressure and free boundaries.

Conclusions The paperboard model with a combination of continuum

model and interface model showed good convergence in

the forming simulations. The paperboard model also

captured more information than the continuum and shell

models. However, the shell model had shorter

computational time, but also more convergence

problems, and since it lacked of out-of-plane properties it

was judged to be inadequate for characterization of

forming behaviour.

In the forming simulations, delamination occurred at

the edge of mould shape and the deepest area of mould

shape; this affected the in-plane strain fields. Also the

boundary conditions had a large effect on forming

performance. It was observed that the strain in MD was

more affected by boundary conditions than strain in CD.

Moreover, it was observed that the material properties

affected the strain field in CD more than in MD for both

MD and CD forming.

If the objective would be to minimize the in-plane

strain components, we could observe that the model

should have free boundaries and CD forming should be

used. Moreover, a graded material with the stiffest part

facing the mould, or a low stiffness material, would be

used. Hence, in the development of experimental

techniques it can be advantageous to utilize delamination

and boundary conditions to improve forming

performance for a given paperboard. In an axi-symmetric

mould could e.g. a combination fixed and free boundary

conditions be used to minimize in-plane strain

components and still enable delamination. If the material

properties can be modified an even better optimization

can be done. Highly anisotropic material property is e.g.

not very suitable for double-curved forming processes.

Reduction of anisotropy could therefore improve

performance.

By just comparing the strain levels achieved in the

simulations if would be difficult to find a real paperboard

that can withstand these strain levels. The fact that a low

stiffness material works well in the forming can be

utilised by using moisture to lower the paperboard

stiffness, which would than lower the straining of the

model. At the same time moisture will improve local

straining of the paperboard, since bond strengths are

locally reduced. Both effects contribute to good forming

performance.

Double-curved forming is a temperature and moisture

dependent process for commercial paperboard. In this

work, only standard climate was assumed during the

numerical and experiment investigation. In future work,

the numerical model can include temperature and

moisture dependencies to mimic more reliable forming

processes.

Acknowledgement

The authors would like to acknowledge Professor Sören Östlund for many valuable discussions. The authors also want to thank BiMaC Innovation and its industrial partners for the financial support.

Literature

ABAQUS (2010). ABAQUS User’s Manual. Abaqus Inc, Providence, RI, USA, 6.10 edition.

Hallbäck, N., Girlanda O. and Tryding J. (2006): Finite element analysis of ink-tack delamination of paperboard, International Journal of Solids and Structures, 43: 899-912.

Xia, Q. S. (2002): Mechanics of inelastic deformation and delamination in paperboard, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, USA.

Stenberg, N., Feller, C., Östlund, S. (2001): Measuring the stress-strain of paperboard in the thickness direction, Journal of Pulp and Paper Science, 27:213-221

Beldie, L., Sandberg G. and Sandberg L. (2001): Paperboard packages exposed to static loads-finite element modelling and experiments, Packag. Technol. Sci. 14:171-178.

Huang, H. and Nygårds, M. (2010): A simplified material model for finite element analysis of paperboard creasing, Nord. Pulp Paper Res. J. 25(4):505.

PAPER PHYSICS


Huang, H. and Nygårds, M. (2011): Numerical and experimental investigation of paperboard folding, Nord. Pulp Paper Res. In press

Nygårds, M., Just, M. and Tryding, J. (2009): Experimental and numerical studies of creasing of paperboard, Int. J. Solids Struct. 46:2493.

Östlund, M., Borodulina, S., and Östlund, S. (2011): Influence of paperboard structure and processing conditions on

forming of complex paperboard structures, Packag. Technol. Sci. 24:331-341.

Nygårds, M. (2008): Experimental techniques for characterization of elastic-plastic material properties in paperboard, Nord. Pulp Paper Res. J. 23(4):432

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Appendix 1: Material model constants Table A1. Parameters in the continuum material model, elastic, initial yield stresses and hardening.

Continuum material model- Elastic constants

Top Ply Middle Ply Bottom Ply

/MPa 5920 /MPa 3203 /MPa 8760

/MPa 2670 /MPa 1233 /MPa 3030

/MPa 228 /MPa 160 /MPa 130

/MPa 1431 /MPa 638 /MPa 1617

/MPa 60 /MPa 30 /MPa 68

/MPa 60 /MPa 30 /MPa 68

0.45 0.47 0.51

0 0 0

0 0 0

Top Ply Middle Ply Bottom Ply

H/MPa 2289 H/MPa 802 H/MPa 4288

/MPa 40

/MPa 22 /MPa 66

/MPa 14.2

/MPa 10.1 /MPa 24

/MPa 14.2

/MPa 10.1 /MPa 24

/MPa 28

/MPa 13.2 /MPa 33

/MPa 4.8

/MPa 0.66 /MPa 3.96

/MPa 3.48

/MPa 0.28 /MPa 2.97

Table A2. Interface material model parameters.

/(MPa/mm) 776 /(MPa/mm) 2235 /(MPa/mm) 1300

/MPa 0.28

/MPa 1.36, 1.06, 1.38 /MPa 1, 0.6, 1.1

/mm 3 α 11.5

* Material model constants marked with gray background were used in Shell-Forming simulation

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