Analysis of length effect dependencies in tensile test for paperboard

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Analysis of length effect dependencies in tensile test for paperboard

Filip Claesson

Mechanical Engineering, master's level (120 credits) 2020

Luleå University of Technology

Department of Engineering Sciences and Mathematics

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ACKNOWLEDGMENTS

This master thesis in mechanical engineering at Lule˚a Technical University has been conducted in col- laboration with Tetra Pak. I would like to thank my supervisors at Tetra Pak, Eric Borgqvist, for his engagement in the project and for sharing his knowledge in material modelling and Johan Tryding, for help with the experimental setup and for sharing his large knowledge in solid mechanics. I would also like to thank my examiner at LTU, J¨orgen Kajberg, for listing to me talking about paper mechanics.

Finally, I would thank the people at Tetra Pak for doing my time there a nice experience.

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ABSTRACT

Paperboard combined with polymer and aluminium films are widely used in food packages. Paperboard is used for the bulk of the package material, and provides the stiffness. Paperboard is a highly anisotropic material, which is affected by how the fibers are orientated. Most fibers are aligned in the machine direction (MD), which is the stiffest direction, perpendicular is the cross-machine direction (CD) where fewer fibers are aligned, and the thickness direction (ZD) which is considerably weaker than in the MD and CD directions. Continuum models are used to describe the material properties to aid the design of package manufacturing processes. In continuum models there are no inherent length scale effects, and the material behaviour is the same regardless of the geometry. For paperboard there have been experimentally observed effects of the gauge length and width of tensile tests. To calibrate and develop these models it is important to observe which effect is a material property, if there is an inherent length scale, and which properties are from the boundary conditions of the experimental setup.

Creasing is a process where the length scale is considerably smaller than at the standard tensile test, where the material deforms plastically to create creasing lines to easier fold the paperboard. The failure properties from standard tensile tests are not a good predictor of failure in creasing, where the length scale is considerably smaller.

To investigate if there is an effect of the length scale, as the length gets smaller, tensile tests have been performed at different gauge lengths. The tensile tests were performed with a width of 15mm and the gauge length was varied in the range 3-100mm in MD and CD. The results from the tensile tests were, the failure strain and failure stress increased as the gauge length of the tests specimens decreased, both in MD and in CD. Initial stiffness decreased as the gauge length decrease (more notable in MD), and there was an increase in hardening at large strains with decreasing gauge length (more notable in CD).

An analytical calculation of the reduction in measured stiffness as the gauge length get smaller was performed, where the decrease in stiffness deemed to be strongly related to the out-of-plane shear modulus.

By fitting the analytical solution the experimental data the shear modulus was approximated to 60MPa.

The shear modulus has been measured for the same paperboard to 70±23MPa.

Simulations of the tensile tests at 5mm did fit the experimental data when the material model was calibrated from the tensile test at 100mm, except the increase in hardening at large strains in CD. It was noted that it was important to use the shear modulus that was inversely calculated by the analytical calculations to get the right initial slope of the simulations of the 5mm tensile tests.

Creasing simulations were performed of a test setup of the creasing procedure. The male die was lowered 0.3mm to perform the creasing, which in the tests setup do not result in failure in the material. From the simulations the stress at the bottom of the paperboard during creasing exceeded the failure stress from the tensile test performed at 100mm. The stress during creasing was biaxial, it has stresses both in MD and CD, with is different compared to the uniaxial tensile tests at 100mm. The stress from the creasing simulation in CD was at a maximum of 40MPa where the 3mm tensile tests in CD resulted in a failure stress at 39MPa. The maximum stress in the MD creasing simulation was 96MPa, where the 3mm tensile test resulted in a failure stress at 69MPa.

The properties from a long span tensile test are not good predictors of failure in creasing, where both stress state and length scale are very different. The failure stress at 3mm tensile tests in CD is close to the maximum stress from creasing simulations, and may be a good indication of failure. The 3mm tensile test in MD resulted in a considerably lower failure stress than the maximum stress in the creasing simulations, which indicates that the 3mm long tensile test is not a good predictor of failure in MD for creasing, where the length scale is even smaller.

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1. INTRODUCTION

The packing industry has a high demand for cheap, sustainable and practical material for food packaging.

Paperboard with metallic and polymer films has suited the needs in many years and is widely used, where paperboard is used for the bulk of the material, and provides the material stiffness (Borgqvist, 2016).

There are manufacturing strategies to optimize the properties of the paperboard to the needs of different containers or packaging solutions. Paperboard is a highly anisotropic material that is affected by the direction it is manufactured in. The properties are divided in the three directions as seen in Figure 1, where MD is in the machine direction, CD cross-machine direction and ZD is in the thickness direction.

The stiffness in MD can be 5 times higher than in CD, and 100 times greater than in ZD, thus creating a highly anisotropic material (Stenberg, 2002). Usually the deformation in the plane spanned by MD and CD are called in-plane and the deformation in ZD as out-of-plane. A typical stress-strain response in MD and CD can be seen in Figure 2, where it can be noted that the paperboard is stiffer in the MD direction and the failure strain is significantly higher in the CD direction.

Figure 1: The coordinate system usually used to describe paper, MD in the machine direction, CD in the cross-machine direction and ZD in the thickness direction (Borgqvist, 2016).

Other parameters affecting the stiffness of a paper material are density, humidity, strain rate, temperature, fiber length and strength (Niskanen et al., 2011; Linvill & ¨Ostlund, 2014; Bergstr¨om, Hossain, & Uesaka, 2019). A paperboard can be constructed of several plies through the thickness (ZD), creating a sandwich structure. By having higher density plies in the top and bottom layers, and lower density in the middle, bending stiffness can be kept high and material use low, utilizing the same principles as an I-beam (H. Huang, Hagman, & Nyg˚ards, 2014).

0 0.01 0.02 0.03 0.04 0.05 0.06

Normalized Displacement, u/l₀ 0

10 20 30 40 50 60 70

Normalized Force, F/A0 [MPa]

MD CD

Figure 2: Typical uniaxial tensile tests results of paperboard in MD (blue) and CD (red). A0 is the initial area, F the force, u the displacement and l0 the initial length.

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1.1 Size effects

Tensile test of paperboard has shown that the size of the test specimen has a significant impact on the results (Vojskovi´c, 2015; H¨agglund, Gradin, & Tarakameh, 2004; Hagman & Nyg˚ards, 2012; Tryding, 1996). In Figure 3, an example of how the force and strain relationship varies with different gauge lengths. Three phenomena has been observed as the gauge length of the test specimen get narrower.

• Apparent stiffness decreases

• Increase in hardening

• Tensile strength increase

As shown by H¨agglund, Gradin, and Tarakameh (2004) both analytically and with finite element analyses, the clamps that hold the paperboard in tensile tests affect the strain distribution, and thus affect the measured stiffness. This effect will be most distinct at shorter gauge lengths, and as H¨agglund et al. (2004) concludes, two paperboard specimens with the same properties, when measured can provide different stiffness if different geometries are used.

0 0.01 0.02 0.03 0.04 0.05 0.06

Strain 0

50 100 150 200 250 300

Force [N]

3mm 5mm 10mm 20mm 100mm

Figure 3: Tensile tests of a paperboard in the MD direction, with different gauge lengths and a constant width.

As shown by Hagman and Nyg˚ards (2012) the length to width ratio highly affect the failure strain at tensile tests for paperboard, both in MD and CD, where a lower length to width ratio increases the strain at failure. The same tendency can be seen in Figure 3. An elasto-plastic continuum model for paperboard (Borgqvist, Wallin, Ristinmaa, & Tryding, 2015) was used by Vojskovi´c (2015) to simulate tensile test at different length to width ratios. The simulation showed some of the same tendencies as experimental (high length to width ratio in a test specimen result in a stiffer behaviour), but did not align with the shorter gauge length tests.

To simulate the behaviour of paperboard material under manufacturing and in use, numerical models are used to describe the material. To calibrate the numerical models, experiments are necessary. When performing the experiments, it is important to know what is results from the mechanical properties of the specimen (stiffness, failure strain, et cetra), and what is an effect of the boundary condition (gauge length, clamp geometry, et cetra). Here is still more to discover in this area. Is there a length scale effect due to the experimental setup? Or is there an inherent length scale in the material, and if so, how to model the behaviour.

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1.2 Creasing

The process of forming a package involves multiple steps. Before a paperboard is folded, it is creased. This is performed to ensure that the folding become straight, and to reduce the risk of cracks in the corners when the paperboard is folded, by locally damage the material to reduce the bending stiffness. This is performed when a paperboard is placed between a rotational male and a female cylindrical dies where the male die presses the paperboard into grooves in the female die, thus creating creasing lines. Since the large deformation of paperboard is complicated to model, the creasing patterns were often designed by experts with empirical knowledge (Nagasawa et al., 2003). To study the paperboard when creasing an experimental creasing setup was created by Nyg˚ards, Just, and Tryding (2009), that compared the experimental results and results from simulations with an elastic-plastic material model. They concluded that the results from the simulations correlated well with the experimental results, and that the material properties affecting the results were the out-of-plane shear stress, out-of-plane compression and the friction between the paperboard and the laboratory setup.

A simulation of an experimental setup of the creasing process can be seen in Figure 4, where in Figure 4a is before the creasing process, Figure 4b during the creasing procedure and Figure 4c after the creasing process, where the paperboard has permanently been deformed.

(a) Before creasing procedure. (b) Creasing is performed, the male die is at its lowest position.

(c) Creasing has been performed, the paperboard is permanently deformed.

Figure 4: Simulation of an experimental setup of the creasing process. The male die (green), female die (red) and the paperboard (blue) can be seen. Where a) uncreased paperboard in the setup, b) male die at the maximum displacement and c) after the creasing procedure where permanent deformation is left in the paperboard.

1.3 Aim and limitations

The aim of this thesis is to investigate the size effect on paperboard, examine how well the current material model catches the size effects, investigate how to model size effects, and see how the failure stress can differ locally and globally when performing creasing. Furthermore, if there exists length scale effects, investigate how it can be incorporated in the current models. The main properties that will be studied are stiffness, hardening and failure stress, how they are affected by the gauge length, are there physical reasons, and how well does the simulations capture these behaviours.

In this thesis the effects of strain rate, humidity, temperature, fiber quality and the stiffness in ZD will not be investigated.

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2. THEORY

2.1 Paper mechanics

Models to predict paper and paperboard behaviour are highly useful to predict the material behaviour when creasing and folding, but trial and error is still widely used (Xia, 2002).

Continuum modelling assumes that the microscopic behaviour can be sufficiently described with macro- scopic characteristics. A paperboard consists of wood pulps, matrix, polymer, etc, there all materials have different properties. A continuum model sees the paperboard as a solid material with certain properties, which will not describe the behaviour of the different sub-components, it can therefore be seen as an average response. This can describe the material behaviour at a larger scale, but can be problematic at a small scale where the microscopic behaviour influence the material behaviour. The smallest volume element which accurately describes a material, is called a representative volume element (RVE). For an anisotropic material like paper, the RVE must include enough fibers, and fiber to fiber bonds, and thus the RVE in-plane MD component can be expected to be larger than its CD component (M¨akel¨a &

Ostlund, 2003), see Figure 5.¨

Figure 5: In plane shape of RVE (Mäkelä & Östlund, 2003).

This implies that the critical width of tensile tests in CD is larger than in MD, where below the critical width the material do not behave as a continuum. Since there are considerably fewer fibers in the thickness direction, the RVE component in ZD can be assumed to be significantly smaller than the in-plane components.

A continuum model for paperboard suggested by Borgqvist (2016) is summarised in this section. For large deformation a transformation from the undeformed body to the deformed state at a certain time.

The position in a body x, can be expressed by the original position X and time t, as x = φ(X, t). The mapping vector F, also known as deformation gradient, is defined as

F = ∇φ. (1)

The mapping vector can be expressed in a permanent part F^p for the plastic deformation and an elastic deformation part, F^e, as

F = F^eF^p. (2)

As large deformation in a highly anisotropic material is considered, there is a need to include local direction vectors that reshape with the deformation in the continuum. The local direction vectors in MD and CD, υ⁽¹⁾ and υ⁽²⁾ respectively, can be assumed to follow the elastic deformation as seen in (3). Where υ⁽¹⁾₀ and υ⁽¹⁾₀ are unit length vectors align with the reference coordinate system in the corresponding direction.

υ⁽¹⁾= F^eυ⁽¹⁾₀ υ⁽²⁾= F^eυ⁽²⁾₀

(3)

To ensure that the ZD direction vector always is perpendicular to the in-plane MD and CD directions, it can be defined as

n⁽³⁾= υ⁽¹⁾× υ⁽²⁾, (4)

(9)

where n⁽³⁾ is the current normal for the ZD direction. By defining the normal in the ZD direction as in (4) it makes certain that the out-of-plane properties always are perpendicular to the in-plane direction vectors. The stress, τ , at elastic strain can be expressed as

τ = τ (Fê, ν⁽¹⁾(Fê), ν⁽²⁾(Fê), n⁽³⁾(Fê)). (5) The stiffness at elastic strain can initially be expressed as

D = ∂τ

∂FF^T

_F=I, (6)

where the stiffness matrix D can be split into an in-plane part D^ip and an out-of-plane part D^op as

D = D^ip+ D^op, (7)

where the in-plane components can be written as seen in (8), where A1, A2, A4 and A5 are in-plane elastic parameters.

D^ip =







2A1+ 2A4+ 2A5 A5− 2A4 0 A5− 2A4 2A2+ 2A4+ 2A5 0

0 0 0

A₅ 0

0







(8)

To obtain more information about how to calibrate the elastic parameters A1, A2, A4, A5 and about the out-of-plane stiffness, cf. Borgqvist et al. (2014).

A yield surface containing 6 sub-surfaces was considered by Xia (2002), and further developed by Borgqvist et al. (2014). In Borgqvist, Wallin, Ristinmaa, and Tryding (2015) 6 new sub-surfaces were added to the model, to include the out-of-plane behaviour. The yield surface, containing 12 sub-surfaces is expressed in (9). Where n^(β)s are the normals to the sub-surfaces planes, τ the current stresses, K^(β) containing hardening parameters, χ^(ν) is a switch function that describes if the sub-surface is potentially active, τ^(ν) represent the distance to yield surface ν, and k changes the shape of the yield surfaces, where a higher k value generate sharper corners between the sub-surfaces, see Figure 6. The operator : is the scalar product of second order-tensor (Menzel, 2019).

f (τ , n^(β)_s , K^(β)) =

12

X

ν=1

χ^(ν) τ : n^ν_s τ^(ν)

!2k

− 1 (9)

The switch function χ is defined as

χ^(ν)=

( 1 if τ : n^ν_s> 0 0 otherwise.

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Figure 6 shows how a larger k generates sharper corners for the sub-surface in the MD-CD stress space.

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0 20 40 60 MD Stress

-20 -10 0 10 20 30

CD Stress ^{k = 2}

k = 3 k = 6 k = 12

Figure 6: Part of sub-surface in MD and CD stress space, with different values for k.

The yield plane normals, n^ν_s, defines the sub-surfaces, and are defined by the dyadic product (Menzel, 2019) of the director vectors and direction normals as

n^(ν)_s =

3

X

i=1 3

X

j=1

N_ij^(ν)υ⁽ⁱ⁾⊗ υ^(j), (11)

where υ⁽ⁱ⁾ represent the normed coordinate vectors, the coordinate vectors can be seen in (3) and (4), and are normed as

υ⁽¹⁾= 1

|υ⁽¹⁾|υ⁽¹⁾

υ⁽²⁾= 1

|υ⁽²⁾|υ⁽²⁾

υ⁽³⁾= 1

|n⁽³⁾|n⁽³⁾.

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N_ij^(ν)contains the coefficients of the yield normals, and are chosen such that the condition q

n^(ν)s : n^(ν)s = 1 is fulfilled. In Figure 7 a part of a yield sub-surface is shown, as well as the normals in the MD-CD stress space, to illustrate how the normals define the yield surface. Figure 8 illustrate how the normal n⁽²⁾s is defined by the components in the N_ij⁽²⁾ matrix, in the MD-CD stress space.

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0 20 40 60 MD Stress

-20 -10 0 10 20 30

CD Stress

, ₁₁

,22

Figure 7: The normals n⁽²⁾s and n⁽¹⁾s defines part of the sub-surface in the MD- CD stress space .

11

C ,

22

Figure 8: The normal n⁽²⁾s and its component of N_ij⁽²⁾.

Hardening of the yield surface can be done individually for the sub-surfaces. It is performed by the use of scaling the distance to the sub-surface τ^(ν) as

τ^(ν)= K₀^(ν)+ K^(ν), (13)

where K₀^(ν) correspond to the initial yield limit in the n^(ν)s direction. The hardening variables K^(ν) are the change in distance to the corresponding sub-surface, with is coupled to an internal variable κ^(ν)that is related to the plastic strain (Lindstr¨om, 2013). The hardening variable is used for the sub-surfaces which describe in plane tension and shear, ν ∈ 1, 2, 3, 6, and for the subsurface corresponding with ZD compression, ν = 7. The hardening variable K^(ν) is defined by

K^(ν)= (

aνln(bνκ^(ν)+ 1) if ν ∈ {1, 2, 3, 6}

a_νκ^(ν) if ν = 7 (14)

It is illustrated in Figure 9 how the yield surface changes with hardening in CD direction in MD-CD stress space.

-20 0 20 40 60 80

MD Stress -10

0 10 20 30 40 50

CD Stress

Ini al Yield surface

Yield surface a er hardening in CD

(2) (2)

Figure 9: Yield surface before and after hardening in the CD stress direction. K₀⁽²⁾ correspond with the yield limit in the n⁽²⁾s direction and K⁽²⁾ is the hardening parameter.

Associated plasticity is assumed for the evolution laws as

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d^p= λ∂f

∂τ

˙κ^(ν)= λ ∂f

∂K^(ν),

(15)

where it assumed that the plastic strain increment is normal to the yield-surface, λ is the Lagrange multiplier, which is chosen such that f = 0.

2.2 Zero span tensile test

The zeros span tensile test is a tensile test with a narrow gauge length, which is used to test maximum fiber strength of pulps (Graminski & Bonin, 1978), and for paper material and the fiber network (Gurnagul, Norayr, & Page, 1989). For paperboard consisting of softwood pulps the strength is typical 1.5 to 3 times higher when measured in zero span tensile test compared to conventional test, and usually greater increase for hardwood pulps (Ek, Gellersted, & Henriksson, 2009).

An experimental analysis of short span tensile test was performed by Gatari (2004), where Gatari shows the effect of clamping pressure on the failure strain, due to slippage. Gatari (2004) also show how the failure stress at short gauge length decreased with increased thickness for paper, due to the non-uniform stress field.

An analysis of the change in apparent elastic modulus is performed by H¨agglund, Gradin, and Tarakameh (2004) and this section is based on their work.

The geometry of the short span tensile test can be seen in Figure 10a, and a fourth of the geometry is to be analyzed which can be seen in Figure 10b. The analysis is 2D, where the width is not taken into account. In Figure 10, σ_c is the clamping pressure, t half the thickness, L the length of the clamps, s is half the gauge length, σ_tis the tension in the specimen, ε_t is the strain and u_tis the elongation.

(a) Geometry of Zero Span Tensile test.

(b) Analysed part of the setup with coordinate system.

Figure 10: Geometry of zero span tensile test: a) full setup, b) geometry to be analysed (H¨agglund, Gradin, & Tarakameh, 2004).

The apparent elastic modulus Eapp, see equation (16), is the measured elastic modulus in a tensile test, which can be compared to the effective elastic modulus Es, see equation (17). The effective elastic modulus Esshould be the measured elastic modulus if it assumed that there is no strain in the z-direction and no stress in the y-direction and it is uniform stress in the x-direction, where Exis the elastic modulus in the x-direction and

Eapp= σt

εt

(16) νzxand νxz are the Poisson’s ratio in the corresponding directions.

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Es= Ex

1 − ν_zxν_xz (17)

The paperboard is assumed to be a continuum, where the stress field depends on the x and y coordinates.

An orthotropic material model is assumed as seen in equation (18)

σx= C11εx+ C12εy

σy= C12εx+ C22εy

τ_xy= C₆₆γ_xy,

(18)

where C₁₁, C₁₂, C₂₂and C₆₆are elastic stiffness constants, σ_xand σ_yare the stresses in the corresponding normal direction, τ_xy is the shear stress in the xy-plane, ε_x and ε_y are the strains in the corresponding direction and γ_xy is the shear strain. The inverse form can be seen in equation (19)

εx= S11σx+ S12σy

ε_y = S₁₂σ_x+ S₂₂σ_y γxy= S66τxy,

(19)

where S11, S12 and S22 are flexibility constants. Boundary conditions for the displacement field can be obtain from Figure 10b. This gives the boundary conditions

uy= 0, 0 ≤ x ≤ L + s, y = 0 (20a)

u_x= ε_ts, x = L + s, 0 < y ≤ t (20b)

ux= 0, 0 ≤ x ≤ L, y = t. (20c)

Boundary condition (20a) states that in the middle of the paperboard (y = 0), there is no displacement in the y-direction due to symmetry. Boundary condition (20b) states that the end of specimen (x = L + s) in Figure 10b, the displacement in the x-direction is uniform across the cross-section area, with the strain ε_t. Boundary condition (20c) states that at top of the paperboard (y=t) there are no displacement in the x-direction where the specimen has contact with the clamp (0 ≤ x ≤ L), thus no slip occur.

A displacement field that fulfil the boundary conditions can be seen in (21), where ux and uy are the displacement in the corresponding direction in the interval 0 ≤ x ≤ L, the function f (x) and the strain εy0 needs to be determined (H¨agglund, Gradin, & Tarakameh, 2004).

ux= f (x)(t²− y²)

u_y= ε_y0y (21)

Rewriting the displacement field to a strain field, using the definition εx = ∂ux/∂x, εy = ∂uy/∂y and γxy= ∂ux/∂y + ∂uy/∂x provides

εx= f⁰(x)(t²− y²) ε_y = ε_y0

γxy = −2yf (x).

(22)

The potential energy U can be expressed as seen in (23), where the first integral is the internal elastic energy in the clamped area, the second integral is the strain energy in the interval L ≤ x ≤ L + s and the last term is the potential energy from the external forces.

U =1 2

L

Z

0 t

Z

0

(σxεx+ σyεy+ τxyγxy)dxdy +sEs

2

t

Z

0

(εt− ux(L, y)/s)²dy − σtεtts (23)

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The best approximation of the displacement field f (x) can be obtain where it minimises the potential energy, δU = 0. By combining equation (20), (22) and (23), an expression for f (x) is obtained, see (24) (H¨agglund, Gradin, & Tarakameh, 2004)

f (x) = Acosh(λx) + Bsinh(λx), (24)

where A and B are integration constants and λ is an eigenvalue, see Appendix A. Also four natural boundary conditions appears, see (25) (H¨agglund, Gradin, & Tarakameh, 2004)

C₁₂I₂(f (L) − f (0)) + C₂₂tLε_y0 = 0 C12I2εy0C11I1f⁰(0) = 0

C₁₂T₂ε_y0C₁₁I₁f⁰(L)E_sI₂ε_t+ E_sI₁f (L)/s = 0 Esstεt− EsI2f (L) = σtst,

(25)

where the definition of I1 and I2can be found in (26) and (27) respectively.

I1=

t

Z

0

(t²− y²)²dy = 8t⁵

15 (26)

I2=

t

Z

0

(t²− y²)dy = 2t³

3 (27)

Note that the term I1 in H¨agglund, Gradin, and Tarakameh (2004) Appendix A1 is written as I1 = Rt

0(t²− y²)²dy = 8t²/15, whereas here it has been changed to the expression in (26).

By inserting (24), (26) and (27) in (25), it can be written in matrix form as

" cosh(λL)−1 sinh(λL) C₂₂tL/C12I₂ 0

0 C₁₁I₁λ C₁₂I₂ 0

(C₁₁λsinh(λL)+E_scosh(λL))/s (C₁₁λcosh(λL)+E_ssinh(λL))/s C₁₂I₂/I₁ −E_sI₂/I₁

cosh(λL) sinh(λL) 0 −st/I2

#"

A B ε_yo

εt

#

=

" ₀

0 0

−σ_tst/E_sI₂

# . (28) If the elastic material parameters and the material Poisson ratios are known, and a value for σtis assumed, the equation system in (28) can be solved. By solving the equation system for different gauge lengths s, the apparent elastic modulus (16), can be compared with the effective elastic modulus (17), an example of this can be seen in Figure 11.

0 10 20 30 40 50 60 70 80 90 100

Normalised gauge length s/t 0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalised apparent E-modulus Eapp/Es

Figure 11: Change of measured elastic modulus at different gauge length, calculated by solving (28) at different gauge lengths s.

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As seen in the figure, the measured elastic modulus decreases with reducing gauge length according to the assumption in this model. As the gauge length approaching zero, the apparent elastic modulus approaches Es1/6 (H¨agglund, Gradin, & Tarakameh, 2004). See Appendix A for the full orthotropic material model and eigenvalues of the solution.

2.3 Gradient Plasticity theory

Continuum modelling of plasticity has no apparent length scale effects. For metals, there have been shown to be a length scale dependency when considering very small length scales. As shown by Fleck, Muller, Ashby, and Hutchinson (1994) the effect of the diameter on thin cooper wires in the range of 12 − 170µm is affecting the plastic response. There was some effect of the length scale in uniaxial tensile tests (negligible by the authors, perhaps due to grain size difference), but a significant distinct dependency of the length scale on hardening behaviour in torsion of the copper wires. This can be modeled by letting the hardening depends on both the plastic strain and the plastic strain gradient. For tension tests, the displacement field is relatively homogeneous, and thus the effect of strain gradient is negligible, but for torsion, a smaller diameter creates a larger strain gradient, thus effecting the material hardening a lot, which correlate to the experimental results by Fleck, Muller, Ashby, and Hutchinson (1994).

Gradient plasticity can also be of interest for aluminium-foil, which is a part of packaging material- structure. A test method for characterising the strain gradient effects in thin foil material was developed by St¨olken and Evans (1998) which utilised a micro bending test. Two length scales, one for stretch and one for rotation were calculated for plasticity in Nickel foil, where the range for the length parameters were in the range 3 − 5µm, which was dominant by the rotational gradient. The test method was developed to determine a rotational plastic length scale and St¨olken and Evans (1998) concludes that the test method can be used for thin foil material that is ductile enough to be rolled over a mandril with a small radius.

Three strain gradient plasticity theories were implemented in Abaqus by Heggelund (2015) to evaluate the different length scale implementing methods. Two suggested way to define plastic gradient, η, by Y. Huang et al. (2004) can be seen in (29) and (30),

η^p= r1

4η^p_ij,kη_ij,k^p , where η^p_ij,k= ε^p_ki,j+ ε^p_jk,i− ε^p_ij,k (29) η^p=p

(ε^p× 5) : (ε^p× 5)) = || 5 ×ε^p|| (30) where εij is the plastic strain tensor, and the plastic strain gradient notation is defined as

ε^p_ij,k= ∂ε^p_ij

∂xk

. (31)

The third definition of the strain gradient implemented by Heggelund (2015) is defined as

η^p=r ∂ε^p

∂x_i

∂ε^p

∂x_i = || 5 ε^p||, (32)

which has the advantage that it only depends on the plastic strain, and not on the plastic strain gradient tensor, ε_ij,k, making the implementation simpler. The strain gradient was used to model the hardening behaviour as

σf low = σy

pf (ε^p)²+ lη^p where f (ε^p) =

1 +Eε^p σ_y

^N

, (33)

where σ_y is the yield stress, σ_{f low} is the stress after hardening from elastic strain and strain gradient, E the elastic modulus and N is a dimensionless constant.

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2.4 Statistical analysis of failure

To analyse failure and fatigue in a material, statistical models are needed to predict when a specimen most likely is to break and how many specimens will most likely sustain a certain load without failure. A commonly used distribution is the Weibull distribution, which is used in failure analyses and can also be implemented in many more areas where a predictable model of a distribution is needed (Weibull, 1951;

Scholz, 1999). The cumulative function F (x), see in equation (34), describes the subset of the distribution that fulfilled < x, divided with the full set. Where β is the shape parameter

F (x) = 1 − e^−(x/α)^β (34)

and α is the scale parameter. The probability density function f (x), see equation (35), is defined as the derivative of the cumulative function, f (x) = ^{dF (x)}_dx .

f (x) = β α

x α

(β−1)

e^−(x/α)^β, 0 if x < 0 (35)

To calculate the possibility that a value in a distribution is between x0 and x1, integrate the probability density function, Rx₁

x₀ f (x)dx = F (x1) − F (x0). A Weibull plot can be used to determine if a set of data is from a Weibull distribution (Park, 2017). Let p = F (xp) (Wilk & Gnanadesikan, 1968; MathWorks, 2019), rewrite the expression and take the logarithmic of the cumulative function two times giving

log − (1 − p) = β log(xp) − β log(α), (36)

which when plotted (logarithmic y-axis) creates a straight line with the slope of β. If the data fits a Weibull distribution it will correlate to the line, see Figure 58a, if the data do not match up with the line, see Figure 58b, the data is not from a Weibull distribution.

10^-1 10⁰

Data 0.02

0.05 0.10 0.25 0.50 0.75 0.90 0.96

Probability

Weibull Probability Plot

(a) Weibull probability plot of data from a Weibull distribution.

10³ 10⁴ 10⁵ 10⁶

Data 0.02

0.05 0.10 0.25 0.50 0.75 0.90 0.96

Probability

Weibull Probability Plot

(b) Weibull probability plot of random data.

Figure 12: Weibull probability plots: a) the data fits a Weibull distribution, b) the data do not fit a Weibull distribution.

It was shown in Roberts, Andersons, and Sparnins (2009) that tensile strength of cellulose fibers at a fixed length followed a two-parameter Weibull distribution. It was also shown in Sia, Nakai, Shiozawa, and Ohtani (2014) that failure strength in oil palm fiber fit a two-parameter Weibull distribution at a certain fiber length, but it did not accurately predict the length effect on tensile strength.

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3. EXPERIMENTS

3.1 Experimental setup

To investigate the size effects on paperboard tensile test with different lengths, thickness, density, man- ufacturer and direction were performed. Clamps were used to hold the specimen in place, see Figure 13, which were controlled with pneumatic, where a maximum pressure was set to 5 bar. The specimens consist of strips, cut out with constant widths of 15mm, but with varying gauge lengths and thicknesses (see ISO:1924-3 for standard uniaxial tensile test for paperboard). A load cell of maximum 10kN was used, and the system and load cell were considered significantly stiffer than the specimens and assumed not to significantly influence the experimental results. All specimens were pre-conditioned in an oven for 30 minutes at 60^◦C, and then stored in 23^◦C and 50% humidity at least 24 hours before the tests were performed. This was done to ensure that all the specimens have the same water concentration and water concentration history. All tests were performed in 23^◦C temperature and 50% relative humidity.

Figure 13: Upper clamp at tensile test. Half is straight and other part has a cylindrical shape (Andersson & Hedberg, 2018).

For all different sets of parameters, 20 tests were performed with the same setup, to get accurate statistic data. To remove the effects of strain rate in the comparison, all test were performed with the same strain rate of 0.01s⁻¹. Four different paperboards were tested, two single-ply and two with multiple plies. The experiments were performed in both the MD and CD direction at different gauge lengths with a constant width of 15mm. An experimental matrix can be seen in Appendix B.

3.2 Analysis of experimental data

The experimental data were analysed with MATLAB 2017b. The outputs of the data sampler were time, displacement and force. The quantities of interests were stiffness, maximum stress, failure strain, yield stress and the distribution of these quantities at different gauge lengths.

The strain ε, was defined as δ/l0 where l0is the initial gauge length and δ is the displacement. A defined way to calculate the stiffness of a specimen was needed, and doing it the same way for all specimens was important to ensure that a comparison of stiffness can be done correctly. This was achieved by fitting the experimental force-strain curves to splines, and define the stiffness as the maximum slope of the curves divided with the width of the specimen, using Shape Language Modeling tool by D’Errico (2020) implemented in MATLAB.

The yield stress was calculated by assuming that the stress is elastic when the ratio ε −_E^σ < x is fulfilled, where ε is the current strain, σ the current stress, E the elastic modulus and x is a constant that was set to 0.002, as seen in Figure 14.

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0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Strain

0 5 10 15 20

Stress F/A[MPa]

E

Approximated yield stress

Figure 14: Approximation of yield stress.

The stress was defined as σ = _wt^F where F is the measured force, w the initial width of the specimen, t the initial thickness. The Elastic modulus is calculated as E = ^k_t where k is the stiffness, note that the stiffness k has the unit N/m and do not take the thickness into account. The failure strain was defined as the strain when the force was at the peak (this is also the definition of failure strain in ISO:1924-3).

After the peak force there was damage in the material and it loses most of its load-bearing capabilities, and local deformation where the paperboard is damaged erupts, the strain after this point can not be measured as ε = δ/l0. These quantities were calculated for all the specimens, and the distributions were calculated with a Weibull distribution, as described in Section 2.4.

3.3 Experimental Results

In this section a selected part of the experimental results are presented, all the experimental results can be seen in Appendix C. All the stress-strain curves that are presented are averages of 20 experiments to get a better overview. In Figure 15 stress-strain curves can be seen in MD and CD for different gauge lengths for a single-ply paperboard with a thickness of 0.26mm.

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0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Strain

0 10 20 30 40 50 60 70 80

Stress [MPa]

100mm 30mm 20mm 15mm 10mm 7mm 5mm 4mm 3mm

(a) Stress-strain curves in MD for different gauge lengths.

0 0.02 0.04 0.06 0.08 0.1 0.12

Strain 0

5 10 15 20 25 30 35 40

Stress [MPa]

100mm 20mm 10mm 5mm 3mm

(b) Stress-strain curves in CD for different gauge lengths.

Figure 15: Experimental stress-strain curves for a single-ply paperboard with the thickness 0.26mm and width 15mm. Experiments in a) are tensile test in the MD-direction and b) are in the CD- direction where all curves are averages of 20 experiments with the same setup.

From the figure some behaviours can be noted, an increase in failure stress as the gauge length decrease, an increase in the plastic hardening with decreasing gauge length (more noticeable in CD) and change in stiffness (more noticeable in MD). The yield stress is calculated for the data in Figure 15 and presented in Figure 16 with the standard deviation plotted for the different gauge lengths.

0 10 20 30 40 50 60 70 80 90 100

Gauge length[mm]

0 5 10 15 20 25 30

R0.2% [MPa]

(a) Yield stress in MD, with standard deviation error bars.

0 10 20 30 40 50 60 70 80 90 100

Gauge length[mm]

0 2 4 6 8 10 12 14

R0.2% [MPa]

(b) Yield stress in CD, with standard deviation error bars.

Figure 16: Calculated yield stress with standard deviation error bars for different gauge lengths in a) MD and b) CD.

As seen in the figure the yield stress seems constant if measured at different gauge length, except for one outlier in the MD direction with a noticeable larger standard deviation. The failure stress in Figure 15 is presented in Figure 17 with error bars to illustrate the spread at different gauge lengths.

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0 10 20 30 40 50 60 70 80 90 100 Gauge length[mm]

0 10 20 30 40 50 60 70 80

Failure Stress [MPa]

(a) Failure stress in MD, with standard deviation error bars.

0 10 20 30 40 50 60 70 80 90 100

Gauge length[mm]

0 5 10 15 20 25 30 35 40 45

Failure stress [MPa]

(b) Failure stress in CD, with standard deviation error bars.

Figure 17: Failure stress with standard deviation error bars for different gauge lengths in a) MD and b) CD.

As seen in the figure the failure stress increases as the gauge length decreases, but this effect appears only for gauge lengths shorten than around 10mm. It can also be noted from the figure that the standard deviation increases as the gauge lengths gets smaller. An interesting observation is that the increase in failure strength in CD is larger than in MD in relative terms, but in absolute terms it seems to be a very similar increase in failure strength as the gauge length decreases. Also, the spread of the failure strength increases as the gauge length decreases (more notable in CD). This can be a result of the inhomogeneity of the paperboard, where there are higher and lower density zones in the paperboard (discussed in Hagman

& Nyg˚ards 2012). The shorter span tensile test will be more affected by the inhomogeneity of the material and a greater spread in failure stress can result from that.

Failure strain can be seen in Figure 18 for the same paperboard.

0 10 20 30 40 50 60 70 80 90 100 110

Gauge length[mm]

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

Failure Strain

(a) Failure strain in MD, with standard deviation error bars.

0 10 20 30 40 50 60 70 80 90 100 110

Gauge length[mm]

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

Failure Strain

(b) Failure strain in CD, with standard deviation error bars.

Figure 18: Failure strain with standard deviation error bars for different gauge lengths in a) MD and b) CD.

The failure strain in both MD and CD increases as the gauge length decrease. In Hagman and Nyg˚ards (2012) it was shown that failure strain drastically increased when the length to width ratio reached values below unity. The same trends can be noted in Figure 18, where the failure strain increases when the gauge length decreases below 15mm.

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In Figure 19 stress-strain curves can be seen from experiments with a single-ply paperboard in the MD direction with a thickness of 0.39mm. The stiffness at different gauge length can be seen in Figure 20.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Strain

0 10 20 30 40 50 60 70

Stress [MPa]

100mm 30mm 20mm 10mm 5mm 3mm

Figure 19: Stress-strain curves for a single-ply paperboard in MD for different gauge lengths with a thickness of 0.39mm .

0 10 20 30 40 50 60 70 80 90 100

Gauge length[mm]

0 500 1000 1500 2000 2500 3000

Stiffness [N/m]

Figure 20: Calculated stiffness and standard deviation for the experimental data shown in Figure 19.

A notable difference can be seen in the figures, where the initial stiffness in Figure 19 is more affected by the gauge length, compared to the thinner paperboard in Figure 16a. As seen in Figure 20 the stiffness measured with a 5mm gauge length is around half the stiffness measured with a gauge length of 100mm.

Failure stress and strain can be seen in Figure 21 and 22 respectively, for the same paperboard.

0 10 20 30 40 50 60 70 80 90 100 110

Gauge length[mm]

0 10 20 30 40 50 60 70

Figure 21: Failure stress for single-ply paperboard with standard deviation error bars at different gauge lengths and a constant width of 15mm in MD.

0 10 20 30 40 50 60 70 80 90 100 110

Gauge length[mm]

0 0.01 0.02 0.03 0.04 0.05 0.06

Failure Strain

Figure 22: Failure strain for single-ply paperboard with standard deviation error bars at different gauge lengths and a constant width of 15mm in MD.

In the figures it can be noted the same behaviour as before, where the failure strain and stress increases as the gauge length decreases, which is more notable when the length to width ratio is below 1, both for stresses and strains. It can also be noted that the deviation increases as the gauge length get smaller.

This can be an effect of local defects in the paperboard, where a shorter span tensile test is more affected if there are any local defects, whereas a long specimen will mostly include some local defects, and the material properties will be averaged out.

The stress-strain response from a paperboard with a thickness of 0.4mm and three plies can be seen in Figure 23 for CD at different gauge lengths.

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0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Strain

0 10 20 30 40 50 60 70

Stress [MPa]

(a) Stress-strain curves in the MD for different gauge lengths.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Strain

0 5 10 15 20 25 30 35 40 45

Stress [MPa]

100mm 10mm 5mm

(b) Stress-strain curves in the CD for different gauge lengths.

Figure 23: Stress-strain curves in a) MD and b) CD for a paperboard with three plies and a thickness of 0.4mm.

It can be seen from the figures that the paperboard with three plies follows the same trend as shown before, with the initial stiffness decrease more noticeable in MD as the gauge length decreases. And the increase in failure stress, both in MD and CD. It can be noted that in CD there is an increase in hardening before failure, which is more notable for the shorter gauge lengths. This could be an effect of reorientation of fibers at large strains, but in Dunn (2000, p.228) observations at a microscopic level showed no observation of change in fiber orientations before failure. In Nyg˚ards and Bonnaud (2010) it is argued that the effect of the fibers increased in hardening (fibers with high fibril angle) after approximately 5% strain can cause an increase in hardening in CD. Since the failure strain increases as the gauge length decreases the effect of increase hardening at large strains for the fibers will affect the short span tensile test, and increase the hardening behaviour for shorter gauge lengths.

The stiffness can be seen in 24 for different gauge lengths.

0 10 20 30 40 50 60 70 80 90 100 110

Gauge length[mm]

0 500 1000 1500 2000 2500 3000

Stiffness [N/m]

(a) Stiffness in MD for different gauge lengths.

0 10 20 30 40 50 60 70 80 90 100 110

Gauge length[mm]

0 100 200 300 400 500 600 700 800 900 1000 1100

Stiffness [N/m]

(b) Stiffness in CD for different gauge lengths.

Figure 24: Stiffness in a) MD and b) CD at different gauge lengths for a paperboard with three plies and a thickness of 0.4mm.

The failure strain can be seen in Figure 25

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0 10 20 30 40 50 60 70 80 90 100 110 Gauge length[mm]

0 0.01 0.02 0.03 0.04 0.05 0.06

Failure Strain

(a) Failure strain in MD for different gauge lengths.

0 10 20 30 40 50 60 70 80 90 100 110

Gauge length[mm]

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Failure Strain

(b) Failure strain in CD for different gauge lengths.

Figure 25: Failure strain in a) MD and b) CD at different gauge lengths for a paperboard with three plies and a thickness of 0.4mm.

As seen in the figure, there is an increase in failure strain as the gauge length decreases. As discussed both in H¨agglund et al. (2004) and Gatari (2004) the influence of slippage between the clamp and the paperboard gets larger as the gauge lengths decrease, and experiments with thicker paperboard has a higher risk of involving slippage, due to the increased tensile force. There can be some slippage between the clamps and paperboard, resulting in some of the increase of failure strain as the gauge decreases. If there is slippage, that should be more notably in MD because of the higher tensile force, in the same manner as the increase of slippage with a thicker paperboard. However, there are no indications (plateau in strain-stress) of slippage in the strain-stress curves in Figure 23.

The failure stress can be seen in 26 at different gauge lengths.

0 10 20 30 40 50 60 70 80 90 100 110

Gauge length [mm]

0 10 20 30 40 50 60 70

(a) Failure stress in MD for different gauge lengths.

0 10 20 30 40 50 60 70 80 90 100 110

Gauge length[mm]

0 5 10 15 20 25 30 35 40 45 50

(b) Failure stress in CD for different gauge lengths.

Figure 26: Failure stress in a) MD and b) CD at different gauge lengths for a paperboard with three plies and a thickness of 0.4mm.

The stress-strain response can be seen for a paperboard with three plies and a thickness of 0.49mm in Figure 27 for MD and Figure 27b for CD.

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0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Strain

0 10 20 30 40 50 60 70

Stress [MPa]

(a) Stress-strain curves in the MD direction at different gauge lengths.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Strain

0 5 10 15 20 25 30 35 40 45

Stress [MPa]

100mm 10mm 5mm

(b) Stress-strain curves in the CD direction at different gauge lengths.

The stiffness at different gauge lengths can be seen in Figure 29

0 10 20 30 40 50 60 70 80 90 100 110

Gauge length[mm]

0 500 1000 1500 2000 2500 3000 3500

Stiffness [N/m]

0 10 20 30 40 50 60 70 80 90 100 110

Gauge length[mm]

0 200 400 600 800 1000 1200

Stiffness [N/m]

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0 10 20 30 40 50 60 70 80 90 100 110 Gauge length[mm]

0 500 1000 1500 2000 2500 3000 3500

Stiffness [N/m]

0 10 20 30 40 50 60 70 80 90 100 110

Gauge length[mm]

0 200 400 600 800 1000 1200

Stiffness [N/m]

Failure stress at different gauge lengths can be seen in Figure 30.

0 10 20 30 40 50 60 70 80 90 100 110

Gauge length[mm]

0 10 20 30 40 50 60 70

(a) Failure stress in MD for different gauge lengths.

0 10 20 30 40 50 60 70 80 90 100 110

Gauge length[mm]

0 5 10 15 20 25 30 35 40 45

(b) Failure stress in CD for different gauge lengths.

Figure 30: Failure stress in a) MD and b) CD for a paperboard with three plies and a thickness of 0.49mm.

Failure stress at different gauge lengths can be seen in Figure 31.

Analysis of length effect dependencies in tensile test for paperboard