Keywords

1 Introduction

As a material that can be used for forming processes, paperboard has very special properties. It is the material with the highest added value within the paper and paperboard system and is most frequently used for packaging and high-quality printed products.

Paperboard is a fibrous material that can be designed by a targeted fiber orientation during production, resulting in varying degrees of anisotropic behavior. Fibers are mainly dispersed in the in-plane direction, and the degree of anisotropy between the machine direction (MD) and the cross-machine direction (CD) is usually from 1 to 5, while it may differ by 50–100 between the through-thickness direction (ZD) and MD [1]. The paper, moreover, has an inhomogeneous structure due to the formation effect. This inhomogeneity is caused, on the one hand, by material variations in the properties of the fibers created in natural processes or by differences in the amount of added fillers and additives. On the other hand, it is due to mechanical effects such as differences in the sheet formation due to filtration and thickening during the production of the pulp in the manufacturing process, as well as to the different tensile forces when passing through the paper machine. The effects of inhomogeneity become noticeable mainly in the form of unstable mechanical properties in the longitudinal and transverse directions or differences in the thickness and density of the material, as well as in surface roughness [2].

Since paper is by nature a heterogeneous material, it exhibits diffuse unstable strain under uniaxial tension, which can be well observed by the Digital Image Correlation (DIC) method. The DIC method only requires a simple experimental setup and has a wide range of measurement sensitivity and resolution, and it provides local strain information by comparing before and after images during the test [3]. In a previous work by Hagman and Nygards [4] speckle analysis was used to observe the localization in the paper when it is plastically stretched. However, a few studies have reproduced the non-uniform strain distribution in numerical simulations.

The Finite Element Method (FEM) is a widely used numerical method for solving scientific and engineering problems. It can be applied to complex geometries and boundary conditions, as well as to time-dependent problems and non-linear material behavior. However, FEM is inherently deterministic and therefore limited to describing the characteristics of a system under prescribed model assumptions. In particular, it cannot directly and reliably represent a system with some degree of uncertainty. The combination of microstructure models with finite element simulations is a well-established approach for investigating the relationship between microstructural geometry and the macroscopic properties of the material. An established way to deal with fiber networks is the micromechanical approach, which can include fiber properties, bonding between fibers, and fiber arrangement in 2D or 3D [5]. However, the complexity of micromechanical models, the difficulty of characterizing fiber properties at the microscopic level, and the excessive measurement and computational costs limit the scope of application of direct simulation of fiber networks [6].

For product and process development, a modeling tool that captures the material behavior with low effort is extremely important. Therefore, the continuum modeling approach with high computational efficiency is widely used to predict mechanical response. When using this approach, it is usually assumed that the materials are uniform, and local inhomogeneities are neglected. However, the stochastic finite element method (SFEM) is an extension of the FEM to include random parameters. The SFEM can represent randomness in one or more of the main components of classical FEM, namely geometry, external forces, and material properties [7]. Despite the inhomogeneity of properties between different regions of the actual material, most finite element analyses of paperboard forming processes use the assumption of an ideal and homogeneous continuum model. An inhomogeneous material model can be built at the microscopic level using various methods, such as cellular automata, Monte Carlo, Voronoi polygons, and crystal plasticity finite element methods [8]. At the macroscopic level, the conventional finite element software also provides the possibility to assume parameter distributions.

For paperboard, inhomogeneity is mainly reflected in three aspects: distribution of thickness, density, and fiber orientation. These three local paper structures can be related to local strains and local material failure using regression models for unbleached cork kraft paper with low grammage according to Lahti et al. [9]. Since all studies (e.g., [4, 9]) are based solely on experimental methods, the results are valid only for the material studied, and the degree of influence of these three major factors is not investigated. In this work, the effect of the aforementioned distributions on the material properties is investigated using numerical methods with the uniaxial tensile test, and the simulation results are validated with experimental data based on the DIC method. A sensitivity study of the influence is also conducted to find out which parameter of inhomogeneity has the greatest influence on the mechanical properties of paperboard.

2 Material and Methods

2.1 Material

The material used in this work is made of 100 % recycled fibers and is usually utilized as printing paperboard. The thickness is 0.3 mm and the grammage, defined as mass per unit area, is 250 \({\text {g/mm}}^2\). It is in the transition zone between paper and paperboard materials and has typical characteristics of paper material, i.e., better tensile strength in the MD direction and better ductility in the CD direction.

2.2 Experimental Tensile Tests

A Zwick/Roell Z100 material testing machine with a video extensometer was used to perform the tensile tests. As for the DIC system, GOM Aramis 5M, an optical system for 3D deformation analysis, provided the information on the strain distribution during the uniaxial tensile test, and Zeiss Quality Suite software was used to analyze the captured images. All the experiments were performed under laboratory conditions (50 % relative humidity and 23 \(^\circ \)C).

The specimen geometry is a simple square with a gripping length of 90 mm and width of 30 mm, and the test speed is 20 mm/min. The tensile specimens are first sprayed with black random spots and then clamped on the tensile test machine (see Fig. 1). From the strain analysis of the example given, it is clear that the distribution is inhomogeneous, which is caused by the inherent inhomogeneity of the material.

Fig. 1
figure 1

Tensile specimens sprayed with black random spots for strain measurement and strain distribution during experiment due to inhomogeneity of paperboard

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2.3 Numerical Simulation

The numerical simulation for the uniaxial tensile test was conducted by ANSYS 2021. As aforementioned, the inhomogeneity of paperboard is mainly due to the disorderly arrangement of the fibers, which macroscopically leads to a local thickness and density heterogeneity of the paperboard. Since the paperboard has strong anisotropy, directional inhomogeneity is also an important factor to be considered in the simulation. These three factors are first taken into consideration separately to study the influence on the mechanical properties of two different paper materials. Then, the simulation including all three factors (label: all inhomo) is also performed to compare the force response and mechanical properties. At the same time, the homogeneous material model (label: homo), which has the average value of the inhomogeneous material model, is always used as a reference.

3 Model, Results, and Discussion

3.1 Material Modeling and Validation

In this work, an elastic-plastic orthotropic model with Hill’s yield criterion [10] is used as the material model. The elastic model is defined by three elastic moduli \(E_1\), \(E_2\), and \(E_3\); three Poisson’s ratios \(\nu _{12}\), \(\nu _{13}\), and \(\nu _{23}\); and three shear moduli \(G_{12}\), \(G_{13}\), and \(G_{23}\). It is assumed that Poisson’s ratios in thickness direction are zero, i.e.,

$$\begin{aligned} \nu _{13}=\nu _{23}=0 \end{aligned}$$
(1)

and

$$\begin{aligned} G_{xz}=G_{yz}=G_{xy} \end{aligned}$$
(2)

Then the required parameters of the orthotropic material model are reduced to

$$\begin{aligned} \begin{bmatrix} \epsilon _{xx} \\ \epsilon _{yy} \\ \epsilon _{xy} \end{bmatrix} = \begin{bmatrix} \frac{1}{E_{x}} &{} \frac{-\nu _{yx}}{E_{y}} &{} 0 \\ -\frac{\nu _{xy}}{E_{x}} &{} \frac{1}{E_{y}} &{} 0 \\ 0 &{} 0 &{} \frac{1}{2G_{xy}} \end{bmatrix} \begin{bmatrix} \sigma _{xx} \\ \sigma _{yy} \\ \sigma _{xy} \end{bmatrix} \end{aligned}$$
(3)

The in-plane shear modulus \(G_{xy}\) is given by the empirical expression [11]:

$$\begin{aligned} G_{xy}= \frac{1}{\frac{4}{E_{45}}-\frac{1}{E_{x}}+2\frac{\nu _{xy}}{E_{x}}-\frac{1}{E_{y}}} \end{aligned}$$
(4)

and Poisson’s ratio \(\nu _{xy}\) is then calculated by the following equation: [12]:

$$\begin{aligned} \nu _{xy}= 0.293\sqrt{\frac{E_{x}}{E_{y}}} \end{aligned}$$
(5)

For the plastic part, Hill’s quadratic yield criterion is used. Since only the in-plane properties of the paperboard are discussed here, the stress value related to the thickness direction is considered to be 0, i.e., \(\sigma _{i3}=0\). For the material model in this work, the y-axis is chosen as the reference direction, i.e., \(R_{yy}\) = 1, so 2D-Hill’s criterion is given by

$$\begin{aligned} f(\sigma ,\sigma _{y}) = \sqrt{\sigma ^{T}M\sigma } -\sigma _{y}=0 \end{aligned}$$
(6)

where \({\textbf {M}}\) is Hill’s orthotropic coefficient matrix given by

$$\begin{aligned} {\textbf {M}} = \begin{bmatrix} \frac{1}{R_{xx}^{2}} &{} -\frac{1}{2R_{xx}^{2}} &{} 0 \\ -\frac{1}{2R_{xx}^{2}} &{} 1 &{} 0 \\ 0 &{} 0 &{} \frac{3}{R_{xy}^{2}} \end{bmatrix} \end{aligned}$$
(7)

and \(\sigma _{y}\) is the yield stress that can, in general, evolve as a function of some material internal variables.

$$\begin{aligned} \sigma _{y} = a(1-e^{-b\hat{\epsilon }_{pl}})+c\hat{\epsilon }_{pl}^{1/d}+\sigma _{0} \end{aligned}$$
(8)

where a, b, c, and d are plastic parameter constants; \(\sigma _{0}\) is the initial yield stress; and \(\hat{\epsilon }_{pl}\) is the equivalent plastic strain. Note that Eq. (8) is fitted to the response in the y-direction since \(R_{yy}\) = 1, the shear parameter in the planer direction is determined with the behavior in the 45\(^\circ \) tensile direction, and the plastic behavior in the thickness direction is assumed to have the same value as in the y-direction \(R_{zz}\) = 1.

Tensile tests at MD, 45\(^\circ \), and CD are used to calculate Hill’s parameters in order to establish the material model, and experiments at 22.5\(^\circ \) and 67.5\(^\circ \) are used for validation. The following material parameters were obtained for the material from the reverse engineering method, as shown in Table 1. The comparison of the experimental and simulation data shows that the material model has a good agreement and can represent the properties very well (see Fig. 2).

Table 1 Material constants and evaluated values
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Fig. 2
figure 2

Validation of the numerical material model (label: SIM) with the experimental data (label: EXP)

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3.2 Analysis of Simulation Results

In the simulation, the three distributions (density, fiber orientation, and thickness) shown in Fig. 3 for MD and CD are assigned to the sample of size \(90\times 30\, {\text {mm}}^2\). The resolution is four elements per \(mm^2\), which means that the sample consists of 10800 elements in total. According to the requirements, each element is assigned a different value of density, orientation, thickness, or both of them. For a better comparison of the degree of influence, they all follow a Weibull distribution with a positive deviation of 20 %, and the mean values (density: 1000 \({\text {kg/m}}^{-3}\); fiber orientation: 90\(^\circ \) for MD and 0\(^\circ \) for CD; thickness: 0.3 mm) are used in the simulation of the homogeneous material model as reference.

Fig. 3
figure 3

Three distributions and the mapping on the a) MD and b) CD tensile samples: a Density [\({\text {kg/m}}^{-3}\)]; b Fiber orientation [\(^\circ \)]; c Thickness [mm]

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3.2.1 Force Response Due to Structural Parameters

Figure 4 shows the force response of the three structural distributions in terms of tensile test in MD and CD, respectively. It is shown that the influences of thickness and density inhomogeneities are in both tensile directions smaller than in the homogeneous material used as a reference for the same displacement. The offset of the curve due to these two factors is small, especially for density, where the effect is almost unnoticeable. This is because, in the simulation, density only affects the gravity of the sample, which can be negligible for the tensile test (see Fig. 5 and derived equation below).

The total force in the z-direction \(F_z\) is the sum of the tensile force F and the gravity of the specimen.

$$\begin{aligned} F_z=F+m\cdot g=F+\rho \cdot V\cdot g \end{aligned}$$
(9)

where the volume V is the multiplication of length L, width w, and thickness t.

$$\begin{aligned} V=L\cdot w \cdot t = L\cdot A \end{aligned}$$
(10)

where A is the cross-section area, and the stress \(\sigma \) is

$$\begin{aligned} \sigma =\frac{F_z}{A}=\frac{F+\rho \cdot V\cdot g}{A}=\frac{F+\rho \cdot L\cdot w\cdot t\cdot g}{w\cdot t}=\frac{F}{w\cdot t}+\rho \cdot L\cdot g \end{aligned}$$
(11)

Since F is at least two orders of magnitude larger than gravity, i.e., \(\rho \cdot V \cdot g\), it is clear that thickness t has a much greater effect here than density \(\rho \). The authors would like to clarify that this is only a derivation of the influencing factors, so that before and after deformation it has no effect on the equation. Since volume constancy is assumed, a 10% increase in thickness, for example, will obviously have a greater effect on the results than a 10% increase in density.

The effect of fiber orientation is more complex. Since the homogeneous material in the MD direction (all fibers oriented at 90\(^\circ \)) has the highest strength, the sample with inhomogeneous fiber distribution results in lower reaction force; conversely, the homogeneous material in the CD direction (all fibers oriented at 0\(^\circ \)) has the lowest strength, so the sample with inhomogeneous fiber distribution requires more reaction force. The results of considering all three inhomogeneities at the same time are consistent with the results of the fiber orientation, further demonstrating the absolute dominance of the influence of fiber orientation in the three inhomogeneous parameters.

Fig. 4
figure 4

Force response in MD and CD

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Fig. 5
figure 5

Force response in MD and CD

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3.2.2 Influence of Structural Parameters on Material Properties

Figure 6 shows the stress-strain curve of the three structural distributions in MD and CD. The conclusions remain consistent with the force-displacement curves, except for the two curves affected by material thickness (thickness and all inhomo). This is due to the relationship between strain, force, and thickness, so that the effect of thickness inhomogeneity on the stress-strain curve is largely eliminated, and the deviation from the homogeneous material curve is reduced compared to the force-displacement curve.

Fig. 6
figure 6

Stress-strain curve in MD and CD

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In order to analyze the elastic properties more specifically, E-modulus and Poisson’s ratio are compared according to different distributions, as shown in Fig. 7. As can be seen from the diagram, the influences of density and thickness distributions can be neglected, while the fiber orientation distribution decreases these two parameters in MD, but increases them in CD.

Fig. 7
figure 7

Comparison of E-modulus and Poisson’s ratio in MD and CD

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As for the plastic part, it is known from the stress-strain curve that the plastic part continues the difference in the elastic part in a consistent manner, affecting in the same direction and to the same extent as the elastic part.

3.2.3 Material Failure Due to Inhomogeneity

Since these simulations use an implicit analysis method, there is no direct visualization of material failure. However, the failure of the paper material can be described with stress-based or strain-based failure criteria. Since the strain-based failure surface, i.e., the forming limit diagram, is more useful in the case of a 3D forming process, the maximum strain value of the entire sample is compared to effectively predict the fracture of the material, as shown in Fig. 8. It is obvious that the inhomogeneous samples have larger maximum strain values both in MD and CD, especially the fiber orientation. Despite some differences in the force-displacement and stress-strain curves in both directions and in the three inhomogeneous parameters in the previous subsection, the conclusions for fracture are consistent. Therefore, it can be concluded that the inhomogeneity leads to an increase in the maximum strain, which in turn results in an earlier failure of the material.

Fig. 8
figure 8

The maximum strain value on the sample at the same moment

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3.3 Discussion

The analysis of the presented simulation results shows that the inhomogeneous nature of the paper material can be well represented numerically. By assigning different local densities, local thicknesses, or local directional tensors to the material, the material response resulting from the inhomogeneity can be obtained. To the best of the authors’ knowledge, most established finite element software can implement this feature. Therefore, this allowed the investigation of materials with inhomogeneities other than paperboard, such as composites with fiber substrates. In other words, as long as the specific inhomogeneous property parameters of the material are obtained by measurement, its mechanical properties can be explored by numerical analysis. Conversely, if the desired mechanical properties are available, it is also possible to invert the desired inhomogeneous parameters and thus make appropriate adjustments in the manufacturing process.

Returning to the paper material, the measurements on the microstructure of the paper and the identification and study of weak spots in the previous literature [9] have shown that the inhomogeneity of the paperboard properties is due to the randomness of the fibers and their arrangement during the production process. Therefore, inhomogeneity has an effect on the fracture stress and the maximum elongation strain. Moreover, the degree of influence of different structural parameters varies for different paperboard materials. For most paperboards, i.e., those with highly anisotropic properties, fiber orientation has the greatest effect. Due to the composition of the paper and the production process, it is not difficult to understand that the fibers themselves and the fiber-to-fiber bond have the most direct impact on the performance of the paperboard. For paperboard with special manufacturing processes or special fibers, this conclusion may not apply and requires a separate investigation.

Numerical analysis shows that the inhomogeneity of the paper structure affects the material properties and failure behavior, since the structural inhomogeneity leads to localized strain overload and premature fracture. In terms of fracture mechanism, it is always the weakest point that causes the specimen to fail. Inhomogeneous materials, although having exactly the same average properties as homogeneous materials, will also fail earlier due to a larger number of weak spots. This is advantageous for predicting the failure of inhomogeneous materials.

4 Conclusion

In this work, the inhomogeneous properties of paper, as a fiber-based inhomogeneous material, are analyzed using numerical simulations. Similar distributions in terms of density, fiber orientation, and thickness are assigned to the elemental points of the sample, and their response to mechanical properties is investigated individually and together. Through this study, it was found that inhomogeneities can be well represented numerically. For paperboard with strong anisotropy, fiber orientation has a dominant effect, followed by thickness, while density has only a relatively minimal effect. In addition, the inhomogeneity leads to element points of increased strain on the specimen, resulting in earlier failure of the specimen compared to a homogeneous material of the same character. This method is of great importance for predicting the failure of inhomogeneous materials and, conversely, for determining production parameters.