Material Parameters Identification of Thin Fiber-Based Materials Using the Method of Machine Learning Exploiting Numerically Generated Simulation Data

Abstract

The determination and validation of material parameters required for finite element simulation of the forming processes of fiber-based materials such as paperboard can be accomplished by strain-based loading of a specimen in combination with a simulation-based reverse engineering approach. Due to the complexity of the material itself, such as anisotropy, the development of such approaches can be very time-consuming and requires programming skills as well as expertise in FEM analysis and optimization. Machine learning methods offer a practical alternative to optimization, parameterization, and reverse engineering approaches, assuming that the data is fully known, generalized, and learned by the machine learning model. More specifically, a machine learning model can compute the material parameters required for a finite element simulation directly from the experimental measurements, if the hypothetical mapping function in this case is learned from the numerical study between material parameters and deformation behavior. In this paper, such data generated by numerical studies are used to train the machine learning model and, based on this, to determine elastic (e.g., Young’s modulus), plastic, and Hill’s parameters.

1 Introduction

Nowadays, industrial products have to meet both technical and environmental requirements, making sustainability a key factor in product design and material selection. Sustainability in the form of recycling, degradability, and circularity has also become very important in the packaging industry. Plastic packaging, which has been used extensively in the past, clearly does not meet this requirement, so paper and paperboard are gaining ground as an alternative that can meet sustainability requirements at low cost. However, since paper is made from natural fibers, the material itself is more complex in nature and much less ductile than plastic products, so there is still much to be investigated about the forming process of paper and paperboard. In particular, an accurate material model is essential for the numerical approach, which offers a great opportunity to improve both the quality and efficiency of production.

Paper consists mainly of natural fibers plus fillers and additives, and the natural fibers form the basic structure of the material. The small fibers, bonded together by hydrogen bonds, form the fiber network, which exhibits heterogeneity, structural disorder, and anisotropy. For this reason, it is difficult to make accurate predictions about the mechanical behavior of paper and paperboard as a fiber-based material [1]. Obviously, the first consideration is modeling at the microscopic level, such as fibers or fiber networks. Despite the gradual maturation with the advancement of optical measurement techniques, especially Computed Tomography (CT), the microstructure measurement of paperboard remains labor-intensive and costly due to the random and dense arrangement of the fibers and the paperboard itself consisting of multiple compressed layers. In addition, the computation of microscopic models is also very time-consuming and has limited use in microstructural studies. Multiscale mechanical modeling of paperboard has also been proposed in recent years, but still relies heavily on X-ray CT measurements and is not yet widely used for the simulation of forming processes. [2]. It can be seen that sheet scale modeling using continuum models is still dominant due to the convenience of modeling and the practicality of computation. In this approach, paperboard can be considered as a homogeneous anisotropic material. To account for the anisotropic nature, three main directions are usually defined as machine direction (MD), cross direction (CD), and thickness direction (ZD).

Various continuum models have been proposed to describe the anisotropic properties of fiber-based materials using an orthotropic yield function that accounts for different yield stress values in different preferential directions [3]. The most commonly used formulation for modeling paperboard is that of Hill [4], which includes six yield stress values. It has been successfully used to study the forming process and was extended by Hoffman to include tension–compression properties [5]. Another well-known method was later proposed by Xia et al. [6], where the yield surface can be defined as the sum of six sub-surfaces, four for tension and compression in MD and CD, while two correspond to shear behavior. Li et al. [7] proposed a modification of Xia’s model by reducing the material parameters from 23 to 10, which includes both isotropic and kinematic hardening behavior. Xia’s model was also extended by Borgqvist et al. [8] to account for the plastic response of the ZD.

As can be seen from these typical continuum models, there are always many parameters involved in modeling paper materials, which means that many experiments are required. The resulting problem is that specialized equipment for testing material properties and tools developed specifically for paper materials are always in demand. In addition, due to the inhomogeneity of paper, it is not easy to obtain reliable experimental data, and more replicates are needed than for homogeneous materials. Thanks to advances in full-field optical measurements, more and more inverse methods are being used for material characterization [9]. There are currently two main approaches. One is Finite Element Model Updating (FEMU), which determines the exact values of material parameters by building a finite element model of the characterization experiment and using simulation iterations. One of the best-known FEMU methods is the inverse parameter identification method, where the corresponding FE simulations are based on the results of material property tests. The calculated strain distribution is compared with the measured strain distribution until the deviation reaches a convergence criterion so that the simulated strain distribution is as close as possible to the measured distribution (see Fig. 1).

Fig. 1

Principle procedure of the inverse parameter identification method

The second method is the fitting procedure or direct evaluation of the measured strain field using the equilibrium equation, and the Virtual Field Method (VFM) is currently the most widely used method based on this approach. However, these methods are difficult, costly, and time-consuming for users.

Machine learning (ML) approaches, as part of artificial intelligence techniques, which build a model based on sample data, known as training data, in order to make predictions or decisions without being explicitly programmed to do so, can simplify this process [10]. This approach provides a new and efficient way for the measurement, calibration, and validation processes to find the material parameters.

In this work, ML models are proposed to determine the material parameters obtained directly from experiments and required for FE simulation. The data generated by purely numerical studies used to establish the hypothetical function between material parameters and deformation behavior is used to train the ML model. From this, the material parameters (e.g., Young’s modulus, yield stress, Hill’s parameters, etc.) of the fiber-based material required for the FEA can be determined.

2 Materials and Methods

2.1 Material

Paperboard used in this study (thickness: 0.3 mm; grammage: 250 ${\text {g/m}}^2$) is an uncoated recycled paper made from 100% recycled fibers, which is on the borderline of the definition of paper and paperboard and is commonly used for printing or forming.

2.2 Experimental Material Characterization

The tensile test is conducted in five directions of the paper samples, namely MD or x-direction (0$^\circ$), 22.5$^\circ$, 45$^\circ$, 67.5$^\circ$, and CD (90$^\circ$ or y-direction), as shown in Fig. 2. The test is strain-controlled and the test speed is 20 mm/min. A Zwick/Roell Z100 materials testing machine equipped with a high-resolution video extensometer was used to perform the experiments. GOM Aramis 5M, an optical system for 3D deformation analysis, provided strain distribution information during the uniaxial tensile test, and Zeiss Quality Suite software was used to analyze the captured images. All tests are performed in a controlled laboratory conditioned at 23 $^\circ$C and 50% relative humidity (RH).

Fig. 2

Tensile samples in five directions

Since the material behavior in the z-direction (ZD) affects the shear behavior in the planar direction (MD-CD), it is necessary to determine the material behavior in the ZD. Due to the delamination in the tensile direction, the compression test was performed to determine the material behavior in the ZD [11]. The compression test is performed under the same conditions as the tensile test, and the machines used are the same. The simple square specimen with a side length of 30 mm is compressed to 20 kN by the Zwick machine and the displacement is recorded by both video extensometer and GOM Aramis. The test speed is 0.36 mm/min.

2.3 Paperboard Material Model and Parameters

Paperboard is an anisotropic material that can be approximated as an orthotropic material. The material exhibits different behavior in response to tension and compression. For this reason, a 2D plane stress state with Hill’s yield criterion [4] and multilinear isotropic hardening law is used.

2.3.1 Elastic Part

The elastic part of the paperboard is described using Hooke’s law $\epsilon = C\sigma$. The elastic model is defined by three elastic moduli $E_x$, $E_y$ and $E_z$, three Poisson’s ratios $\nu _{xy}$, $\nu _{xz}$, and $\nu _{yz}$, and three shear moduli $G_{xy}$, $G_{xz}$, and $G_{yz}$, where x represents MD, y represents CD, and z represents ZD (also see in Fig. 2). It is assumed that Poisson’s ratios in thickness direction are zero, i.e.,

$$\begin{aligned} \nu _{xz}=\nu _{yz}=0, \end{aligned}$$

(1)

and

$$\begin{aligned} G_{xz}=G_{yz}=G_{xy}, \end{aligned}$$

(2)

The orthotropic elastic material properties 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 [12]:

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

(4)

and Poisson’s ratio $\nu _{xy}$ is then calculated following equation [13]:

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

(5)

To determine the $E_{x}$, $E_{y}$, $E_{45}$ and $E_{z}$ for Young’s modulus and the initial yield stress $\sigma _{xx}^{0}$, $\sigma _{yy}^{0}$, $\sigma _{45}^{0}$, and $\sigma _{zz}^{0}$ parameters, a linear regression is used that allows to determine the linearity at the beginning of the stress–strain curve. Another possibility would be to use the method of reverse engineering for the determination of the parameters.

2.3.2 Plastic Part

The constitutive models for elastic–plastic behavior can be written with a decomposition of the total strain into elastic and plastic parts.

$$\begin{aligned} \epsilon = \epsilon ^{el}+\epsilon ^{pl} \end{aligned}$$

(6)

The stress is proportional to the elastic strain $\epsilon ^{el}$

$$\begin{aligned} \sigma = D\epsilon ^{el} \end{aligned}$$

(7)

and the evolution of plastic strain $\epsilon ^{pl}$ is a result of the plasticity model. The plastic behavior of the material is assumed to follow Hill’s yield criterion [4], which is expressed on the basis of the ratio $R_{ij}$ of the yield stress in direction ij with respect to a reference direction. If the y-axis is chosen as the reference direction, i.e., $R_{yy}$ = 1, the 2D-Hill’s criterion is given by

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

(8)

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

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

(9)

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

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

(10)

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. It should be noted that Eq. (10) is fitted to the response in the y-direction since $R_{yy}$ = 1, the shear parameter in planer direction is determined with the properties behavior in 45$^\circ$ tensile direction and the plastic behavior in the ZD was assumed to have the same value as the y-direction $R_{zz}$ = 1.

The following material parameters were obtained for the material from the reverse engineering method, as shown in Table 1.

Table 1 Material constants and evaluated values

The properties of this paperboard in different directions, MD, 22.5$^\circ$, 45$^\circ$, 67.5$^\circ$, and CD, are studied and compared numerical experimental data with numerical experimental data. The results can be seen from Fig. 3, which shows a good agreement.

Fig. 3

Comparison of the numerical material model (Sim) and the experimental data (Exp) for the paperboard

2.4 Machine Learning (ML) for Parameters Identification

This work demonstrates that machine learning can be used to predict the parameters and thus the material properties. In the material model presented in the previous section, the values of the following material parameters are variables (as in Table 2), while their ranges are defined in such a way that the computation load is controllable and the generated datasets contain enough information to perform a feasibility study of the method. In order to determine the material parameters for the forming process of thin-walled fiber-based materials, a total of 11 parameters are required. Using the parameter ranges from Table 2, 10000 data were generated with the advanced Latin hypercube sampling. The strength of many fiber-based materials is highest in the MD direction, followed by 45$^\circ$ and then CD direction. We have applied the following condition $E_{x}$ $\ge$ $E_{45}$, $E_{45}$ $\ge$ $E_{y}$, $\hat{\epsilon }_{pl}$ (The minimal maximum plastic strain has been set to 1.5) and Eq. 4 should always be positive. In total, 2526 generated stress–strain sets were obtained. Both the input (material parameters) and the output (strain values) of the FEM simulation were stored in a database, which is necessary for the subsequent training of the ML model. For the prediction of parameters, the machine model of the SKlearn module in Python was focused due to the fixed number of generated data and the determination of important features for the prediction of material parameters. If the determination of important features was not necessary in the workflow, the Artificial Neural Network (ANN) model would be used for the prediction of material parameters.

Table 2 Range of material parameters for numerical simulations

Fig. 4

Process models with boundary conditions, loads and mesh of a tensile test and b compression test

To perform the FEM study, the Ansys 2021R2 solver was used in a quasi-static mode under implicit analysis. The use of Optislang software was also used to perform the DOE analysis.

The process models of tensile test and compression test, including boundary conditions, loads, and mesh, are shown in Fig. 4. The tensile test uses a shell model, and due to orthogonal symmetry, only a quarter of the samples are simulated to save computation time. The size of the simulated sample is $45\times 15\,{\text {mm}}^2$, and the mesh size is 1 mm. The entire sample is evaluated as in the experiments. In the compression test, a solid model is used to investigate the behavior of the ZD, here only one eighth of the specimens are simulated for the same reason as before. The size of the simulated sample is $15\times 15\times 0.3\,{\text {mm}}^3$. The mesh size in the planar direction is also 1 mm, however, there are five elements in the thickness direction. The material response in each direction (average stress and strain) of the sample surface is evaluated.

3 Results and Discussion

3.1 ML Modeling

In order to predict the material parameters, a large study was conducted in which the material parameters were varied and a total of 2526 different stress–strain curves were generated (see Fig. 5).

Fig. 5

Numerically generated stress–strain curves that can be used for characterization

The generated stress–strain curve (ML-Input) was used as input to identify the material parameters (ML-Output), thus reducing the effort to determine the material parameters. Three different ML models were developed to determine the material parameters. One model to determine the elastic material parameters and the yield point for the reference direction y, one model to determine the plastic material properties for the reference direction y, and one model to determine the Hill’s parameters. To develop the three ML models, a dataset with each having a data length of 50 was used as input (the data length for the ML input is $8\times 50 = 400$): $\sigma _{yy}$, $\epsilon _{yy}$, $\sigma _{xx}$, $\epsilon _{xx}$, $\sigma _{zz}$, $\epsilon _{zz}$, $\sigma _{45}$, $\epsilon _{45}$ (see Fig. 6). It should be mentioned that 0 was always inserted as the first data for the stress and strain in each direction.

To ensure that the method for predicting material parameters is general and works for all curves, the stresses are divided by the maximum stress and all strains are divided by the maximum strain in the main direction (here is y-axis).

$$\begin{aligned} Scaled_{\sigma _{ii}} = \sigma _{ii}/max(\sigma _{y}) \end{aligned}$$

(11)

$$\begin{aligned} Scaled_{\epsilon _{ii}} = \epsilon _{ii}/max(\epsilon _{y}) \end{aligned}$$

(12)

The following scaling was performed for the material parameters (see Table 3)

Table 3 Scaling of the material parameters for the ML model

where S = max($\sigma _{y}$) and E = max($\epsilon _{y}$).

For the prediction of the material parameters, different ML models from the Sklearn library of Python 3.9 were investigated.

Depending on whether the parameters are elastic, plastic, or Hill’s parameters, three different ML models have been developed according to the high accuracy.

Fig. 6

Sequence of values for the ML-Input

Elastic Parameters: For predicting the elastic parameters, the ML model gradient boosting regressor with multi-output regressor was found to give the best results. The accuracy of the ML model is 97.88%. The structure of this model is as follows:

MultiOutputRegressor ( GradientBoostingRegressor (learningrate = 0.1, nestimators = 500, subsample = 1, criterion = friedmanmse, maxdepth = 3))

Plastic Parameters: The ML model multi-output regressor with ExtraTreesRegressor was used to determine the plastic parameters. The model has an accuracy of 92.38%.

To build the model, the following parameters are needed:

MultiOutputRegressor ( ExtraTreesRegressor ( nestimators = 200, criterion = squarederror, minsamplessplit = 2, minsamplesleaf = 1, minweightfractionleaf = 0.0))

Hill’s Parameters: The linear regressor from Sklearn is the best model for predicting Hill’s parameters.

The parameters for the model are shown below:

LinearRegression(fitintercept = True, copyX = True, positive = False) was used. The model has an accuracy of 95.87%.

3.2 Scaling of Parameters and Verification with Experimental Results

In order to predict the material parameters, four tests should be performed from the developed ML method: tensile test in MD, 45$^\circ$, CD, and compression test in ZD. Afterward, the experimental data are interpolated with a data length of 49 and null was inserted as the first data point since the generated data or experimental data frequently do not start with null.

The material parameters predicted from the ML models are shown in Table 4 and compared with the real experimental values.

Table 4 Material constants and evaluated values

The predicted stress–strain curves in different directions are also shown in Fig. 7, which shows a good agreement with the experimental parameters as well.

Fig. 7

Comparison of the predicted material parameters (Pre) and the experimental data (Exp) for the paperboard

3.3 Discussion

The materials used in the forming process are not all homogeneous and isotropic, nor do they possess an ideal stress–strain curve and a simple yield equation. For these materials, there is a similar need for a material model that can accurately describe the material properties for further studies such as forming limits or process optimization. To this end, in this work, the ML method has been developed to determine the parameters using only numerically generated data, which makes this process more available and efficient. Taking the paperboard material as an example, different ML models were tested and compared in terms of accuracy for the elastic part, plastic part, and Hill’s parameters, and it was shown that they all found suitable models with satisfying accuracy.

From this workflow, it can be seen that it is absolutely applicable to other fiber-based materials, composite materials, and orthotropic materials as well. This also greatly simplifies the calculation process from experimental data to the parameters needed for simulation and provides the possibility to reduce the number of experiments. Not only that but also other materials used in the forming process, such as aluminum alloys which are currently being widely used due to their lightweighting advantages, an accurate material model including properties such as anisotropy, temperature dependence, etc., is also a demand. Even some more complex intrinsic structure relationships, or other yield functions, can take this approach to make the building of material models more accurate and efficient.

4 Conclusion

For materials with anisotropy, such as fiber-based materials, building an accurate material model in numerical simulations is not an easy task, partly because of the large number of experiments and partly because of the calculation from experimental data to material parameters. The ML method simplifies this process using numerically generated data and offers the possibility to determine material parameters efficiently and accurately, and can also be applied to modeling other relatively complex materials. In the future, ML will be used to reduce the number of experiments needed to determine material parameters, and the developed method will be validated for other types of materials. In addition, increasing the amount of simulation data used to train the ML model will improve the accuracy of the model in predicting plastic parameters.