Methods for characterization and continuum modeling of inhomogeneous properties of paper and paperboard materials: A review

Abstract

The potential of paper and paperboard as fiber-based materials capable of replacing conventional polymer-based materials has been widely investigated and evaluated. Due to paper’s limited extensibility and inherent heterogeneity, local structural variations lead to unpredictable local mechanical behavior and instability during processing, such as mechanical forming. To gain a deeper understanding of the impact of mechanical behavior and heterogeneity on the paper forming process, the Finite Element Method (FEM) coupled with continuum modeling is being explored as a potential approach to enhance comprehension. To achieve this goal, utilizing experimentally derived material parameters alongside stochastic finite element methods allows for more precise modeling of material behavior, considering the local material properties. This work first introduces the approach of modeling heterogeneity or local material structure within continuum models, such as the Stochastic Finite Element Method (SFEM). A fundamental challenge lies in accurately measuring these local material properties. Experimental investigations are being conducted to numerically simulate mechanical behavior. An overview is provided of experimental methods for material characterization, as found in literature, with a specific focus on measuring local mechanical material structure. By doing so, it enables the characterization of the global material structure and mechanical behavior of paper and paperboard.

Full Article

Methods for Characterization and Continuum Modeling of Inhomogeneous Properties of Paper and Paperboard Materials: A Review

Cedric W. Sanjon,^a,* Yuchen Leng,^b,** Marek Hauptmann,^a,c Peter Groche,^b and Jens-Peter Majschak ^a,d

The potential of paper and paperboard as fiber-based materials capable of replacing conventional polymer-based materials has been widely investigated and evaluated. Due to paper’s limited extensibility and inherent heterogeneity, local structural variations lead to unpredictable local mechanical behavior and instability during processing, such as mechanical forming. To gain a deeper understanding of the impact of mechanical behavior and heterogeneity on the paper forming process, the Finite Element Method (FEM) coupled with continuum modeling is being explored as a potential approach to enhance comprehension. To achieve this goal, utilizing experimentally derived material parameters alongside stochastic finite element methods allows for more precise modeling of material behavior, considering the local material properties. This work first introduces the approach of modeling heterogeneity or local material structure within continuum models, such as the Stochastic Finite Element Method (SFEM). A fundamental challenge lies in accurately measuring these local material properties. Experimental investigations are being conducted to numerically simulate mechanical behavior. An overview is provided of experimental methods for material characterization, as found in literature, with a specific focus on measuring local mechanical material structure. By doing so, it enables the characterization of the global material structure and mechanical behavior of paper and paperboard.

DOI: 10.15376/biores.19.3.Sanjon

Keywords: Paper; Paperboard; Material characterization; Inhomogeneity; Simulation; Modeling; Forming process

Contact information: a: Fraunhofer Institute for Process Engineering and Packaging IVV, Heidelberger Str. 20, D-01189, Dresden, Germany; b: Institute for Production Engineering and Forming Machines, TU Darmstadt, Otto-Berndt-Straße 2, D-64287, Darmstadt, Germany; c: Chair of packaging machines and packaging technology, Steinbeis-University, Ernst-Augustin-Straße 15, D-12489, Berlin, Germany, d: Faculty of Mechanical Science and Engineering Institute of Natural Materials Technology, TU Dresden, D-01062, Dresden, Germany; *Corresponding author. cedric.sanjon@ivv-dd.fraunhofer.de; **Corresponding author. yuchen.leng@ptu.tu-darmstadt.de

GRAPHICAL ABSTRACT

INTRODUCTION

A wide range of fiber-based materials are used in packaging nowadays – from tissue paper for tea bags to heavy-duty cartons for distribution. Cellulose-based fiber materials can provide benefits such as low density, environmentally friendliness, and economy. Thanks to the high recyclability and good sustainability of paper products, they are seen as an excellent solution for the packaging industry, which is reflected in both the raw materials and production processes. Paper designed for routine applications, such as office documentation and newspapers, typically has a thickness of about . The area density of paper products, expressed as mass per unit area, called grammage or basis weight (International Organization for Standardization 1995), typically ranges from 40 to 100 g/m² and it varies depending on the used type of paper. On the other hand, paper materials used for book covers or packaging can exceed 1 mm in thickness. Thick paper grades are known as paperboard, with a typically higher grammage range of 150 to 500 g/m² (Gustafsson and Niskanen 2021).

Despite the many advantages mentioned above, paper and paperboard as fiber-based materials have many limitations in processing, especially in the forming process, in comparison to plastic material, which can be easily formed into complicated shapes. Mechanical forming refers to a manufacturing process involving plastic deformation, wherein the physical shape of the material undergoes permanent alteration while maintaining its mass and material cohesion (Schuler GmbH 1998). Many phenomena are related to the forming process, such as material properties, e.g. elastic and plastic behavior; tribology, including friction, lubrication, and wear; and forming limits, e.g. fracture and defects such as wrinkles. Characteristics of paper and paperboard, such as anisotropy, inhomogeneity, and hygroscopicity within the material, as well as design parameters such as moisture, blank holder force, temperature, punch speed, and stress state during the forming process, all influence the quality of final products. It is therefore important to study the properties and response of the materials, the forming process and the relationship between them.

When it comes to the better control and prediction of forming processes, the finite element method (FEM) serves as a potent numerical technique that has found extensive application in modeling and analyzing diverse engineering processes. To study the process and system parameters and improve the process stability of the forming process, a paper and paperboard material model based on continuum mechanics is preferable, since the micromechanical models are extremely expensive with respect to computations. However, especially in paper and paperboard, many microscopic phenomena occur discontinuously due to the random arrangement of fibers. Therefore, a purely macroscopic description is unlikely to encompass all relevant effects. It follows that the range of details should be extended. The Stochastic Finite Element Method (SFEM) serves as an expansion of the traditional deterministic Finite Element Method (FEM), offering a means to address static and dynamic problems within a stochastic framework. This involves considering stochastic variations in mechanical, geometric, or loading characteristics (Stefanou 2009). While the physically based approach permits the parameter values to be transferred to other processes, it necessitates conducting microscopic-level experimental analyses and applying procedures to manage significant parameter value fluctuations (Volk et al. 2019). In addition to more detailed constitutive models, models for failure prediction with their own model parameters are needed, which also require parameter values. Thus, global mechanical tests and local structural measurements are necessary for the characterization of fiber-based materials as well as for the study of the forming process.

Some reviews are presently accessible, focusing on the modeling of paper materials or forming processes. Östlund (2017) provided an exhaustive examination of the literature regarding 3D forming processes, i.e. deep drawing, hydroforming, and press forming for paper and board. The work was carried out experimentally and numerically, encompassing analyses of deformation mechanisms, damage phenomena, and also friction behaviour. (Fadiji et al. 2018) reviewed the application of FEM in the food-related packaging industry, with an emphasis on corrugated packaging, along with an example of its application in the forming process. More recently, Simon (2021) considered advancements in the material modeling methods of paper and paperboard across multiple scales, from the fiber and network level to the sheet and laminate scale. Multiscale modeling is a potential approach, but continuous models are more realistic for completely stochastic fiber-based materials, especially when it comes to application in forming processes. In order to incorporate inhomogeneity or varying local material structure into continuum modeling of paper and paperboard materials, the characterization of microscopic structure and macro-mechanical properties is of particular importance.

The aim of this work was to facilitate the application of continuous modeling, particularly in the treatment of inhomogeneities. To achieve this, the stochastic modeling methodology, including SFEM, can be applied to enable the stochastic modeling, which will be summarized from the literature firstly. This involves identifying the necessary measurements to quantify inhomogeneity and conducting experiments for material modeling to implement global and stochastic modeling. The required microstructural measurements and the measurement methods used in the literature are presented first, followed by the summarization of experiments used to determine the in-plane and out-of-plane mechanical properties.

BASIC CHARACTERISTICS OF PAPER MATERIALS

Paper is a network of natural fibers bonded together. The mechanical properties of paper and paperboard remain similar despite variations in fiber types and manufacturing processes. Commercial paper is typically characterized as an anisotropic material, with principal directions identified as the machine direction (MD), cross-machine direction (CD), and thickness direction (ZD or out-of-plane direction), as illustrated in Fig. 1. In mass production, the tensile strength is typically higher in the MD than in the CD, with a factor of 1 to 5 due to different manufacturing principles and up to 100 times higher than in the ZD (Stenberg and Fellers 2002), but the elongation at break shows the opposite behavior. It is important to note that some papers may exhibit less anisotropic properties, resulting in minimal differences between MD and CD properties. In some cases, the laboratory paper may even be isotropic.

Fig. 1. Principal directions in paper

In addition to anisotropies, paper materials also manifest structural non-uniformity, referred to as inhomogeneity, leading to fluctuations in local mechanical properties and surface characteristics such as roughness. These fluctuations may arise longitudinally or transversely within the material and particularly in regions characterized by diminished thickness, density, and area-related attributes (Hauptmann 2010). Therefore, it is imperative to quantify this inhomogeneity to facilitate its numerical representation.

NUMERICAL METHODS OF MODELING STOCHASTIC MATERIALS

Progress in understanding material behaviors, achieved through experimental measurements covering both microscopic and macroscopic aspects, finds effective representation in numerical methods. Modeling mechanical, tribological, thermal material properties, or their combination contributes to achieving two main objectives. Firstly, material models are indispensable for predicting material properties, such as optimal parameters in a processing operation, to enhance product quality. Secondly, these material models play a crucial role in exploring the effects that transpire during processing operations and to enable the correlation of these effects with the underlying behavior and involved interdependencies of the components (Wallmeier 2018).

Motivated by these considerations, two distinct models have been devised for fiber-based materials. Continuum models, assuming homogeneity of material on the considered scale, are adept at simulating complex forming processes due to their more straightforward numerical modeling and faster calculations. However, discrepancies arise compared to experimental findings due to inherent inhomogeneity and random material behavior. Material models incorporating microstructural details facilitate the linkage of macroscopic loads to stresses and deformations within individual fibers, fiber-fiber bonds, or fiber networks (Mansour et al. 2019). This process involves the creation of a mathematical model that represents the interactions between fiber-fiber bonds, individual fibers, and either two-dimensional (Bronkhorst 2003) or three-dimensional network (Li et al. 2016b) structures. At scales below the network structure, it is necessary to consider the mechanical properties of individual fibers and the bonds between fibers. In addition, micromechanics has produced models integrating material property variability through randomly generated networks (Kulachenko and Uesaka 2012). Nonetheless, due to their significant computational demands, these models are typically unsuitable for simulating processing operations such as forming processes. Instead, they find utility in micromechanical investigations. To accurately characterize stochastic materials such as paper and paperboard, a numerical method is indispensable, allowing for the integration of their inhomogeneous structural properties with their homogeneously assumed mechanical properties.

Introduction of the Stochastic Finite Element Method

FEM stands as a widely embraced numerical technique for solving scientific and engineering problems, demonstrating its capability to handle intricate geometries with mixed material and boundary conditions. It is also proficient in addressing time-dependent issues and nonlinear material behaviors. However, the inherent determinism of the FEM imposes a limitation in directly dealing with systems containing uncertainties. The direct study of a system with a degree of uncertainty is not feasible using traditional FEM. A well-established approach for studying the relationship between microstructural geometry and material macroscopic properties involves integrating microstructure models with finite element simulations.

SFEM, an extension of the basic FEM, incorporates random parameters to represent uncertainties. SFEM can introduce randomness into one or more of the main components of the classical FEM, including geometry, external forces, and material properties. It can be used to find correlations between microscopic and macroscopic behavior and represent inhomogeneous properties in continuum modeling. A comprehensive review of SFEM is presented in Stefanou (2009).

Random Fields (RFs)

A random field (RF) is a set of indexed random variables that characterize inherent randomness within a system. The indices denote the spatial, temporal, or spatiotemporal positions of these variables (Thomson 1983). RFs are defined by essential statistical information, including mean, variance, probability distribution, auto-correlation function, and other statistical parameters. An ideal random field should capture the main properties of a stochastic system by considering a minimal number of meaningful and quantifiable parameters. Various methods, such as the local average method (Vanmarcke and Grigoriu 1983), turning-bands method (Matheron 1973), Fourier transform method (Yaglom 2004), and local average subdivision method (Fenton and Vanmarcke 1990), have been developed to determine material properties.

For modeling the uncertainty of composites, researchers can integrate RF representation methods using techniques such as the representative volume element (RVE), homogenization, DIC-based characterization, and random media techniques. The RVE, the smallest volume providing a representative overall value when measured (Hill 1963), is particularly significant. The RVE should remain sufficiently small to be treated as a volume element within the framework of continuum mechanics. In random media, the situation is more complex than in periodic materials. Representative properties cannot be defined for volumes smaller than the RVE. Instead, the material must be described using statistical volume elements (SVE) and random fields (RF). To accurately describe random continuum fields below the scale of an RVE, determining the appropriate size of the RVE for deterministic continuum theories is crucial (Ostoja-Starzewski 1998).

Variants of the SFEM Techniques

The Stochastic Finite Element Method (SFEM) employs various techniques to investigate the uncertainty and inherent stochasticity of a system. Three widely accepted variants of SFEM are commonly used: Monte Carlo simulation (MCS) (Astill et al. 1972), perturbation method (Liu et al. 1986), and spectral stochastic finite element method (SSFEM) (Ghanem and Spanos 2003). MCS, the most general and direct approach, is suitable for a wide range of applications, including nonlinear issues. It provides exact approximations when the deterministic solution to the problem is available (La Bergman et al. 1997). Despite demanding high computational power, MCS is widely accepted and frequently used to validate perturbation methods and SSFEM. The perturbation method is a popular and straightforward technique for estimating the statistical moments of response variables. It is applicable to linear, non-linear, and eigenvalue problems, providing distribution-free results (Sudret and Der Kiureghian 2000). This method strikes a balance between complexity and computing load by estimating the effect of the mean, standard deviation, and covariance of the response variable on the structure’s behavior. However, it is generally limited to random variable values not significantly different from the mean.

The Spectral Stochastic Finite Element Method (SSFEM), a recent extension of the SFEM, primarily focuses on representing the stochastic material properties of structures. It has garnered attention for its ability to reduce the computational effort required for analyzing random processes compared to Monte Carlo simulation (MCS). While SSFEM performs well in linear analysis, some researchers question its practicality for nonlinear analysis (Stefanou 2009). Arregui-Mena et al. (2016) provided a comprehensive review of SFEM’s applications in science and engineering. In materials science, SFEM investigates the behavior of complex materials such as composites and fiber structures such as paper. Various technologies have been devised to characterize the stochastic properties of these materials. However, there are limitations to the application of SFEM (Arregui-Mena et al. 2016). One notable challenge is the absence of experimental procedures for measuring the spatial variability of material mechanical properties. The quality of SFEM studies heavily depends on both experimental data and model property assumptions. Furthermore, simulations are rarely validated against experimental data. An issue with data collection is the difficulty in measuring certain variables. In engineering, for example, while material properties are often well established, repositories of material data may not always provide sufficient information to determine the type of random field to be utilized.

Potentials of Stochastic Modeling in Forming Processes

Certainly, the matter of multiscale modeling is of paramount importance, seeking to integrate microscopic and macroscopic material properties, and there have been notable attempts with a few successes in this direction. Alzweighi et al. (2021) recently proposed a multiscale method that combines detailed micromechanical simulations, physical measurements of fiber-level variability, and mesoscale continuum models. This approach aims to quantify the influence of spatial variability in the structural properties of paper and paperboard resulting from the disorder of the fiber network.

The proposed method bridges the gap between intricate, computationally intensive micromechanical simulations and the continuum approach, which may overlook material inhomogeneity. However, there remains significant doubt about whether fiber network simulation can be applied to the forming process, considering computational time and modeling difficulty. While multiscale modeling is a relatively new methodology, there is still much to explore in this area, and there remains a gap in its application to the forming process.

Another attempt came from Lindberg and Kulachenko (2022), using implicit solver and Hill’s plasticity with consideration of subsequent failure evaluation, to model the forming process of paperboard. MCS was applied to simulate the plane stress, where random numbers of the tensile and the compressive stresses were picked from their respective distributions. It was also proposed that this approach was conservative due to a deficit of the data for material size dependency.

LOCAL STRUCTURAL PROPERTY MEASUREMENTS

Image-based methods are frequently used to measure the local structural parameters of fiber-based materials, as well as other technologies. The following section describes common formation-related and fiber-level measurements.

Formation-related Measurements

Formation refers to the quantification of variations resulting from the non-uniform distribution of fibers both within the plane and in the thickness direction of a paper sheet (Bouydain et al. 2001b). The resulting local structural properties of paper and paperboard, including mass distribution, thickness distribution, and density distribution, are examined by various measurement methods.

Local grammage measurement

The main method of measuring the local grammage of paper is by the gravimetric method, in which the weight of a region is divided by its plane area. Several energy sources including light transmission, 𝛽-ray, soft X-ray and electrography have been used to determine the grammage of paper by recording the local transmission within a spatial region. The radiation will interact differently with the sample, depending on its mass distribution, which means absorption of radiation is related to the mass in the region. A comparison of these four paper imaging techniques based on their process parameters and image features was reviewed in Tomimasu et al. (1991).

The 𝛽-radiography technique provides detailed internal imaging and high sensitivity and specificity for detecting variations in material density and structure. The formation data from 𝛽-radiography could be recorded by application of either X-ray film (Tomimasu et al. 1989) or a storage phosphor screen (Keller and Patrice 2001) as the detector. Limitations of using X-ray film to detect transmitted energetic rays are the variability of the film, its development, and the digitization process, especially between laboratories. The use of stored phosphor screens reduces processing time and experimental variables while maintaining a high spatial resolution, which in the range of a few microns compared to X-ray film in the range of tens of microns. Exposure of the storage phosphor screen to irradiation produces a latent image that can be digitized by the scanning system to produce a mass formation map. Obviously, there are some limitations to the use of β-ray, such as necessity for specialized equipment and handling of radioisotopes, and also the limited to laboratory settings, so there have been some studies conducted with the objective of establishing a correlation between transmitted light and β-ray. For example, Raunio and Ritala (2009) devised a novel approach to deduce the basis weight using light transmittance, offering a promising avenue for approximating the basis weight of paper. The investigation delves into the correlation between basis weight and light transmittance, alongside exploring how this correlation evolves across diverse spatial scales. Notably, it was discerned that the strongest correlation manifests at relatively diminutive scales, signifying that fluctuations at larger scales do not disrupt the relationship; instead, the method adeptly filters out measurement noise.

Soft X-ray offers high-resolution imaging of paper and paperboard structure, including surface roughness and internal voids, sufficient to distinguish fiber features through the entire thickness of samples (Abedsoltan et al. 2016), but specialized equipment and expertise for sample preparation and analysis are required. Electrography can provide quantitative measurements of paper and paperboard formation based on toner deposition patterns, but has difficulty measuring grammage greater than 120 g/m² (Keller 1996).

The light transmission technique is the most popular due to its low cost, simplicity, relative safety, and the ability to obtain data with great rapidity (Bouydain et al. 2001a). It offers high-resolution images suitable for observing surface features and internal structures of the sample. However, limitations exist in its sensitivity to the nature of test material, especially the effect of sheet composition, the effect of local density variations, limitations due to sheet opacity, and the inability to reliably determine the actual local basis weight (Abedsoltan et al. 2016).

A comparison of the methods for measuring paper grammage is presented in Table 1. Light transmission involves shining light through a paper sample and interpreting the patterns of transmitted light to determine grammage. This technique offers rapid exposure times, facilitating quick measurements. It is a safe option as it employs non-ionizing light sources. However, the method’s accuracy may be impacted by the paper’s transparency, composition, and moisture content, which can alter light transmission. It works well for lower to moderate grammage papers but is less effective with high grammage samples. Proper calibration and consistent lighting are essential for accurate results.

Table 1. Comparison of Light Transmission, Electrography, 𝛽-ray, and Soft X-ray

Electrography uses an electric field on the paper to measure the charge patterns on its surface. It provides moderate spatial resolution and can measure a wide range of grammages. While generally safe, it may require careful handling of electrical equipment. Electrography can be affected by environmental factors such as moisture, as well as paper density, necessitating precise calibration for reliable outcomes. The β-ray method utilizes beta radiation to assess the absorption and scattering patterns in the paper, providing insights into grammage. This approach has fast exposure times and offers high contrast, distinguishing paper density and composition. It is effective across a range of grammages, though specialized equipment and safety measures are required due to the use of radioactive sources. Soft X-ray analysis uses low-energy X-rays to explore the paper’s internal structure and determine grammage. This technique offers high contrast and spatial resolution, providing detailed internal structural information. It works across a broad range of grammages and can facilitate quick measurements. However, like β-ray, soft X-ray requires specialized equipment and adherence to safety protocols due to potential radiation exposure.

In summary, each method has its own pros and cons when measuring grammage. Light transmission is fast and safe but may face challenges with high grammage papers. Electrography provides versatility in grammage range but needs careful calibration. Both β-ray and soft X-ray methods deliver high contrast and spatial resolution, making them suitable for a range of grammages. Nonetheless, these methods require specialized equipment and strict safety precautions due to radiation exposure. The choice of method will depend on the specific requirements and available resources for each application.

Local thickness measurement

The paper thickness is measured as the perpendicular dimension between the two main surfaces of the paper. It is difficult to measure thickness accurately due to the roughness, web discontinuity, compressibility, and the difficulty of defining the true outer boundaries of paper. Some contact measurement methods were applied to paper thickness measurement in the earlier period, such as hard platen caliper (Fellers et al. 1986), soft platen caliper (Wink and Baum 1983), and opposing spherical platens (Schultz-Eklund et al. 1992). These methods entail applying pressure to the surface of the sample and making contact with it, resulting in a deformation of the surface contours to varying degrees. However, the listed contact measurement methods, although simple in principle, have a slightly insufficient resolution for localized thickness measurements, and another is that they can have a slight effect on the surface of the material.

Fig. 2. Principle of non-contacted laser profilometry instrument

Non-contact local thickness measurements are available using different techniques, including laser, ultrasonic, and X-ray. The concept of non-contact profilometry was extended to encompass the measurement of local thickness by means of a simultaneous scanning of both sides of the paper with laser sensors (see Fig. 2). Izumi and Yoshida (2001) developed a dynamic confocal sensor-based instrument for mapping the local thickness irregularities from surface data of the front and back surfaces of a sample with a mm² area. The instrument has an in-plane spatial resolution of 500 μm and micrometer-scale in the thickness direction. The similar method introduced by (Sung et al. 2005), known as the twin laser profilometer, utilizing triangulation-based sensors, involves the interference of two laser beams to accurately measure the distance to a surface, offering an expanded scope of surface topography. The resolution has been enhanced to 25 µm with the incorporation of high-precision stages.

The above-mentioned thickness assessment techniques are still in the face of some challenges: radiation hazards, contact media required, limitations with respect to measurement speed and depth resolution, or deficiencies in the use of multi-layer board systems. The advent of terahertz (THz) technology offers an innovative method to overcome the limitations, and structures in the order of mm can be detected without great technical effort. Terahertz (THz) radiation falls within the electromagnetic spectrum between millimeter-waves and infrared wavelengths. Employing the short-pulse technique along the beam direction allows for significantly enhanced resolution, enabling the current capability to resolve layer thicknesses of approximately 10 𝜇m (Wietzke 2021). The principle of thickness measurement with THz is shown in Fig. 3, in which each layer interface reflects part of the terahertz pulse, so the thickness can be analyzed by calculating the time difference between the signal returns.

Fig. 3. Principle of thickness measurement with THz

Remarkable strides have been taken in the recent industrialization of this innovative technology. THz time-domain spectroscopy (THz-TDS), employing short pulses of THz radiation to probe material properties, has emerged as a promising spectroscopic technique having high spectral resolution ranging from sub-micrometers to micrometer levels. In comparison to continuous-wave THz spectroscopy, THz-TDS provides ample sample information through pulsed THz excitation, making it a powerful technique for a wide range of applications in materials science, chemistry, biology, and physics. In THz-TDS, a short-duration pulse of terahertz radiation is emitted and directed towards the sample. The interaction of this pulsed terahertz radiation with the sample results in the generation of a time-domain waveform (Withayachumnankul et al. 2014). In a study by Mousavi et al. (2009), a non-contact method was proposed for simultaneous measurements of the thickness and moisture content of paper using THz-TDS.

Local density determination

Paper density can be calculated by dividing its mass by the thickness in the same area. Dodson et al. (2001) employed an opposing laser non-contact thickness tester developed by Izumi and Yoshida (2001) to measure local thickness and density variation maps. Sung et al. (2005) utilized TLP instrumentation to acquire a local thickness map and combined it with a local mass map through storage phosphor 𝛽-radiography. This approach generated localized apparent density image pixels () for analyzing the in-plane inhomogeneity of printing paper’s thickness, grammage, and apparent density. Similarly, Keller et al. (2012) applied this method to study the distribution of local mass, thickness, and density in different nonwoven materials, utilizing a binary mask for segmentation and distinguishing relief features from random background structure.

Given the porous nature of paper, mercury intrusion porosimetry (MIP) serves as a viable method for measuring aspects related to paper and paperboard density. MIP is a commonly used technique for analyzing porous material structures, involving subjecting a sample to controlled pressure while immersed in mercury (Johnson et al. 1999). As the pressure increases, mercury is forced into the pores of the sample. The pores fill with mercury in a sequential manner, starting from the largest pores to the smallest. During the intrusion process, the volume of mercury that enters the sample is measured at regular intervals, while at the same time, the change in volume is recorded as a function of applied pressure. MIP measurements are rapid and straightforward, providing valuable structural parameters such as porosity, pore size, pore volume, pore distribution, and density (Giesche 2006). When coupled with microscopic measurements such as scanning electron microscopy (SEM) and X-ray 3D micro tomography (X-𝜇CT), as will be discussed in the next section, MIP measurements can offer more in-depth information regarding the shapes or spatial distribution of pores within multilayer structures, such as thick-structured paper (Charfeddine et al. 2019).

Fiber Network Structure Measurements

In addition to grammage and thickness measurements, the fiber structure of paper also significantly influences its mechanical behavior. Therefore, the measurement of fiber structure is equally crucial in understanding the mechanical properties of paper materials, which are characterized by a non-unique network of interconnected fibers. Measurements related to fiber structure, including fiber length, width, and orientation, are pivotal for the examination of material properties.

Image acquisition technology

Image acquisition technology has been a significant focus in recent research, particularly concerning the analysis of 3D paper structures. In a study by Chinga-Carrasco (2009), various imaging techniques were assessed for evaluating printing paper structures. These methods included transmission electron microscopy (TEM), focused-ion-beam (FIB), scanning electron microscopy (SEM), light microscopy (LM), confocal laser scanning microscopy (CLSM), and X-ray microtomography (X-𝜇CT). A comparison of diverse image acquisition devices used in paper structure measurement is presented in Table 2.

Table 2. Imaging Systems Employed for the Structural Examination of Paper Materials