*Article* **Large Deformation Finite Element Analyses for 3D X-ray CT Scanned Microscopic Structures of Polyurethane Foams**

**Makoto Iizuka 1,\* , Ryohei Goto <sup>2</sup> , Petros Siegkas <sup>1</sup> , Benjamin Simpson <sup>1</sup> and Neil Mansfield <sup>1</sup>**


**Abstract:** Polyurethane foams have unique properties that make them suitable for a wide range of applications, including cushioning and seat pads. The foam mechanical properties largely depend on both the parent material and foam cell microstructure. Uniaxial loading experiments, X-ray tomography and finite element analysis can be used to investigate the relationship between the macroscopic mechanical properties and microscopic foam structure. Polyurethane foam specimens were scanned using X-ray computed tomography. The scanned geometries were converted to threedimensional (3D) CAD models using open source, and commercially available CAD software tools. The models were meshed and used to simulate the compression tests using the implicit finite element method. The calculated uniaxial compression tests were in good agreement with experimental results for strains up to 30%. The presented method would be effective in investigating the effect of polymer foam geometrical features in macroscopic mechanical properties, and guide manufacturing methods for specific applications.

**Keywords:** polyurethane foam; structure-property relationships; finite element analysis; microscale analysis; X-ray computed tomography

#### **1. Introduction**

Polyurethane foams have many unique properties, such as elasticity, softness, and ease of forming. These properties make polyurethane foams attractive to automotive seat designers since they can effectively support the human body and distribute the body pressure. The improvement of the mechanical properties of the foams is an important challenge. Controlling the mechanical properties of foams would be useful in designing seats that are more comfortable and potentially at lower cost. The mechanical properties of polyurethane foams depend largely on their microstructures (Figure 1). The foam structure consists of a cluster of bubbles and struts at the edges of the cells. Figure 1 shows an example of an open-cell foam in which the bubbles are linked together. The macroscopic stress– strain relationship depends on the mechanical properties of the parent material, of which the struts are made, and the geometrical structure of cells and struts [1]. Understanding the relationships between the microscopic geometrical structures and macroscopic mechanical properties is essential in developing foam products with superior mechanical properties.

Three main regions can be identified in the stress–strain curve for the compressive deformation of elastomeric foams [1]. Figure 2 shows the typical stress–strain curve under the uniaxial compression of foams. Linear elasticity is shown in the small strain region, followed by a collapse plateau, and then densification appears accompanied by a rapid increase in the stress. Firstly, the struts bend and the macroscopically linear elastic behaviour is shown. Next, some of the struts start buckling and the slope of the curve decreases due to the increase of the macroscopic stress. Finally, the slope of the curve

**Citation:** Iizuka, M.; Goto, R.; Siegkas, P.; Simpson, B.; Mansfield, N. Large Deformation Finite Element Analyses for 3D X-ray CT Scanned Microscopic Structures of Polyurethane Foams. *Materials* **2021**, *14*, 949. https://doi.org/10.3390/ ma14040949

Academic Editor: Aleksander Hejna

Received: 31 December 2020 Accepted: 11 February 2021 Published: 17 February 2021

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increases again up to the same value as the matrix material, because of the contact between struts. The contribution of microstructures to macroscopic properties depends on these deformation mechanisms.

**Figure 1.** An example of optical microscope images of polyurethane foams.

**Figure 2.** The typical stress–strain relationship of elastomeric foams under the uniaxial compressive stress.

Cell structure geometries are virtually generated and their deformations are analysed to investigate the effect of microstructures on macroscopic properties [2]. The cells were postulated to have same size and the shape of the Kelvin tetrakaidecahedron. The edges of the polyhedron were assumed to be struts that are represented by Euler–Bernoulli beams and the macroscopic elastic properties were analytically calculated. This approach was also expanded to the large compressive strain range up to 70% [3,4] and creep deformations [5]. Other researchers repeated the calculations of Zhu et al. [2], employing a finite element approach, while still making use of Kelvin's cell shape and Euler–Bernoulli beams [6–10]. Okumura et al. [11] and Takahashi et al. [12] analysed the mechanical responses in the [001], [011], and [111] directions, as the Kelvin's cell has anisotropic mechanical properties. Furthermore, closed cell foams have been analysed with shell elements [13]. Modelling the microscopic structures of polyurethane foam materials using the Kelvin's cell is thought to be a simple and effective way of investigating the deformation behaviour.

The Kelvin cell approach assumes that the microstructure is homogeneous; however, in contrast, cell structures are generally heterogeneous. This is a significant disadvantage of the repeated unit cell modelling approach [14]. To model the inhomogeneous structures

of foams, the 2D and 3D Voronoi tessellations were employed and the Voronoi edges were regarded as struts [14–17]. Moreover, faces in Voronoi polyhedrons were assumed as cell membranes in closed cell foams [18,19]. The elastic properties in the small strain region and the compressive stress–strain curves on the plateau region were calculated by the finite element method while using beam elements. Furthermore, although the cross-sectional area of a strut is often assumed to be constant, the central parts of struts are thinner than other parts. The effect of this necking can be taken into account using solid elements [11,12,20–27] or beam elements with variable cross-sectional properties [23,28–33]. In addition, the curvature of struts were modelled [34]. Models that consider the heterogeneity of foams are thought to show better results than Kelvin cell models with straight struts. Dynamic crushing behaviour [35,36] and multiaxial crushing [37] were also analysed.

The use of X-ray computed tomography (CT) is one effective method for obtaining a more adequate model that represents actual foam microstructures. The X-ray CT has been performed to observe the microstructures of various kinds of porous materials, for example, biomaterial scaffolds [38,39], soil materials [40], and polyurethane foams [41]. Therefore, the X-ray CT has also been used to generate the geometries for finite element analyses. For example, finite element models for the microstructure of a trabecular bone was generated based on micro-CT [42]. For artificial foam materials, Jeon et al. [43] analysed closed-cell aluminium foams with finite element models meshed with solid tetrahedron elements. The compressive stress–strain curves of the foam were calculated and compared to the experimental results and the 20.86% volume error was shown up to 5.31% strain. Similarly, linear elastic properties under the small strain regions were obtained from X-ray CT scanned finite element models for ceramic foams [44] and a rigid organic foam [45]. Models that were obtained from the X-ray CT have been effectively used to investigate the mechanical properties of foams under small deformations.

For cushioning products, such as automotive seat pads or bed mattresses, the mechanical properties in the plateau regions are more important than the linear elastic regions. As the slope of the stress–strain curve decreases in the plateau region, elastic foams soften and help to distribute body pressure. Most of the studies employ tetrahedron meshing due to the complexity of the geometry; however, this makes analysing large deformations difficult. To analyse the deformation within the plateau region, hexahedron meshing is required, as it is more suitable for large deformation problems.

This study aims to use X-ray CT scans of foam specimens in order to construct validated finite element (FE) models that can be used to study and manipulate the foam microstructure for achieving desirable stress–strain behaviour in the plateau region. The microstructures of elastic polyurethane foams for automotive seat pads are scanned using X-ray computed tomography and converted to STL files. The STL files are smoothed and converted to solid CAD files with commercial CAD software, so that they can be meshed with a hexahedron dominant solid mesh. The uniaxial compressive deformation of the models are analysed with a finite element method and then compared with the experimental results.

#### **2. Materials and Methods**

The methodology for analysing the deformation of X-ray CT scanned foam materials and the materials supplied to validate its accuracy are explained here. The specimens were scanned using X-ray CT, converted to CAD models, and then analysed with the implicit finite element method. The tools used for this study are either commercially available CAD or open-source software. Moulded elastic polyurethane foams were investigated using the presented method and physically tested to compare with the result of the analyses.

#### *2.1. Materials*

The tested materials were supplied by Bridgestone Corporation in Tokyo, Japan. Polyols, isocyanates, water, and low amounts of other materials were mixed and poured into a 400 × 400 × 100 (mm<sup>3</sup> ) sized mould and then expanded and polymerized. After

demoulding, the foams were crushed between rollers, so that cell membranes were broken and resulted in open-cell foams. The foams were left at least 24 h before proceeding to any other process of the investigation to let the chemical reactions be completed. The foam materials that were investigated in this study are mainly used for automotive seat pads by moulding in product shaped moulds.

#### *2.2. Scanning by the X-ray Computed Tomography*

Specimens from the centre of larger samples were cut into 5 × 5 × 5 (mm<sup>3</sup> ) sized cubes. The X-ray tomography equipment that was employed for this study was the ScanXmate RA150S145/2Be, a product of Comscantecno Co., Ltd. in Kanagawa, Japan. Figure 3, shows an example X-ray CT scan image of the foam. The white parts indicate the foam struts and the black parts are the pores. The size of the pixel was 7.5 (µm). The cross section images were taken by rotating the specimens every 0.18◦ , so that the cell structures could be observed in three dimensions.

**Figure 3.** An example of the X-ray computed tomography (CT) scanned images for the polyurethane foams.

#### *2.3. Converting the Scanned Images to 3D STL Files*

The cross sectional two-dimensional (2D) images were converted to three-dimensional (3D) STL files by Fiji [46], a distribution of Image J2 [47]. Firstly, the scanned images were binarized to black and white images using a threshold of the brightness. The threshold was determined using Otsu's method [48] and then verified by comparing the relative densities measured with the actual specimen and calculated from the computational models. The borders between the black and white pixels were regarded as the surfaces of the struts. Triangles were then applied to the strut surfaces and the resulting surfaces exported as STL files. Figure 4a shows an example STL file.

(c) T-spline interpolation (d) Boundary representation solid model

**Figure 4.** Conversion from the STL files to the boundary representation solid models.
