**1. Introduction**

Sustainability of polymers has become a significant concern both within the polymer industry and within society, with various researchers focusing on the use of polymer composite structures, specifically plastic lumber, made from various polymer solid wastes (PSW) including Polyethylene and Polypropylene (see e.g., [1–4]). With the intent of reducing waste and also reducing the use of high density wooden structures, themselves a depleting resource, recycled polymer composite structures

are being used as crossties (sleepers), marine pilings, short bridges, and other load bearing applications. Researchers have demonstrated the feasibility of these structures as viable load bearing applications, which are being used all over the world [5–7].

The use of fillers, such as glass fibers and natural fibers, are often integrated into the polymer matrix to enhance stiffness, strength, and impact resistance (see e.g., [8,9]). To reduce manufacturing costs, improve light weighting, and allow easy installation of screw fasteners in crossties, chemical blowing agents (CBA) such as azodicarbonamide (ADC) are introduced into the extrusion process [10]. Kord et al. [11] studied the effect of the blowing agen<sup>t</sup> ADC on composites made from High-Density Polyethylene (HDPE), wood flour, and nanoclay. They observed that the increase of ADC content resulted in increased cell size and average cell density, which decreased the overall density of the structure and also reduced the tensile modulus. Therefore, the amount of ADC induced into the composite needs to be precise to achieve an acceptable density spatial distribution to maximize the strength to weight ratio.

When using chemical blowing agents, melt temperature and pressure are key components for nucleation to occur [12]. As PSWs in their molten state are mixed with ADC, the blowing agents undergo a chemical reaction by dissolving into a gas as the pressure in the melt increases. Due to a pressure differential, the nucleation of the individual cells is caused when the gas is released within the mold. The density of the foam varies based on the gas fraction reacted in the molten PSWs, gas lost into the atmosphere, and the rate at which the gas decompresses [12]. As the nucleated cells achieve their maximum growth, the glass fibers in the mold are forced into the dense outer solid shell adding reinforcement to the shell where the maximum stresses are expected [6]. For the expected loading applications, the foamed core does not yield the same stress requirements as the exterior shell. Ruiz et al. [12] observed that when blowing agents are not activated properly, the inactivated ADC particles can form clusters, inducing poor cell morphology and leading to poor internal load transfer.

The cooling rates of the mold have an impact on the formation of cells as well [13]. When the mold is externally cooled using water channels, the shell layer thickness varies based on the total heat transfer between the mold walls and the molten polymer. The molten reinforced polymer is cooled faster in the exterior region limiting the cell growth to the core with a fully densified outer shell. Yousefian et al. [14] studied the Polypropylene/Nano-crystalline cellulose composite and found that by varying mold temperatures between 30 °C and 80 °C, the ratio between the shell and the core could be altered by 5%. Tissandier et al. [15] studied the effects of temperature, blowing agents, and fiber content on composites made from HDPE, flax fiber, and ADC, which were mixed and manufactured using injection molding. This process included applying a temperature gradient (0–60 °C) inside the mold and achieved acceptable microcellular asymmetric structure with high fiber content, blowing agents, and mold temperature. They showed that if a bar is cooled using a gradient, the cell wall thickness on the cooler side is thicker than the warmer side. In their configuration, they concluded that the cold side of the mold controlled the final morphology of the foaming process.

The mold temperature also influences cell size, cell density, and distribution [16]. It was found that as the mold temperature decreases in injection molded HDPE structural foams foamed with ADC, a small but measurable increase is observed in the average cell density and cell size [14]. Cell growth can be limited by increasing the fiber content, which increases the viscosity of the melt and potentially impacting processability [17]. These variations of cell size and its dependence on material properties, namely elastic stiffness, have been studied by Redenbach et al. [18] by using Laguerre tessellations, a weighted mathematical generalization used to model foams using convex cells. They found that as the cell aspect ratio, defined as the ellipsoidal major to a minor axis, increased from spherical, the effective elastic stiffness decreased.

Cell wall thickness also impacts the material properties of foams. Barbier et al. [19] studied Voronoi closed-cell foams (which is a mathematical partitioning of regions in proximity to grid points in space) under a variety of imposed strains caused by wall stretching, and they found a correlation between the density and cell irregularity on the effective elastic and plastic properties. They concluded

that under elastic deformation, material properties rely only on relative density. Whereas, under plastic deformation, there is a dependence on both relative density and structural deformation.

To calculate the material properties of closed cell foams, many researchers have used various micromechanical approaches to estimate the effective homogenized medium, specifically, the tensile modulus, as either an isotropic or anisotropic structure (see e.g., [20]). The micromechanical approaches do not account for various combinations of closed cells, including cell sizes, cell shapes, and relative density, to predict the material behavior. Zhang et al. [20] have reviewed various micromechanics models (see e.g., [21–25]) using uniform closed cells by studying HDPE foams made by compression molding techniques. They suggested that the differential scheme [24] and the square power-law suggested by Moore et al. [26] are the most accurate within the cell volume fraction of 0–55%. In the present research, the square-power law will be used. Recently, Lo et al. [27] studied transversely isotropic PVC foams and modeled and predicted the elastic stiffness using unit cell representation for the foam microstructure. They modified the Halpin–Tsai equations by introducing an apparent volume fraction to calculate the material properties and replaced the actual resin volume fraction. Tucker and Liang [28] suggested that the Halpin–Tsai equations are most applicable for long fibers, but for the short fiber the reinforced system employed in the present study, the Tandon and Weng [29] model is more appropriate. The present paper uses the closed form solution of the Tandon and Weng model, implied by Tucker and Liang as derived in the work by Zhang [30], for the fiber reinforced micromechanics modeling.

It has been shown that many structural foams have a spatially varying density gradient that often decreases in density from the outer solid shell to the closed cell foamed core [31]. To understand the complex nature of foams with varying cell sizes, density, and cell size distribution, various studies using images from Scanning Electron Microscopy (SEM) [32], digital microscopy [33,34], and Micro-CT/X-ray [35–37] were used. Sadik et al. [33] studied images obtained from a digital microscope at 5× magnification and used the micrographs to binarize cells and measure the void fraction using ImageJ software. The void fraction was then used to calculate linear elastic material properties of the foamed polymer such as tensile, shear, and flexural modulus. These images were analyzed for the foamed structures by using micrographs and calculating the void fractions. Davari et al. [32] studied chemically linked polyethylene foams and the foam morphology using images from SEM. They concluded that density is the dominant parameter for predicting material properties, but when the density is uniform, there is a small but measurable contribution to the material properties as a function of the cell size distribution.

Recently, Zhu et al. [36] used MATLAB image processing techniques to study aluminum foams by examining images obtained from X-ray computed tomography (*μ*CT) and extended the foam characteristics into a Finite Element Model (FEM). They showed that the image processing techniques can be used to study the porosity of aluminum foams and can reasonably predict the compression behavior. In an attempt to characterize porosity using inexpensive and quick modeling techniques, Yunus et al. [38] studied the porosity of polyurethane foams by using a Canon camera, a black box, and a LaserJet scanner. They then processed the resulting images in a custom MATLAB subroutine. They also validated their results for the porosity measured by each instrument using a stereomicroscope and an SEM. They concluded that the black box results were the most comparable to the results obtained from an SEM.

In a recent work by the authors [39], the composite structures made from recycled materials were studied by integrating the fiber aspect ratio, uniform cell density, orientation state of the fibers, and the constitutive properties of the raw materials. These properties were used as inputs into micromechanics models, and the effective elastic moduli of the solid shell outer core of the composite as well as the foamed core were predicted. Stress values of the raw material were measured at a 0.2% reference strain, and the reference stress of the shell and core formed a nonlinear stress–strain relationship using a modified rule of mixtures and micromechanics models. The predicted elastic moduli and reference stresses were then used in a nonlinear finite element model to predict the flexural deflection response of the composite crosstie under a 4-point bend test using ASTM D6109-13. The results from the model were compared to experimental tests from sixteen different samples, and the finite element results were found to be within the experimental variation.

In this paper, a weighted method is proposed and demonstrated to characterize the density of glass fiber reinforced HDPE/PP (High-Density Polyethylene/Polypropylene) foamed core with cells of varying sizes and distribution using a commercial off-the-shelf (COTS) EOS60D Canon Camera (Canon Inc., Tokyo, Japan). This is achieved by taking 2D images of the cross-section, such as that depicted in Figure 1, using the COTS camera and obtaining a density map by measuring areas of cells contained within the image using image processing techniques implemented in MATLAB (2018a, MathWorks, Natick, MA, USA). The cross-sectional area of the cells within a sample image is tabulated and their results are validated against results obtained using a high-resolution VR-3100 3D digital microscope (Keyence Corporation of America, Itasca, IL, USA). The density map is translated to a point by point mapping of the elastic modulus using a differential scheme. A finite element model is then implemented using a gridding technique to map the spatial varying density results obtained from the imaging results, and the solutions are compared to those from the uniform density method. The von-Mises stress is analyzed for comparison purposes and the results that include local variations in the cell density yield a ~11% higher peak stress as compared to that of the uniform density core. This method will allow for safe design considerations for various applications due to the stresses caused by variations in cell size and distributions.

**Figure 1.** Example of a cross-section.
