**3. Results**

#### *3.1. Cell Generation and Cell Size Distribution*

Using the image thresholding discussed previously, along with the automated removal of any false-positives, the image of the cross-section was analyzed, and the results for the three samples shown in Figure 3 are shown in Figure 7a. Notice that each cell region is converted to a circular region with only the center position and radius archived in a look-up table for fast processing. The histogram of the cell effective pixel radius distribution is provided in Figure 7b. Observe that the cell size distribution varies between each of the three samples. Samples 1 and 3 have similar cell size trends, with a number average for the effective radius of 8.4 pixels and 11.04 pixels, respectively, and a weighted average for the effective radius of 17.04 pixels and 22.7 pixels, respectively. Conversely, the cells in sample 2 have a number and weighted average for the effective radius of, respectively, 9.6 pixels and 14.8 pixels. The weighted average of samples 1 and 3 are greater than sample 2 because the effective radius in the samples 1 and 3 have larger cells compared to sample 2 overall.

**Figure 7.** Various stages within the MATLAB image processing technique. (**a**) Cell area identification for samples 1 through 3. (**b**) Histograms of the cell size distribution for samples 1 through 3. (**c**) Spatially varying effective local density for samples 1 through 3 within the foamed core.

Although it is not obvious from visual observations, sample 2 contains fewer cells than sample 1 and 3. However, it is clear from the distribution that there is a more uniform distribution of cell sizes for the cells in sample 2, whereas in samples 1 and 3 there is a higher probability of the smaller cells with *r* < 10 pixels that are more dispersed within throughout the core along with several large cells with *r* > 50 pixels. It is important to note that the overall density of the three samples is similar, with ~14% of the surface area representing a cell, as are the basic manufacturing parameters, but it is clear from Figure 7c that there is some variation in both the size and spatial distribution between the three samples. This variability of cell size and distribution is due to the variability within the use of recycled materials as well as the manufacturing parameters.

The spatially varying density estimation from Equation (5) shows regions of significant density variation in the cross-section for each of the three samples. The point-wise grid with an *R* = 300 pixels gives a homogenized density range from 0–60% relative to that of the fully densified polymer composite. Each of the three samples has substantial variation in the spatial density as observed in Figure 7c. The larger open cells provide more weight in the regional density homogenization, and they affect the regions around them by causing a reduction in the local density. It is worth noting that each sample, although having the same overall density, has a unique spatial distribution of density. Specifically, in sample 1 the cells tend to be well dispersed, whereas in sample 3, there is a clear region in the core with a higher concentration of cells and a slow increase in the density as one moves away from the center region. This dispersion of cells can be observed through the image thresholding approach coupled with the spatially varying density weighting. The spatial varying density obtained in Figure 7c is then saved to a lookup table for later use in a finite element subroutine.

#### *3.2. Finite Element Analysis*

The pointwise Young's modulus was calculated from the spatially varying density using Equation (9) using the data shown in Figure 7c. This spatially varying modulus was then read into COMSOL as discussed in Section 2.4 and the results are presented in Figure 8 for each of the three samples. The Young's modulus was predicted for the shell region as *Ec* = 1.73 GPa [39], where, *Ec* is the Young's modulus of the HDPE, GFPP composite, and is used for the solid outer shell. Figures 7c and 8 are similar in spatial distribution as the localized density relates to the localized stiffness from Equation (9).

**Figure 8.** Spatially varying Young's modulus obtained from image analysis for (**a**) sample 1, (**b**) sample 2, (**c**) sample 3.

At the boundary between the foamed core and uniform density shell, a linear blending technique was implemented to allow for a smooth and continuous transition of the material properties within the finite element model cross-section. This blend is defined as

$$E\_l = E\_s(1 - \alpha) + E\_f \alpha \tag{11}$$

where *Ef* is the foamed core modulus calculated using Equation (9) and *Et* is the modulus approximated near the transition region around the shell and the core. The constant *α* is a normalized scalar that linearly increases from 0 to 1, with 0 corresponding to a point fully within the shell region and 1 corresponding to a point within the foamed core region. For the results presented, a transition region of 0.1 inch was taken with the region centered about the transition between the core and the shell.

Performing the FEA simulation for the model discussed in Section 2.4 with the applied load of 111 kN (25,000 lbs), results were plotted to study the load-deflection and stress behavior of the structure. The deflection of the core structure for sample 3 is shown in Figure 9, and the von-Mises stress on the core is superimposed on the deformed structure.

**Figure 9.** FEA results for core stress under deformation for sample 3.

To compare the load-deflection of the four models, the results from samples 1, 2, and 3 were compared to a model that one would obtain if the core had a uniform distribution. The results are shown in Figure 10a and it is observed that all four scenarios yield equivalent results for the overall load-deflection behavior. Equation (12) is used to convert the load to stress and displacement to strain, and the results are shown in Figure 10b.

**Figure 10.** Results from FEA (**a**) comparing load–deflection from various samples and (**b**) comparing stress–deflection from various samples.

The plotted stress, *S*, value is defined in the ASTM standard D6109 and is plotted against the strain *r*, defined as [44]

$$S = PL/bd^2$$

$$\sigma = 4.70 \text{Dd/} \, ^\circ \text{L}^2 \tag{12}$$

where, *L* is the total span length, *b* is the width, *d* is the depth of the beam as shown in Table 2, *P* is the applied load, and *D* is the deflection due to the applied load.

The real benefit of the spatially varying cell density homogenization approach allows one to look carefully at the point-wise stress distribution within the core region. Notice in Figure 11a that for the uniform core with no spatial variation, the internal von-Mises stress varies linearly in *y* with a zero-stress state at the geometric centroid. This behavior changes quickly for the three samples, with spatial variations in the cell density as shown in Figure 11b–d.

**Figure 11.** Core von-Mises stress at mid-point for each four-point bend simulation in the units of MPa for (**a**) uniform core, (**b**) sample 1, (**c**) sample 2, and (**d**) sample 3.

Notice that the stress state is no longer a linear function in the vertical dimension, and there are regions where the peak stress is measurably higher within the core than previously. The peak von-Mises stress predicted for sample 3 is a stress value of 13.7 MPa at peak loading, whereas the peak stress in the uniform core is 12.3 MPa for the same loading configuration. There is an ~11% variation in maximum stresses of the samples when compared to the uniform core. Variations in the peak von-Mises stress within the core region at the center-plane of the beam loaded in 4-point bending are shown in Figure 12 as a function of increasing deflection.

**Figure 12.** Stress comparison for the various FEA models with image cross-sections and the uniform core.

It is evident that with each of the different cores, the peak von-Mises stress changes as a function of both cross-head displacement and local cell distribution. It is interesting to note in Figure 12 that there is little difference in the results from the three samples with spatially varying density, but there is a difference in the results with and without spatially varying density. Figure 11a shows the von-Mises stresses with a uniform core, while Figure 11b–d show maximum stress with cores from samples 1–3.

Although the stresses with samples 1–3 have higher stresses overall, the main load and stress are distributed within the outer shell. In addition, the distribution has a relatively smooth transition from the shell to the core, even if there is variability within the core, it is carrying a nominal amount of the overall load. The present study focused mainly on the onset of failure. Various researchers have studied advanced methods for tracking crack propagation within the failed composite (see e.g., [45]), and it is worth studying in future works.
