**4. Conclusions**

In this paper, a method to locally estimate the cell density without the full meshing of the individual cells was presented. This local homogenization was performed using images captured using a digital camera, and a procedure was proposed to analyze the correlation of the cell size and cell distributions to the local stiffness. These results were then fed into a finite element model of a structure subjected to four-point bending. Images taken using a digital camera were analyzed using a custom script written in the MATLAB environment, and an exponential function was proposed to establish a weighting method to allow for a smooth continuous function to be used to represent the local cell variations. This weighted method calculates a density of cells based on their distance from the point analyzed. A periodic boundary was used to estimate the density near the boundaries. Once the local densities were calculated, a micromechanics model that included the constitutive properties of the fiber and matrix along with fiber length and volume fraction of fibers was used to calculate the pointwise Young's modulus of the core as a function of cell density.

The spatially varying Young's modulus was then used in a finite element model to analyze the load-deflection of a four-point bend test. These results were compared to a uniform core model previously validated for the macroscopic response to a physical specimen. The macroscopic response for the spatially varying core density and the uniform core density was graphically equivalent. The local von-Mises stress was then analyzed between the two density distribution assumptions, and it was seen that the stresses from the spatially varying cell density resulted in a ∼11% higher stress within the core as compared to the uniform core.

The present study identifies that the spatial variation of the cell distribution within the core impacts the as manufactured performance, but it is unclear from the present work the sensitivity of the final performance on subtle differences in the spatial variations. Future work will consider sectioning multiple regions within the composite structure to study the variability in location, cell size, and distribution, and its impact on final part performance. This method also allows quality control procedures to remain in place without needing an extensive need for equipment and personnel. The ease of taking images and analysis of the proposed method can be extended to other handheld devices like a cell phone or a tablet.

**Author Contributions:** D.A.J. and D.P.P. were instrumental in the conceptualization of this method. D.P.P. implemented the concept and performed the analysis with assistance from D.A.J. D.P.P. developed the finite element model. D.P.P. wrote the original draft and D.A.J. reviewed the manuscript and edited it. D.A.J. supervised all facets of this research. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Axion Structural Innovations.

**Acknowledgments:** We would also like to thank Axion Structural Innovations for providing the manufactured crosstie cross-section samples.

**Conflicts of Interest:** The authors declare no conflict of interest.
