# Experimental and Numerical Investigation of the Mesoscale Size Effect in Notched Woven Composites
## 2 Materials and Methods
### Numerical Model  
A simplified mesoscale numerical model was developed, which accounts for the heterogeneity of the material under study to correctly predict the strain and stress localization. The complex strain distribution observed in woven composites originates from the orthotropic fiber bundles (tows) interlocking in perpendicular directions. This structure, which needs to be described in the model, is visible as it appears on the surface of the specimen in Figure 2a and schematically in Figure 2b. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Specimen** & **1 [mm]** & **w [mm]** & **D [mm]** \\ \hline C20 & 250 & 24 & Unnotched \\ C21 & 250 & 24 & 2 \\ C22 & 250 & 24 & 4 \\ C23 & 250 & 24 & 8 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Specim dimensions. Figure 1: Images of one of the notched specimens (C22) on the test fixture, captured with one of the cameras used for DIC: (**a**) reference image used to build the model; (**b**) speckle image taken during testing. The DIC area-of-interest is overlayed on both images. Composite laminates are often modeled in a simplified form using 2D shell elements. In the model developed, the mesoscale-level heterogeneity has been described by locally changing material directions--or, in other words, dividing the component into two orthotropic materials with identical properties and orthogonal orientations. The 3D woven structure was replaced by two layers with opposite distributions of material, as depicted in Figure 2c. These were implemented as two layers of integration points in a single shell, thus allowing a sizeable reduction in computational effort when compared to more standard multiscale approaches. Properties of the constituent material, while not identifiable through classic mechanical testing, were obtained by minimizing the discrepancy between the experimental and numerical full-field strain. The finite element solver used was LS-Dyna. The two-material model was built using scripts in MATLAB and Python, and the open-source software GMESH v.4.8.4. The process is summarized in Figure 3.  
At first, a map of the vertical and horizontal fiber tows on the outer face of the specimen was built via a MATLAB script by processing the image of the specimen surface taken immediately before the test. The coordinates of every point on the specimen surface were determined with calibration of the stereoscopic DIC. Figure 3a shows a detail of the specimen, while Figure 3b-f illustrates the process developed to build its material map. Thanks to their differences in terms of reflectivity, in the grayscale images, the tows of fibers emerging to the surface appear brighter when oriented along the vertical direction than along the horizontal direction. Using a K-means clustering algorithm operating on the grey level of the image, defined with integer values between 0 (white) and 255 (black), the tows were identified. The most straightforward method was found to divide the image into three clusters, and to flatten the two least numerous into one, giving good tow identification. Figure 3: Stages in the process of building the specimen material map: (**a**) raw.tiff image captured with DIC cameras, (**b**) after K-means clustering and filtering, (**c**) centroid of the tows calculated from the clusters overlaid on the initial image, (**d**) fitting through optimization of a geometric grid (red circles) to the centroids (blue dots), (**e**) selection of the vertical tow area, (**f**) material orientation map used to prepare the FEM model via a Python script. Figure 2: Progressive abstraction of the 2 \(\times\) 2 will weave unit cell: (**a**) imaging of the specimen during testing; (**b**) schematic representation, highlighting the vertical (warp) and horizontal (wref) fiber directions; (**c**) simplified model, with two materials to represent the two orientations and two layers to represent the weave. Global (in black) and local (in orange) material directions are shown. The tensile loading direction is \(y\); 1 is the “fiber direction”, 2 is the “matrix direction”.