**3. Finite element modeling**

*Composite and Nanocomposite Materials - From Knowledge to Industrial Applications*

In order to obtain the in-plane mechanical properties of the present material, the following ASTM D3039 [7] and ASTM D3518 [8] for tensile and shear properties,

The tensile properties of the unnotched samples, such as the laminate Young's modulus *E*1 and *E*2, Poisson's ratio *v*12, and ultimate strength, were measured by static tension testing of longitudinal [0]4 and transverse [90]4 UD samples. The shear modulus of the samples was measured by loading the specimens whose principal axes are on 45°. Four samples were used in the characterization tests. The INSTRON-5969 testing machine was used in the present study in order to

The testing machine is connected with a computer in order to record the stressstrain curves during the tensile tests. As shown in **Figure 2**, the samples were illuminated by ordinary white light during the experiments. During the loading process, high-resolution images were taken using a digital camera. The experimental results obtained in the present study were processed with a 2D-DIC MATLAB code [9]. In order to perform the DIC tests, we replicate three experiments for each sample, and the results are averaged. This procedure was repeated for all samples to obtain correct stress distributions and reduce the errors that can be related to the speckle

**Remark**: The DHS technique is based on the idea of introducing smaller holes (auxiliary holes) on both sides of the main notch, in order to smooth the flow of the principal stress paths past the main notch, and this will reduce the stress concentration developed around the original notch. This process is similar to the topology optimization technique which is based on logic of "material should be removed

**Remark**: The DIC is one of the powerful noncontact techniques used for measuring the deformations. The DIC technique uses images in order to track the relative displacements of a random speckle pattern point. These displacements are calculated between an undeformed image (reference image) and the current one (the deformed image). In the present work, the authors obtained the full-field

conduct the experimental tests on laminated samples (see **Figure 2**).

from the regions that are less essential for carrying the loads."

strain distributions using a 2D-DIC MATLAB code [9].

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pattern.

**Figure 2.**

respectively, have been used.

*Experimental setup for the DIC analysis.*

2D finite element models were developed using the open-source FE software FreeFem++ [10]. The models were developed using a linear triangular element (three nodes with 2 degrees of freedom per node), because these elements are more adaptable for meshing plates with circular notches. In order to validate the experimental results, the dimensions and the mechanical properties of the numerical models are chosen to be the same as the experimental specimens (a total length of *L* = 250 mm and a width of *W* = 25 mm). In addition, various sizes of hole diameters (2.5, 5, 7.5, and 10 mm) are used in order to obtain different diameterto-width (*D*/*W*) ratios. The length and width of the plates are divided into 70 and 5 elements, respectively.

The FE models of all groups of samples are created, and the stress-strain distributions at the vicinity of notches are obtained. Furthermore, in view of the rapid change in the stress-strain fields around the holes, a higher mesh density with smaller finite elements is adopted and a coarse mesh far from the hole region.

A convergence study is carried out to obtain initial appropriate fine mesh in the open hole zone (the initial mesh size was 30 elements around a hole diameter of 2.5 mm), and then automatic parametric program was developed in order to change the notch size and the mesh refinement automatically, because if one keeps the same element number at the vicinity of the notch boundary, the stress-strain distributions will be affected by changing the notch diameter (see **Figure 3a**). All the numerical models are subjected to a tensile load.

On the other hand, the introduction of the DHS is dependent on the logic of adding auxiliary holes in the areas of low stress near the main cutout. The number of the auxiliary holes in this study is two circular holes (see **Figure 3b**). Various finite element models are also developed for different DHS configurations.

#### **Figure 3.**

*Finite element models of laminated plates under a tensile loading. (a) A plate with a single hole and (b) a plate with DHS.*
