Individual re-entrant unit cells were drafted using SOLIDWORKS 2018 software. Unit cells with different internal angles, θ, of 55˚, 65˚ and 75˚ were designed. The internal angle of 75˚ was observed to produce a negative Poisson’s ratio by Li et al. [7]. Since the smaller the internal angle, the more the structure would axially and transversely displace when subjected to tensile force, smaller internal angles of 55˚ and 65˚ were considered with no fillet radius. Tensile force was applied in the CS orientation for the sake of comparison to other research that predominantly applied compressive force in the corresponding direction [1] [6] [7]. The data, shows that internal angle 65˚ produced a comparable negative Poisson’s ratio without failing compared to 75˚ at slightly higher load (5N). The 55˚ angle showed smaller Poisson ratio value but handled higher force before failure.

The unit cell with the internal angle of 65˚ was selected as a base design for the remainder of the study. The forces displayed in brackets under the corresponding Poisson’s ratio values in Table 1 were the forces beyond which failure occurred as per von Mises stress analysis.

The basic re-entrant unit cell geometry of the model was designed such that the internal side “a” is double in length to side “b”, as shown in Figure 2. The extrusion of the unit cell was 1.2 mm, and the cell wall thickness was 0.5 mm. The re-entrant unit cells with internal angles, θ, of 65˚ were used and simulated with fillet radii of 0 mm, 0.5 mm, and 1 mm. As shown in Figure 3. Fillets were added to the corners where the connective segments protrude from the re-entrant structure and to the internal corners of the structure.

FEA was performed using SOLIDWORKS on the unit cells. The XZ plane was used as the “Section Plane”. The “Section Depth” was 1.2 mm. The same size of mesh was used for all unit cells for direct comparison. The simulated material used was acrylonitrile butadiene styrene (ABS), the elastic modulus used was 2.415 GPa, the tensile strength was 44.8 MPa, the yield strength was 42.05 MPa, Poisson’s ratio was 0.4, and the mass density was 1.045 gm/cm³ [9]. The “Model

Type” used was “Plasticity-von Mises” to capture the failure point relative to von Mises values and yield strength limit [10].

Tensile force was applied to each unit cell consisting of the different fillet radii on both the FS orientation and the CS orientation. A range of magnitudes of force were applied to each configuration until failure limit was reached.

The Poisson’s ratio was calculated using the equation

ν = − ε x ε z

where ε x is the strain in the direction perpendicular to the force applied and ε z is the strain in the direction the applied force. The sum of the resultant nodal strains in the corresponding x-and z-directions were obtained from SOLIDWORKS and used as the data for the Poisson’s ratio calculations.

Transverse strain had a consistent second-degree polynomial trend against axial strain for all three fillets as shown in Figure 4. Strain are unitless and the plot of strain at the axial direction versus strain at the transverse direction shows an inverse relationship, up to a certain point, then it reverses. The local minimum point, indicates the initiation of auxetic property. The local minimum point of the polynomial trend for each cell design is calculated for both loading directions, CS and FS. It is observed that the minimum point is achieved at lower strain values for sharper angle with no fillet radius (0 mm), compared larger filler radius (1.5 mm).

The direction of load applied on the structure has a direct effect on the amount of axial strain needed to reverse the transverse strain. As presented in Table 2, strain applied on the CS direction reaches the minimum point at 5 times less than the FS strain direction for the 0- and 1-mm fillet radius cell design. The more generous fillet (1.5 mm) required 8 times axial strain to reverse the transverse strain for FS loading direction, compared to the CS direction. The effect of the fillet radius on the axial strain required to reach the minimum point that would lead to auxetic property increases quadratically for both CS and FS loading directions.

Therefore, to achieve an auxetic property, the smallest fillet radius would be the preferred design.

The force applied to each unit cell was incrementally increased until the model failed. Failure is indicated by exceeding yield stress of the material utilizing von Mises stress calculation. The endpoints of the data collected and depicted in Figure 5 are the maximum force value that would reach the von Mises stress point. As the fillet radius increases, for both the CS and FS orientation, the structure can withstand higher load before failure. The force limit increases 1.5 times when fillet radius increases from 0 to 0.5 mm, and the force limit doubles

between 0.5- and 1-mm fillets. Connective segment (CS) loading direction handles less forces and steep decrease of Poisson’s ratio response is observed.

When 5N of force is applied to a non-filleted unit cell in both the FS and CS orientations, the corners of the cell experience greater displacement for CS orientation in the direction of the applied load, compared to the FS orientation, as shown in Figure 6. At this low load, the Poisson’s ratio exhibited a negative value for the CS orientation, which is promising for exhibiting auxetic properties. The disadvantage of the non filletted design, along with the CS loading direction is that the structure withstands the lowest stress before failure.

For the sake of visualizing stress distribution across the unit cell for the 3 fillet radii under study and direction of loading, a 5 N load simulation model is captured and shown in Figure 7. Although filleted corners reduce stress concentration, the stress distribution are observed in similar locations with decreasing

magnitude and decreasing affected area. A non-filleted unit cell experiences highest stresses around the sharp corners of the connective segments when force is applied in the CS direction. A compromise between stress distribution, maximum allowable load before failure, and strain response needs to be considered in designing auxetic structures.