# Characterization of Shear Strain on PDMS: Numerical and Experimental Approaches
## 3 Materials and Methods
### Numerical Simulation  
The numerical simulation was executed using a commercial finite element method (FEM) software ANSYS(r) R18.1 academic research mechanical license. A model with a geometry similar to the specimens, boundary conditions matching the experimental testing and a discretized domain finite element mesh were required to perform the numerical simulation. The loading and kinematic conditions were identical to those used in the experimental test. Nonlinear hyperelastic behaviors based on the constitutive models of Mooney-Rivlin, Yeoh, polynomial and Gent were considered for the PDMS material. Many authors recommend these models for simulating PDMS materials [19; 42]. The application of these models required determining several constants, which were identified from the experimental curves of the tensile tests. However, the steel bars were considered to have a linear elastic behavior and the mechanical properties used were [PERSON]'s modulus of 210 GPa and Poisson's ratio of 0.3. In the present simulation, the PLANE183 element considering the plane stress condition was used. PLANE183 is a higher-order 2-D, 8-node element with two degrees of freedom at each node and a quadratic displacement behavior well suited for the hyperelastic application and modelling irregular meshes. As with the previous equations, all the present constitutive models need to have specific constants and coefficients determined, and for that, the experimental stress-strain curve is used. The constants and coefficients of the material are estimated from hyperelastic curve-fitting. Figure 5 shows the curve-fitting for the constitutive models used in this work. The black circle represents the experimental data, and the others represent the constitutive models. The geometry of the specimen was experimentally tested to implement the numerical solution. With the same dimensions and conditions as the experimental study, the boundary conditions were the bottom edge clamped and constant displacements in the vertical direction on the upper edge of the model. Table 1 presents the applied displacements in the numerical simulations for both geometries. Numerical simulation 1 corresponds to the 2 mm PDMS thickness and numerical simulation 2 is 4 mm PDMS thickness. The displacement values were the same that were used in experimental tests. For the shear-stress study, the four constitutive models presented before were used to describe the properties of the sample, Mooney-Rivlin 3 parameter, Yeoh 3 rd Order, polynomial 3 rd Order, and Gent. All the models converged to a similar response. For this study, as the focus is on the reaction of the central region of PDMS to the shear test, a path operation command was defined in the numerical simulation. This function makes it possible to trace a path and analyse the mechanical behavior in a specific region. The defined path is shown in Figure 6, where XY shear strain was studied in both samples, on the 41 nodes selected. The numerical simulations were used as a mesh divergence with 20 finite elements horizontally spaced equally in the PDMS region. Five different areas were created in the geometric model to obtain a better result refinement at the sample. In the discretization of the areas adjacent to the PDMS, a 0.5 mm ratio was used for the elements to create a greater convergence in the model. The mesh has 1000 elements and 3161 nodes. \begin{table}
\begin{tabular}{c c} \hline
**Numerical Simulation** & **Displacement [mm]** \\ \hline
1 & 0.5 \\
2 & 0.2 \\ \hline \end{tabular}
\end{table} 
Table 1: Displacements for the numerical simulations. Figure 5: Stress-strain curve solution for different constitutive hyperelastic mathematical models. Figure 6: Discretization of nodes and elements for simulation. In the analysis option, the large displacement static was chosen because the maximum number of sub-steps was defined as 150 and the minimum as 100. It is necessary to define a convergence criterion to solve this type of nonlinear structural analysis.