# Characterization of Shear Strain on PDMS: Numerical and Experimental Approaches
## 2 Numerical Constitutive Models  
Currently, FEM has been used to study the mechanical behavior of hyperelastic materials. These studies are based on mechanical models obtained from experimental uniaxial tests. To numerically reproduce the nonlinear hyperelastic behavior of this material, it is necessary to develop new and more accurate constitutive models. The developments of hyperelastic models are supported in two different theories, the micromechanical and macro mechanical models. Micromechanical models are developed from chemical manufacture and are based on the concept of the unit cell. The second theory, phenomenological models are based on the material behavior observed during experimental tests. For developing these models, it is necessary to know the mechanical behavior of the material through the experimental tensile test [26]. Hyperelastic materials are known to have a nonlinear relationship between stress and strain. Thus, hyperelastic material behavior is usually defined based on strain energy or stored energy. Hyperelastic materials are commonly defined as having nonlinear mechanical properties, presenting high deformation rates. The theory of hyperelastic material behavior, also known as Green elastic material, is defined as a function of the Helmholtz free energy, also called the strain energy or stored energy (\(\Psi\)). This describes the behavior of this class of materials in terms of mechanical energy and can be defined according to the following equation:  
\[P=\frac{\partial\Psi}{\partial F} \tag{1}\]  
or, more generally:  
\[P=-pF^{T}+\frac{\partial\Psi}{\partial F} \tag{2}\]  
where \(P\) is the first stress tensor of Piola-Kirchhoff, \(F^{T}\) is the transposed deformation gradient and \(p\) is a multiple of Lagrange obtained according to the state of tension \(T\). Helmholtz free energy (\(\Psi\)) is a thermodynamic potential that measures the useful work for a closed thermodynamic system with constant temperature and volume [27]. A model of hyperelastic materials depends on the definition of the strain energy function, which assumes different shapes, according to the material or class of materials considered. This function is obtained from symmetry and thermodynamic energy considerations [27]. For simplicity, it is assumed that the material is isotropic and incompressible. As an isotropic material, the strain energy function (\(\Psi\)) depends on the invariants of the deformation:  
\[\Psi_{isotropic}=\Psi(I_{1},I_{2},I_{3}) \tag{3}\]  
where the invariants are defined as:  
\[ \begin{split} I_{1}=\sum\limits_{i=1}^{3}\lambda_{i}^{2}\\ I_{2}=\sum\limits_{i,j=1}^{3}\lambda_{i}^{2}\lambda_{j}^{2}\ i \
eq j\\ I_{3}=\sum\limits_{i=1}^{3}\lambda_{i}^{2}\end{split}  \tag{4}\]  
where \(\lambda 1\), \(\lambda 2\) and \(\lambda 3\) are the principal deformations. If the material is too incompressible, the third invariant, \(I_{3}\), is 1 and Equation (5) is defined as:  
\[\Psi_{1}=\Psi(I_{1},I_{2}) \tag{5}\]  
The constitutive equations of hyperelastic models are determined from the Cauchy tensor equation and the calibration of the main experimental tensile tests (uniaxial and biaxial). There are several constitutive equations for hyperelastic models, the most used models being the Yeoh, Mooney-Rivlin, polynomial, and Gent models. [PERSON]'s constitutive model is the most suitable for situations of an incompressible material, it was first presented in the early 1990s [28]. Moreover, the Yeoh model of hyperelastic materials has proved popular because it depends only on the first invariant deformation, (\(I_{1}\)). The Yeoh energy density function is given by [29]:  
\[\Psi=\sum_{i=1}^{3}C_{i}(I_{1}-3)^{i} \tag{6}\]  
where \(C_{1}\), \(C_{2}\) and \(C_{3}\) are constants of the material that are determined from the experimental tests. The Mooney-Rivlin model is one of the most known and used hyperelastic models, mainly in numerical simulations of rubbers and their derivates. This model is very widespread because it presents good convergence for a relatively large range of deformation [30].