## 3 Finite Element Model
### Model Description
In [PERSON]'s model, the only variable that is considered is I\({}_{1}\) in the case where the material is assumed to be incompressible. In the case of hyperelastic compressible materials, the dependence on I\({}_{3}\) is added. Because only one of the three invariants of the Cauchy-Green strain tensor is considered, [PERSON]'s model is sometimes called a reduced polynomial model. This is because the strain energy density function is represented by a polynomial. In ABAQUS, hyperelastic materials are described in terms of the \"strain energy potential\", which defines the strain energy stored in the material per unit reference volume (volume in the initial configuration) as a function of the strain at that point in the material. Several forms of strain energy potentials are available in ABAQUS to model approximately incompressible isotropic elastomers [45]. The form of the Yeoh strain energy potential is:  
\[U=C_{10}(T_{1}-3)+C_{20}(T_{1}-3)^{2}+C_{30}(T_{1}-3)^{3}+\frac{1}{D_{1}}(J^{el }-1)^{2}+\frac{1}{D_{2}}(J^{el}-1)^{4}+\frac{1}{D_{3}}(J^{el}-1)^{6} \tag{4}\]  
where \(U\) is the strain energy per unit reference volume, and \(C_{i0}\) and \(D_{i}\) are temperature-dependent material parameters; it is the first deviatoric strain invariant defined as:  
\[\overline{I}_{1}=\overline{\lambda}_{1}^{2}+\overline{\lambda}_{2}^{2}+ \overline{\lambda}_{3}^{2} \tag{5}\]  
where deviatoric traits \(\overline{\lambda}_{i}=J^{-\frac{1}{3}}\lambda_{i};J\) is the total volume ratio; and \(J^{el}\) is the elastic volume ratio that relates to the total volume ratio J and the thermal volume ratio \(J^{th}\), according to:  
\[J^{el}=\frac{J}{J^{th}} \tag{6}\]  
with \(J^{th}\) in turn defined as:  
\[J^{th}=(1+\varepsilon^{th})^{3} \tag{7}\]  
where \(\varepsilon^{th}\) is the linear thermal expansion strain obtained from the temperature and isotropic thermal expansion coefficient. The \(\lambda_{i}\), finally, are the main stretches. The initial shear modulus and bulk modulus are given by:  
\[\mu_{0}=2C_{10}\hskip 28.452756 ptK_{0}=\frac{2}{D_{1}} \tag{8}\]  
A convenient way to define a hyperelastic material is to provide ABAQUS with data from experimental tests [45]. ABAQUS then calculates the constants by a least-squares method. The experimental tests for which ABAQUS can fit the data are:  
* Uniaxial tension and compression. * Equi-biaxial tension and compression. * Planar tension and compression (pure shear). * Volumetric tension and compression. Thus, Yeoh parameters were calculated in ABAQUS as C10 = \(-\)2.977 MPa and C20 = \(-\)2.812 MPa, based on uni-axial tensile test curves carried out at room temperature; the values of nominal stress and strain considered are shown in Table 5.  
A further investigation using ABAQUS calculation has been computed by assuming that another compressible hyperelastic Mooney-Rivlin (M-R) model was applied after [46] in form (9). M-R parameters, determined in ABAQUS for different temperature curves of the uni-axial tensile test of polyurethane PS, are listed in Table 6 and were used in further numerical analysis. \[W=C_{10}\big{(}\overline{I}_{1}-3\big{)}+C_{01}\big{(}\overline{I}_{2}-3\big{)} +\frac{1}{D_{1}}\Big{(}J^{el}-1\Big{)}^{2} \tag{9}\]