# Development of an Eco-Friendly Head Impact Protection Device
## 2 Material and Methods
### Constitutive Modelling  
For product development purposes, it becomes important to provide reliable constitutive numerical models to represent the materials tested previously. In this work, all the numerical modelling resorted to finite element techniques within the commercial package Abaqus [29]. An extensive and versatile material library is provided, including material models that can simulate the mechanical behaviour of a great variety of cellular materials. The hyperelastic constitutive model is mostly employed for materials displaying an immediate nonlinear elastic reaction to applied loads, keeping this behaviour to large strains. Well-known examples are latex or any other elastomer [29]. Hyperelasticity is described using strain energy potentials, which in simple terms, defines, per unit volume in the undeformed configuration, the strain energy stored in the material. Usually, the strain energy is a function of the strain components at the material point. However, there are numerous strain energy potentials accessible from materials libraries, able to model isotropic elastomers: the Arruda-Boyce, Marlow, Mooney-Rivlin, Neo-Hookean, Ogden form, polynomial/reduced polynomial, among others [29]. In the present work, two different models were used: the reduced polynomial (Equation (1)) and Ogden form (Equation (4)). The employed reduced polynomial strain energy potential follows as:  
\[U=\sum_{i=1}^{N}C_{i0}(\overline{I}_{1}-3)^{i}+\sum_{i=1}^{N}\frac{1}{D_{i}}(J ^{el}-1)^{2i} \tag{1}\]  
Being \(U\) the strain energy (per reference volume); \(N\) is an integer (the polynomial order); \(C_{i0}\) and \(D_{i}\) are parameters depending on temperature; \(\overline{I}_{1}\) is the first strain invariant (deviatoric):  
\[\overline{I}_{1}=\overline{\lambda_{1}}^{2}+\overline{\lambda_{2}}^{2}+ \overline{\lambda_{3}}^{2} \tag{2}\]  
\(\overline{\lambda_{i}}=J^{-\frac{1}{3}}\lambda_{i}\) refers to the deviatoric stretches whereas \(J\) refers to the total volume ratio; \(J^{el}\) is the elastic counterpart and \(\lambda_{i}\) are the principal stretches. Finally, the initial shear and bulk moduli are:  
\[\mu_{0}=2C_{10},\hskip 14.226378 ptK_{0}=\frac{2}{D_{1}} \tag{3}\]  
Figure 4: Stress-strain curves from dynamic compression tests. The Ogden form of the strain energy potential follows as:  
\[U=\sum_{i=1}^{N}\frac{2\mu_{i}}{\alpha_{i}^{2}}(\overline{\lambda_{1}}^{\alpha_{ i}}+\overline{\lambda_{2}}^{\alpha_{i}}+\overline{\lambda_{3}}^{\alpha_{i}}-3)+ \sum_{i=1}^{N}\frac{1}{D_{i}}(J^{el}-1)^{2i} \tag{4}\]  
where \(\overline{\lambda}_{i}\) points to the deviatoric principal stretches \(\overline{\lambda}_{i}=J^{-\frac{1}{3}}\lambda_{i}\); \(N\) is again(the polynomial order; \(\mu_{i},\alpha_{i},D_{i}\) translate temperature-dependent parameters. The initial shear and bulk moduli in the case of Ogden form follows as:  
\[\mu_{0}=\sum_{i=1}^{N}\mu_{i},\ \ \ \ K_{0}=\frac{2}{D_{1}} \tag{5}\]  
The Hyperfoam material model present in Abaqus library is isotropic and nonlinear. It is typically employed to model elastomeric foams showing hyperelastic behaviour. It is commonly used in finite-strain applications, where compressive deformations can reach up to 90%. Again, it derives from a strain energy potential, as follows:  
\[U=\sum_{i=1}^{N}\frac{2\mu_{i}}{\alpha_{i}^{2}}\left[\hat{\lambda}_{1}^{\alpha _{i}}+\hat{\lambda}_{2}^{\alpha_{i}}+\hat{\lambda}_{3}^{\alpha_{i}}-3+\frac{1 }{\beta_{i}}((J)^{-\alpha_{i}\beta_{i}}-1)\right] \tag{6}\]  
being \(N\) the polynomial order, while \(\mu_{i}\), \(\alpha_{i}\), and \(\beta_{i}\) temperature-related parameters;  
\[\hat{\lambda}_{i}=(J^{th})^{-\frac{1}{3}}\lambda_{i}\longrightarrow\hat{ \lambda}_{1}\hat{\lambda}_{2}\hat{\lambda}_{3}=J^{el}; \tag{7}\]  
where \(J^{el}\) and \(J^{th}\) are the elastic and thermal volume ratios.