# Preliminary Failure Analyses of Loaded Hot Water Bottles
## 2 Methodology
### Experimental Approach
#### 2.2.3 Finite-Element (FE) Analysis
Since no lateral forces were applied during the tensile tests, principal stretches can be simplified to \(\overline{\lambda}_{2}=\overline{\lambda}_{3}=\overline{\lambda}_{1}^{- \frac{1}{2}}\); hence,  
\[U=\sum_{i=1}^{N}\frac{2\mu_{i}}{\alpha_{i}^{2}}\left(\overline{\lambda}_{1}^{ a_{i}}+\overline{\lambda}_{1}^{-\frac{a_{i}}{2}}+\overline{\lambda}_{1}^{- \frac{a_{i}}{2}}-3\right) \tag{11}\]  
\[U=\sum_{i=1}^{N}\frac{2\mu_{i}}{\alpha_{i}^{2}}\left(\overline{\lambda}_{1}^{ a_{i}}+2\overline{\lambda}_{1}^{-\frac{a_{i}}{2}}-3\right) \tag{12}\]  
\[\sigma=\frac{\partial U}{\partial\overline{\lambda}_{1}}=\sum_{i=1}^{N}\frac {2\mu_{i}}{\alpha_{i}^{2}}\left(\alpha_{i}\overline{\lambda}_{1}^{a_{i}}- \alpha_{i}\overline{\lambda}_{1}^{-\frac{a_{i}}{2}-1}\right) \tag{13}\]  
where  
\[\overline{\lambda}_{1}=1+\varepsilon \tag{14}\]  
Figure 5: The hot water bottle geometry was created as a CAD file using (**a**) Creo Parametric 11 software, and its coordinates were represented in (**b**) MATLAB to facilitate geometry meshing. Consequently, the stress-strain association can be defined in a uniaxial approach as follows [20; 21]:  
Figure 7.0  
Considering the rubber nonlinear response to loading, Ogden hyperelastic material models of Orders 1 to 4 were considered for demonstrating the material behaviour in the current study, as shown in Figure 7.1 From the second order (N = 2), the root mean squared (RMS) error was not falling lower to any further extent, indicating that orders higher than two do not improve the model fit to experimental data. Therefore, the hot water bottle material model was set as a hyperelastic material of \(\mu_{1}\) = 0.397, \(\mu_{2}\) = 0.443 MPa and \(\alpha_{1}\) = 1.38, \(\alpha_{2}\) = 1.775 of the second-order (N = 2) Ogden model, with an RMS of 0.014 MPa after 90 iterations. PSO optimisation stopped because the step size was less than the value of the step size tolerance of \(10^{-6}\), and constraints were satisfied to within the value of the constraint tolerance. The material density was derived from weight and volume analysis and set to 1.2 g/cm\({}^{3}\) for the rubber bottle and 7.72 g/cm\({}^{3}\) for mild steel plates. The elastic modulus and Poisson's ratio were set to E = 200 GPa and \(\
u\) = 0.3 for mild steel plates. In the FE modelling process, the two mild steel plates sit each above and below the hot water bottle, mimicking the experimental compression test. For the upper plate, rotations around the principal axes were restricted, and movement was only allowable along the vertical direction (Z-axis). The lower plate was fixed with both translation and rotation constrained in all directions. The simulation was conducted in two steps. The first step was a static step of a normalised time of 1 with an increment of 0.1 to capture the model's behaviour in good resolution. Secondly, a dynamic step was conducted with downward displacement of the upper plate towards the bottle in the same experimental compression rate and displacement. In order to simulate the effect of water inside the hot water bottle, an internal pressure of 50 kPa was applied on its internal surface. The pressure value was selected (via a trial-and-error approach) through a preliminary investigation where the initial model volume approximately matched the experimental hot water bottle volume. In the simulation, a simplified assumption was made by modelling the water inside the bottle as an internal pressure rather than a fluid element to reduce the complexity of the model. Hence, the initial water pressure value was empirically adjusted to inflate the model to  
Figure 7.2 FE model representations of the loading process: (**a**) initial position, (**b**) simulating water filling the bottle, (**c**) simulating upper plate downward displacement, and (**d**) bottle fully compressed. the experimentally recorded bottle volume and compensate for the lack of a fluid element simulation component.