## 2 Materials and Methods
### Thermal and Structural Analysis  
To determine application feasibility, thermal and structural analyses were conducted in 1D, 2D, and 3D. The idea was to use the 1D analysis to obtain a reasonable confidence level in the overall application of the PerFORM, the 2D analysis to iterate on major design features, and the 3D analysis to mitigate risk and obtain detailed stress distributions that would take into account all major aspects of the assembly. Failure criteria were also a unique challenge associated with this assembly. Materials typically experience ductile or brittle fracture when loaded past the ultimate strength. One arbitrary distinction based on elongation at failure states that ductile failure constitutes a strain of at least 0.05 before a fracture occurs [PERSON] (1998); [PERSON] (1998). By examining stress-strain curves, ductile materials might level off more due to yielding while brittle curves will end more abruptly and appear more linear. Ductile materials are typically associated with metals while common brittle materials consist of ceramics. PerFORM is described as a ceramic-like  
Figure 6: Thermal expansion coefficients for 6061 Aluminum and PerFORM. thermost plastic and tends to flake or crack from machining operations, indicating that its nature is likely brittle. However, the stress-strain curve at 200 \({}^{\circ}\)F reaches an ultimate strain of 5% and appears to show some evidence of yielding before fracture. To further complicate the situation, the curves at room temperature and 300 \({}^{\circ}\)F exhibit brittle behavior. It appears that failure near the glass transition temperature would be more ductile in nature compared to other temperatures. To instill confidence and mitigate risk, failure criteria for both ductile and brittle materials were used and the stresses had to be manageable for both analysis methods. Furthermore, the ultimate strength was derived using a tensile test, but significant compressive stresses were expected to be present. Since the ultimate compressive strength (assuming no structural instabilities) is typically higher or similar to the ultimate tensile strength, the latter value was used for the analysis [13,14]. This is especially true for brittle materials as they are typically significantly stronger under compressive loads. Specific failure criteria included the distortion energy theory (von Mises stress) and maximum normal stress theories. The distortion energy theory compares the von Mises equivalent stress to the yield strength (in this case the ultimate tensile strength) to determine a safety factor and is well-regarded for ductile materials. To provide insight into brittle failure, the maximum and minimum normal stresses were compared to the ultimate strength to determine a safety factor. In theory, brittle failure will occur when the maximum normal stress (tension) or the magnitude of the minimum normal stress (compression) exceeds the ultimate strength. As a conservative estimate, the ultimate strength used in the analysis was 8900 psi, which is representative of the failure point from the weakest build direction at 200 \({}^{\circ}\)F.  
The simplest form of structural analysis can be conducted in 1D and assumes the PerFORM inserts are a cohesive cylinder of uniform temperature. These assumptions are not unreasonable in the area at the aft end. In this location, the inserts are constrained by the aluminum and will heat up the most from the total temperature rise generated by the fan. Equation (1) represents the thermal stress, \(\sigma_{th}\), that will accumulate in the inserts due to thermal expansion with no additional gaps between the materials. This assumes the PerFORM will constantly contact the aluminum in this area and the aluminum is rigid. \(E\) is the modulus of elasticity, \(\alpha\) is the coefficient of thermal expansion, and \(\Delta T\) is the difference in temperature between the inserts and the zero-strain reference temperature for the PerFORM (70 \({}^{\circ}\)F, 21 \({}^{\circ}\)C).