1. Modal Analysis of Rectangular Simply-Supported Functionally Graded Plates By Wes Saunders Final Report

2. Purpose To use Finite Element Analysis (FEA) to perform a modal analysis on functionally graded materials (FGM) to determine modes and mode shapes.

3. Background FGMs – FGMs are defined as an anisotropic material whose physical properties vary throughout the volume, either randomly or strategically, to achieve desired characteristics or functionality – FGMs differ from traditional composites in that their material properties vary continuously, where the composite changes at each laminate interface. – FGMs accomplish this by gradually changing the volume fraction of the materials which make up the FGM. Modal Analysis – Modal analysis involves imposing an excitation into the structure and finding when the structure resonates, and returns multiple frequencies, each with an accompanying displacement field.

4. Problem Description Each Case – Frequencies (4) – Mode Shapes (4) – Plates 1m x 1m, 0.025m and 0.05m thick Case A – Compare to theoretical values Case B-D – Compare to isotropic Select Cases – Compare to Efraim formula

5. Methodology FEA Modal analysis performed by COMSOL eigenfrequency module.

6. Methodology (cont.) Mori-Tanaka estimation of material properties for FGMs – Gives accurate depiction of material properties at certain point in the thickness, dependent on volume fractions and material properties of the constituent materials – Get density (ρ), shear (K) and bulk (μ) moduli – Use elasticity to get expressions for E and ν

7. Results - Isotropic Isotropic results matched with theory Reasons for isotropic case – Verify FEA model – Check plate thickness limit – Have baseline for comparison to FGM

8. Results – Linear Represents, on average, a 50/50 metal-ceramic FGM h=0.05m frequencies were bounded by their constituent materials Mode shapes 2a and 2b swapped from where they were in isotropic cases

9. Results – Power Law n=2 Represents, on average, a 67/33 metal-ceramic FGM Frequencies are bounded by their constituent materials Mode shapes are changed by addition of ceramic

10. Results – Power Law n=10 Represents, on average, a 91/9 metal-ceramic FGM, or a metal plate with a thin ceramic coating Frequencies were very close to isotropic metal frequencies Mode shapes 2a and 2b highly distorted due to presence of ceramic

11. Comparison to Efraim Efraim formula is a simple method of determining FGM frequencies by knowing the isotropic frequencies of the constituent materials Method showed results that were 6-11% lower than the FEA results

12. Conclusions Using the isotropic and MT checks, a great level of confidence was achieved in the results. When considering a FGM that is metal and ceramic, the frequency seems to follow the metal while the mode shape seems to the follow the ceramic A FGM should be thick enough so that enough data is able to be extracted from the cross-section of the plate. The Mori-Tanaka estimate is heavy computationally, as demonstrated in Appendix C. For future use, the use of µ and K should stand to simplify the calculations. The FEA results were found to be within 6-11% of the computed values from Efraim. Efraim formula can be used as a starting point reference or sanity check on more complex FGM FEA models.