Algorithm Verification

The following two subsections verify the proposed root solving algorithms by compar- ing the output to the well know Rayleigh-Lamb solution for a single isotropic plate with traction free surfaces.

6.10.1 Brute Force Algorithm

The flow chart for the specific brute force algorithm used in this section is shown in Figure 6.12. After the program has discretized the domain and performed the root finding method for each initial guess, the roots are filtered before plotting the results and saving the data. The exit status from each iteration is saved, and this information is used to decide if the calculated root is correct. Additionally, if the roots fall outside of the domain then they are discarded. Finally, the roots are sorted into real, imaginary and complex results. This is accomplished by setting a tolerance on the definition of a zero value for the real or imaginary component.

In order to verify that this algorithm will work for a multilayer laminate, two test scenarios are constructed and compared. First, the non-dimensional Rayleigh-Lamb solution of equations 4.25 and 4.26 are used in the brute force algorithm. Next, the three-layer non-dimensional solution is implemented with all three layers having the

same material properties. If the algorithm is correct, then both solutions should be the same.

Figure 6.13 shows the real and imaginary non-dimensional wavenumber solutions. Since the root solving method is unconstrained, some curves have a dense collection of roots, while others are missing point along part of the curve. Each approach seems to be good in one section and poor in another, generally these weaknesses and strengths are not the same. Overall, it appears that the multilayer solution has been successfully implemented. The bulk shear wave solution is not fully resolved, but this is partly due to the close proximity of the fundamental modes and the bulk wave mode at low frequencies.

The complex wavenumber perspective view is shown in Figure 6.14. Overall there is a good match. The Rayleigh-Lamb solution does not include the bulk pressure and shear waves, while the laminate model does include parts of this solution. The laminate model provides a more consistent solution along the imaginary root curves, but the Rayleigh-Lamb spears to be clearer along the complex curves.

These results show that the brute force algorithm has been successfully imple- mented and can solve for the roots for multi-layered isotropic laminates; however, some improvements can be made to this basic forceful approach.

6.10.2 Behavioral Algorithm

Again, the same verification process was performed. This time the Raleigh-Lamb equation was not used. Instead only the three-layer setup was implemented with all three layers containing the same material properties, and the previous results can be used for comparison.

The real-imaginary wavenumber view is shown in Figure 6.15. The calculation of this set of roots took about one twenty fifth of the time that was required in the brute force algorithm. Additionally, the domain in these results is much larger than

(a) Rayleigh-Lamb solution

(b) Isotropic laminate model with identical plies

(a) Rayleigh-Lamb solution

(b) Isotropic laminate model

the results presented previously. Another major improvement over the brute force algorithm is that the curve for each mode is knows, as opposed to a random collection of points.

Figure 6.15: Single material, three lamina model solution using the behavioral algo- rithm

The complex wavenumber view of the solution is shown in Figure 6.16. The behavioral algorithm provides clean plot lines for each real mode. It is clear how the fundamental modes converge to the Rayleigh wave speed, while the resonance modes converge on the shear wave speed. Also, the figure shows clearly how the complex modes begin at local maximum and move in the positive frequency direction to a local minimum.

The algorithm was also used to calculate the dispersion curves for a two material and three-layer model. This is the scenario encountered when an adhesive is bonded between two identical isotropic plates. The real-imaginary plot of the solution is shown in Figure 6.17. This solution has some similar properties to the single material model, but changes dramatically at higher frequencies and wavenumbers.

Figure 6.16: Perspective view of the complex single material, three lamina model solution using the behavioral algorithm

Figure 6.17: Two material, three lamina model solution using the behavioral algo- rithm

The complex wavenumber view of the solution is shown in Figure 6.18. There are some clear differences from the single material case. The fundamental modes do not converge to the Rayleigh wave speed; in fact, they do not converge towards each other. The solution also contains the solution for both of the pressure and shear waves. It appears that many of the higher modes will converge towards the shear speed of the outer material, while multiple modes have wavenumbers below the shear speed of the outside material, these are most likely interface modes. There also appears to be an error were one curve jumps from one mode to another. This could possibly be corrected by using a higher order backward difference method.

Figure 6.18: Real perspective view of the complex two material, three lamina model solution using the behavioral algorithm

There are two groups of imaginary curves in this figure. There is a set of curves which form very large peaks and troughs as they move in the imaginary wavenumber direction, and there is a group that forms much tighter and more frequent fluctuations. The curves with large variation most likely correspond to the thickness resonance of the thin inside layer representing the adhesive, while the tighter variations correspond

to the resonances of the outside layers representing the adherends.

In order to give a closer look into the two material results, the solution has been replotted with a smaller domain size in Figure 6.19. In these figures, the lower order modes are easily distinguishable. It is clear how the fundamental modes have changed from the Rayleigh-Lamb solution. Also, from this perspective two of the imaginary modes appear to be nearly horizontal with a wide range of wavenumber solution for limited range of frequencies.

(a) (b)

Figure 6.19: Close up view of the two material, three lamina model solution using the behavioral algorithm; (a) real and imaginary solution; (b) full complex solution