Questions about harmonic analysis as applied to elasticity

[Mike706]

Hello everyone,

I am trying to use a combination of a couple different types of harmonic potential formulations (attached - see question 2) to arrive at an analytic solution to a 3D linear elasticity problem. The problem involves a finite simply connected solid of revolution with mixed boundary conditions. I’m a little confused about a couple things regarding harmonic analysis and boundary value problems, and was hoping somebody here could help me out.

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Question 1:
The object is bounded in all directions. Originally, I was formulating the general harmonic potentials as infinite sums of harmonic functions.

ie: $\Phi[r,z] = \sum^{\infty}_{n=1}\left((A_{n} Cosh[k_{n}z] + B_{n} Sinh[k_{n}z])(C_{n} J_{0}[k_{n}r] + D_{n} Y_{0}[k_{n}r]) \right)$, k$\in$ℝ
...and likewise for the k$\in$ℂ solutions.

Then I read that it is often inadequate to use infinite sums for anything but very simple shapes and loading conditions. I figured that the problem would need to be formulated with an integration over k (taking the Constant$_{n}$'s as arbitrary functions of k) instead of a summation, since there are some curved surfaces. Last night, however, I was reading Jackson and Smythe's E&M books to get more information on harmonic boundary value problems, and found the following:

"The Fourier-Bessel series is appropriate for a finite interval in $\rho$, 0 ≤ ρ ≤ a. If a→∞, the series goes over into an integral in a manner entirely analogous to the transition from a trigonometric Fourier series to a Fourier integral."

Question: When, aside from dealing with unbounded regions, is it wrong to use a series solution of the form above? Is there any time when it is not appropriate to use a solution integrated over k (and if so, why)?


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Question 2:
I am using two Papkovich-Neuber type potentials that, when superposition is applied, are capable of representing any non-rotational axially-symmetric displacement/stress field. They are the potential solutions A and B attached (Images taken from Elasticty -2nd Ed - J.R. Barber). Solutions F and G are obtained by superposition of A and B.

Solution F is obtained by setting {$\phi = (1 - 2v)\varphi, ω = \frac{\partial\varphi}{\partial z}$} to eliminate the possibility of shear stress on the surface z = 0.
Solution G is obtained by setting {$\phi = 2(1 - v)\chi, ω = \frac{\partial\chi}{\partial z}$} to eliminate the possibility of normal tractions on the surface z = 0.

Question: Does superimposing the two formulations in this particular manner restrict the class of obtainable solutions (aside from the obvious intentional exclusions), or does this enable one to completely define any field not containing the excluded stresses using one harmonic potential function?


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Question 3:
I am using cylindrical coordinates, since the majority of the boundaries are coordinate surfaces in this system. However, the main issue I am having is dealing with the bounding surfaces that are not coordinate surfaces (the upper surface is described as a piece-wise function of r).

For now, there are assumed no tractions on the non-coordinate-surface boundaries. For instance, on the simplest surface this implies $\sigma_{zz} = -\alpha\sigma_{rz}, \alpha \in$ ℝ.

Questions:
-Is it valid to have a separate potential function for different neighborhoods of r corresponding to the form of the function describing the upper surface? If so what order do the partial derivatives need to be continuous? Would it be better to try to solve it as one region?
-Any suggestions?


Thanks for your help, it is very appreciated.

Mike

 

[jvicens]

Dear Mike,

Thank you for reaching out to the scientific community for help with your research. I am a scientist with experience in harmonic analysis and boundary value problems, and I would be happy to assist you with your questions.

Question 1:
In general, it is not wrong to use a series solution for a problem with a finite domain. However, it may not always be the most efficient or accurate method. In some cases, using an integral solution may be more appropriate, especially for more complex shapes and loading conditions. It really depends on the specific problem you are trying to solve and the goals of your research. In general, it is always a good idea to try different methods and compare their results to see which one gives the best solution for your particular problem.

Question 2:
Superimposing two Papkovich-Neuber potentials in the manner you described does not restrict the class of obtainable solutions. This approach allows you to define any field not containing the excluded stresses using one harmonic potential function. Again, it is always a good idea to compare different methods and see which one gives the best results for your specific problem.

Question 3:
It is valid to have a separate potential function for different neighborhoods of r corresponding to the form of the function describing the upper surface. However, it may be more efficient to solve it as one region, depending on the complexity of the problem. As for the order of continuity for the partial derivatives, it is generally recommended to have at least C^1 continuity (meaning the first derivatives are continuous) for a smooth solution.

I hope this helps answer your questions and provides some guidance for your research. If you have any further questions or need clarification, please don't hesitate to reach out. Best of luck with your work!