Effective tensor of a polycrystal

[pathfinder]

Hi everybody,
i have a problem that i wanted to share with you
if we consider a polycrystal made of cylindrical fibers following a von mises-fisher distribution equation (17) in http://bit.do/vmisesfisher (called orientation distribution function of fibers) . i must change the probability density in equation (28) http://bit.do/e28 with the von mises-fisher than i must follow the steps listed in the article http://bit.do/effectivetensor1 , http://bit.do/effectivetensor2 , http://bit.do/effectivetensor3 so that by using orientation averaging, i find the effective (elasticity) tensor of the polycrystal
it s an optimization problem
arg min of the integral over rotation group of the von mises-fisher distribution multiplied by the distance between the effective tensor of the polycrystal (what we are looking for) and the one of a single cylindrical fiber (given).

if anyone could give ideas about how can i start solving this optimization problem to find the effective tensor of the polycrystal
.
thank you

 

[Achy47]

Thank you for sharing your problem with us. It sounds like you are trying to find the effective tensor of a polycrystal made of cylindrical fibers using the von Mises-Fisher distribution equation. This is an interesting and challenging problem, and I would be happy to offer some suggestions on how to approach it.

Firstly, it is important to understand the concept of effective tensors in polycrystals. An effective tensor is a way to represent the average mechanical behavior of a material that is made up of multiple individual components with different orientations. In your case, the individual components are the cylindrical fibers and the polycrystal is the material. The effective tensor takes into account the anisotropy of the individual components and their orientations within the material.

To solve this optimization problem, you will need to follow the steps outlined in the articles you mentioned (http://bit.do/effectivetensor1, http://bit.do/effectivetensor2, http://bit.do/effectivetensor3). These steps involve calculating the orientation distribution function of the fibers using the von Mises-Fisher distribution equation, and then using orientation averaging to find the effective tensor of the polycrystal.

One approach to solving this optimization problem is to use a numerical method such as the Monte Carlo method or the gradient descent method. These methods involve randomly generating orientations for the fibers and calculating the corresponding effective tensor. The process is repeated multiple times, and the orientations that give the lowest distance between the effective tensor of the polycrystal and that of a single cylindrical fiber are chosen as the optimal orientations.

Another approach is to use analytical methods, such as the Fourier series method or the Taylor series method, to find the effective tensor. These methods involve solving the equations for the effective tensor using the orientation distribution function and the orientations of the fibers.

In either case, it is important to carefully consider the parameters and assumptions used in the von Mises-Fisher distribution equation and the orientation averaging process. These can have a significant impact on the accuracy of the results.

I hope these suggestions will help you in solving your optimization problem. If you have any further questions or need any clarification, please do not hesitate to ask. Good luck with your research.