... | ... | @@ -3,12 +3,27 @@ Photoelatic images: intensity analysis |
|
|
|
|
|
The fringe pattern, which is due to internal varying stress structure of particles, can be calculated using [theory of elasticity](https://archive.org/details/TheoryOfElasticity). The contact forces can be obtained using this pattern at the points of contacts. In many similar experiments, spatial derivative of image intensity is used to retrieve local stresses. This method requires high resolution imaging of the media of order of 100 pixels per particles diameter in order to distinguish fringes. High resolution imaging is currently possible only by using quasi-static imaging. However, studying fast dynamics such as [impact](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.109.238302), rheology, or [stick-slip](https://arxiv.org/abs/1803.06028) requires high speed imaging and limits the resolution. Consequently, the fringes are not clear enough to be used for calculation and the noise may dominate the calculation. As an alternative, image intensity can be used to quantify the internal stress.
|
|
|
|
|
|
## Experimental evaluation curve
|
|
|
|
|
|
The relation between image intensity and external applied stress can be empirically evaluated. The figure below presents calibration relation for a small chain of five particles as a function of applied external force. The mean image intensity increases linearly with the applied force, showing that image intensity is a measure of pressure in circular particles.
|
|
|
|
|
|
|
|
|
* Experimental calibration curve of mean image intensity for a chain of five photo-elastic disks under applied force, f. Images, corresponding to different forces, are shown on the top of the diagram. The linear relation shows that image intensity can be used as a measure for pressure in the circular particles. The error bars are standard deviation on mean with three measurements for each data point:
|
|
|
|
|
|
![ch2_calibration1](uploads/92a764315e10b1af07a8b82881f5e5fe/ch2_calibration1.png)
|
|
|
|
|
|
## Numerical evaluation curve
|
|
|
|
|
|
In principle it is possible to analytically find a closed form for Cauchy stress tensor inside a disk, and consequently, aggregated intensity. The closed form is rather complicated to derive. However, using the same principles, we numerically solve for stress inside a disk and solve for local intensity. Figure below presents the calibration curve using numerical simulation. The curve is derived using simulation of a single disk internal stress from 20 pixels wide images, similar to the experimental resolution. If the force is small, it is linearly related to mean intensity of image. Mean intensity reaches a flat plateau for large forces as more fringes appear.
|
|
|
|
|
|
|
|
|
* Numerical simulation of mean image intensity for a photo-elastic disks under applied force, f. Images, corresponding to different forces, are shown on the top of the diagram. The curve shows that image intensity can be used as a measure for the pressure in disks up to a threshold. Inset presents heatmap of intensity inside a particle:
|
|
|
|
|
|
![ch2_calibration2](uploads/81ca546e087ad59d6ae23a1f64cee7f9/ch2_calibration2.png)
|
|
|
|
|
|
## Local energy calculation
|
|
|
|
|
|
To relate intensity with local energy, we need to know scaling of the force with particle deformation. Figure below experimentally investigates this relation for the sample circular particles. The relation shows a great fit with Hertzian contact force, $f = k(\delta r)^{3/2}$, as [expected in circular particles](https://books.google.com/books?hl=en&lr=&id=Do6WQlUwbpkC&oi=fnd&pg=PA1&dq=johnson+1987+contact&ots=gqgjhpg96V&sig=JMvJtJKYULMcacedO_4RA-MDL8w#v=onepage&q=johnson%201987%20contact&f=false). Combining Hertzian force relation and the linear relation of force and mean image intensity for a single particle, we can deduce that
|
|
|
|
|
|
|
|
|
[<go back to home](https://git-xen.lmgc.univ-montp2.fr/PhotoElasticity/Main/wikis/home) |
|
|
\ No newline at end of file |