I know that the polarized microscope can observe the location of stressed area of a solid, however, how can it measure it quantitatively? Any formulae?
There is a technique called photoelasticity which is a whole field experimental method for stress analysis using polarised light. This technique gives the principal stress difference and their orientations.
The law that relates the optical information to the stress information is called the 'Stress-Optic Law'. There are a good set of lectures on this topic including all the fundamentals in the following link:
http://nptel.ac.in/courses/112106068/
Alternately, you may also watch it on youtube. Photoelastic techniques are applicable only for certain materials that exhibit stress-induced birefringence like epoxy, polycarbonate, glass etc.
http://nptel.ac.in/courses/112106068/
There is a technique called photoelasticity which is a whole field experimental method for stress analysis using polarised light. This technique gives the principal stress difference and their orientations.
The law that relates the optical information to the stress information is called the 'Stress-Optic Law'. There are a good set of lectures on this topic including all the fundamentals in the following link:
http://nptel.ac.in/courses/112106068/
Alternately, you may also watch it on youtube. Photoelastic techniques are applicable only for certain materials that exhibit stress-induced birefringence like epoxy, polycarbonate, glass etc.
http://nptel.ac.in/courses/112106068/
You need to know the change of the refractive index tensor with strain to understand the image contrast. There data on silicon available, for example.
Keep in mind too that the main challenge in non-uniformly stressed materials is assessing the effective depth of the stress. what you'll measure is the phase retardation due to birefringence. you'll retrieve the principal stress with knowledge of the stress-optic coefficient and the depth of the stressed region.
As already pointed out by vivek, photoelasticity gives the principal stress difference and their corresponding orientations. For common problems involving 2-D stress states involving plane stress assumptions,the use of stress-optic law is valid. But for 3-D problems involving non-uniform stress field one may have to resort to 3-D photoelastic techniques like scattered light and stress freezing.techniques.
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