# Strain optic coefficient definition

1. Jan 11, 2012

### Femme_physics

I can't find the THEORETICAL definition of "strain optic coefficient" online. I googled and wiki'd enough. Can someone provide me with one, please?

2. Jan 11, 2012

### Andy Resnick

Born and Wolf defined stress- and strain-optical constants in terms of stress birefringence (photo-elastic effect). For example, the index ellipsoid of an unstressed material may be written as:

$\frac{x^{2}}{\epsilon_{x}}+ \frac{y^{2}}{\epsilon_{y}}+ \frac{z^{2}}{\epsilon_{z}} = 1$

and applying a stress $\sigma$ with components $\sigma_{xx}, \sigma_{xy}, \sigma_{xz}$, etc changes the ellipsoid to:

$a_{xx}x^{2}+a_{yy}y^{2}+a_{zz}z^{2}+a_{xy}xy+a_{xz}xz+a_{yz}yz+=1$, with the optical-stress coefficients q relating the unstressed and stressed index ellipsoid: for example

$a_{xx}-\frac{1}{\epsilon_{x}}=q_{xxxx}\sigma_{xx}+q_{xxyy}\sigma_{yy}+q_{xxzz}\sigma_{zz}+q_{xxyz}\sigma_{yz}+q_{xxzx}\sigma_{zx}+q_{xxxyx}\sigma_{xy}$.

Similarly, by using the stress-strain relationship $\sigma_{ij} = C_{ijkl}\epsilon^{kl}$.. sorry, 'epsilon' got used twice here... you can generate the strain-optic coefficients.

This subject gets covered in various places- crystal optics, acousto-optics, etc.

3. Jan 12, 2012

### Femme_physics

So, to put things in English :) --> it's the relation between stress and strain of a certain material?

4. Jan 12, 2012

### Andy Resnick

not exactly- it's the relationship between the applied stress and induced birefringence.

5. Jan 12, 2012

### Studiot

Hello again, Femme Physics.

I take it you are now studying photoelasticity?

So you will have seen the striking pictures that photoelastic analysis can generate?

These are alternate regions of light and dark (and sometimes pretty colours) when polarised light is shone through a suitable material undergoing strain. Alternatively if the object is opaque and we coat it with a suitable photoelastic coating then the light passes through the coating is reflected by the substrate and passes back through the coating - a double journey.

Either way the difference in stress between two dark zones, a and b ( is given by the equation

$${\sigma _b} - {\sigma _a} = \frac{{CN}}{t}$$

Where N-1 is the number of dark regions between a and b,
t is the thickness,
C is a material constant which I think (edit: but I am not certain) is your strain optic coefficient.

Typical values are

polyurethane 3 - 5
epoxy 60
pound-fringes per inch (sorry it's imperial)

Last edited: Jan 12, 2012