Strain calculation from stress tensor

[Shan]

Dear sir,

I have applied tensile stress on BN nanowire by increasing the size of supercell in the periodic direction (z-direction).
The computed stress tensor in z-direction before and after applying stress are 0.0377 and 0.152 eV/A^3.

1) How can I calculate the strain tensor in z-direction from the above computed stress tensor?
2) The percentage increase in stress in the above case is 303 %, Is the percentage increase in strain will also be 303 % ? (I read in some sources that strain is directly proportional to stress) 


Please clarify sir.  Thank you very much for your time.

[zh]

1). It is much easier to estimate the strain tensor from the change in the cell length of BN nanowire.
   The conversion from the stress tensor to strain tensor, the stiffness coefficient is required.
2). The change in the stress tensor is  of course not equal to the change in strain tensor, because the stiffness coefficient isn't equal to one in most cases. 

[Jess Wellendorff]

Perhaps this tutorial can help you: http://docs.quantumwise.com/tutorials/elastic_constants.html

[Shan]

Dear sir,

I tried to calculate the stiffness coefficient as shown in the below image,  the procedure adopted is-

stiffness coefficient = (force applied)/(displacement produced),                         [displacement in z-direction = 1.2662 Ang]

The force applied is calculated from the stress tensor, F = (sigma)*(cross sectional area)

1) Please verify the procedure sir... whether the procedure I followed to calculated stiffness coefficient is correct?
2) How can I calculate strain tensor (in Z direction) by using this stiffness coefficient? (I also uploaded the elastic constants and stress tensor results before and after applying stress on the nanowire)

Thank you very much for your time sir.

[zh]

If the length of BN nanowire at equilibrium state (i.e., stress tensor = 0.0 GPa) is  a0 and the length of nanowire at the tensile state is a1,
the amplitude of physics strain along the tensile direction would be  (a1-a0)/a0.

If you know the stress tensor of nanowire at the tensile state ($\sigma_1$), you can estimate the stiffness coefficient (i.e., elastic constant) by the linear elasticity theory:
$sigma_1$ =     c *  (a1-a0)/a0

[Shan]

So...   

(a1-a0)/a0  is strain tensor, a unit less quantity...?

[zh]

strain tensor doesn't have unit.
Recommend you read the following reference to understand the concepts of strain tensor, stress tensor, elastic constants, stiffness:
http://exciting.wdfiles.com/local--files/elastic/cpc-2013.pdf

[Shan]

thank you Dr. Zh