I would simply find components you need without troubles with a transformation matrix as follows. Let S be the stress tensor (matrix) in a Lab Cartesian system and N=(c1,c2,c3) be the unit normal vector (in the same system) to the plane you consider (ci, I=-1,2,3 - directing cosines).

 1) Take the above mentioned N for the ort K of new Z direction: K=N;

2) Take I=k x N / |k х N|   (normalized vectorial product) for the ort of new X direction (on the plane), where k=(0,0,1) is shown also in the lab system and hopefully you can find the vectorial product of these two vectors;

3) Take J=K x I= N x I for the ort of new Y direction (on the plane).

4) Then the stress components in the new Cartesian system associated with the plane are:

normal S33=K*S*K  (scalar products, S and K are still represented in the lab System, to perform operations properly use whether lines or columns for involved vectors);  

S11=I*S*I,   S22=J*S*J, S12=S21=I*S*J  (all in in the plane considered);

 S13=S31=I*S*K, S23=S32=J*S*K  (all containing both the normal N and a direction in the plane).

IF YOU ARE STILL OBLIGED to find the coordinate transformation matrix, just find

A11=I*(1 0 0), A12=I*(0 1 0),  A13=I*(0 0 1), A21=J*(1 0 0), ... etc, where components (cosines) of I, J or K are again given in the old system.

Thank you so much Prof. Zisman. You answered it in a very simplified way. Can you give any reference on this ??

I am not understanding why  I=k x N / |k х N| .

Write the rotation matrix for rotation about one axis then write the matrix for rotation about another axis and multiply them this will give you the rotation matrix for 3D rotation

Dear Debasis!

For particular details you've got my e-mail. As to relevant references, the procedure seems to be very obvious. In other words, you may use it with no reference and no prove should be required when using it in any printed work.    

Dear Debasis!

If you are looking for formulas for converting from the transformation matrix to Euler angles, any book on Texture, crystallography etc. will help. 

Recently, there has been a paper published in MSMSE by well known scientists. It deals about rotations and provides conversion routines, that you are looking for.

http://dx.doi.org/10.1088/0965-0393/23/8/083501

Hi Debasis,

Kindly see the attachment (extract from my thesis). For further details and references, you can see my thesis

http://doc.utwente.nl/82517/

With the fourth order tensor 'Rot' you can rotate any 2nd order tensor (including 3D stress tensor) to any orthogonal coordonate system. You just need to know the direcrion cosines from which you can form the 4th order tensor 'Rot'.

Kind regards

Muhammad

Hello Debasis,

Your question was a little unclear to me, I am going ahead and answering for different possibilities

Assuming that you have the stress tensor with respect to the global coordinate system, and you said that you have the direction cosines for the local coordinate system. Lets call that the R matrix (3*3).

sigma_local_ij = R_ip R_jq sigma_global_pq  ( standard tranformation for a second order tensor)

In matrix form, sigma_local = [R] *[sigma_global]*[R]T ; 

components of sigma local gives the components with respect to the local coordinate system.

Any Continuum Mechanics book will have the above equation  or see this link

http://continuummechanics.org/stressxforms.html

Now coming to the plane that you mentioned. On a given plane, we generally define the normal and shear "tractions"  and not stresses! 

In case, you wanted to calculate the shear and normal traction on a known plane.You may have follow a different procedure

Then, coming to the euler angles. Do you want to define the euler angles to go from global to local , given the direction cosine matrix?

You can do it matlab (given that you have the robotic system tool box ), use the command rotm2eul

For the reference, you may look the lectures from Prof. Tony Rollet !

Hope this helps !

http://continuummechanics.org/stressxforms.html