Stress tensor transformation

[tricha122]

Hi all,

I have been reading up about continuum mechanics recently, and have a question regarding the reduction in stiffness coefficients in the stiffness matrix.

I am aware of how the stiffness matrix is reduced to 21 coefficients. However, in order to reduce it from 21 to 13, one has to assume 1 plane of material symmetry.

When doing so, (say x1 = 0 is the plane of symmetry), in order to reduce the # of coefficients, you have to look at sigma = C*epsilon, and sigma' = C'*epsilon'

where C = C' for 1 symmetric plane.

Transforming sigma and epsilon from OX1X2X3 to OX1'X2X3 you can get the "boundary conditions" you need to zero out certain coefficients of the stiffness matrix and result with 13 (from 21).

My problem is O-X1-X2-X3 to O-X1'-X2-X3 is changing from a RH coordinate system to a LH coordinate system.

a RH to RH system would look like this: O-X1-X2-X3 to O-X1'-X2'-X3.

However, when i attempt the same reduction with that transformation the math doesn't work out correctly.

SHould it work out? Does one always use a RH->LH transformation when using planes of symmetry?

Any help would be greatly appreciated.

Thanks

 

[Valerie Park]

!</code>Yes, the transformation you are using is correct. The transformation from a right-handed coordinate system to a left-handed coordinate system is necessary when dealing with plane symmetry. This is because of the way in which the stress and strain components transform under rotations. The components of stress and strain are always invariant under rotations in the same handedness. Therefore, if we have a right-handed coordinate system, then when we rotate it by 180 degrees (which would be the plane of symmetry for plane strain problems), the stress and strain components would remain the same.However, if we rotate it by 180 degrees in a left-handed coordinate system, the stress and strain components will no longer remain the same. Therefore, we need to use the left-handed coordinate system to properly account for the plane of symmetry. Hope this helps!