Understanding the Cauchy Stress Tensor for Beginners

[member 392791]

Hello,

I am not sure what the first indice in the cauchy stress tensor indicates

For example,


σ_xy means that the stress in the y direction, but does x mean the cross sectional area is normal to the x direction?

 

[SteamKing]

Perhaps this illustration might help:

http://en.wikipedia.org/wiki/Cauchy_stress_tensor

 

[member 392791]

I can't see how σ_xy is dependent on σ_yx, they are perpendicular to each other

 

[SteamKing]

Who said anything about these stresses being dependent on one another?

 

[member 392791]

My textbook said that the entire stress state can be determined with just 6 of the 9 components of the stress tensor. Is that to mean something different than there is a dependency? Or even they are equal?

 

[arildno]

The Cauchy stress tensor must be SYMMETRIC.
The reason for this is apparent when you consider the TORQUES about an infinitesemal square element.
Unless the stress tensor is symmetric, you'll get infinite angular accelerations of the square element.

The symmetry condition on the stress tensor explains why s_xy=s_yx and so on.

It should be mentioned, as is done in the Wikipedia article, that the symmetry of the Cauchy stress tensor is a special case as the Knudsen number goes to 1, so that symmetry is not required generally

 

[Chestermiller]

Hello,

I am not sure what the first indice in the cauchy stress tensor indicates

For example,


σ_xy means that the stress in the y direction, but does x mean the cross sectional area is normal to the x direction?

Yes. There are three components of the stress vector acting on a plane oriented normal to the x direction. The (normal) component in the x direction is σ_xx. The (shear) component in the y direction is σ_xy=σ_yx. The (shear) component in the z direction is σ_xz=σ_zx.