Basic tensors. Drawing and orienting.

[K29]

What is the rule for orienting tensors?

In the above image we have the components of the Cauchy Stress Tensor in 2 dimensions. In the bottom left you see the cartesian co-ordinates are oriented as normal.

How do I know $\sigma_{xy}$ is oriented upwards on the right face and downwards on the left face? Is there a right-hand rule or standard about going anti-clockwise or something?

For example the rule for stresses $\sigma_{ii}$ is it is always normal to its face

 

[TSny]

How do I know $\sigma_{xy}$ is oriented upwards on the right face and downwards on the left face?

Consider a patch of area oriented perpendicular to the x-axis in a stressed material. Let positive x be toward the right as in your figure. $\sigma_{xx}$ is defined to be the x-component of the force per unit area that the material just to the right of the patch exerts on the material just to the left of the patch. Thus if $\sigma_{xx}$ is positive, then the material just to the right of the area is pulling the material just to the left of the patch toward the positive x direction. But by Newton’s third law, that means that the material just to the left of the patch is pulling the material just to the right of the patch toward the negative x direction.

Similarly,$\sigma_{xy}$ is the y-component of the force per unit area that the material just to the right of that same patch of area exerts on the material just to the left. Let the y-axis point upward as in your figure. Thus, if $\sigma_{xy}$ is positive, it means that the material just to the right of the patch is exerting an upward force on the material just to the left of the patch. The third law implies that the material just to the left of the area will exert a downward force on the material just to the right.

Now consider a small cubical element of the material with faces perpendicular to the coordinate axes. Suppose you want to know the forces that the material surrounding the cube is exerting on the surfaces of the cube. For the face at the right, we want to know the force which the material just to the right of the face exerts on the material just to the left. The x-component of this force will be to the right if $\sigma_{xx}$ is positive, as shown on the right side of your figure. For the cube face on the left, we want to know the force which the material just to the left of the face exerts on the material just to the right. As explained above, if $\sigma_{xx}$ is positive, this force will be to the left as shown on the left side of your figure.

Continuing with that line of reasoning, see if you can understand the directions of all of the arrows in your figure.

 

[K29]

Thank you for the detailed explanation. I have worked through each step, and it has been a huge help. I have a very clear understanding now. Thank you.