K29 said:
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.
