State of Stress in rotated 2D system

Question

I'm working on the rotation of the state of stress using the book Geodynamics of Turcotte page 101. The 2D coordinate system $x'$ and $y'$ is inclined at an angle $\phi$ with respect to the $x$, $y$ coordinate system (see Figure 2.14).

The force in the $y$-direction on face $AO$ is

$$ \sigma_{yy}AO $$

and the force in the $y$-direction on face $OB$ is

$$ \sigma_{xy}OB $$

The force in the $y$-direction on face $AB$ is

$$ -\sigma_{x'x'}AB\cdot\sin\theta -\sigma_{x'y'}AB\cdot\cos\theta$$

I don't understand why the force in the $y$-direction on face $AB$ is equal to the above equation?

I think the negative signs are there because the stresses are not oriented in the same direction than the $y$-direction. I don't know why it is necessary to subtract the two lengths.

Answer 1

You are correct in your intuition regarding the signs of the forces. The reason for two terms is due to the direction of the force. The y-direction is not aligned with the face AB (it is aligned with y') so the force on this face is a combination of the forces normal to AB ($-\sigma_{x'x'}AB$) and the force along the face ($-\sigma_{x'y'}AB$). Because the question wants only the force in the y-direction, we need to find the projection of these forces along that direction.

If the angle between the coordinate axes is $\theta$, then the contribution of $-\sigma_{x'x'}AB$ in the y-direction is $-\sigma_{x'x'}AB\sin\theta$ and the contribution of $-\sigma_{x'y'}AB$ in the y-direction is $-\sigma_{x'y'}AB\cos\theta$, thus the total force in the y-direction is the sum of these contributions:

$$-\sigma_{x'x'}AB\sin\theta-\sigma_{x'y'}AB\cos\theta$$

Here is a quick figure to help visualize the geometry: