A plane-strain compression (forging) of a block is shown in the figure. The strain in the z-direction is zero. The yield strength (S_y ) in uniaxial tension/compression of the material of the block is 300 MPa and it follows the Tresca (maximum shear stress) criterion. Assume that the entire block has started yielding. At a point where σx = 40 MPa (compressive) and τ_xy = 0, the stress component σ_y is

                                                              

The correct answer is 340 MPa (compressive).

1. Understanding the Problem and Image Context:
From the provided image, we see a block of material undergoing plane-strain compression (forging) between a downward-moving top platen and a fixed bottom platen. The coordinate system shows the compression axis along the vertical $y$-direction and expansion along the horizontal $x$-direction.
We are given:
- Yield strength in uniaxial tension/compression, $S_y=300\text{MPa}$.
- Plane-strain condition in the $z$-direction, meaning the strain component is zero: ${\epsilon }_z=0$.
- At the point of interest, the horizontal normal stress is compressive: ${\sigma }_x=-40\text{MPa}$.
- Shear stress is zero: ${\tau }_{\mathrm{xy}}=0$.

2. Identifying Principal Stresses:
Since the shear stress ${\tau }_{\mathrm{xy}}=0$, the normal stresses ${\sigma }_x$, ${\sigma }_y$, and ${\sigma }_z$ are the principal stresses.
Under the plane-strain condition (${\epsilon }_z=0$), the normal stress in the constraint direction ($z$-direction) is the average of the other two principal stresses:
${\sigma }_z=\frac{{\sigma }_x+{\sigma }_y}{2}$
Consequently, ${\sigma }_z$ acts as the intermediate principal stress (${\sigma }_2$).

3. Applying the Tresca Yield Criterion:
Because the block is compressed along the $y$-direction and expands along the $x$-direction, the stress in the loading direction (${\sigma }_y$) is the most compressive stress, representing the minimum principal stress (${\sigma }_3$). The stress ${\sigma }_x$ is the maximum principal stress (${\sigma }_1$).
Thus, we define:
${\sigma }_1={\sigma }_x=-40\text{MPa}$
and
${\sigma }_3={\sigma }_y$
The Tresca yield criterion states that yielding occurs when:
${\sigma }_1-{\sigma }_3=S_y$
Substituting the known values:
$-40-{\sigma }_y=300$
Rearranging the equation to solve for ${\sigma }_y$:
${\sigma }_y=-40-300=-340\text{MPa}$
The negative sign indicates that the stress is compressive. Therefore, the stress component ${\sigma }_y$ is $340\text{MPa (compressive)}$.