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 Physical Setup and Coordinates
From the provided image, the block is being compressed along the vertical $y$-axis by a downward-moving platen against a fixed bottom platen. The horizontal axis is denoted as $x$, and the longitudinal axis (out of the page) is $z$.

2. Applying the Plane-Strain Condition
Under plane-strain forging conditions, the strain in the longitudinal direction is zero:
$${\epsilon }_z=0$$
For plastic flow under plane strain, the normal stress in the direction of zero strain is the average of the other two normal stresses:
$${\sigma }_z=\frac{{\sigma }_x+{\sigma }_y}{2}$$
Since ${\sigma }_z$ is the exact average of ${\sigma }_x$ and ${\sigma }_y$, it serves as the intermediate principal stress (${\sigma }_2$). Given that the shear stress ${\tau }_{\mathrm{xy}}=0$, the normal stresses ${\sigma }_x$ and ${\sigma }_y$ are the principal stresses in their respective directions.

3. Determining Principal Stress Order
Because the block is actively compressed by the moving platen along the $y$-axis, the compressive stress in the vertical direction (${\sigma }_y$) will be greater in magnitude (more negative) than the compressive stress in the lateral direction (${\sigma }_x$). Thus:
$${\sigma }_x>{\sigma }_z>{\sigma }_y$$
Therefore, the maximum principal stress is ${\sigma }_{\max }={\sigma }_x$ and the minimum principal stress is ${\sigma }_{\min }={\sigma }_y$.

4. Yielding and the Tresca Criterion
According to the Tresca (maximum shear stress) criterion, yielding occurs when:
$${\sigma }_{\max }-{\sigma }_{\min }=S_y$$
Substituting the principal stresses into the criterion:
$${\sigma }_x-{\sigma }_y=S_y$$
We are given:
• $S_y=300\text{MPa}$
• ${\sigma }_x=-40\text{MPa}$ (compressive)
Substituting these values:
$$-40-{\sigma }_y=300$$
$${\sigma }_y=-40-300=-340\text{MPa}$$
Since the result is negative, ${\sigma }_y$ is compressive with a magnitude of $340\text{MPa}$.