A bolt head has to be made at the end of a rod of diameter d = 12 mm by localized forging (upsetting) operation. The length of the unsupported portion of the rod is 40 mm. To avoid buckling of the rod, a closed forging operation has to be performed with a maximum die diameter of ________ mm.

The correct answer is 18.

In the localized forging (upsetting) operation of a rod, rules are established to prevent buckling of the unsupported portion of the rod. Let:
- The diameter of the rod be _d. Here, _d = 12 mm.
- The length of the unsupported portion of the rod be _L. Here, _L = 40 mm.

First, we compare the unsupported length _L to the rod diameter _d to check the ratio:
$$\frac{L}{d}=\frac{40}{12}\approx 3.33$$

Since _L > 3_d (because 40 mm is greater than 3 × 12 mm = 36 mm), the rod is highly susceptible to buckling if not properly supported during upsetting.

To avoid buckling under these conditions, a closed forging operation must be performed using a die. According to the standard rules of upsetting, when the unsupported length is greater than three times the diameter of the rod (_L > 3_d), the maximum cavity/die diameter must not exceed 1.5 times the rod diameter:
$$D_{\text{die}}=1.5\times d$$

Substituting the given rod diameter _d = 12 mm into the equation:
$$D_{\text{die}}=1.5\times 12=18\text{mm}$$

Therefore, the maximum die diameter required to avoid buckling of the rod is 18 mm.