To calculate the smallest punch diameter that can be designed for piercing sheet metal strip, we need to consider the crushing strength of the punch material, the thickness of the sheet, the factor of safety, and the shear strength of the sheet material.

Given Data:
- Crushing strength of the punch material: 1500 MPa
- Thickness of the sheet: 2 mm
- Factor of safety: 3
- Shear strength of the sheet material: 500 MPa

Let's calculate the maximum shear stress that the punch can withstand using the crushing strength of the punch material and the factor of safety.

Calculation:
Maximum Shear Stress = Crushing Strength / Factor of Safety
= 1500 MPa / 3
= 500 MPa

Now, we can calculate the maximum shear force that the punch can withstand using the maximum shear stress and the area of the punch.

Calculation:
Maximum Shear Force = Maximum Shear Stress x Area of the Punch

To calculate the area of the punch, we need to assume a diameter (D) for the punch.

Area of the Punch = π/4 x D^2

Now, we can rearrange the equation for the maximum shear force to solve for the punch diameter.

Maximum Shear Force = Maximum Shear Stress x Area of the Punch
= Maximum Shear Stress x (π/4 x D^2)

Simplifying the equation, we get:

D^2 = (4 x Maximum Shear Force) / (π x Maximum Shear Stress)

Now, substituting the given values in the equation:

D^2 = (4 x Maximum Shear Force) / (π x Maximum Shear Stress)
= (4 x Maximum Shear Force) / (π x 500 MPa)
= (4 x Maximum Shear Force) / (π x 500 x 10^6 N/mm^2) (1 MPa = 10^6 N/mm^2)

Let's assume the maximum shear force is F.

D^2 = (4 x F) / (π x 500 x 10^6 N/mm^2)

Now, to find the smallest punch diameter, we need to find the smallest value of D that satisfies the above equation.

Since the shear strength of the sheet material is given as 500 MPa, it implies that the maximum shear force should not exceed this value. Therefore, we can substitute the maximum shear force as 500 MPa in the equation and solve for D.

D^2 = (4 x 500 MPa) / (π x 500 x 10^6 N/mm^2)
= (4 x 500) / (π x 500 x 10^6)
= 0.000008

Taking the square root of both sides, we get:

D = √0.000008
= 0.00283 mm

Therefore, the smallest punch diameter that can be designed for piercing the sheet metal strip is approximately 0.00283 mm, which is equivalent to 2.83 µm.