The yield strength of an alloy with an average grain diameter of d1 is given as Yield Stress 1. When the grain diameter is d2, the yield strength is Yield Stress 2. Determine the grain diameter in millimeters at which the yield strength will be 217 MPa.

To determine the grain diameter at which a particular yield strength occurs, you can use the Hall-Petch relationship, which expresses how yield strength (σy) varies with grain size. The relationship is typically formulated as:

σy = σ0 + k * d^(-1/2)

Where: σy = Yield strength σ0 = Material constant (yield strength for a material with very large grains or a single crystal) k = Material's strengthening coefficient (a constant that depends on the material) d = Grain diameter

Assuming that you have two different grain diameters d1 and d2 with corresponding yield strengths Yield Stress 1 and Yield Stress 2, you can set up two equations:

Yield Stress 1 = σ0 + k * d1^(-1/2) (1) Yield Stress 2 = σ0 + k * d2^(-1/2) (2)

Using these two equations, you can solve for the constants σ0 and k if they aren't given, provided you have the values for d1, d2, Yield Stress 1, and Yield Stress 2. If σ0 and k are already known, you can directly calculate the grain diameter for any yield stress, such as 217 MPa.

However, in order to determine the new grain diameter d3 for yield strength of 217 MPa, you would need at least one of the pairs (Yield Stress 1, d1) or (Yield Stress 2, d2) and the values of σ0 and k.

Assuming we know σ0 and k, the corresponding equation for the yield strength of 217 MPa would be:

217 MPa = σ0 + k * d3^(-1/2) (3)

You can rearrange this equation to solve for d3:

d3^(-1/2) = (217 MPa - σ0) / k d3 = (k / (217 MPa - σ0))^2

You would then plug in your known values for σ0 and k to find d3.

Unfortunately, without numerical values for d1, d2, Yield Stress 1, Yield Stress 2, σ0, and k, we cannot numerically solve this problem. You need these values to either solve for the unknown constants or to solve for d3 directly if σ0 and k are known.