Solving for Hall Petch Unknowns

[cperez]

The lower yield point for an iron that has an average grain diameter of 5x10^-2mm is 135 MPa. At a grain diameter of 8x10^-3, the yield point increases to 260 MPa. At what grain diameter will the lower yield point be 205 MPa?

Homework Equations

Hall-Petch Equation:
σy = σo + ky(d^-(1/2))

The Attempt at a Solution

135 MPa = σo + ky(22.36)
260 MPa = σo + ky(89.44)

I know this is going to sound silly, but I don't know to solve for two unknowns. If someone could just show me how to solve for one of them, I could solve for the other and figure the rest out.

 

[berkeman]

The lower yield point for an iron that has an average grain diameter of 5x10^-2mm is 135 MPa. At a grain diameter of 8x10^-3, the yield point increases to 260 MPa. At what grain diameter will the lower yield point be 205 MPa?

Homework Equations

Hall-Petch Equation:
σy = σo + ky(d^-(1/2))

The Attempt at a Solution

135 MPa = σo + ky(22.36)
260 MPa = σo + ky(89.44)

I know this is going to sound silly, but I don't know to solve for two unknowns. If someone could just show me how to solve for one of them, I could solve for the other and figure the rest out.

I'm not familiar with the physics of the problem, but I should be able to help you solve equations. What units are your last two equations in?

 

[cperez]

It's an engineering material science class problem.
The two equations are in MPa and mm.. so I think σy is in MPa/mm

 

[berkeman]

It's an engineering material science class problem.
The two equations are in MPa and mm.. so I think σy is in MPa/mm

I believe those are mixed units. Don't you usually work in the mksA unit system?

 

[cperez]

I don't believe so.

 

[TymeMastery]

135 MPa = σo + ky(22.36)
260 MPa = σo + ky(89.44)

I know this is going to sound silly, but I don't know to solve for two unknowns. If someone could just show me how to solve for one of them, I could solve for the other and figure the rest out.

One way you can do this is substitution. Isolate one of the variables and plug it into the other equation. In this instance σo seems like a good choice.

Using equation one:
σo = 135MPa - ky(22.36)

Plugging into equation two:
260Mpa = 135MPa - ky(22.36) + ky(89.44)

Solve for ky, and use that to get the other constant.