To determine the yield strength of the material using a 0.05% offset, you can follow these steps:
Calculate the elastic modulus (E):
The elastic modulus is the slope of the stress-strain curve at zero strain (i.e., the derivative of the function). In your case, the function is given as σ(ε) = 4[1500ε - (750ε)^2].
Take the derivative of this function with respect to ε:
dσ/dε = 4[1500 - 2 * 750^2 * ε]
Evaluate the derivative at ε = 0 to find the elastic modulus (E):
E = dσ/dε at ε = 0
E = 4 * 1500 = 6000 ksi
Determine the offset stress (0.05% offset):
A 0.05% offset means you're looking for the stress corresponding to a strain of 0.05% (0.0005 in decimal form) offset from the initial linear elastic region.
Calculate the offset stress:
Offset stress = E * Offset strain
Offset stress = 6000 ksi * 0.0005 = 3 ksi
Find the intersection point between the offset stress line (3 ksi) and the stress-strain curve:
Set the stress-strain curve equal to the offset stress and solve for ε:
4[1500ε - (750ε)^2] = 3
This equation may require numerical methods to solve accurately. Using a numerical solver or software, you can find the value of ε when σ(ε) = 3 ksi.
Once you have found ε, you can calculate the yield strength using the equation:
Yield strength = Offset stress + σ(ε)
Substitute the values you've found to get the yield strength of the material