n isotropic alloy contains 5% by volume of a precipitate of radius 10nm. Assume the primary metal is in a simple cubic arrangement. The alloy has a Young's modulus E=70GPa and a Poisson's ratio ν=0.35. If Γ=0.9J/m2 (the energy needed to cut through the precipitate lattice per unit surface area), and b=0.25nm, what is the precipitate spacing L in the alloy?
did anyone got the answer?
To find the precipitate spacing (L) in the alloy, we can use the formula for the interparticle spacing in a simple cubic lattice. The interparticle spacing (a) is related to the precipitate spacing (L) by the formula:
a = L/(n1/3),
where "n" is the number of particles along each side of the simple cubic lattice.
First, let's calculate the number of particles (n) along each side of the simple cubic lattice. Since the alloy contains 5% by volume of the precipitate, we can assume that the remaining 95% is the primary metal. Therefore, the volume fraction of the precipitate is given by:
Volume fraction (Vf) = (Volume of precipitate)/(Total volume of alloy)
Vf = 5% = 0.05
Now, let's assume that the precipitate is evenly distributed throughout the alloy. In this case, the volume fraction is also equal to the number fraction (Nf) of particles (or precipitates) in the lattice.
Nf = 0.05
We know that in a simple cubic lattice, the number of particles is given by:
N = n^3,
where "N" is the total number of particles and "n" is the number of particles along each side.
Substituting the values, we have:
0.05 = n^3
Now we can solve for "n":
n = (0.05)^(1/3)
Next, we need to calculate the interparticle spacing (a) using the formula:
a = L/(n^(1/3))
For the given problem, the precipitate radius is 10 nm. Since the precipitate is assumed to be spherical and evenly distributed, the precipitate diameter is equal to 2 times the precipitate radius:
d = 2 * 10 nm = 20 nm
The interparticle spacing (a) can be calculated as:
a = d/(n^(1/3))
Now we need to convert the interparticle spacing (a) from nanometers (nm) to meters (m) so that we can use the values provided for Γ and b in the same units. 1 nm = 10^(-9) m. Therefore,
a = (20 nm) * (10^(-9) m/nm) = 20 * (10^(-9)) m = 2 * (10^(-8)) m
Now we have found the interparticle spacing (a). Next, we can calculate the precipitate spacing (L) by rearranging the formula:
L = a * (n^(1/3))
Substituting the values, we have:
L = (2 * (10^(-8)) m) * ((0.05)^(1/3))
Finally, we can calculate the value for L using a calculator.