A materials scientist is working on developing a dispersion-hardened aluminum alloy. What is the minimum surface energy required for the dispersion particles to induce dislocations to be pinned at the particles, and bow out around them, rather than to cut through the particles? Aluminum has a Burgers vector of 0.286nm and a shear modulus of 26GPa.
To determine the minimum surface energy required for the dispersion particles to pin dislocations, we can use a mathematical model called the Orowan equation. The Orowan equation relates the applied stress, the Burgers vector, the shear modulus, and the spacing between the dispersion particles.
Here's how you can calculate the minimum surface energy:
Step 1: Find the critical resolved shear stress (CRSS)
The CRSS is the stress required to move a dislocation through a material. It can be calculated using the equation:
CRSS = μ * b / d
where μ is the shear modulus, b is the Burgers vector, and d is the spacing between the dispersion particles.
Given:
Shear modulus (μ) = 26 GPa = 26 * 10^9 Pa
Burgers vector (b) = 0.286 nm = 0.286 * 10^-9 m
Since we don't have the spacing between the particles (d), we cannot calculate the exact value of CRSS, but we can proceed with the equation using an arbitrary value for d.
Step 2: Calculate the minimum surface energy (ΔG_min)
The minimum surface energy required for the dispersion particles to induce dislocations to be pinned at the particles can be calculated using the equation:
ΔG_min = CRSS * b
Given that we don't have the exact CRSS value, we cannot calculate the exact minimum surface energy (ΔG_min). However, we can provide you with the equation and the steps mentioned above, which can be used once the requisite value of d is obtained.
Note: The actual determination of the spacing between the dispersion particles (d) would require experimental measurements or further information from the materials scientist working on the specific alloy.