Frank Read Source Stress Calculation

The stress required to operate a Frank-Read source is inversely proportional to the length of the dislocation segment acting as the source. In a polycrystalline material, the maximum length of such an immobilized segment is related to the grain size ($d$). Therefore, the shear stress ($\tau$) required is proportional to the inverse of the grain size: $\tau \propto \frac{1}{d}$.

Stress Calculation Steps

We can establish a relationship between the initial state (1) and the final state (2):

$ \tau_1 d_1 = \tau_2 d_2 $

Where:

• $ \tau_1 $ is the initial shear stress
• $ d_1 $ is the initial grain size
• $ \tau_2 $ is the final shear stress
• $ d_2 $ is the final grain size

Given Information:

• Initial grain size, $ d_1 = 10 \text{ µm} $
• Initial shear stress, $ \tau_1 = 100 \text{ MPa} $
• Final grain size, $ d_2 = 10 \text{ nm} $

Unit Conversion:

Convert the initial grain size to nanometers (nm) for consistency:

$ d_1 = 10 \text{ µm} = 10 \times 10^3 \text{ nm} $

Calculating Final Shear Stress ($ \tau_2 $):

Rearrange the formula to solve for $ \tau_2 $:

$ \tau_2 = \tau_1 \times \frac{d_1}{d_2} $

Substitute the values:

$ \tau_2 = 100 \text{ MPa} \times \frac{10 \times 10^3 \text{ nm}}{10 \text{ nm}} $

$ \tau_2 = 100 \text{ MPa} \times 10^3 $

$ \tau_2 = 100,000 \text{ MPa} $

Expressing this in scientific notation:

$ \tau_2 = 10^5 \text{ MPa} $

Conclusion:

When the grain size is reduced to $ 10 \text{ nm} $, the shear stress required to operate the Frank-Read source increases significantly to $ 10^5 \text{ MPa} $.