Determining the Peierls stress
The force required to move the dislocation is \(F = \;\frac{{\delta {\rm{\Delta }}{U_T}\left( \alpha \right)}}{{\delta \left( {\alpha b} \right)}} (3)\)
Plotting the derivative of the energy-alpha graph gives a plot of the force required to move a dislocation. The corresponding graph gives the stress required to move the dislocation per unit length of dislocation.
\[\tau = \;\frac{1}{{{b^2}}}\frac{{\delta {\rm{\Delta }}{U_T}\left( \alpha \right)}}{{\delta \left( {\alpha b} \right)}} (4)\]
As we saw in expression (1) on the “Peierls energy” page.
\[\;\Delta {U_T}\left( \alpha \right) = \;\frac{1}{2}\Delta {U_P}\left( {1 - cos4\pi \alpha } \right) (5)\]
Hence,, differentiating, we get \({\tau _P} = \)\(\frac{{2\pi }}{{{b^2}}} \)\(\;\Delta {U_P}\;{\rm{sin}}\left( {4\pi \alpha } \right) (6)\)
The maximum stress is where \(\;\sin \left( {4\pi \alpha } \right) = \;1 \). The expression is often written as
\[\frac{{{\tau _P}}}{G} = \;\frac{{2\pi }}{{G{b^2}}}\;\Delta {U_P} (7)\]
A note on this graph: It's important to remember that even though we are multiplying by Gb2, d/b also affects the dislocation width - increasing d/b increases the dislocation width as seen on the "determining w" graph. So for this graph if we impose a w/b, then we can see the effects of changing the magnitude of Gb2 by adjusting the parameters G and b/d. Likewise if we impose a b/d we can see the effects of altering w/b through changing something else like the Poisson ratio.
The graph shows the stress calculated using the maximum of the graph (expression 4) and also the differentiated expression for ΔU (equation 7). They are very similar, which is expected.
Therefore, using expression (2) from the “Peierls energy” page.
\[\frac{{{\tau _P}}}{G} ∝ \;2\pi \exp ( - \frac{w}{b}) (8)\]
The analytical solution provided by the “continuum elasticity” model (/tlplib/dislocations/slip_via_dislocation.php) gives the constants as:
\[ {\tau _P} = \;3G\exp ( - 2\pi \frac{w}{b}) (9)\]
which is sometimes simplified to \( \frac{G}{{180}}\) if we assume that w = b (which is not in fact true as we have seen!). This solution gives us a theoretical shear stress – i.e. the stress required for uniform slip – but as can be seen from the above graph, it is several orders of magnitude higher than the actual Peierls stress.
The analytical solution is from Peierls 1940 and Nabarro 1947. In essence, they assume that the in-plane strain is the surface of a semi-infinite elastic continuum and use that and the sum of misalignment energies to find the width at the equilibrium point. The elastic continuum (i.e. in-plane energy) is assumed to be fixed as is the width; therefore, the changes in energy are due only to the misalignment summation. This gives a wrong answer and indeed the maximum and minimum positions swap (U-tot Vs U-misalignment are out of phase). Since both the in-plane strain energy and the width change as the dislocation is displaced these should be included.
Compared with real values, the atomistic model is actually a small underestimate of the Peierls stress, but the analytical solution is a vast overestimate, which renders it is not very useful.