Editing Dislocation (section)

=== Forces on dislocations ===
Dislocation motion as a result of external stress on a crystal lattice can be described using virtual internal forces which act perpendicular to the dislocation line. The Peach-Koehler equation<ref>{{Cite journal|last1=Peach|first1=M.|last2=Koehler|first2=J. S.|date=1950-11-01|title=The Forces Exerted on Dislocations and the Stress Fields Produced by Them|journal=Physical Review|volume=80|issue=3|pages=436–439|doi=10.1103/PhysRev.80.436|bibcode=1950PhRv...80..436P}}</ref><ref>{{Cite book|last=Suzuki|first=Taira|title=Dislocation Dynamics and Plasticity|date=1991|publisher=Springer Berlin Heidelberg|others=Takeuchi, Shin., Yoshinaga, Hideo.|isbn=978-3-642-75774-7|location=Berlin, Heidelberg|pages=8|oclc=851741787}}</ref><ref>{{Cite book|last=Soboyejo|first=Winston O.|title=Mechanical properties of engineered materials|date=2003|publisher=Marcel Dekker|isbn=0-8247-8900-8|location=New York|chapter=6 Introduction to Dislocation Mechanics|oclc=50868191}}</ref> can be used to calculate the force per unit length on a dislocation as a function of the Burgers vector, <math>\mathbf{b}</math>, stress, <math>\sigma</math>, and the sense vector, <math>\mathbf{s}</math>.

::<math>\mathbf{f} = (\mathbf{b}\cdot \sigma)\times \mathbf{s}</math>

The force per unit length of dislocation is a function of the general state of stress, <math>\mathbf{F}</math>, and the sense vector, <math>\mathbf{s}</math>.

::<math>
\mathbf{f} = \mathbf{F} \times \mathbf{s} =  
\begin{vmatrix}
\hat\imath & \hat\jmath & \hat k \\
F_x & F_y & F_z \\
s_x & s_y & s_z
\end{vmatrix}</math>

The components of the stress field can be obtained from the Burgers vector, normal stresses, <math>\sigma</math>, and shear stresses, <math>\tau</math>.

::<math>\begin{aligned}
F_x &= b_x\sigma_{xx} + b_y\tau_{xy} + b_z\tau_{xz} \\
F_y &= b_x\tau_{yx} + b_y\sigma_{yy} + b_z\tau_{yz} \\
F_z &= b_x\tau_{zx} + b_y\tau_{zy} + b_z\sigma_{zz}
\end{aligned}</math>