### 5.5 The uniformly moving special case

For the case in which the dislocation moves uniformly with constant speed v, the past history function takes the form l(t) = vt. In that case, it is possible to find an analytical solution for elastodynamic fields of dislocations, either by the equations given in Table 2.2 or by directly solving the governing equations using η(x) = x · d where d = 1/v is the slowness of the dislocation.

In the latter case, one would obtain that the transformed potentials can be written as:

$\text{\Psi}\left(\lambda ,z,s\right)=-\frac{\text{\Delta}u\left({b}^{2}-2{\lambda}^{2}\right)}{{s}^{2}{b}^{2}\beta}\left[{\displaystyle {\int}_{0}^{\infty}{\text{e}}^{-s\left(d+\lambda \right)\xi}\text{d}\xi}\right]{\text{e}}^{-s\beta z}$

(2.71)

$\text{\Phi}\left(\lambda ,z,s\right)=-\frac{2\text{\Delta}u\lambda}{{s}^{2}{b}^{2}}\left[{\displaystyle {\int}_{0}^{\infty}{\text{e}}^{-s\left(d+\lambda \right)\xi}\text{d}\xi}\right]{\text{e}}^{-s\alpha z}$

(2.72)

These expressions can be directly ...