The Force Between Dislocations and Glide Obstacles

Our current ambition is to elucidate one of the microscopic mechanisms that has been charged with giving rise to solution and precipitate hardening. The argument is that by virtue of the elastic fields induced by an obstacle there will be a force on a dislocation which the dislocation must overcome in its motion through the crystal. As a first step towards modeling this phenomenon, we imagine the obstacle to be a spherical disturbance within the material. As was already demonstrated in chap. 7, such an obstacle produces spherically symmetric displacement fields of the form Ur = Ar + bjr.  

In principle, eqn (11.59) is all that we need to go about computing the interaction force between an obstacle and a dislocation. However, we will find it convenient to rewrite this equation in a more transparent form for the present discussion. Using the principle of virtual work, we may rewrite the interaction energy as 

Recall that the obstacle elastic fields may be arrived at by solving a boundary value problem in which three force dipoles are superposed as we did in chap. 7. In particular, we had 

If we now go about expanding the displacement fields about the point x, then we are left with 

Recall from our review of continuum mechanics that the divergence of the displacement field is a measure of the volume change associated with a given deformation. In this case, it is the volume change associated with the dislocation fields at the obstacle that gives rise to the interaction energy. 

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