G. Schoeck, DISLOCATION EMISSION FROM CRACK TIPS AS A VARIATIONAL PROBLEM OF THE CRACK ENERGY, Journal of the mechanics and physics of solids, 44(3), 1996, pp. 413-437
The emission of dislocations from crack tips can be described by the g
radual build-up of a distribution of infinitesimal dislocations ahead
of the crack front. The change in energy results from the work done by
the K-stress, part of which is introduced into the elastic self energ
y of the emanating dislocation (interacting with its image), part into
the atomistic interplanar energy in the glide plane caused by the dis
placement u(x), and part dissipated. The shape of u(x) is determined b
y the condition that the total energy is a minimum. Instead of solving
the resulting (two-dimensional) variational problem with the correspo
nding Euler equations (which leads to generalized Peierls integral equ
ations) we apply the Ritz technique in choosing appropriate trial func
tions for u(x) with adjustable parameters. Physical intuition shows th
at for the displacement u(x) a truncated arctg function with a polynom
ial as argument is an appropriate choice. In introducing a scaling len
gth w and an escape coordinate lambda the stability of the configurati
on can be studied and the condition when it becomes unstable and spont
aneous emission occurs can be identified. Without special assumptions
about u(x) the functional relation between external and material param
eters controlling the emission process can be determined. It turns out
that in equilibrium half of the work done by the K-stress goes into a
tomistic interplanar energy. For shear loading in mode II and mode III
with crack plane and glide plane coinciding (theta = 0) we give a sol
ution taking full account of anisotropy using an arctg type displaceme
nt. The assumption of a geometrically constrained path for u(x) leads
to a simplification but it is not necessary. When a tensile stress exi
sts across the emission plane (mode I and/or a not equal 0) the resist
ance against decohesion and the lowering of the shear resistance owing
to normal displacements has to be included in the energy balance. Oth
erwise the procedure is the same. The method also allows to account fo
r the energy of ledge formation which can influence drastically the em
ission criteria and which is difficult to treat on the stress level.