DISLOCATION EMISSION FROM CRACK TIPS AS A VARIATIONAL PROBLEM OF THE CRACK ENERGY

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.