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

摘要

The emission of dislocations from crack tips can be described by the gradual 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 energy of the emanating dislocation (interacting with its image), part into the atomistic interplanar energy in the glide plane caused by the displacement u(x), and part dissipated. The shape of u(x) is determined by the condition that the total energy is a minimum. Instead of solving the resulting (two-dimensional) variational problem with the corresponding Euler equations (which leads to generalized Peierls integral equations) we apply the Ritz technique in choosing appropriate trial functions for u(x) with adjustable parameters. Physical intuition shows that for the displacement u(x) a truncated arctg function with a polynomial as argument is an appropriate choice. In introducing a scaling length w and an escape coordinate lambda the stability of the configuration can be studied and the condition when it becomes unstable and spontaneous emission occurs can be identified. Without special assumptions about u(x) the functional relation between external and material parameters controlling the emission process can be determined. It turns out that in equilibrium half of the work done by the K-stress goes into atomistic interplanar energy. For shear loading in mode II and mode III with crack plane and glide plane coinciding (theta = 0) we give a solution taking full account of anisotropy using an arctg type displacement. The assumption of a geometrically constrained path for u(x) leads to a simplification but it is not necessary. When a tensile stress exists across the emission plane (mode I and/or a not equal 0) the resistance against decohesion and the lowering of the shear resistance owing to normal displacements has to be included in the energy balance. Otherwise the procedure is the same. The method also allows to account for the energy of ledge formation which can influence drastically the emission criteria and which is difficult to treat on the stress level. [References: 34]