Higher order alignment tensors in continuum dislocation dynamics


Prof. Dr. Thomas Hochrainer
Bremen Institute for Mechanical Engineering BIME

Wednesday, 29. May 2013, 17.00
WW8, Raum 2.018, Dr.-Mack-Str. 77, Fürth

A recently developed continuum dislocation dynamics theory allows for a self-contained dislocation density evolution and overcomes conceptual problems present in approaches based on the distinction of geometrically necessary and statistically stored dislocations. Continuum dislocation dynamics is based on a higher dimensional dislocation density tensor comprised of two distribution functions on the space of local orientations (e.g. the unit circle in a glide plane), which are the density of dislocations per orientation and the density of dislocation curvature per orientation. We expand these functions into infinite series of symmetric traceless tensors (multipole expansion), which we call alignment tensors as in the realm of liquid crystals. The first two terms in the expansion of the density define the total dislocation density and the Kröner-Nye tensor. The first term in the expansion of the curvature density, the scalar total curvature density, turns out to be a conserved quantity; the integral of which corresponds to the total number of dislocations. From the higher dimensional evolution equations we derive an infinite hierarchy of evolution equations for the alignment tensors of all orders. Low order closure approximations needed to arrive at tractable systems of equations are discussed. The theory is applied to small example problems and reproduces the phenomenon of mechanical annealing observed in sub-micrometer compression tests.