Loss of pseudo-momentum, energy-release rate and the effective mass of a moving dislocation.

In a seminal paper in the Philosophical Transactions of the Royal Society (A244, 87–112). Eshelby (Eshelby 1951 Phil. Trans. R. Soc. Lond. A 244, 87–112. (doi:10.1098/rsta.1951.0016)) introduced the concept of 'the force on an elastic singularity' and suggested that the extensions to dynamics include the application of the momentum flux. In this direction, it is shown that, in the non-uniformly motion of a dislocation there is a loss in quasi-momentum (or pseudo-momentum) across the scales ε to ε2, which induces an effective mass of the dislocation, and a loss in kinetic energy across the scales. It is shown through Noether's theorem that the rate of change of quasi-momentum in the volume enclosing the dislocation is equal to the flux of it through the surface minus a quasi-force, which is the dynamic J integral. The connection between the variation in the Hamiltonian and in the Lagrangian relates the quasi-force to the energy-release rate, yielding the same effective mass, while providing physical meaning both through momentum and energy. The effective mass of a dislocation is important in relating the energetics of defects from the micro to the macro scales, and the loss of quasi-momentum can have wider applications in continua. This article is part of the theme issue 'Foundational issues, analysis and geometry in continuum mechanics'. [ABSTRACT FROM AUTHOR]