Evolution equation of moving defects: dislocations and inclusions

Evolution equations, or equations of motion, of moving defects are the balance of the “driving forces”, in the presence of external loading. The “driving forces” are defined as the configurational forces on the basis of Noether’s theorem, which governs the invariance of the variation of the Lagrangean of the mechanical system under infinitesimal transformations. For infinitesimal translations, the ensuing dynamic J integral equals the change in the Lagrangean if and only if the linear momentum is preserved. Dislocations and inclusions are “defects” that possess self-stresses, and the total driving force for these defects consists only of two terms, one expressing the “ self-force” due to the self-stresses, and the other the effect of the external loading on the change of configuration (Peach–Koehler force). For a spherically expanding (including inertia effects) Eshelby (constrained) inclusion with dilatational eigenstrain (or transformation strain) in general subsonic motion, the dynamic J integral, which equals the energy-release rate, was calculated. By a limiting process as the radius tends to infinity, the driving force (energy-release rate) of a moving half-space plane inclusion boundary was obtained which is the rate of the mechanical work required to create an incremental region of eigenstrain (or transformation strain) of a dynamic phase boundary. The total driving force (due to external loading and due to self-forces) must be equal to zero, in the absence of dissipation, and the evolution equation for a plane boundary with eigenstrain is presented. The equation applied to many strips of eigenstrain provides a system to solve for the position/ evolution of strips of eigenstrain.