It was noted that evolution equations, or equations of motion, of moving defects were the balance of the so-called driving forces in the presence of external loading. The driving forces were defined to be configurational forces in the sense of Noether’s theorem governing the invariance of the variation of the Lagrangean of the mechanical system under infinitesimal transformations. For infinitesimal translations, the resultant dynamic J integral equalled the change in the Lagrangean if and only if the linear momentum was preserved. Dislocations and inclusions were defects that possessed self-stresses, and the total driving force for these consisted of only two terms; one expressing the self-force due to the self-stresses, and one the effect of external loading upon the change in 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 equalled the energy-release rate, was calculated. By a limiting process as the radius tended to infinity, the driving force (energy-release rate) of a moving half-space plane inclusion boundary was obtained which was 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 was presented. The equation applied to many strips of eigenstrain provided a system to solve for the position/ evolution of strips of eigenstrain.

Evolution Equation of Moving Defects: Dislocations and Inclusions. X.Markenscoff: International Journal of Fracture, 2010, 166[1-2], 35-40