On energy balance and the structure of radiated waves in kinetics of crystalline defects

Abstract

Traveling waves, with well-known closed form expressions, in the context of the defects kinetics in crystals are excavated further with respect to their inherent structure of oscillatory components. These are associated with, so called, Frenkel–Kontorova model with a piecewise quadratic substrate potential, corresponding to the symmetric as well as asymmetric energy wells of the substrate, displacive phase transitions in bistable chains, and brittle fracture in triangular lattice strips under mode III conditions. The paper demonstrates that the power expended theorem holds so that the sum of rate of working and the rate of total energy flux into a control strip moving steadily with the defect equals the rate of energy sinking into the defect, in the sense of N.F. Mott. In the conservative case of the Frenkel–Kontorova model with asymmetric energy wells, this leads to an alternative expression for the mobility in terms of the energy flux through radiated lattice waves. An application of the same to the case of martensitic phase boundary and a crack, propagating uniformly in bistable chains and triangular lattice strips, respectively, is also provided and the energy release is expressed in terms of the radiated energy flux directly. The equivalence between the well-known expressions and their alternative is established via an elementary identity, which is stated and proved in the paper as the zero lemma. An intimate connection between the three distinct types of defects is, thus, revealed in the framework of energy balance, via a structural similarity between the corresponding variants of the ‘zero’ lemma containing the information about radiated energy flux. An extension to the dissipative models, in the presence of linear viscous damping, is detailed and analog of the zero lemma is proved. The analysis is relevant to the dynamics of dislocations, brittle cracks, and martensitic phase boundaries, besides possible applications to analogous physical contexts which are marked by macroscopic energy release through emission of waves and possibly linear viscous damping.