MODEL OF FORMATION AND PROPAGATION OF SLIP BANDS IN METALS
Authors: Sarafanov G.F. , Utkin A.A. 
Published: April 2023

A theoretical study of the processes of localization of plastic deformation in metals has been carried out. Within the framework of the system of evolutionary equations for dislocation density, taking into account the multiplication and annihilation of dislocations, the possibility of a running solution for the slip strip is established. It is shown that the initial system has two equilibrium states. For the total dislocation density and dislocation charge normalized to a stationary homogeneous solution for dislocation density, these are the states (0,0) and (1,0) on the phase plane of the above variables in dimensionless form. From the analysis of singular points, it follows that the point (0,0) is a stable node, and the equilibrium state (1,0) is a saddle. In this case, the desired solution of the initial system of evolutionary equations is a separatrix going from point (0,0) to point (1,0), which corresponds to solutions in the form of a drop wave for the dislocation density forming the slip band and a momentum for the dislocation charge. It is shown that the dislocation charge propagates in the front of the slip band, which moves at a velocity u = kV (V is the drift velocity of dislocations due to external load, the proportionality coefficient k satisfies the condition 0 < k < 1). After analyzing the existence of self-similar solutions, it follows that non-homogeneous wave solutions occur only at A = t1/t2 > 1, where t1 and t2 are, respectively, relaxation times of the total dislocation density and dislocation charge to a homogeneous state. Estimates show that for a given dislocation kinetics (dislocation generation and annihilation processes) A > 1, i.e. satisfy the conditions for the formation of a slip strip of a given type. The stability of the obtained wave stationary self-similar solutions is considered. Assuming that deviations from stationary solutions for the dislocation density and dislocation charge are limited to a given domain, and considering the deviations small, we obtain the Sturm – Liouville problem for own functions and eigenvalues with zero boundary conditions at the boundary of this domain. The appropriate transformation reduces the problem to an equation of the Schrodinger equation type. It is shown that under certain conditions the spectrum of the Schrodinger operator is in the left half-plane, i.e. deviations of dislocation density and dislocation charge decay exponentially over time and the desired stationary solutions are asymptotically stable. The issue related to the determination of the steady-state velocity of wave propagation is considered. Linearization of the initial system of equations for dislocation density and dislocation charge allowed us to obtain a linear sine-Gordon equation, from the solution of which the steady-state wave velocity is determined. It is shown that for the initial system of the system of equations for A > 1, arbitrary initial distributions of the desired variables acquire the form of self-similar solutions over time and move at the lowest possible speed u = 2VA1/2/(1 + A).