Dislocation dynamics and plastic shock waves

Abstract

In a previous paper (Weertman and Follansbee (1983)) steady-state finite amplitude elastic plastic waves of moderate amplitude were treated as summed steady-state infinitesimal elastic plastic waves, each traveling at slightly different velocities with respect to the crystal lattice but at the same velocity with respect to the laboratory coordinate system. In this paper finite amplitude elastic-plastic waves are again treated as summed infinitesimal elastic-plastic waves. The infinitesimal waves, however, are not steady-state waves. For the constant dislocation density case it is shown that the governing wave equation for the infinitesimal wave is ( ∂/ ∂t){( ∂ 2 σ/ ∂x 2)-(1/ a 0 2)( ∂ 2 σ/ ∂t 2)} = −( a 0/ Λ){( ∂ 2 σ/ ∂x 2)-(1/ a 1 2) ( ∂ 2 σ/ ∂t 2)}, where σ is stress, a 0 is the elastic wave velocity, a 1 is the bulk wave velocity, and Λ is a characteristic length. The steady-state finite amplitude wave profile (for a constant dislocation density) is given by the equation a′( ∂σ x / ∂x) = [ a 1 + u]( ∂σ x / ∂x)- D( ∂ 2 σ x / ∂x 2), where a′ is the finite amplitude wave velocity and μ is the particle velocity (μ is proportional to σ) and D=( a 0 2- a 1 2) a 1 2 Λ/2 a 0 3. Similar equations hold for the case in which the density increases. The tendency of the infinitesimal wave to spread is counteracted by the squeezing effect of the particle velocity to give a steady-state finite amplitude wave. The width of the steady-state finite amplitude wave is inversely proportional to the maximum stress if the dislocation density is constant and inversely proportional to the cube of the stress if the density increases with stress. The finite amplitude wave velocity is slightly greater than a 1 for the constant dislocation density case and lies between a 1 and a 0 when the density is not constant. The plastic deformation that occurs behind a strong shock wave is also treated in this paper.