Unified wave equation and numerical simulation of mechanical wave propagation in alloy solidification

Abstract

A unified wave equation of mechanical wave propagation during solidification of an alloy was established and the numerical solution of the wave equation was obtained. A three-element model (KSLS) used to describe the stress–strain constitutive relation of the alloy in the mushy zone was established by the analysis of rheological characteristics of molten melt during solidification. Based on the KSLS model, we could describe the constitutive relation of the liquid alloy, a Maxwell medium, and of the solidified alloy, an elastic medium, by the various Lame coefficients. The wave propagation during solidification was identified by a unified wave equation coming from a unified integral constitutive equation, to adapt to a variety of viscoelastic media. The wave equations could be solved succinctly by introducing a "memory factor" and the staggered grid finite difference method. The analytical results demonstrated that the unified wave equation could be perfectly applied to the numerical simulation of wave propagation during solidification. The propagation of the P-wave in a one-dimensional alloy was simulated during solidification, obtaining the propagation law of the mechanical wave: the wave with variable wavelength and amplitude propagated with attenuation during solidification; otherwise, as the source excitation force was fixed, the fluctuations caused by the mechanical wave could be more serious when the vibration was applied in the melt directly, which is more conducive to grain refinement.