Simulation of solid deformation during solidification: Compression of a single dendrite

A method is developed to numerically simulate coupled solidification and deformation of dendrites. Dendritic solidification is modeled using the phase-field method. The elasto-viscoplastic deformation of the growing solid is computed using the material point method. The stress analysis assumes a sharp and stress free solid-liquid interface, with the zero contour line of the phase field used to identify the interface. The deformation-induced flow in the liquid is approximated through a zero-gradient extension of the deformation velocities in the solid. Changes in the crystallographic orientation angle and advection of the phase and temperature fields due to solid deformation are all accounted for. Numerous tests are performed to validate the various numerical procedures. The full model is then applied to simulate in two dimensions the compression of a single dendrite of a pure substance growing in an undercooled melt. The development of complex stress and strain distributions is observed. The deformations result in variations in the crystallographic orientation angle within the dendrite that, in turn, affect the subsequent solidification behavior. The modeling of the deformation of polycrystalline solidifying structures, including the formation of grain boundaries, is described in a companion paper.