A funnel-type mold is commonly used to provide necessary clearance for the submerged entry nozzle in the thin slab casting (TSC). The partially solidified shell is subjected to the mechanical deformations, which can lead to the defects formation and, as a results, to a breakout. Traditionally, the results of the flow simulation, performed by the finite volume method (FVM), are fed to the external package for the finite element analysis of stress and strain. A coupled model was assembled using “creeping solid” approach by blending the Norton-Hoff viscoplastic stress for the solidifying shell with the Newtonian viscous stress of the liquid melt. The FVM was used to combine both liquid and solid stress models within a single solver. The iterative procedure based on the improved both side diffusion method was introduced to treat the nonlinear relation between the viscoplastic stress and the strain rate. The modeled shell thickness was verified by previously published breakout measurements and the simulation results. Temperature distribution, obtained during the TSC simulation, dominantly corresponds to the viscoplastic range. Developed numerical approach is robust and has direct industrial application.