Simulation of Shrinkage and Stress in Solidifying Steel Shells of Different Grades

Thermal–mechanical behavior of the solidifying shell is important for design of taper and understanding crack formation and other defects during continuous casting of steel. A transient finite-element model, CON2D, has been developed to simulate the evolution of temperature, stress and strain in the solidifying shell during this process. The model features unified elasticviscoplastic constitutive models for austenite, ferrite, mushy, and liquid steel. The model was validated by simulating an SSCT experiment similar to that of Kurz. CON2D was then applied to investigate the effect of steel grade on thermo-mechanical behavior of a slice domain under realistic heat flux conditions. The shrinkage predicted by CON2D was compared with simpler methods, such as that of Dippenaar. This simple method is found to over-estimate the shrinkage of low carbon steels, where a substantial fraction of soft delta-ferrite exists, but matches reasonably for high carbon steel, containing strong austenite. Implications of the stress and strain profiles in the solidifying steel and practical applications are also discussed. Introduction The shrinkage associated with solidification and cooling is of practical importance for many casting processes, as it affects both the casting dimensions and the formation of hot tear cracks, and other defects in the product. In continuous casting processes, molds are often tapered to match the shrinkage, in order to continuously support the weak shell and avoid defects. Taper design depends on accurately quantifying the fundamental phenomena that govern shrinkage during the formation and cooling of a solidifying shell. This paper summarizes recent work with computational models to predict these phenomena for steel, during the continuous casting process. The model is first validated by comparison with measurements of a "Submerged Split Chill Tensile" (SSCT) test. This important measurement tool was pioneered by Kurz to study mechanical behavior and failure phenomena during solidification. The model is then applied to predict shrinkage during continuous casting of steel, investigating the effect of grade. Finally, simplified models to predict shrinkage are evaluated. Shrinkage Models A finite-element elastic-viscoplastic thermal-stress model, CON2D has been developed to predict thermal-mechanical behavior of steel during continuous casting, including shrinkage of the solidifying steel shell. The model results have many applications, such as for designing the Page 34 Y. Meng, C. Li, J. Parkman, & B.G. Thomas, MCWASP X, TMS, 2004 taper of the narrow faces of a continuous casting mold for steel slabs, in order to accommodate shrinkage of the wide faces, as shown in Fig. 1. The model solves a 2D finite-element discretization of the transient heat conduction equation in a Lagrangian reference frame that moves down through the caster with the solidifying steel shell. Next, the force equilibrium, constitutive, and strain displacement equations are solved under a condition of generalized plane strain in both the width and casting directions. Thermal and mechanical behavior are studied here in a longitudinal slice through the centerline of the shell (Fig. 2). This slice domain has been shown to be an accurate and economical method to approximate shrinkage of the thin solidifying shell in the mold, despite the corner effects. Simple Shrinkage Predictions Owing to the great computational effort required for a complete finite-element stress simulation, and the dominance of thermal strain in the shrinkage, simpler ways are sought to estimate shrinkage from the computed temperature histories of points in the shell. Recent work with CON1D compares two simple methods. First, thermal strain εth1 can be estimated simply from the shell surface temperature, Ts: 1 ( ) ( ) th sol s TLE T TLE T ε = − [1] where TLE is the thermal linear expansion function for a given steel, Fig. 3, calculated from weighted averages of the phases present. Another method, developed by Dippenaar computes the strain εth2, by summing the average TLE of the solid portion of the shell between each pair of consecutive time steps: ( ) ( ) ( ) 2 0 1 1 t solid nodes t t t th i i t i TLE T TLE T i ε +Δ = = ⎛ ⎞ ⎛ ⎞ = − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ∑ ∑ [2] where the shell thickness is divided into i sections. CON2D Stress Model In the elastic-viscoplastic finite-element model, the total strain is decomposed into elastic, thermal, inelastic and flow-strain components. Thermal strain dominates the total, and depends on temperature and steel grade, as shown in Fig. 3. A simple micro-segregation model is adopted to track the weight fractions of each phase. Unified plastic-creep constitutive models are used to capture the temperature, strain-rate, phase fraction, and grade sensitivity of steel strength at high temperature. The instantaneous equivalent inelastic strain rate depends on the current equivalent stress, temperature, the carbon content, and the current equivalent inelastic Fig. 1: Wide face shrinkage and narrow face taper Fig. 2: 1-D Slice Simulation Domain for CON2D ~ 1500 °C