Measuring Mechanical Behavior of Steel during Solidification: Modeling the Sscc Test

The Submerged Split Chill Contraction (SSCC) test can measure forces in a solidifying steel shell under controlled conditions that match those of commercial casting processes. A computational model of this test is developed and applied to increase understanding of the thermal-mechanical behavior during initial solidification of steel. Determining the stress profile is difficult due to the complicated geometry of the experimental apparatus and the non-uniform temperature and strength across the shell. The two-dimensional axisymmetric elastic-viscoplastic finite-element model of the SSCC test features different mechanical properties and constitutive equations for delta-ferrite and austenite that are functions of both temperature and strain rate. The model successfully matches measurements of 1) temperature history; 2) shell thickness; 3) solidification force; and 4) failure location. In addition, the model reveals the stress and strain profiles through the shell and explains what the experiment is actually measuring. In addition to the strength of the shell, the measured force is governed by the strength of junction between the upper and lower test pieces and depends on friction at the shell / cylinder interface. The SSCC test and validated model together is a powerful analysis tool for mechanical behavior, hot tear crack formation and other phenomena in solidification processes such as continuous casting. Introduction Fundamentally-based computational models are useful tools for understanding and solving problems in commercial casting processes . To find the material properties needed for the models, requires controlled laboratory experiments. This work combines these two tools to gain new insight into both the mechanical behavior of a solidifying steel shell, and the experiments used to measure that behavior. The Submerged Split Chill Contraction (SSCC) test is a controlled laboratory experiment to measure the force generated in a steel shell as it solidifies and contracts around a solid cylinder that is suddenly immersed into molten steel [1]. This test, pictured in Figure 1, is a simplified form of the Submerged Split Chill Tensile (SSCT) experiment developed by Ackerman et al, and since applied by [2-4]. The SSCC test body consists of two separate solidsteel pieces shown in Figure 2. Most of the outer surface of both bodies is sprayed with a 0.4 ± 0.02 mm ZrO2 layer to control the heat flux to match the heat transfer conditions found in continuous casting. A steel shell solidifies with the primary dendrite growth direction perpendicular to the interface. The relative vertical position of the tops of the two pieces is held constant by a servo-hydraulic cylinder. During solidification, a load cell measures the vertical force that is needed to maintain the vertical positions of the pieces. This generates membrane stresses in the shell across the dendrites with the same orientation experienced in commercial casting processes. After ~25 sec, the test body is removed from the melt. This causes the axial force to decrease to zero, so further contraction causes shrinkage stresses to increase mainly in the radial and circumferential directions. After cooling to room temperature, the shell is detached, cut, and analyzed metallographically for hot-tear cracks. This test is highly repeatable [8] and has been used to investigate force-time [5] and stress-strain [2-6] relationships, shell strength [7], microstructure morphology [6, 9, 10] , and various defects for many different steel compositions [1], other castable metal alloys [11], and cooling rates [12]. Unfortunately the SSCC experiment yields little information about its fundamental operation, including what force is being measured,and its relationship to the high-temperature thermal-mechanical behavior of the solidifying steel that is sought. The measured force is a single time-varying scalar taken at a single location far away from the steel shell that forms at the interface of a complex 3D structure. How the local temperature, strain, and stress profiles evolve within the shell and lead to hot-tear cracks cannot be determined using only this test. In this paper, a transient finite-element model of the SSCC test is developed and applied to gain better understanding of the thermal-mechanical behavior of steel during initial solidification. Separate constitutive models of the thermal-mechanical behavior for austenite [13] and delta-ferrite [12] were developed in previous by empirically fitting measured data [15, 16] to determine the relation between stress, strain, strain rate, temperature and carbon content. These models for the separate phases were incorporated into an efficient numerical methodology of modeling solidification developed by Koric [17] and implemented into the commercial software ABAQUS [18]. This modeling approach has been used for a wide range of applications including stress development in solidifying steel [17] and the formation of longitudinal face cracks [19]. However, the constitutive equations were developed based on tensile test and creep experiments on solid steel that was reheated after solidification and cooling. The differences relative to the mechanical properties during solidification are unknown. By calibrating the computational model to match the temperature and force measurements of the SSCC test, additional insights can be gained into initial solidification that are more powerful than either of these two tools used separately. Previous Experiments Attempts to understand the behavior of solidifying steel have been ongoing for centuries. Only recently have methodologies been developed to quantify its thermal-mechanical properties and to predict the formation of defects, using both measurements and computational models. Initial efforts applied standard mechanical tests to reheated solid samples, including hightemperature tensile tests [15, 16] and creep tests [20, 21]. Wray conducted tensile tests on steel heated in a vacuum furnace from room temperature for both austenite at 950 – 1350 C [15] and delta-ferrite at 1200-1525 C [16] at strain rates ranging from 10 10 sec. Strength decreased exponentially with increasing temperature towards the solidus and at lower strain rates [15,16] in accordance with observations of other metals [22-27]. Austenite was shown to have much higher strength than delta-ferrite at the same temperature and strain rate [15, 16]. Creep tests on reheated steel samples have been performed primarily to investigate bulging during continuous casting [20, 29]. In creep tests by Suzuki, 5 mm diameter cylindrical test specimens were machined from as cast-steel and subjected to constant stress levels of 4.1 9.8 MPa and temperatures of 1250 1400 C for ~1000 s [30, 21]. The direction of loading with respect to dendrite growth direction did not have an effect on the creep curves [30]. Austenite recrystallization was observed at 1350 C at 7.1 MPa where strains above 0.1 were produced [30]. Because the microstructure experienced during solidification differs from reheated samples, better “in-situ” testing methods were developed to measure mechanical behavior of metal during solidification. The punch press or “melt-bending” test [31-33] uses a mold with a removable water-cooled copper plate as one side. The liquid metal cools to begin forming a shell, The copper plate is then removed and the shell is deformed by a cylinder at strain rates of 10 10 sec for specific durations. The cylinder pressure is measured by a DMS pressure transducer and the cylinder movement by an inductive displacement transducer. However, analysis has been very simplistic, as strain at the solidification front and force are estimated from the imposed deflection using simple beam bending theory 3 max 192 l EI F δ = (1) where F is force, E is the elastic modulus, I is the moment of inertia, max is the maximum displacement, and l is the restrained length. While such in-situ tests are useful for quantifying hot-tear cracking, the forces needed to quantify the stress profiles in the solidifying shell have not been recorded. Moreover, the great range of temperatures across the ingot would cause the measured forces to be dominated by low-temperature mechanical properties. Mizukami [28] used a high-temperature tensile apparatus setup in a vacuum, connected to a load cell and high speed video camera to measure the strength and deformation of steel with varying carbon contents. The samples were reheated, melted, and tested during solidification. The tensile strength was concluded to depend on the phase (austenite or ferrite) present and the strain is concentrated in the phase with the lowest tensile strength and elongation. These observations agree with other researchers [22-27]. An experiment designed to quantify the mechanical behavior of a solidifying shell was developed by Ackermann [11]. He pioneered the two-piece Submerged Split Chill Tensile (SSCT) test, by plunging water-cooled copper cylinders directly into an aluminum melt. After allowing time for the growing shell to reach a desired thickness, the two pieces of the cylinder were separated, which applied a tensile load perpendicular to the direction of primary dendrite arm growth. This allowed measurement of the strength near to the mushy zone. Ackermann found considerable strength for solid fractions greater than 0.95 and virtually no strength below this solid fraction. The most sophisticated experimental apparatus to determine the strength of a solidifying shell was accomplished by Bernhard and coauthors [34-36]. Their SSCT test is a refined version of Ackermann’s experiment with better repeatability. It has also been used to determine the susceptibility of forming hot tear cracks for steels with a wide range of carbon and other alloying content [1, 37, 38]. Bernhard compared SSCT test results to other measurements of thermomechanical behavior of reheated solid samples. The tensile strength near the solidus temperature was found to be about 50% less than other measurements for austenite [39,40] due to the SSCT test being more sensitive to effects of segregation. Tests involving delta-ferrite matched measurements made by Wray [16], which was attributed to the negligible sensitivity of deltaferrite to microstructural effects. All of these tests reveal only a single measurement about the complex multidimensional, multi-scale mechanical behavior during solidification of a steel shell. Only by developing a realistic mathematical model of this behavior, including its spatial and time variations can the experimental measurement be translated into real understanding. Hot Tearing Cracks can occur in steel due to tensile stress combined with any of several different embrittelement mechanisms, which span a wide range of temperatures. Hot tearing is distinct in that it develops near the solidus temperature [41], due to strain concentration in the liquid phase between dendrites which cannot be accommodated by liquid feeding. Hot tearing affects all alloys, but increases with increasing difference between the liquidus and solidus temperatures [42]. It is aggravated by the partitioning of alloying elements via segregation during solidification. Segregation, in turn, is worsened by the slow diffusion in the solid phase, relative to the liquid, and results in small-scale compositional variations. The result is local suppression of the solidus temperature, which increases the size of the mushy zone and makes the steel more susceptible to forming hot tears. Owing to the great complexity of the phenomena that result in hot tearing, a fundamental theory to predict this defect is too difficult. Thus, research has instead focused on developing simple empirical models to predict hot tear formation, based on fitting experimental measurements. Rappaz [43] proposed a hot tearing criterion with the physical basis of when the applied tensile stress causes pressure in the liquid in the mushy zone drops low enough to nucleate voids due to cavitation. Other simpler criteria, such as those postulated by ClyneDavies [44], Feurer [45] and Katgerman [46] include the effects of phase fractions, casting conditions and calculation of the temperature zone most likely to form hot tears. Other criteria are derived from empirically fitting experimental data to determine strain [47, 48-50], strain rate [43, 51-57] and stress [7, 47, 58-60] levels that coincide with hot tear formation. Pierer [7] compared stress based [61], strain based [62], strain rate based [63] and the Clyne-Davies model [64] with experimental data from SSCT tests. He found that even though each criterion approaches the problem from a different perspective, the predictions of cracking susceptibility are nearly the same. Previous Thermal-Mechanical Models Initial endeavors to apply computational modeling to the high temperature thermalmechanical behavior of steel solidification began with a semi-analytical solution of the elasticplastic behavior of a semi-infinite steel plate prevented from bending [65] and models of the thermal stresses that develop in the shell in and below the mold [66, 67]. Computational modeling evolved to more complex behavior including coupling the heat conduction and mechanical equilibrium equations with creep [68, 69] and elastic-viscoplastic behavior [14, 17, 70-73]. Koric, Thomas and coauthors implemented the Kozlowski III model for austenite [13] and the Zhu model for delta-ferrite [14] into both implicit [17] and explicit [77] integration schemes to model the high temperature behavior of steel. These models have been applied to predict crack formation during continuous casting [19]. Several computational models of the SSCT [6] and SSCC [1] tests have been performed to study phenomena such as shell strength [74, 75], grain size [9], hot tearing [1, 38, 10, 76], and to evaluate proposed constitutive models [94]. Due the complexity of the phenomena, many aspects of the test are still not fully understood. Experimental Methodology In this work, Submerged Split Chill Contraction (SSCC) experiments on steel solidification were performed at the Christian Doppler Laboratory, University of Leoben, Austria [1]. A photograph and schematic of the SSCC test body is given in Figure 1. The test body is composed of two main pieces. The “lower part”, consists of two stacked cylinders on a common centerline. The larger lower cylinder has a radius of 26 mm and length of 48 mm and is spray coated with a 0.4 mm thick ZrO2 layer to control the heat flux to levels experienced in commercial continuous casting processes. The smaller upper cylinder of the lower part has a radius of 10 mm and length of 98 mm. The second major piece is a geometrically-complex “upper part”. The dimensions of both parts are given in Appendix A, Figure A-1. The top surface of upper part is welded to a fixed steel plate with a hole through which the smaller upper cylindrical portion of the lower part can move. A servo-hydraulic loader controls the alignment of these two parts. A gap of 4 mm separates the large radius of the lower part from the inner wall of the upper part sleeve. The assembly of the upper and lower parts, initially at room temperature, is immersed into molten steel, causing a shell to solidify normal to the external surface and at the melt – air interface as shown in Figures 2 (a-c). During the experiment, the temperature at the SSCC cylinder-steel melt interface decreases causing a shell to form and subsequently desire to shrink due to thermal contraction. However, being in contact with the heating and expanding test body, shrinkage of the solidifying shell is prevented. The net force exerted by the solidifying shell lifts up on the lower part and pulls down on the upper part, but the servo-hydraulic loader prevents any vertical motion. The force required to maintain the relative positions of the upper and lower parts is applied by the servo – hydraulic loader and is the measured ‘solidification force’. Four thermocouples record temperature histories at different locations: two are located inside the test body 2 mm from the steel melt interface while the other two are located in the steel melt 20~25 mm away from this interface. Each pair of thermocouples is positioned on opposite sides of the test cylinder, 180 apart. The pour temperature was measured prior to immersing the SSCC test body into the steel melt and was superheated 20 C above the liquidus temperature. The test body is immersed for a period of ~ 20-30 seconds. Then the test body along with the attached steel shell is removed from the melt and cooled to room temperature. This process is shown in Figure 2 (a-c). The shell is then detached from the test body and cut into 16 separate pieces, 8 from each half, as shown in Figure 3. The shell thickness is measured at multiple locations per sample and then micro-analyzed to determine if any defects are present. The steel melt alloy analyzed in this study is given in Table 1. Table 1: Composition of Experimental Steel Alloy %C %Mn %S %P %Si %Al %Fe 0.315 1.44 0.0051 0.0039 0.356 0.0715 Balance Computational Model A transient two-dimensional finite-element model of the entire SSCC test has been developed at the Metals Processing Simulation Laboratory at the University of Illinois at Urbana-Champaign, taking advantage of the cylindrical symmetry of the process. It consists of separate heat transfer and mechanical models of both test body pieces, the solidifying steel shell, and the surrounding liquid. Heat Conduction Model Governing Equations Heat transfer in the test body and shell is governed by the energy conservation equation. The transient heat conduction model is two-dimensional axisymmetric ,with a Lagrangian reference frame (no material velocities) or and no heat generation and is governed by eq. (2). ' 1 p T T T c rk k t r r r z z ρ ∂ ∂ ∂ ∂ ∂ = + ∂ ∂ ∂ ∂ ∂ (2) where is the temperature dependent density, p c is the temperature and solid fraction dependent specific heat, T is temperature, t is time, k is the temperature dependent thermal conductivity, r is the radial coordinate and z is the vertical coordinate. The latent heat effects during solidification are included as an adjustment to the specific heat term as a function of solid fraction when cooling between the liquidus and solidus temperatures according to eq. (3). ' s p p f df c = c L dT − (3) where cp is found from eq. 3B in the appendix, Lf is the latent heat, assumed to be 271,000 J/kg and fs is the solid fraction. Mechanical Model Governing Equations Strains that develop during solidification are small, so the small strain assumption is used for formulating the mechanical behavior. The linearized strain tensor is ( ) [ ] T u u ∇ + ∇ = 2 1 ε (4) The spatial displacement gradient r u u ∂ ∂ = ∇ / is small and the Cauchy stress tensor is balanced by the body forces, b, according to the initial configuration of the material as: [ ] b + ⋅ ∇ σ =0 (5) Details of the derivation of these material states are given elsewhere [19]. The total strain rate is represented by summation inclusion of the elastic, inelastic (plastic + creep) and thermal components: thermal inelatic elastic ε ε ε ε + + = (6) The stress rate depends on the elastic strain rate only for this particular case where large rotations are negligible and the material having a linear isotropic behavior ( ) thermal inelastic D ε ε ε σ − − = : (7) where D is the fourth order isotropic tensor of elasticity I I k I D B ⊗ − + = μ μ 3 2 2 (8) with and kB being the temperature dependent shear and bulk modulus respectively, I and I are the fourth and second order identity tensors. Determination of the Thermal Strain Volume changes caused by temperature differences and phase transformations must be considered. The thermal strain tensor is found from eq. (9).