Numerical modeling of thermomechanical processes during continues casting of steel LH15SG for shape rolling

Transcription

1 Andriy Milenin and Henryk Dyja: Numerical modeling of thermomechanical processes during continues casting of steel LH15SG for shape rolling Paper deals with problems of crystallization, uprising and development of thermal stresses as well as bending and unbending of bloom in continues casting machine. To obtain numeric solution the finite element method is used. The mathematical model of thermal stress development in the billet during the continues casting process which takes into account the processes of crystallization, thermal volume changes and phase transformation in the viscoplastic rheological formulation for the steel in the high temperature is developed. The problem is solved in three-dimensional formulation. Constructive features of continues casting machine is taken into account, namely location and parameters of rolls, spray nozzles and cristallizator. It is shown an example of casting process for 300x400 mm blooms of LH15SG steel. The results of dilatometric and plastometric tests as well as results of temperature, strain and stresses modelling during casting process are presented in the paper. The effective solution of a problem manufacture of qualitative cast blooms requires optimization of the metal deformation condition in the continuous casting machine (CCM). The numerical model of thermomechanical processes in metal during continuous casting consist of the following components: model of the thermal and crystallization processes; model of development and relaxation of thermal stresses; model of influence of constructive elements of the CCM on metal. Review of this methods was presented in [1-], most of them is based on finite element method for plane strain or three-dimensional state of deformation. The development of numerical three-dimensional models is directed to the accounting of non-stationary processes, which occur in given volume of metal, as it passes through the CCM. Some of these processes are occurrence of thermal and technological elastic - plastic deformations, stresses relaxation and the temperature changes. A solution of a three-dimensional non-stationary task in full formulation - from the beginning of the casting process up to the establishment of stationary phase, requires long time. That time is increased also due to high gradients of temperature and mechanical properties of metal. It requires a large number of finite elements in zones of metal contact with the mould and rollers of the CCM, as well as on boundary of the different metal phases. The accounting of the features of appropriate materials requires to perform dilatometric and plastometric researches under conditions that are closed to real conditions in the CCM. Thus, in the paper the following tasks are considered. 1. Development of mathematical model of the termomechanic phenomena that occur in metal during continuous casting in the curvilinear CCM type.. Development of a method, which allows to replace a complete three-dimensional non-stationary task by a sequence of -dimensional tasks. 3. Performance of dilatometric and plastometric researches appropriate material and implication of results of these researches into the mathematical model. 4. The experimental researches and modelling of the hardening and softening processes. Mathematical model of heat exchange and metal crystalization In order to model crystallization process it was used the heat transfer equation modified by method of effective specific heat. The method is based on the following equation: dt ceff () t ρ () t div( k() t grad() t ) dτ =, (1) where: ρ (t) metal density, kg/m 3, t temperature, K, τ time, s, k(t) heat conductivity coefficient, W/m K (Fig.1,a), ceff(t) effective specific heat, J/kg K, (Fig.1,b), Fig. 1: Dependences of effective specific heat on the temperature (a); dependences of heat conductivity coefficient on the temperature (b). Professor, D.Sc. Andriy Milenin (Professor), Professor, D.Sc. Henryk Dyja (Professor), Faculty of Materials Processing Technology and Applied Physics, Częstochowa University of Technology, Czestochowa/Poland (004) Suppl. Metal Forming

2 which in the simplest cases is described according to the following order: ceff = c S (t) t < t S, dfs L ceff = cf + L cf + ts < t < t L, dt tl ts ceff = cl t > t L, cf specific heat in the range of the alloy crystallization temperature, cf = fscs + (1-fs)cL, tl liquidus alloy temperature, ts solidus alloy temperature, L hidden crystallization heat, J/kg, fs fraction of solid phase, cs specific heat of solid phase, cl specific heat of liquid phase. An algorithm of 3D solution of heat transfer and crystallization of the casting bloom is built upon the sequent solutions of the plane tasks, which are corresponded to the movement of the cross-section through mould, secondary cooling zone and air cooling zone with the casting speed. In order to solve the equation (1) it is used the variation problem formulation in which it must be fulfilled the minimization of the following functional: 1 t t dt J = k(t) + k(t) c eff (t) ρ(t) t dv V x y dτ α + ( t t ) df, (3) F where: α effective coefficient of heat exchange, W/m K, ( )( ) σ α α 10 rad = conv + t + t t+ t, (4) 8 σ rad coefficient of heat exchange by radiation, W/m K 4 ; α conv coefficient of heat exchange by convection, W/m K; F area of contact metal with tool, m ; V volume, m 3 ; t temperature of environment, K. Derivative of temperature with respect to time is calculated implicit by: dt tτ tτ τ =. (5) dτ τ To solve the task a 6-nodes triangular finite elements are used. () Mathematical model of stresses and strains development Calculation of strains and stresses caused by of thermal loading and ferrostatic pressure. Let's consider a solution of a -dimensional task with linear distribution of deformation in direction of bloom height (along an axis y). The solution is sought from the necessary condition of the minimum of the Lagrange variation functional: 1 3 E J = E εi dv + ( ε0 β t) dv V v1 ν + f p ε β t dv + σ u ds, V ( ) L f 0 i i s where f L fraction of liquid phase, E the modulus of plasticity (correspondent to Young s modulus in elasticity). The value of E is determined by following equation: ( ε, ε,t) σs i i E = εi where σ ( ε, ε,t) (6), (7) s i i dependence of effective stress σ s on effective strain rate ε i, effective strain ε i and temperature t with accounting of stress relaxation. The ferrostatic pressure is calculated according to formulae pf = ρgh= ρgz, where h height of a pole of liquid metal in the CCM. The term of β t in functional (5) can be evaluated by dilatometric test: ( ) l t, t β t =, (8) l0 where lt () specimen elongation with temperature changes, l 0 initial specimen length. The dependence of lt () is determined as a result of experimental (dilatometric) researches. Hardening and softening model. Presently available studies on the theory of dislocation enable one to use a number of assumptions underlying this theory as a basis for the calculation of yield stress during non-monotonic thermal deformation. The basics of the approach taken in the present study were originated in the studies of Taylor [3]. Without going into details, it should be noticed that Taylor obtains the following dependence of yield stress on dislocation density: σs = AGb ρ, (9) where G shear modulus; A factor accounting for the interaction of dislocations (according to Taylor, it is equal to 0.1); b Burgers vector (its value is in the order of m); ρ dislocation density, m -. As follows from formula (9), the yield stress depends directly on the current value of dislocation density. This value can be determined from one of the known models of dislocation generation. A more detailed description of some existing models of dislocation generation during hot deformation is given in references [4-5]. Putting aside other models, we will focus on the approach adopted in reference [5], which, in our view, has the best grounds, and is based on the following equation: dρ ε A3 = i k ρ(τ) ρ(τ)r [ ρ(τ) ρcr], (10) dτ bl D 574 (004) Suppl. Metal Forming 004

3 where l mean length of the free propagation of dislocations, which, for large deformations is determined from the formula of l=aoz -A 1 (for small deformations, we have l=aρ -0.5 ); А3 coefficient of grain boundary mobility; Z Zener Hollomon parameter, k self-diffusion coefficient, ρcr critical dislocation density, above which dynamic recrystallization processes begin to apply; r the function is equal to zero, when ρ<ρcr; otherwise, it is equal to ρ-ρcr ; D grain diameter; Аî, А1, А empirical material coefficients. The analysis of this model, partially performed in work [4], shows that the process of choosing the coefficients in formulae (9) (10) is made very difficult because of the introduction of the discontinuous function r to Equation (10). We will write the proposed model in the following form for easy identification: dρ C dτ = σ 1 εi Cρ, (11) s = C 3 ρ, (1) j = 1j + j εi + 3j, (13) C a a a t where Cj model coefficients, akj matrix of empirical coefficients. For the determination of the coefficients akj data from experimental plastometric tests is used. The accounting of nonlinearity of the rheologycal properties of the metal. The dependence of σs( εi, εi,t, τ) is a nonlinear function, which should be determined by experimental research. The scheme of nonlinear problem solution used in the present work is based on the Iliuszyn`s method [6]. Parameter E is known as modulus of plasticity, which depends on σs, ε i. On the basis of the equations (6)-(7) the iterative process of determination of E is constructed. As a first approximation we accept E = E. After task has solved are new values of parameters σs, ε i are determined. Then according the formula (7) the next approximation of parameter E is calculated. In this work the following criteria of a stop of calculations is used: Definition of resistance to deformation σ SL at presence of liquid phase is carried out by the following formula: σ SL = fsσs. (16) Bending and unbending of metal in the CCM. The thermal stresses is calculated on the basis of the above mentioned equations (equation 6-8). The strain increment ε z changes from one section to the next linearly with the height of section. The increment depends on a location of the section and rollers in the machine: H 1 1 = H H, (17) Ri ε zi + 1 Ri+ 1 where Н height of the casting bloom, R radiuses of bending in the previous (i) and current (i+1) location of the section in the CCM. The geometrical data were shown on the Fig.,a. Radius R for each section of the bloom is determined by coordinates of three rollers, nearest to this section. For this purpose the points of contact of metal with the rollers and equation of a circle, which passes through these points are determined. After the radius of the circle has calculated the radius of a bend as for the given section of metal is considered as current. The results of calculation of deformation by the given algorithm for the zone of bending machine are shown in the Fig.,b. Experimental evaluation of the stress curve and dilatometric research The dilatometer 805A/D was used for plastometric and dilatometric research [7]. The standard specimens from the LH15SG steel were examined. The specimen sizes are: height 10 mm, diameter 5 mm. Temperature was varied from 700 to 100 C, strain rate was 0.001, 0.01 s -1 and 0.1 s -1. The maximal strain was 0.1. The results of experimental evaluation of the hardening Ep Ep + 1 max , (14) E p+ 1 up up+ 1 max (15) up+ 1 Description of dependence σ ( ε, ε,t, τ) only for solid state. s i i can be used Fig. : Geometrical parameters of bending of the metal in the CCM (a) and result of calculation of the field of deformation ε z in the bending zone (b). (004) Suppl. Metal Forming

4 curve for temperature С and strain rates and 0.01 s -1 are shown in Fig 3. In that figure are shown the results of plastic stress calculation by the model according to the dependences (11)-(13), as well. Methodology lies in matching of coefficient Ci in equations (11)-(1) for each curve. Then the dependence coefficient Ci on the temperature and strain rate according to equations (13) is built: C1 = ε i t C = ε i t. (18) C3 = ε i t To estimate model efficacy research was performed with two stage loading. The temperature was С and strain rate 0.01 s -1. The results of the study are shown in Fig. 4. The results of simulation are shown in Fig. 5. The pause was 1 s. From Figs. 4 and 5, it can be concluded that the material behavior in the pause between deformations can be modeled with accuracy allowed by experimental measurements. The segment of curve in Fig. 4 corresponding to the pause dropped to zero because of the experimental appliance. Heating and cooling rate in the dilatometric research was 10 C/min. The results of researches during cooling a sample are shown in figure (Fig.6). During modeling there are used appropriate dilatometric curve [8]. Results of stress and strain calculations by the finite element method The cooling of the rectangular 300x400 mm of the SH15SG steel is presented. The initial temperature is 150 C. Effective heat exchange coefficient in mould is set α = 3000 W/m K. In the task, boundary conditions there were assumed that the length of contact of metal with rollers is equal to 0.1 r, where r radius a roller. On the Fig. 7a location of the rollers in the CCM is shown. Width of a jet of spray nozzles is equal to 15 mm. Effective heat transfer coefficient for contact of the metal with the roller was accepted 500 W/m K, for the zone of spray nozzles cooling 300 W/m K, for the other surfaces contacting to air W/m K. The casting speed was accepted 0.75 m/min. On the Fig. 7b it is shown location the points in the cross-section. The points were used for the control of stresses, strain and temperature. The changes of temperature are presented on the Fig. 7. The change of temperature in control points in the mould zone and zone of secondary cooling are shown in a Fig. 8. The step character curve in points and 3 is connected with alternation of the rollers, cooling devices and cooling on air. The deviation curve in points and 3 is connected to different quantity of rollers on the stretched and compressed side of metal (see Fig. 7 a). Fig. 3: Experimental (1,3) and calculated (,4) hardening curve. Fig. 5: Calculated results hardening curve. Fig. 4: Experimental results of the hardening curve for temperature 900 Ñ and strain rate 0.01 s-1. Fig. 6: Dependences of elongation on the temperature under the cooling. 576 (004) Suppl. Metal Forming 004

5 Fig. 7: Location of rolls in CCM (a) and calculations temperature distribution in metal, (b). The results of account of deformation ε y in control points and 3 are shown in a Fig. 9. An initial parts of curves corresponded to the passage of the section through the mould, are identical. In this zone any mechanical influences except for thermal stress and ferrostatic pressure are not considered. Further, getting in the zone of bending, the curves diverge. On the compressed side of the bloom to compressive deformations from temperature pressure the compressing deformations from a bend (curve in a point ) are added. On the side of a stretching there is a change of a sign of deformations. Further, after the curves do not vary mach and in the zone they meet and diverge again in opposite direction. The analysis of intensity of stress in the mould and the zone of secondary cooling (Fig. 10) shows, that under the given conditions the maximal stresses arise on the exit from the mould. Further stresses change wavy according to alternating cooling of the different zones. Fig. 9: Changes of deformation and 3. ε y in the control points Fig. 8: Temperature distribution in control points in mould zone and secondary cooling zone Fig. 10: Changes of stress intercity in the control points in the mould zone and in the secondary cooling zone. (004) Suppl. Metal Forming

6 The analysis of stresses σ y in the control points (Fig. 11) shows, that on oscillation caused by cyclic temperature influence the stresses from a bend of bloom having monotonous character is imposed. It is necessary to note, that complexity in definition of real borders between the consequent cooling zones and the distributions of intensity cooling in that zones brings in complexity in the model and it requires additional experimental researches. Conclusion The new three-dimensional model of thermomechanical processes occurring in the metal at continuous casting is offered. The model is based on replacement of a threedimensional task by the sequence of d tasks. The processes of occurrence of the thermal stresses, action of ferrostatic pressure, stress relaxation phenomena, bending and unbending of bloom in the CCM are taken into account. It is shown, that the stresses in bloom consist of the cyclic components caused by thermal loadings and monotonous components connected with the bending and unbending of the bloom in the CCM. References [1] J.S. Ha, J.R. Cho, B.Y. Lee, M. Y.Ha Numerical analysis of secondary cooling and bulging in the continuous casting of slab, J. Mat. Proc. Techn. 113 (001), p [] J.R. Boehmer, G. Funk, M. Jordan, F.N.Fett Strategies for Coupled Analysis of Thermal Strain History During Continuous Solidification Processes, Advances in Engineering Sofrware Vol. 9. No p , [3] G.I. Taylor, Proc. Roy.Soc. London, A145, (1934), 36. Fig. 11: Changes of stress σ y in the control points in mould and the secondary cooling zone [4] A.A. Milenin, H. Dyja, S. Mroz Simulation of Metal Forming During Multipass Rolling of Shape Bars, Proc. of the Int. Conf. Advances in Materials and Processing Technologies (AMPT 003), 003, p [5] J. Ordon, R. Kuziak, M. Pietrzyk. History Dependent Constitutive Law for Austenitic Steels. Proc. Int. Conf. Metal Forming 000, Kraków, Poland, (000), p [6] A.A. Iliuszyn, Plasticznost. Мoskwa, Gostechteorizdat, [7] Operating instructions for the Type 805A/D dilatometer, BAHR Thermoanalyse GmbH 1997, 3. [8] A.A. Milenin, H. Dyja, R. Dobrakowski Modeling of Thermal Stresses in the Continues Casting Bloom of LH15SG steel accounting Phase Transformation, Proc. 1 th Int. Sc. Conf. Achievements in Mechanical & Materials Engineering, Gliwice-Zakopane, 003, p (004) Suppl. Metal Forming 004