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A model for prediction of cracks in a solidifying shell

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A model for prediction of cracks in a solidifying shell
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  MA IERIAlS S IEM EIl NGIM RIMe A ELSEVIER Materials Science and Engineering A 413-414 (2005) 37-43 www.elsevier.comllocate/msea A model for prediction of cracks in a solidifying shell A Lagerstedt *, H Fredriksson Casting o Metals Royal Institute o Technology SE-1oo 44 Stockholm Sweden Received in revised form 25 August 2005 Abstract A model coupling temperature and stress calculations with cracking criteria has been developed in order to predict crack positions in a solidifying shell. The model is based on a one-dimensional FDM approach suitable for continuous casting of slabs. The strain/stress model is based on a purely elastic analysis of a solidifying shell giving a straightforward comparison between stresses and crack criteria. This approach makes the model easy to use. The model is numerically evaluated using available material data for Fe-2 Ni with primary ferrite solidification and Fe-lO Ni with primary austenitic solidification. The results of the calculations are discussed and the impact of material behavior as well as process parameters is evaluated. Evaluation of the influence of changes in the heat transfer coefficient shows that the rapid changes introduce stresses large enough to induce crack formation in the solidifying shell. © 2005 Elsevier B Y All rights reserved. Keywords: Cracks; Stress; Solidification 1. Introduction Halfway cracks often appear in continuously cast material as a result of the combined effect of stress build-up during the solidification process, and low strength of the newly solidified material. During solidification, stresses are induced in a solidifying shell due to uneven shrinkage of the metal. The level of the induced stress depends on the cooling shrinkage and the temperature gradient in the shell. In a continuous casting process, the strand is withdrawn during the solidification subjecting the strand surface to different cooling conditions as it leaves the mold, enters different water cooling regions, etc. The reheating of the strand surface due to decreased cooling has been shown to influence the stress distribution in the solidified shell so that tensions arise in the inner parts of the strand [1-3]. External loads are another source for increased stresses in a solidifying shell. External loads include the influence of bulging, roll misalignment and bending and straightening of the strand. Such phenomena have been investigated by several studies [3 6]. Corresponding author. Present address: SSAB Oxeliisund AB, SE-613 80 Oxeliisund, Sweden. Tel.: +46 ISS 254181; fax: +46 ISS 255541. E-mail address:anders.lagerstedt@ssabox.com (A. Lagerstedt). 0921-5093/ - see front matter © 2005 Elsevier B Y All rights reserved. doi: I 0.1016/j.msea.2005.09.oI 7 In order to predict cracking in a solidifying shell, the thermomechanical properties of the material in question have to be known. High temperature tensile testing of in situ solidified samples gives information about the tensile strength and at what temperatures a material behaves brittle or ductile. Such measurements show that brittle behavior may be found well below the solidus temperature for some materials [7,8]. However, existing models are based on the assumption that the cracking may only occur while a liquid is present [9-12]. In addition, the stress or strain in the solidifying material has to exceed a critical value [ ]. The knowledge of thermo mechanical properties from tensile testing together with calculations of temperatures and strain and stress distribution give possibilities to predict the actual location of the cracks in a solidifying strand, not only to identify areas where the risk of cracking is increased. Such a model for prediction of the location of cracks in continuously cast slabs will be presented in this study. 2 Mathematical model 2.1. Thermal model A one-dimensional analysis of the temperatures in a continuously cast strand has been performed. The solution technique applied is a finite difference method used with central  38 A. Lagerstedt, H. Fredrikssonl Materials Science and Engineering A 413-414 2005) 37-43 Fig. 1. Illustration of the computational grid. approximation. A schematic illustration over the position of the calculation is shown in Fig. 1. The one-dimensional approach may be used as long as the geometry of the cross-section fulfills the condition that the length is much longer than the thickness of the strand, which may be expressed as L» according to the notations in Fig. 1. The temperature calculations are based on the Fourier equation in one dimension, a a ( aT p p = -   +q at ax ax 1) where p is the density, p the heat capacity and K is the heat conductivity. The source term, q describes the heat, which is generated within the material. Assuming equilibrium solidification, the source term is expressed, { 0, whenT> h _ dgs a q - p(- lH) dT at when Ts < T < h 0, whenT Ts 2) where the equilibrium value of the latent heat -  IN), is distributed over the solidification range. hand Ts represent the liquidus and solidus temperatures, respectively. The rate of solidification may be expressed in several ways. Here, the change of the fraction-solidified material is calculated by the expression given by Rogberg [13]. (3) Eqs. 1)- 3) are combined in order to find the temperature distribution in the solidified layer. 2.2. Strain model The temperature calculations are used as the starting point for the stress calculations, and are not coupled with the stress calculations. The derivation of the elastic strain is based on an approach made by Kristiansson [14]. He derived an analytical expression for the elastic strain based on the conclusion that the elastic strain was not much different from the total strain in a solidifying shell. The derivation disregards the ferro static pressure at the solidification front and assumes no remelting. The expression is valid as long as the coherent part of the mushy zones does not meet at the centreline of the strand. The elastic strain is expressed [14], 4) t is the time at which the solidification front, XS, passes a certain position. The thermally induced tension, 8th, is expressed, th a a at at 5) where a is the linear expansion coefficient. The strains in the solidifying shell are thereafter converted to stress by means of Hooke s law. Hooke s law will only give valid results within its limits of validity, which is below the yield stress. 2.3. Crack prediction model A crack will form when the stress in the brittle areas of the material exceeds the strength of the material. In ductile areas, the stress will be relaxed by plastic deformation when it reaches the level of the material strength. Hence, in order to form cracks two conditions have to be fulfilled. Firstly, the temperature in the solidified shell has to be higher than the transition temperature from brittle to ductile state, TDB. Secondly, the stress in the material has to exceed the rupture limit of the material, a rI. These cracking criteria of a material are found by performing hot tensile testing on in situ solidified specimens. There are several such experiments made on different steel grades, showing a sharp ductile to brittle transition temperature and an ultimate tensile strength gradually decreasing to zero at the coherence temperature [7,8,15]. These investigations also show that the transition temperature is affected by the cooling rate as well as the strain rate. The calculated stress profile is connected to the thermomechanical properties of the material, which is schematically shown in Fig. 2. If the stress in the solidifying material exceeds the measured maximum tensile strength from the tensile tests and this occurs at a temperature above the ductile to brittle transition temperature, a crack will form. This procedure is illustrated as path A in Fig. 2. On the other hand, if the stress reaches the maximum strength of the material below the transition temperature the material is ductile and will therefore deform plastically, see path B in Fig. 2. Combining the results from the calculation of the temperatures and stresses in the solidifying shell with the crack criteria gives the positions in the solidifying strand where the stress and temperature are sufficient for crack formation.  A. Lagerstedt, H Fredriksson / Materials Science and Engineering A 413-414 2005) 37 43 39 Ul Ul en Stress ? '\ I I \ , ......................................... 7 ... l // I ~~ ~ .. Temperature t x B I S L T DB Temperature Fig. 2. Schematic illustration of how cracking criteria is combined with stress profile to determine crack position. 3. Calculations 3.1. Data used in calculations The theoretical model presented above were used in order to investigate how material dependent data, such as coherence temperature and brittle-to-ductile transition temperature, as well as process dependent parameters such as the heat transfer coefficient influence the cracking behavior of a continuously cast slab. The model is evaluated for the two binary alloys Fe-2 Ni and Fe-lO Ni. The material data used is summarized in Table 1 n cases where actual material data is not available, values for delta ferrite has been used for Fe-2 Ni, and values for austenite has been used for the Fe-lO Ni. Studies of how the change of a specific parameter influences the results are made starting from the base data presented in Table 1 Unless otherwise stated by reference indication, the material properties have been estimated with aid of the commercial software IDS Solidification Analysis Package. Table 1 Material properties used in the calculations Property Symbol Liquidus temperature DC) h Solidus temperature DC) Ts Coherence temperature DC) Tcoh Brittle-ductile transition temperature DC) TD Heat conductivity of solid (W/(m K)) ks Heat conductivity of liquid (W/(m K» kL Heat capacity of solid (W/kg K) c P s Heat capacity of liquid (W kg K) C L P Density of solid (kg/m3) Ps Density of liquid (kg/m3) P Equilibrium heat of fusion (kJ/kg) t.H Elongation coefficient (I/K) a Elastic modulus (GPa) E Thermomechanical properties of Fe with 2 and 10 Ni have been evaluated in high temperature tensile testing on in situ solidified samples by Hansson and Fredriksson [8]. These values are adopted in this investigation and are included in Table 1. It is of interest to note that the transition temperature for Fe-l O Ni is found approximately 200 K below the solidus temperature while the ductile to brittle transition in Fe-2 Ni is found to be less than 50 K below the solidus temperature. In the same investigation the coherence temperature was found to be 151O°C for Fe-2 Ni and between 1465 and 1490 °C for Fe-l O Ni [8], which is somewhat below the solidus temperature for both alloys. In the present study, the temperature under which the material can withstand strain will be set to the solidus temperature. However, in order to see its influence on the stresses, a temperature corresponding to a fraction solidified of 0.6, according to Eq. (3), and the measured coherence temperature will also be used. During a solidification process, the heat transfer coefficient normally varies as a function of time. This is pronounced in a continuous casting process as the strand moves from one cooling zone to another. In order to investigate the impact of a change of heat transfer on the cracking, calculations have been performed for cases with a constant heat transfer coefficient of 800 W/(m 2 K) throughout the solidification process as well as cases where the heat transfer coefficient is changed. In cases with a varied heat transfer coefficient, the change is set to occur after 120 s and different decreases of the heat transfer coefficient has primarily been tested. The aim is to illustrate the changes occurring during a continuous casting operation as the heat removal drastically decreases as the strand leaves the mold. The shell thickness has been set to 0.1 m, the casting temperatures was 10K above the liquidus temperature and ambient temperature 35°C. 3.2. Results of calculations The cooling curves for the two alloys Fe-2 Ni and Fe-lO Ni are shown in Figs. 3 and 4, respectively. The figures show the results in three positions; the surface, the centreline and a point midway between surface and centre. In the first 120 s, Fe-2 Ni 1528 1524 1524 1500 35.2 35 738 813 7287 7057 242 22 x 10- 6 70 Fe-lO Ni 1503 1497 1497 1300 36.7 35 698 798 7368 7139 260 24 X 10- 6 70 Remark Ref. [i7] Ref. [i7] Assumed Ref. [8] Est. from Ref. [18]  40 A Lagerstedt, H Fredriksson / Materials Science and Engineering A 413-414 2005) 37 43 1600r------.------.------ -------r------ 1400 E 1200 ~ ~ ~ 1000 Q 0-   800 I 600 ----Solidification interval HTC = 200 ¥' _HTC=600 HTC = 700 .::::::. HTC = 800 HTC = 900 400~=====c======c=~ __ ______ __ o 200 400 600 800 1000 Time[s] Fig. 3. Cooling curves for Fe-2%Ni at surface, centre and a point between the two. Calculations with different changes of heat transfer coefficients are shown. 1600r------.------.-----~----~.---__ 1400 ~ ~ ~ 1000 ~ E ~ 800 600 ~~~ HTC = 600 ~---HTC = 700 HTC=800 ~HTC=900 ._-_. Solidification interval 400~==============~ ~ ~ o 200 400 600 800 1000 Time[s] Fig. 4. Cooling curves for Fe-JO Ni at surface, centre and a point midway between the two. Calculations with different changes of heat transfer coefficients are shown. all calculations have a heat transfer coefficient of 800 W/(m 2 K and it is thereafter changed resulting in temperatures as shown. Figs. 5 and 6 show the temperature profiles after 150 and 500 s for the two alloys. The stress resulting from three different coherence temperatures and a constant heat transfer coefficient are shown in Figs. 7 and 8 for the two alloys. In the figures, stress profiles 1500 E 1400 Q a 1300 Q 0- E 1200 Q I No HTC change HTC change to 700 HTC change to 600 HTC change to 400 HTC change to 200 -Solidification interval 0.1 Fig. 5. Temperature profiles for Fe-2%Ni 30 and 380 s after change of heat transfer coefficient. 1500 1400 IT ~ 1300 a 1200 0- E ~ 1100 HTC change to 900 No HTC change HTC change to 700 HTC change to 600 HTC change to 400 HTC change to 200 -Solidification interval 0.1 Fig. 6. Temperature profiles for Fe-JO Ni 30 and 380 s after change of heat transfer coefficient. 50r---~--- --- ---- ----, ~ p ~~- I.() en 0 cn LO U 0 /) ~g u ;g ~o 1 9 g 1 0 II ; II ; II II II II -50 ::-...JI..--::--=-=--'-I--=-' '-:::'-'I-~-=---=-=-=:__......:....I-...:-7':~hl-.....::-:........_=_ o 0.02 0.04 0.06 0.08 0.1 Distance from cooling surface [m] Fig. 7. Calculated stress distribution in Fe-2%Ni with different coherence temperatures. for calculations with constant heat transfer coefficient for the times 150 and 500 s are shown. Figs. 9 and 10 show the stress distribution resulting from a change of heat transfer coefficient for the Fe-2 Ni. The calculated stress profiles are shown for the times 150 and 500 s. The ultimate tensile strength for the temperature distribution 50 ------.------.------ -------.------ ~ ~ lOcn Den Orn OU ~9 ~o 0 ~o .... LO ~ g g II ~~ II II II II -50~~-_=~1-~-=--: - 1-~~--J~1-~~--~1-~---~ o 0.02 0.04 0.06 0.08 0.1 Distance from cooling surface [m] Fig. 8. Calculated stress distribution in Fe- \o%Ni with different coherence temperatures.  A. Lagerstedt, H Fredriksson / Materials Science and Engineering A 413-414 2005) 37 43 41 100 Rupture limit HTC change to 900 TC change to 200 0.01 0.02 0.03 0.04 Distance from cooling surface m] Fig. 9. Calculated stress distribution in Fe-2 Ni after 150 s with different heat transfer coefficients. 100 HTC change to 900 o change HTC change to 700 HTC change to 600 HTC change to 400 Rupture limit 0.01 0.02 0.03 0.04 Distance from cooling surface m] Fig. 10. Calculated stress distribution in Fe-2 Ni after 500 s. corresponding to the case with a constant heat transfer coefficient of800W m 2 K) is also shown. Stress profiles for the Fe-lO%Ni alloy are shown in the same manner in Figs. 11 and 12. During each time step, the positions where the calculated stress exceeds the ultimate tensile strength and the temperature at the same time exceeds the ductile to brittle transition temperature ro 150 100 a 50 ~ o HTC change to 600 Rupture limit 0.02 0.04 0.06 0.08 0.1 Distance from cooling surface [m] Fig. 11. Calculated stress distribution in Fe-IO%Ni after 150s with different heat transfer coefficients. HTC change to 200 ';;;' 100 HTC change to 400 ' HTC change to 600 ~ 0 1-   --::om-~~t~~~:~~~~:~ ~~R:u~p~ture limit <Jl ~ n ~ -100 co 'S C, r 0 200 300~ ~ ~ ~ ~~ ~ o o ~ QW QOO QOO 0.1 Distance from cooling surface m] Fig. 12. Calculated stress distribution in Fe-IO%Ni after 500 s. o~~---.-------.------ ------- ------. 100 200 ~ Liquidus front OJ 300 E i= 400 500 600~ __ __ 7 __ ____ __ ~ o 0.02 0.04 0.06 0.08 Distance from cooling surface m] Fig. 13. Predicted cracking (shadowed area) in Fe-2 Ni with constant heat transfer coefficient of 800 W/(m 2 K). Coherence and ductile to brittle transition temperatures are 1524 and 1500°C. are recorded. The result for the entire solidification process is plotted for some performed calculations in Figs. 13-16. The grey surface in these figures shows the time and position in the shell where the conditions for cracking are fulfilled. o 200 -~-' . '''' '' ~~. ~ 400 OJ E i= 600 800 o 0.02 ~ .~.(iqUidUS fronl . . ~   .. - -   Solidus front \. -~\. 0.04 0.06 0,08 0.1 Distance from cooling surfce [m] Fig, 14. Predicted cracking (shadowed area) in Fe-2 Ni with heat transfer change from 800 to 200 W/(m 2 K after 120 s, Coherence and ductile to brittle transition temperatures are 1524 and 1500 DC.
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