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TRANSACTIONS OF MATERIALS AND HEAT TREATMENT Vol.25 No.5 PROCEEDINGS OF THE IFHTSE CONGRESS October 2004 Finite Element Analysis of Residual Stress and Distortion in an Eccentric Ring Induced by Quenching YAO Xin, ZHU Li-hua, LI M. Victor Portland State University, PO Box 751-ME, Portland, OR 97207-075J, USA Abstract: The residual stresses and distortion induced by quenching in an eccentric ring were investigated in this study with finite element method. The ring was made of AISI 52100 steel. A fully coupled 3D temperature-displacement analysis was performed to simulate heat transfer, phase transformations, and mechanical stresses and strains during the heating and subsequent quenching processes. Commercial FEA package ABAQUS/Standard 6.4 was used for the analyses along with user subroutines developed by the authors to model the thermal and mechanical constitutive behavior. The simulation results show that transformation plasticity plays an important role on the residual stress distribution. Key words: Residual Stress, Quenching, Transformation Plasticity, Modeling THERE is a pressing need in the metal manufacturing internal stress evolution of a cylinder, and considered industries to minimize the amount of rework due to the effect of stress on transformation kinetics as well as such unanticipated effects of quenching as distortion transformation plasticity. Inoue [2J developed a CAE and cracking. Computational modeling offers a system "HEARTS" for heat treatment simulation based powerful tool to predict the component response to on Metallo-Thermo-Mechanics theory. Giir[31 applied quenching. However, physics associated with the a thermo-elastic-plastic approach to investigate the quenching process and materials response is very evolution of internal stresses and non-homogeneous complex. There is not yet a complete set of models plastic deformation in the quenching process of an that can fully incorporate all the relating factors axisymmetric component. together and readily duplicate the experimental In the previous work, we developed integrated results."1 modeling procedure for the analysis of heat transfer, Quenching distortion results from the complex microstructure evolution, and mechanical stress and interactions in the materials between temperature field, distortion in quenching process and thermal material metallurgical changes and stress/strain field. Though processing in general, but no transformation the relating theories of heat transfer, physical plasticity was included. This research aims to metallurgy, and solid mechanics have been well incorporate the transformation plasticity effect. established, quenching distortion modeling remains a challenging field because of the coupling effects. In 1. Model Description the quenching process, thermal stresses are induced by 1.1 Finite Element Model and Analysis Procedure the non-uniform temperature distribution depending on An eccentric ring made of AISI 52100 steel was the thermal properties of the components and surface chosen as the test component for the numerical analysis. convection. Phase transformations take place to The ring has an outer diameter of 48 mm and an inner minimize the free energy in the material. They diameter of 30 mm. The eccentric distance was 5.4 usually introduce volumetric changes, transformation mm. The height of the ring was 18 mm. The ring plasticity and changes in mechanical properties. was first heated up to the uniform austenizing Consequently, phase transformations bring in changes temperature of 860 °C, and then quenched in oil to in the stress field. It is also well known that stress and room temperature of 12 °C. The heating and strain have substantial influence on the kinetics of quenching process are subjected to numerical phase transformations. Local plastic flow may also simulation in this study. The shape and finite element occur when the effective stress exceeds the yield model are shown in Figure 1. strength. All these factors interact with each other and A fully coupled temperature-displacement analysis cause a continuously changing internal stress/strain based on the commercial finite element software field. ABAQUS is adopted to simulate the heating and In recent two decades, the modeling of residual quenching cycle. With the degrees of freedom of both stress and distortion in the quenching of steel the displacement and temperature, 3D eight-node components has received considerable attentions. tri-linear elements of the type C3D8T are used to Most of the studies used finite element method to discretize the part. Only one quarter of the ring is compute temperature and stresses during quenching. considered in this study to take the advantage of Denis m used a coupled model to investigate the geometric symmetry. The whole model includes 6912 Vol.25 No.5 TRANSACTIONS OF MATERIALS AND HEAT TREATMENT 747 elements and 8415 nodes. The mesh density is biased user subroutine FILM. Of particular importance for towards the surfaces to improve calculation precision. the accuracy of thermal solutions is the heat transfer coefficient of oil quenching. The heat transfer coefficients of oil quenching are highly temperature dependent. The values used in this research were based on the previous work of the first author. Only martensitic transformation is included in the present model and is represented by XM = l - e x p [ - 0.011 (Ms -7)] (1) where XM is the volume fraction of martensite and T is Figure 1. The geometry and the finite element mesh of the temperature. Ms is the martensitic transformation the eccentric ring temperature and for AISI 52100 steel, Ms=331 °C. 1.3 Thermal Stress Analysis The integrated analysis procedure was developed by The total strain rates can be expressed as the utilizing ABAQUS as the platform, which provides the summation of the following strain rates pre- and post- processors and the solvers for the finite element analyses. Additional functionalities through (2) user subroutines offer the users powerful means to where are the elastic, plastic, accurately simulate the non-linear boundary conditions, thermal, transformation, and transformation plasticity loads, and constitutive behavior of materials. strain rates, respectively. It remains the responsibility and challenge for the analysts to properly incorporate the essential physical C f - phenomena involved in the quenching process. The 3000 quenching process involves three primary physical phenomena: thermal interactions, microstructural 2500 evolution, and mechanical responses. ABAQUS 2000 /Standard is an implicit code, which utilizes a backward (too finite-difference integration scheme between time iriui increments. At each time increment, the nonlinear 500 equation system is solved using Newton's method. 0 The thermal constitutive behavior of the materials is 400 603 defined using a user material subroutine, UMATHT. Temperature (°C) Microstructural constituents are defined as state Figure 2. Heat transfer coefficient of oil quenching variables. They are computed and updated inside UMATHT subroutine. The primary phase User subroutine UEXPAN is coded to compute the transformation during quenching is the martensitic thermal strain rate £th and transformation strain transformation, which is a diffusionless athermal rate£ tr : reaction. In this research, martensitic transformation z'h=(XAaA + XMaM}T-\ (3) is considered using the Koistinen-Marburger model.I61 The mechanical constitutive behavior of the material is defined using a user subroutine UMAT. Classic where , XM are the volume fractions of austenite and theories of isotropic elasticity, Mises plasticity, and martensite, respectively, aA, a» are the thermal isotropic hardening are implemented in the subroutine. expansion coefficients of the corresponding phases, In addition to the thermal strain, elastic and plastic PA-M is the dilatation due to martensitic transformation, strains, transformation strain and transformation 1 is the second order identity tensor. plasticity associated with the martensitic transformation A fully coupled displacement-temperature analysis is are also incorporated into the model. performed to simulate the thermal, metallurgical and Therrno-physical and thermo-mechanical properties are mechanical processes involved in the heating and dependent on both temperature and microstructure and quenching process. At each time increment, nodal were calculated using JMatPro. displacements and temperature are computed. The 1.2 Thermal and Metallurgical Analyses volume fractions of phases and the thermal strain are The heating and cooling cycle is a transient heat calculated using the above-mentioned model. If the conduction problem with convective boundary updated state of stress is within the yield surface, the conditions. A uniform initial temperature of 12 °C material is assumed to have either remained elastic or was assumed to the ring. The convective heat transfer to have unloaded elastically. If Von-Mises yielding boundary conditions are applied on the surfaces using a condition is satisfied, the plastic flow rule associated 748 PROCEEDINGS OF THE IFHTSE CONGRESS October 2004 with the isotropic hardening rule is employed to the transformation plasticity strain rate can be given by determine the incremental plastic strain. When phase transformation occurs, the volume fraction of martensite and its increment are jointed together to where A€A-M is volume variation of austenite and determine whether it is necessary to incorporate the martensite, and a/ the yield stress of austenite. transformation plasticity and if necessary, the following Comparing the predictions by Eq. (6) with some test model is loaded to determine the transformation plastic results, Leblond et al.[16) concluded that at the strain. beginning of transformation, Eq. (6) overestimates the 1.4 Transformation Plasticity Model transformation plasticity. To account for this Transformation induced plasticity (TRIP) is defined discrepancy and also to avoid the singularity at XM= 0, as an anomalous plastic strain observed when they proposed to disregard the transformation plasticity metallurgical transformation occurs under external when < 0.03 for small applied stress. This stresses that are much lower than the yield limit. formulation is then extended to a higher stress level in Recent studies have shown that transformation the following expression: plasticity plays an essential role in the residual stress distribution and distortion in welding and heat treatment processes. Two mechanisms are often credited for the explanation of TRIP phenomena.'91 One is the Greenwood - Johnson mechanism,1101 which corresponds to the accommodation process due to specific volume variation during transformation; and the other is Magee mechanism, which corresponds to the orientation process due to the formation of preferred variants for minimizing the total energy. And also, there are two dominate methods that can be used to introduce transformation plasticity into Oy is the yield stress of the mixed phases and G*q is modeling:112' one includes the use of an unusually low the equivalent stress. yield stress during transformation,1131 and the other uses When classical plasticity and transformation plasticity are considered, the constitutive equation can an additional strain to represent the enhanced dilation or contraction of the material as it transforms in the be described in rate form as presence of an applied stress.1141 (9) In this study, the second method is used to (10) incorporate transformation plasticity into the where I is the fourth order identity tensor, K and G are calculation model because there are several drawbacks the bulk and shear modules, respectively. of the first method. Please refer to Denis* paper "5) for detail. In the case of martensitic transformation, most At each time step, the stress tensor on and strain existing models for TRIP based on the tensor 6n are assumed known at the beginning of the Greenwood-Johnson mechanism is finally time increment. The material constitutive relation is corresponding for a low applied stress and the applied to update an+i, Ejj'+1, E* , and corresponding transformation plastic strain rate can be generalized state variables at the end of the time increment. Purely into [16] elastic material behavior is assumed in the beginning, the trial deviatoric stress tensor can be calculated by S'™'=S n +2GA£ n+1 (]]) where Ktp is the transformation plasticity constant, S the deviatoric stress tensor, and f(XM) a function of where AEn+i is the strain variation during the current martensite volume fraction XM, normalized so that time increment, the deviatoric stress tensor is given by /T0)=0, f(l)=l. S.=0.-p-l 02) In recent years, a number of experimental, theoretical and numerical studies have been performed p is the hydrostatic stress. The equivalent trial stress is on the TRIP phenomenon/17'181 In this research, we defined as used the model developed by Leblond,1191 which is one a*** = ^(3/2)8'™' -S'™' (13) of the most widely used in practical applications. This If the trial elastic stress is larger than the current model is obtained from a micromechanical analysis yield stress (o"ia/>cr>,), plastic deformation is involved. based on the determination of the plastic strain induced According to the isotropic hardening law, the plastic in a spherical parent phase by the growth of a spherical flow rule follows product phase core. According to Leblond's derivation, Vol.25 No.5 TRANSACTIONS OF MATERIALS AND HEAT TREATMENT 749 2.1 Thermal and Microstructural Simulation Results (14) The cooling curves of points A, B, C and D (refer to where The equivalent stress is defined as Figure 1) are given in Figure 3. Among the heating and cooling rates, the highest one locates at point D on ; the equivalent plastic strain rate is the outer surface and the lowest at center point B. Point given by £ p = A on the outer surface has a higher cooling rate than the point C on the inner surface. This is because the outer If martensitic transformation occurs and its volume surface has more space to transfer heat. After oil fraction is greater than 0.03, according to Eq. (5), quenching for about 100 seconds, the temperature transformation plastic flow occurs even the equivalent distribution in the ring evolves to be uniform and the stress is lower than the yield stress. As given in Eq. (5), magnitude is close to the quenchant temperature. The transformation plastic strain flows along the direction volume fractions of martensite versus time of the same of the current derivatoric stress and can be simplified as points are also shown in Figure 3. In accordance with etp=Ctp-S = C'p-(r-h (15) the temperature variation, martensite forms firstly at the where C? a coefficient derived from Eq. (5). outer surfaces and then spreads inwards. After The constitutive formulation can then be written as quenching to room temperature, over 95% austenite has (16) transformed into martensite and less than 5% austenite is retained. (17) 2.2 Stress Analysis Results without Considering Transformation Plasticity The non-uniformities of martensite formation and where (18) temperature distribution in the heating and quenching process introduce thermal and residual stresses. If (19) martensitic transformation is not considered in the model, the problem becomes a typical displacement-temperature coupling problem. Using ABAQUS/Standard analysis, we can get the normal stress variations of surface point C and center point B with time, as given in Figure 4. Figure 4. Predicted stress development without considering phase transformation In the initial stage of oil quenching, the normal stress, , was tensile at the surfaces and compressive in the core. Because of the sudden contraction of the surfaces and the great temperature gradient in the vicinity of the surface, the normal stress at surface point C increases sharply to about 200 MPa, and then Figure 3. Thermal history and microstructure decreases slowly with the decreasing temperature evolution of point A, B, C and D gradient. At the same time, the compressive normal stress at the center point B gradually increases, after 2. Results and Discussion reaching the nadir, also decreases because of the 750 PROCEEDINGS OF THE IFHTSE CONGRESS October 2004 decreasing temperature gradient. During this process, are greatly reduced. the ring undergoes plastic deformation. The plastic deformation finally results that the surface tensile residual stress shifts to compressive and the core compressive stress switches into tensile stress at the later stage of quenching. The final residual stress distribution without considering phase transformation is shown in Figure 5. 40 60 80 Time (s) Figure 7. Equivalent transformation plastic strain Figure 5. Predicted residual stress distributions without considering phase transformation Figure 8. Predicted stress development considering transformation plasticity At temperatures above Ms point, the relationships between stress and strain were identical to those shown Time (*) in Figure 4 and Figure 6. Once transformation began in the presence of internal stresses, transformation Figure 6. Predicted stress development considering plasticity strain was introduced as an additional plastic phase transformation strain. Figure 7 shows the predicted equivalent 2.3 Stress Analysis Results Considering transformation plasticity strain versus time at points B, Transformation Plasticity C, D and E. Among them, point C at the inner surface The predicted stress development with the has the highest TRIP strain (0.52%) while the value at consideration of phase transformation is presented in point B in the center is the lowest (0.15%). As can be Figure 6. The phase transformation greatly influences seen in Figure 8 and Figure 9, the residual stress is the stress and strain histories and distributions. The characterized with compressive stress in the center variations of normal stresses are the same as those (a^ -44 MPa) and tensile stress at the surfaces (a^ without considering phase transformation before 93 MPa). Transformation plasticity has such martensitic transformation starts. When martensitic significant influence on the final stress distribution that transformation begins at the surface at about 24 s, the the signs of residual stress are opposite to Figure 4 and transformation induced dilation makes the compressive Figure 6, which have not taken account of stress to a higher level (-276 MPa). But soon, transformation plasticity. martensitic transformation also takes place in the core After quenching, the inner and outer diameters, as and the transformation dilation nearly counteracts the well as the height of the ring all increased, as shown in tensile stress there. As transformation processes, the Figure 10. The inner and outer surfaces of the compressive stress at the surface is also relaxed eccentric rings are also distorted to become ellipses significantly due to the dilation in the core. The final after quenching, with the long axis of the ellipses residual stress is also compressive at outer and inner parallel to X-axis and short axis parallel to Y-axis. surfaces and tensile in the core, but their magnitudes Vol.25 No.5 TRANSACTIONS OF MATERIALS AND HEAT TREATMENT 751 non-homogeneous plastic deformation in quenching process. Materials Science and Engineering, 2001, A319-312: 164-169 4. Li M V and Totten G E. Computational modeling of quenching and distortion. Proceedings of the fourth international conference on quenching and the control of distortion, 20-23 May, 2003, Beijing. 81-86 5. Li M V, Niebuhr D V, Meekisho L L and Atteridge D G Computational model for the prediction of steel hardenability. Metallurgical and Materials Transactions B, 1998,298:661-672 6. Koistinen D P and Marburger R E. A General equation prescribing the extent of the austenite-martensite Figure 9. Predicted residual stress considering transformation in pure iron-carbon alloys and plain carbon transformation plasticity steels. Acta Metall., 1959, 7: 59 7. Yao X. The application of computer simulation on the quenching of large-sized mould blocks and bearing steel. [Dissertation] Shanghai Jiao Tong University, 2003, March 8. Taleb L and Sidoroff F. A micromechanical modeling of the greenwood-Johnson mechanism in transformation induced plasticity. International Journal of Plasticity, 2003, 19: 1821-1842 9. Fischer F D, Reisner G Werner E, Tanaka K, et al. A new view on transformation induced plasticity. International Journal of Plasticity, 2000, 16: 123-748. Figure 10. Predicted distortion after quenching 10. Greenwood G W and Johnson R H. The deformation of (scale factor = 10) metals under small stresses during phase transformations. Proc. Roy. Soc., 1965, A283: 403-421 11. Magee C L. Transformation kinetics, micro-plasticity and 3. Conclusion aging of martensite in Fe-31Ni. [Desesertation] Carnegie This article presents an example of multidisciplinary Inst. Tech., Pittsburgh, PA, 1966 12. Abbasi A and Fletcher A J. Effect of transformation analysis using ABAQUS/Standard and user material plasticity on generation of thermal stress and strain in subroutines UMATHT and UMAT. By incorporating quenched steel plates. Materials Science and Technology, Leblond's transformation plasticity model into UMAT, 1985,1:830-837 the residual stress and distortion of an eccentric ring 13. Rammerstorfer F G Mitter W and Bartosch H. Numerical during oil quenching were computed. The predicted methods in thermal problems. Swansea, Pineridge Press Ltd., 1983 stresses are then compared with those without 14. Leblond J B, Mottet G and Devaux J C. A theoretical and considering phase transformation and transformation numerical approach to the plastic behavior of steels during plasticity. The comparisons show that transformation phase transformations - I: Derivation of General Relations. plasticity has significant influence on the residual stress J. Mech. Phys. Solids, 1986a, 34: 395 evolution and distribution. The residual normal 15. Denis S, Gautier E, Simon A and Beck G Stress- phase- transformation interactions-basic principles, modeling, and stresses are compressive at the surfaces and tensile in calculation of internal stresses. Material Science and the center without taken account of transformation Technology, 1985, 1:805-814 plasticity, whereas the stress distribution is just 16. Leblond J B, Mottet G, Devaux J and Devaux J C. opposite when considering transformation plasticity. Mathematical models of anisothermal phase Predicted results show that the ring expands and the transformations in steels, and predicted plastic behavior. profiles of inner and outer surface no longer keep circle Materials Science and Technology, 1985, 1: 815-822 17. Fischer F D, Antretter T, Azzouz F, et al. A new view on after quenching. transformation induced plasticity (TRIP). International Journal of Plasticity, 2000a, 16: 723-748 References 18. Han H N and Suh D W. A model for transformation plasticity during bainite transformation of steel under 1. Denis S, Sjostfim S and Simon A. Coupled Temperature, external stress. Acta Materialia, 2003, 51: 4907-4917 stress, phase transformation calculation model numerical 19. Leblond J B, Devaux J and Devaux J C. Mathematical illustration of the internal stresses evolution during cooling modeling of transformation plasticity in steels I: case of of a eutectoid carbon steel cylinder. Metallurgical ideal-plastic phases. International Journal of Plasticity, Transactions A, 1986, ISA: 1203-1212 1989,5:551-572 2. Inoue T and Arimoto K. Development and implementation of CAE system "HEARTS" for heat treatment simulation based on Metallo-Thermo-Mechnics. Journal of Materials Corresponding author: Dr. Yao Xin Engineering and Performance, 1997, 6: 51-60 Email: xyao@pdx.edu 3. Giir C H and Tekkaya A E. Numerical investigation of Mail: PO Box 751-ME, Portland OR 97207-0751,US A