Finite Element Analysis of Residual Stress and Distortion in an by mifei

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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


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