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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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