[Elsevier] Comparison Of Plastic, TYMCZASOWY
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Comput. Methods Appl. Mech. Engrg. 192 (2003) 5189–5208
www.elsevier.com/locate/cma
Comparison of plastic, viscoplastic, and creep models
when modelling welding and stress relief heat treatment
H. Alberg
a,
*
, D. Berglund
a,b
a
Division of Computer Aided Design, Lulea University of Technology, SE-97187 Lulea, Sweden
b
Volvo Aero Corporation, Advanced Manufacturing Technology, SE-46181 Trollhattan, Sweden
Received 24 February 2003; accepted 22 July 2003
Abstract
A major concern when carrying out welding and heat treatment simulations is to accurately model material be-
haviour as it varies with temperature and composition. Early in the product development process, a less sophisticated
material model may be suitable to compare different concepts where less accuracy in deformation and residual stress is
acceptable. At later stages in the product development process, more sophisticated models may be used to obtain more
accurate predictions of deformations and residual stresses. This paper presents a comparison of five different material
models applied to the simulation of a combined welding and heat treatment process for a fabricated martensitic
stainless steel component.
2003 Elsevier B.V. All rights reserved.
Keywords: Welding; Heat treatment; Viscoplasticity; Plasticity; TRIP
1. Introduction
Welding and heat treatment are widely used in the aerospace industry. However, these manufacturing
processes can generate unwanted stresses and deformations, a fact that has to be taken into consideration
when designing or changing the sequence of manufacturing operations for a given component. Previous
experience can offer some help, but costly and time consuming experiments are often required to evaluate
component design, material selection, and manufacturing schedules. One way to decrease cost and reduce
product development time is to use numerical simulation which can reliably predict the final properties and
shape of a component based upon the manufacturing processes used. This kind of simulation can also help
predict the effect of the chosen manufacturing sequence on the final component performance.
Simulation techniques for manufacturing processes are less well developed than the simulations used
when designing components. This is because simulation of manufacturing processes are more di
5
cult to
perform and require more expertise from the user and more powerful hardware and software. Another
*
Corresponding author.
0045-7825/$ - see front matter 2003 Elsevier B.V. All rights reserved.
doi:10.1016/j.cma.2003.07.010
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H. Alberg, D. Berglund / Comput. Methods Appl. Mech. Engrg. 192 (2003) 5189–5208
significant di5culty when carrying out manufacturing simulations is modelling of boundary conditions and
material behaviour, both of which are crucial for a successful simulation.
This paper is concerned with material modelling. There are many constitutive models but these require
expensive and time consuming material tests to acquire material data for the chosen constitutive model.
Even given this data, it can be di5cult to determine the material parameters for the chosen constitutive
model. It istherefore important chose a constitutive models that capture the behaviour of the material to the
required level of accuracy, but at minimum cost. For example, during the early stages of the product de-
velopment process, a less sophisticated model may be suitable to study different concepts, but with less
accuracy as far as predicting, say, deformation and residual stresses. At a later stage of development, a more
sophisticated model may be used to obtain more accurate results.
This paper presents a comparison of five material models for use when simulating a combined welding
and heat treatment process for a martensitic stainless steel component. A key dimension was used to
compare the results from the different material models. The material models that were compared are (1)
rate-independent plasticity model including simplified model for phase changes described in Berglund et al.
[9], (2) rate-independent plasticity including microstructure calculation, (3) rate-independent plasticity in-
cluding microstructure calculation and transformation induced plasticity (TRIP), (4) rate-dependent
plasticity (viscoplasticity) including microstructure calculation and (5) rate-dependent plasticity including
microstructure calculation and TRIP. Two numerical methods to model creep during the heat treatment
process were also compared. The heat treatment simulation included the use of computational fluid dy-
namics to obtain an approximation of the heat transfer coe5cient of the components surface during the
cooling part of the heat treatment cycle, Berglund et al. [9].
2. Background
The finite element method (FEM) has been used since the early 1970s to predict stresses and defor-
mations resulting from welding processes, Hibbit and Marcal [14] and Ueda et al. [26,37]. Welding simu-
lation has been further investigated by many researchers, Goldak et al. [13], Oddy et al. [20], and Yang et al.
[38] and several review articles on welding have been written, e.g. Lindgren [17–19].
Heat treatment simulations to predict the residual stresses and microstructure in a solid rail wheel have
been carried out by Donzella et al. [31] amongst others. Whilst Thuvander [36] computed distortion due to
quenching. However, simulation of the combined effect of welding and heat treatment are not common in
the literature. Josefson [15] calculated the residual stresses after post weld heat treatment of a thin wall pipe
and Wang et al. [28] simulated local post heat treatment of a pipe with different heated bands and used a
power creep law when simulating the heat treatment process.
2.1. Material modelling
A major concern when simulating welding and heat treatment is being able to model the varying material
behaviour. This is further complicated by temperature and rate effects and also the evolution of micro-
structure during the various processes, which will change the material properties in response to the thermo-
mechanical history of the material and can cause transformation plasticity. A fully coupled thermal,
metallurgical and mechanical coupling has been applied to a single pass weld by Inoue and Wang [32].
Borjesson and Lindgren [12] used phase-dependent material properties when simulation multipass welding.
Rammerstorfer et al. [24] performed a thermo-elastic–plastic analysis including phase changes, transfor-
mation plasticity and creep. However, creep and transformation plasticity were not included in the same
simulations. Ronda and Oliver [25] used different viscoplastic constitutive equations in welding simulations.
H. Alberg, D. Berglund / Comput. Methods Appl. Mech. Engrg. 192 (2003) 5189–5208
5191
This work included deriving a consistent tangent matrix for each model but did not draw any conclusions
as to which model was the most suitable for welding simulation.
A large number of physical processes involving varying strains, strain rates, and temperatures can lead to
inelastic deformations. The range of processes potentially involved is one of the reasons for the large
number of material models that have been developed; even when limiting the scope of the models to metals,
Stouffer and Dame [8]. Different phenomena and models used to model plasticity are discussed by Lemaitre
and Chaboche [5] and Miller [6]. These models ranging from the deviatoric, ideal plasticity model using the
von Mises yield condition and associated flow rule to complex sets of equations such as the MATMOD-
model developed by Miller [6].
Rate independent plasticity, viscoplasticity and creep have been studied, modelled and used for many
years. Many models have been proposed, some using a threshold, usually called a yield limit, to separate the
elastic and the inelastic behaviour whilst others use no such threshold. Zienkiewicz and Cormeau [29] used
a viscoplastic constitutive model similar to the one used in this paper which set the yield limit to zero in
order to model creep behaviour. They also developed a strategy for determining the time steps to be used
based on stability considerations but did not include viscoplasticity and creep in the same simulation.
Bodner and Partom [10,11] did not use a threshold but used the same constitutive model for viscoplasticity
and stress relaxation (creep).
2.2. Heat input modelling
Another important issue when modelling both welding and heat treatment is how well the boundary
conditions in the model correlate with the actual process. The heat input from the welding process was
simulated in this study as a moving heat source where the energy input is distributed as a double ellipsoid; a
method first proposed by Goldak et al. [13]. The thermal load in the heat treatment simulation is modelled
as heat transfer from the surroundings over the boundary of the component. The amount of energy
transferred depends on two major parameters; the temperature difference between the component surface
and its surroundings and the heat transfer coe5cient of the surface. Lind et al. [34] used computational fluid
dynamics (CFD) simulations to obtain an approximate distribution of the surface heat transfer coe5cient
when quenching a steel cylinder in a gas cooled furnace. Berglund et al. [9] also used CFD simulations to
approximate the distribution of the surface heat transfer coe5cient when an aerospace component was
cooled in a heat treatment furnace.
3. Welding and heat treatment of an aerospace component
The component and process studied in this work, as well as the simplifications made and the boundary
condition applied on the numerical model are presented below.
The simulation presented in this paper is a part of an aerospace component called the Turbine Exhaust
Case (TEC) which is a fabricated structure made of a martensitic stainless steel manufactured by Volvo
Aero, Sweden. The inner part of the TEC is called the Hub, Fig. 1,
1
and was the main focus of this work.
Simulation of the welding and heat treatment of the Hub was presented in Berglund et al. [9]. The work
in the present paper concerns additional simulations on the same component using different material
models and their affects on the key dimension.
The Hub is fabricated by welding two discs, the front and rear supports, between the bearing housing
and inner ring. Many different manufacturing steps are involved in manufacturing the Hub is made,
1
For interpretation of color in Fig. 1, the reader is referred to the web version of this article.
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H. Alberg, D. Berglund / Comput. Methods Appl. Mech. Engrg. 192 (2003) 5189–5208
Fig. 1. Part of the TEC V2500, (a) back view, (b) front view.
however, as in the paper by Berglund et al. [9], the work presented here concentrates on the welding and
heat treatment processes.
The component is welded using gas tungsten arc welding (GTAW) with additional filler material.
Fabrication begins by welding the rear support, welds A and B in Figs. 1a and 2b, followed by welds C and
D on the front support, Figs. 1b and 2b. Parts are first tack-welded together before the final welds are made.
Any changes in material behaviour, residual stresses and deformations due to the tack-welds have not been
taken into account. The component is heat treated after welding in order to reduce residual stresses.
A key dimension on the Hub is the axial distance a between the bearing housing and the flange on the
inner ring, Fig. 2a. Different simulations were used to investigate how the choice of material model affects
the key dimension, a, and the residual stresses.
The finite element programme MSC.MARC [40] was used for the thermo-mechanical analyses of the
welding and heat treatment. An axisymmetric model, with 1315 elements and 1862 nodes was used. The
methods and boundary condition used for the welding and heat treatment simulations are the same as those
used by Berglund et al. [9]. Special attention was paid to the thermal boundary condition during cooling in
the heat treatment process. Heat transfer during cooling is mainly due to convective effects. A fluid mech-
Fig. 2. (a) Cross section of the Hub showing the key dimension a, (b) axisymmetric FE-model.
H. Alberg, D. Berglund / Comput. Methods Appl. Mech. Engrg. 192 (2003) 5189–5208
5193
anics analysis to estimate this heat transfer was performed using the finite volume based software FLUENT
[39] before the thermo-mechanical analysis. Its results transferred to the model used in the FE-simulation,
this procedure is also described in Berglund et al. [9].
4. Material models and properties
The material used to manufacture the component investigated in this study is a martensitic stainless steel.
The material properties and models used are described below. The material model includes components for
calculation of the development of microstructure during the process, and transformation induced plasticity
strain (TRIP) combined with the use of elasto-plastic/viscoplastic/creep constitutive relations.
A fully coupled thermal, metallurgical and mechanical analysis of a single pass weld was made by Inoue
and Wang [32]. In the present work, a so-called staggered approach was used to couple the thermal and
mechanical fields as shown in Fig. 3. The thermal field is first calculated followed by the microstructural
evolution and finally the mechanical quantities is determined. The geometry in the thermal analysis is
updated one time step behind. Fig. 3 shows the involved variables.
A change in microstructure affects mechanical properties and also results in a volumetric transformation
strain e
tra
and a deviatoric strain e
tp
(TRIP). The latter exists only if a stress field is present during the
martensite phase change. A more detailed description of microstructural, mechanical and thermal prop-
erties of the material are described in subsequent sections. The effect of the thermal field and microstruc-
tural evolution on the mechanical solution are discussed later.
4.1. Microstructure evolution and its influence on material properties
The initial microstructure of the material consists of a mixture of ferrite and pearlite. The start tem-
perature for the ferrite/pearlite to austenite transformation is 700 C(A
e1
-temperature) and the highest
temperature for stable ferrite is 880 C(A
e3
-temperature). The austenite begins to transform to martensite at
250 C(M
s
) if the cooling rate is su5ciently high. It has been observed experimentally that if the cooling
rate exceeds 0.4 C/s when cooling from the austenite start temperature down to a temperature below 500
C, the material becomes fully martensitic. The minimum cooling rate in the heat affected zone of the weld
is greater then 10 C/s in this temperature range and therefore that all austenite is assumed to transform to
martensite during cooling. The calculation of austenite transformation for an arbitrary thermal history is
based on the theory presented by Kirkaldy and Venugopalan [33]. The rate of transformation is described
in Eq. (1), according to the expression proposed by Oddy et al. [20] for low carbon steel,
z
eu
ð
T
Þ
z
eu
ð
T
Þ
z
a
ð
t
Þ
n
1
n
z
eu
ð
T
Þ
z
a
s
ð
T
Þ
:
z
a
ð
t
Þ¼
n
ln
ð
1
Þ
Fig. 3. Coupling between thermal field, microstructural evolution, and mechanical field. T is the temperature, z
a
is the volume fraction
of austenite, z
m
is the volume fraction of martensite, and u
ij
is the displacement field.
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