Thermomechanical Coupling Model for a Stainless Steel-Clad Plate on Heat Treatment
Quick and correct prediction of the internal stress and deformation of stainless steel-clad plates on heat treatment is a problem of specific interest. A thermomechanical coupling model based on the engineering elastic-plastic theory is detailed. The discretization method and difference equations us...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
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irk-123456789-1883022023-02-23T01:27:17Z Thermomechanical Coupling Model for a Stainless Steel-Clad Plate on Heat Treatment Guan, B. Zang, Y. Yang, F. Yang, X.Y. Qin, Q. Научно-технический раздел Quick and correct prediction of the internal stress and deformation of stainless steel-clad plates on heat treatment is a problem of specific interest. A thermomechanical coupling model based on the engineering elastic-plastic theory is detailed. The discretization method and difference equations used simultaneously provide effective numerical calculations for the model. The generalized finite element model verifies relevant simplification conditions and efficiency of the numerical calculation logic. Heat treatment experiments of clad plates are also designed and presented, and the ability of the model to predict the stress and deformation behavior of the clad plate in the actual heat treatment process is evaluated. The calculation logic of the model is reasonable, and the prediction error of deformation and internal stress of the plate is ~15%. The model exhibits an extremely high computational efficiency and can meet the requirements of on-line analysis for heat treatment processes. Быстрое и точное прогнозирование внутреннего напряжения и кривизны деформации листов, плакированных нержавеющей сталью, при термообработке остается актуальной проблемой в этой области исследования. Рассматривается модель термомеханической связи листа, основанная на инженерной теории упругих и пластических деформаций. Для реализации эффективного численного расчета модели совместно используются метод дискретизации и разностные уравнения. Применение обобщенной конечноэлементной модели позволяет подтвердить соответствующие условия упрощения и рациональность логики численного расчета модели. Планируются и описываются эксперименты по термообработке таких плакированных листов и анализируется способность модели прогнозировать характер напряжения и деформации плакированного листа в реальном процессе термообработки. Логика расчета модели обоснована, при этом погрешность прогнозирования деформации и внутреннего давления листа составляет ~15%. Модель также имеет очень высокую вычислительную эффективность и может удовлетворять требованиям анализа в режиме он-лайн для процессов термообработки. Швидке і точне прогнозування внутрішніх напружень і кривизни деформації листів, плакованих нержавіючої сталлю, при термообробці залишається актуальною проблемою в цій області дослідження. Розглядається модель термомеханічного зв'язку листа, заснованої на інженерної теорії пружних і пластичних деформацій. Для реалізації ефективного чисельного розрахунку моделі спільно використовуються метод дискретизації і різницеві рівняння. Застосування узагальненої скінченноелементної моделі дозволяє підтвердити відповідні умови спрощення і раціональність логіки чисельного розрахунку моделі. Плануються і описуються експерименти по термообробці таких плакованих листів і аналізується здатність моделі прогнозувати характер напружень і деформації плакованного листа в реальному процесі термообробки. Логіка розрахунку моделі обгрунтована, при цьому похибка прогнозування деформації і внутрішнього тиску листа складає ~ 15%. Модель також має дуже високу обчислювальну ефективність і може задовольняти вимогам аналізу в режимі он-лайн для процесів термообробки. 2019 Article Thermomechanical Coupling Model for a Stainless Steel-Clad Plate on Heat Treatment / B. Guan, Y. Zang, F. Yang, X.Y. Yang, Q. Qin // Проблеми міцності. — 2019. — № 1. — С. 48-64. — Бібліогр.: 25 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/188302 539.4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| institution |
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English |
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Научно-технический раздел Научно-технический раздел |
| spellingShingle |
Научно-технический раздел Научно-технический раздел Guan, B. Zang, Y. Yang, F. Yang, X.Y. Qin, Q. Thermomechanical Coupling Model for a Stainless Steel-Clad Plate on Heat Treatment Проблемы прочности |
| description |
Quick and correct prediction of the internal stress and deformation of stainless steel-clad plates on heat treatment is a problem of specific interest. A thermomechanical coupling model based on the engineering elastic-plastic theory is detailed. The discretization method and difference equations used simultaneously provide effective numerical calculations for the model. The generalized finite element model verifies relevant simplification conditions and efficiency of the numerical calculation logic. Heat treatment experiments of clad plates are also designed and presented, and the ability of the model to predict the stress and deformation behavior of the clad plate in the actual heat treatment process is evaluated. The calculation logic of the model is reasonable, and the prediction error of deformation and internal stress of the plate is ~15%. The model exhibits an extremely high computational efficiency and can meet the requirements of on-line analysis for heat treatment processes. |
| format |
Article |
| author |
Guan, B. Zang, Y. Yang, F. Yang, X.Y. Qin, Q. |
| author_facet |
Guan, B. Zang, Y. Yang, F. Yang, X.Y. Qin, Q. |
| author_sort |
Guan, B. |
| title |
Thermomechanical Coupling Model for a Stainless Steel-Clad Plate on Heat Treatment |
| title_short |
Thermomechanical Coupling Model for a Stainless Steel-Clad Plate on Heat Treatment |
| title_full |
Thermomechanical Coupling Model for a Stainless Steel-Clad Plate on Heat Treatment |
| title_fullStr |
Thermomechanical Coupling Model for a Stainless Steel-Clad Plate on Heat Treatment |
| title_full_unstemmed |
Thermomechanical Coupling Model for a Stainless Steel-Clad Plate on Heat Treatment |
| title_sort |
thermomechanical coupling model for a stainless steel-clad plate on heat treatment |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| publishDate |
2019 |
| topic_facet |
Научно-технический раздел |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/188302 |
| citation_txt |
Thermomechanical Coupling Model for a Stainless Steel-Clad Plate on Heat Treatment / B. Guan, Y. Zang, F. Yang, X.Y. Yang, Q. Qin // Проблеми міцності. — 2019. — № 1. — С. 48-64. — Бібліогр.: 25 назв. — англ. |
| series |
Проблемы прочности |
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| first_indexed |
2025-07-16T10:18:16Z |
| last_indexed |
2025-07-16T10:18:16Z |
| _version_ |
1837798362487717888 |
| fulltext |
UDC 539.4
Thermomechanical Coupling Model for a Stainless Steel-Clad Plate on Heat
Treatment
B. Guan,
1
Y. Zang, F. Yang, X. Y. Yang, and Q. Qin
School of Mechanical Engineering, University of Science and Technology Beijing, Beijing, China
1 niuben57@163.com
Quick and correct prediction of the internal stress and deformation of stainless steel-clad plates on
heat treatment is a problem of specific interest. A thermomechanical coupling model based on the
engineering elastic-plastic theory is detailed. The discretization method and difference equations
used simultaneously provide effective numerical calculations for the model. The generalized finite
element model verifies relevant simplification conditions and efficiency of the numerical calculation
logic. Heat treatment experiments of clad plates are also designed and presented, and the ability of
the model to predict the stress and deformation behavior of the clad plate in the actual heat treatment
process is evaluated. The calculation logic of the model is reasonable, and the prediction error of
deformation and internal stress of the plate is ~15%. The model exhibits an extremely high
computational efficiency and can meet the requirements of on-line analysis for heat treatment
processes.
Keywords: stainless steel-clad plates, heat treatment, thermomechanical coupling model,
deformation and internal stress prediction.
Introduction. A stainless clad plate is a kind of laminated composite composed of a
stainless steel plate and low-carbon steel or low-alloy steel plate. Such plates have the
favorable physical and chemical properties of stainless steel plates, as well as the excellent
mechanical properties of the base plate, enabling their use in the nuclear energy, shipbuilding,
and oil pipeline industries, among others [1–4]. In order to ensure and improve the material
physical and mechanical properties for each layer in the stainless clad plate, heat treatment
processes such as controlled cooling and solution treatment are required during the
manufacturing of different stainless clad plates [5–7].
The material properties and functional requirements are different for stainless steel
and low-carbon steel; the best heat treatment process for each type of layer is different.
When designing a heat treatment process for the stainless clad plate, the characteristics of
the two different metals must be considered to obtain the best comprehensive service
performance. The effects of heat treatment on the material microstructure and mechanical
properties of the stainless clad plate have been studied extensively [8–11]. For example,
Rao et al. [8] studied the mechanical properties and microstructure of explosively clad
HSLA steel with AISI 304L grade under typical quenching and tempering treatment
process. Jiang et al. [9] used X-ray diffraction and transmission electron microscopy to
examine the microstructure and mechanical properties of Ti-steel explosive-rolling clad
plate after different heat treatment processes.
Stainless steel and carbon steel have different thermal conductivities and coefficient of
thermal expansion (CET) (among other physical parameters), which can cause asymmetric
temperature changes and inconsistent thermal strain during heat treatment. These differences
can lead to large deformation with a high macroscopic curvature and high thermal stress
levels within the clad plate (Fig. 1), which affects the service performance of the clad plate.
Therefore, the heat treatment process significantly affects the microstructure properties of a
material, as well as a non-negligible effect on the macroscopic geometrical and mechanical
state of the clad plate. In addition to the resulting material microstructure, the thermal stress
© B. GUAN, Y. ZANG, F. YANG, X. Y. YANG, Q. QIN, 2019
48 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2019, ¹ 1
level and macroscopic curvature deformation must be fully predicted when designing a heat
treatment process for the clad plate. The operation speed of the prediction model must meet
the requirements of online heat treatment process analysis. Therefore, an efficient thermo-
mechanical coupling model for the clad plate during heat treatment process must be
established.
In previous studies, the finite difference method was widely used as an efficient
numerical method for online analysis and prediction modeling of the heat treatment process
[12–14]. Mukhopadhyay and Sikdar [15] established a one-dimensional online temperature
prediction model for the steel plate coiling process with the finite difference method.
Saboonchi and Hassanpour [16] simulated the temperature field during the forced cooling
process of the hot-rolled coil with the implicit finite difference method, enable them to
obtain the temperature distribution in the radial direction of the steel coil. Mansouri et al.
[17] studied the effects of heat-retaining conditions on the temperature distribution when
the hot-rolled strip passes the transfer table with the finite difference method. Chen et al.
[18] proposed an improved finite difference model to predict the temperature changes
during ultra-fast cooling more accurately. Due to the symmetrical structure parameters,
deformation of traditional plates as caused by thermomechanical coupling effects during
heat treatment is not prominent. Therefore, most of these studies focused only on the
prediction of a single temperature field during heat treatment; the above models cannot
predict the large curvature deformation and internal stress of the clad plate. Some scholars
have also used the finite element (FE) method to study the temperature, stress, and phase
transition coupling effects of single plates during heat treatment [19–22]. Sun et al. [19]
presented a FE model coupled analysis of the thermal and metallurgical behavior of the
strip occurring on the run out table cooling. Then, Cho et al. [21] built a three-dimensional
FE model to predict the edge wave behavior of hot rolled steel during run out table cooling,
which considered the coupling of thermal, metallurgical, and mechanical effects. This kind
of multi-field coupling finite element model is very powerful, but the extremely large scale
of its calculations makes it unfeasible for analyzing the processing strategy in real time.
In this study, an efficient thermomechanical coupling numerical model was established
based on the elastoplastic theory to predict the thermal stress and deformation of stainless
clad plates during heat treatment. A finite element model was then built to verify the
relevant simplified conditions and the numerical calculation logic rationality of the
established thermomechanical coupling numerical model. Heat treatment experiments for
the clad plates were designed and conducted, and the ability of the established thermo-
mechanical coupling numerical model to predict the stress and deformation behavior was
analyzed by comparing the predictions to the actual heat treatment results. The results
obtained by the finite element model and the experimental data were compared with the
results predicted by the model established in this paper. The latter not only predict accuracy
for the stress and deformation behavior of clad plates during heat treatment with high
accuracy, but also has a high computational efficiency. Thus, it can realize the online
analysis of heat treatment processes.
Thermomechanical Coupling Model ...
ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2019, ¹ 1 49
Fig. 1. Deformation of stainless clad plate after heat treatment.
1. Establishment of Model.
1.1. Model Structure. By analyzing the physical process of stainless clad plate heat
treatment, the change of the temperature field of the clad plate leads to a change in the
physical parameters of the material, affecting the mechanical behavior of the plate such as
its deformation and internal stress behavior. Hence, the calculation of the mechanical
behavior of the clad plate during heat treatment is based on the calculation results of the
temperature field. However, the mechanical behavior has little effect on the heat transfer
boundary condition and temperature field. Therefore, the heat treatment process of a clad
plate is a typical one-way mechanical and thermal coupling problem.
The most efficient way to analyze a one-way coupled physical process is the
sequential coupling method. As such, the model in this paper is constructed based on
thermomechanical sequential coupling principles, for both the heat transfer model and the
mechanical model of the clad plate. In each incremental step, the temperature field of the
clad plate is first obtained with the heat transfer model, and then the mechanical parameters
and physical parameters related to the material are accordingly fed into the calculation
results of the temperature field of the clad plate. Finally, these parameters are input into the
mechanical model to solve the deformation and stress field of the clad plate. During
solving, the clad plate will be first discretized along the thickness direction. The heat
transfer model and the mechanical model will be calculated with the same discretization
nodes, which can guarantee effective data transfer between different physical fields and
greatly improve the physical field data transfer efficiency. The calculation flow is shown in
Fig. 2.
1.2. Basic Assumptions of the Model. Based on the deformation and heat transfer
characteristics of heat treatment process for the stainless clad plate heat treatment process,
two basic assumptions were made in the present study:
1) Plane section assumption: An arbitrary section of the plate remains flat after
deformation. During the heat treatment of the stainless clad plate, the bending deflection of
the plate is far smaller than its length, and the height-to-span ratio remains less than 1:5,
thus, the plane section assumption is applicable.
B. Guan, Y. Zang, F. Yang, et al.
50 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2019, ¹ 1
Fig. 2. Calculation flow chart of thermomechanical coupling model.
2) One-dimensional heat transfer assumption: Compared with a single-material plate,
the heat transfer process in a composite metal plate has a key unique feature, namely, the
base and cladding materials (which have different thermal physical properties) are laminated
along the thickness direction of the plate. This results in obvious property differences in the
thermal properties along the thickness direction of the clad plate during heat transfer. This
is also the primary cause of the large deformation of the plate after heat treatment. Other
conventionally observed deformations in the clad plate are far smaller than the deformation
caused by this difference in the thermal properties. Thus, in order to analyze this key
feature of the clad plate and to simplify the model, it is reasonable to establish a
one-dimensional heat transfer model along the thickness direction of the plate and ignore
the special heat transfer conditions at the top and tail regions of the plate.
1.3. Heat Transfer Model.
1.3.1. Establishment of the Model. The physical model is shown in Fig. 3a. The total
thickness of the clad plate is h, while the thickness of the cladding is hc , thus, t the
thickness of the base is h hc� . Based on the basic principles of heat transfer [23], there is
no heat source in the clad plate during the heat treatment process. The one-dimensional
thermal conduction differential equation is
� � �
�
�
� �
�
c c cz c z
dT z
d
z
d T z
dz
h z h( , ) ( , )
( , )
( , )
( , )
(� � �
2
2
2 2�
�
h
z c z
dT z
d
z
d T z
dz
hb b b
1
2
2
2
),
( , ) ( , )
( , )
( , )
( , )
(� � �
�
�
� �
�
� � ��
�
�
h z h1 2),
(1)
where T z( , )� is the temperature field of the clad plate at any time, � is the unit time, and
� �c z( , ) and � �b z( , ) are the thermal conductivity of the cladding and the base material,
respectively. The density of the cladding and the base material is � �c z( , ) and � �b z( , ),
respectively, and c zc ( , )� and c zb ( , )� are the specific heat of the cladding and the base
material, respectively.
ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2019, ¹ 1 51
Thermomechanical Coupling Model ...
a b
Fig. 3. Schematic diagram of the heat transfer physical (a) and discrete (b) models.
Before the heat treatment process, the metal plate is in a certain soaking state T0. With
that in mind, the initial condition for the differential equations is
T z T( , )0 0� ( ).0� �z h (2)
During the heat treatment, there is heat transfer between the clad plate surface and the
ambient medium, meaning that the convective heat transfer boundary conditions are shown
in Eqs. (3) and (4) below. Due to the effective metallurgical bonding at the bimetallic
bonding interface of the hot-rolled clad plate, there is no thermal resistance at the bimetallic
bonding interface and the temperature is continuously distributed within the clad plate.
According to the initial conditions of Eq. (2) and the boundary conditions of Eqs. (3) and
(4), the definite solution of the thermal conductivity differential Eq. (1) can be obtained
� � �� �
�
�c c fcz
dT z
dz
a T T z( , )
( , )
( ( , )) ( , ),z h� �2 0� (3)
� � �� �
�
�b b fbz
dT z
dz
a T T z( , )
( , )
( ( , )) ( , ).z h�� �2 0� (4)
1.3.2. Numerical Solution. In order to solve the clad plate heat transfer model
established above, the heat transfer model of n time increment steps is numerically solved
with the numerical difference method by discretizing the plate along the thickness direction.
The specific numerical solution process can be divided into the following two steps.
Discretization of plate: As shown in Fig. 3b, the stainless clad plate is divided into m
discretization nodes along the thickness direction. Position coordinates of the discretization
nodes are z zm1 , ... , , respectively. The cladding is divided into m1 discretization nodes,
and the base is divided into m m� 1 discretization nodes. Meanwhile, the base and the
cladding share m1 discretization nodes at the bonding interface to ensure the continuity of
the temperature field.
z is the distance between two discretization nodes, while
� is the
time length of the increment step. During the numerical calculation process, the temperature
T T
n
m
n
1
( ) ( ), ... , of the discretization nodes are used instead the temperature T zn( ) ( ), which
is continuously distributed along the thickness direction. The thermophysical parameters
�( ) ( ),n z �( ) ( ),n z and c zn( ) ( ) of the material are thus replaced by � �1
( ) ( ), ... ,
n
m
n ,
� �1
( ) ( ), ... , ,
n
m
n and c c
n
m
n
1
( ) ( ), ... , , respectively.
Establishment of the thermal and boundary difference equations: Based on the
coordinates of the discretization nodes, the thermal conduction differential equation of the
clad plate and the difference equations of the boundary conditions are established. The
thermal conduction differential equation is as follows:
�
�
�
j
n
j
n
j
n
j
n
j
n j
n
jc T T T T
( ) ( ) ( ) ( )
( )
( ) (
( )
� � �
� ��
�
�1 1 1
1 1
n
j
n j
n
j
n
z
T T
z
)
( )
( ) ( )
( ) ( )
,
2 1
1 1
1
2
�
�
�
� �
�
�
j m� 1, ... , .
(5)
The difference equations of heat transfer boundary condition are as follows:
�
�
�1
1
1
1
1 1
1
1
1
2
( ) ( ) ( ) ( ) ( )
( )( )
(
n n n n n
nc T T z
z
T
� � � ��
� �
T T T
c T T
n
c fc
n n
m
n
m
n
m
n
m
n
1 2
1 1
( ) ( ) ( )
( ) ( ) ( ) (
) ( ),
(
� �
�� � �
�
� 1
1
1
1
) ( )
( ) ( ) ( ) ( ))
( ) (
z
z
T T T T
m
n
m
n
m
n
b fb
n
m
n
�
�
�� � � ��
�
� ).
�
�
(6)
52 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2019, ¹ 1
B. Guan, Y. Zang, F. Yang, et al.
Setting
�
�
j j
j
c z
K
� and
� j
j
z
A
� , Eqs. (5) and (6) can be derived into the following
difference equations:
( ) ,
( ) ( ) ( ) ( )� �1 1 1 1 1 2 1 1
1
1
� � � � �
�
�
�
K A T A T K T T
A T
n n n
c fc
n
j j� � �
�� � � � � �1 1 1
1
2 3
( ) ( ) ( ) ( )
( ) , ( ,
n
j j j j
n
j j
n
j j
n
K A A T A T K T j , ... , ),
( )
( ) ( ) (
m
A T K A T K Tm m
n
m m m m
n
m m
n
�
� � � � �� � � �
�
1
1 1 1 1� 1) ( )
.�
�
�
�b fb
n
T
(7)
By rearranging Eq. (7), a matrix is obtained as follows:
�c K A A
A K A A A
A K A A A
� � �
� � � �
� � � �
1 1 1
1 2 2 1 2
2 3 3 2 1
0 0 0
0 0
0 0
�
�
�
� 0
0 0
0 0
2 1 1 2 1
1
� � � �
�
� �
� � � �
� � �
� � � � �
� �
A K A A A
A K A
m m m m m
m b m m� 1
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
T
T
T
T
T
Kn
n
n
m
n
m
n
1
2
3
1
1
( )
( )
( )
( )
( )
�
T T
K T
K T
K T
K
n
fc
n
n
n
m m
n
m
1
1
1
2 2
1
3 3
1
1 1
1
( ) ( )
( )
( )
( )
�
�
�
� �
�
��
�
T Tm
n
m fb
n( ) ( )
.
�
��
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�1
1�
(8)
This matrix is a three diagonal matrix, which is usually solved numerically using the
chase method. By solving Eq. (8), the internal temperature field of the clad plate after
�
can be obtained according to the result of the upper incremental step and the heat transfer
boundary condition. Hence, the heat transfer result can be obtained through iterative
calculation using this method.
1.4. Mechanical Model.
1.4.1. Establishment of Model. As shown in Fig. 4a, the temperature and expansion
coefficient both vary along the thickness direction of the clad plate. This produces a
nonlinear distributed thermal strain � th along the thickness direction of the plate, which
forms an uncoordinated thermal strain state. Based on the plane section assumption, in
order to make the plate section meet the deformation coordination relationship, internal
stress must be produced to form a bending deformation with a certain curvature.
The specific establishment process of the mechanical model of this process is as
follows. The internal strain and bending curvature accumulated in the plate after the
(n�1)th increment step are � ( )n�1 and C n( )�1 , respectively. In the nth increment step, the
expansion coefficient in each position of the plate thickness direction at this time, � ( ) ( )n z ,
can be determined based on the heat transfer model T zn( ) ( ). Based on the temperature
field calculation results, T zn( ) ( ) for the nth increment step and T zn( ) ( )�1 for the (n�1)th
increment step, the thermal strain � th
n
z
( )
( ) in each position of the plate under free
conditions can be obtained as follows:
ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2019, ¹ 1 53
Thermomechanical Coupling Model ...
� �th
n n n nz z T z T z
( ) ( ) ( ) ( )( ) ( )( ( ) ( )).� � �1
(9)
However, because of the plane section assumption and deformation coordination
relationship, a linear distributed strain �h
n
z
( )
( ) along the thickness direction of the plate (as
shown in Fig. 4b) will be produced in the entire clad plate. The strain is defined as follows:
� � �h
n n nz z
( ) ( ) ( )( ) tan ( ).� � (10)
In the above equation, �( )n is the distance between the strain neutral surface of �h
n( )
and the geometric neutral layer of the plate, � ( )n is the bending curvature accumulated
during the nth increment step of the plate; �( )n and � ( )n are unknowns to be solved. Thus,
the actual strain �n
n z( ) ( ) produced in the nth increment step is a linear superposition of
� th
n
z
( )
( ) and �h
n
z
( )
( ), namely:
� � �n
n
h
n
th
n
z z z( ) ( ) ( )
( ) ( ) ( ).� � (11)
Similarly, according to the linear superposition theory of strain, the total strain
� ( ) ( )n z on the clad plate accumulated within the nth step is a linear superposition of the
total strain � ( )n�1 within the (n�1)th step and the actual strain �n
n z( ) ( ) for the nth step, as
� � �( ) ( ) ( )( ) ( ) .n
n
n nz z� � �1
(12)
According to the stress–strain relationship � �� f T( , ) for the base and cladding
materials, the stress distribution � ( ) ( )n z is obtained using � ( ) ( )n z and T zn( ) ( ). � ( ) ( )n z
must fulfill the following static force and moment equilibrium equations:
54 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2019, ¹ 1
B. Guan, Y. Zang, F. Yang, et al.
a
b
Fig. 4. Schematic diagram of mechanical model: (a) uncoordinated thermal strain �th
n( )
; (b) relationship
between � �h
n
th
n( ) ( )
, and � �th
n
n
n( ) ( ), .
� ( )
/
/
( ) ,n
H
H
z dz �
�
� 0
2
2
(13)
� �( ) ( )
/
/
( )( ) .n n
H
H
z z dz� �
�
� 0
2
2
(14)
The two unknowns �( )n and � ( )n can be obtained from Eqs. (13) and (14).
Furthermore, � ( ) ( )n z can be obtained by combining Eqs. (10), (11), and (12), after which
the stress distribution � ( ) ( )n z can be obtained. The bending curvature C n( ) of the plate at
the nth step is given as
C Cn n n( ) ( ) ( ) ,� ��1 � (15)
� ( ) ( )n z and C n( ) will be taken as the initial conditions for the (n�1)th increment step.
1.4.2. Numerical Simulation. According to the above mechanical analysis model, the
mechanical model of the clad plate at the nth time increment step is numerically solved
with the same discretization method. The specific numerical solution also can be divided
into two steps.
Plate discretization: The clad plate is discretized into 1, ... , m nodes, which are
consistent with the discretization node coordinates z zm1 , ... , of the heat transfer model.
The temperature field calculation results of each node are then transmitted directly to the
mechanical model. During the numerical calculation process, the continuously distributed
strain � ( ) ( )n z and stress � ( ) ( )n z along the thickness direction are replaced by strain
� �1
( ) ( ), ... ,
n
m
n and stress � �1
( ) ( ), ... ,
n
m
n of the discretization nodes.
Iterative solution of unknown stress and strain of discretization nodes: The temperature
field calculation results T T
n
m
n
1
( ) ( ), ... , and T T
n
m
n
1
1 1( ) ( ), ... ,
� � for step n and step n�1
are read into the mechanical model. Using Eq. (9), the thermal strain � th
n
1
( )
, ..., � thm
n( )
in free
conditions of each discretization node can then be obtained.
The iteration error limits of the static equilibrium equation are set as eps1 and eps2
for the static moment equilibrium equation, while the number of limit iterations is I.
Supposing iteration time i� 1, and given a certain initial �1
( )n
and �1
( )n
, the mechanical
model results � �1
1 1( ) ( ), ... ,
n
m
n� � of step n�1 are read into the mechanical model. The
total strain � �1i
n
mi
n( ) ( )
, ... , of each node accumulated within the nth increment step is then
obtained based on Eqs. (9)–(12).
Through interpolation calculations of the input stress–strain relationship data � �
� f T( , )� of the base and cladding material to the position ( , ),
( ) ( )�1 1i
n n
T ..., ( , ),
( ) ( )�mi
n
m
nT
the stress � �1i
n
mi
n( ) ( )
, ... , of each discretization node can be obtained. The static force and
moment equilibrium equations (16) of the discretization nodes can be derived from Eqs.
(13) and (14). One should check whether � �1i
n
mi
n( ) ( )
, ... , meet the following criteria using
Eqs. (16) or (17):
� ji
n
j
m
eps
( )
,
�
� �
1
1 (16a)
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Thermomechanical Coupling Model ...
� �ji
n
j i
n
j
m
z eps
( ) ( )
( )� �
�
�
1
2, (16b)
i I� . (17)
If � �1i
n
mi
n( ) ( )
, ... , meet any of the above two conditions, � i
n( )
, � i
n( )
, � �1i
n
mi
n( ) ( )
, ... , ,
and � �1i
n
mi
n( ) ( )
, ... , are put out as the solution of the unknown stress and strain in the nth
increment step. The calculation result C n( )�1 of the (n�1)th step is then read into the
mechanical model. The curvature C n( ) after n increment steps is obtained with Eq. (15). If
the above conditions are not satisfied, � i
n( )
and � i
n( )
need to be reset based on
two-dimensional search methods. Meanwhile, setting i i� �1, the above operations are
performed by iterating until the output conditions are satisfied. The logical frame of the
numerical analysis process is shown in Fig. 5.
56 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2019, ¹ 1
B. Guan, Y. Zang, F. Yang, et al.
Fig. 5. Logical frame of the mechanical model calculation.
2. Validity Analysis and Experimental Verification of Model. In order to validate
the simplification and numerical calculation logic of the established model, a two-
dimensional thermomechanical coupling finite element model (FEM) is established with
ABAQUS to analyze the controlled cooling process. The FEM results are compared to the
calculated results of the proposed model. Then, in order to verify the accuracy of the
established model in predicting the stress and deformation behavior during heat treatment, a
stainless clad plate heat treatment experiment was designed to comparatively analyze the
experimental data and prediction results.
2.1. Mechanical and Physical Properties of Materials. Accurate mechanical and
physical properties are vital to reliable model calculation. Accordingly, the mechanical and
physical parameters of the clad plate used in the experimental and numerical simulations
were accurately determined using tests or material properties simulation methods.
The base layer is composed of carbon steel Q345R and the clad layer is stainless steel
316L. The chemical compositions of the two materials are shown in Table 1. The expansion
coefficients are the most important material parameters affecting the deformation and
residual stress of the plate. They are influenced by phase changes, composition etc. Thus,
they were precisely measured using a DIL801 thermal dilatometer, and the results are
shown in Fig. 6a. The elastic modulus and flow stress are two other important parameters.
They describe the stress–strain relationships of the material under elastic and plastic
conditions. These two parameters were calculated using the JMatPro software based on the
components of two materials, and the result are shown in Fig. 6b–d. Other physical
parameters such as the specific heat, conductivity, and Poisson’s ratio were also calculated
using the JMatPro software.
2.2. Validity Analysis of Model. The FE model was established based on a stainless
clad plate 500 mm in length and 30 mm in thickness, including 5 mm of cladding (316L)
and 25 mm for the base, as shown in Fig. 7. The initial temperature of the plate was set to
900�C and the cooling process lasted for 30 s. The heat transfer coefficient for both the top
and bottom surfaces was 500 W/(m K2 � ). The cooling medium temperature was 30�C.
CPS4T, the element type of the thermomechanical coupling, was chosen to mesh the plate
in the FEM. The cladding is divided into 10 layers, while the base is separated into 20
layers; interfaces between the base and cladding share the same node. The model contains
3000 elements. In the thermomechanical coupling model established here, the total number
of discrete nodes m is 180. The cladding is separated into 60 nodes, and the base is
discretized into 120 nodes. The time increment step of
� is 0.2 s.
Figure 8 shows the results obtained by using the FE model and the proposed model.
As can be seen here, the temperature and stress distribution obtained with the proposed
model exhibited good agreement with the FEM results, with a maximum difference of
�5%. This shows that the related simplifications and numerical calculation logic of the
proposed model are correct and effective. In terms of computational efficiency, the FE
model required approximately 30 min of computing time, while the proposed model only
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Thermomechanical Coupling Model ...
T a b l e 1
Element Composition of Materials (mass fraction, %)
Steel C Si Mn P S Cr Ni Mo Al Ti Fe
Q345R
(base)
0.17 0.27 1.40 0.011 0.007 0.023 0.02 �0.01 �0.01 �0.01 Bal.
316L
(cladding)
0.017 0.42 1.26 0.031 0.003 16.03 10.03 1.95 �0.01 �0.01 Bal.
needed 10 s under the same hardware conditions. Because of its extremely high
computational efficiency, the proposed model can rapidly predict stress and deformation
behavior, as well as online process strategy analysis, for clad plate heat treatment.
58 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2019, ¹ 1
B. Guan, Y. Zang, F. Yang, et al.
a b
c d
Fig. 6. Mechanical and physical properties of materials: temperature dependences for expansion
coefficient (a) and elastic modulus (b); stress–strain curves at strain rate 0.001 s�1 for 316L (c) and
Q345 (d) steels.
a
b
Fig. 7. Two-dimensional finite element model of thermomechanical coupling: (a) finite element
mesh; (b) calculation result of temperature (cooling 20 s).
2.3. Experimental Verification of the Model.
2.3.1. Experimental Design. The stainless composite plate used in the experiment is
shown in Fig. 9a. The thickness of the base and cladding and their material properties are
the same as the FE model (Section 1.1). The concrete dimensions of the test pieces are
1320 350 30� � mm (length�width�thickness). Two temperature measuring points are
arranged in the inner plate using a thermocouple (Fig. 9b). The 1# point is located at the
interface between stainless steel and carbon steel, while the 2# point is located in the center
of the clad plate.
The heat treatment of the plate during the experiments was divided into three steps as
shown in Fig. 10a–c. The plate is first put into the heating furnace, and the bonding
interface is heated to 950�C and held for 20 min. Next, the plate is removed from the
furnace and the stainless steel side of the clad plate undergoes solution treatment with water
spraying until the temperature at the bonding interface is reduced to 400�C. Finally, the
plate is air cooled to room temperature. After the heat treatment process, the residual
stresses of both sides of the plate are determined using a blind-hole method. The
deformation curvature is measured by a laser 3D scanning method (Fig. 10d).
2.3.2. Experimental Results and Analysis. The temperature data acquired during the
heat treatment at the temperature measuring points by the thermocouple are shown in
Fig. 11a. The average cooling rate during the spray solution treatment is approximately
3.36 �C/s at the center and 3.98 �C/s at the bonding interface. Because the heating and
cooling device used in the heat treatment experiments is relatively simple, it is difficult to
control the heat transfer process of the plate surface with the required accuracy and
ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2019, ¹ 1 59
Thermomechanical Coupling Model ...
a b
Fig. 8. Comparison of FE and thermomechanical coupling model results: temperature (a) and internal
stress (b) distributions in the plate.
a b
Fig. 9. Experimental plate (a) and location about temperature measuring points (b).
60 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2019, ¹ 1
B. Guan, Y. Zang, F. Yang, et al.
Fig. 10. Clad plate heat treatment experimental process: (a) heating and heat preservation; (b) solution
treatment; (c) air cooling; (d) detection.
a
b
Fig. 11. Temperature changes with time during experimental heat treatment: (a) experimental process
temperature curve; (b) change of surface heat transfer coefficient with time.
stability. Therefore, based on the solution of the inverse thermal conductivity problem, [24]
and using the temperature data clad plates measured during experiments, the surface heat
transfer coefficient of the clad plate is solved inversely using the heat transfer model
established in this article (Fig. 11b). Furthermore, the temperature evolution of each
position on the clad plate can be obtained.
The surface heat transfer coefficient obtained by inverse solution is then plugged into
the thermomechanical coupling model established in this article, and the residual stress
distribution and deformation curvature of the clad plate can be acquired after heat
treatment. The residual stresses calculated with the model are compared to the residual
stress data of the two surfaces of the clad plate obtained using the blind hole detection
method (Fig. 12). According to the calculated results of the model, the center of the plate is
in the compressive stress state. However, both surfaces are in the tensile stress state,
consistent with the test results. The residual stress calculated value of the base (Q345R) is
closer to the detected value with an error of approximately 14.4%; the error is likely to be
produced by the detection process. The residual stress error on the cladding surface of 316L
between the calculated values and the detection values is relatively larger (27.1%). On the
one hand, the error is partly due to the complex heat transfer conditions of the cladding
surface. There are significant differences between the ideal calculation model and the actual
situation. On the other hand, error arises because the residual stress on the cladding surface
exceeds the material yield limit, and the blind hole detection method has a comparatively
large error under plastic deformation conditions [25]. The detected results demonstrate that
the residual stress on the cladding surface indeed exceeds the material yield limit, agreeing
with the calculated results.
The deformation cloud diagram and the deformation curve of the clad plate acquired
by laser 3D scanning are shown in Fig. 13. Clearly, plate deformation occurs in a single
one-dimensional curvature state, consistent with the model assumptions. The deformation
curvatures of the clad plate obtained through the model calculation and the test data are
both shown in Table 2. The results obtained using these two different methods are close to
each other, with 15.1% error between them. The measured curves results are larger than the
calculated results; the main reason for the error is that the proposed model established in
this paper is one-dimensional along the thickness of the plate. This does not consider the
special heat transfer conditions at the top and tail regions of the plate. This simplification is
Thermomechanical Coupling Model ...
ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2019, ¹ 1 61
Fig. 12. Residual stress after heat treatment of the plate.
reasonable during the actual production process, but the length-thickness ratio of the plate
used in this experiment is much greater than those of the plates in actual production
environments. As such, the rapid temperature drop at the top and tail of the experimental
plate would significantly affect its entire deformation. This can reduce the thermal
deformation recovery during the cooling process, thereby increasing the final residual
deformation of the plate.
C o n c l u s i o n s
1. Based on engineering elastoplastic theory and heat transfer theory, a thermo-
mechanical coupling model aiming at predicting the stress and deformation of a stainless
clad plate during heat treatment is established. The discretization method and differential
operation are used in combination to implement the efficient numerical calculation of the
model.
2. The solving results and computation time of the model established in this paper and
the FE model are compared with each other. The difference in solving results between the
two models is �5%. This indicates that the relevant simplification and numerical solution
logic of the established model are reasonable. This model has a higher computational
efficiency than the general finite element model, enabling it to meet the requirements of the
online process strategy analysis for clad plate heat treatment.
3. The model calculation results and experimental data are comparatively analyzed.
The model has high prediction accuracy for the stress and deformation behavior of stainless
clad plate during actual heat treatment. The prediction error of the deformation curvature
and internal stress is approximately 15%.
B. Guan, Y. Zang, F. Yang, et al.
62 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2019, ¹ 1
T a b l e 2
The Deformation Curvature of Plate after Heat Treatment
Measured curvature (m 1� ) Calculated curvature (m 1� ) Error
0.1199 0.1019 15.01%
Fig. 13. Plate deformation after heat treatment.
4. Further studies can improve the structure of the model proposed in this paper.
Firstly, some simple microstructural prediction models can be incorporated into this model
to comprehensively analyze the micro- and macro-states of the clad plate during the heat
treatment. Secondly, based on this model, a finite strip element model can be established to
study the shape wave defection problem of clad plates under two-dimensional conditions.
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Received 15. 03. 2018
B. Guan, Y. Zang, F. Yang, et al.
64 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2019, ¹ 1
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/NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
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/ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
>>
/Namespace [
(Adobe)
(Common)
(1.0)
]
/OtherNamespaces [
<<
/AsReaderSpreads false
/CropImagesToFrames true
/ErrorControl /WarnAndContinue
/FlattenerIgnoreSpreadOverrides false
/IncludeGuidesGrids false
/IncludeNonPrinting false
/IncludeSlug false
/Namespace [
(Adobe)
(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AddBleedMarks false
/AddColorBars false
/AddCropMarks false
/AddPageInfo false
/AddRegMarks false
/ConvertColors /ConvertToCMYK
/DestinationProfileName ()
/DestinationProfileSelector /DocumentCMYK
/Downsample16BitImages true
/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements false
/GenerateStructure false
/IncludeBookmarks false
/IncludeHyperlinks false
/IncludeInteractive false
/IncludeLayers false
/IncludeProfiles false
/MultimediaHandling /UseObjectSettings
/Namespace [
(Adobe)
(CreativeSuite)
(2.0)
]
/PDFXOutputIntentProfileSelector /DocumentCMYK
/PreserveEditing true
/UntaggedCMYKHandling /LeaveUntagged
/UntaggedRGBHandling /UseDocumentProfile
/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice
|