Constitutive Modeling for Al–Cu–Mg Alloy in Creep Aging Process
The aim of this paper is to develop a set of creep aging constitutive equations for Al–Cu–Mg alloys containing plate- or rod-like precipitates. Average length, aspect ratio and relative volume fraction are introduced to quantitatively analyze precipitates evaluation of such alloy in creep aging proc...
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Yang, Y.L. Zhan, L.H. Xu, X.L. 2020-12-03T19:56:58Z 2020-12-03T19:56:58Z 2016 Constitutive Modeling for Al–Cu–Mg Alloy in Creep Aging Process / Y.L. Yang, L.H. Zhan, X.L. Xu // Проблемы прочности. — 2016. — № 1. — С. 29-38. — Бібліогр.: 30 назв. — англ. 0556-171X https://nasplib.isofts.kiev.ua/handle/123456789/173414 539.4 The aim of this paper is to develop a set of creep aging constitutive equations for Al–Cu–Mg alloys containing plate- or rod-like precipitates. Average length, aspect ratio and relative volume fraction are introduced to quantitatively analyze precipitates evaluation of such alloy in creep aging process. The strong interaction between creep deformation and aging treatment is considered by the intermediate state variables of dislocation density and precipitate characteristic dimension. A unified creep aging constitutive equation is derived, in which the correlations between microscopic characteristics and macroperformances of material are linked by the yield strength of the material. Using AA2124 as subject, a series of uniaxial tensile creep tests are carried out at 185°C for 12 h under different stresses. The material constants within constitutive models are determined with the experimental data. A good agreement between experimental and computed values confirms that the established constitutive equations can well characterize the creep behaviors. This research was supported by the National Basic Research Program of China (Grant No. 2014CB046602), the Key Program of National Natural Science Foundation of China (Grant No. 51235010), and Ph.D. Programs Foundation of Ministry of Education of China (Grant No. 20120162110003). en Інститут проблем міцності ім. Г.С. Писаренко НАН України Проблемы прочности Научно-технический раздел Constitutive Modeling for Al–Cu–Mg Alloy in Creep Aging Process Определяющее моделирование сплава Al-Cu-Mg в процессе старения при ползучести Article published earlier |
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Constitutive Modeling for Al–Cu–Mg Alloy in Creep Aging Process |
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Constitutive Modeling for Al–Cu–Mg Alloy in Creep Aging Process Yang, Y.L. Zhan, L.H. Xu, X.L. Научно-технический раздел |
| title_short |
Constitutive Modeling for Al–Cu–Mg Alloy in Creep Aging Process |
| title_full |
Constitutive Modeling for Al–Cu–Mg Alloy in Creep Aging Process |
| title_fullStr |
Constitutive Modeling for Al–Cu–Mg Alloy in Creep Aging Process |
| title_full_unstemmed |
Constitutive Modeling for Al–Cu–Mg Alloy in Creep Aging Process |
| title_sort |
constitutive modeling for al–cu–mg alloy in creep aging process |
| author |
Yang, Y.L. Zhan, L.H. Xu, X.L. |
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Yang, Y.L. Zhan, L.H. Xu, X.L. |
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Научно-технический раздел |
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Научно-технический раздел |
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2016 |
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English |
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Проблемы прочности |
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Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Article |
| title_alt |
Определяющее моделирование сплава Al-Cu-Mg в процессе старения при ползучести |
| description |
The aim of this paper is to develop a set of creep aging constitutive equations for Al–Cu–Mg alloys containing plate- or rod-like precipitates. Average length, aspect ratio and relative volume fraction are introduced to quantitatively analyze precipitates evaluation of such alloy in creep aging process. The strong interaction between creep deformation and aging treatment is considered by the intermediate state variables of dislocation density and precipitate characteristic dimension. A unified creep aging constitutive equation is derived, in which the correlations between microscopic characteristics and macroperformances of material are linked by the yield strength of the material. Using AA2124 as subject, a series of uniaxial tensile creep tests are carried out at 185°C for 12 h under different stresses. The material constants within constitutive models are determined with the experimental data. A good agreement between experimental and computed values confirms that the established constitutive equations can well characterize the creep behaviors.
|
| issn |
0556-171X |
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https://nasplib.isofts.kiev.ua/handle/123456789/173414 |
| citation_txt |
Constitutive Modeling for Al–Cu–Mg Alloy in Creep Aging Process / Y.L. Yang, L.H. Zhan, X.L. Xu // Проблемы прочности. — 2016. — № 1. — С. 29-38. — Бібліогр.: 30 назв. — англ. |
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| first_indexed |
2025-11-25T23:10:33Z |
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2025-11-25T23:10:33Z |
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| fulltext |
UDC 539.4
Constitutive Modeling for Al–Cu–Mg Alloy in Creep Aging Process
Y. L. Yang,
a,b,c
L. H. Zhan,
a,b,c,1
and X. L. Xu
a,b,c
a State Key Laboratory of High-Performance Complex Manufacturing, Central South University,
Changsha, China
b School of Mechanical and Electrical Engineering, Central South University, Changsha, China
c Collaborative Innovation Center of Advanced Nonferrous Materials and Manufacturing, Central
South University, Changsha, China
1 yjs-cast@csu.edu.cn
The aim of this paper is to develop a set of creep aging constitutive equations for Al–Cu–Mg alloys
containing plate- or rod-like precipitates. Average length, aspect ratio and relative volume fraction
are introduced to quantitatively analyze precipitates evaluation of such alloy in creep aging process.
The strong interaction between creep deformation and aging treatment is considered by the
intermediate state variables of dislocation density and precipitate characteristic dimension. A unified
creep aging constitutive equation is derived, in which the correlations between microscopic
characteristics and macroperformances of material are linked by the yield strength of the material.
Using AA2124 as subject, a series of uniaxial tensile creep tests are carried out at 185�C for 12 h
under different stresses. The material constants within constitutive models are determined with the
experimental data. A good agreement between experimental and computed values confirms that the
established constitutive equations can well characterize the creep behaviors.
Keywords: constitutive modeling, Al–Cu–Mg alloy, creep aging, microstructure evolution.
Introduction. With increasing market demands for cost reduction and good
comprehensive performances, aluminum alloys are extensively used in the industries [1–3].
For example, truss structures constructed of AA6061-OA/T6 (Al–Mg–Si alloys) are
equipped with good dynamic crush resistance and energy absorbing capability when
undergoing uniaxial impact loading [4–7]; automobile parts made of aluminum alloy can
make automobile lightweight and accomplish energy-saving and emission reduction [8].
Al-Cu-Mg alloys are typical deformable and hardenable alloys. They are intensively
adopted to manufacture large aircraft integral panels or structures due to high strength to
weight ratio, lightweight and high resistance to corrosion with proper heat treatment [9–11].
However, how to accurately deform aluminum alloy to obtain large integral component
with desired configuration and properties is an extremely difficult problem. Either high
residual stress exists in formed part or final mechanical properties of the part are not
satisfied. Therefore creep age forming (CAF), a combined process of creep forming and
aging treatment, is a favored forming process and becomes a research focus.
In CAF process, component to be formed remains in full contact with tool surface by
pressure at a certain temperature in the autoclave. After a given time, the component is
shaped due to creep deformation and the material’s properties are enhanced owing to
artificial aging. The deformation history and ultimate mechanical properties of the formed
component can be predicted by creep aging constitutive equations of the alloy. So
establishing constitutive equations of material is indispensable and significant.
With the objective of investigating constitutive models, lots of efforts have been made
over last twenty years. Sallah et al. [12] proposed a simplified inelastic constitutive
equation for autoclave age forming using a cylindrical tool shape and a good agreement
was obtained between the predictions and Textron data. Kowalewski et al. [13] developed a
unified mechanisms-based constitutive equation with three internal state variables, which
© Y. L. YANG, L. H. ZHAN, X. L. XU, 2016
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 1 29
can model the primary, secondary, and tertiary creep. In recent years, more and more
attention has been paid to microstructure evolution because the size, volume fraction,
morphology and distribution of precipitates exert key influence on mechanical properties
and marc-creep behaviors of aluminum alloy. Ho et al. [14] constructed a set of physically
based unified aging-creep constitutive equations for AA7010. Zhan et al. [15] established a
creep aging constitutive models for AA7055 which takes the normalized radius, and
normalized dislocation density into account. The investigated precipitates in their works are
the spherical shape. For plate-shaped precipitates, Li et al. [16] studied the mean radius and
volume percentage of precipitates and correspondingly derived the constitutive equations
for AA7B04. Zhang et al. [17] preliminarily explored the microstructure variation in terms
of dislocation density, the size, volume fraction and aspect ratio of precipitates and hence
proposed a constitutive equation set. However, although the precipitate variation has been
studied by various characteristic parameters, the interaction between them and their effects
on creep deformation and age hardening have not fully or properly formulated. Further
efforts are required to develop mechanism-based creep aging constitutive equations for
aluminum alloys where the correlations between precipitate evolution, creep deformation
and aging hardening are coupled adequately.
The aim of this paper is to develop a set of creep aging constitutive equations for
Al–Cu–Mg alloys. The main strengthening precipitates of Al–Cu–Mg alloys in aging
process are �S phases (Al2CuMg) or �� phases (Al2Cu). They are either plate or rod
shape. In first part, using AA2124 as material (one of Al–Cu–Mg alloys), a series of
uniaxial constant-stress creep tests are carried out at the temperature of 185�C under the
stresses of 200, 225, and 250 MPa for 12 h. In the second part, the average length L, aspect
ratio q, and relative volume fraction fv are introduced to quantitatively determine the
evolution of precipitates. In the third part, a unified mechanisms-based creep aging
constitutive equation set is established for Al–Cu–Mg alloys under isothermal aging
condition. In last part, with the data from experimental tests, material constants in
constitutive equation set are determined for AA2124 and then predicted values calculated
through models are compared with experimental data.
1. Experimental Methods.
1.1. Test Materials. The studied material is a commercial AA2124-T851, which was
supplied by Southwest Aluminum Co. Ltd. The normal chemical composition of
AA2124-T851 is 4.67% Cu, 1.46% Mg, 0.63% Mn, 0.18% Fe, 0.12% Si, 0.04% Zn, 0.01%
Ti, 0.01% Ni, (bal.) Al. Creep specimens were machined out in the rolling direction from
the 2124-T851 plate as shown in Fig. 1. After solution treatment (490�C/50 min), the
specimens were immediately subjected to water quenching. Subsequently the specimens
were kept in the refrigerator (� �18 C) to reduce natural aging. Prior to tests, all specimens
should be polished by abrasive papers of #400 and #600 to eliminate surface defects.
1.2. Test Procedures. Uniaxial constant-stress tensile creep tests were carried out on
specimens at 185�C for 12 h under different stresses. The yield strength of the tested alloy
at 185�C was determined to be 225 MPa. To examine the creep deformation behaviors of
Y. L. Yang, L. H. Zhan, and X. L. Xu
30 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 1
Fig. 1. Geometry of high-temperature creep specimen (unit: mm).
the material within and beyond the elastic range, the applied stresses were selected for 200,
225, and 250 MPa. Creep tests under each condition were not less than three times to
precisely describe the creep behaviors of the material. Room temperature tensile tests were
conducted on creep-aged specimens with a universal testing machine to obtain yield
strength. Microstructure observations were performed with a JEOL-2010 transmission
electron microscope operating at 200 kV. TEM samples were first machined down to 80 �m
in thickness, followed by standard twin-jet electropolishing using 70% methanol and 30%
nitric acid solution at �35 to � �25 C cooled by liquid nitrogen and lastly by anhydrous
alcohol cleaning for 2 min. The geometrical parameters of precipitates were statically
obtained by methods of the reference [18] and the standard variances of the aspect ratio,
mean length and relative volume fraction of precipitate, yield strength and creep strain of
the material corresponded to 1.68, 6.76, 3.77, 3.51, and 0.11 respectively.
2. Results and Discussion.
2.1. Creep Aging Behaviors. Creep strain-time curves of the alloy are drawn in Fig. 2.
From the figure it can be clearly seen that each curve underwent typical primary and
secondary creep stages. By comparison it is found that increasing stress level can shorten
the duration of secondary creep but advance the advent of tertiary creep. Figure 3 illustrates
the yield strength variation of material in CAF process. As the aging time prolonged, the
yield strength increased first and then decreased gradually after reaching the peak value.
Furthermore, compared with stress-free aging, creep aging not only enhances the yield
strength but also shortens the time required to reach the maximum strength.
In Al–Cu–Mg alloys, the strengthening precipitate model can be simplified in Fig. 4a
for plate shape and in Fig. 4b for rod shape [19]. For such precipitates, two characteristic
parameters, i.e., average length L and aspect ratio q are chosen to describe the dimension
evolution of precipitates in this paper. The aspect ratio equals to L h, where h is the mean
thickness of plate-like precipitate or mean diameter of rod-like. Relevant research have
found that dominate strengthening precipitates in creep aging of AA2124 are plate-like �S
phases (Al2CuMg), as shown in Fig. 4c, because the ratio between the content of Cu and
Mg is less than 4 [17, 20]. For simplicity, it is assumed here that only phases precipitate in
CAF process. The general accepted precipitation sequence is SSS (supersaturated solid
solution) � GP zone � �-Al ��S � �-Al �S � �-Al S [21, 22]. The change trend of
mean length of strengthening phase is depicted in Fig. 4d. The average length of
Constitutive Modeling for Al–Cu–Mg Alloy in Creep Aging Process
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 1 31
Fig. 2 Fig. 3
Fig. 2. Creep strain–time curves of AA2124 specimens aged for 12 h at 185�C under different
stresses.
Fig. 3. Variation of yield strength of AA212 specimens aged at 185�C under different stresses.
precipitates increased monotonously with increasing aging time but the growth rate reduced
gradually. This trend can be explained by the fact that the diffusion rate of solute atoms to
form precipitates drops due to decreasing concentration of solute in the matrix. Besides, at
high stress levels, the precipitate length was shorter. This is because more dislocations
produced by the higher external load in aging process would cause acceleration in the
heterogeneous nucleation of precipitates and increase precipitate density, which results in a
low growth rate of precipitates to some extent.
2.2. Constitutive Model Establishment. Creep aging consists of creep deformation
and aging heat treatment. In CAF process, the elastic strain is in part converted to
permanent creep strain and this creep strain is responsible for retaining the final
configuration of the part. Aging is a process that can increase the strength of a metal.
Although some studies have been carried out the creep and aging as individual processes,
when the two processes are combined in CAF, the interaction between them is highly
sophisticated. On the one hand, massive dislocations are created due to external stress.
These dislocations do affect precipitation process of the material, stress-induced orientation
effect [23, 24] for instance, and eventually change material strength. On the other hand, the
fine and dense precipitates formed in aging process improve the strength of the material
[25]. The increase in strength of the alloy in turn impedes the creep deformation/rate.
Consequently, material strength plays the role as a bridge between microscopic factors and
macrocreep strain/rate.
Based on the understanding of creep aging and the knowledge of continuum
mechanics and thermodynamics, a set of constitutive equations for Al–Cu–Mg alloys under
isothermal aging treatment has been determined to describe the creep history, properties
hardening and the evolution of microscopic factors. For ease of description, the equations
are first listed as below:
32 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 1
c d
Fig. 4. A simplified mathematic model for plate-like precipitate (a), rod-like precipitate (b), TEM
image of AA2124 at 185�C for 8 h under 200 MPa (c), average length of precipitate in specimen aged
at 185�C under different stress levels: 200 and 250 MPa (d).
a b
Y. L. Yang, L. H. Zhan, and X. L. Xu
where A, B,
0, k1, k2, Css , C ppt , Cdis , C1, C2, C3, C4, C5, m1, m2, m3, m4, m5,
n1 , n2, n3, n4, n5, L* , and t* are material constants.
Equation (1) represents the evolution of creep strain under isothermal aging condition.
Referring to Eq. (1), the change of creep strain is affected by the contradictory effects
between externally applied stress
and material’s yield strength
y . These effects are
controlled by the exponent (m1) in sinh function. The yield strength varies with the
microstructure evolution in the isothermal aging process (see Fig. 3). So the yield strength
can be regarded as a junction between microstructure evolution and creep strain rate.
Equations (2)–(5) depict the strengthening responses of material in creep aging. The
contributions to the yield strength of the material in Eq. (2) are composed of the Al matrix
strength (
0), solution strengthening (
ss), aging hardening (
ppt ) and dislocation
strengthening (
dis) [26, 27]. A schematic for strengthening responses in CAF process is
plotted in Fig. 5. Upon aging (stage I), the initial yield strength reveals the contributions
from the intrinsic strength (Al matrix strength) and solute hardening because there is
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 1 33
� sinh�
c
y
m
A
B
�
�
�
�
�
�
�
�
1
(1) Creep rate
Yield strength evolution
y ss ppt dis� 0 , (2)
ss ss v
n
C f� �( ) ,1 3 (3)
ppt ppt
m
v
n
C L f q� 5 50 5. , (4)
�dis dis
m
C� 4 , (5)
Precipitate evolution
�
( ) ,f
C L
q
fv v
m n� �1
3
1 2 1� (6)
� ( ) ( ),*L C L L k
m n� � �2 1
3 21 � (7)
q C k t t� �3 2
2exp[ ( ) ],*
(8)
� ( )| � | ,� � � �� � �C Cc
n
4 51 4 (9)
�
� �
Fig. 5. Contributions of solid hardening, aging hardening, and dislocation hardening to yield strength.
Constitutive Modeling for Al–Cu–Mg Alloy in Creep Aging Process
insufficient time for precipitate to nucleate in the water-quenched aluminum alloy. At the
under-aged period (stage II), transient second phases begin to form and dislocation density
increases quickly leading to an increase in the yield strength. During this stage, the solute
hardening effect decreases with decreasing solute concentration in the matrix. However, the
decrease in solute hardening is less than offset by the increase owing to precipitate and
dislocation hardening. So the overall strength increases as the precipitate grows. At stage
III, yield strength reaches maximum value as there is no further decrease in solute
hardening and no increase in aging hardening. With increasing aging time (stage IV), the
yield strength begins to decrease because of the coarsening of precipitates.
For solid solution strength (
ss), it can be calculated by the average concentration of
solute in the matrix [28]:
�ss t
n
C� 3 , where � and n3 are material constants related to
the discrepancy in crystal lattice between the solute atoms and matrix, and Ct is the
current average concentration of solute. Shercliff and Ashby [28] found the correlation
between solute concentration and relative volume fraction of precipitates, which can be
expressed as f C C C Cv l t l e� � �( ) ( ), where Cl is the current solute concentration in
the matrix and Ce is the equilibrium solute concentration and approximates to zero
theoretically, fv denotes the relative volume fraction of precipitates that will be discussed
later. Combining above two equations gives the expression of solid solution strength
ss ss v
n
C f� �( )1 3 [Eq. (3)].
Precipitation (aging) hardening embodies the contribution of precipitates to yield
strength of the material. In this paper, the main strengthening precipitates ( �S phases) are
always assumed to be unshearable. So these precipitates are bypassed by dislocations by
the Orowan bowing process. Many strengthening models related to precipitates are made
based on the above assumptions. Through analyzing the interaction force between
precipitates and dislocations together with the mean distance between precipitates, Li et al.
[16] deduced the expression of precipitation hardening as
ppt ppt
m
vC L f� 5 0 5. . However,
ppt only involves effects of mean length and relative volume fraction without consideration
of the variation of precipitate thickness. Because of the non-coherent interface between
precipitates and matrix on the peripheral plane of precipitate, the ability to receive atoms
from the matrix to form second phases is strong. While for the plate planes of the
precipitate, this ability is relatively weak due to the semi-coherent/coherent relationship.
Although the growth rate along the length direction is larger than that of the thickness
direction, the morphology of precipitate varies along length and thickness direction
simultaneously. So the aspect ratio is employed in Eq. (4) to describe precipitation
strengthening more accurately.
Dislocation strengthening is also called work hardening. It is essentially caused by
increasing dislocation density in deformation process. The formula of dislocation
strengthening is illustrated in Eq. (5), which is a function of normalized dislocation density
( )� defined by [15]: � � � �� �( )i m, where � is the current dislocation density of Al
matrix, � i is the dislocation density for the virgin material (the initial state), and �m is
the maximum dislocation density that the material could have. When � �i m�� , the
average dislocation density � varies from 0 to 1. The evolution of average dislocation
density is summarized in Eq. (9). The first term in Eq. (9) reveals the development of
dislocation density due to creep deformation and the dynamic recovery. The second term
gives the effect of static recovery in the dislocation density at higher temperature.
Equations (6)–(8) denote the evolution of microprecipitates. Three parameters
including relative volume fraction fv , average length L, and aspect ratio q are adopted to
quantitatively characterize the precipitates. Relative volume fraction is defined by
f f fv v v� * , where fv is the current volume fraction of precipitates and fv
* is maximum
volume fraction. Based on the understanding of precipitation kinetic, the evolution of
34 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 1
Y. L. Yang, L. H. Zhan, and X. L. Xu
relative volume fraction is established in Eq. (6). The model not only includes the effects of
precipitate growth, but also the effect of dislocation density.
Equation (7) refers to the development of the mean length of precipitates. The form of
Eq. (7) consists of two parts. One part is that precipitates grow under a pure aging
condition when dislocation density � approaches to zero. The other is that when external
stress is applied, the dislocation density could have impacts on precipitates length, which is
controlled by two parameters k1 and n2. Moreover, the parameter, L* , the peak length of
precipitate, is introduced to avoid the infinite increase of precipitate. This is consistent with
the fact that precipitate in the matrix cannot grow continually because of the limitation of
depletion of solute atoms from the aluminum matrix to precipitates. It is found that the
value of aspect ratio q of precipitates is not always the constant in aging process [29, 30].
The change of aging processing conditions (aging temperature, time, material’s constituent,
etc.) has significant influences on aspect ratio and hence on yield strength. So the variation
of aspect ratio during the isothermal aging is built in Eq. (8), i.e., q C k t t� �3 2
2exp[ ( ) ],*
where t* is the time to obtain maximum aspect ratio, t is the current aging time, and C3
is material constituent dependence of constant.
2.3. Constitutive Model Verification. To verify the accuracy of the above constitutive
equation set, the material constants for 2124 aluminum alloy are determined with the
experimental results from the test. However, it is difficult to directly determine the material
constants because there are too many constants and the number of constants is more than
constitutive equations. It is better to use an optimization algorithm to get a set of
approximate optimal solutions. In this study, an optimization algorithm called particle
swarm optimization (PSO) was adopted to solve the material constants. Nevertheless, it is
found that the fitness process often converges to local best but rarely to global best.
Therefore, a step-by-step fitting process is used to calculate the material constants from
microstructures to macroproperties aspect. First, the material constants related to aspect
ratio, mean length and relative volume fraction were determined respectively. Second, the
material constants related to macroperformances were determined that are yield strength
and creep strain.
The derivation of material constants from the experimental data is evaluated by the
fitness value when using PSO algorithm. The form of fitness value is listed as
Fit
n
x x
x
T
Ti
n
�
�
�
�
1
1
,
where x and xT denote the computed and experimental data respectively, and n is the
number of data points.
The less the fitness value, the less the derivation of material constants from the
experimental data. The values of all material constants are listed in Table 1 and the
comparisons of experimental values and fitness curves are illustrated in Fig. 6 corresponding
to (a) aspect ratio, (b) mean length, (c) relative volume fraction, (d) yield strength, and
(e) creep strain with fitness values being respectively 6.80, 4.65, 8.98, 1.73, and 14.25%.
With regard to Fig. 6e, the discrepancy between the fitting curve and experimental data in
the latter stage of creep tested under 250 MPa/185�C is relatively large. This is because that
the specimen exhibits the tertiary creep at high temperature and high stress level, while the
modeling process is not taken the nucleation and growth of microvoid that can bring about
creep damage into account. Overall, the good agreement between experimental data and
computed values confirms that the established constitutive models can well describe
macrocreep behaviors, the variation of yield strength and precipitates evolution for
Al–Cu–Mg alloys.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 1 35
Constitutive Modeling for Al–Cu–Mg Alloy in Creep Aging Process
36 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 1
T a b l e 1
Material Constants within Constitutive Equation for 2124 Aluminum Alloy at 185�C
Constant Value Constant Value Constant Value
A 8 64 10 6. � � h�1 C2 0.067 n1 1.09
B 1.64 C3 27.2 n2 1.21
0 93 MPa C4 179 n3 1.96
k1 1.45 C5 0.01 n4 0.76
k2 �0 03. h�2 m1 3.72 n5 0.32
Css 132 MPa m2 1.95 L* 623 nm
C ppt 27.8 MPa m3 0.95 t* 8 h
Cdis 129.3 MPa m4 1.14
C1 6 95 10 5. � � m5 0.21
a
c
e
Fig. 6. Comparisons of experimental and
computed aspect ratio (a), mean length (b),
relative volume fraction (c), yield strength (d),
and creep strain (e).
b
d
Y. L. Yang, L. H. Zhan, and X. L. Xu
C o n c l u s i o n s
1. With increasing aging time, the mean length of strengthening precipitates increases
gradually, the aspect ratio varies in a low-to-high-to-low manner and precipitates volume
fraction increases to the peak value and remains unchanged.
2. A unified mechanism-based creep constitutive equation set for Al–Cu–Mg alloys is
established, which is a sinh function of externally applied stress and yield strength of
materials. The material’s yield strength links the microstructures with macroperformances.
The models are not only suitable for plate-like precipitate but also suitable for rod-like
one.
3. The good agreement between experimental data and computed values obtained by
constitutive equation set confirms that the established constitutive models can well describe
creep behaviors for Al–Cu–Mg alloy.
Acknowledgments. This research was supported by the National Basic Research
Program of China (Grant No. 2014CB046602), the Key Program of National Natural
Science Foundation of China (Grant No. 51235010), and Ph.D. Programs Foundation of
Ministry of Education of China (Grant No. 20120162110003).
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Received 03. 08. 2015
38 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 1
Y. L. Yang, L. H. Zhan, and X. L. Xu
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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
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/IncludeSlug false
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(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AddBleedMarks false
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/AddRegMarks false
/ConvertColors /ConvertToCMYK
/DestinationProfileName ()
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/Downsample16BitImages true
/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements false
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/MultimediaHandling /UseObjectSettings
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/PDFXOutputIntentProfileSelector /DocumentCMYK
/PreserveEditing true
/UntaggedCMYKHandling /LeaveUntagged
/UntaggedRGBHandling /UseDocumentProfile
/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice
|