Aluminum microstructure evolution and

effects on mechanical properties in

quenching and aging process

by

Guannan Guo

A dissertation submitted to the faculty of the

Worcester Polytechnic Institute
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
in
Manufacturing Engineering
by

——————————————

August 2017

Approved：
————————————————

Richard D. Sisson, Advisor

George F. Fuller Professor Mechanical Engineering
Director of Manufacturing and Materials Engineering

Yiming Rong, Co-Advisor

John W. Higgins Professor of Mechanical Engineering
Associate Director of Manufacturing and Materials Engineering Program
Director of Computer-aided manufacturing Laboratory (CAM Lab.)
Abstract
High strength aluminum alloys are recently widely used in aircraft, automobile and
construction industry fields. Typical T6 heat treatment process can be applied to
improve the heat treatable aluminum alloy in order to facilitate the formation of prime
strengthening precipitate phases. Critical steps in T6 heat treatment process include
solution treatment, quenching and aging. Due to high thermal gradients in quenching
process and aging process, large thermal stress will remain in the matrix and may
bring unexpected deformation or distortion in further machining. Therefore, in order
to predict the thermal stress effects, constitutive model and precipitate hardening
model are needed to simulate the mechanical properties of alloy.

In this dissertation, an optimized constitutive model, which is used to describe the

mechanical behavior during quenching and intermediate period of quenching and
aging process, was given based on constitutive models with Zenor-Holloman
parameter. Modification for constitutive model is based on the microstructure model,
which is developed for the quenching and aging processes. Quench factor analysis
method was applied to describe the microstructure evolution and volume fraction of
primary precipitate phases during quenching process. Some experimental phenomena
are discussed and explained by precipitate distributions. Classical precipitate
hardening models were reviewed and two models were selected for Al-Cu-Mn alloy
aging treatment. Thermal growth model and Euler algorithm were used to improve the
accuracy and the selected precipitate hardening models were validated by yield stress
and microstructure observations of Al-Cu-Mn aging response experiments.
Acknowledgements
Firstly, I would like to express my sincere gratitude to my advisor Prof. Richard D. Sisson
for the continuous support of my PhD study and related research, for his patience,
motivation, and immense knowledge. He helped me to finish my final defense and his
guidance helped me in all the time of research and writing of this thesis. I could not have
imagined having a better advisor and mentor for my PhD study. I also thank Prof. Rong
who accepted me as PhD candidate firstly and gave me opportunities to work in General
Motors and Tsinghua University, where I accomplished my most important projects of
my PhD thesis.

Besides my advisors, I would like to thank the rest of my thesis committee: Prof. Liang
Prof. Wang, for their insightful comments and encouragement, but also for the hard
questions which incented me to widen my research from various perspectives.

My thanks also goes to Dr. Wang from General Motor who provided me an opportunity to
join their team as intern, and who gave access to the laboratory and research facilities.
Without they precious support it would not be possible to conduct this research.

I thank my fellow colleagues in for the stimulating discussions, my friends and all people
who help me in research, lab and academic works. In particular, I am grateful to my wife
Jia Wang and my family for supporting me spiritually throughout writing this thesis and
my life in general.
Contents

Introduction ......................................................................................................................... 5
Literature reviews ........................................................................................................... 10
1 Additional elements effects on microstructure and mechanical properties of Al-Cu
system alloy ...................................................................................................................................10
2 Heat treatment processes effects on microstructure and strengthening mechanism
of aluminum alloy ........................................................................................................................11
3 Strengthening mechanism of deformation and constitutive models ............................16
4 Precipitate hardening models of aluminum alloys modified by thermal growth
model...............................................................................................................................................23
Competitive relationship between thermal effect and grain boundary
precipitates on the ductility of an as-quenched Al-Cu-Mn alloy ..................... 30
Temperature-dependent constitutive behavior with consideration of
microstructure evolution for as-quenched Al-Cu-Mn alloy .............................. 55
A BRIEF REVIEW OF PRECIPITATION HARDENING MODELS FOR ALUMINUM
ALLOYS ................................................................................................................................ 76
Modeling the yield strength of an A356 aluminum alloy during the aging
process ................................................................................................................................ 84
Contribution ...................................................................................................................... 96
Introduction
With the development of aerospace and aircraft technology, widely applications of
high-strength aluminum alloy can satisfy the demand of high strength, high corrosion
resistance and high toughness. [1] Since most of the applied aluminum alloys are
heat-treatable, appropriate heat treatment processes can dramatically improve their
strength, ductility and other mechanical properties. Al-Cu-based alloys have good
mechanical properties at both low and high temperature ranges and are thus widely
used in manufacturing industries. Several high-tech components, particularly in
aerospace and aircraft industries with high standards of structural stability and
mechanical properties under severe conditions, are composed of Al-Cu-based alloy.
Heat-treatable Al-5%Cu-0.4%Mn alloys, denoted as ZL205A, not only improve the
ductility, but also increase the yield strength by refining the primary strengthening ’
phase (Al2Cu) size and by homogenizing the distribution of precipitate particles in the
T6 heat treatment process. Nowadays, ZL205A are widely used to make structural
components in aerospace, aircraft and automobile industries. The common failure of
work piece made by ZL205A is crack propagation in quenching process, where local
deformation expands beyond maximum ductility. Microstructure variation in heat
treatment process including recovery, recrystallization and rearranged dislocations
can increase the ductility, which is usually accompanied with reduction of strength.
[2-6] Quenching process can also affect the microstructure of ZL205A, and the
morphology and distribution of precipitates formed in this process also have
significant effects on ductility. The primary phases in ZL205A include Al2Cu,
Al20Cu2Mn3 and Al3Ti phases, where Al2Cu are generally considered to be primary
strengthening phases in most Al-Cu based alloys. In order to obtain fine and uniform
distribution Al2Cu precipitates particles; Mn additions introduced into system are
used for refining small Al2Cu particles by increasing nucleation sites for nucleus.
Besides, it can also accelerate the formation of Al3Ti by improving recrystallization
resistance, which in turn leads to good strength at elevated temperatures. [7,8]
Moreover, different intermetallic phases have different impact on the ductility of
materials. [9-11]

Since ductility has closed relationship with precipitates under different temperatures,
the outcome of microstructure characteristics of heat treatment process should be
quantified in order to better predict and monitor mechanical behaviors. Several
researchers have studied the effects of the T6 heat treatment on the yield strength of
aluminum alloys with similar chemical concentration and their mechanical behavior
at both high and low temperatures. [12,13] More works should have discussed
mechanical properties such as strain and flow stress variation during the quenching
process, where the factor of large temperature gradients should be considered. The
cooling rate varies a lot among different locations due to the large and complex
structure of the components, creating additional thermal stresses in the matrix.
Although maintaining materials at low temperature would eliminate the effects of the
natural aging process on the precipitates, the reheating process to the artificial aging
temperature may allow some precipitate particles to nucleate and grow ahead of the
aging process. Previous studies of flow stress in quenching ignored the phase effects
on aluminum alloy, while assuming microstructure was identical and homogeneous.
[14, 15] However, the precipitation process can be initiated during quenching process
for aluminum alloy, [16] which cannot be avoided in all alternative methods. [17]
Al2Cu precipitates nucleation occurred from super-solid solution due to different
solubility of solute elements and aluminum, which was accelerated by inherited
vacancies from casting process. The vacancies aggravated the reduction of elongation
and ductility of materials. [18] It is also reported that Al2Cu phase formed in interface
layer severely affected bond strength of clad composite and a number of micro-cracks
caused by hard and brittle Al2Cu phase, which made a contribution to low elongation.
[19] Besides, T dispersoids have great thermal stability and easily nucleate during
casting and solutionizing periods, which hardly dissolve into the matrix during
consequent reheating and aging process. Due to complexity of structure and actual
quenching methods, formation of T dispersoids before and during quenching process
can hardly be avoided. [20] Therefore, there is an attempt to clarify the microstructure
evolution of ZL205A during the reheating process and reveal the comprehensive
effects of microstructure on deformation and fracture behaviors in this thesis.

The first component of this thesis is concerned of the mechanical properties of

ZL205A (Al-Cu-Mn alloy) such as flow stress over a wide temperature range. A
modified constitutive model has been developed in terms of temperature effects on
phase transformation. Microstructure observations of dislocations and precipitate
phases at different periods of heat treatment process are used to determine the effects
on flow stress and ductility. Isothermal tensile tests have been accomplished over a
wide range of applicable temperatures and strain rates. Arrhenius-type constitutive
model has been developed and calibrated with experimental data. Therefore, a better
constitutive model can be used to simulate ZL205A mechanical behaviors under wide
temperature ranges and to monitor quenching deformations. Besides, the
corresponding microstructure observation and analysis helps explain the strain
sensitivity and strain hardening trends varied at low or high temperature, and
unexpected variation of the ductility of the alloy.

Another component of this thesis is focused on the precipitate hardening models of

A356. Since A 300 series, aluminum–silicon series alloys have good fluidity, high
strength and maintaining reasonable ductility, this kind of alloys are widely used in
automobile manufacturing fields such as engine block, cylinder head and suspension
components. Typically, silicon gives good fluidity and magnesium provides high
strength by forming precipitates through proper heat treatment. T6 heat treatment is
generally used in A356 production in order to obtain higher yield strength, while with
the scarification of ductility. Besides, after aging process, the sizes of components
made by A356 could expand, which leads to serious distortion. One possible reason is
the growth of precipitate particles and the phase transformation of precipitate phase
from coherent phases to incoherent stable phases. Therefore, a microstructure model
of precipitate particles should be established. Many well-known precipitate-hardening
models followed similar strengthening mechanism that the interactive motion
behaviors of mobile dislocations and precipitate particles determine the contribution
to final strengthening effects. [21-27]The particles could be sheared or surrounded by
motive dislocations, which leads to the increase of dislocation density. The increasing
dislocation density provides more inhibitions to plastic deformation. Since the criteria
of separating these two strengthening modes are the precipitate particle size, accurate
microstructure models of the precipitate particles should be studied through aging
process.

During the aging process, the aging temperature and time are two critical values to
simulate precipitate particles evolution under thermal dynamic principles and volume
diffusion of solute element. For the microstructure part of these precipitate hardening
models, there are two major methods to deal with precipitate particles evolution.
When assuming all particles as spherical shape, the radius of precipitate particles can
either obtained as representative of mean value of all particles, which is called mean
value approach method; [21, 23, 25, 26] or categorized particles into small groups
with similar radius, which is called discrete value approach method. [28] Besides,
when actual morphology of precipitate particles was considered, there would be
different strengthening effects on different orientations, and different growth kinetics
on different orientations which makes calculation of single particle more complicated
than spherical particle assumption. The volume fraction of primary precipitate phase
is usually increased exponentially and remains constant when reaching peak aging
state. However, the prolonging aging process can activate the phase transformation of
precipitate particles from semi-coherent phase to incoherent stable phase, which
reduces the strengthening effects. Therefore, there is a need to review classical
precipitate hardening models and find appropriate models to model the distortion of
component made of A356 during aging. In this thesis, a microstructure model of
precipitate phase of A356 is discussed when applying thermal growth model to the
primary strengthening phases. The results of modified volume fraction of precipitate
phases are used in the selected precipitate hardening models and a thorough
comparison with experimental data of A356 aging data is given.

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Literature reviews

1 Additional elements effects on microstructure and mechanical properties

of Al-Cu system alloy
Since precipitate phases are critical factors related with final yield strength and other
properties, optimizing morphology and distribution of precipitate phase facilitate well
strengthening effects on Al-Cu alloy. Additional elements such as Mg, Mn, Si and Sn
are introduced into Al-Cu system can adjust Al2Cu nucleation and growth by
affecting nucleation kinetics, volumetric misfit strain, shear strain and interfacial
energy of θ’ precipitate phases. For example, {1 0 0}α plate of metastable Al2Cu in
2XXX series aluminum alloy play the dominate role in strengthening, and {1 1 1}α
plates of metastable Al2Cu are founded in Al-Cu alloys microalloyed with Mg and Ag.
Li-modified Al-Cu alloys has {1 1 1}α plates of T1 (Al2CuLi) to strength and AA2195
is considered to be strengthened by uniform distribution of T1 plates of large aspect
ratio. [1, 2, 3] Sn (roughly 0.01~0.02at%) is introduced into Al-Cu system to increase
the transaction from θ’’ to θ’ and perform rational orientation and uniform θ’
precipitate phases. It was reported that for Al-1.7at%Cu, 0.01at% Sn dramatically
shorten the time to reach peak hardness and increase peak hardness, and solute Sn is
more effective to involve into θ’ nucleation than Sn phases. [4] Sc is also an important
solute element that is used to improve Al-Cu mechanical properties. Al3Sc dispersoids
are specially referred to the Al3Sc particles formed in high temperature processing of
the alloy, such as homogenization, hot-rolling or extrusion, which are considered to
stabilize the grain boundary of Al and hence inhibit the recrystallization. Al3Sc
precipitates referred to those formed during aging process after solution treatment,
however, due to the limited decomposition of Al3Sc from super solid solution, the
strengthening contribution from Al3Sc usually be restricted, and Al3Sc hardly
coexisted with other strengthening precipitates due to its temperature regime. [5]

Among the additional elements, manganese is introduced into Al-Cu system to obtain
fine-grain polycrystalline and improve its strength especially at high temperature. Mn
addition had already been proved to refine θ’-Al2Cu and retard Orowan coarsening.
Related DSC analysis of compare between Al-Cu alloy with and without Mn addition
is accomplished by previous researchers, which proved the formation of θ’-Al2Cu is
largely postponed but dominantly enhance the strength just before peak aged.
Diffusivity of Mn is closed to Cu that leads to easily attracting Cu atoms toward Mn
atoms. Thus, quenched vacancy or clusters formed after quenching or beginning of
aging does not efficiently work to form GP zones. [6] T dispersoids have great
thermal stability and provide preferred nucleation sites for other possible precipitate
phases. Even though addition of Mn can efficiently reduce the coarsening stage of θ’-
Al2Cu, consideration of phase transformation from θ’-Al2Cu to stable phase losing its
coherency and strengthening should be included. However, the introduction of Mn
into system also brings undesirable effects on mechanical properties, especially on the
resistance to crack growth. [7, 8] T dispersoids formed as grain boundaries also
decrease the binding capacity of neighbor grains, which leads to micro voids or cracks
easily occurred. Thus, for solution treatment, quenching and aging periods, the
evolution of T dispersoids or precipitates should be taken into consideration in order
to predict deformation behaviors of ZL205 alloy.

2 Heat treatment processes effects on microstructure and strengthening

mechanism of aluminum alloy
1) Casting

Typical heat treatment process concludes casting, solutionizing, quenching and aging
steps for most of heat treatable aluminum alloy. Casting process is critical to the
whole heat treatment process since most of microstructure characteristics of matrix
are set in this period. Secondary dendrite arm spacing, SDAS, is often applied to
evaluate the coarseness of the microstructure after casting. [9] Dendrite structure has
lower copper content in central part. Moving toward the outside of the arms, which
corresponds to metal freezing later, the copper content increases and combines with
Al to form the Al-α + Al2Cu eutectic mixture. [10] DAS can also affect micro-
segregation of alloy element and distribution of second precipitate phase or micro-
porosity. Cooling rate of solidification process is the critical parameter that
determines length of DAS. Proper selection of pouring temperature, cooling
conditions and chemical distribution should be considered in order to guarantee
satisfied casting quality. Besides, due to long freezing range of Al-Cu based casting
alloys, micro-porosities are easily formed during solidification. The different cooling
rates of raw casting component lead to different thermal shrinkages. Micro-pores
cannot be filled with liquid alloy immediately and be left in the as-cast structure. With
the help of proper squeeze casting method, pressure are applied during casting process
to accelerate liquid alloy flow into micro-pores and fill them, which can dramatically
reduces such defects. [11]

For Al-Cu based alloy, homogeneous Cu distribution is important for Al-Cu based
alloy. Due to gravity casting and solubility of copper element, copper segregation
serious, while Mn or Mg segregation hardly occurs because of fully dissolved in α-Al
matrix. Inverse segregation is usually found in cast Al-Cu alloys owing to long
freezing range, which leads to large dendrite arm spacing with slow cooling rate. Low
temperature gradients also lead to coarsening dendrite skeleton and shrinkage induced
fluid transports the solute rich liquid away from the center of the casting back to the
surface. Such casting microstructures as dendrite arm and additional element
segregations will further affect following up mechanical properties in quenching and
aging such as precipitate phase morphology and distribution. For as-cast aluminum
alloys, yield strength is believed to be affected by copper content in α-Al grains and
increased amounts of intermetallic compounds, especially the fine intermetallic
presented at the α-Al grain boundaries. Since present project focus on quenching and
aging effects on mechanical properties, fine α-Al grains and short DAS are needed to
eliminate casting defects and negative effects on following heat treatment process.
Casting process parameters such as temperature gradient and cooling rate should be
well controlled.

2) Solutionizing
Method that introducing additional elements into the aluminum matrix in order to
produce second precipitate phases is termed as solutionizing. Particles formed at
casting and homogenization process will dissolve and additional solute elements will
diffuse into the matrix in solution treatment. These additional elements can affect the
pre-quenching process microstructure and also play as nucleation sites for further
precipitation in aging to strengthen the aluminum alloys leading to the significant
increase of ductility and decrease of yield strength because of less pining of the fine
intermetallic presented at the α-Al phase boundaries. It is worth to note that the solute
content in the primary α-Al phase is increased after solutionizing, which means
solution strengthening is enhanced. However, the alternations in microstructure from
other phases and at grain boundaries are more significant and effective, resulting in a
reduction of the yield strength and an increase of elongation. [12] In order to obtain
the fine super solid solution for quenching and homogenization nucleation sites for
aging, the effects of solutionizing on microstructure and mechanical properties should
be discussed and the process parameters of solution treatment should be optimized.

The objective of choosing proper solution process parameters is to maximum solute

element in aluminum alloys in the aluminum matrix. Solution temperature, which is
determined by solute element content, should be in the range that avoids overheating
or local melting and accelerates the dissolution of particles formed in pre-heating and
casting process as much as possible. Upper limit of the solution temperature is usually
closed to the eutectic temperature when considering the effects of grain growth,
surface effects and economy of operation. Solution time required in solutionizing
should guarantee sufficient dissolution and make the maximum solute elements solve
into matrix. It is dependent on the types of products, alloy systems, casting or
fabricating procedures used and thickness insofar as it influences pre-existing
microstructure. From the microstructure aspect, the coarseness of as-cast
microstructure determines the time needed for solution treatment. The finer as-cast
microstructure, the shorter solution time is needed. From the macrostructure aspect,
the larger the section thickness is, the longer holding time required, while thin
products just need several minutes to finish the solution treatment. Excessive
diffusion should be avoided when prolonging the solution time and particular
mechanical properties decreased with increasing solution time should be also
controlled. Since current project aims at studying thin wall components made by
ZL205, the solution process will be finished very soon to avoid unexpected
coarsening and mechanical properties reduction.

3) Quenching
Quenching process is used to obtain super saturated solid solution in order to facilitate
the further precipitation in aging. Different quenchants such as oil, water or polymer
solution are used to cooling materials to room temperature when providing suitable
cooling rate. Microstructure formation and distribution occurred in varied quenching
conditions have great effects on aluminum quenching behaviors and further effects on
aging process. Fink and Willey pioneered attempts to describe the effect of quenching
on properties of aluminum alloys. [13] They applied isothermal quenching techniques
to develop C-Curve for particular aluminum alloys. Even though their method worked
well when the cooling rates are uniform, it is failed to predict the effect of quenching
on aluminum alloys properties when the cooling rate varied considerably during the
quench. Since the properties are highly dependent on the microstructure evolution,
and such microstructure evolution determined by the material, quenching temperature
and time needs to be predicted by quenching process parameters. [14, 15, 16] Quench
factor analysis method is put forward to predict the volume fraction of precipitate
phase in quenching process instead of average cooling rate method. Time elapsed
during non-isothermal quenching process are divided into several short time steps,
which can be considered as isothermal quenching process. [17-21] Since quenching
process typically lasts very short, the growth or coarsening of these intermetallic
dispersoids or precipitate particles can be ignored. However, lack of considering
dispersoids formed in quenching process brings overestimation of primary precipitate
hardening phase’s volume fraction in simulation and negative effects on mechanical
properties.

4) Aging
Result of strengthening is dislocation motion is blocked by the formed precipitate
particles during the aging process due to change of solubility of alloy element. Since
the mechanical properties are concerned and highly dependent on the precipitate
phase distribution, the size and volume fraction of precipitate phases should know at
the peak-aged state. T6 heat treatment is widely used in heat treatable aluminum alloy
production to obtain peak yield strength. Binary Al–Cu alloy has following aging
sequence: α → α + GP zones → α + θ″ → α + θ′ → α + θ [22, 23, and 24]. The
aluminum solid solution is indicated by α, the metastable phases are indicated by θ′, θ″
and the stable precipitates by θ.
 Figure 2.1 The aluminum rich end of Al-Cu phase diagram
The super saturation of vacancies allows solute elements diffusion and lead to the
formation of GP zones. GP zone is firstly nucleated from the super solid solution; the
dislocation among the matrix can offer proper nucleation sites for GP zone, also the
grain boundary and other defects in the matrix can be available nucleation sites.
Typically the thickness of the GP zone is one or two atom layers and GP zone is fully
coherent with the matrix. After the formation of the GP zone, the next step is the
formation of transit phase. Some reports show that there exist GP II zone between GP
zone and the transit phase based on different material. GP II zone will generally
follow GP zones by forming a second layer parallel to the GP zone on the plane. [25]
Other authors consider GP II zones as an ordered phase with two Cu layers separated
by three Al layers. Mentioning about transit phase, there will be not only one phase
formation at this period, and the previous transit phase can transform to another type
of transit phase, with the change of composition and morphology. Meanwhile, the
orientation and morphology of the transit phase also alter their contribution to
precipitate strengthen. [26, 27] To simulate precipitate hardening the assumption of
the particle size should not be spherical shape, size and volume fraction of these
transit phases should be modeled precisely. They may alter the shape of precipitates
phase, magnitude and anisotropy of the interfacial energy, the different elastic
constants of matrix and precipitate and crystal structure. Detailed simulation models
will be given in the next section and compared.
Transit phase is considered to generate from GP zone and consume the original GP
zone space at the same time. The coherency relationship with the matrix now turns to
semi-coherent, which will produce large lattice deformation and strengthen the
material. The final phenomena in the aging process is the stable second phase
nucleation, growth and coarsening. Transit phase cannot maintain stable at room
temperature or in-service condition and spontaneously transform to the stable phase.
The non-coherent boundary and complicate semi-coherent boundary with the matrix
decrease the strength of the material. The formation of the stable phase will consume
the precipitate phase and the strength effect is reduced.

3 Strengthening mechanism of deformation and constitutive models

Deformation of ZL205A is one of the concerns in my proposal especially in heat
treatment process. Dislocation motions in matrix allow for deformation of material.
When the dislocation motion is inhibited and dislocation density is increased, the
movement of dislocation will be restricted due to the pile up of dislocation. Any
strengthening method is aiming at either improving the dislocation density or
hindering moving dislocations. [28] Work hardening mechanism, solid solution
mechanism, precipitate hardening mechanism and grain boundary mechanism are
general strengthening mechanisms, which will be discussed with actual heat treatment
periods to tailor their effects on yield strength and flow stress.

Work hardening method is usually used to improve strength and hardness in cold
deformation by introducing more dislocation that increasing dislocation density. The
interactive action with encountered dislocations will impede dislocation motion by
stress field generated by dislocations. Besides, cross dislocation lines may play as pin
point that also increase the barrier to the motion of dislocations. Cold working in the
interval between quenching and aging is considered to accelerate the aging response,
providing numerous sites for heterogeneous nucleation of precipitates. [29] In current
proposal, a cold working is avoided in order not to introduce work hardening effects
into heat treatment process from solutionizing to reheating process. Therefore, the
variation of aluminum grain size and precipitation formed in natural aging or
reheating process will be only related with temperature variations.
Solid solution strengthening is important for aluminum alloy since lots of additional
elements can be introduced into aluminum matrix to form substitution or interstitial
solid solution. The solute atoms can cause lattice distortion to increase yield strength
by impeding dislocation motion. Meanwhile, the stress fields caused by solute atoms
can interact with dislocations. Depends on actual size of solute elements, they can
interfere with neighbor dislocations by playing as potential obstacles. During solution
treatment process of current proposal, the major solute elements are Cu and Mn atoms.
T dispersoids, which has chemical compositions of Al20Cu2Mn3, are easily formed
with great thermal stability making them hardly dissolve in further reheating process.
Cu atoms can be observed to fully dissolve into matrix to obtain fine and
homogeneous solid solution。 Segregation of Cu and some vacancies can be expected
to be eliminated during solution treatment. Therefore, solution-strengthening effects
on yield strength and flow stress may be highly related with T dispersoids size and
volume fraction.

Precipitate hardening methods are widely used to strengthening heat treatable

aluminum alloy by precipitating second phases from super solid solution, when
solubility of solute element reduce during aging process. Precipitate particles play the
similar functions as solutes that pinning the motion of dislocations. With the
continuous of aging process, the precipitate phases will lose their coherency
relationship with the matrix so that the strengthening effects are weakened. Besides,
there are two interactive motions between precipitate particles and dislocations. When
the size of particles are small, the dislocation line will be cut when passing by the
particles; and when the size of particles are relative large, dislocation lines will be
looped or bowed leaving a closed dislocation ring. Both methods increase the
dislocation density that also increases the barrier of dislocation motion. In this project,
working hardening mechanism can be neglected because there is no significant cold
working occurred before or after the experiments. Grain boundaries effects on
strengthening are also remain the same based on the observation of stabilized
aluminum grain size during heat treatment process and isothermal tensile tests. Thus,
solid solution strengthening especially in quenching and reheating process and
precipitate hardening strengthening in aging process are major strengthening
mechanism that governing yield strength and flow stress.
In this study, constitutive models are studied to provide accurate deformation at wide
range of strain rates and temperatures when considering solid solution and precipitate
strengthening, and dynamic recrystallization and recovery softening mechanism.
Therefore, a proper constitutive model that combines the effects of process
parameters and thermal dynamic would be useful. Strain, strain rate and temperature
are three critical parameters determine softening and hardening effects on
deformation behaviors of aluminum alloy. Thermal softening and strain hardening are
considered separately so that the production of both factors and strain factor gives the
variation of flow stress. In this situation, the work of plastic deformation all converses
to heat. However, high strain rate leads to insufficient time for inner heat to disperse,
which leads to the competence between the thermal softening and strain hardening at
particular temperature and strain rate. Johnson-Cook empirical constitutive models
are selected to predict flow stress of ZL205 at high strain rate with wide temperature
range, which is proved to be successful in predicting such deformation behaviors,
while some of parts are not fully discussed. [30, 31] Related researches pointed out
that at higher strain rate, the flow stress would monotonously increase as temperature
rises, which is opposite to the variation at lower strain rate. [32] Dynamic strain aging
mechanism is used to explain this unusual phenomenon considering the new-formed
solute atoms that affect dislocation motion. This similar phenomenon founded in
ZL205 proved sensitivities of ZL205 to deformation time since the deformation
period is much longer than high strain rate situations, which does not have plenty time
for aging. Solute atoms transportation at some particular states may also be controlled
by forest dislocation, which is usually observed at low temperature deformation
circumstance. Therefore, the microstructure state of ZL205 alloy in selected strain
rate and temperature range should be examined firstly, and temperature terms in
constitutive models may be more complicate than the ones descripted in Johnson-
Cook model.

At low temperature range, the flow stress of materials usually has proportional
relationship with dislocation density, and mobile dislocation are easily tangled or
seized with forest dislocation thus reduce ductility and improve strength. The
microstructure at low temperature range is usually considered to be stable and no
phase transformation or precipitation is considered in this circumstance. While at high
temperature range, the factors that affect flow stress can be divided into two
subclasses, one is related with temperature and another is independent on temperature.
The maximum threshold stress is presented as maximum glide resistance force. When
the stress is larger than maximum threshold stress, continuous glide will occur. Strain
rate at this condition can be obtained from the product of Burgers vector, dislocation
density and velocity. However, the stress could be smaller than maximum threshold
stress and jerky glide, which is considered to be discrete glide compared with
continuous glide, will dominate the mechanical behaviors. Under this circumstance,
the total mobile dislocation density is composed of mobile dislocation and potential
dislocation, which will be activated by thermal fluctuation. The increment of strain
contributed by thermal fluctuation can be given as [33]:
𝛿𝛾 = 𝛾0 𝛿𝑃𝑡 (2.1)
where 𝛾0 is average strain together with dislocations taking part in process and 𝛿𝑃𝑡
represents the possibility of released contributed by thermal fluctuation. Then the
product of frequency factor and Boltzmann factor presented as Arrhenius term can be
obtained as:
∆𝐺
𝑃𝑡 = 𝑣𝐺 exp (− ) (2.2)
𝑘𝑇
Equation 2.2 follows the assumption that thermal fluctuation exceeding in magnitude
∆𝐺
is only described by Boltzmann distribution and 𝑘𝑇 should be small in order to neglect

the possibility of backward jumps. At last, thermal activation process is a statistical

process, which means the average value when counting dislocation density is
meaningful. Other factors like stress released by recovery and inhibited by
mechanical stress are also make a contribution to the increment of strain. When the
value of thermal fluctuation term is larger than other factors, the stress-strain curve
can directly give information about ∆𝐺 effects. In addition, ∆𝐺 used in this section is
not as the same as the common symbol G in thermodynamics. ∆𝐺 is called activation
free enthalpy and defined by:
∆𝐺 = ∆𝐹 − ∆𝑊 (2.1)
where ∆𝐹 is the Helmholtz free energy necessary for activation and ∆𝑊 is the
additional work that brought dislocation from stable position to unstable position.
 Figure 2.2 A glide diagram explanation of ∆𝐺, ∆𝐹, and ∆𝑊quantities.

Phenomenological description of activation energy when consider all obstacles are

box-shaped can be given as:
𝜎
∆𝐺 = 𝐹0 (1 − ) (2.2)
𝜏̂
where 𝐹0 represents total energy to activate glide without external energy. Short-range
obstacles are especially sensitive to thermal activation, therefore above equation can
be replace by a useful form:
𝜎
∆𝐺 = 𝐹0 {(1 − ( )𝑝 )}𝑞 (2.3)
𝜏̂
where two adjust parameters p and q are used to describe the profile of stress.
Empirically, 1/2 is chosen for p and 2/3 is chosen for q. In discrete glide system, the
glide resistance may be assumed to be proportional to the shear modulus and
activation area proportional to the square of the Burgers vector, thus the value of
activation energy ∆𝐺 can be calculated as:
𝜎
∆𝐺 = 𝜇𝑏 3 𝑔 ( ) (2.4)
𝜇
where 𝜇 is the shear modulus, which can be taken as a function of temperature and b
is Burgers vector. Above all, the strain rate is highly depended on activation energy
for hot deformation when thermal activation mechanism is the primary cause during
high temperature deformation of materials:
 ∆𝐺
𝛾̇ = 𝛾0 𝑣𝐺 exp (− ) (2.5)
𝑘𝑇
Then, when activation energy of hot deformation can be assumed to be constant
above half of the melting point temperature, and pre-exponential term of dislocation
participated into process is considered at various flow stress state, the above equation
can be written as:
𝑄
𝜀̇ = 𝐴𝐹(𝜎)exp (− ) (2.6)
𝑅𝑇
𝜎𝑛 𝛼𝜎 < 0.8
𝐹(𝜎) = { 𝑒𝑥𝑝(𝛽𝜎) 𝛼𝜎 > 1,2 (2.7)
[sinh(𝛼𝜎)]𝑛 𝑓𝑜𝑟 𝑎𝑙𝑙 𝛼𝜎
And Zener-Holloman parameter is used to represent the relationship of strain rate and
temperature:
𝑄
𝑍 = 𝜀̇ exp (− ) (2.8)
𝑅𝑇
Since constitutive equations used Zener-Holloman parameters well describe the
mechanical behaviors at different stress level by empirical relationship and apply
thermal activation mechanism to quantify the mechanical behaviors at high
temperature, it will be a good choice to apply in Al-Cu aluminum alloy system since
there will be precipitation and recovery occurred at high temperature deformation that
brings extra obstacle to dislocation motion, which is needed thermal activation
mechanism to predict the effects of precipitates and Al grain size. Arrhenius-type
equations have been applied to describe the compression deformation behaviors of
aluminum alloys especially at high temperature deformation or forming process. [34-
38] Hyperbolic law in Arrhenius-type equations gives both excellent descriptions at
high or low flow stress. Zener-Hollmon parameter gave the temperature compensate
strain rate in an exponent-type that helps give a comprehensive description of all
stresses level. [39] The model has been widely used to predict aluminum alloy such as
2124-T851, A356, Al-Cu-Mg alloy and Sn modified Al-Cu-Mg alloy. However,
tensile properties of aluminum alloy are lack of discussion. Previous study have
already pointed that the hot tensile deformation is the competence of work hardening,
dynamic softening and develops of microvoids or cracks. Microvoids or intermetallic
compounds may form before quenching which play as possible crack source and large
thermal gradient in quenching process can produce residual stress that destroying the
thermal stability of aluminum components. At low deformation temperature, dynamic
recrystallization softening hardly occurs and non-uniform microstructure combined
with large dislocation clusters caused by strain hardening will bring serious stress
concentration. While at high deformation temperature, dynamic recrystallization
softening will be enhanced by thermal activated mechanism.

For ZL205 aluminum alloy used in my project, Al grains, Al2Cu precipitate phase,
Mn and Zr additions formed as dispersoids may have effects on mechanical properties
such as fracture, hardness and strength. The variation change in microstructure will
dramatically affect the deformation mechanism and activation energy of hot
deformation should be related with temperature and strain rate. [40-43] In as-cast Al-
Cu aluminum alloy, larger grains usually bring coarser boundary, which leads to
significant deduction of ultimate tensile strength. Eutectic phase will produce fracture
soon because of fragility and thickness. Deformation can increase as the grain size
increase when the particles are small because not enough coarsening, however, larger
particles under other casting processes such as rheocast leads to excessively
coarseness of grain boundaries and therefore fracture occurs before large deformation.
Elongation properties also share the similar behaviors as deformation induced by
variation of grain boundaries or sizes. [44] Different distribution of Al2Cu precipitate
phase caused by different casting methods or additional elements also has an effect on
mechanical properties. Discontinuous but better distribution of the eutectic phase
caused by dendritic material microstructure presents better mechanical properties due
to finer control of this hard and fragile phase than smaller size but continuous and
thick eutectic boundaries. T dispersoids formed in casting and solutionizing also play
important role on mechanical properties in Al-Cu alloy with Mn additions by
stabilizing grain size at elevated temperatures and retarding recovery. [45] Formation
of T dispersoids at grain boundaries decreases the binding capacity of neighbor grains,
which leads to micro voids or cracks easily occurred. Such micro voids or cracks may
play as deformation source during following tension test and easily get fracture. It had
been reported that addition of Mn is used to form Al20Cu2Mn3 dispersion to increase
the resistance to recrystallization and improve damage tolerance by help homogenize
slip. [46] T dispersoids phases also entangled with dislocations and could be potential
dislocation source. Zr additions can form Al3Zr that also inhibit recrystallization.
Previous studies have already proved the T phase formed in Al-Cn-Mn system at
grain boundaries dramatically reduce the binding capacity, which leads to easily crack
formation and propagation, and mathematical model of T phase distribution and
morphology should be investigated and obtained by hot deformation process
parameters. Therefore, combined effects of above microstructure on mechanical
behaviors of ZL205 in whole temperature range should be studied.

4 Precipitate hardening models of aluminum alloys modified by thermal

growth model
Many experiment data have been studied to discover the relationship between the
precipitate and the component of alloy, aging time and temperature. In order to get a
comprehensive law of precipitate hardening mechanism of aging, some previous
models developed to describe the thermodynamic of precipitate particles’ size,
morphology and characteristics, for the size and volume fraction of precipitate phases
directly impede the dislocation motions. Shercliff H.R and Ashby.M.F firstly made an
attempt to combine the process parameters of aging process, such as composition of
alloy, time and temperature, to predict the yield strength, and this method is
successfully applied on 6000 series aluminum alloy. In the model, strengthening
contribution is mainly composed of solute atoms, shearable precipitates and non-
shearable precipitates. [47] A.Deschamps and his partners introduced microstructure
evolution into the precipitate hardening model, described in detail the nucleation,
growth and coarsening of the precipitate particles. [48] Because dislocations are
preferred sites for transit phase nucleation and growth; the precipitates at dislocation
were also discussed to modify the microstructure evolution model. Basic strength
mechanisms are classified as precipitation shearing and bypassing based on the
different size of the particles. And the morphology of the particles is assumed to be
spheroid for convenient calculation. Following fundamental equation are mostly used
in many precipitate hardening models for yield strength calculation:
𝑀𝐹̅
𝜎𝑝 = (2.9)
𝑏𝑙
Where M is the Taylor factor, b is the magnitude of Burgers vector and l is the mean
effective particle spacing in the slip plane along the bending dislocation. [49] Based
on dislocation strength mechanics and morphology, the main two parameters to
determine the final yield strength after aging come to the radius and the volume
fraction of the precipitates. And from these two parameters we can calculate the 𝑙 and
F then get the value of yield strength.
B. Raeisinia and W.J. Poole and etc. used the volume fraction of the different
dominated precipitate strength particles to calculate yield strength. [50] However,
they ignored the size of these particles and assume the homogeneous size of the
precipitates phase. S. Esmaeili and D.J. Lloyd calculated the relative volume fraction
of precipitates with aging time, and then dedicated the value of radius of particles.
This paper considered the actual morphology of precipitates so that the simulation
will be more accurate. [51] On the other hand, since the actual morphology of
precipitates are not usually spheroid, G.Liu gave a more accurate and realistic
description of the precipitates morphology, separating into plate/dis and rod/needle
shape, then applied the peak-aged state to derive the volume fraction of these
practices. And this method was proved to work well when applying to Al-Cu-Mg
alloy and Al-Mg-Si alloy. [52] In M. Song paper, he put forward that at the first of the
aging process, the dislocation cutting mechanism might play a prime role on
dominating the strength. [53] Therefore, he made a modification of G. Liu model and
considers the increment of yield strength is composed of cutting and bypassing
mechanism.

However, in the precipitate hardening model, the microstructure evolution is assumed

to be no contact with other growing atoms and therefore the growth will not affect the
neighbor particles growth. In reality, at the beginning of the aging process, the solute
atom has the maximum super solution and the diffusion rate of solute atom is large.
With prolonging the aging time, the concentration of solute atoms and the precipitate
phase growth rate will decrease, and this will bring the deviation of the model and
experiment. Therefore, we need the accurate prediction of the growth rate of the
precipitates in the aging process. In Deschamps and Brechet, they separated the whole
microstructure evolution into nucleation, growth and coarsening three steps. They
calculated different nucleation rates and growth rates for each step and use them to
get the density and the radius of the precipitate particles. Although they gave a criteria
to identify the critical radius to separate nucleation, pure growth and coarsening, it is
difficulty in determining when the growth and coarsening happening. In my project, I
firstly made attempt to follow the method described in Ford’s patent studying Al-Cu
system. In the patent, the growth rate can be obtained from the volume change and the
phase fraction. The transformation among precipitate phases can produce the volume
change because of the diffusion of precipitate element from the different transit
phases. When applied T6 heat treatment, thermal growth of the prime strengthen
phase Al2Cu (termed with θ’ in the patent) is not completely growth to its maximum
size when it turns to stable phase (termed with θ). And with the time prolonging, the θ
will coarse and consume θ’, which lead to reduce the thermal growth effect. This
model can also predict the non-isothermal aging process. In the end, the dimension
change vs. Cu fraction can be determined from the model. The relative parameters can
get from TTT diagram and equilibrium state phase fraction. [54] In this project, S.
Esmaeili’s model and Myhr’s model have been chosen to be the reference models and
try to modify them with thermal growth model. As mentioned before, the data needed
of both models are the volume fraction of the precipitates, which we can realize by
TEM or SEM data.

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Mechanics, 1056789516683540.

Competitive relationship between thermal effect and grain

boundary precipitates on the ductility of an as-quenched Al-
Cu-Mn alloy

Wenguang Wang1,2; Gang Wang1,2,1; Guannan Guo3; Yiming Rong4

Abstract
The ductility of an Al-Cu-Mn alloy is typically characterized by fracture strain, and is
influenced by experimental temperature and its microstructure. Previous researches
show that the ductility increases with the temperature and decreases with the strain
rate. However, based on the results of isothermal tensile tests of as-quenched Al-Cu-
Mn alloy in this paper, it was found that the ductility decreased apparently
(approximately 90% under strain rate of 0.001/s) at a medium temperature range
(573K – 673K), and gradually reincreased to its original level at higher temperature.
A competitive relationship between temperature softening and grain boundary T
precipitation was proposed to account for the unusual variation of ductility. In
addition, a ductility model based on the competitive relationship was deduced to
quantify the evolution of the fracture strain for the as-quenched Al-Cu-Mn alloy, and
validated by the experimental results.

Keywords
Ductility; Grain boundary precipitation; Thermal softening; Quenching; Al-Cu-Mn
alloy

1
State Key Laboratory of Tribology & Institute of Manufacturing Engineering, Department of
Mechanical Engineering, Tsinghua University, Beijing 100084, PR China
2
Beijing Key Lab of Precision/Ultra-precision Manufacturing Equipments and Control, Tsinghua
University, Beijing 100084, China
3
Department of Manufacturing Engineering, Worcester Polytechnic Institute, Worcester, MA,
01609, USA
4
Mechanical and Energy Engineering Department, South University of Science and Technology of
China, Shenzhen, 518055, China
Corresponding author:
Gang Wang, State Key Laboratory of Tribology, Tsinghua University, Lee Shau Kee S&T Building
A1003-3, Beijing, China. Email: gwang@tsinghua.edu.cn
Introduction
The Al-Cu-Mn aluminum alloy exhibits satisfactory mechanical properties at different
temperatures, which makes it widely used as a structural component in the automobile
and aero-astronautic industries (Belov et al., 2014; Li et al., 2011; Toleuova et al.,
2012; Wang et al., 2016). In the production process, workpieces of Al-Cu-Mn cast
alloy often fail due to the propagation of quenching cracks, where local deformation
expands beyond fracture strain (Chen et al., 2012; Ye et al., 2014).

Ductility is generally affected by the experimental temperature, which is defined as

the fracture strain. Deformed grains at an elevated temperature can be recovered and
recrystallized with rearranging dislocations, thus generating a new set of un-deformed
grains, which are normally accompanied by a reduction in the strength of the material
and a simultaneous increase in the ductility(Besson, 2009; Lin and Chen, 2011; Liu et
al., 2003; Westermann et al., 2016; Zhou et al., 2012). In addition, the microstructures
of alloys, especially the pattern and precipitation position of intermetallic phases,
have a significant influence on the ductility(Chen et al., 2015; Cvijović et al., 2011;
Horstemeyer et al., 2000; Poole et al., 2005; Westermann et al., 2014). In general, the
fracture strain increases with increasing temperature and decreases with increasing
strain rate (Liu and Fu, 2014; Raj and Ashby, 1975; Rakin et al., 2004; Wang et al.,
2010).

The main phases of Al-Cu-Mn alloy include Al2Cu, Al20Cu2Mn3, and Al3Ti phases.
Al2Cu phases are the primary strengthening precipitate phases with fine and uniform
distribution (Gao et al., 2016; Samuel, 1998; Samuel et al., 1995). Mn additions in the
alloy can dramatically refine phases, retard Orowan coarsening, and improve
recrystallization resistance by forming T dispersoids (Al20Cn2Mn3)(Li et al., 1992;
Zupanič et al., 2015). The trialuminide intermetallic Al3Ti, which has a tetragonal
structure and low symmetry, is one of the reasons for the poor ductility of the material,
although it may improve the strength of Al alloys at elevated temperatures(Birol,
2007; Chang and Muddle, 1997). Generally, as a price to pay for strengthening the
alloy by inhibiting the movement of dislocations, precipitate phases usually lead to a
slight decline in the fracture strain. Moreover, different intermetallic phases have
different impacts on the ductility of materials (Simmons et al., 1978; Vasudevan and
Doherty, 1987; Zehnder and Rosakis, 1990).
In this paper, the stress-strain curves of the as-quenched Al-Cu-Mn cast alloy at
different temperatures and strain rates were studied. An interesting experimental
phenomenon was observed. The ductility of the alloy, beyond all expectation, did not
increase as the temperature increased but showed some regularity. Therefore, this
study attempted to explain the unexpected variation of the ductility of the alloy. The
variation was explained through an experimental study and theoretical analysis on the
different effects of temperature and intermetallic phases.

Materials and Methods

Al-5%Cu-0.4%Mn (called ZL205A via the Chinese national norm) is a typical cast
Al-Cu-Mn aluminum alloy with high strength and ductility (tensile strength
Rm  512 MPa ; elongation at fracture A5  7.0% ) after heat treatment. The detailed

chemical compositions are given in Table 1. Inevitably, there are some impurities,
including Fe, Si, Mg, and Zn.
Table 1. Main chemical composition of ZL205A alloy (in wt.%)
Element Cu Mn Ti Zr Cd B V Al
wt.% 4.6- 0.3- 0.15- 0.05- 0.15- 0.005- 0.05- Bal.
5.3 0.5 0.35 0.20 0.25 0.006 0.3
The material used for the tensile tests was produced by low-pressure die casting,
which was then examined by non-destructive X-ray detection to guarantee that no
serious casting defects would occur. Qualified samples were heat-treated, as shown in
Figure 1. The samples were machined into rod shapes with two screw thread ends.
The detailed dimensions are given in Figure 2. The samples were firstly solid
solutionized at 813K for 10 hours in a Muffle furnace and then quenched at 298K
(room temperature) in 11% UCON™ Quenchant A to obtain a supersaturated α-Al
solid solution. Secondly, the samples were heated to test temperatures and held for
five minutes to make the temperature uniform. Afterward, the samples were stretched
in a tensile machine. The tensile tests were conducted at six different temperatures
from room temperature to 773K (298K, 373K, 473K, 573K, 673K, 773K) and under
three different strain rates (0.001/s, 0.01/s, 0.1/s). The parameter scopes were
basically in accordance with the practical situation of heat treatment process. Three
samples were used to guarantee good repeatability under every experimental
condition.
After drawn to fracture, the samples were cooled down quickly at cold water to retain
the microstructure during the tests. Then, the tensile samples were cut into proper
dimensions and washed using an ultrasonic wave cleaning machine. Subsequently, the
tensile fracture surfaces and inner microstructures were observed by a scanning
electron microscope (SEM). The surfaces for inner microstructure observations over a
distance of 5 millimeters from the fracture surfaces were polished without any
chemical etching.

Figure 1. Thermal history used in the tensile tests Figure 2. Dimensions of the tensile samples

An INSTRONTM 5985 equipment was used for the tensile tests, and a FEI QuantaTM
200 FEG machine equipped with Energy Disperse Spectroscopy (EDS) was employed
for the inner microstructure observation and fractographic examination.

Results
3.1 Stress-strain curves and fracture strain variations
The true strain εt during the tensile deformation can be calculated as
A0
et = ln (1)
A
where A0 is the initial cross-sectional area and A is the instantaneous cross-sectional
area. The uniform strain εu describes global deformation before necking of samples
under uniaxial tension.
A0
eu = ln (2)
Au
where, Au is the cross-section area before necking. When fracture occurs, Af is the
cross-section area at fracture. Therefore, the fracture strain εf is defined as
A0
e f = ln (3)
Af

The local deformation that occurs post necking is much larger than the global
response. Therefore, the true stress-strain curves were obtained in consideration of
necking modification(Siebel and Schwaigerer, 1948; Majzoobi et al., 2015).

As shown in Figure 3(a), the true stress-strain curves of ZL205A under a strain rate
of 0.001/s are significantly different in the temperature range from 298K to 773K.
The flow stresses of the material obviously decrease as the experimental temperature
increases. Furthermore, the fracture strains and uniform strains of the samples also
vary hugely with the experimental temperature. Figure 3(b) demonstrates the change
of the fracture strains and the uniform strains with error estimation. The error bars are
obtained based on at least three credible repeated experimental tests under certain
conditions. The overall shape of the fracture strain resembles a “spoon” shape. More
precisely, the curve falls to a minimum value of only 2.55% at approximately 573K,
while it is above 30.94% at the lower temperatures of 289K-373K and approximately
18% at a higher temperature of 737K.

Generally, three typical stages are observed in the curve, which are the low
temperature stage (LTS) at 298K-373K, the middle temperature stage (MTS) at
573K-673K, and the high temperature stage (HTS) at 737K, as well as two
transitional periods, i.e., one from LTS to MTS and the other from MTS to HTS. For
test samples under the strain-rate of 0.001/s, the fracture strains are reduced by 28%
from LTS to MTS and then rise sharply by approximately 15% from MTS to HTS.
Therefore, the temperature played a significant role on the variation of ductility.
(a) True stress-strain curves (b) Uniform strain and fracture strain
Figure 3. Stress-strain curves of ZL205A at different temperatures with a strain rate of 0.001/s

The true stress-strain curves of ZL205A at different temperatures under strain-rates of

0.01/s and 0.1/s are given in Figure 4(a) and Figure 5(a), and their fracture strain and
uniform strain curves are provided, respectively in Figure 4(b) and Figure 5(b). The
flow stress, similar to that under the 0.001/s strain rate, becomes smaller with
increasing temperature, except for at 573K. At that temperature, yield strengths and
elastic moduli of the material are remarkably higher than at other temperatures.
Additionally, the fracture strain patterns under 0.01/s and 0.1/s strain rates are also of
spoon-shapes and reach a minimal value at 573K. Certainly, there are several thought-
provoking differences among the fracture strain curves. The major difference among
them is the variation of the depth of the “spoon” shape. The fracture strain minimum
increases from 2.55% to 7.05% and to 12.2% with the strain rate increasing from
0.001/s to 0.01/s and to 0.1/s. Moreover, the fracture strain declines from LTS to MTS
at the strain rate of 0.001/s, 0.01/s and 0.1/s is 28.39%, 20.04% and 10.36%,
respectively. The fracture strain recovery from MTS to HTS basically declines
slightly, i.e., 15.45%, 13.85%, and 11.3%. A higher strain rate increases the minimum
of the fracture strains; meanwhile, the pattern of the fracture strains remains the same.
Therefore, it is highly likely that temperature has a dominating effect on the variation
of fracture strain; in addition, the influence of strain rate should not be ignored.
(a) True stress-strain curves (b) Uniform strain and fracture strain
Figure 4. Stress-strain curves of ZL205A at different temperatures with a strain rate of 0.01/s

(a) True stress-strain curves (b) Uniform strain and fracture strain
Figure 5. Stress-strain curves of ZL205A at different temperatures with a strain rate of 0.1/s

Previous researchers(Estey et al., 2004; Shi et al., 2014; Yang et al., 2013) usually
studied flow stresses of aluminum alloys through stress-strain curves under uniaxial
compression because compressive tests can be easily conducted under precisely
controlled heating and cooling rates. As a result, the variation in the fracture strains of
alloys was rarely observed. Newman et al.(Newman et al., 2003) also studied the
tensile behavior of the as-quenched W319 aluminum alloy. However, as their samples
were not stretched to fracture, they also did not observe the unusual ductility variation
phenomenon. In this study, the variation of the ductility of the ZL205A cast
aluminum alloy with the temperature and strain rate was observed, analyzed, and
explained through fractography analysis and microstructure observations in the
following sections.

3.2 Fractography analysis

Figure 6 and Figure 7 show fracture surfaces of test samples after fracture at LTS,
MTS, and HTS. At LTS, the tensile sample exhibited “necking” before fracture and
showed a smooth fracture feature at 45 degrees to the loading direction, as shown in
Figure 6(a). Accordingly, the fracture surface at LTS shown in Figure 7(a) is mainly
composed of dense small dimples, with some obvious and large dimples. These
features are all typical characteristics of ductile fracture. The fracture feature plays an
important role in the ductility (fracture strains) of a material, so that as-quenched
ZL205A exhibits great ductility (fracture strain > 30.9%) and good strength at LTS.

(a)

(b)

(c)
Figure 6. The samples after fracture under various tensile test conditions
(a) LTS; (b) MTS; (c) HTS
The fracture features of samples at MTS and HTS exhibit some similar characteristics
but also have differences. The samples at both MTS and HTS show irregular serrated
fracture edges, with no obvious “necking”, as shown in Figure 6(b) and 6(c).
Moreover, the fracture surfaces, as shown in Figure 7(b) and 7(c), are clearly
composed of rock candy patterns and cleavage patterns at both stages, which indicate
intergranular fracture. However, the fracture strains at both stages vary greatly; i.e.,
the fracture strain is only 2.55% at MTS, but nearly 18% at HTS. The grain surfaces
are flat and smooth; and fracture cleavage and grain boundaries are greatly evident
and can be clearly recognized on the fracture surfaces at MTS. These are typical
features of grain boundary brittle fracture (GBBF). On the contrary, at HTS, the grain
surfaces are somewhat rough, and the boundaries are less clear than at MTS. Figure 8
shows a grain surface of the fracture surface under high magnification at HTS. There
are many tiny dimples on the surface, indicating ductile fracture. Therefore, as-
quenched ZL205A exhibits grain boundary ductile fracture (GBDF), which is
different from the other temperature stages.

(a) At LTS (1000x, 200x, 32x)

(b) At MTS (1000x, 200x, 32x)

 (c) At HTS (1000x, 200x, 32x)
Figure 7. Fracture surfaces of test samples by SEM

Figure 8. Fracture surfaces of test samples at HTS under high magnification SEM
3.3 Precipitation analysis and microstructure comparison
Various fracture modes in the test samples at different temperature stages, as
described in Section 3.2, lead to significantly different fracture strains at the different
temperature stages. As a typical high strength heat-treatable Al-Cu-Mn alloy, as-
quenched ZL205A is likely to exhibit different precipitation behaviors at different
temperatures, resulting in different microstructures and mechanical properties. Al, Cu,
Mn, and Ti are the four most common alloying elements. The main intermetallic
phases of ZL205A can be categorized into precipitation strengthening phases, e.g.,
Al2Cu and Al20Cu2Mn3 (T phase), and stable phases, e.g., Al3Ti, which form upon
solidification. The fracture surfaces and inner microstructures were observed using
SEM by applying SE (second electron) and BSE (back-scatter electron) methods at
the LTS, MTS and HTS stages to compare the microstructure and precipitation and
clarify reasons for the variations in the fracture strain and ductility. 错误!未找到引用
源 。 shows the high magnification fracture surface of ZL205A at LTS. White
particles are found inside small dimples, where the matrix is homogeneous, without
significant Cu or Mn aggregation. These white particles are determined to be Al3Ti by
EDS analysis. Apart from large vacancies caused by casting defects, Al3Ti particles
can reduce the bonding capacity of neighboring grains and, in turn, provide
preferential positions of microvoids and crack nucleation(Birol, 2007; Milman et al.,
2001).

The inner structure of the samples near the fracture surface at LTS is shown in Figure
9. The general matrix (Zone I) is homogeneous, while a few precipitate phases, such
as tiny Al2Cu particles, are randomly distributed in the matrix. The loose structure
(Zone II) observed at the grain boundary encompasses several white particles, which
are also Al3Ti particles. The inner microstructure observation is in good agreement
with the fracture microstructure. The matrix at LTS is homogeneous and uniform with
little precipitation, except for several white Al3Ti particles accompanied by a loose
structure. The “loose structure” is distributed around grain boundaries, without
obvious directionality. Therefore, the “loose structure” around Al3Ti particles is
positively shrinkage micro-holes. As a result, in terms of the ductility, ZL205A at
LTS performs well.
 Figure 9. The inner microstructure of test samples at LTS
The inner structure of the samples near the fracture surface at LTS is shown in Figure
9. The general matrix (Zone I) is homogeneous, while a few precipitate phases, such
as tiny Al2Cu particles, are randomly distributed in the matrix. The loose structure
(Zone II) observed at the grain boundary encompasses several white particles, which
are also Al3Ti particles. The inner microstructure observation is in good agreement
with the fracture microstructure. The matrix at LTS is homogeneous and uniform with
little precipitation, except for several white Al3Ti particles accompanied by a loose
structure. The “loose structure” is distributed around grain boundaries, without
obvious directionality. Therefore, the “loose structure” around Al3Ti particles is
positively shrinkage micro-holes. As a result, in terms of the ductility, ZL205A at
LTS performs well.

Figure 10. Microstructure observations of the fracture surface at MTS

The microstructure of the samples at MTS, as shown in Figure 11, shows the
distribution of Al2Cu dispersoids, T phases, and Al3Ti particles. Al2Cu dispersoids
homogeneously precipitate from the supersaturated solid solution, while T phases
dominantly form at grain boundaries. Large T phase particles gather at the intercept
point of grain boundaries, and the rest of the T phases still align with the grain
boundaries. Al3Ti particles are still associated with loose structures and vacancies.
Moreover, the fracture strain dramatically decreases to 2~3% at MTS under the strain
rate of 0.001/s. Al3Ti, as mentioned before, will not change during heat treatment and
therefore could not be the reason for the variation in the fracture strain and ductility.
Al2Cu, the main strengthening precipitation phase, can increase the strength of the
alloy, while slightly decreasing its ductility. However, this reduction cannot possibly
be the cause of the sharp drop in the fracture strain observed in the experiments. After
the exclusion of other possibilities, T phases, gathering along the grain boundaries,
are most likely to be the cause of the variation in the fracture strain. The grain
boundary with T precipitates is an initial source of cracks and easily gives rise to
microvoids, which is thus detrimental to the toughness and ductility of the alloy.

Figure 11. The inner microstructure of the test samples at MTS

The ductility of the alloy increases again to 18% under the 0.001/s strain rate at HTS.
Accordingly, the microstructure of the samples at HTS is observed and shown in
Figure 12. The fracture surface and metallographic microstructure are basically in
agreement with the observations at MTS and include a rock-pattern fracture surface,
Al3Ti particles along with loose structures, homogeneously precipitated Al2Cu
dispersoids in the matrix, and obvious T phases in the grain boundaries. The Al2Cu
dispersoids and grain boundaries with T phases both become more obvious and coarse.
Based on the analysis of the fracture strains at MTS, more grain boundary precipitates
lead to worse ductility of the alloy. However, the fracture strains of samples at HTS
actually increase by more than 15%. Moreover, the brittle facture at the grain
boundary at MTS transforms to a ductile fracture at HTS. The different fracture
modes indicate that the mechanism of ductility is not the same.

Figure 12. The inner microstructure of test samples at HTS

Discussion
4.1 Competitive effect of temperature and precipitation on ductility
The Al-Cu-Mn phase diagram in the Al-rich region is shown in Figure 13. The
aluminum corner contains Al2Cu, Al6Mn and a ternary compound usually designated
as the T phase (Al20Cu2Mn3). According to previous data on casting alloys containing
approximately 5% Cu, the concentration of manganese in the solid supersaturated
solution during solidification can reach 2%(Belov et al., 2005), which is much higher
than 0.5% Mn in ZL205A. The major deviation from equilibrium during solidification
is due to the formation of non-equilibrium (Al) + Al2Cu eutectics and a supersaturated
solid solution of Mn in (Al). The decomposition of the latter during the reheating
process to over 573-773K leads to the formation of Mn-containing precipitates,
represented mainly by Al20Cu2Mn3 (T phase).
 Figure 13. Phase diagram of Al-Cu-Mn at the aluminum rich corner of solidus(Belov et al.,
2005)
Based on the Al-Ti phase diagram(Witusiewicz et al., 2008), Al3Ti particles can
easily sink and aggregate during solidification, leading to the segregation of white
Al3Ti. Al3Ti segregation is difficult to avoid and is generally accompanied by
microvoids and micro-porosity(Wang et al., 2014). Al3Ti particles can almost not be
solid solutionized and will not change during the heat treatment process. Therefore, it
could not be the reason for spoon-shaped variation of ductility. Al2Cu phases are
homogeneously precipitated and uniformly distributed in the matrix. The amount of
Al2Cu precipitates increases gradually with experimental temperature. Hence, the
existence of Al2Cu also cannot cause the phenomenon of ductility variation.
Al20Cu2Mn3 phases do not exist at LTS, and precipitate considerably on grain
boundaries at MTS and HTS, weakening the bonding of grain boundaries.
Consequently, it is almost certain that the grain boundary T phases dominate the huge
decline of ductility of as-quenched ZL205A alloy.

Without the influence of grain boundary precipitates, the ductility of alloys gradually
increases with the increase of temperature and decreases with the increase of strain
rate(Li and Ghosh, 2003). Because of dynamic recovery and dynamic recrystallization,
the strength of 21/4Cr-1Mo steel(Booker et al., 1977) declines greatly at elevated
temperatures, and the ductility obviously increases, as shown in Figure 15. The
ZL205A alloy is softened with the increase of temperature, thus improving the
ductility of the alloy. Dynamic recovery and dynamic recrystallization play an
increasingly important role on the deformation behavior of the alloy when the
samples are stretched at elevated temperatures. Especially when the temperature
increases to near the solid-solution temperature, the ductility greatly improves. The
grain boundary is traditionally considered to be a strengthening factor, in other words,
the bonding force in grain boundaries is higher than in the matrix. Therefore,
intergranular fracture could occur if only grain boundaries are weakened. Generally,
there are two basic reasons for weakened grain boundaries(Vasudevan and Doherty,
1987): 1) the presence of microstructures of the alloy and 2) the influence of high
temperature and conditions.

Figure 14. Effect of test temperature on ductility and tensile strength of steel

The obvious grain boundary T phases at MTS, i.e., the discrete rod-like particles, are
the main reason for the weakened bonding force of the grain boundary, leading to
huge decline of fracture strain and the grain boundary brittle fracture (GBBF). On the
other hand, although the grain boundary precipitates at HTS are coarse and reduce the
fracture strains of samples considerably, temperature has a more important impact on
softening the alloy and increasing its fracture strain and ductility. Besides, the high
temperature close to solution temperature at HTS hugely weakens the bonding of
grain boundaries, leading to the grain boundary ductile fracture (GBDF). Therefore,
the variation of ductility of as-quenched ZL205A alloy results from the combined
effects of both temperature and grain boundary precipitates.

Figure 15 shows that the fracture strains change not only with the test temperature
but also with the strain rate. The fracture strain decreases as the strain rate increases at
LTS, while the fracture strain increases with the increase in the strain rate at MTS and
HTS. This observation can be explained by considering grain boundary precipitates.
At LTS, grain boundary precipitates are truly little and negligible; as a result, a higher
strain rate allows dislocations to generate and glide more easily and quickly, so
ductility becomes worse in agreement with previous studies(Chen et al., 2013; Zhang
et al., 2007). In addition, the fracture strain of samples slightly increases with the
temperature at LTS, as a result of dynamic recovery. At MTS and HTS, a higher
strain rate means a shorter experiment time, resulting in less precipitates being
nucleated, especially grain boundary T phases. Therefore, the fracture strains are
higher when the strain rate is larger. This phenomenon verifies the hypothesis that the
combined effect of thermal softening and grain boundary precipitates is the primary
reason for the variation in the fracture strain and ductility of as-quenched ZL205A.

Figure 15. Fracture strain curves under different strain rates

4.2 A mathematical model of ductility on temperature and strain rate
Under the experimental conditions, it is believed that the competitive effect between
thermal softening and grain boundary precipitates leads to the variation in the fracture
strain. With the increase of temperature, on one hand, dynamic recovery and dynamic
recrystallization reduce the strength of the matrix and increase the ductility of the
matrix; on the other hand, an increasing amount of T phases nucleate and grow on the
grain boundaries, resulting in a great decline of ductility and strength of the grain
boundary. Typically, a large fracture strain of over 18% at HTS is due to the increase
of ductility of the matrix, while the rock-pattern fracture surface is due to the decline
of ductility of the grain boundary.

The ductility of as-quenched ZL205A is expressed by the fracture strain, denoted as εf

(T, 𝜀̇ ). In the experiments, temperature and strain rate are independent variables.
According to the competitive hypothesis of thermal effects and grain boundary
precipitation, the ductility εf (T,𝜀̇) can be expressed as Equation 3:
(4)

where f(T) characterizes the positive influence of temperature on the ductility of the
matrix, and g(T, 𝜀̇ ) expresses the negative correlation between grain boundary
precipitations with the ductility of grain boundary. Under higher temperatures, grain
boundary precipitates nucleate more easily, and at a higher strain rate, the amount of
grain boundary precipitates decrease due to a shorter time in the tensile process. The
three strain rates adopted in the experiments (0.001/s, 0.01/s, 0.1/s) are represented
as 𝜀̇1 , 𝜀̇2 , and 𝜀̇ 3 . The ratio function w is proposed as shown in Equation 4, which is
only dependent on temperature T.

(5)

The w(T) curve is given in Figure 16, and the error bars are computed by using the
error transfer formula. The values of w at all temperature levels basically remain
constant at 1.0. When w is constant, indicating its independence of temperature, the
influence of grain boundary precipitation on ductility could be decomposed into two
parts. Therefore, the function g(T, 𝜀̇ ) could be expressed as the product of a
temperature function k(T) and a strain rate function h(𝜀̇). Then, the ductility εf (T, 𝜀̇)
can be redefined as shown in Equation 5.
 (6)

Figure 16. The relationship between w and elevated temperatures

It is worth noting that the constant relationship, w = 1.0, means a linear relationship
between h( 𝜀̇ ) and ln(𝜀̇) . Therefore, it is reasonable to assume the correlation
. Then,

(7)

where f*(T)=f(T)+B.k(T) and k*(T)=A.k(T).

As a result, the temperature function k*(T) can be derived as shown in Equation (5).

. (8)

k*(T) and f*(T) can be plotted as shown in Figure 17. The function k*(T) characterizes
the ability of precipitation to nucleate at a certain temperature. Thus, the trend of k*(T)
increasing with the temperature conforms to expectations. Effect of grain boundary
precipitation on ductility at 0.001/s is given as , shown in Figure

17(a). g1* (T ) gradually declines with the temperature, and tends to steady after 600K.

Accordingly, the function f*(T) could be represented by

, as shown in Figure 17(b). The function f*(T)
characterizes the thermal effects for improving the ductility of as-quenched ZL205A,
and overall, its value increases with the temperature. However, there are some
fluctuations in the values of f*(T). The values of f*(T) gradually increase at LTS, which
are in accordance with the dynamic recovery at LTS. Then, f*(T) slightly declines at
MTS, which indicates that the grain boundary precipitates inhibit recovery and
recrystallization in the material. After that, f*(T) increases sharply at HTS because
dynamic recrystallization and creep behavior are dominant in the material behavior.
Through comparative analysis of the plots of g1* (T ) and f*(T), it is confirmed that the
dramatic drop of fracture strain at MTS mainly results from the effect of grain
boundary precipitation, and the recovery of fracture strain at HTS is mostly on
account of the high experimental temperature.

(a) k*(T) and g*(T) (b) f*(T)

Figure 17. The relationship between the variables in the ductility model and
temperature

Further experiments and relevant theoretic analysis are required to specifically

determine f*(T) and k*(T). Additionally, the amount of T phases and Al2Cu dispersoids
during the experiments can be characterized using the TTT diagram of ZL205A,
which can be used to define k*(T). Further research on this topic not only contributes
to the explanation of the variation in the fracture strain and ductility of as-quenched
ZL205A, but is also of great importance to comprehensively understand the
relationship between the ductility of materials and process conditions.

Conclusions
In this paper, the ductility of as-quenched ZL205A was investigated in a temperature
range of 298-773K and a strain rate range of 0.001-0.1/s. The fracture strain, which
characterizes the ductility of as-quenched ZL205A, varied with temperature in “spoon”
shape. The phenomena were quite unusual compared with traditional observations.
Through analyzing the corresponding microstructure observations of fracture surfaces
and inner microstructures, the ductility behavior at each temperature stage was
determined by the combined effect of temperature and grain boundary precipitation.

At LTS, ZL205A exhibited good strength and ductility mainly due to a homogeneous
solid-solution matrix and few precipitates. The samples at LTS showed typical ductile
fracture with an orientation of 45 degrees to the loading direction, with a fracture
strain of over 30%. At MTS, ZL205A lost its ductility and presented characteristics of
brittleness. The formation of T phases (Al20Cu2Mn3) at the grain boundaries played a
primary role in intergranular fracture. The fracture strains still reached a minimal
value of approximately 2%. At HTS, because it was close to the solid-solution
temperature, dynamic recrystallization and creep behavior played a dominant role on
the material behavior. The fracture strains increased noticeably to about 18%, even
though the grain boundary T phases grew larger, which was detrimental to the
ductility. The competitive relationship between the temperature and grain boundary
precipitates was validated by the fracture strains under different strain rates. A higher
strain rate meant that there was less time for the grain boundary T phases to nucleate
and grow; thus, the smaller the amount of grain boundary T phases, the higher the
fracture strains. The ductility model was proposed based on the analysis of
experimental data and the linear relationship assumption between h(𝜀̇) and . The

ductility, εf (T, 𝜀̇), was deduced as follows:

The curves of f*(T) and k*(T) were plotted with the temperature.

Acknowledgements
This work was supported by the National Natural Science Foundation of China [grant
number U1537202]. The authors declare that there is no conflict of interest.

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Temperature-dependent constitutive behavior with

consideration of microstructure evolution for as-quenched
Al-Cu-Mn alloy
Wenguang Wang1, Gang Wang1,2, Yisen Hu1, Guannan Guo2, Tingting Zhou3,
Yiming Rong4

1 State Key Laboratory of Tribology, Tsinghua University, Beijing,

100084, China
2 Department of Manufacturing Engineering, Worcester Polytechnic
Institute, Worcester, MA, 01609, USA
3 Science and Technology on Surface Physics and Chemistry
Laboratory, Jiangyou, 621908, China
4 Mechanical and Energy Engineering Department, Southern
University of Science and Technology of China, Shenzhen,
518055, China

Abstract

It is difficult to predict the deformation behavior of Al-Cu-Mn alloy during a

quenching process due to the complex hardening mechanisms. In this paper,
isothermal tensile tests were conducted under controlled experimental conditions (298
- 773 K and 0.001 - 0.1 /s strain rate). Observations on the microstructure of the alloy
and tensile test analyses on stress-strain curves both verified that the deformation
mechanisms differed drastically at 298 - 473 K and 573 - 773 K. Therefore, a
temperature-dependent constitutive model was established to characterize the
divergent flow behaviors of as-quenched Al-Cu-Mn alloy. In addition, the activation
energy, Q, in the model is determined by the combined effect of dislocation forests
and precipitate phases, various with different experimental conditions.

Keywords

2
Corresponding author:
Gang Wang, State Key Laboratory of Tribology, Tsinghua University, Lee Shau Kee S&T Building
A1003-3, Beijing, China. Email: gwang@tsinghua.edu.cn
As-quenched Al-Cu-Mn alloy; Precipitation; Dislocation forest; Arrhenius model;
Activation energy

Introduction
Al-Cu-Mn cast alloy is widely used to produce large, thin-wall structures in the
automobile and aerospace industries because of its light weight and high strength
[1]. Before using the alloy in practical industry, proper heat treatment processes are
usually applied to improve the mechanical properties of the Al-Cu-Mn alloy.
Because of structural complexity and inevitable numerous casting micro-defects,
e.g., micro-porosity and micro-segregation [2], workpieces of the alloy are very
prone to heavy distortion during heat treatment, especially in the quenching process.
Therefore, a step-quenching method is generally employed to decrease the
quenching distortion of large workpieces [3]. Previous studies tried to predict
quenching deformation via building a constitutive model of as-quenched Al-Cu-Mn
alloy in a high temperature range [4-6]. Owing to the rapid cooling rate and wide
temperature range of the quenching process, existing constitutive models are hardly
sufficient to characterize the constitutive behavior of the as-quenched alloy in all
temperature ranges [7, 8].

The Arrhenius model is a semi-physical constitutive model commonly used in the hot
working field. The model was proposed by Sellars and Mctegart [9, 10] and
developed from the empirical Zener-Hollomon model [11], which combines
temperature and stress influences on dislocation motion. The thermal activation
mechanism, a fundamental of the Arrhenius model, interprets the statistic process of
dislocation thermal release over glide resistance. Recent studies have demonstrated
that the hyperbolic-sine Arrhenius model is appropriate for Al alloys in a temperature
range from 0.5Tm to Tm during quenching [12, 13], but the model fails to agree with
experimental results in the low temperature range. In the Arrhenius model, the Zener-
Hollomon parameter is proposed to characterize the activated effects of temperature
and stress with the assumption of constant activation energy. Therefore, other glide
mechanisms of dislocations, such as strain and precipitation hardening, are not taken
into account in this model. Liu et al. revised the model by compensating for strain and
strain rate and fitting the activation energy, Q, as a polynomial function of strain to
describe the strain hardening influence on the activation energy in engineering[14].

The activation energy reflects the energy required for dislocations to glide across
obstacles, e.g., intermetallic particles or grain boundaries. The classical Arrhenius
model only considers the effect of dynamic recrystallization and recovery, and the
changing grain boundary was supposed to act as the primary obstacle to dislocations.
As a result, the classical model is mostly effective for the aluminum alloys which
satisfy the assumption of supersaturated homogeneous solid-solution state. For
workpieces of Al-Cu-Mn alloy, precipitation during the quenching process is difficult
to avoid, if the temperature is not cooled quickly enough. The difference of
mechanical behaviors of the alloy at the high temperatures and low temperatures is
also observed in Yang’s research [15]. However, the relationship between
microstructures and mechanical property variation with temperature is lacking in
further investigation. More importantly, how microstructures affect the model
parameters is particularly conducive to predicting quenching deformation of the alloy
better and improving applicability of the constitutive model.

Estey et.al. [16]proposed that the diversity of mechanical behaviors of as-quenched

alloys is probably relevant to the slight precipitation during experiments. The pinning
effect of small precipitated particles on dislocations increases the energy of
dislocation gliding, which results in variations in the activation energy [17, 18]. In
addition, the different deformation mechanisms of aluminum alloys over different
temperature ranges have an important impact on the activation energy, such as
dynamic recovery at low temperatures and dynamic recrystallization at high
temperatures. Sellars stated that the complexity of an aluminum alloy or the formation
of a precipitate phase among the matrix may affect the activation energy of hot
deformation [19]. Chemical concentration effects [20] and initial structure [21] have
also been proven to have an effect on the activation energy. Therefore, the activation
energy cannot be considered to be constant if the applicable scope of the constitutive
model needs to be extended.
In this paper, a modified Arrhenius model has been used to predict the flow stress of
the Al-Cu-Mn alloy over a wide temperature range (298 - 773 K). Additionally, the
initial structures at different quenching temperatures and interactive motions of
dislocation and precipitate phases at different temperatures were observed to
determine microstructure effects on flow stress. The activation energy of hot
deformation is given as a function of temperature and strain.

Experiments
Al-5%Cu-0.4%Mn is used in this experiment, and the detailed composition is given
in table 1. The experimental samples were machined into a rod shape with two
screw thread ends. At first, all samples were heated in a Muffle furnace at 813 K
for 10 hours in order to obtain full solution treatment. After solution treatment,
samples were taken out of the furnace and quenched in 11% UCONTM quenchant.
Next, tensile samples were stretched on an Instron 5985 at experimental
temperatures (298 K, 373 K, 473 K, 573 K, 673 K, 773 K). Because the heat-up
rate was great enough, the material state at experimental temperatures was able to
simulate the conditions of step-quenching. The strain rates were 0.001 /s, 0.01 /s
and 0.1 /s, which covered the possible values of strain rate during a quenching
process[22].
Table 2 Main element composition of Al-5%Cu-0.4%Mn
Element Cu Mn Ti Zr Cd B V Al
4.6 - 0.3 - 0.15 - 0.05 - 0.15 - 0.005 - 0.05 -
wt% Bal.
5.3 0.5 0.35 0.20 0.25 0.006 0.3
After tensile testing, specimens for TEM were prepared by mechanical and ion
thinning methods. Their microstructure was examined on a high-resolution
transmission electron microscope, TECNAI G2 20.

Results

Stress-strain curves
Figure 1 gives the stress-strain curves for as-quenched Al-Cu-Mn alloy in a
temperature range of 298 to 773 K and strain rate range of 0.001 to 0.1 /s. The stress-
strain curves show great differences at 298 - 473 K and 573 - 773 K. At low
experimental temperatures (298 - 473 K), the flow stress is less sensitive to
temperature and strain rate, but strain hardening behavior can be observed. However,
at high experimental temperatures (573 - 773 K), the flow stress greatly declines with
an increase in temperature and rises with an increase in strain rate. In addition, at high
temperatures, the alloy behaves as a steady flow beyond yield strength during the
tensile tests without a typical strain hardening stage.

Figure 1. Stress-strain curves of as-quenched Al-Cu-Mn alloy

Figure 2(a) shows a plot of the strain hardening rate (SHR), θ(=dσ/dε), vs. true strain,
ε, at different temperatures. The SHR θ drops dramatically with strain and remains
roughly stable with large strain (> 0.01). It is interesting that the SHR at 298 - 473 K
is much higher than the value at 573 - 773 K. The SHR at 298 - 473 K finally stays in
a constant state around zero, which indicates the alloy reaches a steady flow. However,
the SHR at 573 - 773 K reaches a constant level at approximately 2500 MPa, which
indicates strain hardening behavior in the strain range of 0 - 0.02.

To quantify the strain hardening response, the strain hardening exponent (SHE), n,
was used to fit the tensile curves based on the Ludwik equation [5, 23]:
 𝑛
𝜎 = 𝜎𝑦 + 𝐾(𝜀 − 𝜀𝑦 ) (1)
where, σy and εy are yield strength and strain of the material, and K is the strength
coefficient. The n plot is shown in Figure 2(b) for all experimental conditions. A
linear correlation between SHE and temperature was found for both 298 - 473 K and
573 - 773 K with reasonable approximation. However, the fitted lines were distinctly
separate from each other in the temperature ranges. The SHE n at 298 - 473 K is
about twice as large as that for 573 - 773 K. The results for SHR θ and SHE n
highlight that there is a drastic difference in the strain hardening mechanism of as-
quenched Al-Cu-Mn alloy at 298 - 473 K and 573 - 773 K.

Figure 2. Strain hardening behavior of as-quenched Al-Cu-Mn alloy

The effects of strain rate and temperature on material flow behavior are illustrated in
Figure 3. Strain rate sensitivity (SRS), m, is another important parameter that reflects
the flow behavior of materials. The m values can be evaluated based on the following
equation, which is quite similar to the Lindholm equation [24]:
σ/σmin = 𝑚log𝜀̇ (2)
The flow stress is normalized by σmin, the minimum value of flow stresses at the
corresponding temperatures. The SRS plot of σ0.01/σmin vs. log𝜀̇ is given in Figure 3(a),
and the slope is the SRS m value. At 298 - 473 K, the SRS plots slowly shift, and the
values of σ0.01/σmin are close to one. This finding indicates that flow stress is less
sensitive to strain rate at low temperatures. It is worth noting that the flow stress shifts
downward at 298 K. This unusual phenomenon is quite complex and will be
discussed later. At 573 - 773 K, σ0.01/σmin increases dramatically at larger strain rates
as the temperature increases. Therefore, flow stress at high temperatures is
increasingly sensitive to the strain rate. The relationship between SRS, m, and
temperature, T, is shown in Figure 3(b). Correspondingly, the trends at 298 - 473 K
and 573 - 773 K completely differ from each other. At 298 - 473 K, the SRS is low
and increases slowly, while at 573 - 773 K, the SRS climbs sharply with increasing
temperature.

Figure 3. Strain rate sensitivity of as-quenched Al-Cu-Mn alloy

The strain hardening exponent, n, and the strain rate sensitivity, m, play very
important roles in characterizing the deformation behavior, which is closely related to
the microstructures. The distinct differences in the SHE and SRS plots imply
dissimilarities in the low temperature range (298 - 473 K) and high temperature range
(573 - 773 K). Therefore, to better represent the stress-strain behavior of as-quenched
Al-Cu-Mn alloy, the constitutive model should be analyzed separately at 298 - 473 K
and 573 - 773 K. The constitutive behavior in the transitive temperature range (473-
573 K) should be a linear combination of the behavior at 473 K and 573 K.

Constitutive model
The total constitutive model for the whole temperature range (298 - 773 K) is
proposed to be the following:
𝐹1 (𝜎, 𝜀, 𝑇) 298𝐾 ≤ 𝑇 ≤ 473𝐾
𝜀̇ = {𝑎 ⋅ 𝐹1 (𝜎, 𝜀, 𝑇) + 𝑏 ⋅ 𝐹2 (𝜎, 𝜀, 𝑇) 473𝐾 < 𝑇 < 573𝐾 (3)
𝐹2 (σ, ε, T) 573𝐾 ≤ 𝑇 ≤ 773𝐾
where, a and b can be determined by the corresponding proportion of the specific
temperature in the 473 - 573 K range and F1(σ,ε,T) and F2(σ,ε,T) are the Arrhenius
models at 298 - 473 K and 573 - 773 K, respectively. The Arrhenius model is useful
for predicting the flow stress of aluminum alloy during a hot working process. The
Zener-Hollomon parameter combines the effects of strain rate and temperature on
deformation in an exponent-type equation. The hyperbolic law between the Zener-
Hollomon parameter is employed for better approximations at all stress levels. Thus,
the flow stress can be written as a function of the Zener-Hollomon parameter. The
basic Arrhenius model can be represented as
𝑄
𝜀̇ = 𝐴[sinh(𝛼𝜎)]𝜂 exp (− ) (4)
𝑅𝑇
𝑄
Z = 𝜀̇ exp (𝑅𝑇) (5)

1 𝑍 1/𝜂 𝑍 2/𝜂
σ = α ln [(𝐴) + √(𝐴) +1] (6)

in which, 𝜀̇ is the strain rate (s-1), R is the universal gas constant (8.31 J⋅ mol-1K-1), T
is the absolute temperature (K), Q is the activation energy of hot deformation
(kJ⋅ mol-1), σ is the flow stress (MPa) for a given strain, and A, α and η are the
material constants.

To obtain exact parameter values for the Arrhenius model, the natural logarithm of
Equation (4) is given as:
ln 𝑍 = ln 𝐴 + 𝜂 ln[sinh(𝛼𝜎)] (7)
Figure 4 shows the plots of lnZ vs. ln[sinh(ασ)] for 298 - 473 K and 573 - 773 K,
respectively. The completely separate lines confirm the difference in the constitutive
behaviors of as-quenched Al-Cu-Mn alloy in the two temperature ranges.
 Figure 4. Variations of the Zener-Hollomon parameter with flow stress
Finally, at 298 - 473 K, the Arrhenius model, F1(σ,ε,T), of as-quenched Al-Cu-Mn
alloy at 0.2% strain was obtained and is given as Equation (8). The parameters of the
model at strain rates of 0.2 - 2.0% were obtained as polynomial functions of strain,
using Lin’s method [14].
1.360 × 103
𝜀̇ = 8.54 × 10−5 [sinh(0.0084𝜎0.2 )]24.71 exp (− ) (8)
8.314𝑇
The values calculated at 298 K, 373 K and 473 K based on the model agreed with the
experimental results with a mean relative error of 4.3%, shown in Figure 5. The
calculated values cannot describe the negative strain rate sensitivity of stress-strain
behavior at 298 K, as shown in Figure 5(a). The calculated values at 0.001 /s and 298
K are in good agreement with the experimental results. However, as the strain rate
increases, the calculated values increase based on an Arrhenius-type prediction, which
is contrary to the experimental results. At other temperatures (373 K and 473 K), the
model predicts the experimental results well.
(a) 298 K

(b) 373 K
 (c) 473 K
Figure 18. Comparison of calculated values with experimental results at 298 - 473 K
At 573 - 773 K, the Arrhenius model, F2(σ,ε,T), of as-quenched Al-Cu-Mn alloy at
0.2% strain was obtained and is given as Equation (9). Accordingly, the model
parameters at a strain rate of 0.2% to 2.0% were also obtained as polynomial
functions of strain.
3.98 × 105
𝜀̇ = 2.89 × 1028 [sinh(0.0152𝜎0.2 )]8.145 exp (− ) (9)
𝑅𝑇
The comparison of calculated values and experimental results at 573 - 773 K with a
mean relative error of 2.1% is shown in Figure 6.

(a) 573 K
 (b) 673 K

(c) 773 K
Figure 19. Comparison of calculated values with experimental results at 573 - 773 K
Therefore, for the whole temperature range (298 - 773 K), a constitutive model was
built and is shown as Equation (3). The model is comprised of two separated
Arrhenius models, (F1(σ,ε,T) for 298 - 473 K and F2(σ,ε,T) for 573 - 773 K, and a
transition model for 473 - 573 K.

Discussion

Microstructure evolution
The change in the constitutive behaviors of as-quenched Al-Cu-Mn alloy over the
temperature ranges is closely related to the variation in the microstructures of the
alloy. Remarkably diverse microstructures of as-quenched Al-Cu-Mn alloy are
shown in Figure 7 at different temperatures. At 373 K, the matrix exhibits a
homogeneous state with rare second precipitation particles, and numerous
entangled dislocation forests can be clearly observed. At 573 K, tiny precipitation
particles are evident, and the dislocation forests are significantly reduced. The tiny,
acicular particles were shown to be Al2Cu precipitation by EDS analysis, shown in
Figure 8. The blocky particles were shown to be intermetallic Al3Ti, which could
exist in the form of a twin, as shown in Figure 7(d). Al3Ti is a type of impurity
generated during the casting process and does not change during heat treatment
[25]. At 773 K, there is distinct rod-like precipitation in the matrix and almost no
visible dislocation forests. Therefore, as the experimental temperature increases
from low to high, the alloy microstructures gradually transform from a dislocation
forest dominant state into a precipitation dominant state.

(a) 373 K, 0.001/s (b) 573 K, 0.001/s

 (c) 773 K, 0.001/s (d) HRTEM of Al3Ti particles
Figure 7. TEM observation of as-quenched Al-Cu-Mn alloy microstructures

Figure 8. EDS analysis of the tiny acicular particles at 573 K

Dislocation forests and precipitation hardening are the two major hardening
mechanisms of as-quenched Al-Cu-Mn alloy. When a gliding dislocation encounters a
forest or other dislocation, the dislocation junctions will give a temperature
independent contribution to the flow stress. The components of the Burgers vector do
not play a role in the flow stress for repulsive dislocation trees [26]. Precipitation
strengthening is achieved by producing a particulate dispersion of obstacles to the
dislocation movement. The flow stress increases beyond the resistance, and
dislocations will bypass the particles either by Orowan looping or cross-slip [27]. The
flow stress increment by dislocation bowing leads to the Orowan equation, ∆τ=Gb/L,
which is linearly dependent on the Burgers vector. As a result of different hardening
mechanisms, resulting from distinct microstructures, the deformation behavior of as-
quenched Al-Cu-Mn alloy exhibits various trends at different temperatures.

Temperature-dependent thermal activation energy

The dramatic difference between the thermal activation energy Q of the alloy at 298 -
473 K and 573 - 773 K is displayed in Figure 9. Therefore, the assumption that the
activation energy Q is independent of temperature is not applicable to as-quenched
Al-Cu-Mn alloy. The activation energy Q is the energy required when a thermal
release of dislocations occurs at obstacles (thermal activation). Under many
circumstances, the activation energy is closely related to temperature [28].
For as-quenched Al-Cu-Mn alloy, the activation energy Q reaches more than 3 × 105
J at 573 - 773 K and only approximately 1 × 104 J at 298 - 473 K. Indeed, the
simplified assumption for the activation energy Q in the Arrhenius model makes it
easier to obtain approximate values from experimental data. However, the activation
energy Q is affected by experimental conditions and is most sensitive to temperature.
The activation energy Q is essentially determined by the microstructures of the
material [13], such as dislocation forests and precipitation.

Figure 9. Thermal activation energy in F1(σ,ε,T) and F2(σ,ε,T)

The comparison between microstructures in Figure 7 and activation energy in Figure
9 indicates that precipitation leads to an increase in the activation energy of the
material, and dislocation forests decrease the activation energy. Upon accumulation of
plastic strain, activation energies at 298 - 473 K and 573 - 773 K begin to drastically
diverge. One possible explanation for this is that a larger plastic strain means longer
experimental time, thus, the microstructure of the material gradually changes at the
same time. At 298 - 473 K, because of high super-saturation in the matrix, some tiny
precipitation may nucleate over a longer period of time, leading to a slow increase in
the activation energy. At 573 - 773 K, the precipitation barely increases with low
super-saturation in the matrix. However, massive dislocations are activated and
entangled with new dislocation forests at higher plastic strains, resulting in a gradual
decline in the activation energy. Presently, the competitive relationship between the
effects of the dislocation forests and the precipitation on the activation energy is quite
complex and still needs further study.
For complex commercial aluminum alloy, e.g., the Al-Cu-Mn alloy in this paper, the
analysis of the macroscopic and microscopic aspects of deformation is still
challenging. The activation energy is a significant physical parameter of the material,
but, in most cases, it is calculated based on idealized assumptions, such as in the
Arrhenius model. Kocks[29] proposes the activation area, Δa, to describe thermal
activated dependence on obstacle hardening mechanisms. The activation area is
defined as
1 𝜕Δ𝐺
∆𝑎 ≡ − 𝑏 (10)
𝜕𝜎

where ΔG is the activation free enthalpy, b is the Burgers vector, and σ is the applied
stress.

Use of the b2/Δa vs. σ plot to determine the dominant strengthening mechanism of a
material is illustrated in Figure 10. The slope of the b2/Δa plot is proportional to the
activation work for dislocations sliding across obstacles [30] and is sensitive to
dislocation forests. The steep slope at low temperatures indicates that the dislocation-
dislocation interactions act pivotal parts in the hardening mechanism. At 298 - 473 K,
the dislocations are relatively athermal with steep slopes, and, therefore, the flow
stress is less sensitive to strain rate and temperature. Along with the experimental
temperature, the effect of dislocation forest hardening gradually decreased. At 573 -
773 K, the activation work for dislocation strengthening is likely to be much smaller,
as a result of dominant precipitation hardening.
 Figure 10. Effect of temperature on the slope of a b2/Δa plot

Strain rate sensitivity

Figure 11 shows variations in the parameter η of F1(σ,ε,T) for 298 - 473 K and
F2(σ,ε,T) for 573 - 773 K. The parameter exhibits completely opposite trends with
strain in different models, which again verifies the huge difference between the
deformation at 298 - 473 K and 573 - 773K. 1/η characterizes strain rate sensitivity in
the Arrhenius model and is defined as follows:
∂ln[sinh(𝛼𝜎)] 1
| =𝜂 (11)
𝜕 ln 𝜀̇ 𝑇,𝜀

The value of η at 298 - 473 K is far higher than the value at 573 - 773 K, indicating
the alloy is more sensitive to strain rate at higher temperatures. The result corresponds
with the analysis in Figure 3. However, the SRS |m| of as-quenched Al-Cu-Mn alloy
is not constant but increases with the experimental temperature. Although the
parameter η in the Arrhenius model cannot change, variation of η with different
temperature ranges is in agreement with the SRS |m|. The reason why the Arrhenius
model cannot describe the finding is because the activation energy Q in the Arrhenius
model is supposed to be a constant. If the activation energy Q were a function of
strain rate, Q=Q(ln𝜀̇), then,
∂ln[sinh(𝛼𝜎)] 1 1 𝜕𝑄
| = (1 + ) (12)
𝜕 ln 𝜀̇ 𝑇,𝜀 𝜂 𝑅𝑇 𝜕 ln 𝜀̇

Therefore, the activation energy Q should decline along with strain rate. A larger
strain rate leads to denser dislocation forests, which bring about a decline in the
activation energy.
Figure 11. Comparison of parameters in F1(σ,ε,T) for 298 - 473 K and F2(σ,ε,T) for 573-773 K
The negative strain rate sensitivity in Figure 1 and Figure 2 is also observed in other
aluminum alloys [31, 32]. Such behavior may reduce the ductility of materials and
affect its formability. The negative strain rate sensitivity is generally explained by
dynamic strain aging (DSA). The microscopic mechanism of DSA has been proposed
by Picu [33] based on the concept of strength variation in the dislocation junctions
due to the presence of solute clusters on forest dislocations. At high strain rates, when
the average time is short, the clusters are too small to produce an effective
enhancement of the obstacle strength. Because it is related to solute diffusion, DSA is
thermally activated. The transition temperature from negative to positive |m| is
between 298 K and 373 K. This phenomenon corresponds with the observations of
Picu et al. [34] and Ling et al. [35]. The negative strain rate sensitivity does not
appear at high temperatures due to structural changes, such as precipitation, which
could change the features of the dislocation motion rate controlling obstacles.

Conclusions
A set of isothermal tensile tests on as-quenched Al-Cu-Mn alloy were conducted over
a range of temperatures (298 - 773 K) and strain rates (0.001 - 0.1 /s), which cover the
actual ranges in practice. Based on observations of the alloy microstructures and
analysis on the stress-strain curves, different microstructures (dislocation forests are
dominant at low temperatures, while precipitation is dominant at high temperatures)
result in the divergence of hardening mechanisms and deformation behaviors over
different temperature ranges.

Finally, an Arrhenius-type constitutive model was proposed for the whole temperature
range and was in good accordance with the experimental data. The model includes
three parts, F1(σ,ε,T) for 298 - 473 K, F2(σ,ε,T) for 573 - 773 K and a transition model
for 473 - 573 K.
1.360 × 103
𝜀̇298−473𝐾 = 8.54 × 10−5 [sinh(0.0084𝜎0.2 )]24.71 exp (− )
8.314𝑇

28 8.145
3.98 × 105
𝜀̇573−773𝐾 = 2.89 × 10 [sinh(0.0152𝜎0.2 )] exp (− )
𝑅𝑇
The strain hardening and strain rate sensitivity behaviors of the alloy at low and high
temperatures were also reflected in variations of the parameter, η, in different models.
In this paper, the activation energy Q was not constant and varied with temperature,
strain rate and plastic strain. Experimental conditions influence the microstructures of
the alloy and thus affect the activation energy value. It is reasonable to conclude that
the activation energy has a positive correlation with precipitation and a negative
correlation with dislocation forests.

Acknowledgments
This work was supported by the National Natural Science Foundation of China [grant
number U1537202].
The authors declare that there is no conflict of interest.

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Computational Materials Engineering (ICME). Springer, Cham, 2013.

A BRIEF REVIEW OF PRECIPITATION HARDENING

MODELS FOR ALUMINUM ALLOYS
Guannan Guo1, Qigui Wang 2, Gang Wang3, Yiming Rong1,3

1
Worcester Polytechnic Institute, Manufacturing Engineering, Worcester, MA 01609
USA
2
GM Powertrain Materials Technology, Pontiac, MI, USA
3
Tsinghua University Precision, Instruments and Mechanology, Beijing China

Keywords: A356, aging, yield strength, modeling, precipitate hardening

Abstract

This paper briefly reviews the precipitation hardening models in aluminum alloys.
Several well-accepted precipitation and strengthening models are compared with
experimental data of aluminum A356 alloy. The differences among various models
are presented and further improvements of precipitation hardening models are
discussed.

1 Introduction

Aluminum alloys are increasingly used in structural applications because of their

lightweight, relatively low manufacturing cost, and high strength to weight ratio
particularly after heat treatment. Most aluminum alloys, like A356 used for critical
structures are usually subjected to aging precipitation hardening. The main reason for
aluminum alloy strengthening is the formation of the precipitates, act as point obstacle
to inhibit the motion of the dislocation. The early period of aging is governed by the
dislocation mechanism of shearing while the dislocation mechanism of bypassing
dominates the later period of aging. The type, size and volume fraction of precipitates
depend upon the alloy compositions and heat treatment conditions. In Al-Mg-Si
system, like A356 alloy, Mg/Si precipitates are the dominated strengthening phases
after aging.

Modeling of precipitation hardening has been extensively studied in past years [1-7].
Several well-known strengthening models for aluminum alloys are reproduced in this
paper. The model predictions are compared with experimental data of A356
aluminum alloy. The differences among various precipitation and hardening models
are presented and further improvement of hardening models are proposed.

2 Microstructure Models

Mean value approach and discrete value approach are two types of models in the
literature to predict the size or volume fraction of the precipitate particles. The mean
value method does not consider the particle size distribution, [1, 3, 5, 6], taking all the
particles as the same. The discrete value approach considers the particle size
distribution based on selected radius classes. [9]

2.1 Mean Value Approach

In the mean value approach, the modeling of precipitates follows the classical
nucleation and growth theory. [4] The basic principle for the growth of precipitate
particles is diffusion mechanism of solution element. In each period, volume fraction
and the mean radius of particles follow different growth kinetics.
Table 3 Input Data for Figure 1
C0 Ce Cp D(Diffusion γ(surface V(volume a(lattice T(K)
2 2 3
coefficient m /s) energy J/m ) per atom m ) parameter nm)

0.06 0.01 1 5.0*10-20 0.13 1.6*10-29 0.404 433

Table 1 gives the experiment data for mean value approach method, where C0, Ce, Cp
is the initial solute concentration, equilibrium solute concentration and solute
concentration in precipitate. Figure 1 gives the result.
 Figure 1 The mean particle size and volume fraction predicted by A.Deschamps’s
model
The nuclei remain at the critical radius at the nucleation stage. And the nucleation rate
drop to zero when no extra solute element remains in the solid solution. Growth
period is corresponded with the dramatically increased mean radius of precipitates,
meanwhile the volume fraction increases. In coarsening period, the radius slowly
increases and the volume fraction almost remain the same at peak value.

2.2 Discrete value approach

New formed nuclei at each time step are grouped and the size evolution of each group
is tracked. The following plots (Figure 2) showing the changes of mean radius,
volume fraction and particle size distribution during aging are based on this
approach.[10]

(a) (b)
Figure 2 The mean radius, critical radius, density distribution and volume fraction of
aluminum alloy A356 at 443K, and different aging time, predicted by Myhr’s model

With the density distribution at each radius group, the total density at each time and
the mean radius can be calculated by summing up all groups. The volume fraction can
be then derived based on the assumption of spherical precipitate particles.

3 Yield Strength Model

3.1 Either shearing or bypassing mechanism considered

Considering the particle radius, there are two types of dislocation hardening
mechanisms- shearing and bypassing. Both mechanisms follow the similar
strengthening prediction, which is given below [10]:
𝑀𝐹
σ= = 𝐶𝑟 𝑚 𝑓 𝑛 (1)
𝑏𝐿
where M is the Taylor factor, F represents the average obstacle force, b is the burgers
vector and L is the average space of particles. C is the coefficient decided by material
and aging conditions, r and f represent the mean radius and volume fraction of the
precipitates, respectively. m and n are different for shearing and bypassing
mechanism. In Liu’s model, particles are considered to strengthen the matrix via
bypassing mechanism.

3.2 Combined shearing and bypassing mechanism

Ashby and Shercliff combined shearing and bypassing mechanism using the harmonic
value of shearing and bypassing strength. [1] Deschamps’ model and Myhr’s model
separate the shearing mechanism and bypassing mechanism with critical radius,
applying corresponding equations in different periods. [4] At the beginning of the
aging process, the particles are small and coherent with matrix; the dislocation can
shear these particles. [14] At peak aging and over-aged conditions, particle size is
large and incoherent with matrix and bypassing mechanism dominates deformation.
[13]

4 Results and Discussion

4.1 Modeling of precipitate evolution during aging

Difficulty in building good microstructure model is how to quantify the nucleation,

growth and coarsening period. Critical radius, defined as the minimum radius for
stable particle, is one way to separate growth and nucleation periods, which is widely
used in many models. However, it is hard to identify when the coarsening period
begins since the radius grows continuously with no obvious change in short period of
time. Deschamps defines coarsening portion to calculate the growth and coarsening
particles fraction: [4]
𝑅 𝐶
𝑓𝑐𝑜𝑎𝑟𝑠𝑒 = 1 − erf (4 ( log ( ) − 1)) (2)
𝑅0 𝐶𝑒

Volume fraction can be another way to build the microstructure model without
considering radius. Figure 3 gives the volume fraction evolution and yield strength
change curve predicted by Ashby’s model. Lloyd also predicted volume fraction by
JMAK model which was calibrated using TEM.

Figure 3 Yield strength and volume Figure 4 Shearing and bypassing

fraction for Al-Mg-Si at 443K mechanism for A6111 aging at 453K

4.2 Modeling of yield strength

As mentioned above, the shearing and bypassing mechanisms are strongly related
with the radius of particles. Lloyd made a comparison when considering only shearing
mechanism or bypassing mechanism. [12] In Figure 4, the experimental data lie
between the two prediction lines, which indicate that there should be a method to
combine two dislocation mechanisms in order to make the prediction more reliable.
Ashby’s model takes the harmonic value of shearing strength and bypassing strength
which matches well with experimental data before peak-aging, but not good in
overaging period.

The orientation and the shape of precipitates also affect the yield strength. [13] Liu
considers this effect in his model when predicting Al-Mg-Si alloy aging behaviors, [7]
following the method given by Zhu et al. to evaluate the yield strength based on
bypassing mechanism. [8]

(a)
(b)
Figure 5 (a) Ashby’s model;(b) Lloyd, Liu, Deschamps & Myhr’s model for A356 at aging
temperature 443K, the green square dots are experiment data;

Figure 5 compares the yield strength predictions from various models including
Ashby, Loyld, Liu, Deschamps and Myhr’s model with experimental data of A356
alloy aged at 443K. It can be seen that Liu’s model has the largest deviation from the
experimental data and Ashby and Myhr’s models match well with the experimental
data.

5 Future works

Microstructure model is critical to the precipitate hardening prediction. Mean or

discrete value approaches are developed well for the spherical shape precipitates.
However, the actual shape of the precipitate particles in aluminum alloys is not
spherical. Following the principle of the discrete value approach, it seems that the
length and the radius of the precipitate particles can be considered to be two-axis
coordinate to classify the group of particles. A better method to distinguish the growth
and coarsening periods in order to separate the shearing and bypassing mechanisms is
needed. In discrete value approach, each group can be considered as a unity to
analyze its contribution to the yield strength by comparing with the critical radius.

6. Acknowledgement

This work is financially supported by Materials Technology Department of GM

Global Powertrain Engineering.

Reference:
1 H.R Shercliff and M.F. Ashby, Acta mater, Vol 38, No. 10, pp. 1789-1802, 1990

2 M. Perez, M. Dumont and D. Acevedo-Reyes, Acta mater 56(2008) 2119-2132

3 A. Deschamps, Acta mater, Vol 47, No. 1, pp 281-292, 1999

4 A. Deschamps, Acta mater, Vol 47, No. 1, pp 293-305, 1999

5 S. Esmaeili, Acta mater 51 (2003) 2245-2257,

6 A. Bahrami, Metallurgical and Materials Transactions A, 09 August 2012

7 G. Liu, G.J. Zhang, X.D. Ding, J. Sun, K.H. Chen,. Materials Science and
Engineering A

8 A.W Zhu, E.A Starke, Jr, “Stress aging of Al-Cu Alloys: Computer Model”, Acta
mater. 49(2001)3063-3069

9 S. Esmaeili, D.J. Lloyd, Metallurgical and Materials Transactions A, Vol 34A,

March 2003.

10 O.Myhr, Acta mater. 48(2000)1605-1615,

11 S. Esmaeili, D.J. Lloyd, Acta mater 51(2003) 2243-2257,

12 M. Ferrante and R.D. Doherty, Acta mater Vol. 27 pp. 1603-1614

13 O.R.Myhr, S.J. Andersen, Acta mater.49(2001) 65-75

14 M. Song, Metallurgical and Materials Transactions A, 443(2007) 172-177

To be submitted to Materials Performance and Characterization

Modeling the yield strength of an A356 aluminum alloy

during the aging process

Guannan Guo1, Qigui Wang 2, Yiming Rong1,3

1
Worcester Polytechnic Institute, Manufacturing Engineering, Worcester, MA 01609
USA
2
GM Powertrain Materials Technology, Pontiac, MI, USA
3
Mechanical and Energy Engineering Department, Southern University of Science
and Technology of China, Shenzhen 518055, China

1 Introduction of the precipitate hardening models

Aluminum alloys are widely used in automotive industry because of their high
mechanical performance and lightweight. The precipitate hardening process is an
important heat treatment technique to improve the yield strength of aluminum alloys.
It is generally acknowledged that the precipitate phase formed during the aging
process strengthens the matrix by blocking the dislocation motions. [1] Mobile
dislocations can shear coherent or semi-coherent smaller precipitate particles or
bypass the larger incoherent precipitate particles, which makes different strength
contributions to the matrix. Thus, the dislocation density will be increased by the
interactive motion of precipitate phase and dislocations. [2] Most aging hardening
models have applied process parameters, such as aging conditions, to relate with the
final mechanical properties, such as hardness or yield stress. In these simulations, the
process parameters are first used to calculate the precipitate phase morphology and
volume distribution, and then these microstructure data are used to calculate the yield
stress. [3, 4]

Precipitate phase formation is affected by aging conditions and aluminum alloy

components. In general, one primary precipitate phase may make a dominant
contribution to strengthen the matrix, which is usually fully coherent within the
matrix and has the dominant volume fraction at the peak-aged state. [1] For a giving
alloy composition, the aging sequence under different aging conditions can be
characterized by TEM and DSC curves to determine the prime strengthening
precipitate phase. Based on the experimental data, the crystallographic parameters,
orientations, dimensions and particle space distributions of the precipitate phase are
obtained, and the related thermodynamic parameters can be derived based on first
principles theory. [5, 6]

The chemical concentration, aging time and temperature are input data to evaluate the
volume fraction of the precipitate phase. The variation of the solute element
concentration can change the final yield stress under the same aging conditions. The
activation energy and equilibrium volume fraction for the precipitate phase is a
function of the temperature, which inhibits or stimulates the formation of the
precipitate phase. TEM and isothermal calorimetric experiments can be used to obtain
the volume fraction of the prime precipitate phase. While the TEM image only
reflects the volume distribution at a selected section, the DSC curve for a specific
material gives the exothermic and endothermic peaks and the heats released or
absorbed, corresponding with the precipitation or dissolution, respectively. [11] After
obtaining the activation energy for the prime precipitate phase, the classic JMAK
equation could be adapted to calculate the volume fraction of the precipitate phase:

𝑓 = 𝑓𝑒𝑞 𝑒𝑥𝑝(−𝑘(𝑇)𝑡)𝑛 1

Where 𝑓𝑒𝑞 is equilibrium volume fraction, T is temperature, 𝑘(T) is transformation

kinetic as a function of temperature and 𝑄𝑎 is active energy. 𝑡 is transformation time
and 𝑛 is a materials relate parameter. The mathematical presentation of 𝑘(T) is given
as followed:
𝑄𝑎
𝑘(𝑇) = 𝑘0 𝑒𝑥𝑝 (− ) 2
𝑅𝑇
When taking all the particles as spheroids for simplification, the volume of the
precipitate particles can be easily calculated. The volume fraction increases very
slightly at the beginning part of the aging process; meanwhile, the precipitate particles
nucleate during this period. Then, the volume fraction rapidly rises to the peak value,
which represents the pure growth period of the particles. After the peak-aged state, the
volume fraction is assumed to maintain the peak value. The smaller particles dissolve
into the matrix and larger particles continue to grow, which leads to the total volume
fraction remaining the same during the overaged period. However, the volume of the
prime precipitate phase will decrease during the transformation from the coherent and
semi-coherent stage to the incoherent stable phase. The volume fraction calculations
in previous models are all concentrated on the prime precipitate phase without
considering the phase transformation from primary precipitate phase to other phases.
In this paper, a new aging model of A356 alloy is proposed consider the volume
fraction change of primary strengthening precipitate phases. The new model will
fulfill at least the following functions:
a) To determine the volume fraction transformation from the metastable phase to the
stable phase and obtain the volume fraction evolution of each phase;
b) To modify the volume change due to the loss of the metastable precipitate phase.
The volume fraction of the metastable precipitate phase during the aging process
should be the original value minus the fraction transformed to the stable phase.

2 Experiments
In this project, A356 is chosen as test materials to accomplish aging experiments. The
chemical content of A356 is given as followed:
Table 1 A356 chemical composition

Element Cu Mg Si Ti Mn Zn
Wt% ~0.2(max) 0.2~0.4 6.5~7.5 0.25 0.1 0.1
The heat treatment process of A356 is typical T6 heat treatment. Samples are heated
to 540C solution temperature and keep for 10 hours, then take out the samples and
quickly quench to room temperature. The as-quenched samples are kept in fridge for
avoiding natural aging. Then samples are taken out to reheat various aging
temperature: 150C, 160C and 170C for 10min, 20min, 30min, 1 hour, 2 hours, 5
hours, 10 hours, 20 hours, 50 hours and 100 hours. The aged samples quench in air to
room temperature after aging process and conduct quasi-tensile tests. The 0.2% offset
yield strength of samples under different aging conditions are obtained for
experimental fitting and validation.

3 Precipitate hardening model of A356 alloy

Before establishing precipitate hardening model of A356 alloy, the primary
strengthening precipitate should be determined in order to obtain related materials
parameters. In this section, the aging sequence and primary strengthening precipitate
of A356 are discussed and confirmed. Then, applying JMAK equations, the volume
fraction of primary strengthening precipitate is used to predict the yield strength of
A356 under different aging conditions. After obtaining needed microstructure
parameters, the calculation of yield strength is given as consequence. The
strengthening mechanism is obeyed bypassing mechanism corresponded with strong
obstacle particles during underaged and peak aged states.

3.1 A356 aging sequence and the primary strengthening precipitate

There is a generally accepted aging sequence for casting the A356 aluminum alloy:
SSS(supersaturated solid solution )α →GP zones (spheres or needles) →β”(needles)
→β’ (rods) →β (plates, Mg2Si or non-stoichiometric MgxSiy). At the peak aged stage,
the primary strengthening precipitate in the cast A356 aluminum alloy is the Mg5Si6
(β”) precipitate phase, which is different from the Mg2Si (β’) phase previously
reported. Therefore, in order to apply the thermal growth model, the precipitate phase
lattice parameter and related thermal constant should be calculated based on the β”
precipitate phase.

Mg2Si rods such as the β’ phase precipitate from β”; meanwhile, volume loss during
the transformation because of the phase change will occur. Considering different
precipitate phases may exist during the aging process, we should obtain knowledge of
those possible precipitate phases. The following table gives the lattice parameters of
possible precipitate phases in the Al-Mg-Si alloy system:
Table 2 Possible precipitate phases of Al-Mg-Si alloy aging process

Phase Lattice parameter (nm)

GP-zones, Mg4AlSi6 a=1.48, b=0.405, c=0.648, β=105.3°
β”, Mg5Si6, Monoclinic a=1.516, b=0.405, c=0.674, β=105.3°
β’, Mg1.8Si, Hexagonal a=b=0.715, c=0.405, γ=120°
β, Mg2Si, Cubic a=0.6354
U1, MgAl2Si2 a=b=0.405, c=0.674, γ=120°
U2, Al4Mg4Si4 a=0.675, b=0.405, c=0.794
Based on the aging sequence, the β” phase precipitates from the GP-zones. Thus, the
volume changes contributed by β” are calculated directly by the volume loss from the
GP-zones to β”. However, there will be some β’ phase formulated from β”, which
means the volume of β” should be subtracted from the transformed β’ phase to modify
the value before calculating the growth contribution of the β” phase. Additionally, the
volume of the β’ phase also needs to be subtracted from the volume of the β phase,
since in the latter periods of the aging process, the β’ phase will turn into the β phase.
In addition to the aging sequence, the Mg/Si content ratio and aging conditions also
affect the precipitate phase distribution. The concentration of Mg is approximately
0.31%~0.34%, while the Si concentration is 0.7%. Some works have also noted the
phase fraction of these precipitate phases may be changed because of the chemical
concentration and aging conditions. In the T6 heating condition, the β” phase is the
prime precipitate phase with a high volume fraction when the aging time is less than
half an hour, while the β’ phase only forms slightly. When the aging time prolongs 3
hours, the prime precipitate phase is the β’ phase, and almost no β” phase exists. The
following chart given by C.D. Marioara in 2006 shows the phase volume fractions at
different aging times of these precipitate phases:
Table 3 Si/Mg ratio and aging treatment effect on the prime precipitate phase volume fraction

Si/Mg ratio Aging treatment Prime precipitate phase

2 3h 97% U1 phase(Al2MgSi2)
1.25 10 min 76% β”
3h 26% β’, 65% U2(Al4Mg4Si4)
0.8 10 min 100% β’
3h 97% β’
In this paper, the volume loss of primary strengthening precipitates is the phase
transformation from β” phase to β’ phase. Since T6 aging process is applied in this
experiment, β phase which is usually formed in over aging process is not included in
phase transformation. This phase transformation also explains the overestimation of
yield strength at peak aging.

3.2 Volume fraction of primary strengthening precipitates

The primary strengthening precipitate is β” of A356 alloy in aging process. In this
work, the volume fraction of the precipitate phase is used as an input to predict the
final yield stress. The basic equation of the volume fraction applied JMAK equation:
𝑓𝑟 = 1 − 𝑒𝑥𝑝(−𝑘(𝑇)𝑡)𝑛 3
This equation can be applied to the pre-aged aluminum alloys in isothermal or non-
isothermal aging processes. For the present work, with our A356 experiment, the
isothermal aging process is mainly taken into consideration. Parameter n in equation 3
is determined based on the shape and composition of the precipitates. Parameter k is a
temperature dependent constant, which is calculated by a form of the Arrhenius-type
relationship. The JMAK equation is adapted to predict the relative volume fraction
change of the precipitates during the nucleation and growth process.

Based on JMAK equation’s assumption, the volume fraction of primary strengthening

precipitates remains constant at peak state and after. However, with prolonging the
aging process, some of the meta-stable precipitates will form from primary
strengthening precipitates, which reduces the volume fraction of primary
strengthening precipitates. Ford’s patent proposed a method to optimize heat
treatment via predicting thermal growth of the precipitate phases.

The second term is the volume fraction of the precipitate phase, which is calculated
based on the JMAK equation. This term is determined by the thermally dependent
equilibrium phase fraction of the precipitate phase 𝑓𝑒𝑞 (𝑇) , and the temperature-
dependent kinetic growth coefficient. The function 𝑓𝑒𝑞 (𝑇) utilizes the computational
thermodynamic method, considering the complexities of the precipitation-hardened
alloys, and is only dependent on temperature.

The temperature-dependent kinetic growth coefficient k(T) is used to predict the

volume at different heating times. The following equations show k values of the
primary precipitate phase 𝛽 ′ ′ and transformed phase 𝛽 ′ of A356 alloy:
90000
𝑘𝛽′ = 8750 𝑒𝑥𝑝 (− ) 4
𝑅𝑇
76000
𝑘𝛽′′ = 19000 𝑒𝑥𝑝 (− ) 5
𝑅𝑇
After obtaining the above terms, the volume fraction of the precipitate phase, after
considering the transformation from the precipitate phase to the stable phase, can be
modified as:

𝛽 ′′
𝑓𝑟 = 1 − 𝑒𝑥𝑝[−𝑘𝛽′′ (𝑇)𝑡] 6
𝛽′
𝑓𝑟 = 1 − 𝑒𝑥𝑝[−𝑘𝛽′ (𝑇)𝑡] 7
𝛽 ′′ 𝛽′
𝑓𝑟 = 𝑓𝑟 − 𝑓𝑟 8
Since the precipitate phases will tend to grow or transform before being stabilized and
bring detritus deformation to the alloy system, the volume fraction for the primary
precipitate phase is not constant as previous assumption claimed. The combination of
the volume evolution and transformation among the precipitate phases during the
aging process can reduce the volume fraction of primary strengthening precipitate.
Therefore, the relative volume fraction of primary strengthening precipitate, which is
used as input data, is obtained from the difference between the value calculated from
equation 6 and 7.

3.3 Modeling of yield strength of A356 in underaged process

The yield strength model of A356 is composed of three terms: the intrinsic strength of
aluminum (𝜎𝑖 ); the solid solution strength (𝜎𝑠𝑠 ) and precipitate strength (𝜎𝑝𝑝𝑡 ). They
are summarized by following equation:
𝜎 = 𝜎𝑖 + 𝜎𝑠𝑠 + 𝜎𝑝𝑝𝑡 9
Generally, the intrinsic yield strength of aluminum matrix is treated as constant during
aging process. The strength contribution of solid solution is affected by the
transformed volume fraction of primary strengthening precipitates, which are
nucleated from solid solution. Esmaeili gave the mathematical relationship between
solid solution strengthening and volume fraction of primary strengthening phase:
𝜎𝑠𝑠 = 𝜎0𝑠𝑠 (1 − 𝑓𝑟 )2/3 10
The strengthening mechanism of A356 in aging process is governed by the interactive
motion of precipitate particles and dislocations. Based on different size of precipitate
particles played as obstacles to mobile dislocation, the strengthening mechanism can
be categorized as strong or weak strengthening depending on dislocation breaking
angle. When the breaking angle is smaller than 120°, the mobile dislocation could by-
passing this kind of precipitate particle (which is called strong obstacles) and leave a
dislocation loop behind so that increasing dislocation density. When the breaking
angle is larger than 120°, the mobile dislocation will be cut when encountering with
precipitate particles, in this circumstance, such particles are called small obstacles.
During underaged period, the precipitate strengthening mechanism is governed by
strong obstacles strengthening. The contribution of precipitate strengthening is
calculated by following equation:
𝑀𝐹
𝜎𝑝𝑝𝑡 = 11
𝑏𝐿
where M is the Taylor factor, b is magnitude of the Burgers vector, F is the average
force of a particle and L is the average space of the particles. The average obstacle
strength F has a simple linear relationship with the average radius of the differently
shaped or oriented precipitates, and the average spacing L is defined as the planar
center-to-center distance between the strong particles or adapted to Freidel statistics
for a triangular array of weak obstacles, which is related to the volume fraction of the
precipitate phase. [7-9] At peak-aging, Equation 11 may be rewritten as:
1
𝜎𝑝𝑝𝑡 = 𝐴𝑓𝑟 2 12
where A is a constant related to the material, and 𝑓𝑟 represents the relative volume
fraction of the precipitate phase. [10] Under this condition, the yield stress is only
dependent on the volume fraction, regardless of the morphology of the precipitate
phases. Constant A will be calibrated by yield strength at peak aged state. The
average obstacle strength is taken as a simple linear relationship with the average
radius of the differently shaped or oriented precipitates, and the average spacing is
defined as the planar center-to-center distance between the strong particles or adapted
by Freidel statistics for a triangular array of weak obstacles. Therefore, the total yield
strength is obtained as followed:
1
𝜎 = 𝜎𝑖 + 𝜎0𝑠𝑠 (1 − 𝑓𝑟 )2/3 + 𝐴𝑓𝑟 2 13
The flow chart of modified precipitate hardening model for A356 is pictured as
followed:
 Figure 1 Flow chart of modified precipitate hardening model for A356

3.4 Simulation results compare between original precipitate hardening model and
modified model
Based on original precipitate hardening, the volume fraction of primary strengthening
precipitates remains constant at and after peak aged state. The strengthening
mechanisms at under aged and over aged period are different. In this project, the
under aged aging process is concerned. Figure 2 gives the simulation results obtained
from original model. It can tell from the figure that the simulation results predict large
underestimation at 150C compared with experimental data while give overestimation
at 170C. Since during under aged state, the strong strengthening mechanism is used to
calculate the yield strength, the volume fraction of primary strengthening phase
should be the only reason to cause this derivation. If the volume fraction of primary
strengthening precipitate is the only concerned parameter, the deviation of
experimental results and simulation results cannot be eliminated.

Figure 2 Yield strength prediction results by original model at different aging conditions

When applying the optimized model, the difference kinetic parameters of primary
strengthening precipitate 𝛽 ′ ′and transformed precipitate 𝛽 ′ can help to improve the
accuracy of volume fraction value at both lower temperature and high temperature.
The needed parameters for optimized model are given in Table 2 and the simulation
results for A356 are given in Figure 3:
Table 2 The value of parameters used in optimized model

Adjusted parameters used in optimized model Value

𝜎𝑖 10MPa
𝜎0𝑠𝑠 42MPa
A 240MPa
 Figure 3 Yield strength predicted by optimized model for A356 at different aging temperatures

It can be seen from the figure that the predicted results match well with the
experimental data. While the previous precipitate hardening model is overestimated
during the over-aging period, the modified model shows the drop of the yield stress
since the difference volume fraction of primary precipitate and transformed
precipitate reduce at 150C. At 170C, the overestimation of original simulation results
also eliminate because of more primary precipitate occurred in under aged state than
transformed precipitate compared with low temperature.

4 Summary
The precipitate hardening model based on JMAK equation is reviewed and applied to
predict A356 aging response behaviors. The volume fraction is used as an input to
study the microstructure evolution, according to the JMAK equation. It is concluded
that applying the JMAK equations is much easier to do, and a satisfactory simulated
result can be obtained from the original model. However, the simulated result is
overestimated in the over aged period at high temperature while underestimated in
under-aged period. The volume fraction of the prime strength precipitate for
hardening is considered to remain the same since there is no newly formed precipitate
particles in pure growth period in under-aged period and large precipitate particle will
consume small particles in over-aged period. While in reality, the transformation
among the prime precipitate phase and the other precipitate phases or stable phases
can lead to extra volume losses during the whole aging process. Therefore, the
modification of the volume fraction should be considered. In this project, the
optimized method of volume fraction is learnt from thermal growth model based on
Ford’s patent. By applying this model, the volume change due to the newly formed
precipitates in the aging process and the volume fraction of each phase can be
calculated. The differences in volume transformation kinetics of primary
strengthening precipitate 𝛽 ′ ′ and transformed precipitate 𝛽 ′ can adjust such derivation.
The optimized precipitate hardening model is presented in this report.

References
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Contribution
In the paper of “Competitive relationship between thermal effect and grain boundary
precipitates on the ductility of an as-quenched Al–Cu–Mn alloy”, the experiments
including tensile tests and observations of fracture surface under SEM are conducted.
The phenomena that ductility of Al-Cu-Mn alloy change a lot when at low
temperature and high temperature ranges is observed in tensile tests during different
temperatures. Then, a series of tensile tests under different strain rates and
temperatures are conducted, which proves temperature has dominant impact on
ductility behaviors. The observations of fracture surface via SEM show there are
precipitates gathered at the grain boundary, and such precipitates, which are identified
with EDS, are confirmed to reduce bonding energy of neighbor grains. Thus, the
ductility reduction at high temperature range can be explained by precipitates
formation.

In the paper of “Temperature-dependent constitutive behavior with consideration of

microstructure evolution for as-quenched Al-Cu-Mn alloy”, a modified constitutive
model is proposed to describe mechanical behaviors of as-quenched Al-Cu-Mn alloy
from room temperature to high temperature (up to 500C). Stress-strain curves
obtained tensile tests behaves different pattern at the low temperature range and the
high temperature range. Arrhenius type constitutive model with Zenor-Hollomon
parameters is modified since the classical version of this model is only valid at high
temperature range with the assumption that hot deformation energy remains constant.
My contribution in this paper is release the assumption so that hot deformation energy
is a function of temperature. At low temperature, the forest dislocations dominates the
deformation properties and precipitates formed at high temperature interact with
mobile dislocation and reduce bonding energy of grains affect deformation behaviors
at high temperature range. At last, the final temperature-dependent constitutive
models are established in three temperature ranges, low temperature range, transition
temperature range and high temperature range.
In the paper of “A brief review of precipitate hardening models for aluminum alloy”,
my work is focus on previous classical precipitate hardening models for aluminum
alloy and categorized them into mean value approach and discrete value approach
based on the simulation method of morphology and volume fraction of precipitate
phases. The regeneration of classical models is accomplished with experiment data of
A356 offered by General Motors Company.

At last, a modified precipitate hardening model is put forward to obtain more accurate
simulation results of A356 aging response behaviors. The volume fraction of primary
strengthening precipitate 𝛽 ′ ′ is adjusted by considering the volume loss caused by
phase transformation. The 𝛽 ′ phase will nucleate from 𝛽 ′ ′ phase so that the total
volume fraction of 𝛽 ′ ′ is reduced. The optimized mode gives perfect fitting results
compared with experiment data.