Calculation of the γ-TiAl Lattice Resistance
The dislocation width and lattice resistance (Peierls stress) of a γ-TiAl alloy are calculated by the density ratio method. The lattice resistance is shown to decrease with the dislocation width. The relationship between the Peierls stress and dislocation width variation is defined by theoretical de...
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irk-123456789-1883032023-02-23T01:27:21Z Calculation of the γ-TiAl Lattice Resistance Feng, R.C. Li, L.L. Li, H.Y. Wang, Z.M. Zhu, Z.X. Научно-технический раздел The dislocation width and lattice resistance (Peierls stress) of a γ-TiAl alloy are calculated by the density ratio method. The lattice resistance is shown to decrease with the dislocation width. The relationship between the Peierls stress and dislocation width variation is defined by theoretical derivation. The yield stress is negatively correlated with the shear stress of the material. It can become a useful tool for choosing an appropriate shear stress under deformation. Методом отношения плотностей рассчитаны ширина дислокации и сопротивление решетки (напряжение Пайерлса). Результаты показывают, что сопротивление решетки уменьшается с увеличением ширины дислокации. Путем расчета получена зависимость между напряжением Пайерлса и дислокационным упрочнением, свидетельствующая о том, что предел текучести отрицательно коррелирует с напряжением сдвига материала. Поэтому она может дать полезную информацию для выбора соответствующего напряжения сдвига в процессе растяжения. Методом відносини щільностей розраховані ширина дислокації і опір решітки (напруження Пайерлса). Результати показують, що опір решітки зменшується зі збільшенням ширини дислокації. Шляхом розрахунку отримана залежність між напруженням Пайерлса і дислокаційним зміцненням, яка свідчить про те, що межа плинності негативно корелює з напруженням зсуву матеріалу. Тому вона може дати корисну інформацію для вибору відповідного напруження зсуву в процесі розтягування. 2019 Article Calculation of the γ-TiAl Lattice Resistance / R.C. Feng, L.L. Li, H.Y. Li, Z.M. Wang, Z.X. Zhu // Проблеми міцності. — 2019. — № 1. — С. 65-71. — Бібліогр.: 21 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/188303 539.4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Научно-технический раздел Научно-технический раздел Feng, R.C. Li, L.L. Li, H.Y. Wang, Z.M. Zhu, Z.X. Calculation of the γ-TiAl Lattice Resistance Проблемы прочности |
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The dislocation width and lattice resistance (Peierls stress) of a γ-TiAl alloy are calculated by the density ratio method. The lattice resistance is shown to decrease with the dislocation width. The relationship between the Peierls stress and dislocation width variation is defined by theoretical derivation. The yield stress is negatively correlated with the shear stress of the material. It can become a useful tool for choosing an appropriate shear stress under deformation. |
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Feng, R.C. Li, L.L. Li, H.Y. Wang, Z.M. Zhu, Z.X. |
| author_facet |
Feng, R.C. Li, L.L. Li, H.Y. Wang, Z.M. Zhu, Z.X. |
| author_sort |
Feng, R.C. |
| title |
Calculation of the γ-TiAl Lattice Resistance |
| title_short |
Calculation of the γ-TiAl Lattice Resistance |
| title_full |
Calculation of the γ-TiAl Lattice Resistance |
| title_fullStr |
Calculation of the γ-TiAl Lattice Resistance |
| title_full_unstemmed |
Calculation of the γ-TiAl Lattice Resistance |
| title_sort |
calculation of the γ-tial lattice resistance |
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Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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2019 |
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Научно-технический раздел |
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http://dspace.nbuv.gov.ua/handle/123456789/188303 |
| citation_txt |
Calculation of the γ-TiAl Lattice Resistance / R.C. Feng, L.L. Li, H.Y. Li, Z.M. Wang, Z.X. Zhu // Проблеми міцності. — 2019. — № 1. — С. 65-71. — Бібліогр.: 21 назв. — англ. |
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Проблемы прочности |
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AT fengrc calculationofthegtiallatticeresistance AT lill calculationofthegtiallatticeresistance AT lihy calculationofthegtiallatticeresistance AT wangzm calculationofthegtiallatticeresistance AT zhuzx calculationofthegtiallatticeresistance |
| first_indexed |
2025-07-16T10:18:21Z |
| last_indexed |
2025-07-16T10:18:21Z |
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1837798367576457216 |
| fulltext |
UDC 539.4
Calculation of the �-TiAl Lattice Resistance
R. C. Feng,
a,b
L. L. Li,
a,b
H. Y. Li,
a,b,1
Z. M. Wang,
a,b
and Z. X. Zhu
a,b
a Key Laboratory of Digital Manufacturing Technology and Application, The Ministry of Education,
Lanzhou University of Technology, Lanzhou, China
b School of Mechanical and Electronical Engineering, Lanzhou University of Technology, Lanzhou,
China
1 cfrcly76@126.com
The dislocation width and lattice resistance (Peierls stress) of a �-TiAl alloy are calculated by the
density ratio method. The lattice resistance is shown to decrease with the dislocation width. The
relationship between the Peierls stress and dislocation width variation is defined by theoretical
derivation. The yield stress is negatively correlated with the shear stress of the material. It can
become a useful tool for choosing an appropriate shear stress under deformation.
Keywords: �-TiAl alloy, lattice resistance, dislocation, crystal interplanar spacing.
Introduction. The progress in aerospace and automotive industries require a higher
level of comprehensive properties and lower weight materials compared to the currently
used ones [1]. Because of its advantages, TiAl alloy has become a new type of
high-temperature structure material with huge development potential and application
prospects at high temperature [2, 3]. TiAl-based alloys have attracted wide attention [4].
However, as an intermetallic compound, low ductility and fracture toughness are its main
weakness [5, 6]. So, it is very vital for improving the low temperature ductility to make
these materials useful for industrial applications [7].
For high-temperature materials, the elastic modulus and yield strength are important
mechanical properties of TiAl alloy [8]. The binary aluminum–titanium phase diagram is
quoted in Fig. 1 [9] to show the intermetallic parts that we are focusing in this work. The
plastic response of crystalline materials to an applied stress is determined by dislocation
motion. Processes such as fatigue, dynamic recovery, and all stages of the plastic
deformation curve (in particular, stage III) are controlled by the dynamic behavior of
dislocation, including motion, cross-slip, multiplication and annihilation [10]. The stress–
strain curve of single crystal material often has the form shown in Fig. 2. The curve in
Fig. 2 consists of three phases: stages I, II, and III. Dislocation motion is a direct cause of
crystal deformation. It can be activated only under certain stress conditions. There are many
resistances need to be overcome in the process of the movement of dislocation. For
example, precipitates, grain and twin boundaries [11]. The mobility of these defects largely
determines the mechanical properties of the materials. Therefore, accurate prediction of
lattice resistance for a dislocation is important to understand the mobility of the dislocation
[12]. Zhang et al. [13] described the relationship between dislocation slip and fracture. But
the relevant calculations is not given for the Peierls stress. In this paper, the width of
dislocation and lattice resistance of �-TiAl alloy is calculated, the facts indicate the lattice
resistance decrease with the increase of the width of dislocation. In addition, the relationship
between critical resolved shear stress and yield strength is obtained by the theoretical
derivation.
1. The Method of Calculation the Crystal Interplanar Spacing. The method of
density ratio is used to calculate the crystal interplanar spacing [14]. The areal density � is
defined as the number of the total atoms of per unit area of crystal; the bulk density � is
© R. C. FENG, L. L. LI, H. Y. LI, Z. M. WANG, Z. X. ZHU, 2019
ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2019, ¹ 1 65
defined as the number of the total atoms of a unit volume of crystal. Thus, the interplanar
spacing of cubic lattices can be evaluated by
dhlk �
�
�
, (1)
where dhlk is crystal interplannar spacing, � is areal density, and � is bulk density.
2. Calculation of the Crystal Interplanar Spacing with Face Centered Tetragonal
(FCT) Structure. The crystal structure of �-TiAl alloy is shown in Fig. 3 [15]. According
to Fig. 3 the volume of unit cell is abc, including four atoms. So the bulk density of lattice
of �-TiAl is expressed by formula (2)
��
4
abc
. (2)
The calculation of areal density is quite complex than volume density. The arrangement
of atoms in different crystal interplanar is shown in Fig. 4. Taking an example for (100)
R. C. Feng, L. L. Li, H. Y. Li, et al.
66 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2019, ¹ 1
Fig. 1. The binary aluminum–titanium phase diagram.
Fig. 2. Stress–strain curves of single crystal. Fig. 3. The L10 structure of �-TiAl.
plane, which the area is ac. It contains two atoms. So the areal density can be represented
by formula (3)
� �
2
ac
. (3)
As is known above, the crystal interplanar spacing of (100) plane can be written as
d
b
( ) .100 2
� (4)
Other crystal interplanar spacing can be obtained by referring to the above method.
The results are listed in Table 1.
For the edge dislocation, the width of dislocation is a ( ),1� � where a is crystal
interplanar spacing and � is Poisson’s ratio. Through the above description, it is easy to
obtain the width of dislocation in different slip plane. Therefore, the dislocation width are
2.6, 1.82, and 3.03(Å) in (100), (110), and (111) plane, respectively. However, the width of
dislocation of screw dislocation is a. The width of dislocation on different slip planes is
2.0005, 1.41, and 2.24(Å), respectively. It is obvious that the width of dislocation on the
(111) plane is the largest.
3. Theoretical Calculations of Peierls–Nabarro Stress. Peierls and Nabarro first
estimated this fundamental quantity of the dislocation by using a continue model [16]. The
Peierls stress is given as
�
�
P N P N
b
E
G
b
� �� �
�
�
�
�
�
�
�
2 2
1
2
2 ( )
exp , (5)
Calculation of the �-TiAl Lattice Resistance
ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2019, ¹ 1 67
a b c
Fig. 4. The arrangement of atoms in different crystal interplanar: (a), (b), and (c) represent (100),
(110), and (111) plane, respectively.
T a b l e 1
Crystal Interplanar Spacing of FCT Structure
Crystal
face
Graphics area Number
of atoms
Areal density
(�)
Bulk density
(�)
Interplanar
spacing (d)
(100) ac 2 2 ac 4 abc b 2
(110) 2ac 2 2 2ac 4 abc 2 4b
(111) 2
2
22 2a
a c�
2 4
2 22 2a a c�
4 abc bc
a c2 22 2�
where G is shear modulus, E is the Young modulus, � is Poisson’s ratio, b is Burgers
vector, and
is the width of dislocation. Table 2 is the physical parameter of TiAl alloy.
From the classical dislocation theory, the most favorable slip systems should contain
the close-packed lattice planes and the Burgers vectors with shortest shear displacements.
Thus, dislocation slip in the L10 FCT structure favors the (111) planes along close or
relatively close packed directions. According to the definition of the Burgers vector, the
shortest lattice translation vectors which join two points in the lattice is choose as the
Burgers vector. Substituting in Eq. (5)
�
P N� �
�
�
�
��
�
�
�
��
2 70
1 023
2 303
2 8.
exp
.
.
. (6)
The Peierls–Nabarro stress on (111) plane is 0.2041 GPa. It is the smallest lattice
resistance in the TiAl alloy. It is because that the atomic spacing (Burgers vector) of close
packed plane is least but crystal interplanar spacing (the width of dislocation) is largest.
From the above calculations, it is clear that Peierls–Nabarro stress is low in TiAl alloy. In
addition, the calculation of Peierls stress lead to an exponential dependence of � p on the
dislocation width. Thus, the lattice resistance decrease with the increase of the dislocation
width. The Peierls stress of other crystal interplanar is listed Table 3.
4. Relationship between the Peierls Stress and Dislocation Strengthening.
Dislocation strengthening is a very important strengthening mechanism in metal. It can be
caused by the increase in glide resistance that occurs when dislocations move, interact and
change their distribution and density. The general form of dislocation strengthening is given
by [19]:
� � � �� �0
1 2Gb / , (7)
where � 0 is the Peierls stress, G is shear modulus, � is mobile dislocation density, � is
a constant equaling about 0.3, and b is Burgers vector. The Peierls stress is a constant and
the value is relatively small in TiAl alloy. So the dislocation density play mainly role for
the dislocation strengthening. When considering hardening, the strain rate, dislocation
density and dislocation rate have a relationship as following [19]:
� .� �� mb v (8)
68 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2019, ¹ 1
R. C. Feng, L. L. Li, H. Y. Li, et al.
T a b l e 2
The Physical Parameter of TiAl Alloy
Alloy Shear modulus
(GPa)
Elastic modulus
(GPa)
Poisson’s ratio Reference
Ti–49.5Al 70 176 0.23 [17]
Ti–25.5Al 56 144 0.28 [18]
T a b l e 3
The Peierls Stress of Other Crystal Interplanar
Alloy (100) (GPa) (110) (GPa) (111) (GPa)
Ti-49.5Al 0.5329 3.0609 0.2041
Ti-25.5Al 0.3064 1.9104 0.1419
Substituting into Eq. (7)
� � �
�
� �
�
�
�
�
�0
1 2
Gb
mbv
�
,
/
(9)
where v is dislocation rate and m is Schmid factor.
Studies on face-centered cubic (FCC) and hexagonal close-packed (HCP) crystals
have shown that at the critical resolved shear stress (CRSS) for macroscopic slip satisfies
[16]
v A n� � . (10)
Thus, the Eq. (9) can become
� � �
�
�
� �
�
�
�
�
�0
1 2
Gb
mbA n
�
,
/
(11)
where A is a material constant and n is a constant.
During the tensile deformation, the strain rate can be treated as a fixed value.
Therefore, only when the CRSS is the smallest, the effect of strengthening is most obvious.
In polycrystal, the relationship between the yield stress and the dislocation is the same as
that observed for single crystal [18]. But the individual grains usually have a random
orientation with respect to one another. So grain boundaries have a very important role in
the deformation of polycrystalline materials. From experiment measurement of the yield
stress of poly aggregates in which grain size d is the only material variable, it has been
found that Hall–Petch relationship is satisfied [19]:
� �y kD� �0
1 2/ . (12)
where � y is the yield stress, � 0 is a frictional stress required to move dislocation, k is
the Hall–Petch slope, and D is the grain size. This indicates that the mechanical properties
of TiAl alloy can be improved by the refinement of grain size.
5. Results and Discussion. The Peierls stress arises as a direct consequence of the
periodic structure of the crystal lattice and depends sensitively on the form of the
force–distance relation among individual atoms. But the dependence of � p on temperature
is complex. Dietze [20] took account of thermal vibrations by smearing the sampling points
with a Gaussian distribution corresponding to the thermal vibrational motion. As a result, if
the dislocation is
( )T Burgers vector wide at a temperature of T [
�( )T b� 2 ], it is
related to its width at 0 K,
0, by [21]
( ) ( )exp( ),T T Tm� 0 3 (13)
where Tm is the melting point and � is half-width of the dislocation.
Substituting this relationship into Eq. (5), one obtains the � p at temperatures above
the Debye temperature, � p T( ),
�
�
�p
T T
p
T T
G G
m m
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�1
1
3 3/ exp( / )
, (14)
where G is shear modulus at 0 K.
Equation (14) can be used to probe the temperature dependence of � p .
ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2019, ¹ 1 69
Calculation of the �-TiAl Lattice Resistance
Conclusions. To summarize, the crystal interplanar spacing of �-TiAl alloy is
obtained by the method of density ratio in this paper. Besides, the width of dislocation and
lattice resistance of �-TiAl with a L10-type ordered structure is calculated. The calculation
shows that there is a clear correlation between � p and
b for the �-TiAl alloy. This is
consistent with the characteristics of other metal. Through the calculation of the Peierls
stress of �-TiAl. It can provide an effective evidence for the movement of dislocation and
the activation of the slip system. In addition, the relationship of critical resolved shear stress
and yield stress is obtained by the theoretical derivation. It indicates that only when the
critical resolved shear stress is the smallest, the reinforcement is most obvious.
Acknowledgments. This work was supported by a grant from the National Science
Fundation of China (No. 51665030) and the Program for ChangJiang Scholars and
Innovative Research Team in University of Ministry of Education of China (No. IRT_15R30)
and Doctoral research Foundation of Lanzhou University of Technology. The authors wish
to thank Engineering Research center of Nonferrous Metallurgy’s New Equipment,
Ministry of Education, Lanzhou University of technology for providing help.
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Received 15. 03. 2018
ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2019, ¹ 1 71
Calculation of the �-TiAl Lattice Resistance
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/ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
>>
/Namespace [
(Adobe)
(Common)
(1.0)
]
/OtherNamespaces [
<<
/AsReaderSpreads false
/CropImagesToFrames true
/ErrorControl /WarnAndContinue
/FlattenerIgnoreSpreadOverrides false
/IncludeGuidesGrids false
/IncludeNonPrinting false
/IncludeSlug false
/Namespace [
(Adobe)
(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AddBleedMarks false
/AddColorBars false
/AddCropMarks false
/AddPageInfo false
/AddRegMarks false
/ConvertColors /ConvertToCMYK
/DestinationProfileName ()
/DestinationProfileSelector /DocumentCMYK
/Downsample16BitImages true
/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements false
/GenerateStructure false
/IncludeBookmarks false
/IncludeHyperlinks false
/IncludeInteractive false
/IncludeLayers false
/IncludeProfiles false
/MultimediaHandling /UseObjectSettings
/Namespace [
(Adobe)
(CreativeSuite)
(2.0)
]
/PDFXOutputIntentProfileSelector /DocumentCMYK
/PreserveEditing true
/UntaggedCMYKHandling /LeaveUntagged
/UntaggedRGBHandling /UseDocumentProfile
/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice
|