Study on elastic deformation of substitution alloy AB with interstitial atom C and BCC structure under pressure
TẠP CHÍ KHOA HỌC SỐ 20/2017
55
STUDY ON ELASTIC DEFORMATION OF SUBSTITUTION
ALLOY AB WITH INTERSTITIAL ATOM C AND BCC STRUCTURE
UNDER PRESSURE
Nguyen Quang Hoc1, Nguyen Thi Hoa2 and Nguyen Duc Hien3
1Hanoi National University of Education
2University of Transport and Communication
3Mac Dinh Chi High School
Abstract: The analytic expressions of the free energy, the mean nearest neighbor distance
between two atoms, the elastic moduli such as the Young modulus E, the bulk modulus K,
the rigidity modulus G and the elastic constants C11, C12, C44 for substitution alloy AB
with interstitial atom C and BCC structure under pressure are derived from the statistical
moment method. The elastic deformations of main metal A, substitution alloy AB and
interstitial alloy AC are special cases of elastic deformation for alloy ABC. The theoretical
results are applied to alloy FeCrSi. The numerical results for alloy FeCrSi are compared
with the numerical results for main metal Fe, substitution alloy FeCr, interstitial alloy FeSi
and experiments.
Keywords: Substitution and interstitial alloy, elastic deformation, Young modulus, bulk
modulus, rigidity modulus, elastic constant, Poisson ratio.
Email: hoanguyen1974@gmail.com
Received 02 December 2017
Accepted for publication 25 December 2017
1. INTRODUCTION
Thermodynamic and elastic properties of interstitial alloys are specially interested by
many theoretical and experimental researchers [1-7, 10, 12, 13].
In this paper, we build the theory of elastic deformation for substitution alloy AB with
interstitial atom C and body-centered cubic (BCC) structure under pressure by the statistical
moment method (SMM) [8-10].
56
TRƯỜNG ĐẠI HỌC THỦ ĐÔ HÀ NỘI
2. CONTENT OF RESEARCH
2.1. Analytic results
In interstitial alloy AC with BCC structure, the cohesive energy of the atom C (in face
centers of cubic unit cell) with the atoms A (in body center and peaks of cubic unit cell) in
the approximation of three coordination spheres with the center C and the radii r ,r 2,r 5
1
1
1
is determined by [8-10].
ni
1
2
1
2
u
r
2 r 4
r
2 8
r
1
5
AC
0C
i
AC
1
AC
1
AC
i1
r 2
r
2 4
r
1
5 ,
(2.1)
AC
1
AC
1
AC
n
where
is the interaction potential between the atom A and the atom C, is the number
AC
i
th
r(i 1,2,3),
of atoms on the i coordination sphere with the radius
r º r r y0 A (T) is
i
1
1C
01C
1
the nearest neighbor distance between the interstitial atom C and the metallic atom A at
r
temperature T,
is the nearest neighbor distance between the interstitial atom C and the
01C
metallic atom A at 0K and is determined from the minimum condition of the cohesive energy
u0C
, y0 A (T ) is the displacement of the atom A1(the atom A stays in the body center of cubic
1
unit cell) from equilibrium position at temperature T. The alloy’s parameters for the atom C
in the approximation of three coordination spheres have the form [8-10].
2
1
AC
2
16
(2)
AC
(1)
AC
(1)
AC
4 2C
,
kC
r
r 2
r 5 ,
2
C
1C
1
1
1
ui2
r
1
5 5r
i
eq
1
4
ui4
1
1
2
1
4 5
1
(4)
(2)
(1)
(4)
(3)
AC
1C
(r )
(r 2)
(r )
(r 2)
(r 5),
AC
1
AC
1
AC
1
AC
1
AC
1
48
24
8r2
16r3
150
125r
1
i
eq
1
1
6
4AC
1
1
5
2
(3)
AC
(2)
AC
(1)
AC
(3)
AC
2C
(r )
(r )
(r )
(r 2)
1
48
1
1
1
ui2 ui2
4r
4r2
8r3
8r
1
i
1
1
1
eq
1
2
25
3
2
3
(2)
(4)
AC
(3)
(2)
(1)
AC
(r 2) (r 5)
(r 5)
(r 5)
(r 5),
AC 1C
1
AC
1
AC
1
1
8r2
25r2
25r 5
25r3 5
1
1
1
1
(2.2)
where(i) (r ) 2AC (r ) / r2 (i 1,2,3,4),, x, y, z, and ui is the displacement
AC
i
i
i
of the ith atom in the direction .
TẠP CHÍ KHOA HỌC SỐ 20/2017
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The cohesive energy of the atom A1 (which contains the interstitial atom C on the first
coordination sphere) with the atoms in crystalline lattice and the corresponding alloy’s
parameters in the approximation of three coordination spheres with the center A1 is
determined by [8-10]
u0A u0A AC
r
,
1A
1
1
2
AC
1
5
(2)
AC 1A
(1)
AC 1A
2
kA kA
kA
r
r
, 4 2 A
,
A
1A
1
ui2
2r
1
1
1
1
1
i
1A
eq
1
rr
1A
1
4
AC
1
1
24
1
1
(4)
AC 1A
(2)
AC 1A
(1)
AC 1A
1A 1A
1A (r )
(r )
(r ),
48
ui4
8r2
8r3
1
1
1
1
i
1A
1A
1
eq
1
rr
1A
1
4AC
ui2 ui2
1
3
4r2
1A
1
3
6
(3)
AC 1A
(2)
AC 1A
(1)
AC 1A
2A 2 A
2 A
(r )
(r )
(r )
4r3
1
1
1
1
48
2r
i
1A
1A
eq
1
1
rr
1A
1
(2.3)
where r r is the nearest neighbor distance between the atom A1and atoms in crystalline
1A
1C
1
lattice.
The cohesive energy of the atom A2 (which contains the interstitial atom C on the first
coordination sphere) with the atoms in crystalline lattice and the corresponding alloy’s
parameters in the approximation of three coordination spheres with the center A2 is
determined by [8-10].
u0A u0A AC
r
,
1A2
2
2
1
4
r
1A
2
(2)
AC 1A
(1)
AC 1A
kA kA
kA 2
r
r
, 4 2A
,
AC
2
A
1A
2
u2
2
2
2
2
2
i
i
eq
rr
1A
2
4
1
1
24
1
1
1
(4)
AC 1A
(3)
AC 1A
(2)
AC 1A
(1)
AC 1A
1A 1A
1A (r )
(r )
(r )
(r ),
AC
48
u4
4r
8r2
8r3
1A
2
2
2
2
2
2
i
i
1A
1A
eq
2
2
rr
1A
2
6
4AC
1
1
3
3
(4)
AC 1A
(3)
AC 1A
(2)
AC 1A
(1)
AC 1A
2A 2A
2A (r )
(r )
(r )
(r )
48 u2 u2
8r2
8r3
2
2
2
2
2
8
4r
1A
2
i
i
i
1A
1A
eq
2
2
rr
1A
2
(2.4)
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TRƯỜNG ĐẠI HỌC THỦ ĐÔ HÀ NỘI
where r r y0C (T),r isthe nearest neighbor distance between the atom A2 and
1A
01A
01A
2
2
2
atoms in crystalline lattice at 0K and is determined from the minimum condition of the
cohesive energy u0 A , y0C (T) is the displacement of the atom C at temperature T.
2
u0A,kA,1A,2A
In Eqs. (2.3) and (2.4),
are the coressponding quantities in clean metal
A in the approximation of two coordination sphere [8-10]
The equation of state for interstitial alloy AC with BCC structure at temperature T and
pressure P is written in the form
1 u0
1 k
Pv r
xcth x
.
(2.5)
1
6 r
2k r
1
1
At 0K and pressure P, this equation has the form
u0 0 k
Pv r
.
(2.6)
1
r
4k r
1
1
,
If knowing the form of interaction potential
eq. (2.6) permits us to determine the
i0
nearest neighbor distance r P,0 X C, A, A , A at 0K and pressure P. After knowing
2
1X
1
r
1X
P,0 , we can determine alloy parametrs k (P,0), (P,0), (P,0), (P,0) at 0K
X
1X
2X
X
and pressure P. After that, we can calculate the displacements [8-10].
2 X (P,0) 2
y0X (P,T)
AX (P,T)
X
,
3kX3 (P,0)
i
5
YX
X
A a
aiX ,kX mX2 , xX
,a 1
,
(2.7)
X
1X
1X
2
k
2
2
i2
X
With aiX (i 1,2..., 5) are the values of parameters of crystal depending on the structure
of crystal lattice [10].
From that, we derive the nearest neighbor distance r P,T at temperature T and
1X
pressure P:
r (P,T) r (P,0) yA (P,T), r (P,T) r (P,0) yA (P,T),
1C
1C
1A
1A
1
r (P,T) r (P,T), r (P,T) r (P,0) yC (P,T).
(2.8)
1A
1C
1A2
1A2
1
Then, we calculate the mean nearest neighbor distance in interstitial alloy AC by the
expressions as follows [8-10].
TẠP CHÍ KHOA HỌC SỐ 20/2017
59
C 1A
r (P,T) r (P,0) y(P,T ), r (P,0) 1 c r (P,0) c r (P,0),
C
1A
1A
1A
1A
r (P,0) 3r (P,0), y(P,T) 1 7c y (P,T) c y (P,T) 2c y (P,T) 4c y (P,T),
C
1A
1C
A
C
C
C
A
C
A2
1
(2.9)
where r (P,T) is the mean nearest neighbor distance between atoms A in interstitial alloy
1A
AC at pressure P and temperature T, r (P,0) is the mean nearest neighbor distance between
1A
r (P,0)
atoms A in interstitial alloy AC at pressure P and 0K,
is the nearest neighbor
1A
r (P,0)
distance between atoms A in clean metal A at pressure P and 0K,
is the nearest
1A
neighbor distance between atoms A in the zone containing the interstitial atom C at pressure
P and 0K and cC is the concentration of interstitial atoms C.
In alloy ABC with BCC structure (interstitial alloy AC with atoms A in peaks and body
center, interstitial atom C in facer centers and then, atom B substitutes atom A in body
center), the mean nearest neighbor distance between atoms A at pressure P and temperature
T is determined by:
BTAC
BT
BTB
BT
aABC (P,T,cB ,cC ) cAC aAC
cBaB
, BT cAC BTAC cB BTB ,
1
1
cAC cA cC ,aAC r (P,T), BTAC
, BTB
,
1A
TAC
TB
3
aAC (P, T,cC )
a0 AC (P, 0,cC )
TAC (P, T,cC )
,
2
3
1
AC
2P
4aAC (P, T,cC ) 3N aA2C
T
2
2
2
2
2
2AC
A
A
1
2
17c
C
cC
2c
4c
,
C
C
AC
A
C
2
a2
a2
a2
aA
aA
2
2
T
T
T
r (P,T)
T
AC
A
C
T
T
1
2
1A
2
1 2u0 X X kX
1
kX
2
2
1
X
,a º r (P,T).
(2.10)
X
1X
3N aX2
6 aX2
4kX aX2 2kX a
T
X
The mean nearest neighbor distance between atoms A in alloy ABC at pressure P and
temperature T is determined by:
B0TAC
B0T
B0TB
B0T
a0 ABC (P, T,cB ,cC ) cACa0 AC
cBa0B
.
(2.11)
60
TRƯỜNG ĐẠI HỌC THỦ ĐÔ HÀ NỘI
c c c
The free energy of alloy ABC with BCC structure and the condition
has
C
B
A
the form:
ABC AC c TS AC TScABC
B A
,
B
c
AC 1 7c c 2c 4c TScAC
,
C
A
C
C
C
A
C
A2
1
2
21X
3
XX
X U0X 0X 3N
2X XX2
1
kX2
2
23 4
XX
XX
22X XX 1
2 2 2
1
1 X
,
X
2X
1X
1X
kX4
3
2
2
2xX
0X 3N xX ln(1 e
) , X X º xX coth xX .
(2.12)
is the free energy of interstitial alloy AC, ScAC
AC
where
is the free energy of atom X,
X
is the configuration entropy of interstitial alloy AC and ScABC is the configuration entropy
of alloy ABC.
The Young modulus of alloy ABC with BCC structure at temperature T and pressure P
is determined by:
EABC c E E E c E c E c c E E E c c E E ,
B
A
B
B
B
AC
B
B
A
A
A
A
AC
AB
A
A
AC
2 A
2 A
2C
2
1
2
2
4
2
2
1
E c E c E ,
E
,
EAC E 1 7cC cC
,
AB
A
A
B
B
A
A
2 A
2
.r A
1A 1A
2 A2 2
1
A
2
1
A
1
1 xActhx 1 x cthx , xA
,
A
1A
A
A
kA
kA4
2
2
2
2
2 X
2
1 U
3 X kX
1
kX
4r2
0X
01X
2 r2
4 kX r2
2kX r
1X
1X
1X
1 U0X
3
1 kX
X
2
kX
X cthxX
2r01X , xX
,X
,
(2.13)
2 r
2
2kX r
m
1X
1X
where is the relative deformation, EABC EABC (cB ,cC , P,T), EAB EAB c , P,T is the
B
Young modulus of substitution alloy AB and EAC EAC c , P,T is the Young modulus of
C
interstitial alloy AC.
TẠP CHÍ KHOA HỌC SỐ 20/2017
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The bulk modulus of alloy ABC with BCC structure at temperature T and pressure P
has the form:
EAB c , c , P,T
.
B
C
KABC c , c , P,T
(2.14)
B
C
3(1 2 )
A
The rigidity modulus of alloy ABC with BCC structure at temperature T and pressure P
has the form:
EABC c , c , P,T
B
C
GABC c , c , P,T
.
(2.15)
B
C
2 1
A
The elastic constants of alloy ABC with BCC structure at temperature T and pressure P
has the form:
EABC c , c P,T 1
B
C
A
C11ABC c , c , P,T
,
(2.16)
(2.17)
(2.18)
B
C
1
1 2
A
A
EABC c , c , P,T
B
C
A
C12 ABC c , c , P,T
,
B
C
1
1 2
A
A
EABC c , c , P,T
B
C
C44 ABC c , c , P,T
.
B
C
2 1
A
The Poisson ratio of alloy ABC with BCC structure has the form:
ABC cAA cBB cCC cAA cBB AB.
(2.19)
,
respectively are the Poisson ratioes of materials A, B and C and are
where
and
A
B
C
determined from the experimental data.
When the concentration of interstitial atom C is equal to zero, the obtained results for
alloy ABC become the coresponding results for substitution alloy AB. When the
concentration of substitution atom B is equal to zero, the obtained results for alloy ABC
become the coresponding results for interstitial alloy AC. When the concentrations of
substitution atoms B and interstitial atoms C are equal to zero, the obtained results for alloy
ABC become the coresponding results for main metal A.
2.2. Numerical results for alloy FeCrSi
For alloy FeCrSi, we use the n-m pair potential
62
TRƯỜNG ĐẠI HỌC THỦ ĐÔ HÀ NỘI
m
D
r
r
n
0
0
(2.20)
(r)
m
n
,
n m
r
r
where the potential parameters are given in Table 1 [11].
Table 1. Potential parameters m, n, D, r0 of materials
16
10
Material
m
n
D 10 erg
m
r 10
0
Fe
Cr
Si
7.0
6.0
6.0
11.5
15.5
12.0
6416.448
6612.96
2.4775
2.4950
2.2950
45128.24
Considering the interaction between atoms Fe and Si in interstitial alloy FeSi, we use
the potential (2.20) but we take approximately
Therefore,
D DFe DSi , r r0Fer0Si .
0
m
D
r
r
n
0
0
(2.21)
Fe-Si (r)
m
n
,
n m
r
r
where
FeSi are taken as in Table 2 [10].
Table 2. Potential parameters
m n
and are determined empirically. The potential parameters for interstitial alloy
r
m n
, , 0 , D of alloy FeSi
16
10
m
n
Alloy
D 10 erg
m
r 10
0
FeSi
2.0
5.5
17016.5698
2.3845
According to our numerical results as shown in figures from Figure 1 to Figure 6 for
alloy FeCrSi at the same pressure, temperature and concentration of substitutrion atoms
when the concentration of interstitial atoms increases, the mean nearest neighbor distance
also increases. For example, for alloy FeCrSi at the same temperature, concentration of
substitution atoms and concentration of interstitial atoms when pressure increases, the mean
nearest neighbor distance descreases. For example for alloy FeCrSi at T = 300K, cCr = 10%,
cSi = 3% when P increases fro 0 to 70 GPa, r1 descreases from 2.4715A0 to 2.3683A0.
For alloy FeCrSi at the same pressure, temperature and concentration of interstitial
atoms when the concentration of substitution atoms increases, the mean nearest neighbor
distance descreases. For example for alloy FeCrSi at T = 300K, P = 50 GPa, CSi = 5% when
CCr increases from 0 to 15%r1 desceases from 2.4216 A0to 2.4178A0.
For alloy FeCrSi at the same pressure, concentration of substitution atoms and
concentration of interstitial atoms when temperature increases, the mean nearest neighbor
TẠP CHÍ KHOA HỌC SỐ 20/2017
63
distance increases. For example for alloy FeCrSi at P = 0, CCr = 10% và CSi = 3% when T
increases from 50K to 1000K, r1 increases from 2.4687A0 to 2.4801A0.
For alloy FeCrSi at the same pressure, temperature and concentration of substitutrion
atoms when the concentration of interstitial atoms increases, the elastic moduli E, G, K
increases. For example for alloy FeCrSi at T = 300K, P = 10GPa and CCr = 10% when CSi
increases from 0 to 5%, E increases from 18.4723.1010 Pa to 30.0379.1010Pa.
For alloy FeCrSi at the same temperature, concentration of substitution atoms and
concentration of interstitial atoms when pressure increases, the elastic moduli E, G, K
increases. For example for alloy FeCrSi at T = 300K, CCr = 10%, CSi = 1% when P inceases
from 0 to 70GPa, E inceases from 15.2862.1010Pa to 48.0400.1010Pa.
For alloy FeCrSi at the same pressure, temperature and concentration of interstitial
atoms when the concentration of substitution atoms increases, the elastic moduli E, G, K
desceases. For example for alloy FeCrSi at T = 300K, P = 30GPa, CSi = 5% when CCr tăng
từ 0 đến 15%, E desceases from 39.38931010 Pa to 39.2128.1010Pa.
For alloy FeCrSi at the same pressure, temperature and concentration of substitutrion
atoms when the concentration of interstitial atoms increases, the elastic constants
C11 , C12
,C44 increases. For example for alloy FeCrSi at T = 300K, P = 10GPa, CCr = 10% when CSi
inceases from 0 to 5%,
increases from 23.7286.1010 Pa to 38.5851.1010 Pa.
C11
For alloy FeCrSi at the same temperature, concentration of substitution atoms and
concentration of interstitial atoms when pressure increases, the elastic constants
C11 , C12
,C44 increases. For example for alloy FeCrSi at T = 300K, CCr = 10%, CSi = 1% when P
increases from 0 to70GPa,
increases from 14.6358.1010 Pa to 61.7096.1010 Pa.
C11
For alloy FeCrSi at the same pressure, temperature and concentration of interstitial
atoms when the concentration of substitution atoms increases, the elastic constants
C11 , C12
,C44 descreases. For example for alloy FeCrSi at T = 300K, P = 30GPa, CSi = 5% when CCr
increases from 0 to 15%
desceases from 51.6175.1010 Pa to 49.8943.1010 Pa.
C11
When the concentration of substitution atoms and the concentration of interstitial atoms
are equal to zero, the mean nearest neighbor distance, the elastic moduli and the elastic
constants of alloy FeCrSi becomes the mean nearest neighbor distance, the elastic moduli
and the elastic constants of metal Fe. The dependence of mean nearest neighbor distance,
the elastic moduli and the elastic constants on pressure and concentration of interstitial atoms
for alloy FeCrSi is the same as the dependence of mean nearest neighbor distance, the elastic
moduli and the elastic constants on pressure and concentration of interstitial atoms for
interstitial alloy FeSi. The dependence of mean nearest neighbor distance, the elastic moduli
and the elastic constants on pressure and concentration of substitution atoms for alloy
64
TRƯỜNG ĐẠI HỌC THỦ ĐÔ HÀ NỘI
FeCrSi is the same as the dependence of mean nearest neighbor distance, the elastic moduli
and the elastic constants on pressure and concentration of substitution atoms for substitution
alloy FeCr.
Table 3 gives the nearest neighbor distance and the elastic moduli of Fe at T = 300K,
P = 0 according to the SMM and the experimental data [12, 13].
3. CONCLUSION
The analytic expressions of the free energy, the mean nearest neighbor distance between
two atoms, the elastic moduli such as the Young modulus, the bulk modulus, the rigidity
modulus and the elastic constants depending on temperature, concentration of substitution
atoms and concentration of interstitial atoms for substitution alloy AB with interstitial atom
C and BCC structure under pressure are derived by the SMM. The numerical results for alloy
FeCrSi are in good agreement with the numerical results for substitution alloy FeCr,
interstitial alloy FeSi and main metal Fe. Temperature changes from 5 to 1000K, pressure
changes from 0 to 70 GPa, the concentration of substitution atoms Cr changes from 0 to 15%
and the concentration of interstitial atoms Si changes from 0 to 5%.
Table 3. Nearest neighbor distance and elastic moduli E, G of Fe at P = 0, T = 300K
according to SMM and EXPT [12, 13]
E 1010 Pa
G 1010 Pa
a(A0 )
Method
SMM
EXPT
2.4298
20.83
8.27
2.74[12]
20.98[13]
8.12[13]
55
50
45
40
35
30
25
20
15
10
5
70
E
G
K
C11
C12
C44
65
60
55
50
45
40
35
30
25
20
15
10
5
0
10
20
30
40
50
60
70
0
10
20
30
40
50
60
70
p (GPa)
p (GPa)
Figure 1. Dependence of elastic moduli E, G,
K (1010Pa) on pressure for alloy
Figure 2. Dependence of elastic constants
C11, C12, C44(1010Pa) on pressure for alloy
Fe-10%Cr-5%Si at T = 300K
Fe-10%Cr-5%Si at T = 300K
TẠP CHÍ KHOA HỌC SỐ 20/2017
65
55
50
45
40
35
30
25
20
15
10
5
40
E
G
K
C11
C12
C44
35
30
25
20
15
10
0
1
2
3
4
5
0
1
2
3
4
5
Nong do Si (%)
Nong do Si (%)
Figure 3. Dependence of elastic moduli E, G,
K (1010Pa) on concentration of Si for alloy
Fe-10%Cr-xSi at P = 30GPa and T = 300K
Figure 4. Dependence of elastic constants
C11, C12, C44 (1010Pa) on concentration of Si
for alloy Fe-10%Cr-xSi at P = 30GPa and
T = 300K
50
60
C11
C12
C44
E
G
K
55
50
45
40
35
30
25
20
15
10
45
40
35
30
25
20
15
10
0
5
10
15
0
5
10
15
Nong do Cr(%)
Nong do Cr(%)
Figure 5. Dependence of elastic moduli E, G,
K (1010Pa) on concentration of Cr for alloy
Fe-xCr-5%Si at P = 30GPa and T = 300K
Figure 6. Dependence of elastic constants
C11, C12, C44 (1010Pa) on concentration of Cr
for alloy Fe-xCr-5%Si at P = 30GPa and
T = 300K
REFERENCES
1. K. E. Mironov (1967), Interstitial alloy. Plenum Press, New York.
2. A. A. Smirnov (1979), Theory of Interstitial Alloys, Nauka, Moscow, Russian.
3. A. G. Morachevskii and I. V. Sladkov (1993), Thermodynamic Calculations in Metallurgy,
Metallurgiya, Moscow, Russian.
4. V.V.Heychenko, A.A.Smirnov (1974), Reine und angewandteMetallkunde in
Einzeldarstellungen 24, pp.80-112.
5. V. A. Volkov, G. S. Masharov and S. I. Masharov (2006), Rus. Phys. J., No.10, 1084 .
6. S. E. Andryushechkin, M. G. Karpman (1999), Metal Science and Heat Treatment 41, 2 80.
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7. M.Hirabayashi, S.Yamaguchi, H.Asano, K.Hiraga (1974), Reine und angewandteMetallkunde
in Einzeldarstellungen 24, p.266.
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162(1990), p.379.
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NGHIÊN CỨU BIẾN DẠNG ĐÀN HỒI CỦA HỢP KIM THAY THẾ
AB CẤU TRÚC LẬP PHƯƠNG TÂM KHỐI CÓ NGUYÊN TỬ C
XEN KẼ DƯỚI TÁC DỤNG CỦA ÁP SUẤT
Tóm tắt: Áp dụng phương pháp thống kê mô men vào nghiên cứu biến dạng đàn hồi của
hợp kim thay thế AB cấu trúc lập phương tâm khối có nguyên tử C xen kẽ, chúng tôi thu
được các biểu thức giải tích cho phép xác định các đại lượng: năng lượng tự do, khoảng
lân cận gần nhất giữa hai nguyên tử, mô đun Young E, mô đun khối K, mô đun trượt G và
các hằng số đàn hồi của các hợp kim này dưới tác dụng của áp suất. Các kết quả lý thuyết
được áp dụng tính số với hợp kimFeCrSi. Trong trường hợp giới hạn, các kết quả tính số
được so sánh với các số liệu thực nghiệm của kim loại Fe, hợp kim thay thế FeCr và hợp
kim xen kẽ FeSi.
Từ khóa: Hợp kim thay thế, hợp kim xen kẽ, biến dạng đàn hồi, mô đun Young, mô đun
khối, mô đun trượt, hằng số đàn hồi, hệ số Poisson.
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