Build theory of nonlinear deformation for BCC and FCC substitutional alloys AB with interstitial atom C under pressure

HNUE JOURNAL OF SCIENCE
DOI: 10.18173/2354-1059.2019-0030
Natural Sciences, 2019, Volume 64, Issue 6, pp. 45-56
This paper is available online at

BUILD THEORY OF NONLINEAR DEFORMATION FOR BCC
AND FCC SUBSTITUTIONAL ALLOYS AB
WITH INTERSTITIAL ATOM C UNDER PRESSURE

Nguyen Quang Hoc1, Nguyen Thi Hoa2 and Nguyen Duc Hien3
1

Faculty of Physics, Hanoi National University of Education
2
University of Transport and Communications
3
Mac Dinh Chi High School, Chu Pah, Gia Lai

Abstract. Analytic expressions of characteristic nonlinear deformation quantities
such as the density of deformation energy, the maximum real stress and the limit of
elastic deformation for bcc and fcc substitutional alloys AB with interstitial atom C
under pressure are derived by the statistical moment method. The nonlinear
deformations of the main metal A, the substitutional alloy AB and the interstitial
alloy AC are special cases for nonlinear deformation of substitutional alloy AB with
interstitial atom C and the same structure.
Keywords: Interstitial and substitutional alloy, binary and ternary alloys, nonlinear
deformation, density of deformation energy, maximum real stress, limit of elastic
deformation, statistical moment method.

1. Introduction
Thermodynamic and elastic properties of metals and interstitial alloys are specially

interested by many theoretical and experimental researchers [1-14]. For example in [1],
strengthening effects interstitial carbon solute atoms in (i.e., ferritic of bcc) Fe-C alloys
are understood, owning chiefly to the interaction of C with crystalline defects (e.g.
dislocations and grain boundaries) to resist plastic deformation via dislocation glide.
High-strength steels developed in current energy and infrastructure applications include
alloys where in the bcc Fe matrix is thermodynamically supersaturated in carbon. In [2],
structural, elastic and thermal properties of cementite (Fe3C) were studied using a
Modified Embedded Atom Method (MEAM) potential for iron-carbon (Fe-C) alloys.
The predictions of this potential are in good agreement with first-principle calculations
and experiments. In [3], the thermodynamic properties of binary interstitial alloys with
bcc structure are considered by the statistical moment method (SMM).

Received April 28, 2019. Revised June 22, 2019. Accepted June 29, 2019.
Contact Nguyen Quang Hoc, email address:
45


Nguyen Quang Hoc, Nguyen Thi Hoa and Nguyen Duc Hien

The analytic expressions of the elastic moduli for anharmonic fcc and bcc crystals are
also obtained by the SMM and the numerical calculation results are carried out for
metals Al, Ag, Fe, W and Nb in [4].
In this paper, we build the theory of nonlinear deformation for bcc and fcc
substitutional alloys AB with interstitial atom C by the SMM [3, 4, 15, 16].

2. Content
In the case of interstitial alloy AC with bcc structure (where the main atoms A stay
in body center and peaks, the interstitial atom C stays in face centers of cubic unit cell),
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) and the alloy’s parameters in the

approximation of three coordination spheres with the center C and the radii
r1bcc , r1bcc 2 , r1bcc 5 are determined by [3, 15]

u 0bcc
C 



1   2 AC

2 i  ui2

k Cbcc 

  4 ACF
1

48 i  u i4

 

bcc 3
1

16 r

2
bcc
1

8r







( 3)
 AC
r1bcc 2 

3
bcc
1

25 5r










1
( 4)
  1  AC

r1bcc 

8 r1bcc
 eq 24

 



(1)
 AC
r1bcc 2 

  4 AC
6

48 i  ui2 ui2





 

2







(1)


2 (1) bcc
16
( 2)
(1)
bcc
   AC
r1bcc  bcc  AC
r1
2 
 AC
r1bcc 5 ,  Cbcc  4  1bcc
C   2C ,
bcc

r1
5 5r1
 eq

 1bcc
C 

 2bcc
C 



1 ni
 AC (ri )   AC (r1bcc )  2 AC r1bcc 2  4 AC r1bcc 5 ,

2 i 1



 

 

8r



( 3)
 AC
r1bcc 5 









 

( 2)
 AC
r1bcc 2 

2

 

bcc 2
1

25 r





( 2)
 AC
r1bcc 2 



 

bcc 2
1





1 ( 4) bcc
4 5
( 3)
 AC r1
2 
 AC
r1bcc 5 ,
150
125r1bcc


1
( 3)
  1  AC
r1bcc 
bcc

4 r1bcc
 eq 4r1

1

2



2

1

 

bcc 3
1

8r





( 2)
r1bcc  
 AC



5

 

bcc 3
1

8r



(1)
 AC
r1bcc 2 

( 2)
 ACF
r1bcc 5 

3

 

bcc 3
1

25 5 r

(1)
r1bcc  
 AC





2 ( 4) bcc
 AC r1 5 
25





(1)
 AC
r1bcc 5 ,

(2)

where  AC is the interaction potential between the atom A and the atom C, ni is the
number of atoms on the ith coordination sphere with the radius ri (i  1, 2,3),
bcc
bcc
r1bcc  r1bcc
C  r01C  y 0 A1 (T ) is the nearest neighbor distance between the interstitial atom

C and the metallic atom A at temperature T, r01bccC is the nearest neighbor distance
between the interstitial atom C and the metallic atom A at 0K and is determined from
bcc
the minimum condition of the cohesive energy u 0bcc
C , y0 A1 (T ) is the displacement of the
atom A1(the atom A stays in the
bcc unit cell)
from equilibrium position
at temperature T,
46


Build theory of nonlinear deformation for bcc and fcc substitutional alloys AB with interstitial…
( m)

 AB
  m AC (ri ) / ri m (m  1,2,3,4,  ,   x, y.z,    and ui is the displacement of
the ith atom in the direction  .

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 [3, 15].

 





bcc
bcc
bcc
bcc
bcc
u0bcc
A1  u 0 A   AC r1 A1 ,  A1  4  1 A1   2 A1 ,

k

bcc
A1

bcc
 1bcc

A   1A 
1

 2bccA   2bccA 
1

k

bcc
A

1   2 AC
 
2 i  u i2


  4
1
  AC
48 i  u i4


  4
6
AC


48 i  u i2 u i2


 
5
( 2)
(1)
 
k Abcc   AB
r1bcc
 AC
r1bcc
A1 
A1 ,
bcc
 
2r1 A1
 eq  r r bcc

 

 

1 A1

 
1 ( 4) bcc
1
 
 1bcc
  AC
r1 A1 
A

 
24
8 r1bcc
 eq  r r bcc
A1

 

( 2)
r1bccA  
 AC

 

2

1

1

 

bcc 3
1 A1

8r

1 A1

 

1
3
( 3)
 
 2bcc
 AC
r1bcc
A 
A1 
bcc
 
2
r
4 r1bcc
1 A1
 eq  r r bcc
A1

 

 

2

( 2)
r1bccA  
 AC
1

1 A1

3

 

bcc 3
1 A1

4r

(1)
r1bccA ,
 AC
1

(1)
r1bccA ,
 AC

(3)

1

bcc
where r1bcc
is the nearest neighbor distance between atom A1 and atoms in
A1  r1C
crystalline 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 [3, 15]

 





bcc
bcc
bcc
bcc
bcc
u0bcc
A2  u 0 A   AC r1 A2 ,  A2  4  1 A2   2 A2 ,

k

bcc
A1

k

bcc
A

bcc
 1bcc
A   1A 

2

1   2 AC
 
2 i  u i2


  4
1
AC


4

48 i  u i



1

 

bcc 2
1 A2

8r



bcc

2 A2



bcc
2A

3

 

bcc 2
1 A2

8r

 

 

1 A2

 
1 ( 4) bcc
1
( 3)
 
 1bcc
  AC
r1 A2  bcc  AC

r1bcc
A
A2 
 
24
4
r
1
A
 eq  r r bcc
2

 

 

1 A2

( 2)
r1bccA  
 AC
2

  4
6
   2 AC2
48 i  u i u i



 
4 (1) bcc
( 2)
 
k Abcc  2 AC
r1bcc
A2  bcc  AC r1 A2 ,
 
r1 A2
 eq  r r bcc

1

 

bcc 3
1 A2

8r

(1)
r1bccA ,
 AC
2

 
1 ( 4) bcc
1
( 3)
 

 2bcc
 AC
r1bcc
A   AC r1 A2 
A2 
bcc
 
8
4r1 A2
 eq  r r bcc

 

 

1 A2

( 2)
r1bccA  
 AC
2

3

 

bcc 3
1 A2

8r

(1)
r1bccA ,
 AC
2

(4)
47


Nguyen Quang Hoc, Nguyen Thi Hoa and Nguyen Duc Hien
bcc
bcc
bcc
where r1bcc
between the atom
A2  r01A2  y 0C (T ), r01A2 is the nearest neighbor distance
A2and atoms in crystalline lattice at 0K and is determined from the minimum condition
bcc
of the cohesive energy u 0bcc
A2 , y 0C (T ) is the displacement of the atom C at temperature T.
bcc
bcc
bcc
In Eqs. (3) and (4), u0bcc
A , k A ,  1 A ,  2 A are the coressponding quantities in clean bcc
metal A in the approximation of two coordination sphere [3, 15, 16].
In the action of rather large external force F, the alloy transfers to the process of
nonlinear deformation. When the bcc interstititial alloy AC is deformed, the nearest
neighbour distance r1bccF

X (X  A, A1 , A 2 , C) at temperature T has the form
bcc
bcc
bcc
bcc
r1bccF
 r1bcc
X
X  r01X   r01X  1     r1 X  r01X  2   ,

where  

E



(5)

bcc
(  is the stress and E is the Young modulus), r1bcc
X  r1 X ( P, T ) is the nearest

neighbour distance in bcc alloy before deformation. When the alloy is deformed, the
mean nearest neighbour distance r01bccF
X at 0K has the form
bcc
r01bccF
X  r01X 1   .

(6)

The equation of state for bcc interstitial alloy AC at temperature T and pressure P is
written in the form [3]

 

 1 u 0bcc
1 k bcc  bcc 4 r1bcc
bcc
bcc
, v 
Pv  r 


x
cthx
.
bcc
2k bcc r1bcc 
3 3
 6 r1
At 0K and pressure P, this equation has the form
3

bcc

bcc
1

(7)

 1 u 0bcc 0bcc k bcc 
Pv bcc  r1bcc 
 bcc bcc .
bcc
4k r1 
 6 r1

(8)

If we know the interaction potential i0 , the equation (8) permits us to determine
the

nearest neighbour distance r1bcc
X ( P,0)(X  A, A1 , A 2 , C) at pressure P and

temperature 0K.After finding r1bcc
X ( P,0), we can determine



bcc
2
r1bccF
X P,0  r1 X P,0 1  2  



(9)

( P,0),  ( P,0),  ( P,0) at
and then determine
the parameters k ( P,0), 
pressure Pand 0K for each case of X when alloy is deformed. Then, the displacement
y0bccF
X P, T  of atom X from the equilibrium position at temperature T and pressure P is
calculated a in [3, 15].
bccF
X

bccF
1X

bccF
2X

bccF
X

When alloy is deformed, the nearest neighbour distance r1bccF
X ( P, T ) is determined
by [3]
bccF
bccF
bccF
r1bccF
( P, T )  r1bccF
( P,0)  y bccF
( P, T ),
C

C
A1 ( P, T ), r1 A ( P, T )  r1 A ( P,0)  y A
bccF
bccF
bccF
r1bccF
( P, T ), r1bccF
( P, T ).
A1 ( P, T )  r1C
A2 ( P, T )  r1 A2 ( P,0)  yC

48

(10)


Build theory of nonlinear deformation for bcc and fcc substitutional alloys AB with interstitial…

When alloy is deformed, the mean nearest neighbour distance r1bccACF
( P, T ) has the
A
form [3]
r1bccACF
( P, T )  r1bccACF
( P,0)  y bccACF ( P, T ),
A
A







r1bccACF
( P,0)  1  cC r1bccF
( P,0)  cC r1AbccACF ( P,0) 1  2   2 , r1AbccF ( P,0)  3r1bccF
( P,0),
A
A
C
bccF
y bccACF ( P, T )  1  7cC y bccF
( P, T )  cC yCbccF ( P, T )  2cC y bccF
A
A1 ( P, T )  4cC y A2 ( P, T ), (11)

where r1bccACF
( P, T ) is the mean nearest neighbor distance between two atoms A in the
A
deformed bcc interstitial alloy AC at pressure P and temperature T, r1bccACF
( P,0) is the
A
mean nearest neighbor distance between two atoms A in the deformed bcc interstitial
alloy AC at pressure P and temperature 0K, r1bccF
A ( P,0) is the nearest neighbor distance
between two atoms A in the deformed bcc clean metal A at pressure P and temperature
0K, r1AbccACF ( P,0) is the nearest neighbor distance between two atoms A in the zone
containing the interstitial atom C when the bcc alloy AC is deformed at pressure P and
temperature 0K and cC is the concentration of interstitial atomsC.

In the case of fcc interstitial alloy AC (where the main atom A1 stay in face
centers, themain atom A2 stay in peaks and the interstitial atom C stays in body center of
cubic unit cell), the corresponding formulas are as follows [3, 15]

u
k Cfcc 

fcc
0C



1   2 AC

2 i  u i2

 





  4 AB
1


48 i  ui4





 2fcc
C 



 



2



 

  4 AC
6

48 i  ui2 ui2







2

(12)



 

2

 

81 r1

4r

fcc 3





 

bcc 3
1











1 ( 4) fcc
2 (3) fcc
7
  AC
r1
2  fcc  AC
r1
2 
4
8r1
8 r1 fcc

 

 

2



2

(1)
r1 fcc  
 AC





(1)
 AC
r1 fcc 3 

( 2)
 AC
r1 fcc 5 

 



1

2 3


3
( 3)
  1  AC
r1 fcc 
fcc

4 r1 fcc
 eq 2r1







( 2)
r1 fcc  
 AC

( 2)
 AC
r1 fcc 3 

8 5
1
( 3)
 AC
r1 fcc 5 
fcc
125r1
25 r1 fcc





8 5 (1) fcc
fcc

 AC r1 5 ,  Cfcc  4  1fcc
C   2C ,
5r1 fcc

 





 


1
( 4)
  1  AC
r1 fcc 

4 r1bcc
 eq 24

2 3 (3) fcc
2
 AC r1 3 
fcc
27r1
27 r1 fcc





2 (1) fcc 4 ( 2) fcc
8 3 ( 2) fcc
( 2)
   AC
r1 fcc  fcc  AC
r1   AC r1
3  fcc  AC
r1
3 

3
r
9
r
1
1
 eq

( 2)
 4 AC
r1 fcc 5 

 1fcc
C 



1 ni

  AC (ri )  3 AC (r1 fcc )  4 AC r1 fcc 3  12 AC r1 fcc 5 ,
2 i 1

5

 

125 r1

fcc 3

( 2)
r1 fcc  
 AC



( 2)
 AC
r1 fcc 2 

1 ( 4) fcc
 AC r1 3 
54





17 ( 4) fcc

 AC r1 5 
150





(1)
 AC
r1 fcc 5 ,

3

 

4 r1

fcc 3

7 2

 

16 r1

fcc 3

(1)
r1 fcc  
 AC





(1)
 AC
r1 fcc 2 

49


Nguyen Quang Hoc, Nguyen Thi Hoa and Nguyen Duc Hien











4 ( 4) fcc
26 5 (3) fcc
3
 AC r1 5 
 AC r1 5 
fcc

25
125r1
25 r1bcc

 

2





( 2)
 AC
r1bcc 5 

 





3 5

 

125 r





(1)
 AC
r1bcc 5 , (13)

bcc 3
1

fcc
u0fccA1  u0fccA   AC r1Afcc1 ,  Afcc
 4  1fcc
A1   2 A1 ,
1

k Afcc
 k Afcc 
1

fcc
 1fcc
A   1A 
1

 2fccA   2fccA 
1

  4
6
AC



48 i  u i2 u i2


1   2 AC

2 i  u i2


  4
1
AC


4

48 i  u i


 
( 2)
 
k Afcc   AC
r1 Afcc1 ,
 
 eq  r r fcc

 

1 A1

 
1 ( 4) fcc
 
 1fcc
 AC r1 A1 ,
A 
 
24
 eq  r r fcc

 

1 A1

 
1
1
( 3)
 
 2fccA  fcc  AC
r1 Afcc1 
 
4r1 A1
2 r1 Afcc1
 eq  r r fcc

 

 

2

( 2)
r1Afcc  
 AC
1

1 A1

 



1

 

fcc 3
1 A1

2r

(1)
r1Afcc , (14)
 AC
1



fcc
u0fccA2  u0fccA   AC r1Afcc2 ,  Afcc2  4  1fcc
A2   2 A2 ,

k Afcc
 k Afcc 
1

fcc
 1fcc
A   1A 
2

1   2 AC

2 i  u i2


  4
1
  AC
48 i  u i4




2

 

fcc 2
1 A2

9r

 2fccA   2fccA 
2



 

fcc 2
1 A2

27 r

 

1 A2

 

 

1 A2

( 2)
r1Afcc  

 AC
2

(1)
r1Afcc ,
 AC

2

 

fcc 3
1 A2

2

9r

 
1 ( 4) fcc
4
( 3)
 
 2fccA   AC
r1 A2 
 AC
r1 Afcc2 
fcc
 
81

27r1 A2
 eq  r r fcc

( 2)
r1Afcc  
 AC
2

 

 
1 ( 4) fcc
2
( 3)
 
 1fcc
 AC r1 A2  fcc  AC
r1 Afcc2 
A 
 
54
9
r
1 A2
 eq  r r fcc

  4
6
 2 AC2


48 i  u i u i


14

 
1 ( 2) fcc
23 (1) fcc
 
k Afcc   AC
r1 A2  fcc  AC
r1 A2 ,
 
6
6
r
1 A2
 eq  r r fcc

 

 

1 A2

14

 

fcc 3

1 A2

27 r

(1)
r1Afcc ,
 AC

(15)

2

r1XfccF  r1Xfcc  r01fccX   r01fccX  1     r1Xfcc  r01fccX  2   ,
fcc
r01fccF
X  r01X 1   ,

 1 u 0fcc
1 k fcc
fcc
fcc
Pv fcc  r1 fcc 


x
cthx
bcc
2k fcc r1 fcc
 6 r1

(17)
 fcc
, v 



 1 u 0fcc 0fcc k fcc
Pv fcc  r1 fcc 

fcc
6

r
4k fcc r1 fcc
1

50

(16)


,



 

2 r1
2

fcc 3

,

(18)
(19)


Build theory of nonlinear deformation for bcc and fcc substitutional alloys AB with interstitial…





r1XfccF P,0  r1Xfcc P,0 1  2   2 ,
fccF
1C

r

( P, T )  r

fccF
1C

( P,0)  y

bccF
A1

fccF
1A

( P, T ), r

(20)

( P, T )  r

fccF
1A

( P,0)  y

fccF
A

( P, T ),

r1 AfccF
( P, T )  2r1CfccF ( P, T ), r1 AfccF
( P, T )  r1 AfccF
( P,0)  yCfccF ( P, T ),
1
2
2

(21)

r1AfccACF ( P, T )  r1AfccACF ( P,0)  y fccACF ( P, T ),







r1 AfccACF ( P,0)  1  cC r1AfccF ( P,0)  cC r1AfccACF ( P,0) 1  2   2 , r1AfccF ( P,0)  2r1CfccF ( P,0),
y fccACF ( P, T )  1  15cC y AfccF ( P, T )  cC yCfccF ( P, T )  6cC y AfccF
( P, T )  8cC y AfccF
( P, T ).
1
2

(22)

The mean nearest neighbor distance between two atoms A in the deformed bcc
substitutional alloy AB with interstitial atom C at pressure P and temperature T is
determined by [3, 15].
bccF
a bccF
ABC  c AC a AC

bccF
BTAC
bccF
T

B

 c B a BbccF

bccF
BTB
bccF
T

B

bccF
bccF
, BTbccF  c AC BTAC
 c B BTB
, c AC  c A  cC ,

bccABCF
a bccF
( P, T ), a bccF
 r1bccACF
( P, T ), a BbccF  r1bccF
( P, T ),
ABC  r1 A
AC
A
B

2P 
bccF
BTAC


1



bccF
  2 AC
 bccF 2
 a
 AC

bccF
TAC



bccF
3 3 1   2 AC

2
3N  a bccF
4a bccF
AC
AC

 a bccF
AC
3 bccF
 a0 AC


  2 AbccF
  1  7cC  bccF
2

 a A
T






3




T

2P 
bccF
, BTB


  2 CbccF

  cC  bccF
2
T

 aC

1



bccF
TB



 a bccF
3 BbccF
 a0 B

  2 AbccF

  2cC  bccF1 2

 a A
T
1










T

3


  2 AbccF
2
  4cC 
2

 a bccF
T
 A2


 ,

T


, X  A, A1 , A2 , B, C.

(23)
The mean nearest neighbor distance between atoms A in the deformed bcc substitutional
alloy AB with interstitial atom C at pressure P and temperature T = 0K is determined by
bccF
bccF
bccF B0TAC
bccF B0TB

bccF
a0bccF

c
a

c
a
, B0bccF
 c AC B0bccF
ABC
AC 0 AC
B 0B
T
TAC  c B B0TB ,
bccF
bccF
B0T
B0T
1   2 XbccF

2
3N  a bccF
X


 XbccF
1  2 u 0bccF
X
 


bccF 2
4k XbccF
 T 6 a X

  2 k bccF
1
X
 bccF
 bccF
2
2k X
 a X

3 3 1   2 BbccF

4a BbccF 3N  a BbccF 2

 k XbccF
 bccF
 a X





2

bccABCF
bccACF

a0bccF
( P,0), a0bccF
( P,0), a0bccF
 r1bcc
ABC  r1 A
AC  r1 A
B
B ( P,0).

(24)

The Helmholtz free energy of bcc substitutional alloy AB with interstitial atom C
before deformation with the condition cC  c B  c A has the form [3]
bcc
bcc
 ABC
  AC
 c B  Bbcc  Abcc   TScbccAC  TScbccABC ,

51

,


Nguyen Quang Hoc, Nguyen Thi Hoa and Nguyen Duc Hien
bcc
 AC
 1  7cC  Abcc  cC Cbcc  2cC Abcc  4cC Abcc  TScbccAC ,
1

2

2

   bcc bcc
 2 X YX
bcc 2

k
 X 

bcc
 Xbcc  U 0bcc
X   0 X  3N 





 



2 3  4 bcc bcc  YXbcc 
  2  1bcc
  2 X YX 1 
X
bcc 4 3
2
kX





 





bcc
 0bcc
X  3N x X  ln 1  e

2 x bcc
X

,Y



2



2



 YXbcc 

2 1bcc
X
1 
 
3 
2 





,

YXbcc 
bcc 

 1  YXbcc
 2 1bcc

1

X 2X 
2



bcc
X



 


bcc
 x bcc
X coth x X ,

(25)

where  Xbcc is the Helmholtz free energy is an atom X in clean metals A, B or interstitial
alloy AC before deformation, S cbccAC is the configuration entropy of bcc interstitial alloy
AC before deformation and S cbccABC is the configuration entropy of bcc alloy ABC
before deformation.
In the case of fcc interstitial alloy AC, the corresponding formulas are as follows
[3, 15]:
fccF
fccF
a ABC
 c AC a AC

fccF
BTAC
fccF
T

B

 c B a BfccF

BTBfccF
fccF
T

B

fccF
, BTbccF  c AC BTAC
 c B BTBfccF , c AC  c A  cC ,

fccF
fccF
a ABC
 r1 AfccABCF ( P, T ), a AC
 r1AfccACF ( P, T ), a BfccF  r1BfccF ( P, T ),

2P 
fccF
BTAC


1



fccF
 TAC

fccF
  2 AC


 a fccF 2
 AC

2
a

fccF
AC

fccF
1   2 AC

fccF 2
3N  a AC

 a fccF
3 AC
fccF
 a0 AC







  2 AfccF
  1  15cC 
fccF 2


 a A
T

1   2 XfccF

3N  a XfccF 2

a

3

 c AC a

fccF
0 AC

B0fccF
TAC
fccF
0T

B

2P 
1

, BTBfccF 

  2 CfccF


  cC 
fccF 2
T
 aC


 XfccF
1  2 u 0fccF
X
 

fccF 2
4k XfccF
 T 6 a X

fccF
0 ABC




T

 TBfccF

 cB a

B0fccF
TB

fccF
0T

B

a

fccF
B

1   2 BfccF

3N  a BfccF 2

 a fccF
3 BfccF
 a0 B

  2 AfccF

1
  6cC 

 a AfccF 2
T
1


  2 k fccF
1

X
 fccF
 fccF
2
2k X
 a X
fccF
0B



2

 k XfccF
 fccF
 a X





2

3


  2 AfccF
2
  8cC 


 a AfccF 2
2
T


,


 ,

T


, X  A, A1 , A2 , B, C.

(26)

fccF
, B0fccF
 c AC B0fccF
T
TAC  c B B0TB ,

fccABCF
fccACF
a0fccF
( P,0), a0fccF
( P,0), a0fccF
 r1BfccF ( P,0).
ABC  r1 A

AC  r1 A
B

fcc
fcc
 ABC
  AC
 cB  Bfcc  Afcc   TScfccAC  TScfccABC ,

fcc
 AB
 1  15cB  Afcc  c B Bfcc  6c B Afcc  8cB Afcc  TScfccAC ,
1

52








T

2

(27)

Build theory of nonlinear deformation for bcc and fcc substitutional alloys AB with interstitial…
2

   fcc fcc
 2 X YX
fcc 2

k
 X 

 Xfcc  U 0fccX   0fccX  3N 


2 3  4 fcc fcc  YXfcc
 2 X YX 1 
4 
2
k Xfcc  3


 





 

 




  2  1fcc
X


 0fccX  3N x Xfcc  ln 1  e 2 x

fcc
X

,Y



2

fcc
X

2



2 1fcc
X
3

 YXfcc

1 
2




YXfcc
fcc 

 2 1fcc

1

X 2X 
2



 



 1  YXfcc






,

 


 x Xfcc coth x Xfcc .

(28)

When the process of nonlinear deformation in both fcc and bcc alloy happens, the
relationship between the stress and the strain is decribed by

σ1 ABC

ε Fα ABC
 σ oABC
.
1  εF

(29)

Here, oABC and ABC are constant depending on every interstitial alloy. We can find the
strain F corresponding to the maximum value of the real stress through the density of
deformation energy.
In order to determine the stress - strain dependence according to the above formula,
it is necesary to determine two constants oABC and ABC for every intestitial alloy.
Therefore, we can calculate the density of deformation energy of substitutional alloy
AB with interstitial atom C in the form
F
Ψ ABC Ψ ABC
1
f ABC (ε)  F



VABC
N
VABC

 Ψ AF1 Ψ A1
 c A1  F 
 v ABC v ABC


F
 Ψ ABC
 F
 v ABC



Ψ ABC
v ABC


ΨF
Ψ
  c A  A2  A2
F
2

 v ABC v ABC





 


1
N

  Ψ AF
c A  F
  v ABC



ΨA
v ABC


 F
  cB  Ψ B  Ψ B
 vF

 ABC v ABC



 Ψ AF
  cC  F


 v ABC



ΨA
v ABC


ΨF
Ψ
  cB  F A  A

v

 ABC v ABC


 



.



cA  1  cB  7cC , cA1  2cC , cA2  4cC for bcc alloy,
cA  1  cB  15cC , cA1  6cC , cA2  8cC for fcc alloy.

(30)

Since  is very small ( << 1), we can expand the expression of the Helmholtz free
energy Ψ XF (X  A, A1 , A 2 , B, C) in terms of the strain  in the form of series and
approximately,
 XF
 ( )  X  
 
Applying the following formulas:
F
X


1   2 XF
   
2   2
T

 2
  .
T

XF XF r1FX  2 XF
  XF r1FX   2 XF


 F
,


  r1FX   r1FX2
r1 X 

 2

r1FX
2r F
 2r01X 1     2r01F X , 12X  2r01X .



(31)

 r1FX

 

2


 F  2 r1FX
  FX
, (32)
2

r


1X


(33)

Therefore,
53


Nguyen Quang Hoc, Nguyen Thi Hoa and Nguyen Duc Hien
f ABC (ε) 

1   1
1  2εr01F A  Ψ AF 
ε2
c A Ψ A  F 
 F  F   F
N   v AB v AB  v AB  r1 A T 2v AB



cC
N







  1
1
Ψ B  F 
v
v

AB
  AB

c A1   1
1
Ψ A  F 
N  1  v AB
v AB


c A2 
 1
1
Ψ A2  F 
N 
 v AB v AB


cB
N

 2εr01FB
 F
 v AB

  1
1
Ψ B  F 
  v AB v AB



F

F

 2εr01FB
  F
 v AB



 Ψ BF 
ε 2   2Ψ BF 

2r01FB
 F 
F 
F2 

r
2
v

r
AB 
 1B T
 1B T

 2εr01 A1

 F
v AB


 2εr01A2
 F

v AB


  2Ψ AF 
F
 F 2  2r01 A
 r1 A T

 Ψ AF1
 F
 r1 A
1


 Ψ AF2

 r1FA
2


 Ψ BF
 F
 r1B


ε2
  F
 2v AB
T

2

  ε
F

 T 2v AB


ε2
  F
 T 2v AB

  2Ψ AF
 F 21
 r1 A1





 
 Ψ F 
  FB  2r01FB   

 r1B T
 

2


F
 2r01 A1

T



  2Ψ AF
 F 22
 r1 A2


 2r01F A
2

T

  2Ψ BF
 F 2
 r1B


 2r01FB
T





2

 Ψ AF
  F1
 r1 A
1


 Ψ AF
 F2
 r1 A
2






 
 Ψ F 
  FA  2r01F A   
 
 r1 A T

2

2



2

 

F
 2r01 A1   

 
T



 2r01F A   
2

 
T


 Ψ F
  FB
 r1B

 


 2r01FB   
 
T

  1
 
 Ψ F 
2
1  2εr01F A  Ψ AF 
ε 2   2Ψ F 
  F  F   F  F 2A  2r01F A   FA  2r01F A  . (34)
Ψ A  F 
  v AB v AB  v AB  r1 A  T 2v AB  r1 A  T
 r1 A  T
 
Similar to metal when the deformation rate is constant, the density of deformation
energy of alloy has the form
fABC() = CABC.ABC.,
(35)
where CABC is a proportional factor.




cB
N



F

The function fABC() gets its maximum at the strain ε ABC
. This means that





F
F
f ABC ε ABC
 f ABC max  C ABC σ ABC max ε ABC
.

(36)

then, we can find the maximum stress ABCmax and the maximum real stress 1ABCmax
f
σ
f ABC max
(37)
σ ABC max  ABC max
,σ1 ABC max  ABC Fmax 
.
F
F
F
C ABC ε ABC
1  ε ABC C ABC ε ABC
1  ε ABC



 σ
From the maximum condition of stress  1 ABC
 ε





F

 0, we determine the strain ε ABC
 F
 ABC

corresponding to the maximum value of the real stress as follows:
ε

F
ABC





α ABC

α ABC
εF


 σ1 ABC max  σ 0 ABC ABC F .
1  α ABC
1  ε ABC

(38)

The proportional factor CABC is determined from the experimental condition of the
stress 0,2ABC in alloy in the form
f ε 
C ABC  ABC 0.2 .
(39)
σ 0.2 ABC ε 0.2
In substitutional alloy AB with interstitial atom C, if the concentration cC of
interstitial atoms is equal to zero, we obtain the expression of the density of deformation
energy for substitutional alloy AB. In substitutional alloy AB with interstitial atom C,
54


Build theory of nonlinear deformation for bcc and fcc substitutional alloys AB with interstitial…

if the concentration cB of substitutional atoms is equal to zero, we obtain the expression
of the density of deformation energy for interstitial alloy AC. In substitutional alloy AB
with interstitial atom C, if both concentrations cB and cC are equal to zero, we obtain the
expression of the density of deformation energy for main metal A.
F
After having the value of the strain ε ABC
corresponding to the maximum value of
the density of deformation energy, we can find the expression to describe the
relationship between the stress and the strain in the process of nonlinear deformation of

bcc and fcc substitutional alloys AB with interstitial atom C

3. Conclusions
The analytic expressions of the Helmholtz free energy, the mean nearest neighbor
distance and the characteristic nonlinear deformation quantities such as the density of
deformation energy, the maximum real stress and the limit of elastic deformation for
bcc and fcc substitutional alloys AB with interstitial atom C under pressure are derived
from the statistical moment method. From that, we obtain the theory of nonlinear
deformation for binary substitutional alloy, binary interstitial alloy and main metal. We
will carry out the numerical calculations for real alloys in next paper.
Acknowledgement: This paper is completed with financial support of the project titled
“Study on some deformation properties of interstitial alloy” (code: T2019-CB-009) from
The University of Transport and Communications.
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