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.
Trang 1Natural Sciences, 2019, Volume 64, Issue 6, pp 45-56
This paper is available online at http://stdb.hnue.edu.vn
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: hocnq@hnue.edu.vn
Trang 2The 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
5 ,
2
1
bcc bcc
bcc
r
r
r are determined by [3, 15]
2 ) ( )
( 2
1
1 1
1 1
0
bcc AC bcc
AC bcc
AC n
i
i AC bcc
u
i
(1)
5 5
16 2
2 2
1
2 1 1
) 1 (
1 1
) 1 (
1 1 ) 2 ( 2
2
bcc C bcc C bcc C bcc AC bcc bcc
AC bcc bcc i
AC eq i
AC bcc
r
r r
r u
8
1 24
1 48
1
1 ) 2 ( 2 1 1
) 4 ( 4
4
1
bcc AC bcc
bcc AC i
eq i
ACF bcc
r
r
125
5 4 2 150
1 2 16
2
1 ) 3 (
1 1
) 4 ( 1
) 1 ( 3 1
bcc AC bcc bcc
AC bcc
AC
r r
r r
AC bcc
bcc AC bcc
bcc AC i
bcc eq
i i
AC bcc
r
r r
r r
u
) 1 ( 3 1 1
) 2 ( 2 1 1
) 3 (
1 2
2 4
2
8
5 4
1 4
1 48
25
2 2 8
1 2
8
1 2
8
2
1 ) 4 ( 1
) 1 ( 3 1 1
) 2 ( 2 1 1
)
3
(
1
bcc AC bcc
AC bcc
bcc AC bcc
bcc AC
r
r r
r
5 25
3 5
25
2 5
5
25
3
1 ) 1 ( 3 1 1
) 2 ( 2 1 1
) 3 (
1
bcc AC bcc
bcc ACF bcc
bcc AC
r
r r
r
(2)
where AC is the interaction potential between the atom A and the atom C, n is the i
number of atoms on the ith coordination sphere with the radius r i i( 1, 2,3),
) (
1
0 01 1
r bcc bcc C bcc C bcc A is the nearest neighbor distance between the interstitial atom
C and the metallic atom A at temperature T, r01bcc Cis the nearest neighbor distance between the interstitial atom C and the metallic atom A at 0K and is determined from the minimum condition of the cohesive energyu0bcc C , y0bcc A1(T) is the displacement of the atom A1(the atom A stays in the bcc unit cell) from equilibrium position
at temperature T,
Trang 3
(AB m) m AC(r i)/r i m(m1,2,3,4, , x,y.z, and u i 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]
1 , 1 4 1 1,
0
bcc A bcc A bcc
A bcc A AC bcc A bcc
2
5 2
1
1 1
1
1
) 1 (
1 1
) 2 ( 2
2
bcc A AC bcc A
bcc A AB bcc A i
r r eq i
AC bcc
A bcc
r r
k u
k k
bcc A
8
1 8
1 24
1 48
1
1
1 1
1 1
1
) 1 ( 3 1 1
) 2 ( 2 1 1
) 4 ( 1
4 4
1
1
bcc A AC bcc A
bcc A AC bcc A
bcc A AC bcc
A i
r r eq i
AC bcc
A
bcc
r
r r
r u
bcc A
,
4
3 4
3 2
1 48
6
1
1 1
1 1
1 1
) 1 ( 3 1 1
) 2 ( 2 1 1
) 3 ( 1 2 2
2 4 2
2
bcc A AC bcc A
bcc A AC bcc A
bcc A AC bcc A
bcc A i
r eq i i AC bcc
A
bcc
r
r r
r r
u u
bcc A
(3)
where r1bcc A r1bcc C
1 is the nearest neighbor distance between atom A1 and atoms in 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]
2 , 2 4 2 2,
0
bcc A bcc A bcc
A bcc A AC bcc A bcc
2 2
1
2 2
2
2
) 1 (
1 1 ) 2 ( 2
2
bcc A AC bcc A
bcc A AC bcc
A i
r r eq i
AC bcc
A bcc
r r k
u k
k
bcc A
bcc A AC bcc A
bcc A AC bcc
A i
r r eq i
AC bcc
A bcc
r
r u
bcc A
2 2
2
2
) 3 (
1 1
) 4 ( 1
4
4
1 1
4
1 24
1 48
8
1 8
1
2 2
2 2
1 ) 1 ( 3 1 1
) 2 ( 2 1
A AC bcc A
bcc A AC bcc A
r r
r r
bcc A AC bcc A
bcc A AC bcc
A i
r r eq i i
AC bcc
A bcc
r
r u
u
bcc A
2 2
2
2 1
) 3 (
1 1
) 4 ( 2
2 2
4
2
2
4
1 8
1 48
8
3 8
3
2 2
2 2
1 ) 1 ( 3 1 1
) 2 ( 2 1
bcc A AC bcc A
bcc A AC bcc A
r r
r r
(4)
Trang 4wherer bcc A r bcc A y bcc C T r bcc A
2 2
1 ( ), is the nearest neighbor distance between the atom A2and atoms in crystalline lattice at 0K and is determined from the minimum condition
of the cohesive energy 0 , 0 ( )
u bcc A bcc C is the displacement of the atom C at temperature T
In Eqs (3) and (4), u0bcc A,k bcc A ,1bcc A ,2bcc 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 1bccF(XA,A1,A2,C)
X
r at temperature T has the form
1 1 01 2 ,
01 01
1
X bcc X bcc
X bcc
X bcc X bccF
where
E( is the stress and E is the Young modulus), r1bcc X r1bcc 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
1
01
X bccF
X r r
(6)
The equation of state for bcc interstitial alloy AC at temperature T and pressure P is
written in the form [3]
3 3
4 ,
2
1 6
1 1
0 1
bcc bcc
bcc bcc bcc bcc bcc bcc
bcc bcc
v r
k k cthx x r
u r
At 0K and pressure P, this equation has the form
4
6
1
1 0
1
0
1
bcc bcc bcc bcc bcc
bcc bcc
bcc
r
k k r
u r
(8)
If we know the interaction potential i0, the equation (8) permits us to determine the nearest neighbour distance r1bcc X (P,0)(XA,A1,A2,C)at pressure P and
temperature 0K.After findingr1bcc X (P,0),we can determine
1
1 P,0 r P,0 12
r bccF X bcc X (9) and then determine the parametersk bccF X (P,0),1bccF X (P,0),2bccF X (P,0),bccF X (P,0)at
pressure Pand 0K for each case of X when alloy is deformed Then, the displacement
P T
y bccF0X , of atom X from the equilibrium position at temperature T and pressure P is
calculated a in [3, 15]
When alloy is deformed, the nearest neighbour distance r1bccF X (P,T)is determined
by [3]
), , ( )
0 , ( )
, ( ), , ( )
0 , ( )
,
)
, ( )
0 , ( )
, ( ), , ( )
, (
2 2
r bccF A bccF C bccF A bccF A C bccF (10)
Trang 5When alloy is deformed, the mean nearest neighbour distance r1bccACF A (P,T)has the form [3]
, ) , ( )
0 , ( )
,
r bccACF A bccACF A bccACF
)
0
,
r bccACF A C bccF A C A bccACF A bccF bccF C
1 7 ( , ) ( , ) 2 ( , ) 4 ( , ), )
,
(
2
y c T P y c T P y c T
P
y bccACF C bccF A C C bccF C bccF A C bccF A (11) wherer1bccACF A (P,T)is the mean nearest neighbor distance between two atoms A in the
deformed bcc interstitial alloy AC at pressure P and temperature T, r1bccACF A (P,0) is the 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, r1A bccACF(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 c C 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]
4 ) ( 3 ) ( 2
1
1 1
1 1
0
fcc AC fcc
AC fcc
AC n
i
i AC fcc
u
i
(12)
9
3 8 3 3
4 2
2
1
1 ) 2 (
1 1
) 2 ( 1
) 1 (
1 1 ) 2 ( 2
2
fcc AC fcc fcc
AC fcc
AC i
fcc fcc AC eq i
AC fcc
r r
r r
r u
5
5 8 5
1 1
) 2
C fcc C fcc
C fcc
AC fcc fcc
r
54
1 4
1 4
1 24
1 48
1
1 ) 4 ( 1
) 1 ( 3 1 1
) 2 ( 2 1 1
) 4 ( 4
4
1
fcc AC fcc
AC bcc fcc
AC bcc fcc
AC i
eq i
AB fcc
r
r r
r
150
17 3 81
3 2 3 27
2 3
27
3
2
1 ) 4 ( 1
) 1 ( 3 1 1
) 2 ( 2 1 1
)
3
(
1
fcc AC fcc
AC fcc
fcc AC fcc
fcc AC
r
r r
r
125
5 5
25
1 5
125
5 8
1 ) 1 ( 3 1 1
) 2 ( 2 1 1
) 3 (
1
fcc AC fcc
fcc AC fcc
fcc AC
r
r r
r
AC fcc
fcc AC fcc
fcc AC i
fcc eq
i i
AC fcc
r
r r
r r
u
) 1 ( 3 1 1
) 2 ( 2 1 1
) 3 (
1 2
2 4
2
4
3 4
3 2
1 48
16
2 7 2 8
7 2
8
2 2
4
1
1 ) 1 ( 3 1 1
) 2 ( 2 1 1
) 3 (
1 1
)
4
AC fcc
fcc AC fcc
fcc AC fcc fcc
r
r r
r r
Trang 6 5,
125
5 3 5 25
3 5
125
5 26 5 25
4
1 ) 1 ( 3 1 1
) 2 ( 2 1 1
) 3 ( 1 1
)
4
AC bcc
bcc AC bcc
fcc AC fcc fcc
r
r r
r r
1 , 1 4 1 1,
0
fcc A fcc A fcc
A fcc A AC fcc A fcc
, 2
1
1
1
) 2 ( 2
2
fcc A AC fcc A i
r r eq i
AC fcc
A fcc
u k
k
fcc A
, 24
1 48
1
1
1
) 4 ( 1
4 4
1 1
fcc A AC fcc
A i
r r eq i
AC fcc
A fcc
u
fcc A
,
2
1 2
1 4
1 48
6
1
1 1
1 1
1 1
) 1 ( 3 1 1
) 2 ( 2 1 1
) 3 ( 1 2 2
2 4 2
2
fcc A AC fcc A
fcc A AC fcc A
fcc A AC fcc A
fcc A i
r eq i i AC fcc
A
fcc
r
r r
r r
u u
fcc A
(14)
2 2 2
2
0
fcc A fcc A fcc
A fcc A AC fcc A fcc
6
23 6
1 2
1
2 2
2
2
) 1 (
1 1
) 2 ( 2
2
fcc A AC fcc A
fcc A AC fcc
A i
r r eq i
AC fcc
A fcc
r r
k u
k k
fcc A
fcc A AC fcc A
fcc A AC fcc
A i
r r eq i
AC fcc
A fcc
r
r u
fcc A
2 2
2
2
) 3 (
1 1
) 4 ( 1
4 4
1 1
9
2 54
1 48
9
2 9
2
2 2
2 2
1 ) 1 ( 3 1 1
) 2 ( 2 1
A AC fcc A
fcc A AC fcc A
r r
r r
fcc A AC fcc A
fcc A AC fcc
A i
r r eq i i
AC fcc
A
fcc
r
r u
u
fcc A
2 2
2
2
) 3 (
1 1
) 4 ( 2
2 2 4
2
2
27
4 81
1 48
27
14 27
14
2 2
2 2
1 ) 1 ( 3 1 1
) 2 ( 2 1
fcc A AC fcc A
fcc A AC fcc A
r r
r r
(15)
1 1 01 2 ,
01 01
1
X fcc X fcc
X fcc
X fcc X fccF
1 ,
01
X fccF
X r r
(17)
, 2
2 ,
2
1 6
1 1
0 1
fcc fcc
fcc fcc fcc fcc fcc bcc
fcc fcc
v r
k k cthx x r
u r
, 4
6
1
1 0
1
0
fcc fcc fcc fcc fcc fcc fcc fcc
r
k k r
u r
(19)
Trang 7 ,0 1 ,01 2 2,
r fccF X fcc X (20)
), , ( )
0 , ( )
, ( ), , ( )
0 , ( )
,
), , ( )
0 , ( )
, ( ), , ( 2 )
,
(
2 2
r fccF A C fccF fccF A fccF A C fccF (21)
, ) , ( )
0 , ( )
,
r fccACF A fccACF A fccACF
)
0
,
r A fccACF C A fccF C A fccACF A fccF C fccF
) ,
(
2
y c T P y c T P y c T
P
y fccACF C A fccF C C fccF C A fccF C A fccF (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]
, ,
, T bccF AC TAC bccF B TB bccF AC A C
bccF T
bccF TB bccF B B bccF T
bccF TAC bccF AC AC
bccF
B
B a c B
B a c
), , ( ,
) , ( ,
) ,
r
a bccF ABC bccABCF A bccF AC bccACF A B bccF bccF B
, 3
3
1 4
3 3 2 1 ,
3
3
1 4
3 3 2 1
3
0
2 2
3
0
2 2
bccF B
bccF B
T bccF B
bccF B bccF
B bccF
TB
bccF TB
bccF AC
bccF AC
T
bccF AC
bccF AC bccF
AC bccF
TAC
bccF
TAC
a a
a N a
P B
a a
a N a
P B
2
2 2
2 2
2 2
2
2
2 2
1 1
T
bccF A
bccF A C
T
bccF A
bccF A C
T
bccF C
bccF C C T
bccF A
bccF A C
T
bccF
AC
bccF
AC
a
c a
c a
c a
c
, , , , ,
2
1 4
6
1 3
1
2 1 2
2 2
2 0 2
2
2
C B A A A X a
k k
a
k k
a
u a
bccF X bccF X bccF
X
bccF X bccF
X
bccF X bccF
X
bccF X
T
bccF
X
bccF
(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
,
0
0 0 0
0 0 0
bccF TB B bccF TAC AC bccF T bccF T
bccF TB bccF B B bccF T
bccF TAC bccF AC AC bccF
B
B a c B
B a c
)
0 , ( ,
) 0 , ( ,
) 0 ,
1
a bccF ABC bccABCF A bccF AC bccACF A bccF B bcc B (24) The Helmholtz free energy of bcc substitutional alloy AB with interstitial atom C before deformation with the condition c C c B c A has the form [3]
B bcc bcc A c bccAC c bccABC,
B bcc AC bcc
ABC c TS TS
Trang 81 7 C bcc A C C bcc 2 C bcc A1 4 C bcc A2 c bccAC,
bcc
AC c c c c TS
2
1 3
2
2 2 2
0 0
bcc X bcc
X bcc
X bcc X bcc X
bcc X bcc X bcc X
Y Y
k N
2 1 2
2 2
1 3
4 2
2 1 2 1 2
4 3
X
bcc X bcc
X bcc X bcc
X
bcc X bcc
X bcc X bcc
X
Y Y
Y Y
k
0
bcc X bcc
X bcc X x bcc
X bcc
X N x e bcc X Y x x
where bcc X is the Helmholtz free energy is an atom X in clean metals A, B or interstitial alloy AC before deformation,S c bccACis the configuration entropy of bcc interstitial alloy
AC before deformation and bccABC
c
S 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]:
, ,
, T bccF AC TAC fccF B TB fccF AC A C
fccF T
fccF TB fccF B B fccF T
fccF TAC fccF AC AC
fccF
B
B a c B
B a c
), , ( ,
) , ( ,
) ,
r
a ABC fccF A fccABCF AC fccF A fccACF B fccF B fccF
, 3
3
1 2 2
1 ,
3
3
1 2 2
1
3
0
2 2
3
0
2 2
fccF B
fccF B
T
fccF B
fccF B fccF
B fccF
TB
fccF TB
fccF AC
fccF AC
T
fccF AC
fccF AC fccF
AC fccF
TAC
fccF
TAC
a a
a N a
P B
a a
a N a
P B
2
2 2
2 2
2 2
2
2
2 2
1 1
T
fccF A
fccF A C
T
fccF A
fccF A C
T
fccF C
fccF C C T fccF A
fccF A C
T
fccF
AC
fccF
AC
a
c a
c a
c a
c
, , , , ,
2
1 4
6
1 3
1
2 1 2
2 2
2 0 2
2
2
C B A A A X a
k k
a
k k
a
u a
fccF X fccF X fccF
X
fccF X fccF
X
fccF X fccF
X
fccF X
T
fccF
X
fccF
(26)
,
0
0 0 0
0 0 0
fccF TB B fccF TAC AC fccF T fccF T
fccF TB fccF B B fccF T
fccF TAC fccF AC AC fccF
B
B a c B
B a c
)
0 , ( ,
) 0 , ( ,
) 0 ,
1
a fccF ABC A fccABCF fccF AC fccACF A fccF B B fccF (27)
B fcc A fcc c fccAC c fccABC,
B fcc AC fcc
ABC c TS TS
1 15 B A fcc B B fcc 6 B A fcc1 8 B A fcc2 c fccAC,
fcc
AB c c c c TS
Trang 9
2
1 3
2
2 2 2
0 0
fcc X fcc
X fcc
X fcc X fcc X
fcc X fcc X fcc X
Y Y
k N
2 1 2
2 2
1 3
4 2
2 1 2 1 2
4 3
X
fcc X fcc
X fcc X fcc
X
fcc X fcc
X fcc X fcc
X
Y Y
Y Y
k
0
fcc X fcc
X fcc X x fcc
X fcc
X N x e X fcc Y x x
When the process of nonlinear deformation in both fcc and bcc alloy happens, the relationship between the stress and the strain is decribed by
1
1
F
α F oABC ABC
ε
ε σ
(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
2 2
2 1
1
1
ABC
A F
ABC
F A B ABC
B F
ABC
F B B ABC
A F
ABC
F A A ABC
A F
ABC
F
A
A
v
Ψ v
Ψ c v
Ψ v
Ψ c v
Ψ v
Ψ c v
Ψ v
Ψ
c
c c c c c c c for bcc alloy,
c A 1 c B15c c C, A1 6c c C, A2 8c C for fcc alloy (30) Since is very small ( << 1), we can expand the expression of the Helmholtz free
X XA,A1,A2,B,C in terms of the strain in the form of series and approximately,
2
1 )
2
T
F X T
F X X
F
X
(31) Applying the following formulas:
,
2
1
2 1 2 1
2 1
1 2
2 1
X F X
F X F
X F
X
F X F
X F X
F X F
X F
X F X
F X F
r
r r
r r
r
1 2 , 2
2
01 01
1
X
F X F
X X
F X
r
r r
r r
(33)
Therefore,
Trang 10
2 2
2 01
2
2 01
1
2
2 2
2
2
2
2
2 2
2 01
01 2
2
2
T
r
A T
F A
F A F
A T
F A
F A F
AB T
F A
F A F
AB
F A AB
F AB A
A
r r
Ψ r
r
Ψ v
ε r
Ψ v
εr v
v
Ψ
N
c
2 2
2 2
2 2
2
2 2 2
2
01 1
2 01 2
1
2 2
1
01
2 2
2
2 1 1
B T
F B
F B F
B T
F B
F B F
AB T
F B
F B F
AB
F B AB
F
AB
B
r
Ψ r
r
Ψ v
ε r
Ψ v
εr v
v
Ψ
N
c
01 1
2 01 2
1
2 2
1
2
2 1 1
2
2 1 1
01 1
2 01 2
1
2 2
1
01
A T
F A
F A F
A T
F A
F A F
AB T
F A
F A F
AB
F A AB
F AB A
B
r r
Ψ r
r
Ψ v
ε r
Ψ v
εr v
v
Ψ
N
c
(34)
Similar to metal when the deformation rate is constant, the density of deformation energy of alloy has the form
f ABC () = C ABC ABC , (35)
where C ABC is a proportional factor
The function f ABC () gets its maximum at the strain ε ABC F This means that
ABC F ABCmax ABC ABCmax F ABC
f (36) then, we can find the maximum stress ABCmax and the maximum real stress 1ABCmax
1 1
max max
max 1 max
ABC F
ABC ABC
ABC F
ABC
ABC ABC
F ABC ABC
ABC ABC
ε ε
C
f ε
σ ,σ
ε C
f σ
From the maximum condition of stress 1 0,
F ABC
ε
σ ABC
we determine the strain ε ABC F
corresponding to the maximum value of the real stress as follows:
1
ABC
α F ABC ABC ABC
ABC
ABC F
ABC
ε
ε σ σ
α
α ε
ABC
(38)
The proportional factor C ABC is determined from the experimental condition of the stress 0,2ABC in alloy in the form
2 2 2
ABC
ABC ABC
ε σ
ε f
C (39)
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,