Tóm tắt tiếng anh: Nghiên cứu biến dạng đàn hồi – phi tuyến của kim loại, hợp kim xen kẽ hai và ba thành phần.

MINISTRY OF EDUCATION AND TRAINING
HANOI NATIONAL UNIVERSITY OF EDUCATION

NGUYEN DUC HIEN

STUDY ON ELASTIC AND NONLINEAR DEFORMATION
OF METALS, BINARY AND TERNARY INTERSTITIAL
ALLOYS
Speciality: Mathematical and Theoretical Physics
Code: 9.44.01.03

SUMMARY OF PhD THESIS

HANOI – 2023


This work is completed at Hanoi National University of Education
Scientific supervisor 1: Assoc. Prof. PhD. Nguyen Quang Hoc
Scientific supervisor 2: Assoc. Prof. PhD. Hoang Van Tich

Reviewer 1: Prof. PhD. Bach Thanh Cong – University of Science,
Vietnam National University, Hanoi
Reviewer 2: Prof. PhD. Vu Van Hung – University of Education,
Vietnam National University, Hanoi
Reviewer 3: Assoc. Prof. PhD. Nguyen Hong Quang – Institute of
Physics - Vietnam Academy of Science and Technology

The thesis will be defended at the University-level thesis evaluation
Council at Hanoi National University of Education at the hour...
date...month... year...

The thesis is available at
- Hanoi National Library
- Library of Hanoi National University of Education


INTRODUCTION
1. Reasons for choosing the topic
Metals and alloys are materials with a long history of
development and are widely used in industries and practical daily
life. There are two types of alloys as substitutional alloys and
interstitial alloys. Steel is a typical form of exportation with special
importance in construction, transportation, machine manufacturing....
An interstitial alloy Fe is FeC. It is called carbon steel and accounts
for a large proportion in the steel industry. Fe and its interstitial
alloys such as FeSi, FeC, FeH make up most of the Earth's core and
celestial bodies. Their thermodynamic, elastic and melting properties
provide us with information about the composition, structure,
evolution ... of the Earth, celestial bodies. Information about metals
such as Au, Cu, Fe, Cr... and some of their alloys are especially
important for designing and manufacturing equipments to serve the
needs of human life.
The study of mechanical properties including the deformation
process of materials such as metals and alloys, including interstitial
alloys, has attracted the attention of many researchers, both
theoretically and experimentally. These research results play a very
important role in science, technology and human life.
Statistical moment method (SMM) has been successfully applied
to study the deformation of metals and binary substitutional alloys.
However, the study of elastic and nonlinear properties of cubic

ternary and binary interstitial alloys by SMM is still an open
problem.
For the above reasons, we choose the topic “Study on elastic and
nonlinear deformation of metals, binary and ternary interstitial
alloys”.
2. Purpose, object and scope of research
The purpose of the thesis is to develop SMM to study elasticnonlinear deformation, effect of strain on diffusion for ternary and
binary interstitial alloy taking into account the influence of
temperature, pressure, concentrations of substitutional and interstitial
atoms.
The object of the study is the elastic-nonlinear deformation, the
effect of the strain on the diffusion of cubic ternary and binary
interstitial alloy.

1


Research scope: Studied temperature range from zero to the
melting temperature of the main metal; the studied substitutional
atomic concentration range from zero to 10%; the studied interstitial
atomic concentration range is from zero to 5% and the studied
pressure range is from zero to 100 GPa.
3. Research methods
The main research method is the statistical moment methold
(SMM). In addition, in numerical calculations we use the Maple
sofware and approximate methods.
4. Scientific and practical significance
The results from the thesis provide the information about the
nonlinear-elastic deformation and the influence of the deformation on
the diffusion for metals and alloys. The thesis contributes to the

development of SMM in studying properties of materials in general
and in studying the deformation and the influence of the deformation
on the diffusion for interstitial alloy materials in particular. Some
numerical results of the thesis can be references for forecasting and
orienting experimental studies.
5. New contributions of thesis
Build analytic expressions for nonlinear and elastic deformation
quanties of cubic ternary and binary interstitial alloy using SMM.
The thesis contributes to supplementing and perfecting the
deformation theory and cubic interstitial alloy.
Apply the obtained theory to calculate numerically for some
metals and alloys. The obtained numerical results are compared with
the experimental data and the results calculated by other theoretical
methods. Some numerical results can be predictive and guide future
experiments.
6. Thesis outline
In addition to the Introduction, the Conclusion, the References
and the Appendix the thesis content is presented in 4 chapters.
The content of the thesis has been reported at 3 national and
international specialized scientific conferences and published in 7
articles in domestic and international scientific journals.

2


CHAPTER 1: OVERVIEW ON DEFORMATION OF METALS
AND ALLOYS
1.1. Interstitial alloys
1.2. Deformation theory
1.2.1. Elastic deformation

Under the action of an external force, a solid is deformed, that is,
it changes shape and size. When deformed, the points or atoms of the
solid move. Elastic deformation is the deformation of a solid object
under the action of an external force that, after unloading, the solid
body returns to its original shape and size.
1.2.2. Nonlinear deformation
Nonlinear strain (inelastic strain, residual strain) is the
deformation of a solid body under the action of an external force, but
after unloading, the deformation is not lost and the solid body does
not return to shape, size and shape. original size. That happens when
the external force (load) must be large enough. Nonlinear strain does
not change the deformation volume.
1.2.3. Elastic wave in solids
The longitudinal and transverse wave velocities have the form
Vd 

2C44  C12
C
, Vn  44 



(1.28)

1.2.4. Influence of deformation on diffusion
The dependence of the diffusion coefficient D on the biaxial
tensile stress has the form
 2 r
m
m

   3 V  V  V/ /  
,
Dx ( )  Dx ( 0) exp  
kBT







3

(1.31)


where Dx ( ) is the diffusion coefficient in the x direction of the
r
system subjected to biaxial stress  , V is the volume of recovery
m

and V/ / is the volume component displaced in the direction parallel
to the direction of diffusion..
1.3. Some major methods of research
There are many different theoretical methods in studying
nonlinear and elastic deformation of metals and alloys such as the
harmonic theory, the pseudoharmonic theory, the molecular
dynamics method, the finite element method, the calculations from
first principles (ab initio), the tight-binding Hamiltonian method, the
density functional theory, the machine learning method, the lattice

Green function method, the calculation of phase diagram, the
modified embedded atomic method, etc.
Although there are many theoretical methods used to study the
properties of objects during elastic and nonlinear deformation
process, these methods are still limited in one aspect or the other.
Most of the methods do not take into account the anharmonicity
effect in the lattice vibrations, not to mention the effect of pressure
on the deformation processes of the object, not to mention the
dependence of the deformation quantities on the concentrations of
substitutional and interstitial atoms, the results of deformation studies
are mostly for metals and limited for alloys, not taking into account
the effect of stress or strain on the diffusion in the alloy.
1.4. Statistical moment method
Kˆ n 1

a

 Kˆ n

a

Qˆ n 1

a



 Kˆ n
an 1

a





B2 m  i  
2m  !   
m 0 



2m

Kˆ n (2 m )
an 1

.  1.57 
a

Conclusion of Chapter 1
In chapter 1, we present an overview of strain theory and methods
of studying elastic - nonlinear deformation of metals and alloys,
which refers to the theory of elastic deformation, nonlinear strain,
and elastic deformation. linear, elastic wave velocity and effect of
strain on diffusion in the material; Basic content and advantages and
disadvantages of popular deformation research methods. Introduce a
traceability formula that relates higher moments to lower moments
and use it to determine the free energy of the system.

4


CHAPTER 2: STUDY ON NOLINEAR AND ELASTIC
DEFORMATION OF INTERSTITIAL ALLOY AC
2.1. Alloy model and Helmholtz free energy

Fig. 2.1. Model of interstitial alloy AC with BCC structure (a) and và FCC
structure(b)

c 

 AC 

X

X

 TScAC ,

(2.1)

X

2  Y  
  2 
 X U 0 X   0 X  3  2  2 X YX2  1X  1  X   
3 
2 
 k X 

2 3
k X4

4 2
 
 YX 
 YX 
2
 3 2 X YX  1  2   2 1X  21 X 2 X  1  2   1  YX    ,





 
 2 xX 

 0 X 3 x X  ln 1  e
,

 YX  x coth x, X  A, C , A1 , A2 , (2.2)
c A 1  7cC , c A1 2cC , c A2 4cC
c A 1  15cC , c A1 6cC ,










for BCC lattice,

c A2 8cC

for FCC lattice
2.2. Cohesive energy, alloy parameters and mean nearest
neighbor distance between two atoms
For BCC lattice,
u0 A  4 AA  r1 A   3 AA  r2 A  , r2 A 
kA 

2
r1 A ,
3

(2.4)

2
2
4 d  AA  r1 A 
8 d AA  r1A  d  AA  r2 A 



3
3r1A
dr1 A

dr12A
dr22A



5

2 d AA  r2 A 
,
r2 A
dr2 A

(2.5)


1 A 

4
3
2
1 d  AA  r1A 
2 d  AA  r1A 
2 d  AA  r1A 
2 d  AA  r1A 

 3
+
4
3
2

2
54
9r1 A
dr1 A
dr1 A
dr1 A
9r1 A
dr1 A
9r1 A

+

4
2
1 d  AA  r2 A 
1 d  AA  r2 A 
1 d  AA  r2 A 

- 3
,
4
2
2
24
dr2 A
dr2 A
4r2 A
dr2 A
4r2 A
4

2 A 

2

(2.6)
3

1 d  AA  r1A 
2 d  AA  r1 A  2 d AA  r1A 
1 d  AA  r2 A 
 2
- 3
+
.  2.7 
4
2
9
dr1A
2r2 A
dr1 A
3r1 A
dr1 A
3r1 A
dr23A
u0C  AC  r1C   2 AC  r2C  , r2C 

2r1C ,

(2.8)

2

kC 

1 d AC  r1C  d  AC  r2C 
1 d AC  r2C 


,
2
r1C
dr1C
r2C
dr2C
dr2C

C 4  1C  2C  ,
2

1C 


(2.10)
4

1 d  AC (r1C ) 1 d AC (r1C ) 1 d  AC (r2C )
- 3



dr1C
48
8r12C
dr12C
8r1C
dr24C

1 d 3 AC (r2C )
3 d 2 AC (r2C )
3 d AC ( r2C )

,
3
2
2
8r2C
dr2C
dr2C
16r2C
dr2C
16r23C

2C 
+

(2.9)

(2.11)

1 d 3 AC (r1C ) 1 d 2 AC (r1C )

1 d AC (r2C )
- 2
 3
+
4r1C
dr1C
dr13C
2r1C
dr12C
2r1C

1 d 3 AC (r2C )
1 d 2 AC (r2C )
1 d AC (r2C )

 3
,
3
2
2
4r2C
dr2C
dr2C
4r2C
dr2C
4r2C

 

u0 A1 u0 A  3 A1C r1A1 ,

k A1  k A 

 

d 2 A1C r1 A1
dr12A1





(2.13)

 

2 d A1C r1A1
,
r1A1
dr1 A1



A1 4 1 A1  2 A1 ,
4

1 A1 1 A 

(2.12)

(2.14)

(2.15)

3

1 d  A1C
1 d  A1C


4
24 dr1 A
6 r1A1 dr13A
1
1
2
3 d  A1C
3 d A1C
 2
 3
,
2
4r1 A1 dr1 A1
4r1 A1 dr1 A1

6

(2.16)


γ 2A1  2 A 

3
1 d  A1C ( r1A1 )
,
4r1 A1
dr13A1

(2.17)

 

u0 A2 u0 A  6 A2 C r1A2 ,
d 2 A2 C r1 A2

 

k A2  k A  2

dr12A2



(2.18)

 

4 d A2 C r1A2
,
r1 A2
dr1 A2

(2.19)



A2 4 1 A2  2 A2 ,
1 A2 1A 

(2.20)

4
1 d  A2 C r1A2

24
dr14A

 
2



2 A2  2 A 

2
15 d  A2 C (r1A2 ) 15 d A2 C (r1 A2 )

,
dr1A2
dr12A2
4r12A2
4r13A2

(2.21)

4
3
1 d  A2 C (r1A2 )
1 d  A2 C (r1A2 )


8
4r1 A2
dr14A2
dr13A2



2
3 d  A2 C (r1A2 ) 3 d A2C (r1A2 )
- 3
,
dr1 A2
dr12A2
8r12A2
8r1A2

(2.22)

For BCC lattice,
u0 A  6 AA  r1 A   3 AA  r2 A  , r2 A 
2

kA  2

1 A 

d  AA  r1 A 
dr12A

2r1 A ,

(2.23)

2



4 d AA  r1 A  d  AA  r2 A 
2 d  AA  r2 A 


,
2
r1A
dr1 A
r
dr2 A
dr2 A
2A

(2.24

)

4
3
2
1 d  AA  r1A 
1 d  AA  r1A  1 d  AA  r1A 
1 d AA  r1A 

 3
+
4
3
2
2
24
4r1A
dr1 A
dr1 A
dr1A
8r1 A
dr1 A
8r1 A

+

4
2
1 d  AA  r2 A 
1 d  AA  r2 A 

1 d AA  r2 A 

- 3
,
4
2
2
24
dr2 A
dr2 A
4r2 A
dr2 A
4r2 A

2 A 

(2.25)

4
3
2
1 d  AA  r1 A 
1 d  AA  r1A 
3 d  AA  r1A 
3 d  AA  r1 A 


 3
+
4

3
2
2
8
4r1A
dr1 A
dr1A
dr1 A
8r1A
dr1A
8r1A

7


+

3
2
1 d  AA  r2 A 
3 d  AA  r2 A 
3 d  AA  r2 A 
 3
.
3
2
2
2r2 A
dr2 A
dr2 A

4r2 A
dr2 A
4r2 A

u0C 3 AC  r1C   4 AC  r2C  , r2C  3r1C ,

kC 

d 2 AC  r1C 
dr12C



(2.27)

2
2 d AC  r1C  4 d  AC  r2C 
8 d AC  r2C 



,
2
r1C
dr1C
3
3r2C
dr2C
dr2C

C 4  1C  2C  ,
1C 

(2.26)

(2.28)
(2.29)

1 d 4 AC (r1C )
1 d 2 AC (r1C ) 1 d AC (r1C ) 1 d 4 AC (r2C )
 2
- 3
+

4
24
dr1C
54
dr1C
4r1C
dr12C
4r1C
dr24C
2 d 3 AC (r2C )
2 d 2 AC (r2C )
2 d AC (r2C )
- 2
 3
,
3

2
9r2C
dr2C
dr2C
9r2C
dr2C
9r2C

2C 

(2.30)

1 d 3 AC (r1C ) 3 d 2 AC (r1C )
3 d AC (r1C )
- 2
 3

3
2
2r1C
dr1C
dr1C
4r1C
dr1C
4r1C
+

1 d 4 AC (r2C )
2 d 2 AC (r2C )
2 d AC (r2C )


- 3
,
4
2
2
9
dr2C
dr2C
3r2C
dr2C
3r2C

 

u0 A1 u0 A   A1C r1A1 ,
k A1  k A 

 

(2.31)
(2.32)

1 d A1C r1A1
,
r1 A1
drr1 A

(2.33)



(2.34)

1



A1 4 1 A1  2 A1 ,
1A1  1A 
2 A1 2 A 

2
1 d  A1C (r1A1 ) 1 d A1C (r1A1 )
- 3
,
dr1 A1
8r12A1
dr12A1
8r1A1

(2.35)

3
2
1 d  A1C (r1 A1 )
1 d  A1C (r1A1 )
1 d  A1C (r1A1 )

 3

,
3
2
2
4r1 A1
dr1 A1
dr1A1
2r1 A1
dr1 A1
2r1 A1

(2.36)

 

uOA2 uOA  4 A2 C r1 A2 ,

8

(2.37)


k A2

 

 

2
4 d  A2 C r1A2

8 d A2C r1A2
 kA 

,
3
3r1A2
dr1 A2
dr12A2



(2.38)



A2 4 1 A2  2 A2 ,
1A2 1A 

4
3
1 d  A2 C (r1A2 )
2 d  A2C (r1A2 )

54
9r1A2
dr14A2
dr13A2



2 A2  2 A 

(2.39)

2
2 d  A2 C (r1A2 )
2 d A2 C (r1 A2 )
 3
,
2
2
dr1 A2
dr1 A2
9r1A2
9r1A2

(2.40)

4
2
1 d  A2 C (r1A2 )
2 d  A2 C (r1A2 )
2 d A2 C (r1A2 )

- 3
,  2.41
4
2
2
9

dr1 A2
dr1 A2
3r1 A2
dr1 A2
3r1A
2

 1 u0
1 k 
Pv  r1 
 θxxx coth x
,
2k r1 
 6 r1

(2.42)

 1 u0 ω0 k 
Pv  r1 

(2.43)

4k r1 
 6 r1
r1C ( P,T ) r1C (P, 0 )  y A1 (P,T), r1 A (P,T) r1 A (P, 0 )  y A (P,T),

(2.44)
(2.45)

r1 A1 ( P, T ) r1C ( P, T ),

r1 A2 ( P, T ) r1 A2 ( P, 0)  yC ( P, T ).

r1 A (P,T) r1A (P, 0 )  y(P,T),r1A (P, 0 )  1  cB  r1A (P, 0 )  cB r1A (P, 0),

(2.46)

y(P,T) c A y A (P,T)  cB y B (P,T)  c A1 y A1 (P,T)  c A2 y A2 (P,T),

r1A ( P,0)  3r1C ( P, 0), c A 1  7cC , c A1 2cC , c A2 4cC for BC lattice,
r1A ( P, 0)  2 r1C ( P, 0) , c A 1  15cC , c A1 6cC , c A2 8cC

for

FCC

lattice.
2.3. Elastic deformation
EYAC  EYA

c
X

X

2 X
 2

2 A
 2

,

(2.51)

Determining Young's modulus, we can also determine other
elastic modules, elastic constants and elastic wave propagation speed

9


according to formulas (2.52) to (2.59).
2.4. Nonlinear deformation
r1FX  P, 0  r1 X  P, 0   1    .

(2.61)

r1FX  P, T  r1 X  P, T    .r1X  P, 0   2    .

(2.62)

F
 AC


F
X

c 
X

 TScACF ,

(2.63)

X

 AC  0 AC



f AC ( ) 

X



2
F
2v AC



c X  X



 1
1
 F 
v

v
AC
 AC

(2.64)

  AC
,
1
 2 r01F X
  F
v AC




f AC ( F )  f AC max C AC  AC max F .

 ACmax 

1 AC max 


 
T




 2r01X   ,

 
T


  2 F 
2   F
F
  F X2  2r01
  FX
X
 r
  r1 X  T
 1X
f AC ( ) C AC AC  ,



  XF
 F
 r1X

f ACmax
,
C AC  F

(2.70)
(2.72)
(2.73)
(2.74)

 AC max
f AC max


1F
C AC  F (1   F )

(2.75)


 F  AC ,
1   AC

(2.76)

 lAC max  0 AC
C AC 

  F 

1F

f AC ( 0,2 )

,

(2.77)

,

(2.78)

 e AC
,
1  e

(2.80)

 AC 0,2 . 0,2

 ACe  0 AC

AC

EYAC  e  0 AC

 e AC

1  e

(2.81)

2.5. Numerical results and discussion for metals and alloys
Using the Mie-Lennard-Jones n-m, the Finnis – Sinclair and the
Morse potentials

10


The numerical results for Fe, FeSi, FeH, FeC, Au, Cu, and CuSi

are summarized in Table 2.2 to 2.10, Table 2.13 to 2.20, illustrated
from Figure 2.2 to 2.19.
7000
6800
6600

Vd (ms-1)

6400
6200
6000
5800

SMM
Experiments of Shibazaki at al. (2016)
Experiments of Antonangeli and Ohtani (2015)
Experiments of Decremps at al. (2014)
Experiments of Liu at al. (2014)

5600
5400
5200
5000

0

2

4

6

8

10

P (GPa)

Figure 2.3. Vd(P) for Fe at T = 300 K calculated by SMM and from
experiments of Antonangeli (2015)[11], Decremps (2014)[25], Liu (2014)
[80] và Shibazaki (2016)[111]

According to Table 2.3 for FeSi at P = 0 and the same temperature
when increasing Si concentration, the elastic modulus, elastic
constant and elastic wave velocity all decrease sharply. For example,
for FeSi at P = 0 and T = 1000K with increasing Si concentration
from 0 to 5%, E decreases 65.48%.
For FeSi at P = 0 and the same Si concentration, with increasing
temperature, the elastic modulus, elastic constant and elastic wave
velocity all decrease. For example, for FeSi at P = 0 and cSi = 5%
with increasing temperature from 100 to 1000K, E decreases 61.88%.
According to Table 2.4 for FeSi at T = 300K and the same Si
concentration with increasing pressure, the elastic modulus, elastic
constant and elastic wave velocity all increase. For example, for FeSi
at T = 300K and cSi = 5% with increasing pressure from 0 to 10GPa,
E increases 35.85%
Table 2.7. EY(cC) (GPa) for FeC at P = 0 and T = 300 K calculated by SMM
and from experiments [115]
E c (10
cC (%)

0
1
1,4
2
2,3
10
Pa/%)

11


SMM
Experiments[115]
δ (%)

208.2
208.2
0

198.5
204.8
3.1

194.6
201.5
3.4

188.8
197.9
4.6

185.9
193.8
4.1

-0.6309
-0.6180
2.1

Bảng 2.8. EY(cH) (GPa) for FeH at P = 0 and T = 0 K calculated by SMM
and ab initio of Psiachos at al (2011)[104]
cH (%)
PPTKMM
ab initio[104]
 (%)

0
222.8
229.2
2.79

1
216.9
225.9
3.98

2
211.1
222.6
5.17

3
205.4
219.4
6.38

4
199.8
216.1
5.01

5
194.2
212.8
8.74

180

PPTKMM
Experiments of Santra at al. (2014)
Experiments of Ledbetter and Naimon (1974)

160

EY (GPa)

140

120

100

80
0

1

2

3

4

5

cSi(%)

Figure 2.11. EY (cSi ) for CuSi at T = 300K and P = 0 calculated by SMM
and from experiments of Ledbetter and Naimon (1974)[75], Santra at al.
(2014)[109]
18

1200

16

cSi = 0
cSi = 2%
cSi = 5%

14

800

1(MPa)

f(GPa)

12

1000

10
8

400

6
4

Fe, SMM
Fe, XEXPT of Smith et al.(2020)
Fe, DEXPT of Smith et al.(2020)
FeSi2%, SMM
FeSi5%, SMM

200

(a)

2
0

600

0

0

1

2

3

4

5

6

7

0

1

2

3

4

(%)

(%)

12

5

6

7


Figure 2.13. (a) f ( , cSi ) and (b)  1 ( , cSi ) for FeSi at T = 300K, P = 0
calculated by SMM and from experiments of Smith at al. (2020)[113]

Conclusion of Chapter 2
In chapter 2, we build the theory of elastic and nonlinear strain of
interstitial alloy AC with cubic structure in which general analytic
expressions of free energy, displacement of particles from lattice are
drawn, mean nearest neighbor distance between two atoms, elastic
deformation quantities such as isothermal and adiabatic elastic
moduli, Young's modulus, mass compression modulus, shear
modulus, elastic constants, longitudinal wave velocity, transverse
wave velocity, nonlinear strain quantities such as strain energy
density, maximum real stress, elastic strain limit to determine stress strain curve. Analytical results of the resulting strain-specific
quantities depend on the temperature, pressure and the interstitial

atom concentration.
The obtained theoretical results are numerically calculated for
metals Fe, Au, Cu and interstitial alloy FeSi, FeH, FeC, AuSi, CuSi.
The law of deformation depends on temperature and pressure of
interstitial alloy is consistent with the law of deformation depending
on temperature and pressure of the main metal in interstitial alloy.
Young E modules, mass compression modulus K, shear modulus G,
elastic constants C11, C12, C44 and elastic wave propagation velocities
Vd, Vn of metal and interstitial alloy decrease with increasing
temperature and increase with increasing temperature. pressure
increase. The elastic strain quantities of interstitial alloy all decrease
with the increase of interstitial atom concentration. The maximum
real stress and elastic strain limit of metal and interstitial alloy
decrease with the increase of temperature and increase with the
increase of pressure. The maximum real stress and elastic strain limit
of interstitial alloy decrease with the increase of interstitial atom
concentration. Many numerical results obtained by SMM, especially
elastic and nonlinear deformation of metals, have been compared
with numerical results by other theoretical methods and experimental
data and have some very consistent results. good. Many numerical
results obtained by SMM on elastic and nonlinear strain of interstitial
alloy are new, predictive, experimentally oriented.

13


CHAPTER 3: STUDY ON NOLINEAR AND ELASTIC
DEFORMATION OF INTERSTITIAL ALLOY ABC
3.1. Alloy model

Fig. 3.1. Model of substitutional and interstitial alloy ABC with BCC
structure (a) and FCC structure(b)

3.2. Helmholtz free energy
 ABC  AC  cB  B   A   TScAC  TScABC ,

(3.1)

3.3. Mean nearest neighbor distance
a ABC c AC a AC

BTAC
B
 cB aB TB ,
BT c AC BTAC  cB BTB .
BT
BT

(3.2)

3.4. Elastic deformation
EYABC  EYAC  cB  EYA  EYB  ,

(3.9)
Determining Young's modulus, we can also determine other
elastic modules, elastic constants and elastic wave propagation speed
according to formulas (3.12) to (3.19).
3.5. Nonlinear deformation
(3.19)
r1FX  P, 0  r1 X  P, 0   1    ,

r1FX  P, T  r1X  P, T    .r1 X  P , 0   2    ,





F
F
 ABC
 AC
 cB  BF   AF  TScACF  TScABCF ,

 ABC  0 ABC

  ABC
,
1 

14

(3.20)
(3.21)
(3.22)




f ABC ( ) 

X



2
F
2v ABC


c X  X


  2 F
  F X2
  r1X

 1
1
 F 
v
v
ABC
 ABC


F
 2r01 X
T





2


1
  1
 cB  A  F 
v
v
ABC
  ABC




2
F
2v ABC

 2 r01F X
  F
 v ABC

  F
  FX
 r
 1X

 2 r01F A

  F
 v ABC

  AF
 F
 r1A




 
T

 

 2r01A   ,
 
T

f ABC ( F )  f ABC max C ABC ABC max F .

(3.26)
(3.27)
(3.28)

f ABCmax
,
C ABC  F

(3.29)

 ABC max
f ABC max


1F
C ABC  F (1   F )

(3.30)

 ABCmax 
1 ABC max 


 


 

 2r01 X   
 
T

  2 F 
2   F
  F A2  2r01F A   FA
 r
  r1 A  T
 1A
f ABC ( ) C ABC ABC  ,



  XF
 F
 r1X


 F  ABC ,
1   ABC

  F 

 lABC max  0 ABC
C ABC 

AC

1F

f ABC ( 0,2 )

,

(3.32)

,

(3.33)

 e ABC
,
1 e

(3.35)

 ABC 0,2 . 0,2

 ABCe  0 ABC

(3.31)

EYABC  e  0 ABC

 e ABC

1  e

(3.36)

3.6. Numerical results and discussion for alloys
The numerical results for FeCrSi, AuCuSi are summarized in
Table 3.1 to Table 3.18 and illustrated from Figure 3.3 to Figure
3.10.

15


340

240

320

220

300
280

200

K (GPa)

240
220
200
SMM
Ab initio of Zhang (2010)
Ab initio of Olsson (2003)
Experiments of Heintze (2009)
Experiments of Speich (1972)

160
140

(a)

120
100

180
160
140

180

0

2

4

6

8

SMM
Ab initio of Zhang (2010)
Ab initio of Olsson (2003)
Experiments of Heintze (2009)
Experiments of Specich (1972)

120

(b)

100
80

10

0

2

4

cCr (%)

6

8

cCr (%)

E c
K c
Figure 3.2. (a)  Cr  and (b)  Cr  for FeCr at T = 298K, P = 0
calculated by SMM, ab initio of Olsson [100], ab initio of Zhang [131] and
from experiments of Heintze [40] and Specich [115]

35

cSi = 0
cSi = 1%
cSi = 3%
cSi = 5%

30

EY (1010 Pa)

EY (GPa)

260

25

20

15

10

5

0

10

20

30

40

50

60

70

P (GPa)

Figure 3.3. EY(P, cSi) of AuCuSi at cCu = 10% and T = 300K

16

10


18
750

16

cSi=0
cSi=2%
cSi=5%

14

700

f(GPa)

f(GPa)

12
10

8

650

600

6

cSi=0
cSi=2%
cSi=5%

550

4

(a)

2
0

0

1

2

3

(b)

500

4

5

6

0

7

1

2

3

Figure 3.4. (a)

4

5

6

7

(%)

(%)

f ( , cSi ) and (b)  1 ( , cSi ) for FeCrSi at cCr =10%,
T = 300K and P = 0

25

800

P=0
P = 2,55 GPa
P = 4,88 GPa
P = 9,47 GPa
P = 18,78 GPa

20

600
500

1(MPa)

f(GPa)

15

P=0
P = 2,55 GPa
P = 4,88 GPa

P = 9,47 GPa
P = 18,78 GPa

700

10

400
300
200

5
100

(a)
0

0

0

2

4

6

8

10

12

(b)
0

2

4

6

8

10

12

(%)

(%)

 ( , P) for AuCuSi at cCu =10%,
Figure 3.9. (a) f ( , P) and (b) 1
cSi = 1% và T = 300K
Conclusion of Chapter 3
In chapter 3, we build the theory of elastic-nonlinear deformation
of substitutional and interstitial ternary alloy ABC. with the cubic
structure, which derives the general analytic expressions of free
energy, magnitude displacement of the particle from the lattice,

average nearest neighbor distance between two atoms, elastic
deformation quantities such as isothermal and adiabatic elastic
moduli, Young's modulus, mass compression modulus, shear
modulus, constants elastic, longitudinal wave velocity, transverse

17