Academic Journal of Science and Technology 
ISSN: 2771-3032 | Vol. 11, No. 1, 2024 

 

36 

Effect	of	Hydrogen	Embrittlement	on	Dislocation	
Emission	from	A	Bifurcation	Crack	Tip	in	Nano‐metallic	
Materials	
Xiaoya Song, Min Yu 

College of Civil Engineering, Central South University of Forestry and Technology, Changsha 410004, PR China 

 

Abstract: A theoretical model was established to investigate the interaction between hydrogen clusters and edge dislocation 
near bifurcation crack tip in deformed nano-metallic materials. The model’s solution was obtained by using the complex method, 
and the influence of hydrogen concentration, temperature, Relative crack length, material constants of nano-metallic materials 
and the dislocation emission angle on the critical stress intensity factor (SIFs) corresponding to the first dislocation emission 
from the crack tip was investigated through numerical analysis. The results show that dislocations are easy to emit from the crack 
tip at high hydrogen concentrations. However, under the influence of hydrogen clusters, the high temperature will make 
dislocation emission more difficult. At the same time, when relative cracks become longer, it becomes more difficult for 
dislocations to emit from the crack tip. As the angle of dislocation increased, the SIF first decreased and then increased. And as 
the angle between the main crack and the bifurcation crack continues to increase, the dislocation emission behavior at the tip of 
the bifurcation crack will be significantly promoted. 

Keywords: Hydrogen clusters; complex variable method; dislocation emission; stress intensity factor; bifurcation cracks. 
 

1. Introduction 
Metals are commonly used materials in industry, and their 

mechanical properties can significantly affect their service 
life, which has received extensive attention from scholars. 
Grain size, as an important internal structural parameter of 
metallic materials, affects the mechanical properties of the 
materials. Especially, when the grain size of metallic 
materials is reduced to the nanometer order of magnitude, the 
mechanical properties of the materials will be significantly 
changed. In general, nano-metallic materials with a grain size 
of 100 nm or less are referred to as nano-metallic materials 
[1-3]. Cracks are very common in nano-metallic materials and 
there are many typical shapes of crack, such as surface semi-
elliptical cracks, cross cracks, interfacial co-linear cracks, and 
elliptical blunt cracks. When the stress near the crack is high 
enough, dislocations are created near the crack tip. The 
internal stresses generated by dislocations emitted from the 
crack tip can harmonize the stress intensity provided by the 
applied load, which in turn produces toughness in the material. 
During the processing and use of metal materials, hydrogen 
gas, known as hydrogen clusters, inevitably enters, which has 
a serious impact on the mechanical properties of the material. 
In many real-life situations, such as petrochemicals and 
aerospace, hydrogen embrittlement is much more common 
and often results in serious accidents. 

After the phenomenon of hydrogen embrittlement was first 
discovered by Johnson [4] in 1874, it has triggered the 
enthusiasm of many scholars to study the phenomenon, and 
then found that the phenomenon of hydrogen-induced 
embrittlement is universal. Robertson [5] proposed more 
detailed hydrogen embrittlement mechanism. On the basis of 
hydrogen embrittlement mechanism, Oriani, Hirth et al [6-13] 
conducted a detailed study of hydrogen embrittlement 
phenomenon in iron, and different high strength metallic 
materials, etc. Tabata and Birnbaum [14] experimentally 
observed the effect of hydrogen on the velocity of dislocation 

motion and proposed a theory of hydrogen-enhanced 
localized plasticity. This theory states that hydrogen increases 
the rate of motion of dislocations under stress, thus affecting 
the plasticity and toughness of the material. Sirois [15] 
proposed the mechanism of hydrogen around dislocations and 
other elastic stress field centers to explain the effect of 
hydrogen on dislocation motion. Song [16-17] and others 
proposed a model of hydrogen clusters consisting of hydrogen 
atoms aggregated near the crack tip and found that hydrogen 
hinders dislocation emission at the microscopic crack tip, 
which in turn hinders blunting and ductile fracture at the crack 
tip, thus promoting crack tip brittle fracture. Yang [18] 
proposed a coupled mechanical diffusion model to describe 
the effect of stress on hydrogen diffusion in titanium. Huynh 
[19] characterized the microscopic features of hydrogen-
induced delayed crack extension in thin plates of single-
crystal Fe-3wt % Si alloys.  

As technology advances, computer simulations are used to 
conduct research, which can provide microscopic resolution 
that cannot be achieved by experiments at the current 
techniques. Wang and Nowak [20-21] investigated the effect 
of hydrogen on the dislocation motion of surface single-
crystal Fe by nanoindentation experiments and discrete 
dislocation dynamics simulations. There are several papers 
[22-25] that investigated the effect of hydrogen on the 
emission mechanism of α-Fe dislocations using methods such 
as molecular dynamics simulations and atom modeling. Zhao 
[26] utilized a molecular dynamics simulation method to 
investigate the emission of crack tip dislocations in Ni single 
and crystalline samples in a hydrogen environment. Chen [27] 
systematically investigated the quantitative characterization 
of hydrogen-influenced interactions between dislocations and 
grain boundaries through molecular dynamics simulations. 
On this basis, Charles [28] predicted hydrogen concentration 
for various boundary value problems through finite element 
simulation modeling. 

Liu [29] developed a theoretical method for calculating the 



 

37 

stress intensity factors (SIFs) of branching cracks. An 
asymptotic approximation of the conformal mapping was 
proposed by employing the Schwarz-Christoffel mapping and 
Muskhelishvili's method, which was able to derive the critical 
stress intensity factors for arbitrary branching cracks. Also, 
the convenience of this analytical method in obtaining SIFs 
for bifurcated cracks and four-branch cracks is further 
demonstrated. The interaction between dislocations and 
cracks and hydrogen clusters has been studied 
overwhelmingly as a separate study of dislocations and cracks 
or dislocations and hydrogen clusters. But in reality, all three 
phenomena often co-exist and interact with each other. 
Therefore, it is particularly important to investigate the 
mechanism of hydrogen clusters’ effects on dislocation 
emission from crack tips in nano-metallic bi-materials and 
explore their impact on the toughness of nanometallic 
materials. 

So this paper is dedicated to study the effect of hydrogen 
clusters on dislocation emission from interfacial co-linear 
crack tips in nanometallic materials. Based on the observed 
results, a theoretical mechanical model to describe the 
interaction between dislocations emission and hydrogen 
clusters near an interfacial co-linear crack tip in nano-metallic 
materials is established and the analytical solution of the 
model can be obtained through the complex variable method 
of elasticity. Finally, the effects of hydrogen concentration, 
temperature, and crack length on the intensity factor of 
dislocation emission stress at the tip of interfacial co-linear 
linear cracks are discussed. These discussions contribute to a 
deeper understanding of the toughening mechanism of 
nanometallic materials and inform the design and application 
of this material. 

2. Establishment of the Theoretical 
Model 

When hydrogen seeps into metals, it causes a variety of 
reactions that make the material more vulnerable to failure. In 

metallic systems such as high-strength steels and nickel, the 
presence of hydrogen often embrittles the material by causing 
a sharp transition from ductile fracture (i.e. micro-void 
coalescence) to brittle intergranular fracture, accompanied by 
a drastic loss in toughness and ductility. At the microscopic 
level, when the temperature and strain rate are kept in a 
certain range, the presence of hydrogen in the solid solution 
decreases the barrier of dislocation emission, which leads to 
greater deformation in the local area near the fracture surface. 
In addition, hydrogen embrittlement occurs mainly at the 
atomic level, and to investigate the properties of hydrogen 
embrittlement more easily, we introduced a hydrogen cluster. 

As shown in Fig.1, he nanometallic material is a 
homogeneous and isotropic elastic medium. Under this 
assumption, the mechanical properties of the material can be 
described by its shear modulus μ and Poisson's ratio ν. The 
nano-metallic material contains an internal crack consisting 
of the main crack and its finite-length branch cracks, with the 
length of the main crack set to be a, the length of the branch 
cracks set to be b, and the angle between the two cracks set to 
be (1-m)×π. The material is subjected to far-field type I and 
type II loads. For the sake of generality, the branch crack 
length is assumed to be equal to or less than the main crack 
length, i.e., b≤a. Then, introduce a polar coordinate system 

 ,dr  , where right endpoint of crack L0 is set as the origin. 

An edge dislocation with Burgers vector 1 x yb b ib    is 

emitted from the co-linear crack interface to the point Z0. At 
the same time, a circular hydrogen cluster with a dislocation 
as the center is generated near the tip of the interface crack, 
with a radius of R = 20br. Another polar coordinate system (r1,
  ) is established based on the position of the dislocation, i.e. 

the center of the circle, as the coordinate origin. In order to 
study the effect of hydrogen clusters on dislocations, a micro-
element area dS is taken from the circular hydrogen cluster at 
any point. 

 

 

 
Figure 1. Mechanical model for the interference between hydrogen clusters and dislocation emissions at the tip of a bifurcated 

crack 



 

38 

 

3. Derivation of Theoretical Solutions 
For plane strain problems, the stress field can be expressed 

using two of Muskhelishvili’s complex potential functions 
[30]: 

 

   

       

2xx yy

yy xy

z z

i z z z z z

   

     

      




     
  (1) 

 
Assuming the crack face is free, the boundary conditions 

for the crack face can be denoted as: 
 

   i 0yy xyt t   , t∈Blunt crack surface     (2) 

 
To solve the problem more easily, the following mapping 

function is introduced [31]: 
 

 
1 1

(1 )
2

(ζ tan ) (ζ+ tan )
2 2z=ω ζ =2R cos )

ζ 1

m m

i me 

 


 







(1+ (3) 

 
Where, ζ= + ζi . 
Using Eq. (3), the surrounding domain of the crack in the 

z-plane is mapped to the plane. In addition, 
1 1

4 cos cos
2 2

m m

a R
   

 
              

      
，

1 1

4 sin sin
2 2

m m

b R
   

 
              

      
（ b≤a ） ，

sin = sinm   (0≤m≤1). The parameters α and β can be 
determined by assigning values to the factor m and the length 
b/a. 

Derivation of equation (3) gives: 

 
(1 ) 1

2

2R( cos ) (1 cos )(ζ tan ) (ζ+ tan )
2 2z =ω ζ =

ζ 1

i m m me
' '

    



1+ +
 

                 

 

(1 ) 1 1

22

4Rζ (1 cos )(ζ tan ) (ζ+ tan )
2 2

ζ 1

i m m me     




+
 

                  

 

 

(1 ) 1

22

2R 1 (1 cos )(ζ tan ) (ζ+ tan )
2 2

ζ 1

i m m mm e      




+
 

             =

 
(1 )

22

1
[4Rζ (1 cos )(ζ tan ) (ζ+ tan )

2 2ζ 1

i m m me     


+  

 

    2(ζ tan ( ζ tan ))]
2 2

m m m
 

           (4) 

 

  2 3 2 4
2 3

1
z =ω ζ = [ 4 ζ(ζ 1) tan( ) (1 3ζ ) tan( ) )

(ζ 1 2 2
'' '' m

 
    

 ）
 

                
4 2 4 2 2 2 22 ζ(ζ 1) tan ( 1 4ζ 3ζ 2 (1 ζ ) ) tan( )

2 2
m m

 
         

(1 ) 1 1 2 24 R(1 cos )(ζ tan ) (ζ+ tan ) (ζ ( 1 3ζ )]
2 2

i m m me          +   

(5) 
 
The elastic stress field generated by the emission of 

dislocations from the bifurcation crack tip can be accurately 
expressed by the complex potential function and as: 

 

     *
0e e e                  (6) 

 

     *
0 e e e                    (7) 

 

where,  0 0 0ln( ) / ( )e H           ，

( ) / [4 (1 )]y xb ib      ， 

 0 0ln( )e      ， 0 0( ) / ( )H '    。 

The complex potential function can be derived as follows: 
 

  0 0

0

ln( ) ln( )e

W       
 

     


 (8) 

 

  0 0
0

ln( ) ln( )e

H       
 

    


        

 

        2
0 0

( )
[ ]

( ) ( )

W

'

   
     

 
 

      (9) 

                                      

where, 0 0 0[ ( ) ( )] / ( )W '       . 

The force acting on the dislocation at the crack tip consists 
of three main components: firstly, the dislocation's own like 
force, which is an internal force generated by the dislocation's 
own structure; secondly, the force induced by the hydrogen 
cluster, which is a force exerted on the dislocation by some 
mechanism due to the presence of hydrogen clusters in the 
vicinity of the crack tip; and lastly, the force generated by the 
applied load, which is a force generated by the externally 
applied mechanical loads transferred through the crack tip to 
the dislocations. 

Firstly, the image force df  acting on the dislocation itself 

is calculated. The image force acting on the dislocation is 
obtained according to the Peach-Koehler formula while 
considering Eq. (6)-(9): 

       0 0 0 0ˆ ˆ ˆi ˆid xe ye xy x yy y xx x xy yf f f b b b b                    
 

 
         * * * *

0 0 0 0 0Φ Φ Φ Ψ

4 1
e e e e     

   

  
  

   
(10) 

where, ˆ xy  ,  ˆ yy   and  ˆ xx  are the perturbation stress 

components,      * *'
0 0 0Φ /e e        and 

     * *'
0 0 0Ψ /e e      . 



 

39 

Then, the force hf   induced by the hydrogen cluster is 

calculated. As a result of this elastic interaction, the 
concentration of hydrogen is described by [32]: 

 

0

0

1 exp( )
1

int
h

c W
c

c kT
  


             (11) 

 
where hc  is the hydrogen concentration near the 

dislocation core, 0c   is the hydrogen concentration in the 
absence of applied stress; k is the Pozierman constant, and T 
is the absolute temperature. intW   is the interaction energy 
between the hydrogen cluster and the dislocation in a semi-
infinite plane, which can be expressed as [32]: 

 

  1

1 ν sin

3 1
r

int

b
W

r

  
 


  


（ ）
             (12) 

 
where rb  is the Burgers vector,   is the angle between 

the coordinate axis 2x  and the position vector, and 1r  is the 
distance from the center of the dislocation to the center of the 
micro-element dS . 

Therefore, in polar coordinates, the stress induced by a 
hydrogen cluster is given by the following equation: 

 

  2

R 2

0

Ω 2
d d

2 1
r

h r h hr

b
f b c sin r

r

  
 

  
     (13) 

 
where    is the partial molar volume and 2  r  is the 

dislocation core radius of the dislocation. During the analysis, 
it is assumed that the dislocation core radius is equal to the 
Burgers vector rb , R  is the radius of the hydrogen cluster 
with the dislocation as the center of the circle. 

Finally, the force acting on the edge dislocations due to the 
applied loads is: 

 
2

0 0

1 3
( sin cos ) cos sin cos
2 2 2 2 2 22 2

N N
r r

r a

b K b K
f b

r r

    
 
 


     
 

  (14) 

 

where N
ΙCK  and N

ΠCK  correspond to model I and model 

II critical stress intensity factors. 
Thus, combining Eqs. (13), (16), and (17), the combined 

force acting on the dislocation emitted from the tip of the 
colinear linear crack can be obtained as: 

 
   ( cos sin ) Re cos Im sinemit x y d h d hf f f f f f f f f           (15) 

4. Discussion and Analysis 
We adopt the generally accepted criterion for dislocation 

emission. The condition for dislocation emission from crack 
tips is that the sum of all forces acting on dislocations is zero, 
and the distance between dislocations and cracks should be 
greater than the dislocation core radius [38]: 

Combining Eq. (15) and the dislocation emission condition 

0emitf  , the critical stress intensity factor for dislocation 

emission can be calculated as: 
 

N 0K   ,

0N 2 2
(Im[ ]sin Re[ ]cos )

sin cos
2 2

d h d h

r

r
K f f f f

b


        (16) 

 
N 0K  , 

0N

2

2
(Im[ ]sin Re[ ]cos )

3
(cos sin cos )

2 2 2

d h d h

r

r
K f f f f

b

        


(17) 

 
Eqs. (16-17) can be used to analyze the effect of hydrogen 

clusters on the critical stress intensity factor for dislocation 
emission from the colinear crack tip. To facilitate the analysis, 

we dimensionless the stress intensity factors N
ΙCK  and N

ΠCK  

as 0 N
ΙC ΙC / rK K b   and 0 N

ΠC ΠC / rK K b  , 

respectively. The material parameters of the nanometallic 
material Ni are used for the lower half plane of the crack, with 
relative shear modulus 2 73Gpa   , poisson's ratio 

ν2=0.31. In addition, the Burgers vector rb  =0.25nm is 

assumed for edge dislocation. The hydrogen-related 

parameters are set as 3Ω 3.39 Å , 2 rr b  and R=20 rb . 

As shown in Fig. 2: when θ = 0.5 rad, T = 200 °C, b/a = 0.3, 
m = 1/3, and C0 = 0.01 mol, the graphs depicting the changing 
patterns of the two SIFs with the relative position of the 
hydrogen cluster to the crack tip are shown. From the figure, 
it can be seen that as the relative position of the hydrogen 

cluster to the crack tip increases, 0
ΙCK  and 0

ΠCK  then 

decreases, so that the hydrogen cluster relatively closer to the 
bifurcation crack tip hinders the emission of dislocations. 

When the relative position is certain, 0
ΙCK ＞

0
ΠCK  , 

indicating that the type II applied load is more likely to emit 
dislocations from the crack tip than type I. The relative 
position of the hydrogen cluster to the crack tip increases with 
the increase of the relative position of the hydrogen cluster to 
the crack tip. 

As shown in Fig.3, the two dimensionless critical SIFs 
corresponding to the blade-type dislocation emission vary 
with the temperature T when θ = 0.5 rad, b/a = 0.3, and C0 = 
0.01 mol. As can be seen from the figure, the critical stress 
intensity factor increases with increasing temperature under 
the influence of the hydrogen mass, so that the increase in 
temperature hinders the emission of cleavage-tip dislocations, 
which leads to a decrease in the toughness of the material 
induced by the dislocation emission. When the temperature 
takes a certain value, with the increase of m, the critical stress 
intensity factor decreases, that is, when the angle between the 
main crack and the bifurcation crack increases, the tendency 
of dislocations to be launched from the tip of the bifurcation 
crack will be more obviously and easily, and the selection of 
a suitable angle helps the launch of dislocations at the tip of 
the crack, which improves the toughness of the material due 
to the launch of dislocations. 

As shown in Fig. 4, the variation of the dimensionless 
critical SIF corresponding to the edge dislocation emission 
with the hydrogen concentration C0 is plotted when θ=0.5 rad, 
T=200°C, and b/a=0.3. From the figure, it can be seen that 
when different values of m are taken, the critical SIF 
decreases with the increase of hydrogen concentration, i.e., 



 

40 

the relatively high hydrogen concentration can promote the 
emission of dislocations at the crack tip. The critical SIF 
decreases with the increase of m when the hydrogen 
concentration takes a certain value, that is, the larger the angle 
between the main crack and the bifurcation crack, the easier 
the dislocations at the tip of the bifurcation crack will be 
emitted. 

 

25 30 35 40 45 50
0.402

0.404

1.374

1.376

1.378

1.380

h (nm)

 K0
ⅠC

 K0
ⅡC

0
CK  

 
Figure 2. Dependence of normalized critical SIFs with 

different hydrogen cluster relative position to the fracture tip 
 

0 50 100 150 200 250 300 350 400
1.0

1.2

1.4

1.6

1.8

2.0

T (℃)

 m=1/5
 m=1/3
 m=2/3

0
ΙCK

 
Figure 3. Dependence of normalized critical SIFs on the 

temperatures T with different m values 
 

As shown in Fig. 5: a plot of the dimensionless SIF of the 
edge dislocation emission as a function of temperature T 
when C0 = 0.01 mol, θ = 0.5 rad, and m = 1/3. increases with 
increasing temperature. This implies that the emission of 
cleavage-tip dislocations is hindered in a high temperature 
environment. When the temperature takes a certain value, the 
dislocation emission from the bifurcated crack tip becomes 
progressively more difficult as the relative crack length 
continues to increase, making it more difficult. 

As shown in Fig. 6, when C0 = 0.01 mol, T = 200 °C, and 
m = 1/3, the dimensionless two SIFs of the edge dislocation 
emission vary with the dislocation emission angle θ. As can 
be seen from Fig. (a), with the increase of the blade 
dislocation emission angle, it decreases and then increases, 
and there is a very small value (i.e., the easiest dislocation 
emission angle θmin). When the relative crack length is taken 
as b/a=0.2, 0.3, 0.5, 0.7 and 0.9, the corresponding dislocation 
easiest firing angle is θmin =1.1 rad; and when the dislocation 
firing angle is taken as a certain value, it increases with the 
increase of temperature, which leads to more difficult 
emitting of crack tip dislocations, and thus reduces the 

toughness of the material due to dislocation emitting. 
 

0.000 0.002 0.004 0.006 0.008 0.010
0.6

0.8

1.0

1.2

1.4

C0 (mol)

 m=1/5
 m=1/3
 m=2/3

0
ΙCK

 
Figure 4. Dependence of normalized critical SIFs on the 

hydrogen concentrations C0 with different m values 
 

0 50 100 150 200 250 300 350 400
5

10

15

20

25

30

35

T (℃)

 b/a=0.2
 b/a=0.3
 b/a=0.5
 b/a=0.7
 b/a=0.9

0
ΙCK

 
Figure 5. Dependence of normalized critical SIFs on the 
temperatures T with Different relative crack lengths b/a 

 
As can be seen from Fig. 6(b), the dislocation emission 

angle of a finite positive value starts to increase rapidly until 
it tends to infinity, and then becomes negative. When negative 
dislocations are considered, the absolute value of negative 
SIF shows a tendency to decrease and then increase with the 
increase of the dislocation emitting angle, and there also 
exists an easiest emitting angle for negative dislocations. 
When the relative crack lengths are taken as b/a=0.3, 0.5, 0.7 
and 0.9, the most probable emitting angles of the most 
corresponding dislocations are θmin =2.0 rad; and when 
b/a=0.2, θmin =1.8 rad. 

As shown in Fig. 7, when C0=0.01mol, T=200℃, b/a=0.3, 
the dimensionless two SIFs of the edge dislocation emission 
are plotted with the change of the dislocation emission angle 
θ. The SIFs of the edge dislocation emission are shown in Fig. 
7. From Fig. 7(a), it can be seen that with the increase of the 
blade dislocation emission angle, followed by the first 
decrease and then increase, so there exists an easiest 
dislocation emission angle. When taking m=1/5, 1/3, and 2/3 
respectively, the corresponding easiest dislocation launch 
angle is θmin=1.4 rad; and when the dislocation launch angle 
is certain, it decreases with the increase of m, i.e., the larger 
the angle between the main crack and the bifurcation crack, 
the easier it is to emission dislocations from the tip of the 
bifurcation crack, which improves the toughness of the 
material induced by the dislocation emission. 

From Fig. 7(b), it can be seen that the increase of the 



 

41 

dislocation emission angle starts from some finite positive 
value, then increases rapidly until it tends to infinity, and then 
becomes negative. Therefore, the most probable dislocation 
emission angle for a positive edge dislocation is θmin=0 rad. 
Considering the negative dislocations, the most probable 
emission angle for the negative dislocations is θmin=1.6 rad for 
m=1/5, 1/3, and 2/3. For a certain dislocation emission angle, 
the emission angle decreases with the increase of m. The 
dislocation emission angle of a positive edge dislocation is 
θmin=1.6 rad. 

 

0.0 0.5 1.0 1.5 2.0 2.5
0

20

40

60

80

 (rad)

 b/a=0.2
 b/a=0.3
 b/a=0.5
 b/a=0.7
 b/a=0.9

0
ΙCK

 

(a) 
0
ΙCK  

0.0 0.5 1.0 1.5 2.0 2.5
-20

-10

0

10

20

30

 b/a=0.2
 b/a=0.3
 b/a=0.5
 b/a=0.7
 b/a=0.9

 (rad)

0
ΠCK

 

(b) 
0
ΠCK  

Figure 6. Dependence of normalized critical SIFs on the 
emission angle θ with different relative crack lengths b/a for 

(a) 0
ΙCK ,(b) 0

ΠCK  
 

0.0 0.5 1.0 1.5 2.0
0.0

0.5

1.0

1.5

2.0

2.5

(rad)

 m=1/5
 m=1/3
 m=2/3

0
ΙCK

 

(a) 
0
ΙCK  

0.0 0.5 1.0 1.5 2.0
-1.2

-0.9

-0.6

0.6

0.9

(rad)

 m=1/5
 m=1/3
 m=2/3

0
ΠCK

 

(b) 
0
ΠCK  

Figure 7. Dependence of normalized critical SIFs on the 
emission angle θ with different m values (a) 0

ΙCK , (b) 0
ΠCK  

5. Conclusion 
A theoretical model is established to investigate the 

interaction between the hydrogen clusters and edge 
dislocation near an bifurcation crack tip in nano-metallic bi-
materials. The analytic expression of critical SIFs 
corresponding to dislocation emission is deduced by the 
complex potential method in elasticity, and the effect of the 
hydrogen concentration, temperature, relative crack length, 
the angle between the main crack and the bifurcation crack 
and emission angle on the critical SIFs are discussed. The 
conclusions are summarized as follows:  

(1) With the concentration of the applied hydrogen cluster 
changes, the SIFs under applied loading all decrease with 
increasing hydrogen concentration. Therefore, dislocations at 
high hydrogen concentrations are more likely to emit from 
crack tips. 

(2) As the relative crack length increases, dislocation 
emission from the crack tip is hindered. At the same time, the 
increase in temperature is also unfavorable to the edge 
dislocation emission, i.e., high temperatures inhibit the 
emission of dislocations from the bifurcation crack tip under 
the influence of hydrogen gas clusters. 

(3) The critical SIF under both applied loads decreases 
with the increase of the angle between the main crack and the 
bifurcation crack, so the larger the angle is the more favorable 
for the dislocations to be emitted from the bifurcation crack 
tip.  

References 
[1] Abrikosov A. Fundamentals of the Theory of Metals. Phys. 

Today. Vol. 43 (1990) No. 11, p73-76. 

[2] M. S. Muhd Norhasri, M. S. Hamidah, A. Mohd Fadzil. 
Applications of using nanomaterial in concrete: A review. 
Constr. Build. Mater. Vol. 133 (2017), p91-97. 

[3] Meyers M A, Mishra A, Benson D J. Mechanical properties of 
nanocrystalline materials. Prog. Mater. Sci. Vol. 51 (2006), 
p427-556. 

[4] Johnson W.H. On some remarkable changes produced in iron 
and steel by the action of hydrogen and acids. Proc. R. Soc. 
London. Vol. 23 (1874). No. p156-163, p168-179. 

[5] Robertson M I, Sofronis P, Nagao A, et al. Hydrogen 
Embrittlement Understood. Metall. Mater. Trans. B. Vol.46 
(2015) No. 6, p2323. 



 

42 

[6] Oriani R A. Hydrogen embrittlement of steels. Annu. Rev. 
Mater. Sci. Vol. 8(1978), p327. 

[7] Hirth J P. Effects of hydrogen on the properties of iron and steel. 
Metall. Trans. 11A, Vol. 861 (1980). 

[8] Venezuela J, Liu Q, Zhang M, et al. A review of hydrogen 
embrittlement of martensitic advanced high-strength steels. 
Corros. Rev. Vol.34 (2016) No. 3, p153-186. 

[9] Koyama M, Akiyama E, Lee Y, et al. Overview of Hydrogen 
Embrittlement in High-Mn Steels. Int. J. Hydrogen Energy. 
Vol. 42 (2017) No. 17, p12706. 

[10] Panday G A. Hydrogen induced cracking of pipeline and 
pressure vessel steels: A review. Eng. Fract. Mech. Vol. 199 
(2018), p609. 

[11] Ohaeri E, Eduok U, Szpunar J. Hydrogen related degradation 
in pipeline steel: A review. Int. J. Hydrogen Energy. Vol. 43 
(2018) No. 31, p14584. 

[12] Liu Q, Atrens A. A critical review of the influence of hydrogen 
on the mechanical properties of medium-strength steels. Corros. 
Rev. Vol. 31 (2013) No.3-6, p85-103. 

[13] Dwivedi S K, Vishwakarma M. Effect of hydrogen in advanced 
high strength steel materials. Int. J. Hydrogen Energy. Vol. 
44(2019) No.51, p28007-28030. 

[14] Tabata T, Birnbaum H K. Direct observations of hydrogen 
enhanced crack propagation in iron. Scr. Metall. Vol. 18 (1984) 
No. 3, p231-236. 

[15] Sirois. Effects of hydrogen and carbon on thermally activated 
deformation in nickel. Acta Metall. Mater. Vol. 40 (1992) No.6, 
p1377-1385. 

[16] Song J, Curtin W A. A nanoscale mechanism of hydrogen 
embrittlement in metals. Acta Mater. Vol.59 (2011) No.4, 
p1557-1569. 

[17] Song J, Curtin W A. Atomic mechanism and prediction of 
hydrogen embrittlement in iron. Nat. Mater. Vol. 12 (2013) No. 
2, p145-151.  

[18] Yang F Q, Zhan W J, Yan T, et al. Numerical Analysis of the 
Coupling between Hydrogen Diffusion and Mechanical 
Behavior near the Crack Tip of Titanium. Math. Probl. Eng. 
Vol. 2020 (2020), p1-15.  

[19] Huynh T T, Koyama M, Takahashi Y, et al. Plastic deformation 
sequence and strain gradient characteristics of hydrogen-
induced delayed crack propagation in single-crystalline Fe–Si 
alloy. Scr. Mater. Vol. 178 (2020), p99-103. 

[20] Wang J, Zhao L, Huang M S, et al. Effect of Hydrogen on 
Dislocation Nucleation and Motion: Nanoindentation 
Experiment and Discrete Dislocation Dynamics Simulation. 
Acta Mech. Solida Sin. Vol. 35 (2021) No. 1, p1-14. 

[21] Nowak C, Zhou X W. An interplay between a hydrogen 
atmosphere and dislocation characteristics in BCC Fe from 
time-averaged molecular dynamics. Phys. Chem. Chem. Phys. 
Vol. 25 (2023) No. 12, p3290-3299. 

[22] Kapci Mehmet Fazil, Schön J. Christian, Bal Burak. The role 
of hydrogen in the edge dislocation mobility and grain 
boundary-dislocation interaction in [formula omitted]-Fe. Int. 
J. Hydrogen Energy. Vol. 46 (2021) No. 64, p32695-32709. 

[23] Matsumoto Ryosuke, Oyinbo Sunday T, Mugilgeethan 
Taketomi Shinya. Hydrogen Effect on the Mobility of Edge 
Dislocation in α-Iron: A Long-Timescale Molecular Dynamics 
Simulation. I S I J International. Vol. 62 (2022) No. 11, p 2402-
2409. 

[24] Xing X, Yu M, Chen W, et al. Atomistic simulation of 
hydrogen-assisted ductile-to-brittle transition in α-iron. 
Comput. Mater. Sci. Vol. 127 (2017), p211-221. 

[25] Wan L, Geng W T, Ishii A, et al. Hydrogen embrittlement 
controlled by reaction of dislocation with grain boundary in 
alpha-iron. Int. J. Plast. Vol. 112 (2019), p206-219. 

[26] Zhao K, He J, Zhang Z. Effect of grain boundary on the crack-
tip plasticity under hydrogen environment: An atomistic study. 
J. Appl. Phys. Vol. 127 (2020) No.1, p015101. 

[27] Chen J W, Zhu Y X, Huang M S, et al. Study on hydrogen-
affected interaction between dislocation and grain boundary by 
MD simulation. Comput. Mater. Sci. Vol. 196 (2021), p110562. 

[28] Charles Yann, Mougenot Jonathan, Gaspérini Monique. 
Modeling hydrogen dragging by mobile dislocations in finite 
element simulations. Int. J. Hydrogen Energy. Vol. 47 (2022) 
No. 28, p13746-13761. 

[29] Liu Z-E, Wei Y. A semi-analytical solution to the stress 
intensity factors of branched cracks[J]. Journal of the 
Mechanics and Physics of Solids. Vol. 179 (2023), p105351. 

[30] Xianghua P, Min Y, Youwen L. Effect of nanotwin and 
dislocation pileup at twin boundary on dislocation emission 
from an interfacial collinear crack tip in nanocrystalline 
materials. Mech. Adv. Mater. Struct. Vol. 27 (2020) No. 12, 
p965-974. 

[31] Muskhelishvili N I. Some Basic Problems of the Mathematical 
Theory of Elasticity. The American Mathematical Monthly Vol. 
74 (1967) No. 6, p752. 

[32] Sih G C. Stress Distribution Near Internal Crack Tips for 
Longitudinal Shear Problems[J]. Journal of Applied Mechanics, 
Vol. 32 (1965) No.1, p51–58. 

[33] Rice J R, Thomson R. Ductile and Brittle Behavior of Crystals. 
Philos. Mag. Vol. 19(1974) No.1, p73. 

[34] P. Sofronis, H.K. Birnbaum. Mechanics of the hydrogen-
dislocation-impurity interactions—I. Increasing shear modulus. 
J. Mech. Phys. Solids. Vol. 43 (1995) No. 1, p49-90.