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

 

265 

Free	vibration	Analysis	of	Beams	with	Functional	
Gradient	Materials	with	Cracks	
Zhenxing Pei* 

School of Mechanical and Power Engineering, Henan Polytechnic University, Jiaozuo 454000, China 
* Corresponding author Email: 1009399503@qq.com 

 

Abstract: The article presents the development of a dynamic model for a functionally graded material (FGM) beam 
incorporating cracks. Initially, it assumes an exponential distribution of material properties along the thickness direction of the 
beam and simulates the opening crack using a zero-mass rotational spring model. This approach enables the calculation of 
bending stiffness and local flexibility at the cracked section. Subsequently, drawing upon Timoshenko beam theory and von 
Kármán geometric nonlinear theory, the study formulates the energy equation of the beam and establishes the partial differential 
control equations for the cracked FGM using Hamilton's principle. The method of separation of variables is employed to 
discretize the partial differential motion equations into ordinary differential motion equations. Beam functions serve as mode 
functions, whose unknown coefficients are determined according to the boundary and continuity conditions, thereby yielding the 
natural frequencies and mode shapes of the cracked FGM beam. Numerical analysis is conducted to evaluate the impact of 
boundary conditions, the relative position of cracks, and the length-to-thickness ratio on the natural frequencies of the cracked 
FGM beam. 

Keywords: Functional gradient materials, crack, Free vibration. 
 

1. Introduction 
Since the 1930s, the vibration problem of beams with 

cracks has drawn the attention of researchers, and extensive 
research has been conducted in this field over time. The 
exploration process typically includes selecting the method 
for modeling cracks, establishing dynamic equations, 
applying various algorithms, and investigating vibration 
characteristics, excitation responses, stability, and crack 
parameter identification. 

Irwin GR[1] analyzed the dynamic characteristics of metal 
plates with cracks through the theory of local flexibility, 
laying the foundation for subsequent research. Sourki R et 
al.[2] utilized the classical Euler-Bernoulli beam theory and 
different methods such as the modified stress couple theory, 
Fourier series, and Galerkin method to analyze the free lateral 
vibration of cracked beams, obtaining the frequency 
equations for cracked beams. By employing Hamilton's 
principle, they derived control equations and related boundary 
conditions, taking into account the additional strain energy 
caused by cracks and the discontinuity in bending slope due 
to cracks. The study examined the impact of different crack 
positions, crack depths, material length scale parameters, and 
Poisson's ratio on the system's natural frequencies. Based on 
the first-order shear deformation theory, Torabi K et al.[3] 
modeled the free lateral vibration of cracked beams using a 
generalized delta function and variation coefficients. They 
derived the control equations for cracked beams under both 
symmetric and asymmetric boundary conditions and 
analytically solved these equations using basic standard 
trigonometric and hyperbolic functions, analyzing the free 
vibration characteristics of the system and investigating the 
effects of different system conditions on its dynamic behavior. 
Taima MS and El-Sayed TA et al.[4] studied the lateral 
vibration of isotropic and thick beams with cracks, comparing 
the analysis of Timoshenko beam theory and Reddy beam 
theory for beams with cracks. By dividing the beam into 

multiple elements and deriving stiffness and mass matrices 
for each beam element based on Reddy beam theory 
equations, they simulated the impact of cracks and analyzed 
the effects of boundary conditions, slenderness ratio, crack 
position, and depth on lateral vibration. The accuracy of the 
model results was validated through experiments and finite 
element analysis. Zhao et al.[5] used the Green's function 
method and the superposition principle, combined with the 
separation of variables method, Laplace transforms, and other 
mathematical methods, to obtain explicit expressions for the 
steady-state forced vibration of fixed cross-section and 
variable cross-section Euler-Bernoulli beams with multiple 
cracks under harmonic loads. They analyzed the results, 
demonstrating that the geometric shape of the crack (depth 
and position) and section parameters significantly impact the 
system's dynamic behavior. Based on this, X. Zhao et al.[6] 
studied the steady-state forced vibration of Timoshenko 
beams with multiple cracks, verifying that the influence of 
shear deformation and rotational inertia on beams with a 
certain thickness under forced vibration cannot be ignored. 
Loya JA et al.[7] and Chen et al.[8] established crack beam 
models based on Winkler foundation, deriving the differential 
equations for free and forced vibration of beams with multiple 
cracks through corresponding boundary conditions and 
compatibility conditions at the crack sections. They examined 
the impact of different support types, support stiffness, crack 
position, and crack depth on the system's natural frequencies. 

All the above studies focus on the dynamic analysis of 
beams with cracks made of homogeneous materials. This 
article will analyze the free vibration characteristics of 
functionally graded materials with cracks. 

2. Material Properties of Functionally 
Graded Beans and Eauivalent 
Spring Models  

This functionally graded material is composed of two 



 

266 

different constituent materials, with the Young's modulus 
E(z)and ρ(z)density varying exponentially in the thickness 
direction (z-direction) based on the volume fraction of each 
material component. This gradient variation can be expressed 
specifically as: 

 

0 0( ) , ( )z zE z E e z e              (1) 

 

In the equation, E0 and ρ0 represent the Young's modulus 
and density at the mid-plane (z=0) of the functionally graded 
beam, while E1 and E2 correspond to the Young's modulus of 
the upper and lower materials, respectively. Similarly, ρ1  and 
ρ2 correspond to the density of the upper and lower materials. 
The constant denotes the gradient change of material 
properties along the thickness direction, and the beam is 
assumed to be isotropic and homogeneous. Since the 
Poisson's ratio 𝜈 has a minimal effect on the stress intensity 
factor at the crack tip, 𝜈 is taken as a constant. 

The cracked beam can be divided into two sub-beams at the 
crack location, with the two sub-beams connected by a zero-
mass, zero-length rotational spring at the crack cross-section. 
Therefore, the crack cross-section is modeled as a zero-mass 
elastic rotational spring, as shown in Figure 1. The beam 
contains a crack of depth 𝑎 located at a distance 𝐿1 from the 
left end, with a bending stiffness of: 

 

1
TK G
                     (2) 

 

Where G represents the local flexibility caused by the crack. 

 

 
Figure 1. Functional graded beam model with crack 

 

According to fracture mechanics theory, the flexibility 𝐺 
caused by the presence of a crack is related to the stress 
intensity factor at the crack tip. The additional strain energy 
𝑈 caused by the crack can be expressed as: 

 

 2

2
1

1
 d

( )A
U K A

E a


                (3) 

 

Where 𝐴 is the cross-sectional area, 𝐾𝐼 represents the stress 
intensity factor (SIF) at the crack tip under bending load in a 
mode I crack state, and 𝑀 is the bending moment at the crack 
cross-section. The local flexibility 𝐺 at the material’s crack 
cross-section can be expressed as: 

 
2

2

U
G

M





                  (4) 

 

For a functionally graded beam system with a mode I crack, 
the magnitude of the stress intensity factor is obtained from 
the data provided by Erdogan and Wu[9] using Lagrange 
interpolation techniques. 

 

1 2

6
( ), ( 0.7)

M h a
K F

hh

 
        (5) 

3. The Equations of Motion for 
Functionally Graded Beams with 
Crack  

According to Timoshenko theory, the displacements along 
the x-axis and z-axis at any point in a functionally graded 
beam with cracks can be expressed: 

 

( , , ) ( , ) ( , )

( , , ) ( , )

U x z t U x t z x t

W x z t W x t

  


           (6) 

 

Where 𝑢0 and 𝑤0 are the longitudinal and lateral 
displacement components at the mid-plane (z=0), and 𝜙 is the 
rotation angle of the beam cross-section, with 𝑡 representing 
time. 

According to von Kármán’s nonlinear strain-displacement 
relationship, the normal strain and shear strain can be derived 
as follows: 

 
2

1
,

2x xz

U W W
z

x x x x
               

   (7) 

 

According to the linear elastic constitutive relations, the 
normal stress 𝜎 and shear stress 𝜏 can be expressed as: 

 
2

11

55

1
( ) ,

2

( )

xx

xz

U W
Q z z

x x x

W
Q z

x





             
    

      (8) 

 

Where: 

 

11 552

( ) ( )
( ) , ( )

2(1 )1

E z E z
Q z Q z

vv
 


        (9) 

 

The axial force 𝑁, bending moment 𝑀, and transverse shear 
force 𝑉 acting on the functionally graded material beam with 
cracks can be expressed in the following forms: 

 
/2

/ 2

/2

/ 2

/ 2

/2

d

zd

d

h

xxh

h

xxx h

h

xz

x

x h

N z

M z

Q z





 

















               (10) 

 



 

267 

By substituting equation 6 into equation 11, we obtain: 

 
2

11 11 11

2

11 11 11

55

1

2

1

2

i i i
i

i i i
i

ix
i

x

x

i

U W
N A A B

x x x

U W
M B B D

x x x

W
Q A

x


         

         
     

   (11) 

 

In the equation, 𝜅κ represents the shear correction factor 
for the Timoshenko beam, the precise value of which is 
calculated as a function of material properties and beam 
cross-sectional parameters and requires complex 
computations. For functionally graded material beams with 
rectangular cross-section, the value is typically taken as 𝜅=5/6.  

The stiffness coefficients and inertia terms for functionally 
graded Timoshenko beams with cracks can be defined as: 

 

   / 2 2
11 11 11 11/2

, , ( ) 1, , d
h

h
A B D Q z z z z


   

/ 2

55 55/ 2
( )d

h

h
A Q z z


               (12) 

 

   / 2 2
1 2 3 /2
, , ( ) 1, z, z d

h

h
I I I z z


   

Based on Hamilton's principle, continuity conditions of 
force and displacement, and the relationship between crack-
induced additional rotation, and by non-dimensionalizing the 
equations, the system's free vibration equation is derived as: 

 
2 2

11 11 12 2 2

2 2

22
i i i iU U

a Ib I


 


 
 




  


  

255 1

22

2
i i iWW

t
a I


 


  

 


  
        (13) 

 
2 2

11 11 5

2 2

2 32 252 2
i i i

i
iiU W

b Id a
U

I
t t

 
  

 
   

      






 
 

The boundary conditions corresponding to three different 
support types can be stated as follows: 

(a) One end complained, one end free (C-F): 

 

1 1 1

2 2 2

0 0, 0, 0

1 0, 0, 0x x x

u w

N Q M

 

   
   

在 处：

在 处：
   (14) 

 
(b) Hinged at both ends (H-H): 
 

1 1 1

2 2 2

0 0, 0, 0

1 0, 0, 0
x

x

u w M

u w M



   
   

在 处：

在 处：
  (15) 

 
(c) Both ends complained (C-C): 
 

1 1 1

2 2 2

0 0, 0, 0

1 0, 0, 0

u w

u w

 
 
   
   

在 处：

在 处：
    (16) 

By considering different boundary conditions and 
combining the expressions for 𝑁, 𝑀, and Q, 12 homogeneous 
nonlinear algebraic equations can be obtained, which can be 
written in matrix form as: 

 

( ) H χ 0                 (17) 

 

In the equation, H(𝜔) is a matrix that is associated with the 
system's natural frequencies 𝜔 and exhibits nonlinear 
characteristics, while χ represents a vector composed of the 
12 unknown coefficients, defined, In order for the equations 
to have a nontrivial solution, the determinant of H(𝜔) must be 
zero, as: 

 

det[ ( )] 0 H               (18) 

4. Numerical Results and Discussion  
In the numerical simulations presented in this paper, the 

following geometric and material parameters were used: the 
thickness of the beam is ℎ=0.1 m, and the ratio of elastic 
moduli 𝐸2 /𝐸1 is set to 0.2, 1, and 5. The case 𝐸2 /𝐸1=1 
corresponds to an isotropic homogeneous beam. The upper 
surface of the beam consists of 100% aluminum, with the 
material properties being. 

According to Figure 2, when the functionally graded beam 
has material parameters 𝐸2 /𝐸1=0.2, E2 /E1 =0.2, a length-
to-width ratio of 8, and a relative crack depth of 0.2, the 
impact of different crack positions and different boundary 
conditions on the first three natural frequency ratios of the 
system can be observed. From Figure 2(a), it can be seen that 
under one fixed end and one free end boundary conditions, 
the first natural frequency ratio of the system increases as the 
relative position of the crack shifts from the fixed end towards 
the free end, reaching its maximum when the crack is entirely 
at the free end. The second and third natural frequency ratios 
show an irregular fluctuation trend as the crack position 
increases, reaching their maximum values when the relative 
crack position 𝐿1 /𝐿 is at 0.2, 1 and 0.3, 0.5, 1 respectively. 
The natural frequency ratios for the first three modes are all 
minimized when the crack is located at the fixed end, with the 
first mode being the smallest and the second mode following 
closely. 

From Figures 2(b) and 2(c), it can be observed that when 
the functionally graded beam with cracks is under either both 
ends fixed or both ends hinged boundary conditions, the 
curves of the natural frequency ratios with respect to the crack 
position exhibit symmetry. For functionally graded beams 
under both ends fixed conditions, the natural frequency ratios 
for the first three modes are minimized when the crack is 
located at the fixed ends, following a similar trend to the one 
fixed end condition. For the both ends hinged boundary 
condition, the first natural frequency ratio of the system is 
minimized when 𝐿1 /𝐿=0.5. The second natural frequency 
ratio is minimized at 𝐿1 /𝐿=0.25 and 0.75, while the third 
natural frequency ratio is minimized when L1 /L=0.15 and 
0.85. In all three boundary conditions, the pattern remains 
consistent: the first mode frequency ratio is the smallest, 
followed by the second mode frequency ratio. 

 



 

268 

 

 

 
Figure 2. The first three natural frequency ratios of functionally 

graded material (FGM) beams with edge cracks at different 
positions  

 

As seen from Figure 3, when the elastic modulus ratio of 
the functionally graded beam with cracks is 5 and the relative 
crack depth is 0.2, the impact of different length-to-thickness 
ratios on the third mode natural frequency ratio under 
different boundary conditions and crack positions can be 
observed.  

In Figure 3(a), under the boundary condition of one fixed 
end and one free end, the natural frequency ratio is minimized 
when the crack is located at the fixed end. As the crack 
position shifts towards the free end, the frequency ratio 
increases steadily. In Figure 3(b), under both ends fixed 
boundary conditions, the natural frequency ratio is also 
minimized when the crack is at either fixed end, with a 
noticeable increase in the minimum frequency ratio compared 
to the one fixed end condition. The frequency ratio fluctuates 
as the relative crack position increases, reaching a minimum 
when the crack is at the center of the beam. From Figure 3(c), 

under both ends hinged boundary conditions, the natural 
frequency ratio is minimized when the crack is at the center 
of the beam. The natural frequency ratio is maximized when 
the crack is at either end of the beam. As the relative crack 
position increases, the natural frequency ratio for different 
length-to-thickness ratios first decreases and then increases. 
When the functionally graded beam with cracks is under 
either both ends hinged or both ends fixed boundary 
conditions, the curves of different relative crack depths and 
crack positions show geometric symmetry. 

 

 

 
Figure 3. Variation in the first mode natural frequency of cracked 

functionally graded material (FGM) beams with different length-to-
thickness ratios  

 

From Figure 3, under three different boundary conditions, 
when the crack is at any position, the system with a higher 
length-to-thickness ratio tends to have a lower natural 
frequency ratio. When the length-to-thickness ratio of the 
system decreases from 18 to 8, there is a significant decrease 

0 0.2 0.4 0.6 0.8 1
0.85

0.9

0.95

1

L1/L

ω
n/
ω

n0

ω1

ω2

ω3

( a ) C-F

0 0.2 0.4 0.6 0.8 1
0.9

0.92

0.94

0.96

0.98

1

L1/L

ω
n/
ω

n0

ω1

ω2

ω3

( b ) C-C

0 0.2 0.4 0.6 0.8 1
0.9

0.92

0.94

0.96

0.98

1

L1/L

ω
n/
ω

n0

ω1

ω2

ω3

( c ) H-H

0 0.2 0.4 0.6 0.8 1
0.94

0.95

0.96

0.97

0.98

0.99

1

L/h=8
L/h=18
L/h=28

L1/L
ω

1/
ω

10

( a ) C-F

0 0.2 0.4 0.6 0.8 1
0.95

0.96

0.97

0.98

0.99

1

L/h=8
L/h=18
L/h=28

L1/L

ω
1/
ω

10

( b ) C-C

0 0.2 0.4 0.6 0.8 1
0.965

0.97

0.975

0.98

0.985

0.99

0.995

1

L/h=8
L/h=18
L/h=28

L1/L

ω
1/
ω

10

( c ) H-H



 

269 

in the natural frequency ratio, indicating that the first mode 
natural frequency of functionally graded beams with lower 
length-to-thickness ratios is more sensitive to cracks. 

5. Conclusion  
This paper studies the nonlinear static response of 

functionally graded material Timoshenko beams with cracks. 
A nonlinear mechanical model of functionally graded 
material Timoshenko beams with cracks was established 
using the first-order shear deformation theory and von 
Kármán nonlinear theory. The influence of crack position and 
beam length-to-thickness ratio on the system's free vibration 
characteristics under different boundary conditions was 
analyzed. Numerical results indicate: 

Under three different boundary conditions, the first mode 
natural frequency is always less than the second mode natural 
frequency, which in turn is less than the third mode natural 
frequency for different relative crack positions in the cracked 
beam. Additionally, when the relative crack position takes 
specific values, the natural frequency ratio equals 1, 
indicating that the crack at the current position does not 
impact a certain mode of natural frequency in the beam. The 
increase in the length-to-thickness ratio also does not change 
the relative crack positions at which the minimum and 
maximum natural frequency ratios occur for each mode. 

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