PSI - Issue 52

J.C. Wen et al. / Procedia Structural Integrity 52 (2024) 625–646

628

4

Author name / Structural Integrity Procedia 00 (2019) 000 – 000

heat conductivity, nonlinear contact static and dynamic problems. The advantage of FBM is that it only needs to discretize the collocation points in few blocks, which is simpler than the grid generation of FEM, and each block uses mapping technology, which can easily derive the partial differential operator of node values. Since the physical domain is assembled from multiple blocks, the continuity of stress and displacement is considered at the interface. In this paper, Chebyshev polynomial interpolation in the mapped domain is used. The main objective of this paper is to develop a contour integral method to evaluate the stress intensity factors for two-dimensional cracked plates in functionally graded materials under both static and dynamic loadings. The contour integral is path independent. The numerical implementation of variational techniques utilizing the finite block method is presented in this work. The accuracy degree has been demonstrated by rectangular and circular sheets containing central/edge cracks, as well as plate with slant cracks subjected to static and dynamic loads. Excellent agreements with finite element method and analytical expression. 2. Finite block method and its algorithm 2.1. Strong form governing equations for FGMs Functionally graded materials are designed to exhibit a particular spatial variation of their properties and the stresses for 2-D elasticity plane stress problem are given by Hook's law

u

u

u







  

  

  

  

  

  

u

u

u







y

y

y

( ) x

( ) x

( ) x

,

,

,

1 k E

1 k E

2 k E

(1)

x 









=

+

=

+

=

+

x

x

x

y

xy

x

y

x y

y x





 

 

in which x u and y u are displacements, ( ) E x is the Young’s modulus,  is the Poisson’s ratio. In this work, we assume that the Young’s modulus is function of coordinate and Poisson’s ratio is constant. For a 2D dynamic problem, the governing equation is as follows ( , ) x y = x ,

( ) x

( ) x

E

E





              

2

2

u

u

u

u









( ) x

( ) x

1 k E

2 k E

k

k

+

+

+

+

x

x

x

x

1

2

2

2

x x

y y

x

y

 

 





( ) x

( ) x

2

u

u

u







E

E





( ) x

( ) x

y

y

y

, ,

k

k

3 k E

X t



+

+

=

1

2

x y

y x

x y

 

 

 

( ) ,

( ) x

( ) x

(2)

, k

x y 

2

2

u

u

u

u









E

E





( ) x

( ) x

y

y

y

y

2 k E

1 k E

k

k

+

+

+

+

2

1

2

2

x x

y y

x

y

 

 





( ) x

( ) E u  x

E





2

u

u





( ) x

( ) x

, ,

k

k

3 k E

Y t



+

+

=

x

x

x

1

2

y x

x y

x y

 

 

 

where parameters

1

1

1

,

,

,

k

k

k

=

=

=

( 2 1

)

( 2 1

)

1

2

3

2 − and ( ) , X t 1 

+

−





2 u t b  =   − x , ( ) , Y t u t b  =   − x , in which x b and y b donate the components of body force per unit volume,  is mass density. In the case of static problem, the accelerations in Eq. (2) should be omitted or density 0  = . For both of static and dynamic problem, two kinds of boundary conditions are specified on the boundary for (1) Displacement boundary condition 2 / x x 2 2 / y y