PSI - Issue 28

R.M. Zhabbarov et al. / Procedia Structural Integrity 28 (2020) 1768–1773 Author name / Structural Integrity Procedia 00 (2019) 000–000

1772

5

σ 11

σ 11

40

100

N=1

20

0

50

Abaqus

-20

0

-40

Abaqus

N=2

-60

-50

-80

-100

-100

θ

θ

-2

1

-2

1

2

2

-1

0

-1

0

3

3

-3

-3

0.199 r cm  : the red curve shows the one-term

Fig. 5. Distribution of the stress component 11  at the distance from the crack tip

asymptotic solution (left), the two-term asymptotic solution (right)

σ 11

σ 11

40

40

20

20

0

0

Abaqus

Abaqus

-20

-20

N=5

N=3

-40

-40

-60

-60

-80

-80

-100

-100

θ

θ

-2

1

-2

1

2

2

-1

0

-1

0

3

3

-3

-3

0.199 r cm  : the red curve shows the three-term asymptotic

Fig. 6. Distribution of the stress component 11  at the distance from the crack tip

solution (left), the five-term asymptotic solution (right).

σ 11

σ 11

20

40

20

0

N=7

Abaqus

0

-20

Abaqus

-20

N=7

-40

-40

-60

-60

-80

-80

-100

θ

θ

-100

-2

1

2

-2

1

-1

0

-3

3

-1

0

-3

3

2

0.199 r cm  (left) and distribution of the stress component

Fig. 7. Distribution of the stress component 11  at the distance from the crack tip

0.299 r cm  (right): the red curve shows the seven-term asymptotic solution

11  at the distance from the crack tip

One can see from fig. 3-7 that the higher order terms play a significant role in the crack-tip stress field and the classical one- or two-parameter fracture mechanics concept can’t be applied at large distances from the crack tip. Thus, it is necessary to have coefficients of the higher-order terms in the WE. One can represent eqs. (1) in the matrix form С A   ,where  is the vector consisting the numerical data, A is the unknown coefficients, C is the known matrix consisting of the radial and angular functions from (1). One can find the closed form solution for the unknown vector of parameters   1 1 1 1 2 , ,...., K A a a a  .   1 T T A C C C    .The problem can be reduced to the problem of minimizing the objective function     ( ) (1 / 2) T J A CA CA      .This approach provides the coefficients of the WE for the rectangular plate with the small crack which coincide with the theoretical solution (2,3). Having obtained the