PSI - Issue 52

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

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

5. Numerical examples

In the following examples, the Poisson’s ratio is taken to 0.3 and Young’s modulus at origin 0 E is one unit. The plane strain assumption is considered in this paper. Without special specification, the integral contour is defined as boundary  , numbers of Chebyshev polynomial terms 21 M N = = and the normalised variation of the crack length is defined as 0.001. 5.1. Central-cracked rectangular plate

0 

0 

B

B

B

B

y y

y

y

A

A

x

x

x

E

x

E

w E

I

w E

II

2 h

2 h

A

A

2 a

2 a

0

0

2 w

2 w

2 w

2 w

0 

0 

(a)

(b)

Fig. 4. Rectangular plate containing central crack. (a) Geometry and (b) quarter of the plate with two blocks.

A rectangular plate, containing a central crack of length 2 a , height 2 h and width 2 w , is subjected to a uniformly distributed normal load 0  on the top and bottom surfaces, as shown in Fig. 4(a). Due to the symmetry of the problem, a quarter of the plate is modeled as shown in Fig. 3(b) with only two blocks. An exponential variation of Young’s modulus 0 ( , ) E x y E = is considered, 2 2 2 2 exp( / / ) x w y h   + , where 0 ln( / ) w E E  = , 0 ln( / ) h E E  = , 0 E denotes the Young’s modulus at origin, h E and w E are Young’s modulus at point A and B in Fig. 4 (a). To observe the effect of crack length a  by variation of crack on the numerical results. Table 1 shows the normalised stress intensity factors versus the different ratio / a a  . In this test, the geometric parameters are selected as / 0.5 a w = , / 2 w h = 0 / 4 w E E = and 0  = . The boundary  is selected as an integral contour. Apparently, when / 0.01 a a   , the convergent solution can be obtained. The second test is to observe the degree of convergence of GFBM. Normalised stress intensity factors I 0 / K a   versus the numbers of Chebyshev polynomial are presented in Fig. 5, where homogenous material is considered, and the parameters of geometry are the same as specified in the first test. In this case, the normalised stress intensity factor is given I 0 / 1.96 K a   = in a handbook (Rooke, 1976) . The convergence is demonstrated for the same problem with different integral contours (radius R in the mapping domain). The normalised stress intensity factor I 0 / K a   against the dimensionless radius is shown in Fig. 6. It shows that the maximum relative error is around 1.5% in the region of 0 0.3 R   . To observe the effect with material gradient coefficients  and  , the stress intensity factor has been observed with different material gradients and crack length / a w .

/ a  .

Table1. Variation of stress intensity factors versus the increment