PSI - Issue 52

Akihide Saimoto et al. / Procedia Structural Integrity 52 (2024) 323–339 A. Saimoto et al. / Structural Integrity Procedia 00 (2023) 000–000

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for crack tip at s k = − b k . That is, the SIFs at the crack tips do not require further extrapolation or energy separation procedure, but are obtained directly from the value of the weight function at the crack tip.

3. Numerical results

This chapter presents numerical results for 2 kinds of basic crack interference problems in a rectilinearly anisotropic infinite plate subjected to a uniform stress at infinity. Five types of material are considered, as listed in Table 1. These material settings are taken from Sollero and Aliabadi (1993). The angle between the material principal axis and the physical coordinate system α (see Fig.1) was varied from 0 to 180 degrees and the calculations were performed systematically by increasing the number of division of each crack line n . Fig.5 shows the treated problems.

3.1. Tension of an infinite plate of rectilinearly anisotropy with parallel 2 cracks

The first numerical example dealt with the case of two parallel, equal-length cracks in a position to shield each other(Fig.5(a)). Due to the symmetry of the problem, the mode I SIFs are equal at all crack tips, regardless of material properties and the angle α . The mode II SIFs are equal at crack tips A and A’ (and tips B and B’). Furthermore, the mode II SIFs at crack tips A and B are characterized by equal absolute values, di ff ering only in sign. In this problem, the mode I SIF is smaller than the case of a single crack because one crack would shadow the other. In addition, from a numerical standpoint, the calculation accuracy is easily degraded due to severe interference e ff ects between the cracks as h / b becomes small. Therefore, it is suitable as a benchmark example to measure the strength of the method for calculation and numerical program. In this calculation, each crack is equally divided ( n equal-length segments) and the weight function of the body force doublets (the distributed pair of point forces is called body force doublets in this paper) is determined based on the resultant force method. The number of division n is initially set to 30 and is doubled successively, and the convergence of the solution is tabulated. The smallest value of h / b was set 0.1 and was increased up to 10. If the symmetry of the SIF is broken, the analysis is considered incomplete

Table 1. Properties of treated materials (Sollero and Aliabadi (1993)). Material Type E 1 E 2

G 12

ν 12

µ m 1

µ m 2

(GPa)

(GPa)

(GPa)

I-1 ( φ = 1 . 05) I-2 ( φ = 10 . 0) I-3 ( φ = 100 . 0)

Hypothetical Hypothetical Hypothetical Boron-epoxy Graphite-epoxy

G 12 ( φ + 2 ν 12 + 1) G 12 ( φ + 2 ν 12 + 1) G 12 ( φ + 2 ν 12 + 1)

E 1 /φ E 1 /φ E 1 /φ

6.0 6.0 6.0

0.03 0.03 0.03

1 . 024695 i 3 . 162278 i 10 . 00000 i 3 . 364481 i 3 . 696471 i

0 . 9999999 i 1 . 0000000 i 1 . 0000000 i 0 . 1689822 i 0 . 9517087 i

II

55.16 144.8

170.65

4.83 9.66

0.036

III

11.7

0.21

(a) shield cracks

(b) step cracks

Fig. 5. Interference problem between two parallel isometric cracks in an infinite plate of rectilinearly anisotropy under far-field tension.