PSI - Issue 6

Giulio Zuccaro et al. / Procedia Structural Integrity 6 (2017) 236–243 Author name / Structural Integrity Procedia 00 (2017) 000–000

241

6

2

2

2

2

2

2

1.8

1.5

1.5

1.5

1.5

1.5

1.6

1

1

1

1

1

1.4

0.5

0.5

0.5

0.5

0.5

1.2

0

1

0

0

0

0

y

y

y

0.8

x (Maugis, 1992) -0.5

y (Maugis, 1992) -0.5

-0.5

-0.5

-0.5

|u| (Maugis, 1992)

u

u

0.6

-1

-1

-1

-1

-1

0.4

-1.5

-1.5

-1.5

-1.5

-1.5

0.2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x -2

0

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x -2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x -2

-2

-2

(c) u = u 2

2 2

(a) u 1

(b) u 2

1 + u

2

2

2

2

2

2

1.8

1.5

1.5

1.5

1.5

1.5

1.6

1

1

1

1

1

1.4

0.5

0.5

0.5

0.5

0.5

1.2

0

1

0

0

0

0

y

y

y

x (Present)

y (Present)

0.8

|u| (Present)

u

u

-0.5

-0.5

-0.5

-0.5

-0.5

0.6

-1

-1

-1

-1

-1

0.4

-1.5

-1.5

-1.5

-1.5

-1.5

0.2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x -2

0

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x -2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x -2

-2

-2

(f) u = u 2

2 2

(d) u 1

(e) u 2

1 + u

Fig. 1. Displacement fields produced by uniform loading. Solution by Maugis (1992) (above), and by the proposed approach (below).

Thus, in order to retain this basic property and to fully exploit the generality allowed by our treatment, we have decided to approximate the boundary of the elliptical inclusion by a polygon and to numerically ascertain the number of edges required to get entries of the Eshelby tendor S coincident, to within a given tolerance, with the exact ones. The results are reported in Table 1 and are almost independent from the aspect ratio of the ellipse.

Table 1. Exact vs. approximate components of Eshelby tensor obtained by modelling the elliptic boundary as an n–slides polygon. S 1111 S 2222 S 1122 S 1212

time (sec)

exact n = 50

0.30667 0.30651 0.30665 0.30667 0.30668

0.90667 0.90679 0.9067 0.90667 0.90664

-0.040000 -0.040031 -0.040015 -0.040000 -0.039985

0.39333 0.39335 0.39333 0.39333 0.39334

0.00243 33.2589 39.6976 43.3077 46.8195

n = 100 n = 125 n = 150

Although approximation of the ellipse by a polygon clearly requires an increased computational burden with re spect to the analytical formula, this drawback is more than compensated from the simplicity of the whole treatment since it has been numerically proved that tensor S in (13) is constant within the inclusion while, outside the inclusion, it depends upon the point at which it is evaluated. In particular, Formula (15) numerically yields the same value as the analytical one. Furthermore, the displacement field u provided by Eq. (7) has a very compact expression, holding either in side and outside the inclusion, that depends only upon the position vectors defining the vertices of the polygon boundary, see, e.g., Eq. (8). Summing to (7) the displacement field induced by a uniform remote loading, namely u ◦ α = (1 − 2 ν ) σ ◦ x α / 2 µ ( α = 1 , 2), one easily obtains the displacement field illustrated in Figure 1 (below) that can be