PSI - Issue 26

Merdaci Slimane et al. / Procedia Structural Integrity 26 (2020) 35–45 Slimane et al. / Structural Integrity Procedia 00 (2020) 000 – 00

42 8

Table 2. Comparison of fundamental frequency ( ω ) a rectangular FG plate Al/Al 2 O 3

P=1

Mode (m,n)

a/h

Model

ξ=0

ξ=0.1

ξ=0.2

Mouaici et al. (2016)

2,6476 2,6475 2,7937 2,7937 4,0782 4,0781 4,4193 4,4192 6,2664 6,2663 7,0516 7,0515

2,5934 2,5420 2,7328 2,6802 3,9982 3,8355 4,3243 4,1434 6,1508 5,9051 5,9821 6,6163

2,5150 2,4247 2,6452 2,5539 3,8821 3,5340 4,1875 3,8014 6,9033 5,4555 6,6891 6,0764

5

Present

(1,1)

Mouaici et al. (2016)

10

Present

Mouaici et al. (2016)

5

Present

(1,2)

Mouaici et al. (2016)

10

Present

Mouaici et al. (2016)

5

Present

(1,3)

Mouaici et al. (2016)

10

Present

In table.3, four fundamental frequencies of simply supported rectangular plate FGP with even porosity distribution are compared with their counterpart in (Rezaei et al. (2017) and Askari et al. (2018)). It can be shown that the frequencies are increasing with mode variation. The good agreement between the results shows the accuracy of the present approach. Table 3. Comparison of fundamental frequency ( ) ω for simply supported square (b=a=1) plate FG (P=1 and h/a=0.05). Porosity Distribution ξ Method Mode (m,n) (1, 1) (1, 2) (2, 2) (1, 3)

Rezaei et al. (2017) Askari et al. (2018)

0,0217 0,0217 0,0213 0,0210 0,0210 0,0203

0,0538 0,0537 0,0527 0,0520 0,0520 0,0502

0,0851 0,0850 0,0833 0,0824 0,0823 0,0794

0,1057 0,1055 0,1034 0,1024 0,1022 0,0985

ξ=0.1

Present

(Even)

Rezaei et al. (2017) Askari et al. (2018)

ξ =0.2

Present

In fig 3. shows the influence of thickness ratio (a/h), on the dimensionless frequency parameter of FGP perfect and imperfect plates ( ξ =0; ξ =0.1 and 0.2) for the distribution of even porosity, It can be seen that the ratio (a/h) has a considerable effect on the dimensionless frequency of the FGP, the later decreases with the increase of this ratio. In fig. 4. One studies variation of the dimensionless frequency as a function of the geometric ratio (b/a) for the distribution of the even porosity. Decreasing said ratio increases the dimensionless frequency. In addition to the observation mentioned in figure.5 which states that the increase of the ratio (b/a) reduces the dimensionless frequency, it is noted that the ratio (b/a) has little influence on the variation of the frequency. Also from the ratio (b/a=2.5) this parameter has practically no effect on the natural frequencies and this for the distribution of the even porosity. In fig.5. The variation of the dimensionless frequency as a function of the thickness ratio (a/h) for the distribution of the even porosity is represented. There is a rapid variation in the frequency for the low values of the ratio (a/h) (ie for a / h <10) where the plate is considered thick. Exceeding this ratio, the dimensionless frequencies keep a more or less constant pace whatever the chosen distribution. In addition, it should be noted that the distribution even porosity gives higher values.