Table 7 Thermal critical buckling load \({\mathfrak{T}}_{cr}\) of square perfect and imperfect FG plates exposed to nonlinear thermal loading for various values of plate thickness ratios \(a/h\), porosity parameter \(\tau\), thermal exponent \(\gamma\) and volume fraction indices \(\mathfrak{r} (\gamma =2)\).

\(\mathfrak{r}\)	\(a/h\)	Theory	\(\mathcal{q}\)	Perfect	Imperfect I (even)		Imperfect II (uneven)		Imperfect III (logarithmic-uneven)
\(\mathfrak{r}\)	\(a/h\)	Theory	\(\mathcal{q}\)	\(\tau =0\)	\(\tau =0.1\)	\(\tau =0.2\)	\(\tau =0.1\)	\(\tau =0.2\)	\(\tau =0.1\)	\(\tau =0.2\)
0	5	HPT¹⁸	\(=0\)	16.73	21.06	28.42	18.49	20.58	18.45	20.37
		Present	\(=0\)	16.73	21.06	28.41	18.44	20.37	18.44	20.37
		Present	\(\ne 0\)	17.61	22.17	29.91	19.41	21.43	19.41	21.43
	10	HPT¹⁸	\(=0\)	4.84	6.09	8.22	5.36	5.99	5.35	5.93
		Present	\(=0\)	4.84	6.09	8.22	5.35	5.93	5.35	5.93
		Present	\(\ne 0\)	4.91	6.19	8.35	5.43	6.02	5.43	6.02
	20	HPT¹⁸	\(=0\)	1.24	1.57	2.13	1.38	1.55	1.38	1.53
		Present	\(=0\)	1.24	1.57	2.13	1.38	1.53	1.38	1.53
		Present	\(\ne 0\)	1.25	1.58	2.14	1.39	1.54	1.39	1.54
1	5	HPT¹⁸	\(=0\)	7.45	8.18	9.03	7.85	8.27	7.84	8.23
		Present	\(=0\)	7.45	8.17	9.02	7.84	8.23	7.84	8.23
		Present	\(\ne 0\)	7.99	8.81	9.797	8.42	8.84	8.42	8.84
	10	HPT¹⁸	\(=0\)	2.10	2.30	2.52	2.22	2.35	2.22	2.34
		Present	\(=0\)	2.10	2.30	2.52	2.22	2.34	2.22	2.34
		Present	\(\ne 0\)	2.19	2.41	2.67	2.32	2.45	2.32	2.45
	20	HPT¹⁸	\(=0\)	0.53	0.58	0.64	0.56	0.60	0.56	0.59
		Present	\(=0\)	0.53	0.58	0.64	0.56	0.59	0.56	0.59
		Present	\(\ne 0\)	0.55	0.60	0.67	0.58	0.62	0.58	0.62
5	5	HPT¹⁸	\(=0\)	5.36	5.31	4.71	5.41	5.36	5.41	5.37
		Present	\(=0\)	5.37	5.32	4.72	5.42	5.39	5.42	5.39
		Present	\(\ne 0\)	5.76	5.76	5.21	5.82	5.80	5.82	5.80
	10	HPT¹⁸	\(=0\)	1.59	1.57	1.37	1.63	1.65	1.63	1.65
		Present	\(=0\)	1.59	1.57	1.37	1.63	1.65	1.63	1.65
		Present	\(\ne 0\)	1.67	1.68	1.51	1.72	1.75	1.72	1.75
	20	HPT¹⁸	\(=0\)	0.41	0.40	0.35	0.42	0.43	0.42	0.43
		Present	\(=0\)	0.41	0.40	0.35	0.42	0.43	0.42	0.43
		Present	\(\ne 0\)	0.42	0.43	0.38	0.44	0.45	0.44	0.45

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