Machine learning assisted coupled frequency analysis of skewed multi-phase magnetoelectric composite plates with temperature and moisture dependent properties

Figures & data

Table 1. Material properties corresponding to different volume fraction V_f of BaTiO₃–CoFe₂O₄ [26].

Material property	Material constants	0 Vf	0.2 Vf	0.4 Vf	0.5 Vf	0.6 Vf	0.8 Vf	1 Vf
Elastic constants (GPa)	C11=C22	286	250	225	220	200	175	166
	C12	173	146	125	120	110	100	77
	C13=C23	170	145	125	120	110	100	78
	C33	269.5	240	220	215	190	170	162
	C44=C55	45.3	45	45	45	45	50	43
	C66	56.5	52	50	50	45	37.5	44.5
Piezoelectric constants (C/m^2)	e31	0	−2	−3	−3.5	−3.5	−4	−4.4
	e33	0	4	7	9.0	11	14	18.6
	e15	0	0	0	0	0	0	11.6
Dielectric constant (10^−9 C^2/Nm^2)	ε11=ε22	0.08	0.33	0.8	0.85	0.9	1	11.2
	ε33	0.093	2.5	5	6.3	7.5	10	12.6
Magnetic permeability (10^−4 Ns^2/C^2)	μ11=μ22	−5.9	−3.9	−2.5	−2.0	−1.5	−0.8	0.05
	μ33	1.57	1.33	1	0.9	0.75	0.5	0.1
Piezomagnetic constants (N/Am)	q31	580	410	300	350	200	100	0
	q33	700	550	380	320	260	120	0
	q15	560	340	220	200	180	80	0
Magneto-electric constant (10^−12Ns/VC)	m11=m22	0	2.8	4.8	5.5	6	6.8	0
	m33	0	2000	2750	2600	2500	1500	0
Density (kg/m^3)	ρ	5300	5400	5500	5550	5600	5700	5800

Figure 1. Schematic of (a) ME composite plate subjected to hygrothermal environment (b) geometrical skewness.

Figure 2. The schematics of (a) basic elements constituting an ANN model (b, c) architecture of the ANN model developed to predict vibration characteristics of SMEE plates.

Table 2. Different parameters and their ranges selected for the development of ANN model.

Sl. No	Parameters	Range
1	Volume fraction (Vf)	0, 20, 40, 60, 80, 100%
2	Skew angle	0, 15, 30, 45, 60
3	Empirical constant (γ, β)	± 0.0025, ± 0.005, ± 0.01
4	Material property relation	MP-1, MP-2, MP-3, MP-4
5	Temperature and moisture gradients	ΔT = 0–100 (in intervals of 10) Δm = 0–2% (in intervals of 0.05%)
6	Boundary condition	CCCC, SSSS, FSFS, CFFF, CCFF, FCFC, SFSF, SSSF, SCSF, SCSS, SCSC, CFCF, CCCF
7	a/h ratio	10-100 (in intervals of 10)
8	a/b ratio	0.5-3 (in intervals of 0.5)

Figure 3. (a) Comparison of the neuron number on the overall value of ‘R’ of the ANN model (b) regression plots for training, and testing of the optimum ANN model with thirty neurons and Levenberg-Marquardt algorithm.

Figure 4. The plot of (a) Error histogram (b) performance of the ANN model developed to predict vibration characteristics of ME plates.

Table 3. The performance parameters of the ANN model.

Parameters
No of neurons	5	10	15	20	25	30
Computational time (in minutes)	3.28	11.32	18.45	31.22	72	138
RMSE (training)	10.22%	7.81%	4.75%	3.23%	2.77%	1.95%
RMSE (testing)	15.43%	12.17%	7.36%	4.89%	3.74%	2.3%
R	0.9929	0.9982	0.9992	0.9994	0.9994	0.9998
Epoch	423	611	935	1000	1000	1000

Table 4. Convergence of natural frequency of MEE plate with mesh size.

Mode No.	Mesh size				Moita et al. [78]
	4 × 4	6 × 6	8 × 8	10 × 10
1	2422.14	2428.60	2432.14	2432.21	2449.7
2	6255.74	6264.67	6268.31	6268.40	6280.6
3	6255.74	6264.67	6268.31	6268.40	6280.6

Figure 5. Influence of volume fraction and elastic stiffness material property relation on the fundamental frequency of ME plate subjected to hygrothermal environment.

Figure 6. Influence of (a) volume fraction and (b) skew angle on the fundamental frequency of ME plate.

Figure 7. Influence of empirical constant γ (positive) on the fundamental frequency of ME plate with (a) MP-1 (b) MP-2 (c) MP-3 (d) MP-4 material property relations and volume fractions.

Figure 8. Influence of empirical constant γ (negative) on the fundamental frequency of ME plate with (a) MP-1 (b) MP-2 (c) MP-3 (d) MP-4 material property relations and volume fractions.

Figure 9. Influence of empirical constant β (positive) on the fundamental frequency of ME plate with (a) MP-1 (b) MP-2 (c) MP-3 (d) MP-4 material property relations and volume fractions.

Table 5. Verification of the present FE formulation for natural frequencies of layered ME plates (h = 0.3 m; a = b = 1m).

Stacking sequence	Mode No.	Non-dimensional frequency		% Error
		Chen et al. [79]	Present
BFB	1	1.3434	1.3482	0.3601
	2	2.2199	2.2279	0.3597
	3	2.2199	2.2279	0.3597
FBF	1	1.4463	1.4346	−0.8109
	2	2.3602	2.3390	−0.8979
	3	2.3602	2.3392	−0.8905

Figure 10. Influence of (a) thermal (b) moisture gradient on the fundamental frequency of ME plate with different material property relation and positive empirical constants.

Figure 11. Influence of (a) thermal (b) moisture gradient on the fundamental frequency of ME plate with different material property relation and negative empirical constants.

Figure 12. Integrated effects of volume fraction, material property relation and geometric skewness on the fundamental frequency of ME plate with (a) φ = 0^ο (b) φ = 15^ο (c) φ = 30^ο (Δm = 2%; CCCC; γ = β = 0.05).

Figure 13. Integrated effects of volume fraction, different empirical constants and geometric skewness on the fundamental frequency of ME plate with (a) MP-1 (b) MP-2 (c) MP-3 (d) MP-4 material property relation (ΔT = 25; Δm = 2%; CCCC).

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Data availability

The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.