Figure 1 denotes the geometrical parameters of the FG porous metal foam doubly-curved panel reinforced by GPLs, where h is the thickness of the shell structure, R₁ and R₂ are the radiuses of curvature, θ 1   and θ 2   represent the span angles of a doubly curved shell, respectively. Also, four distinct porosity patterns combined with five GPL dispersion functions are depicted in Figure 1.

Four different porosity functions are supposed via the thickness of the doubly curved panel. (See Figure 1) and their relation to changing mechanical properties including Young modulus and density along the structure thickness is shown in Eqs 1–4. Besides, five GPL dispersion functions through the thickness of a doubly curved panel are shown in Figure 1; Eq. 16 (Anirudh et al., 2019; Li and Zheng, 2020; Moradi-Dastjerdi and Behdinan, 2020; Zhao et al., 2020).

Nonlinear symmetric porosity function 1: E ϒ = E * 1 − e 0 1 ⁡ cos π ϒ G ϒ = G * 1 − e 0 1 ⁡ cos π ϒ ρ ϒ = ρ * 1 − e 0 1 ⁡ cos π ϒ (1) where ϒ = z/h

Nonlinear symmetric porosity function 2: E ϒ = E * e 0 2 1 − ⁡ cos π ϒ G ϒ = G * e 0 2 1 − ⁡ cos π ϒ ρ ϒ = ρ * e m 2 1 − ⁡ cos π ϒ (2)

Nonlinear asymmetric porosity function 3: E ϒ = E * e 0 3 1 − ⁡ cos π ϒ / 2 + π / 4 G ϒ = G * e 0 3 1 − ⁡ cos π ϒ / 2 + π / 4 ρ ϒ = ρ * e m 3 1 − ⁡ cos π ϒ / 2 + π / 4 (3)

Uniform porosity pattern 4: E = E * e 0 4 G = G * e 0 4 ρ = ρ * e m 4 (4)

΄Where E * , G * , and ρ * are the corresponding material properties of nanocomposite doubly curved panels reinforced with GPL nanofillers but without internal cavities. Also, e 0 1 , e 0 2 , e 0 3 , and e 0 4 (the amounts of them are between zero and one) represents the coefficients of porosity for distribution functions 1, 2, 3, and 4, respectively. e m 1 , e m 2 , e m 3 , and e m 4 are related to the mass density coefficient for patterns 1, 2, 3, and 4, respectively.

Mass density and modules of elasticity for open-cell metal foam such as used in this research are dependent as presented in the below Eq. (Gibson and Ashby, 1982; Ashby et al., 2000; Asgari et al., 2022). E ϒ E * = ρ ϒ ρ * 2 (5)

Eq. 4 is utilized to denote the dependency between the mass density and porosity coefficients for each porosity function as following relations: 1 − e m 1 ⁡ cos π ϒ = 1 − e 0 1 ⁡ cos π ϒ e m 2 1 − ⁡ cos π ϒ = e 0 2 1 − ⁡ cos π ϒ e m 3 1 − ⁡ cos π ϒ / 2 + p i / 4 = e 0 3 1 − ⁡ cos π ϒ / 2 + p i / 4 e m 4 = e 0 4 (6)

For comparing the stiffness of different distributions, the analyses should be implemented for the shells with identical masses. Hence, it is supposed that the mass of doubly curved shell panels with different porosity functions and nanofiller dispersion functions are similar: ∫ − h / 2 h / 2 1 − e 0 1 ⁡ cos π ϒ   d ϒ = ∫ − h / 2 h / 2 e 0 2 1 − ⁡ cos π ϒ d ϒ = ∫ − h / 2 h / 2 e 0 3 1 − ⁡ cos π ϒ / 2 + p i / 4   d ϒ = e 0 4 (7)

Based on Eq. 7, the amounts of e 0 2 , e 0 3 and e 0 4 may be evaluated with a known value of e 0 1 . Details of these coefficients are presented in Table 1. When e 0 1 reaches 0.6, e 0 2 (=0.9612) is near to the upper bound. This justifies the selection of e 0 1 ∈ [0,0.6] hereafter.

The Young’s modulus of the doubly curved panel fabricated by metallic nanocomposite without porosity based on the Halpin-Tsai micromechanics model (Choi and Lakes, 1995; Liu, 1997; Arshid et al., 2020; Arshid et al., 2021; Ebrahimi et al., 2021) is calculated as the following: E * = 3 8 1 + ε L G P L η L G P L V G P L 1 − η L G P L V G P L E m + 5 8 1 + ε W G P L η W G P L V G P L 1 − η W G P L V G P L (8) in which ε L G P L = 2 l G P L t G P L (9) ε W G P L = 2 w G P L t G P L (10) η L G P L = E G P L − E m E G P L + ε L G P L E m (11) η W G P L = E G P L − E m E G P L + ε W G P L E m (12)

The mechanical properties of GPLs are shown with subscripts of GPL. Additionally, the subscripts of m are utilized for showing the corresponding mechanical properties of matrix material. The volume amount of nanofillers is indicated with V G P L . l G P L , w G P L , and t G P L symbols are hired for showing the length, width, and thickness of nanofillers, respectively (Arshid et al., 2020; Zhao et al., 2020; Ebrahimi et al., 2021).

According to the rule of mixture, mass density and Poisson's ratio of the shell are obtained as below Eqs (Guo et al., 2021; Babaei, 2022): ρ * = ρ G P L V G P L + ρ m 1 − V G P L (13) v * = v G P L V G P L + v m 1 − V G P L (14)

Accordingly, the shear modulus of the shell is expressed below: G * = E * 2 1 + v * (15)

Also, the V G P L for various GPL dispersion functions varies through the shell’s thickness and may be obtained by using the below Eq (see also Figure 1): V G P L z = t i 1 1 − ⁡ cos   π ϒ   G P L −   X t i 2 cos   π ϒ   G P L −   O t i 3   G P L −   U D t i 4 1 − ⁡ cos   π 4 − π 2 ϒ   G P L   − A t i 5 cos   π 4 − π 2 ϒ   G P L −   V   (16)

Where t i 1 , t i 2 , t i 3 , t i 4 , and t i 5 denote the upper limit of the V G P L , and subscript i =1, 2, 3, and 4 denote corresponding parameters for porosity functions 1, 2, 3, and 4 within each pattern. V G P L T is the total volume amount of nanofillers and it is obtained by substituting the GPLs weight fraction Δ G P L into Eq. 16. Hence, t i 1 , t i 2 , t i 3 , t i 4 , and t i 5 may be obtained by Eq. 18 (Esmaeili et al., 2022). V G P L T = Δ G P L ρ m Δ G P L ρ m + ρ G P L − Δ G P L ρ G P L (17) V G P L T ∫ − h / 2 h / 2 ρ ϒ ρ c d ϒ = t i 1 ∫ − h / 2 h / 2 1 − ⁡ cos π ϒ ρ ς ϒ ρ c d ψ t i 2 ∫ − h / 2 h / 2 cos π ϒ ρ ϒ ρ c d ψ t i 3 ∫ − h / 2 h / 2 ρ ϒ ρ c d ϒ t i 4 ∫ − h / 2 h / 2 1 − ⁡ cos   π 4 − π 2 ϒ   ρ ϒ ρ c d ϒ t i 5 ∫ − h / 2 h / 2 cos   π 4 − π 2 ϒ   ρ ϒ ρ c d ϒ (18)

FSDT shell theory is hired to present the displacements of doubly curved shell panels as follows: u = u 0 + z   α   v = v 0 + z   β   w = w 0 (19)

In Eq. 2, u, v, and w, are displacements along the x, y, and z axes, respectively, while u 0 , v 0 , and w 0 are the same displacements at the mid-plane of the shell. Also, α and β are normal transverse rotations around y and x, respectively. Hence, the strain field of the doubly curved shell panel is as follows: ε x = ε x 0 + z   k x , ε y = ε y 0 + z   k y   γ x y = γ x y 0 + z   k x y , γ x z = γ x z 0 , γ y z = γ y z 0 (20) where ε x 0 = ∂ u 0 ∂ x − w R 1 , ε y 0 = ∂ v 0 ∂ y − w R 2 k x = ∂ α ∂ x , k y = ∂ β ∂ y , k x y = ∂ α ∂ y + ∂ β ∂ x γ x z 0 = ∂ w ∂ x + α , γ y z 0 = ∂ w ∂ y + β , γ x y 0 = ∂ v 0 ∂ x + ∂ u 0 ∂ y   −   2 w R 12 (21)

Therefore, the matrix form of Eq. 21 will be as: ε x ε y γ x y = ε x 0 ε y 0 γ x y 0 + z k x k y k x y = ∂ ∂ x 0 − 1 R 1 z ∂ ∂ x 0 0 ∂ ∂ y − 1 R 2 0 z ∂ ∂ y ∂ ∂ y ∂ ∂ x − 2 R 12 ∂ ∂ y ∂ ∂ x u 0 v 0 w 0 α β = d 1 Q γ x z 0 γ y z 0 = 0 0 ∂ ∂ x 1 0 0 0 ∂ ∂ y 0 1 u 0 v 0 w 0 α β = d 2 Q (22) where Q = u 0 v 0 w 0 α β , d 1 = ∂ ∂ x 0 − 1 R 1 z ∂ ∂ x 0 0 ∂ ∂ y − 1 R 2 0 z ∂ ∂ y ∂ ∂ y ∂ ∂ x − 2 R 12 ∂ ∂ y ∂ ∂ x , d 2 = 0 0 ∂ ∂ x 1 0 0 0 ∂ ∂ y 0 1 ε x 0 ε y 0 γ x y 0 = ∂ ∂ x 0 − 1 R 1 0 0 0 ∂ ∂ y − 1 R 2 0 0 0 ∂ ∂ y ∂ ∂ x − 2 R 12 0 0 u 0 v 0 w 0 α β = d 3 Q k x k y k x y = 0 0 0 ∂ ∂ x 0 0 0 0 0 ∂ ∂ y 0 0 0 ∂ ∂ y ∂ ∂ x u 0 v 0 w 0 α β = d 4 Q where d 3 = ∂ ∂ x 0 − 1 R 1 0 0 0 ∂ ∂ y − 1 R 2 0 0 0 ∂ ∂ y ∂ ∂ x − 2 R 12 0 0 , d 4 = 0 0 0 ∂ ∂ x 0 0 0 0 0 ∂ ∂ y 0 0 0 ∂ ∂ y ∂ ∂ x

The constitutive relations of a porous FG-GPL doubly curved shell panel are: σ x σ y τ x y = C 11 C 12 0 C 12 C 22 0 0 0 C 66   ε x ε y γ x y τ x z τ y z = C 44 0 0 C 55 γ x z γ y z C 11 = C 22 = E * 1 − v * 2 , C 12 = v * E * 1 − v * 2 , C 44 = C 55 = C 66 = G * (23)

By integrating the stress field along the thickness direction, resultant moment and force will be: N x N y N x y = ∫ − h 2 h 2 σ x σ y τ x y d z , M x M y M x y = ∫ − h 2 h 2 σ x σ y τ x y   z d z Q x Q y = K 2 ∫ − h 2 h 2 τ x z τ y z d z (24)

In Eq. 24, K is the shear correction factor and equals 5/6.

Simplified form of Eq. 24 is as follows: N x N y N x y M x M y M x y Q x Q y = A 11 A 12 0 B 11 B 12 0 0 0 A 12 A 22 0 B 12 B 22 0 0 0 0 0 A 66 0 0 B 66 0 0 B 11 B 12 0 D 11 D 12 0 0 0 B 12 B 22 0 D 12 D 22 0 0 0 0 0 B 66 0 0 D 66 0 0 0 0 0 0 0 0 K 2 A 44 0 0 0 0 0 0 0 0 K 2 A 55 ε x 0 ε y 0 γ x y 0 k x k y k x y γ x z 0 γ y z 0 (25) where: A i j , B i j , D i j = ∫ − h 2 h 2 C i j 1 , z , z 2 d z (26)

The strain and kinetic energies of the doubly curved panel may be presented as the below Eqs. δ U = δ U 1 = 1 2 ∭ ε T σ   d V = ∬ N x ε x 0 + N y   ε y 0 + N x y γ x y 0 + M x K x + M y K y + M x y K x y + Q x γ x z + Q y γ y z d x d y = ∫ d 3 Q T A T + d 4 Q T B T d 3 δ Q + d 3 Q T B T + d 4 Q T D T d 4 δ Q + d 2 Q T e T d 2 δ Q d x d y δ T = ∫ − h 2 h 2 ρ u ¨ δ u + v ¨ δ v + w ¨ δ w d V (27) where: δ u = δ u 0 + z δ α δ v = δ v 0 + z δ β δ w = δ w 0 u ¨ = ∂ 2 u 0 ∂ t 2 + z ∂ 2 α ∂ t 2 v ¨ = ∂ 2 v 0 ∂ t 2 + z ∂ 2 β ∂ t 2 w ¨ = ∂ 2 w 0 ∂ t 2 δ T = ∫ ∫ − h 2 h 2 ρ ∂ 2 u 0 ∂ t 2 + z ∂ 2 α ∂ t 2 δ u 0 + z δ α + ∂ 2 v 0 ∂ t 2 + z ∂ 2 β ∂ t 2 δ v 0 + z δ β + ∂ 2 w 0 ∂ t 2 δ w 0 d z d A (28)

Replacing Eqs 27, 28 Hamilton’s principle, we have: ∫ t 1 t 2 ∬ − h 2 h 2   ρ ∂ 2 u 0 ∂ t 2 + z ∂ 2 α ∂ t 2 δ u 0 + z δ α + ∂ 2 v 0 ∂ t 2 + z ∂ 2 β ∂ t 2 δ v 0 + z δ β + ∂ 2 w 0 ∂ t 2 δ w 0 d z d A + ∫ d 3 Q T A T + d 4 Q T B T d 3 δ Q + d 3 Q T B T + d 4 Q T D T d 4 δ Q + d 2 Q T e T d 2 δ Q d x d y   d t = 0 (29)

In this section, for solving the governing motion equations of the shell, the graded FE method is used. In conventional FEM, the material property is constant through the element. In GFEM, to treat the material heterogeneity, in addition to the displacement field, the material properties of the FG-GPL porous doubly curved structure could also be determined from their nodal values. This approach leads to a smooth change of the properties along the structure and also obtains more precise results than discretizing the structure into elements with constant properties. By using the interpolation functions of cubic ten nodded triangular element, the displacement field, and material properties of individual element e in terms of the nodal displacement vector q, nodal elasticity modulus E i and mass density ρ i and shape function matrix Ψ are as: Q e = Ψ 1 ⋯ 0 ⋮ ⋱ ⋮ 0 ⋯ Ψ 1 … Ψ 10 ⋯ 0 ⋮ ⋱ ⋮ 0 ⋯ Ψ 10 u 01 v 01 w 01 α 1 β 1 ⋮ u 0   10 v 0   10 w 0   10 α 10 β 10 = Ψ q e E * = ∑ i = 1 10 E i ψ i = Ψ Ξ , ρ * = ∑ i = 1 10 ρ i ψ i = Ψ R (30)

where Ψ n , n = 1 , 2 , … , 10 are the approximation functions and presented in the Appendix. u 0 i , v 0 i , w 0 i , α i and β i are nodal DOFs, Ξ and R are respectively vectors containing elasticity modulus and mass density of each node, and are as: ψ = ψ 1 ψ 2 ψ 3 ψ 4 ψ 5 ψ 6 ψ 7 ψ 8 ψ 9 ψ 10 , Ξ = E 1 E 2 E 3 E 4 E 5 E 6 E 7 E 8 E 9 E 10 T R = ρ 1 ρ 2 ρ 3 ρ 4 ρ 5 ρ 6   ρ 7 ρ 8 ρ 9 ρ 10 T (31)

Substituting Eqs 30, 31 in Eq. 29 can be rewritten as ∫ Ω 0 e d 3 Ψ T A T d 3 Ψ + d 4 Ψ T B T d 3 Ψ + d 3 Ψ T B T d 4 Ψ + d 4 Ψ T D T d 4 Ψ + d 2 Ψ T e T d 2 Ψ q + Ψ T I Ψ q ¨   d x d y = 0 (32) where d 2 Ψ = B 2 , d 3 Ψ = B 3 , d 4 Ψ = B 4 . ∫ Ω 0 e B 3 T A T B 3 + B 4 T B T B 3 + B 3 T B T B 4 + B 4 T D T B 4 + B 2 T e T B 2 q + Ψ T I Ψ q ¨   d x d y = 0 (33)

Rearranging Eq. 33, the FE model of porous FG-GPL RC doubly curved panel element will be as follows: k 1 + k 2 + k 3 e q e + M e q ¨ e = 0 k 1 e = ∫ Ω 0 e B 3 T A T + B 4 T B T B 3 d x d y k 2 e = ∫ Ω 0 e B 3 T B T + B 4 T D T B 4   d x d y k 3 e = ∫ Ω 0 e B 2 T e T B 2   d x d y M e = ∫ Ω 0 e Ψ T I Ψ   d x d y (34)

Where in Eq. 34 and mass matrix of element, [I] may be evaluated as: I = I 0 0 0 I 1 0 0 I 0 0 0 I 1 0 0 I 0 0 0 I 1 0 0 I 2 0 0 I 1 0 0 I 2 (35) where I i , i = 0 , 1 , 2 are I 0 I 1 I 2 = ∫ − h 2 h 2 1 z z 2 ρ d z (36)

By assembly of mass, stiffness, and force matrices of each element, the FE motion equations of the FG-GPL RC doubly curved panel are as k 1 + k 2 + k 3 q + M q ¨ = 0 (37)

The natural frequency problem of the structure may be derived by the solution of the eigenvalue model as follows: k 1 + k 2 + k 3 − M ω 2 q = 0 (38) where ω is the circular natural frequency and q is the mode shapes of free vibrations.

The shell is fully clamped at its all edges as: u 0 , v 0 , w 0 , α , β = 0 (39)

To validate the obtained results of the present research, the first six natural frequencies of isotropic homogenous doubly curved shell panels with clamped edges are extracted employing commercial FEM software ANSYS-WORKBENCH, and the results are compared with the results of the present research. Therefore, one may consider e0=0 and Δ G P L = 0 w t % . Also, the dimensions and material properties of the shell are chosen as the following: Geometry: R1=0.225 m, R2=0.4 m, h=0.025m, θ 1 = 120 ° , θ 2 = 60 ° and Mechanical properties: E=200 GPa, υ = 0.3 , ρ = 7800 k g / m 3

A comparison of the results of the present research with natural frequencies of ANSYS WORKBENCH is presented in Table 2, and it indicates excellent concordance between the results.

The influence of coefficients of porosity and distributions of porosity, GPL pattern, and weight fraction of nanofillers and span angles on the natural frequency characteristics of an FG-GPL porous doubly curved shell panel is investigated in this section. Therefore, the below material properties and dimensions are employed:

Geometry: R1=0.225 m, R2=0.4 m, h=0.025m, θ 1 = 120 ° , θ 2 = 60 ° .

Material property: E m = 130GPa, ρ m = 8960 kg/m3, υ m = 0.34 for copper²⁸, and E G P L = 1.01 TPa, ρ G P L = 1062.5 kg/m3, υ G P L =0.186 , w G P L = 1.5 μm, l G P L = 2.5 μm, t G P L = 1.5 nm for GPLs (Arshid et al., 2020; Zhao et al., 2020; Ebrahimi et al., 2021).

Table 3 describes the influences of nanofiller patterns on the free vibrations of the porous FG-GPL doubly curved shell panel ( θ 1   = 120 ° , PD1,e₀=0.2, Δ G P L = 0.05 w t % ). As it is obvious in this table, the maximum and minimum fundamental frequencies belong to GPLX and GPL-UD, respectively. Concentrating more nanofiller near the upper and lower surfaces of the doubly curved shell panel will result in more stiffness of the shell and consequently higher natural frequencies will be obtained. In addition, for uniform dispersion of nanofillers along the thickness of the structure, the minimum stiffness of the shell and also the lowest natural fundamental frequencies will be created. Also, the results of this table indicate that fundamental frequencies for GPL-A and GPL-V have approximately the same values. Comparing the results of Table 3 shows that the maximum difference between the fundamental frequencies with the change of nanofiller distributions is about 13%. In addition, the results of this table denote that by increasing the span angle θ 2 from 60 ° to 150 ° the natural frequency of the shell rose considerably, by 30%. This is due to the fact that by increasing the span angle, the ratio of stiffness to the mass of the shell increases. The influences of various porosity distributions are reported in Table 4 ( θ 1   = 120 ° , GPLX,e₀=0.2, Δ G P L = 0.05 w t % ). The maximum and minimum fundamental frequencies are estimated for PD1 and PD3, respectively. This means that PD1 provides higher rigidity for the shell while PD3 leads to a lower stiffness of the doubly curved shell panel. Comparing the results of Table 4 shows that the maximum difference between fundamental frequencies considering the effect of porosity distributions is about 43%. Also, the results of this table denote that when the distribution of pores is symmetric and their size is more around the mid-pane of the shell, the stiffness of the shell is greater and for the asymmetric distribution of pores, the stiffness of the shell will be lower. The impacts of the weight fraction of nanofillers on the natural frequencies of the porous FG-GPL structure ( θ 1   = 120 ° , GPLX, e₀=0.2, PD1) is reported in Table 5. By changing the weight fraction of nanofillers from 0 to 1%, the fundamental frequencies of doubly curved shell panels considerably increase (approximately 80%). The impact of porosity coefficient on the free vibrations characteristics of porous FG-GPL doubly curved shell ( θ 1   = 120 ° , GPLX, Δ G P L = 0.05 w t % , PD1) are indicated in Table 6. This table denotes that by increasing the porosity of the shell, the fundamental frequencies of FG-GPL porous doubly curved shell panels for PD1 decrease by approximately 22%. This is due to the fact that both the mass and stiffness of the structure decrease as the size of pores increases, the decrease of the mass of the shell is more remarkable than its stiffness. Comparing the results of Tables 3–6 illustrates that the natural frequencies are less influenced by GPL distribution than other parameters affecting the frequency of the shell. The first six free vibrations mode shapes of porous FG-GPL doubly curved shell panels for different span angles θ 2   = 60 ° , 90 ° , and 150 ° are shown in Figures 2–4. As it can be seen in these figures, it is obvious that by increasing the span angle of the doubly curved shell panel, higher free vibration mode shapes with more strain energies are obtained.