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Modeling and dynamic analysis of functionally graded porous spherical shell based on Chebyshev–Ritz approach

  • Jili Lu , Qingyun Yang , Zhongliang Meng , Kang Yang , Wei Xu and Ching Vincent Chiu EMAIL logo
Published/Copyright: September 26, 2023
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Science and Engineering of Composite Materials
From the journal Science and Engineering of Composite Materials Volume 30 Issue 1

Abstract

This study proposes a unified modeling method to investigate the dynamic behaviors of the functionally graded porous (FGP) spherical shell with elastic boundary conditions. First, three kinds of FGP distributed patterns are defined. Then, the first-order shear deformation theory is selected to build the governing equations of the spherical shell with elastic boundary conditions, which can be solved by the Rayleigh–Ritz approach. Moreover, Chebyshev polynomials of the third kind are selected as an admissible function to express the motion equation. With the constructed model, the correctness is verified by comparing the natural frequency and forced response obtained from both open literature and finite element method. Ultimately, the parameter study is conducted to conclude the effect of the design parameter on the dynamic characteristics of the spherical shell.

Keywords: dynamic analysis; functionally graded porous material; spherical shell; elastic boundary conditions

Nomenclature

notation

definition

A ij

stretching stiffness

B ij

stretching-bending coupling stiffness

D ij

bending stiffness

e 0

porosity coefficient

e m

mass density coefficient

E ( z )

Young’s modulus

E 1

maximum values of Young’s modulus

E 2

minimum values of Young’s modulus

G ( z )

shear modulus

G 1

maximum values of shear modulus

G 2

minimum values of shear modulus

q 0

amplitude of loading pulse

t

time

u

displacement of meridional direction

U m / V m / W m / ψ φ m / ψ θ m

the unknown coefficients of different displacement functions

v

displacement of circumferential direction

w

displacement of normal direction

z

normal direction

ρ ( z )

mass density of FGP material

ρ 1

maximum values of mass density

ρ 2

minimum values of mass density

ψ φ

rotations relative to φ coordinate

ψ θ

rotations relative to θ coordinate

ε φ φ 0 / ε θ θ 0 / ε φ θ 0

strains of surface

κ φ φ / κ θ θ / κ φ θ

curvatures

θ

circumferential direction

φ

meridional direction

φ 0

top boundary of spherical shell

φ 1

bottom boundary of spherical shell

1 Introduction

Spherical shell is a kind of fundamental structure with high stiffness and light weight, which has been widely used in many fields such as architecture, turbine, and launch vehicle. In practical applications, spherical shell may produce vibration, structural fatigue, and noise because of the excitations of the external environment [1]. These undesirable effects can be suppressed and reduced by constructing a spherical shell with excellent vibration characteristics.

To improve the vibration characteristics of spherical shells, many studies have been carried out [2,3,4]. In the study by Adin and Adin [5], mechanical properties of composite materials produced from woven jute type were investigated. In the study by Song and Ge [6], a joined composite shell was presented, and its dynamic response was explored.

In the study by Adin and Kılıçkap [7], considering the effects of high satiety, fracture strength, and good dimensional stability, glass fiber-reinforced composite materials have widespread uses. In the study by Hamit et al. [8], considering the effect of materials, the fatigue behavior of composite patched and non-patched aluminum plates was numerically investigated. The study revealed that the composite patch’s contribution is very important. In the study by Panda and Singh [9], laminated composite was introduced into a spherical shell, and its nonlinear free vibration was studied. In the study by Mahapatra and Panda [10], a laminated composite spherical shell under hygrothermal loading was presented, and its nonlinear free vibration was investigated. The fiber-reinforced composite materials possess higher strength and modules than general laminated composite materials. Hence, a fiber-reinforced spherical shell was prepared by Ram and Babu [11], and its free vibration was studied. Besides, functionally graded (FG) material is also of interest to some researchers. In the study by Su et al. [12], the vibration behavior of the FG spherical shell with arbitrary boundary conditions was investigated. In the study by Duc et al. [13], the nonlinear dynamic behaviors of the FG spherical shell were investigated, and the effect of elastic foundations and temperature was studied. Due to the significant practical application of such structures, some researchers studied their mechanical behaviors via different approaches. In the study by Duc et al. [13], the vibrational behavior of FG spherical shells resting on an elastic substrate was studied. In the study by Xie et al. [14], the effects of elastic boundary conditions on the free vibration behavior of thin spherical shells have been examined. In the study by Duc et al. [13], nonlinear dynamic behaviors of annular spherical shells made of nano-composite materials have been studied. In the study by Li et al. [15], the authors provided a semi-analytical approach for examining the linear vibration behavior of annular spherical shells under different edge conditions. Nevertheless, the aforementioned studies only focus on the improvement of the stiffness and neglected design for light weight, which results in higher weight. Within the framework of the thin-walled shell theory, Li et al. [16,17] investigated the buckling behaviors of confined functionally graded porous (FGP) rectangular arches reinforced with graphene platelets and subjected to a concentrated load and uniform pressure. Their results show that the critical buckling load of porous arches is considerably affected by the porous coefficient and graphene platelets (GPL) weight fraction. Li [18] subsequently extended this work to the partially confined FGP–GPL reinforced composite arches and suggested that a greater contact area with the medium gives a higher buckling load. Liu et al. [19] presented a buckling analysis of FGP circular shallow arches based on the Euler–Bernoulli hypothesis and found that the FGP arches have a good performance in weight reduction and a high strength-to-weight ratio. Therefore, the study of the FGP spherical shell is significant to the development of technology and engineering application. However, there are few discussions about the FGP spherical shell and its vibration characteristics. In this context, the vibration characteristics of the FGP spherical shell are studied in this work.

To better understand the vibration characteristics of the FGP structures, the primary task is to investigate its free vibration by utilizing appropriate methods [20–24]. There were some relative works about FGP plates recently. The higher order shear deformationtheories combined with the finite element method were used to investigate FGP plate vibrations [25]. The vibration study on the FGP structure elements of revolution with general elastic boundary conditions was published [26]. Duc et al. [27] analyzed the nonlinear buckling and post-buckling characteristics of conical sandwich shells with FGP sandwich materials. Van et al. [28] published an article on the free and forced vibration study about truncated conical FGP panel using analytical method. Duc et al. [29] studied the nonlinear dynamic vibration of FGP shells with elastic foundation based on first-order shear deformation theory (FSDT) in the same year. The free vibration of the FGP beam was studied by utilizing a modified Fourier series method [30]. The free vibration of the FGP rotary structure was examined by combining the FSDT and the Rayleigh–Ritz method [31]. The vibration behavior of FGP micro shells with classical boundary conditions was explored by Salehipour et al. [32]. The limitation of opening studies is increasingly apparent. In engineering applications, forced vibration is the main form of structural vibration. However, most of the current research on FGP spherical shell focus on free vibration and lack of attention to forced vibration. To provide a reference for engineering applications, not only the free vibration, but also the forced vibration of the FGP spherical shell should be further studied.

Previous literature shows that structural vibration behaviors are significantly affected by boundary conditions [33]. A Jacobi–Ritz approach was presented to investigate the effect of arbitrary boundary conditions on the dynamic behaviors of the FGP beam by Qin et al. [34]. Considering the effect of the clamped free boundary, the first three out-of-plane bending free vibration of symmetric laminated beams is studied by Timoshenko’s FSDT by Cünedioğlu and Beylergil et al. [23]. Free vibration responses of damaged beams are examined using Euler–Bernoulli beam and classical lamination theories [24]. Considering the effect of general boundary condition, the vibration behavior of the FGP cylindrical shell was investigated by Li et al. [35]. The buckling behavior of FG nanoplates with three kinds of arbitrary restraint was studied by Karami et al. [36]. For various work environments, the boundary conditions are different. To satisfy the application requirements, the effect of boundary conditions on the vibration behaviors of the FGP spherical shell should be considered.

The contribution of this study can be summarized as follows: (1) the FGP is introduced in the spherical shell to enhance its dynamic performance and decrease its structural weight; (2) a dynamic model of the FGP spherical shell based on FSDT and three types of Chebyshev polynomials is proposed to improve the modeling precision of the FGP spherical shell; (3) various boundary conditions and porosity distributions are systematically considered in the dynamic characteristics of the FGP spherical shell to explore the effect of the environment and materials. Similar to the aforementioned analysis, this work mainly investigates the vibration behaviors of the FGP spherical shell. The framework of subsequent work is shown as follows: First, the mathematical model of dynamic analysis of spherical shells is established and solved. Next, the numerical result is discussed. Finally, the conclusion of this work is given.

2 Theoretical formulation

2.1 FGP spherical shell

Figure 1 shows the FGP spherical shell model with elastic boundary conditions. R φ = R sin ( φ 0 + φ ) indicates the distance from the central axis to each point. The essential properties of FGP material are defined as follows:

(1a) E ( z ) = E 1 Θ i ( z ) ,

(1b) G ( z ) = G 1 Θ i ( z ) ,

(1c) ρ ( z ) = ρ 1 Ξ i ( z ) ,

Figure 1 Schematic diagram of FGP spherical shell with elastic boundary conditions.
Figure 1

Schematic diagram of FGP spherical shell with elastic boundary conditions.

where Θ i (z) and Ξ i (z) (i = 1, 2, 3) denote the three kinds of porosity distributions along the z coordinate, and they can be further expanded as follows:

(2a) Θ ( z ) = 1 − e 0 cos ( π z h ) Type 1 1 − e 0 cos ( π z 2 h + π 4 ) Type 2 1 − e 0 α Type 3 ,

(2b) Ξ ( z ) = 1 − e m cos ( π z h ) Type 1 1 − e m cos ( π z 2 h + π 4 ) Type 2 1 − e 0 α Type 3 .

The functions of e 0 and e m are defined as follows:

(3a) e 0 = 1 − E 2 E 1 = 1 − G 2 G 1 ( 0 ≤ e 0 ≤ 1 ) ,

(3b) e m = 1 − ρ 1 ρ 2 ( 0 ≤ e m ≤ 1 ) .

The relation equation of e 0 and e m is given as follows:

(4) e m = 1 − 1 − e 0 .

2.2 Energy equations

In this study, the displacement components of the FGP spherical shell are derived by utilizing the FSDT as follows:

(5a) U ( φ , θ , z , t ) = u ( φ , θ , t ) + z ψ φ ( φ , θ , t ) ,

(5b) V ( φ , θ , z , t ) = v ( φ , θ , t ) + z ψ θ ( φ , θ , t ) ,

(5c) W ( φ , θ , z , t ) = w ( φ , θ , t ) .

The normal and shear strains of the FGP spherical shell are obtained by considering the small deformation assumption, and its expressions are given as follows:

(6a) ε φ φ = ε φ φ 0 + z κ φ φ ,

(6b) ε θ θ = ε θ θ 0 + z κ θ θ ,

(6c) γ φ θ = γ φ θ 0 + z κ φ θ ,

(6d) γ φ z = ψ φ − u R + 1 R ∂ w ∂ φ ,

(6e) γ θ z = ψ θ − v R + 1 R φ ∂ w ∂ θ .

These strains and curvatures can be expanded as follows:

(7a) ε φ φ 0 = 1 R ∂ u ∂ φ + w R ,

(7b) ε θ θ 0 = 1 R φ ∂ v ∂ θ + u R R φ ∂ R φ ∂ φ + w R ,

(7c) γ φ θ 0 = 1 R ∂ v ∂ φ − v R R φ ∂ R φ ∂ φ + 1 R φ ∂ u ∂ θ ,

(7d) κ φ φ = 1 R ∂ ψ φ ∂ φ ,

(7e) κ θ θ = 1 R φ ∂ ψ θ ∂ φ + ψ φ R R φ ∂ R φ ∂ φ ,

(7f) κ φ θ = 1 R ∂ ψ θ ∂ φ − ψ θ R R φ ∂ R φ ∂ φ + 1 R φ ∂ ψ φ ∂ θ .

The relationship between strains and stresses can be defined as follows:

(8) σ φ φ σ θ θ τ φ θ τ φ z τ θ z = Q 11 Q 12 0 0 0 Q 21 Q 22 0 0 0 0 0 Q 66 0 0 0 0 0 Q 66 0 0 0 0 0 Q 66 ε φ φ ε θ θ γ φ θ γ φ z γ θ z ,

where Q ij (i, j = 1, 2, 6) are the stiffness coefficients, which can be further expressed as follows:

(9) Q 11 = Q 22 = E ( z ) 1 − v 2 ( z ) , Q 12 = Q 21 = v ( z ) E ( z ) 1 − v 2 ( z ) , Q 66 ( z ) = E ( z ) 2 [ 1 + v ( z ) ] .

The resulting force and moment of the FGP spherical shell are obtained as follows:

(10a) N φ φ N θ θ N φ θ M φ φ M θ θ M φ θ = A 11 A 12 0 B 11 B 12 0 A 21 A 22 0 B 21 B 22 0 0 0 A 66 0 0 B 66 B 11 B 12 0 D 11 D 12 0 B 21 B 22 0 D 21 D 22 0 0 0 B 66 0 0 D 66 ε φ φ 0 ε θ θ 0 γ φ θ 0 κ φ φ κ θ θ κ φ θ ,

(10b) Q φ z Q θ z = 5 6 A 66 0 0 A 66 γ φ z γ θ z .

The stiffness coefficients of the FGP spherical shell can be defined as follows:

(11) ( A i j , B i j , D i j ) = ∫ − h / 2 h / 2 Q i j ( z ) ( 1 , z , z 2 ) d z .

Based on the aforementioned formulations, the strain energy of the FGP spherical shell is given as follows:

(12) U = 1 2 ∭ v N φ φ ε φ φ 0 + N θ θ ε θ θ 0 + N φ θ γ φ θ 0 + M φ φ κ φ φ + M θ θ κ θ θ + M φ θ κ φ θ + Q φ z γ φ z + Q θ z γ θ z × R R φ d φ d θ d z .

Next, the kinetic energy can be written as follows:

(13) T = 1 2 ∫ φ 0 φ 1 ∫ 0 2 π { I 0 [ ( u ̇ ) 2 + ( v ̇ ) 2 + ( w ̇ ) 2 ] + 2 I 1 ( u ̇ ψ ̇ φ + v ̇ ψ ̇ θ ) + I 2 [ ( ψ ̇ φ ) 2 + ( ψ ̇ θ ) 2 ] } R R φ d φ d θ .

In this work, the mathematical model of elastic boundary conditions is conducted through the virtual spring technique, and it can be expressed as follows:

(14) U b = ∫ c ( k u , 0 u 2 + k v , 0 v 2 + k w , 0 w 2 + k φ , 0 ψ x 2 + k θ , 0 ψ θ 2 ) R R φ d θ ∣ φ = 0 + ∫ c ( k u , 1 u 2 + k v , 1 v 2 + k w , 1 w 2 + k φ , 1 ψ x 2 + k θ , 1 ψ θ 2 ) R R φ d θ ∣ φ = φ 1 − φ 0 ,

where k i,j (i = u, v, w, φ, θ and j = 0, 1) are the virtual springs. To describe the continuity of shell segments, the potential energy is derived as follows:

(15) U ss = 1 2 ∫ 0 2 π ∫ − h / 2 h / 2 k u ( u i − u i + 1 ) 2 + k v ( v i − v i + 1 ) 2 + k w ( w i − w i + 1 ) 2 + k φ ( ψ φ i − ψ φ i + 1 ) 2 + k θ ( ψ θ i − ψ θ i + 1 ) 2 R φ d θ d z .

Therefore, the boundary and continuity conditions can be expressed by the potential energy equations as follows:

(16) U BC = U b + U ss .

2.3 Admissible functions and solution procedure

To obtain a unified solution for the spherical shell, the displacements and rotations should be expanded as admissible displacement functions. The admissible functions should be suitable to different boundary conditions and can provide excellent convergence rates and accuracy. Chebyshev polynomials show great computational efficiency as a kind of admissible function. Therefore, the Chebyshev polynomials of the third kind are employed to build the discrete equations of the spherical shell. The recurrence formulations of the Chebyshev polynomials of the third kind are given as follows:

(17a) P 0 ( φ ) = 1 ,

(17b) P 1 ( φ ) = φ ,

(17c) P i ( φ ) = ( 2 i − 2 ) ( 2 i φ ( 2 i − 2 ) ) 2 i 2 ( 2 i − 2 ) P i − 1 ( φ ) − ( i − 1 ) 2 i 2 ( 2 i − 2 ) P i − 2 ( φ ) ,

where i = 2, 3, …, N.

The linear and rotational displacement components of the spherical shell can be rewritten as follows:

(18a) u = ∑ m = 0 M U m P m ( φ ) cos ( n θ ) e i ω t ,

(18b) v = ∑ m = 0 M V m P m ( φ ) sin ( n θ ) e i ω t ,

(18c) w = ∑ m = 0 M W m P m ( φ ) cos ( n θ ) e i ω t ,

(18d) ψ φ = ∑ m = 0 M ψ φ m P m ( φ ) cos ( n θ ) e i ω t ,

(18e) ψ θ = ∑ m = 0 M ψ θ m P m ( φ ) cos ( n θ ) e i ω t .

The Lagrange function of the spherical shell can be expressed as follows:

(19) L = U + U BC − T .

The derivatives of the Lagrange function with respect to the unknown coefficients are calculated, making the results equal to zero, and then the expression is obtained as follows:

(20) ∂ L ∂ ϑ = 0 ϑ = U m , V m , W m , ψ φ m , ψ θ m .

Substituting equations (12)–(19) into equation (20), the eigenvalue equation is obtained as follows:

(21) ( K − i ω C − ω 2 M ) P = F ,

where K represents the stiffness matrix, M indicates the mass matrix, C is the damping matrix, P is the unknown coefficient vector, and F denotes the external force vector. Then, the spherical shell’s natural frequency and mode shape can be obtained by solving equation (21).

For the forced vibration of the spherical shell, four kinds of loading pulses are considered as follows:

Rectangular f r ( t ) = 1 , 0 ≤ t ≤ τ , 0 , otherwise, (22a)
Triangular f t ( t ) = 2 t τ , 0 ≤ t ≤ τ 2 , 1 − 2 τ t − τ 2 , τ 2 ≤ t ≤ τ , 0 , otherwise, (22b)
Half-sine f h ( t ) = sin π t τ , 0 ≤ t ≤ τ , 0 , otherwise, (22c)
Exponential f e ( t ) = e − ξ t , 0 ≤ t ≤ τ , 0 , otherwise, (22d)
where τ and t are the pulse width and the time variable, respectively. The expression of external force loading is F i ( t ) = q 0 f i ( t ) , where i takes r, t, h, or e. Furthermore, the governing equation of forced vibration of the spherical shell is given as follows:

(23) M u ̈ + C u ̇ + ku = F ,

where u ̈ , u ̇ , and u are the acceleration, velocity, and displacement vectors, respectively. Based on the aforementioned formulations, the forced vibration characteristics of the spherical shell can be calculated by utilizing the Newmark-β approach as follows:

(24a) u ̇ t + Δ t = u ̇ t + [ ( 1 − γ ) u ̈ t + γ u ̈ t + Δ t ] Δ t ,

(24b) u t + Δ t = u t + u ̇ t Δ t + − 1 2 − β u ̈ t + β u ̈ t + Δ t Δ t 2 ,

where γ is set as 1/2, and β is set as 1/4.

3 Analysis and discussion

In this section, the primary goal is to investigate the factors, which can influence the vibration characteristics of the FGP spherical shell. To obtain a satisfactory converged solution, the correlative parameters should be determined first. It is observed that the present literature [37] has accomplished the convergence study of the spherical shell. To simplify the article structure, the number of segments and truncations are set as 5 and 8, and the coupling parameters are equal to 1014. Besides, the different boundary conditions at each edge of the spherical shell (i = 0, 1) are achieved by setting the spring stiffness values as follows: Completely free (F): k u,i = k v,i = k w,i = k φ,i = k θ,i = 0; Completely clamped (C): k u,i = k v,i = k w,i = k φ,i = k θ,i = 1014; Simply supported (S): k u,i = k v,i = k w,i = k θ,i = 1014, k φ,i = 0; Elastic constrain 1 (E1): k u,i = k v,i = 108, k w,i = k φ,i = k θ,i = 1014; Elastic constrain 2 (E2): k u,i = k v,i = k φ,i = k θ,i = 1014, k w,i = 108; and Elastic constrain 3 (E3): k u,i = k v,i = k w,i = 108; k φ,i = k θ,i = 1014.

3.1 Free vibration

Before conducting the parametric study, the correctness of the results obtained from the proposed method should be verified by comparing it with the published literature. The relevant parameters of the FGP spherical shell are set as follows: R = 2 m, φ 0 = 30°, φ 1 = 90°, h = 0.1 m, ρ 1 = 7,850 kg/m3, E 1 = 200 GPa, and μ = 0.3. If there is no special reminder in the following study, the parameters remain unchanged. The dimensionless frequency is defined as follows: Ω = ω R ρ 1 / E 1 / h . Through a comparison study, Table 1 illustrates that the results calculated by the proposed model are in good agreement with the existing data, and the maximum error is less than 1%. Therefore, the proposed model can be used to investigate the vibration behaviors of the spherical shell.

Table 1

Comparison of the first four frequencies Ω = ω R ρ 1 / E 1 / h of Type 1 FGP spherical shell with various boundary conditions (R = 2 m, φ 0 = 30°, φ 1 = 90°, h = 0.1 m, e 0 = 0.2, ρ 1 = 7,850 kg/m3, E 1 = 200 GPa, μ = 0.3)

e 0 Mode Boundary conditions
CC SS CF
Ram and Babu [11] Present Error (%) Ram and Babu [11] Present Error (%) Ram and Babu [11] Present Error (%)
0.2 1 21.15 21.15 0.00 18.11 18.11 0.00 2.251 2.246 0.24
2 21.68 21.68 0.01 19.95 19.95 0.02 3.205 3.199 0.20
3 21.77 21.77 0.02 20.95 20.94 0.03 4.820 4.814 0.12
4 22.39 22.37 0.08 21.35 21.35 0.00 5.691 5.678 0.22
0.4 1 20.38 20.38 0.02 18.30 18.30 0.01 2.167 2.162 0.21
2 20.88 20.88 0.00 19.14 19.13 0.07 3.082 3.075 0.22
3 20.98 20.97 0.03 20.22 20.22 0.01 4.647 4.642 0.12
4 21.55 21.53 0.08 20.61 20.61 0.01 5.472 5.459 0.23
0.6 1 19.55 19.55 0.01 17.38 17.38 0.00 2.068 2.063 0.24
2 20.01 20.01 0.02 18.23 18.22 0.04 2.916 2.910 0.22
3 20.15 20.14 0.03 19.46 19.45 0.03 4.463 4.458 0.12
4 20.58 20.56 0.09 19.82 19.82 0.01 5.177 5.165 0.23

Tables 2 and 3 investigate the effect of boundary conditions and porosity distributions on the vibration behaviors of the FGP spherical shell. Through these tables, it is obvious that the largest frequency appears in the clamped boundary condition, and the frequency parameters under the elastic constraints change irregularly relative to the stiffness value. To clarify this phenomenon, the effect of boundary parameters on the free vibration of the FGP spherical shell is further investigated in Figure 2. The clamped condition and elastic constraints, respectively, are employed at φ = φ 0 and φ = φ 1. In this work, only one of the spring types is changed, and others remain at 1014. It is easy to discover that the frequencies increase rapidly with the increase of the stiffness of boundary conditions, and sufficiently converged solutions are obtained when the stiffness value exceeds 1012. Based on this find, it can be considered that the boundary condition beyond the threshold is clamped boundary. Hence, the vibration behaviors can be regulated by changing the stiffness value. It is also directly observed that the effect of linear springs is significant to the fundamental frequencies, and the rotational springs are unnoticeable. Furthermore, Table 4 shows the fundamental frequencies of the FGP spherical shell with the hybrid boundary of classical-elastic conditions. The parametric study as mentioned earlier is unpublished, and it could provide future worker standard data as a reference.

Table 2

Frequencies Ω = ω R ρ 1 / E 1 / h of the FGP spherical shell considering different classical boundary conditions and porosity distribution (R = 2 m, φ 0 = 30°, φ 1 = 150°, h = 0.1 m, e 0 = 0.2, ρ 1 = 2,702 kg/m3, E 1 = 70 GPa, μ = 0.3)

n m Type 1 Type 2 Type 3
CS CF SF CS CF SF CS CF SF
0 1 13.593 2.300 1.999 13.551 2.292 1.975 13.328 2.249 1.959
2 13.771 13.342 13.026 13.761 13.329 12.990 13.508 13.085 12.778
3 19.253 17.797 17.585 19.193 17.767 17.547 18.843 17.446 17.240
4 21.054 19.119 19.032 20.892 19.070 18.973 20.521 18.726 18.638
1 1 15.962 3.643 3.636 15.943 3.568 3.560 15.660 3.495 3.487
2 18.246 16.598 16.589 18.217 16.573 16.560 17.886 16.275 16.265
3 20.627 18.611 18.541 20.553 18.563 18.504 20.167 18.227 18.166
4 22.568 21.097 20.832 22.388 20.966 20.729 21.969 20.581 20.340
2 1 17.462 9.450 9.450 17.436 9.274 9.274 17.125 9.088 9.088
2 19.394 17.680 17.679 19.336 17.643 17.642 18.984 17.326 17.325
3 21.507 19.752 19.722 21.391 19.665 19.644 20.987 19.308 19.283
4 24.369 22.258 22.066 24.148 22.077 21.916 23.676 21.667 21.495
Table 3

Frequencies Ω = ω R ρ 1 / E 1 / h of the FGP spherical shell considering different elastic boundary conditions and porosity distribution (R = 2 m, φ 0 = 30°, φ 1 = 150°, h = 0.1 m, e 0 = 0.2, ρ 1 = 2,702 kg/m3, E 1 = 70 GPa, μ = 0.3)

n m Type 1 Type 2 Type 3
E2E2 E3E3 E1E2 E2E2 E3E3 E1E2 E2E2 E3E3 E1E2
0 1 13.021 2.594 3.552 13.014 2.593 3.548 12.777 2.549 3.485
2 13.484 3.293 13.150 13.478 3.289 13.138 13.232 3.231 12.898
3 19.117 15.345 17.961 19.081 15.333 17.919 18.736 15.058 17.597
4 19.565 18.117 20.019 19.497 18.084 19.937 19.146 17.760 19.572
1 1 15.843 6.815 10.589 15.827 6.781 10.547 15.543 6.659 10.354
2 18.282 7.943 17.526 18.248 7.899 17.481 17.919 7.754 17.166
3 20.751 17.931 19.234 20.661 17.881 19.176 20.285 17.559 18.828
4 21.887 19.399 22.403 21.776 19.346 22.261 21.376 18.992 21.850
2 1 17.441 13.303 16.012 17.415 13.217 15.959 17.103 12.974 15.671
2 19.409 13.990 18.473 19.345 13.885 18.409 18.996 13.628 18.076
3 21.596 18.687 20.802 21.466 18.608 20.673 21.072 18.270 20.294
4 24.288 21.320 23.924 24.079 21.161 23.691 23.630 20.770 23.248
Figure 2 Effect of boundary stiffness on dynamic characteristics of FGP spherical shell (R = 2 m, φ 0 = 30°, φ 1 = 150°, h = 0.15 m, e 0 = 0.5).
Figure 2

Effect of boundary stiffness on dynamic characteristics of FGP spherical shell (R = 2 m, φ 0 = 30°, φ 1 = 150°, h = 0.15 m, e 0 = 0.5).

Table 4

Frequencies Ω = ω R ρ 1 / E 1 / h of the FGP spherical shell considering different classical-elastic boundary conditions and porosity distribution (R = 2 m, φ 0 = 30°, φ 1 = 150°, h = 0.1 m, e 0 = 0.2, ρ 1 = 2,702 kg/m3, E 1 = 70 GPa, μ = 0.3)

Porosity distribution Mode number Boundary conditions
n m C-E1 C-E2 C-E3 S-E1 S-E2 S-E3
Type 1 1 1 3.779 13.318 3.627 3.592 13.135 3.441
2 13.766 13.804 13.472 13.418 13.627 13.144
3 18.538 19.209 18.165 18.265 19.059 17.911
2 1 10.589 15.906 7.287 10.587 15.895 7.284
2 17.586 18.284 17.037 17.583 18.243 17.029
3 19.340 20.765 18.895 19.137 20.622 18.776
Type 2 1 1 3.771 13.308 3.621 3.575 13.112 3.427
2 13.743 13.792 13.456 13.371 13.605 13.105
3 18.479 19.162 18.124 18.195 19.023 17.861
2 1 10.547 15.891 7.250 10.546 15.875 7.246
2 17.543 18.249 17.008 17.538 18.215 16.995
3 19.267 20.671 18.841 19.083 20.550 18.736
Type 3 1 1 3.703 13.065 3.557 3.524 12.887 3.379
2 13.490 13.539 13.210 13.153 13.368 12.891
3 18.143 18.816 17.796 17.879 18.675 17.551
2 1 10.354 15.606 7.118 10.353 15.593 7.114
2 17.227 17.920 16.702 17.224 17.884 16.692
3 18.916 20.294 18.499 18.732 20.165 18.393

Table 5 shows the frequencies of the FGP spherical shell under the C–C boundary with different porosity coefficients. From the results shown in Table 5, it is noteworthy that frequencies gradually decrease with the increase of porosity coefficient, whatever be the type of porosity distribution. This is because the internal pore leads to the structural stiffness decrease. It can be seen that the FGP porosity distributions have a significant effect on the frequencies. Besides, the growth of the porosity coefficient has less effect on the FGP spherical shell with porosity distribution T – 1. The reason is that the symmetric distribution of porosity is hypersensitive to the change in the size of porosity. Therefore, the porosity distribution T – 1 has better stability under the variation of the porosity coefficient. A creditable explanation is that the porosity distribution T – 1 has higher stiffness than the others. In other words, the FGP spherical shell with porosity distribution T – 1 can achieve the best mechanical performance and possess the highest dimensionless frequencies.

Table 5

Frequencies Ω = ω R ρ 1 / E 1 / h of the C–C FGP spherical shell concerning various porosity ratios and porosity distribution (R = 2 m, φ 0 = 30°, φ 1 = 150°, h = 0.1 m, ρ 1 = 2,702 kg/m3, E 1 = 70 GPa, μ = 0.3)

Porosity distribution n m e 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Type 1 1 1 13.950 13.717 13.478 13.235 12.991 12.751 12.530 12.363 12.362
2 14.236 14.010 13.780 13.547 13.314 13.089 12.886 12.742 12.777
2 1 16.245 15.975 15.699 15.418 15.136 14.860 14.606 14.414 14.419
2 18.584 18.286 17.982 17.674 17.367 17.069 16.801 16.610 16.653
Type 2 1 1 13.947 13.712 13.470 13.223 12.974 12.728 12.499 12.320 12.300
2 14.223 13.982 13.734 13.478 13.217 12.956 12.705 12.492 12.418
2 1 16.238 15.961 15.675 15.383 15.088 14.795 14.521 14.301 14.267
2 18.567 18.250 17.922 17.586 17.243 16.900 16.571 16.298 16.219
Type 3 1 1 13.828 13.460 13.068 12.646 12.187 11.681 11.110 10.442 9.605
2 14.101 13.726 13.326 12.895 12.428 11.912 11.329 10.648 9.795
2 1 16.102 15.674 15.217 14.725 14.191 13.602 12.937 12.159 11.185
2 18.411 17.921 17.399 16.837 16.226 15.552 14.792 13.903 12.788

Table 6 shows the frequencies of the FGP spherical shell under the C–C boundary with various thickness–radius ratios. It is worth noting that the frequencies drop with the increase of the thickness–radius ratio. The result is because the value of the stiffness matrix grows with the increase of thickness–radius ratio. This finding suggests that the higher thickness ratios of the FGP spherical shell should be selected to improve the structural stability. To better understand the vibration behavior, the first four mode shapes of the FGP spherical shell with different boundary conditions are displayed in Figure 3.

Table 6

Frequencies Ω = ω R ρ 1 / E 1 / h of the C–C FGP spherical shell concerning different thickness ratios and porosity distribution (R = 2 m, φ 0 = 30°, φ 1 = 150°, h = 0.1 m, ρ 1 = 2,702 kg/m3, E 1 = 70 GPa, μ = 0.3)

Porosity distribution h/R e 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Type 1 0.02 33.118 32.568 32.005 31.433 30.857 30.294 29.776 29.385 29.393
0.05 13.562 13.341 13.115 12.886 12.656 12.432 12.228 12.078 12.094
0.1 7.040 6.926 6.806 6.685 6.563 6.444 6.334 6.251 6.253
0.15 4.735 4.657 4.577 4.495 4.413 4.333 4.259 4.204 4.205
0.2 3.573 3.514 3.454 3.392 3.331 3.270 3.215 3.173 3.175
Type 2 0.02 33.078 32.484 31.869 31.236 30.588 29.935 29.302 28.751 28.507
0.05 13.535 13.283 13.021 12.750 12.471 12.186 11.903 11.645 11.496
0.1 7.017 6.881 6.739 6.590 6.436 6.275 6.112 5.957 5.847
0.15 4.735 4.656 4.574 4.491 4.406 4.321 4.204 4.080 3.981
0.2 3.574 3.516 3.456 3.394 3.332 3.270 3.210 3.155 3.061
Type 3 0.02 32.828 31.954 31.022 30.020 28.932 27.730 26.374 24.789 22.801
0.05 13.439 13.082 12.700 12.290 11.844 11.353 10.798 10.148 9.335
0.1 6.972 6.787 6.589 6.376 6.145 5.890 5.602 5.265 4.843
0.15 4.693 4.568 4.435 4.292 4.136 3.965 3.771 3.544 3.260
0.2 3.542 3.447 3.347 3.239 3.121 2.992 2.845 2.674 2.460
Figure 3 First four mode shapes of FGP spherical shell under different boundary conditions (Type 1, R = 2 m, φ 0 = 30°, φ 1 = 90°, h = 0.1 m, e 0 = 0.2). (a) E2–E2 and (b) E3–E3.
Figure 3

First four mode shapes of FGP spherical shell under different boundary conditions (Type 1, R = 2 m, φ 0 = 30°, φ 1 = 90°, h = 0.1 m, e 0 = 0.2). (a) E2–E2 and (b) E3–E3.

3.2 Forced vibration

After the dynamic analysis of the FGP spherical shell, the forced vibration response is investigated in the following study. In this section, the steady-state and transient responses of the FGP spherical shell are discussed in detail.

3.2.1 Steady-state response

The validity of the approach proposed in this work needs to be studied before carrying out the relevant research. Figure 4 shows the comparison study of the proposed and finite element modeling (FEM) methods. The loading pulse can be further expressed as f w = f ¯ w δ ( φ − φ 0 ) δ ( θ − θ 0 ) , and it is point loading with an amplitude of −1N. The geometric parameters are set as R = 2 m, φ 0 = 30°, φ 1 = 150°, h = 0.1 m, and e 0 = 0.2. The load position is φ = π/2, θ = 0, and two observed points are constructed: A ( π / 2 , 0 ) and B ( 2 π / 3 , 0 ) . The sweep frequency range is f = 0–400 Hz, Δf = 2 Hz. The boundary condition is the C–C boundary. From Figure 4, it can be seen that the result obtained from the proposed approach is consistent with the FEM result. Hence, the correctness of the proposed model is verified, and based on the verification, the following investigation is presented.

Figure 4 Comparison of both proposed method and ABAQUS software on steady response of T-1 FGP spherical shell (R = 2 m, φ 0 = 30°, φ 1 = 150°, h = 0.1 m, e 0 = 0.2, load position: (π/2, 0), f = 0–400 Hz, Δf = 2 Hz, CC boundary condition).
Figure 4

Comparison of both proposed method and ABAQUS software on steady response of T-1 FGP spherical shell (R = 2 m, φ 0 = 30°, φ 1 = 150°, h = 0.1 m, e 0 = 0.2, load position: (π/2, 0), f = 0–400 Hz, Δf = 2 Hz, CC boundary condition).

Figure 5 illustrates the effect of boundary conditions on steady-state response. This figure illustrates that the boundary conditions of the FGP spherical shell have a direct effect on the steady-state response. For example, in the specific frequency range, the steady-state response curve appears with eight peaks of the wave under the C–F boundary but six peaks of the wave under the E4–E4 boundary. It can be attributed that the spring matrix of the FGP spherical shell is different under different boundary conditions.

Figure 5 Effect of boundary conditions and porous types on steady response of FGP spherical shell (R = 2 m, φ 0 = 30°, φ 1 = 150°, h = 0.1 m, e 0 = 0.2, load position: (π/2, 0), observation location: (2π/3, 0), f = 0 Hz∼400 Hz, Δf = 2 Hz). (a) C–F, (b) C–S, (c) E2–E2, and (d) E4–E4.
Figure 5

Effect of boundary conditions and porous types on steady response of FGP spherical shell (R = 2 m, φ 0 = 30°, φ 1 = 150°, h = 0.1 m, e 0 = 0.2, load position: (π/2, 0), observation location: (2π/3, 0), f = 0 Hz∼400 Hz, Δf = 2 Hz). (a) C–F, (b) C–S, (c) E2–E2, and (d) E4–E4.

Figure 6 carries out the relevant numerical analysis to further investigate the effect of the boundary spring parameters on the steady-state response of the FGP spherical shell. In this case, the variable of the boundary spring is set to only one by taking 103, 108, or 1014, and others are constantly equal to 1014. Through Figure 6, it is apparent that the linear springs affect the steady-state response more remarkably than the rotational springs, and the finding from forced vibration is similar to the free vibration.

Figure 6 Effect of boundary stiffness coefficient on steady response of T-3 FGP spherical shell (R = 2 m, φ 0 = 30°, φ 1 = 150°, h = 0.1 m, e 0 = 0.2, load position: (π/2, 0), observation location: (2π/3, 0), f = 0–400 Hz, Δf = 2 Hz).
Figure 6

Effect of boundary stiffness coefficient on steady response of T-3 FGP spherical shell (R = 2 m, φ 0 = 30°, φ 1 = 150°, h = 0.1 m, e 0 = 0.2, load position: (π/2, 0), observation location: (2π/3, 0), f = 0–400 Hz, Δf = 2 Hz).

Figure 7 investigates the effect of the thickness coefficient on the steady-state response. From the data in Figure 7, it is worth noting that the growth of the thickness ratio leads to the decline of the steady-state response peak, and the peak of the response curve is gradually moving to the right side. The reason may lie in that the increase in the thickness ratio causes the mass matrix of the FGP spherical shell to increase.

Figure 7 Effect of thickness ratio on steady response of the FGP spherical shell (R = 2 m, φ 0 = 30°, φ 1 = 150°, e 0 = 0.2, load position: (π/2, 0), observation location: (2π/3, 0), f = 0–400 Hz, Δf = 2 Hz, E2–E3 boundary condition).
Figure 7

Effect of thickness ratio on steady response of the FGP spherical shell (R = 2 m, φ 0 = 30°, φ 1 = 150°, e 0 = 0.2, load position: (π/2, 0), observation location: (2π/3, 0), f = 0–400 Hz, Δf = 2 Hz, E2–E3 boundary condition).

Figure 8 provides the effect of porosity coefficients on the steady-state response of the FGP spherical shell. It can be found that as the porosity coefficient increases, no matter the porosity distribution types, the peaks of the response curve are slightly moving to the left. This is because the stiffness falls with the increase of the porosity coefficient. Hence, lower porosity coefficients are selected in engineering applications.

Figure 8 Effect of porosity ratios on steady response of FGP conical shell (R = 2 m, h = 0.1 m, φ 0 = 30°, φ 1 = 150°, load position: (π/2, 0), observation location: (2π/3, 0), f = 0–400 Hz, Δf = 2 Hz, S–S boundary condition).
Figure 8

Effect of porosity ratios on steady response of FGP conical shell (R = 2 m, h = 0.1 m, φ 0 = 30°, φ 1 = 150°, load position: (π/2, 0), observation location: (2π/3, 0), f = 0–400 Hz, Δf = 2 Hz, S–S boundary condition).

3.2.2 Transient response

Figure 9 displays the comparison of both the proposed method and ABAQUS software on the transient response of the FGP spherical shell with the C–C boundary condition. The contrastive study illustrates that the proposed method has sufficient accuracy in predicting the transient response. After the verification, the parametric study can proceed.

Figure 9 Comparison of both proposed method and ABAQUS software on transient response of T-1 FGP spherical shell (R = 2 m, φ 0 = 30°, φ 1 = 150°, h = 0.1 m, e 0 = 0.2, load position: (π/2, 0), t = τ = 10 ms, Δt = 0.01 ms, C–C boundary condition).
Figure 9

Comparison of both proposed method and ABAQUS software on transient response of T-1 FGP spherical shell (R = 2 m, φ 0 = 30°, φ 1 = 150°, h = 0.1 m, e 0 = 0.2, load position: (π/2, 0), t = τ = 10 ms, Δt = 0.01 ms, C–C boundary condition).

Figure 10 investigates the effect of thickness ratio on the transient response of the FGP spherical shell. The load position of the pulse is (π/2, 0), and the observation location is (2π/3, 0). As can be seen in Figure 10, the thickness ratios have a remarkable effect on the peaks of the transient response. The peak of the transient response grows up with the increase of the thickness ratio, and this phenomenon becomes more obvious when the thickness ratio reaches 0.05. A possible explanation for this may be that the spherical shell with a higher thickness ratio performs better shock resistance.

Figure 10 Effect of thickness ratio on transient response of FGP spherical shell (R = 2 m, φ 0 = 30°, φ 1 = 150°, e 0 = 0.2, load position: (π/2, 0), observation location: (2π/3, 0), t = τ = 10 ms, Δt = 0.01 ms, S–S boundary condition).
Figure 10

Effect of thickness ratio on transient response of FGP spherical shell (R = 2 m, φ 0 = 30°, φ 1 = 150°, e 0 = 0.2, load position: (π/2, 0), observation location: (2π/3, 0), t = τ = 10 ms, Δt = 0.01 ms, S–S boundary condition).

Figure 11 explores the effect of porosity ratios on the transient response of the FGP spherical shell. It is worth noting that the growth of the porosity coefficient cannot delay the arrival time of the shock wave peaks but can raise the amplitude of the shock wave peaks. This is because the porosity coefficient only affects the stiffness and has no connection with the pulse frequency. Hence, reducing the porosity ratios is an effective way to improve the stability performance of the FGP spherical shell under the random shock wave.

Figure 11 Effect of porosity ratios on transient response of FGP spherical shell (R = 2 m, h = 0.1 m, φ 0 = 30°, φ 1 = 150°, load position: (π/2, 0), observation location: (2π/3, 0), t = τ = 10 ms, Δt = 0.01 ms, S–S boundary condition).
Figure 11

Effect of porosity ratios on transient response of FGP spherical shell (R = 2 m, h = 0.1 m, φ 0 = 30°, φ 1 = 150°, load position: (π/2, 0), observation location: (2π/3, 0), t = τ = 10 ms, Δt = 0.01 ms, S–S boundary condition).

Figure 12 investigates the transient response of the FGP spherical shell with various load types. The triangular pulse can markedly delay the arrival time of the peaks of the loading pulse and decline the loading pulse peaks. This is because the shape and duration of the shock waves have a significant effect on the dynamic response of the FGP spherical shell. The spherical shell’s transient response can be regulated by controlling the type of loading pulse so that the risk of damage to the structure, under external load, can be reduced.

Figure 12 Effect of load types on transient response of FGP spherical shell (R = 2 m, h = 0.1 m, φ 0 = 30°, φ 1 = 150°, load position: (π/2, 0), observation location: (2π/3, 0), t = 10 ms, τ = 5 ms, Δt = 0.01 ms, E1–E2 boundary condition).
Figure 12

Effect of load types on transient response of FGP spherical shell (R = 2 m, h = 0.1 m, φ 0 = 30°, φ 1 = 150°, load position: (π/2, 0), observation location: (2π/3, 0), t = 10 ms, τ = 5 ms, Δt = 0.01 ms, E1–E2 boundary condition).

4 Conclusion

In this study, the dynamic behaviors of the FGP spherical shell with elastic boundary conditions were investigated. Based on the aforementioned work, the findings of this study are summarized as follows:

  1. The calculated results of the model proposed by the comparative study agree well with the available data, with a maximum error of less than 1%. The study reveals that the intrinsic frequency increases rapidly along with the increase of the boundary stiffness and the decrease of the porosity. Besides, compared with the FEM, the proposed method has sufficient accuracy in predicting the transient response.

  2. The boundary conditions significantly affect the steady-state response. Specifically, fluctuations at the E4–E4 boundary condition reach about 10% less than at the other boundary conditions. Also, increasing the thickness ratio increases the stiffness of the structure and leads to a decrease in the peak steady-state response. The dynamic performance of the structure can be improved by adjusting the constraint conditions.

  3. As the thickness ratio increases, the peaks of the transient response gradually decrease, and a higher thickness ratio provides better impact resistance. However, an increase in porosity cannot delay the arrival time of the shock wave peak but increase the amplitude of the shock wave. To suppress the structure vibration, a large thickness ratio and lower porosity are suggested to be considered.

Although FGP is widely used due to its excellent stiffness-to-weight ratio, the structural stiffness weakened by internal pores always is an inevitable problem. Hence, looking for an optimal solution will be the key point in future work.

  1. Funding information: This study was supported by the Natural Science Foundation of Shandong Province (Grant No. ZR2022QE086); Natural Science Foundation of Shandong Province under Grant ZR2023ME133.

  2. Conflict of interest: The authors state no conflict of interest.

  3. Ethical approval: This article does not contain any studies with human participants or animals performed by any of the authors.

  4. Data availability statement: Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.

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Received: 2022-12-20
Revised: 2023-05-31
Accepted: 2023-06-07
Published Online: 2023-09-26

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/secm-2022-0214
Keywords for this article
dynamic analysis; functionally graded porous material; spherical shell; elastic boundary conditions
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