Nonlinear dynamic responses of sandwich functionally graded porous cylindrical shells embedded in elastic media under 1:1 internal resonance

In this article, the nonlinear dynamic responses of sandwich functionally graded (FG) porous cylindrical shell embedded in elastic media are investigated. The shell studied here consists of three layers, of which the outer and inner skins are made of solid metal, while the core is FG porous metal foam. Partial differential equations are derived by utilizing the improved Donnell’s nonlinear shell theory and Hamilton’s principle. Afterwards, the Galerkin method is used to transform the governing equations into nonlinear ordinary differential equations, and an approximate analytical solution is obtained by using the multiple scales method. The effects of various system parameters, specifically, the radial load, core thickness, foam type, foam coefficient, structure damping, and Winkler-Pasternak foundation parameters on nonlinear internal resonance of the sandwich FG porous thin shells are evaluated.


Introduction
Compared with ordinary metal materials, the metal foam has many voids within the material, which results in notable physical and mechanical features involving low weight and high specific surface area, good impact energy absorption, excellent noise attenuation, high thermal conductivity combined with high gas permeability, machinability, and weld ability. Based on these excellent properties, the porous metal foam structure has been widely employed in various engineering applications, such as tissue engineering [1] , shock absorbers [2] , energy absorbers [3] , porous electrodes [4] , and produce filters [5] , which is becoming a kind of multifunctional engineering structure [6] .
In practical engineering applications, those metal foam structures are commonly subjected to severe dynamic loads, and prone to unexpected nonlinear vibrations and complex internal resonances, which may affect their normal operation or even damage the structures. The occurrence of nonlinear internal resonances could lead to large amplitude vibrations, irrecoverable deformations, noises, cracks, and even the failure of structures. Therefore, knowledge and prediction of resonant characteristics of porous metal foam structures are essential for their reliability and security.
Thin shell configurations are commonly employed under various dynamic loading conditions; for instance, they can serve as key components for aircraft, rocket propulsion systems, and large deployable space loop antennas. Thus, understanding the nonlinear internal resonant behaviors of thin cylindrical shells, to ensure their reliability, is critical. Therefore, some theoretical studies have been carried out to clarify the internal resonance of thin shells. Breslavsky and Amabili [7] studied the multiple internal resonances of thin cylindrical shells under multiharmonic excitation. Under the non-normal boundary condition, the 1: 2 internal resonance of thin shells was investigated by Yang et al. [8] . Zhang et al. [9] carried out a study on the nonlinear resonant response of rotary thin shells. Considering the influence of initial geometric imperfections, Rodrigues et al. [10] examined the 1 : 1 : 1 : 1 internal resonant behavior of thin cylindrical shells.
Despite the necessity of comprehensive understanding of nonlinear resonant response, up to now, only the linear responses of functionally graded (FG) porous structures have been reported in a few references. Liu et al. [11] gave an analytical solution to the impact response of sandwich FG porous shells. Dong et al. [12] studied the linear vibration of graphene platelet reinforced FG porous metal foam shells with spinning motion. The dynamic stability of graphene platelet reinforced sandwich porous plates via the classical plate theory was studied by Li et al. [13] . Chen et al. [14] considered two different porosity distributions to study the free and forced vibrations of porous metal foam beams. The wave propagation in graphene platelet reinforced FG porous metal foam plates was studied by Gao et al. [15] . Liu and Wang [16] investigated the thermoelectro-mechanical vibrations of porous piezoelectric nanoshells.
Literature review shows that no study has been reported on nonlinear internal resonance analysis of sandwich FG porous thin cylindrical shells. Therefore, the present work attempts to provide a solution approach for this issue and investigate 1 : 1 internal resonant response of sandwich FG porous thin shells. The dynamic model is established by using the improved Donnell's nonlinear shell theory. Hamilton's principle is used to derive the governing equations. Afterwards, the nonlinear solutions are discretized by the Galerkin method. The multiple scales method is used to obtain an approximate analytical solution to the multi-degree-offreedom systems. Finally, discussion is presented to carry out the nonlinear dynamic analysis and parametric study for the sandwich FG porous shells' nonlinear internal resonance.

Theoretical formulation
As shown in Fig. 1, considering an embedded sandwich FG porous thin shell with thickness h, length L, and radius R, the outer and inner layers are pure metal, and the core is FG porous metal foam. It is assumed that the porous core and surface layers are perfectly bonded. The elastic media are characterized by the Winkler-Pasternak foundation model, where the Winkler and Pasternak foundation parameters are k w and k p , respectively. The shell is subjected to a harmonic point excitation in the radial direction. The displacement components of points at the shell mid-plane are denoted by u, v, and w in the x-, θ-, and z-directions, respectively.
As seen in Fig. 2, three types of porous metal foam distributions are considered for the FG porous core as porous metal foam-I, porous metal foam-II, and porous metal foam-III. Varying material properties of three metal foam distribution types are given as follows [17][18][19] :  Porous metal foam-I, Porous metal foam-II, Porous metal foam-III, Here, ρ 1 , E 1 , and G 1 denote the mass density, Young's modules, and shear modules of pure steel, respectively, ζ m , ζ * m , and ϑ represent coefficients of mass density for porous metal foam-I, porous metal foam-II, and porous metal foam-III, respectively, and ζ 0 , ζ * 0 , and ϑ are foam coefficients of the corresponding porous metal foam distributions, respectively.
The relation between E and ρ, for open-cell metal foams, is given by [14,17,20] E(z) The relations between foam and mass density coefficients are expressed as Assume that the masses of all types of porous core are equivalent, namely, By using Eq. (12), ζ * 0 and ϑ can be determined with a given ζ 0 . The strain components at an arbitrary point of the embedded sandwich FG porous thin shell could be written as [21]  in which ε 0 x , ε 0 θ , and γ 0 xθ define the strains of the middle surface; k x , k θ , and k xθ are the middle-surface curvature and torsion, respectively.
To overcome the inaccuracy of Donnell's nonlinear shell theory at small circumferential wave number, the nonlinear strain-displacement relations, based on the improved Donnell's nonlinear shell theory, are developed as The stress-strain relations are  where The strain energy U s1 of the embedded sandwich FG porous thin shell is The forces and moments are given by  The strain energy U s2 induced by elastic media can be defined as The kinetic energy of embedded sandwich FG porous shell is in which f denotes the excitation amplitude; Ω is the circular excitation frequency; δ is the Dirac delta function; and (x 0 , θ 0 ) denotes the location of the excitation. The work performed by external excitation F can be given by By using Hamilton's principle and applying Eqs. (17), (20), (21) and (23) in Eq. (24), the equations of motion for the embedded sandwich FG porous shell can be obtained as follows: where c is the structure damping coefficient. The simply supported boundary conditions are

Solution method
By defining resonant mode with half-wave m and circumference wave n as mode (m, n) for simplicity, an excitation in the neighborhood of resonant mode (1,n) is considered here. The following three displacements are expanded by trigonometric series, which satisfy the boundary conditions [21] , where  (27) and ignoring the effect of inertia in the inplane direction, and subsequently applying the Galerkin's method [22][23][24][25] , the ordinary differential equations for the variables W A 1,n and W B 1,n can be obtained, where κ 1 , κ 2 , a 1 , a 2 , a 3 , and a 4 denote constant coefficients. Then, the multiple scales method [26][27] is employed to solve Eq. (32). First, the scaled time is assumed in the form in which ε denotes a small dimensionless parameter. The time derivatives in terms of T n become in which D n is defined as The response, which is the function of different scaled times, can be written as follows: By inserting Eq. (36) into Eq. (32), and defining κ i = ε 2 κ i0 and f /I 10 = ε 3 f 1,n , we get and The solution to Eq. (39) could be assumed in the form where Λ 1 and Λ 2 denote the conjugate functions of Λ 1 and Λ 2 , respectively. Due to the resonance of the system occurs at Ω ≈ ω 1 and ω 1 ≈ ω 2 , two detuning parameters σ 1 and σ 2 are used to measure the frequency distance, which can be written as follows: Substituting Eqs. (39)-(40) into Eq. (38) and letting the coefficients of the secular terms equal to zero yield To write the modulation equations in polar coordinates, the amplitude functions are set as in which b j and θ j denote the real numbers of T 2 . Substituting Eq. (42) into Eq. (41) and separating the real and imaginary parts yield where The frequency response equations, by utilizing b j = γ j = 0 (j = 1, 2), can be obtained as The stability of the system could be determined by the eigenvalues of the Jacobian matrix of Eq. (45).

Validation
To verify the correctness of derivation, the natural frequencies of an isotropic thin shell available in literature and those obtained in the present study are compared initially as listed in Table 1. The results show good agreement, where the small difference might be mainly caused by the different shell theories. Considering that there is no appropriate result from literature on internal resonance of isotropic thin shells for comparison studies, the present results are compared with numerical results via the Pseudo-arclength continuation method on the platform of MATLAB [29] . By adopting the material and geometrical parameters listed in Table 1, the nonlinear resonant responses of thin cylindrical shell are calculated, where the external excitation amplitude is f 1,n = 0.000 2hω 2 1,n , the location of the excitation is x 0 = L/2, θ 0 = 0, the damping ratio is 2κ i = 0.001, k w = k p = 0, and ε = 0.01. The mode (1, 7) is considered here. Owing to the axial symmetry of the shell, there are two conjugate modes with n = 7 associated with the same natural frequency, that is, one described by cos(7θ) and the other by sin(7θ). Hence, there exists a 1:1 internal resonance between the two conjugate modes. A comparison of the frequency-response curve is shown in Fig. 3. One can see that the present results match those obtained via the numerical method very well, bespeaking the validity of the present study.

Results of embedded sandwich FG porous shell
In what follows, the nonlinear dynamics analysis of an embedded sandwich FG porous shell is investigated, and the following parameters, if not specified, are considered as E = 200 GPa, The effect of external excitation on frequency-response curves of the embedded sandwich FG porous shell with porous metal foam-I distribution is presented in Fig. 4, where the dashed and solid lines are the unstable and stable analytical solutions, respectively. The response magnitude of the driven mode is about twice that of the companion mode. Additionally, one can find that the system exhibits hardening-spring behavior. With the increase in the excitation frequency from low level to high level, the coupled response loses its stability via a bifurcation. Furthermore, as the amplitude of external excitation increases, the response magnitude increases obviously. Meanwhile, the coupled interaction between driven and companion modes becomes more and more evident at the same time.  Figure 5 shows the effect of porous metal foam distribution type on the frequency-response curves of the embedded sandwich FG porous shell. It is noted that the embedded sandwich FG shell with porous metal foam distribution-II has the most evident hardening-spring behavior, while the porous metal foam-I shell has the weakest one. Besides, the resonant amplitude of the embedded sandwich FG porous shell changes slightly when the porous metal foam distribution type alters, indicating the porous metal foam distribution type has insignificant effect on the vibration amplitude of the system.  The frequency-response curves of the embedded sandwich FG porous shell under different foam coefficients are shown in Fig. 6. Four cases of foam coefficient ζ 0 = 0, ζ 0 = 0.2, ζ 0 = 0.4, and ζ 0 = 0.6 are considered. As can be seen, the resonance domain of the system moves to the low frequency ratio region with the increase in the foam coefficient; in fact, the large foam coefficient could decrease the natural frequencies of the system. In addition, with the increase in the foam coefficient, the hardening-spring behavior of the system becomes weaker and weaker.  In order to discuss the role of elastic foundations in the dynamic behaviors more clearly, Figs. 7-8 show the effect of Winkler and Pasternak foundation parameters on the frequencyresponse curves of the embedded sandwich FG porous shell. It can be found that an increase in Winkler or Pasternak foundation parameters results in the weakening of the hardening-spring characteristics of the system. For driven and companion modes, the resonance domain narrows as the Winkler or Pasternak foundation parameters increase. Furthermore, it is clear that the hardening nonlinearity behavior of system with Pasternak foundation parameter is less than the Winkler foundation parameter.     Figure 9 highlights the effect of damping ratio on the frequency-response curves of the embedded sandwich FG porous shell. As expected, the embedded sandwich FG shell with a smaller damping ratio has a larger resonant amplitude, since the damping ratio reduces the dynamic response of the embedded sandwich FG porous shell.
The effect of core-to-face ratio on the frequency-response curves of embedded sandwich FG porous shell is plotted in Fig. 10. Examining the figure shows that a larger core-to-face ratio makes the resonance domain of both modes move towards the direction of high frequency ratio, which indicates that an increasing core-to-face ratio could generate more notable hardeningspring behavior of the system. At a given excitation frequency ratio, the increase in the coreto-face ratio reduces the response magnitude of the embedded sandwich FG porous shell. Figure 11 gives the effect of radius-to-length ratio on the frequency-response curves of the embedded sandwich FG porous shell. In this case, one can find that an increase in radiusto-length ratio weakens the hardening-spring behavior. For driven and companion modes, the resonance domains narrow as the radius-to-length ratio increases.

Conclusions
In this paper, the nonlinear internal resonance of sandwich FG porous cylindrical shells embedded in elastic media is investigated, where three types of porosity distribution are considered. The governing equations are derived by using the improved Donnell's nonlinear shell  theory and Hamilton's principle, and then transformed into a set of ordinary differential equations through the Galerkin method. Afterwards, an approximate analytical solution is obtained by means of the multiple scales method. Finally, parametric studies of embedded sandwich FG metal foam porous shells are conducted in detail. Main highlights and conclusions are stated as follows. An improved nonlinear displacement expansion method, which requires no Airy stress function, is adopted to solve the nonlinear forced vibrations of the present shells. Compared with the traditional nonlinear governing equations including the Airy stress function, the difficulty of solving the nonlinear vibration issue of thin shells is reduced effectively. The correctness of the method is proved by the numerical verification.
The nonlinear tendency of the system exhibits hardening-spring behavior. The response magnitude of the driven mode is about twice that of the companion mode. The embedded sandwich FG shell with porous metal foam distribution-II has the most evident hardening-spring behavior, while the porous metal foam-I shell has the weakest one. In addition, enlarging the core-to-face ratio yields more notable hardening-spring behavior of the system. Higher values for foam coefficient and Winkler/Pasternak foundation parameters correspond to the weakening of the hardening-spring behavior of the embedded sandwich FG porous thin shell.