Internal Resonance of Hyperelastic Thin-Walled Cylindrical Shells under Harmonic Axial Excitation and Time-Varying Temperature Field

In this paper, the internal resonance characteristics of hyperelastic cylindrical shells 19 under the time-varying temperature field are investigated for the first time, and the evolution of the 20 isolated bubble is carried out. Through the analysis of the influences of temperature on material 21 parameters, the hyperelastic strain energy density function in the unsteady temperature field is 22 presented. The governing equations describing the axisymmetric nonlinear vibration are derived 23 from the nonlinear thin shell theory and the variational principle. With the harmonic balance 24 method and the arc length method, the steady state solutions of shells are obtained, and their 25 stabilities are determined. The influences of the discrete mode number, structural and temperature 26 parameters on the nonlinear behaviors are examined. The role of the parameter variation in 27 evolution behaviors of isolated bubble responses is revealed under the condition of 3:1 internal 28 resonance. The results manifest that both structural and temperature parameters can affect the 29 resonance range of the response curve, and the perturbed temperature has a more significant effect 30 on the stable region of the solution.

Keywords Hyperelastic cylindrical shell · axisymmetric nonlinear vibration · 32 harmonic balance method · internal resonance · isolated bubble 33 1 Introduction 1 Due to the light weight, high strength and stiffness ratio, cylindrical shells are 2 widely applied in many important engineering fields, such as the energy 3 transportation, shipbuilding and aerospace. The pipeline conveying fluid and the 4 spacecraft or missile, both can be modeled as the cylindrical shell for simplifying 5 analysis [1][2][3] . In addition, the mechanical behavior of biological tissues such as 6 bones and blood vessels can also be simulated with the cylindrical shell. However, 7 it should be noted that for those structures, hyperelastic material models should be 8 adopted to describe the nonlinear mechanical behaviors [4,5] . Generally, 9 cylindrical shells are usually subjected to different loads (such as the axial force 10 and temperature), which are prone to induce the large amplitude vibration, and 11 then destroy the structural safety. Therefore, it is essential and necessary to 12 investigate the nonlinear vibration characteristics of hyperelastic thin-walled 13 cylindrical shells subjected to external loads for the optimization of design and 14 improvement of safety. 15 The thin-walled shell is usually the best and only choice for those aerospace 16 and navigation structures, as well as engineering structures such as vehicle and 17 architecture building. Due to the complexity of the vibration behavior of the 18 continuum structure, the discretization method is widely applied to simplify the 19 continuum into a system with finite degree of freedom. This method is also very 20 common in the vibration analysis of beams, and there is a solid theoretical 21 foundation for the related analysis methods. However, shells especially exhibit 22 certain effects that are not presented in beams or even plates, and cannot be 23 interpreted by engineers who are only familiar with the beam-type vibration 24 theory [6] . Thus, it is helpful to improve the understanding of basic phenomena in 25 vibrations of plates and shells, and it will be useful in explaining experimental 26 measurements or the results of the finite element programs. Radwanska et al. [7] 27 presented a comprehensive introduction to the elastic plate and shell theory, 28 formulations and solutions of fundamental mechanical problems (statics, stability 29 and free vibrations) using the exact approaches and approximate computational 30 methods, and also emphasized the modern capabilities of the finite element 31 technology. With the load increases, the shell behavior will show some essential 32 geometric nonlinearity who plays a key role in the structural safety. Therefore, the 33 influence of nonlinearity has also been paid more and more attention in the 1 dynamic analysis of thin-walled shells. Meanwhile, with the development of 2 material science, the combination between shells and new materials is getting 3 more and more inseparably associated, especially, for plates and shells composed 4 of composite materials. The monograph contributed by Reddy [8] systematically 5 introduced various aspects related to laminated composite structures, including the 6 virtual work principle, variational method, anisotropic elasticity, plate and shell 7 theory (including classical theory, first-order and third-order shear deformation 8 theory), geometric nonlinearity and finite element analysis. Shen [9,10] dedicated to 9 the investigation of the geometrically nonlinear problems of inhomogeneous 10 isotropic and functionally graded plates and shells, which includes the large 11 deflection, post-buckling and nonlinear vibration. Amabili [11] studied the plates 12 and shells composed of composite materials, soft materials and biological 13 materials, and researched the hyperelastic, viscoelasticity and nonlinear damping, 14 which are pioneering works considered both material and geometric 15 nonlinearities. 16 The researches on nonlinear dynamic behaviors of cylindrical shells have 17 attracted much attention from a large number of scholars. Ye and Wang [12] 18 analyzed the nonlinear forced vibration of the functionally graded graphene 19 platelet-reinforced metal foam thin-walled cylindrical shell, and found that the 20 inclusion of graphene platelets in the shells weakens the nonlinear coupling effect. 21 Based on the von Kármán geometric nonlinear strain-displacement relationship 22 and the first-order shear deformation theory, Zhang et al. [ [2] investigated the nonlinear breathing vibration of an eccentric rotating 32 composite laminated cylindrical shell subjected to the lateral and temperature 33 excitations. Parvez et al. [16] presented the nonlinear dynamic response of 1 laminated composite cylindrical shells under periodic external forces, and 2 explored the parameters influencing the transition between the softening and 3 hardening nonlinear behaviors of cylindrical shells. Shen et al. [17] investigated the 4 nonlinear flexural vibrations of carbon nanotube-reinforced composite laminated 5 cylindrical shells with negative Poisson's ratios in thermal environments. Wang et 6 al. [18] investigated the strongly nonlinear traveling waves in a thermo-hyperelastic 7 cylindrical shell. 8 Compared with those structures composed of composite materials or other new 9 materials, there are relatively few researches on the dynamics of hyperelastic 10 structures, and most of them are also focused on relatively simple structures, such 11 as the membrane [19][20][21] or the beam [22][23][24] contributed by Soares, Gonçalves, Zhao, 12 Chen and Wang. Dong et al. [25] proposed a novel approach to tune the resonance 13 frequency of circular hyperelastic membrane-based energy harvesters via different 14 stretch ratios applied to membranes, which provided an alternative tuning strategy 15 to enable energy harvesting from different ambient vibration sources in various 16 environments. Wang and Zhu [26] investigated the nonlinear vibrations of a 17 hyperelastic beam under time-varying axial loading are derived via the extended 18 Hamilton's principle. Iglesias et al. [27] studied the large-amplitude axisymmetric 19 free vibration of an incompressible hyperelastic orthotropic cylinder, and analyzed 20 the influence of initial conditions, structural and material parameters on the 21 dynamic behavior. With the Gent-Gent hyperelastic model, Alibakhshi and 22 Heidari [28]  Amabili et al. [30] conducted the numerical analysis and experimental research on 33 the hyperelastic behavior of a thin square silicone rubber plate, obtained a good 1 agreement between the first four natural modes and frequencies numerically and 2 experimentally. 3 It is not difficult to find that there are numerous researches on the nonlinear 4 dynamics of cylindrical shells, however, few researchers pay attention to the 5 dynamic behaviors of hyperelastic cylindrical shells considering the material 6 nonlinearity, and fewer works considering the temperature effect. Therefore, this 7 paper studies the nonlinear dynamic behaviors of the hyperelastic thin-walled 8 cylindrical shells under harmonic axial excitation and time-varying temperature 9 field. The remainder of this paper is organized as follows: Section 2 introduces the 10 thermo-hyperelastic constitutive relationship; Section 3 gives the strain-11 displacement relationship of the thin-walled cylindrical shell, and derives the 12 nonlinear governing differential equations; Section 4 analyzes the natural 13 frequency characteristics of the shells and calculates the corresponding internal 14 resonance parameters; Section 5 presents the solution of nonlinear equations and 15 the stability of periodic solutions; Section 6 discusses the influences of the 16 discrete mode number, structural and temperature parameters via the numerical 17 simulation; Section 7 draws the conclusions. 18

Thermo-hyperelasticity and strain-displacement
is the Green strain. The principal invariants of the right 7 Cauchy-Green deformation tensor are 8 Based on the Kirchhoff-love hypothesis (that is, the stress components 15 perpendicular to the mid-surface are negligible, which is a very good 16 approximation for thin-walled shells) and the assumption of axisymmetric 17 vibrations (the circumferential displacement is zero), the displacements of a 18 generic point on the cylindrical shell are given as follows 19 Additionally, the geometrical nonlinearity of the shell yields the following strain-2 displacement relationships [8] For incompressible thermo-hyperelastic materials, the thermal effect is mainly 5 reflected in their strain energy functions. In this paper, the 8-chain non-Gaussian 6 network model, i.e. the Arruda-Boyce model [31] is adopted, as follows, 7 1 m ln 3 sinh where T  is the linear thermal expansion coefficient, t T is the environmental 10 temperature, where A T is the average temperature part of the environmental temperature and 20 P T  is the amplitude of the perturbed temperature part ( T  is a very small 21 parameter, and TP T   is also a small quantity). In this case, 22 Moreover, the series expansion form for Eq. (7) (keeping the first five terms) is 5 often adopted as follows, 6 ( ) The material parameters required for the calculation are listed in Table 1.  9 Table1. Material parameters [31,32] where  is the mass density, t is the time,  is the strain energy function, 5 and the dot denotes the derivative with respect to t . 6 The Ritz method is employed to approximate the axial displacement u and the 7 deflection w . The boundary conditions should be satisfied at the shell ends with 8 the assumed modes which are axisymmetric. Then, the displacements are 9 expanded by using the eigenmodes as follows 10 where m is the longitudinal half wave number, and M is the truncation order. 12 It should be noted that under the assumption of axisymmetric motion, the 13 circumferential wave numbers is 0, is expressed as follows 20 where  is the Dirac delta function, and ( ) x F t is the net force at the end. 22 Therefore, the virtual work done by the external force can be expressed as 23 The other part of the virtual work is done by the non-conservative damping force. 1 Assuming that the non-conservative damping force is viscous type, Rayleigh's 2 dissipation function is adopted to obtain the virtual work 3 where ij c is damping parameter related to the natural frequency of the shell and 5 can be evaluated from experiments. The Lagrange equation describing the 6 axisymmetric motion for a hyperelastic thin-walled cylindrical shell is given by 7 where i q is the generalized displacement and i is the mode number. The form of 9 external load is expressed as follows 10   1  T  0T  2  0  0  0   2  1  1  1  1  1  2  2  3  3  2  2  2  0  0  0  0   11  ,  ,  ,  ,  ,   11 , , , q Cq K q K q q K q q q F δ Kq δ K q q δ K q q q (25) 3 In summary, the axisymmetric nonlinear differential equations describing the 4 motion of the thin-walled cylindrical shell in the time-varying temperature field 5 subjected to the axial excitation have been derived. It is not difficult to find that 6 the system of nonlinear equations present the influences of the parametric 7 excitation and external excitation. sensitive to the variation of structural parameters when the length-diameter ratio is 20 small. That is, in this situation, a small change in the structural parameters will 21 lead to a significant change in the natural frequency, which should result to 22 extremely complex and unexplained internal resonance behavior [33] . In order to 23 avoid this problem, this paper will only discuss the larger length-diameter ratio 24 which satisfies the internal resonance condition. There are four structural 25 parameters satisfying the condition and those parameters are greater than 3, which 26 are listed in Table 2. 27 The internal resonance behavior is very common in various structures, such as 28 typical suspended cables, arches, beams, plates and shells [12,14,[34][35][36] . Meanwhile, 29 the internal resonance leads to mode interactions, an energy exchange or a 30 coupling among the modes [37] , thus numerous researchers applied it to the energy 31 harvest, and revealed that the working frequency range can be effectively 32 broadened [38][39][40][41][42] . 33  As shown in Fig. 3, the natural frequency increases with the increasing average 1 temperature. The reason is that the constitutive model adopted in this paper 2 gradually hardens as the temperature rises. Moreover, it can be inferred from Eq. 3 (8) that the shear modulus of the material is proportional to the temperature, that 4 is, the higher the environmental temperature, the greater the stiffness, which leads 5 to a greater natural frequency. Meanwhile, it is identified that the variation of 6 average environmental temperature has no effect on the natural frequency ratio, as 7 shown in Fig. 4. Additionally, the internal resonance behavior is very strongly 8 dependent on the natural frequency ratio. Therefore, a preliminarily conclusion 9 can be drawn that the average environmental temperature may not have a 10 significant effect on the axisymmetric nonlinear vibration of the cylindrical shell. 11

12
From the foregoing analysis, when the length-diameter ratio equals to the 13 parameters given in Table 2, the mode frequencies of the shell are commensurable 14 or nearly commensurable, which may lead to the internal resonance. Therefore, 15 this article will consider the nonlinear vibration behavior in a periodic uniform 16 temperature field when the parameter is approximately or exactly taken as the 17 values in Table 2. It is not difficult to find that Eq. (25) is a system of nonlinear differential 27 equations with quadratic and cubic nonlinear terms. For this kind of equations, the 28 analytical method usually fails, and the common strategy is to adopt the numerical 29 integration method or the approximate analytical method. Moreover, in order to 1 reveal the influence of structural parameters, the variation should result to some 2 stiff equations which leads to extremely high computational effort or low 3 accuracy. Fortunately, the approximate analytical methods overcome these 4 shortcomings. For the case that only the steady-state periodic solution is required, 5 the harmonic balance method is an extremely excellent choice. In the field of the 6 research on the nonlinear dynamic system, Nayfeh and Mook [43] are the pioneers 7 who adopt the harmonic balance method, and a more systematic introduction to 8 this method can be found in the works of Krack and Gross [44] . 9 Employing the harmonic balance method to solve Eq. (25), it is necessary to 10 assume that the solutions of the Eq. (25) can be approximated by the following 11 Fourier series 12 where k is the longitudinal half wave number, j is the order of harmonic. 15 1 i = represents the axial direction, while 2 i = represents the radial direction. 16 For the investigation of the resonance, the frequencies of external excitation 17 and temperature are both close to the fundamental frequency, i.e., 0T    sorting out the coefficient matrix of different harmonics, and then solving it by the 25 arc length method [33] , the periodic solution of the equation can be obtained. 26 6. Analysis and discussion 1 The discrete mode method is used to approximate the nonlinear vibration of the 2 shell, in general, it is necessary to discuss the influence of the discrete mode 3 number on the convergence of the solution. 4

Influence of discrete mode number 5
Before analyzing the influence of the discrete mode number, this subsection first 6 analyzes the different influence between odd and even order modes.  and 4  given in Table 2 which are the frequency ratio of even 10 order mode to fundamental mode, will not be discussed and analyzed further. 11 Since the internal resonance condition is satisfied or nearly satisfied, it can be 12 observed that there are many nonlinear peaks in the response curve, which reveals 13 the complex energy transfer behavior, and more details in the parameter analysis 14 will be discussed later. In addition, it is not difficult to find that the amplitude 15 magnitude of the high-order mode is far less than that of low-order mode, which 16 indicates that most energy of the shell vibration converges in the lower frequency 17 region. Therefore, for the analysis of shells, it is essential to focus on the low 18 order modes which possess smaller longitudinal half wave number. The case that 19 the parameter equals to 5  would not be analyzed further, which means this 20 paper mainly focuses on the case that the structural parameter is somewhere 21 around 3  . In this situation, the frequency ratio of the fundamental mode to its 22 same order mode satisfies the 3:1 internal resonance condition. As shown in Fig. 5, it can be found that when the axial excitation frequency is 5 close to the fundamental frequency, the low-frequency mode is directly excited, 6 while the high-frequency mode is hardly to be excited. Figure 5(a) clearly shows 7 that the low-order mode which has the largest axial displacement is dominant, 8 while the high-order modes present a very significant difference in the amplitude 9 magnitude. Interestingly, when the external excitation frequency is close to the 10 high-frequency mode, as shown in Fig. 6 (a), it can be found that the response of 11 the low-order mode is no longer dominated by the axial displacement, and the 12 multiple peak phenomenon no longer appears. There is only a left-bent peak 13 which shows the strong softening behavior. The reason is that when the length-14 diameter ratio is large, the radial displacement of the high-order mode is dominant 15 [33] , which means the most energy is distributed in the radial direction, and the 16 energy of the high-order mode is easily transferred to the low-order mode. Figure  17 6(a) clearly shows that the radial displacement of low-order mode is largest, 18 which is different from Fig. 5(a). 19 The previous analysis manifests that, for the same computational complexity, to 20 keep only odd order modes in discrete modes is more efficient. Additionally, the 21 influence of the discrete mode number on the structural response is analyzed in 22 the following part. It is not difficult to find that the dynamic response of the shell can be accurately 9 described by using two odd order discrete modes, while only one odd order 10 discrete mode should make the response curve tend to be the hardening behavior. 11 Figure 7 also shows that when the condition of 3:1 internal resonance is satisfied 12 approximately, the axial and radial motions of the shell are both dominated by the 13 first and third harmonics which possess similar curve shapes in two directions and 14 the different amplitude magnitude. Additionally, another distinction is that the 15 second harmonics possess different curve shapes in two directions, and the 1 softening behavior is more obvious for the radial direction. Moreover, the energy 2 distribution between two directions is different, i.e., the most energy of axial 3 direction converges in the first harmonic which is the low frequency region, while 4 the more energy of radial direction converges in the third harmonic which is the 5 high frequency region. 6

Influence of structural parameters 7
In this subsection, the value of the length-diameter ratio which satisfy the 3: 1 8 internal resonance is taken as the critical parameter, i.e., 3  = , and the 9 influence of the ratio on the response curve is discussed from the two cases that 10 the ratio gradually decreases or increases. Meanwhile, the stability of the solution 11 is determined, which is based on the eigenvalues of the Jacobian matrix, that is, if 12 the real parts of all eigenvalues are negative, the solution is stable. If there is an 13 eigenvalue with the positive real part, then the solution is unstable [33] . As shown 14 in Figs  . In this situation, the shell response has three peaks 7 named as the left peak, the middle peak and the right peak respectively, which is 8 more clearly distinguished from Figs. 8(e) and 8(f) (i.e., the third harmonic). 9 Obviously, the left is a typical softening peak with both stable and unstable parts, 10 and there will be a jumping near the frequency where the stability changes. Based 11 on this feature, it is not difficult to infer that for the first and second harmonics, 12 the positions of the left and the middle are exchanged. The middle is a completely 13 unstable peak, which is similar to the case of 2:1 internal resonance [45] , while the 14 right is a completely stable peak. With the decreasing ratio, the stable region on 15 the right side of the left peak will gradually decrease until it disappears, and the 16 middle peak will gradually shift to the left until merge with the left peak, which 17 will generate a new left peak, however, it is completely unstable. As the parameter 18 decreases further, the new left peak will shrink between the tip and the valley, and 19 generate an isolated bubble response which is completely unstable. As the ratio 20 decreases further, the isolated bubble will gradually shrink until it disappears 21 completely, and so does the left peak. For this situation, there will be only a 1 completely stable right peak. 2 Additionally, when the length-diameter ratio is in a certain frequency range, 3 there is no stable solution (such as  = 3.900740, the corresponding frequency 4 range is about [0.9995, 0.9998]), which indicates that the chaotic response may 5 occur in this range. As the ratio decreases, the parameter condition of internal 6 resonance is no longer satisfied gradually, then the frequency domain of response 7 will decrease gradually. It is necessary to state that, during the process of the 8 parameter evolution, the stability of the right peak remains, but the amplitude 9 changes. (e) (f) 1 Fig. 9 Influence of increasing structural parameters on frequency amplitude relationship 2 Figure 9 mainly shows the case that the structural parameters are slightly larger 3 than the critical parameter 3  . In this situation, the stable region of the left peak 4 will gradually increase with the increasing ratio, and the unstable middle peak will 5 gradually shift to the right and merge with the right peak. As the ratio increases, 6 the middle peak and the right peak will generate an isolated triangular bubble 7 response, where the bottom of the triangle is composed of the unstable middle 8 peak, and most of the others are composed of the stable right peak. Obviously, the 9 evolution process of the 3-order harmonic demonstrates this statement. 10 Meanwhile, since there are stable periodic solutions in the isolated response, there 11 is a phenomenon of coexistence of stable solutions in the corresponding frequency 12 range, that is, one excitation frequency should correspond to two stable periodic 13 solutions, and there are also some jumping phenomena. It is also worth pointed 14 out that for the 1-order and 2-order harmonics, the isolated triangular bubble 15 response may not be obvious due to the existence of the intersection points, 16 however, those triangle bubbles cannot be traced by the arc length method 17 directly, which indicates those bubbles are isolated with the main curves. In order 18 to obtain the isolated bubbles, it is necessary to adopt the strategy mixing the 19 perturbation method and arc length method. As the ratio increases further, the 20 isolated triangular bubble will shrink until it disappears completely. Finally, only 21 the left peak which characterizes the softening remains. 22 In summary, with the increasing ratio, the parameter condition of internal 23 resonance is no longer satisfied gradually, however, it is different from the case of 24 Fig. 8  between temperature and excitation. The influence of these 30 parameters will be discussed in this subsection. The discussion above mentioned 31 manifests that the responses are dominated by the first and third order harmonics 1 for the 3:1 internal resonance, and the shapes of the axial and radial response 2 curves are similar, while the amplitude magnitudes are different. Therefore, this 3 subsection only presents the first and third order harmonics of the radial 4 displacement, and the length-diameter ratio is taken as 3.902195  = . 5 6 (a) (b) 7 As shown in Fig. 10, the influence of average temperature on frequency 10 amplitude relationship is presented. Obviously, the higher the average 11 temperature, the smaller the response amplitude. The reason is that the shear 12 modulus is directly proportional to temperature which is shown in Eq. (8). Thus, 13 the higher the temperature, the greater the shear modulus, which means the greater 14 the stiffness of the shell, and the stronger the resistible capacity to the 15 deformation, i.e., the smaller the response amplitude.  Figure 11 shows the influence of the perturbed temperature on the vibration 1 response. With the increasing perturbed temperature, the middle peak will 2 gradually distorted and merges with the left peak, which is kind of similar to the 3 situation with gradually decreasing length-diameter ratio (as shown in Fig. 8). 4 However, there is no bubble response during the evolution process, and the right 5 peak will change more significantly. Meanwhile, the softening characteristics of 6 the cylindrical shell will be suppressed gradually with the increasing perturbed 7 temperature, while the hardening characteristics will be enhanced significantly, 8 which is more obvious for the first harmonic. When the perturbed temperature is 9 large enough (for example, 25 P TK = , the black line in Fig. 11), the 10 hyperelastic thin-walled cylindrical shell will present a typical hardening peak. Obviously, when the phase difference is  , the response presents the largest 18 amplitude (the magenta lines), and the amplitude of the three peaks are enhanced 19 significantly. That is, the shell response can be adjusted by controlling the phase 20 difference between the excitation and the temperature. 21 In this subsection, it can be found that the influence of temperature parameters 22 on the response is similar to that of the length-diameter ratio. However, the 23 hardening characteristic caused by the increasing temperature makes some 24 difference. The frequency amplitude response curve evolutes more smoothly with 25 the structural parameter compared with the temperature parameter. Thus, the 26 isolated bubbles only occur during the evolution process of the structural 1 parameter. Comparing Fig. 8(a) and Fig. 11(a), the evolution behavior of 2 frequency amplitude curve manifests that the temperature (especially the 3 perturbed temperature) will distort the response curve and lead to a more irregular 4 result. Meanwhile, the distortion will enhance the hardening significantly. 5 Briefly, the variation of the structural parameter and the temperature both play 6 an important role in the shell response, they bear relations as well as distinctions. 7 Therefore, it is necessary and essential to conduct the comprehensive analysis, 8 which leads to a better understanding and prediction for the nonlinear dynamic 9 behavior of the hyperelastic thin-walled cylindrical shell. 10

11
In this paper, the axisymmetric nonlinear vibration of the hyperelastic thin-walled 12 shell is investigated by the harmonic balance method and the arc length method, 13 and the 3:1 internal resonance of the shell is analyzed. Firstly, the system of 14 nonlinear governing differential equations with time-varying parameters are 15 derived based on the variational method. Then the periodic solution is calculated 16 via using the harmonic balance method and the arc length method, and the 17 stability is determined. Finally, the influences of structure and temperature 18 parameters on the evolution of nonlinear vibration are discussed. The mainly 19 conclusions are drawn as follows: 20 (1) For axisymmetric nonlinear vibrations of hyperelastic thin-walled cylindrical 21 shells, it is inaccurate to carry out the analysis with a single mode. Meanwhile, 22 since the even order modes will not be excited, it is more effective to discretize 23 the model with only the odd order mode. 24 (2) For hyperelastic thin-walled cylindrical shells with 3:1 internal resonance, the 25 length-diameter ratio has an extremely important impact on the shell resonance. 26 The reason is that this parameter plays a critical role in the frequency ratio which 27 is the necessary condition for internal resonance. Near the internal resonance 28 parameters, there are abundant nonlinear dynamic behaviors, including the 29 jumping, softening and hardening, and isolated bubble response. Specially, the left 1 peak will gradually shrink, and the isolated bubble occurs when the decreasing 2 length-diameter ratio is smaller than the critical parameter; while the right peak 3 will gradually shrink, and the isolated bubble occurs when the increasing length-4 diameter ratio is larger than the critical parameter. 5 (3) The influence of temperature parameters on the nonlinear dynamic behavior of 6 the shell is similar to that of the length-diameter ratio. However, the bubble will 7 not occurs during the temperature parameter varies. In addition, the perturbed 8 temperature has a more obvious effect on the stability of the solution. 9