Advanced Frequency Study of Thick FGM Cylindrical Shells by Using TSDT and Nonlinear Shear

: For the advanced frequency study of thick functionally graded material (FGM) circular cylindrical shells, it is interesting to consider the extra effects of nonlinear coefficient term in third-order shear deformation theory (TSDT) of displacements on the calculation of varied shear correction coefficient. The formulation for the advanced nonlinear shear correction coefficient are based on the energy equivalence principle. The values of nonlinear shear correction coefficient are usually functions of nonlinear coefficient term of TSDT, power-law exponent parameter and environment temperature. The free vibration frequencies of thick FGM circular cylindrical shells are investigated with the simply homogeneous equation by considering that simultaneous effects of the TSDT, the nonlinear shear correction coefficient of transverse shear force and the two direction of mode shapes. The novelty is more important and reasonable for the thick FGM circular cylindrical shells, especially for the ratio of length to thickness is five by considering the effects of advanced nonlinear shear correction coefficient and nonlinear coefficient term of TSDT on the advanced calculation of fundamental first natural frequencies.


Introduction
There are some traditional numerical investigations in the frequency study of free vibration for the functionally graded material (FGM) cylindrical shells and panels.Zhang et al. [1] presented the isogeometric numerical method and first-order shear deformation theory (FSDT) of displacements to study the natural frequencies and mode shapes for the carbon nanotubes reinforced (CNTR) FGM cylindrical shells.Liu et al. [2] presented the wave based method (WBM) and FSDT of displacements to study the natural frequency for the FGM cylindrical shells by considering the constant value of shear correction factor equal to 5/6 for the transverse shear force.Baghlani et al. [3] presented the Euler-Lagrange equations and higher-order shear deformation theory (HSDT) of displacements to study the natural frequency for the fluid-filled FGM cylindrical shells surrounded by Pasternak elastic foundation.Shahbaztabar et al. [4] presented the eigenvalue equation and FSDT of displacements to study the natural frequency for the fluid-filled FGM cylindrical shells surrounded by Pasternak elastic foundation by also considering the constant value of shear correction factor equal to 5/6 for the transverse shear force.Babaei et al. [5] presented the two steps perturbation technique and HSDT of displacements to study the natural frequency for the FGM cylindrical panels resting on nonlinear elastic foundation.Zhang et al. [6] presented the modified Fourier cosine series method and FSDT of displacements to study the natural frequency for the moderately thick FGM cylindrical shells by also considering the constant value of shear correction factor equal to 5/6 for the transverse shear force.Fan et al. [7] presented the Walsh series method (WSM) and FSDT of displacements to study the natural frequency for the FGM cylindrical shells by also considering the constant value of shear correction factor equal to 5/6 for the transverse shear force.Awrejcewicz et al. [8] presented the variational Ritz method, the R-functions method (RFM) and FSDT of displacements to study the natural frequency for the shallow FGM cylindrical shells by also considering the constant value of shear correction factor equal to 5/6 for the transverse shear force.Liew et al. [9] presented the eigenvalue equation and FSDT of displacements to study the natural frequency for the coating-FGM-substrate cylindrical shells by also considering the constant value of shear correction factor equal to 5/6 for the transverse shear force.
For the frequency study of free vibration in the thick FGM cylindrical shells, it is usually considered the shear correction factor effect in the transverse shear force.The author has some investigations in the computed and varied values for the shear correction factor.Hong [10] presented the preliminary studies in free vibration frequency of thick FGM circular cylindrical shells without considering the effects of nonlinear coefficient TSDT term on the varied shear correction coefficient calculation.Hong [11] presented the preliminary studies of the varied shear correction and FSDT effects on the vibration frequency of thick FGM circular cylindrical shells in unsteady supersonic flow.In the advanced study for the vibration frequency of thick FGM circular cylindrical shells with the simply homogeneous equation, it is interesting to consider the simultaneous effects of the TSDT of displacements, the nonlinear shear correction coefficient of transverse shear force and the two directions of mode shapes.The vibration frequency results vs. shear correction coefficient  k values, nonlinear coefficient term 1 c of TSDT, FGM power-law exponent parameter and environment temperature are studied, respectively under four main cases of (a) advanced nonlinear  k , 1 c = 0.333333/mm 2 ; (b) linear  k , 1 c = 0/mm 2 ; (c) constant

Formulation for the advanced nonlinear k α
For a two-material thick FGM circular cylindrical shells problem model under environment temperature T with thickness 1 h of FGM constituent material 1 and thickness 2 h of FGM constituent material 2, respectively in the thickness direction of the cylindrical coordinate systems are shown in Fig. 1.The properties i P of individual constituent material are in functions of T for the power-law function type of FGMs [12].The time dependent of nonlinear displacements ( u , v and w ) at any point ( x , , z ) of thick FGM cylindrical shells are assumed in the nonlinear vs.  3 with coefficient  1 term of TSDT equations [13] as follows, where  0 and  0 are the tangential displacements in the in-surface coordinates x and  axes direction , respectively.
w is the transverse displacement in the out of surface coordinates z axis direction of the middle- plane of shells.  and   are the shear rotations.R is the middle-surface radius at any point ( x , , z ) of the FGM cylindrical shells.t is the time.The coefficient for  1 = 4/(3ℎ * 2 ) is given in the TSDT approach, in which * h is the total thickness of FGM circular cylindrical shells.
For the normal stresses ( The stiffness integrations with z for the in-plane force resultant, moment and transverse shear force are expressed in the following equations, In the previous work of preliminary investigations for the computed and varied values of  k are usually functions of total thickness of shells, FGM power law index and environment temperature [16].For the advanced thick FGM cylindrical shells study, it is interesting to consider the extra effects of nonlinear coefficient term of TSDT displacements on the calculation of varied shear correction coefficient.The advanced shear correction factor  k would be nonlinear with respect to  1 can be obtained by using the energy equivalence principle.Let the total strain energy defined by the shear forces and transverse shears stresses, respectively in the form along the length of cylindrical shells by Whitney [15].It is reasonable to assume that ).With the same procedure as in the thick FGM plates by substituting the shear forces and transverse shear stresses equations, respectively into the total strain energy equation, thus the advanced  k can be obtained as follows for the thick FGM circular cylindrical shells, where  = ( −  1 )
In Fig. 3b, * f = 20.434654 is obtained at m= n= 1 under linear  k , 1 c = 0/mm 2 .In Fig. 3d, * f = 20.131851 is obtained at m= n= 1 under constant  k = 5/6, 1 c = 0/mm 2 .There are small effects of nonlinear coefficient term 1 c and  k on the value of fundamental first natural frequencies by using the approaching of simply homogeneous equations.In the linear case 1 c = 0/mm 2 , the values of * f are overestimated.It is reasonable to consider the effect of nonlinear varied values  k and 1 c on the advanced calculation of natural frequencies.
The article is presented for adding the effect of advanced nonlinear  k as a continuation of a previous work [10] which was only using the effect of linear  k .The advanced nonlinear  k is expressed in eq. ( 3), in which the fraction parameters  and  are in nonlinear functions of  1 value.For the linear  k is preliminary expressed in eq. of the work [16], in which did not consider the effect of  1 term in the calculation directly.The impact of the advanced nonlinear  k on results are in clarification with previous work [10].
Comparing the fundamental value of * f results in both articles they are calculated to differ by approximately 0.2%, when  1 is not equal to zero, this corresponds to the nonlinear case.Notably, there is a discrepancy in results when  1 =0 (linear case), varying up to 40%.These compared values of * f vs. T conclusion draws are shown in Fig. 3e for nonlinear  k in this paper and linear  k referred to [10].
For the more supplement of FGM and composited material analyses can be referred further in the fields of thermal analysis of cracked FGM plates by Do et al. [19], finite element method (FEM) applied in cracked nanoplates with flexo-electric effects by Doan et al. [20], and FEM used for triple-layer composite plates under moving load by Nguyen et al. [21].

Conclusions
The advanced frequency values of free vibration are computed by using the simply homogeneous equation and advanced nonlinear  k values for the thick FGM circular cylindrical shells.There are four parameters effects of nonlinear coefficient term 1 c , shear correction coefficient  k , power-law exponent parameter

Figure 1 .
Figure 1.Two-material thick FGM circular cylindrical shells problem model of FGM shells. k is the advanced shear correction coefficient.

 11 f 1 c 1 
] can be applied directly to calculate the free vibration mn  with considering the advanced nonlinear  k , where subscript m is the axial half-waves number, n is the number of circumferential waves, and dz z is the density of ( k )th constituent ply.The FGM temperature dependent constituent materials at inner material 1/outer material 2 i.e., SUS304/Si 3 N 4 layers are used in the frequency computations of thick circular cylindrical shells.The advanced nonlinear values of  k are usually functions of  1 , n R and T .For geometric values of / = 1, , in which L is the length of FGM cylindrical shells, the varied values of advanced nonlinear are increasing with respect to n R .Thus, advanced nonlinear values of are used for frequency mn  computations of the free vibration including the coefficient  1 term.For the non-dimensional frequency parameter  * = 4 11 √ 2 / 11 values under the effects of 1 c = 0.925925/mm 2 and 1 c = 0/mm 2 for /ℎ * = 5, 8 and 10 are shown in Table1a, where is values under advanced  k , environment temperature T and effects are not greater than 10.076400, which is in smaller value than 13.538765 in the preliminary  k study case[10].Another non-dimensional frequency parameter Ω = ( 11  2 / * h )√ 1 / 1 values under the cases of 1 c = 0.925925/mm 2 and 1 c = 0/mm 2 for /ℎ * = 5, 8 and 10 are shown in Table 1b, in which is the density of FGM constituent material 1.The presented values under advanced  k , environment temperature T and 1 c effects are not greater than 26.945133, which is in smaller value than 32.380783 in the preliminary  k study case [10].The presented values of * f vs. * h under /ℎ * =10, 300K, advanced nonlinear  k and 1 c effects are shown in Table 1c.The presented value * f = 8.429713 at 1 c = 0.333333/mm 2 , * h = 2mm, n R = 0.5 is found.The presented values of vs. * h under /ℎ * =10, 700K, advanced nonlinear  k and 1 c effects are shown in Table 1d.The presented value = 1.999438 at 1 c = 6.584362/mm 2 , * h = 0.45mm, n R = 0.5 is found.The natural frequency mn  (1/s) values with subscript mode shapes m and n are presented.The presented 11  vs.
nonlinear  k , environment temperature T and 1 c = 0.925925/mm 2 for /ℎ * = 5 and 10 are shown in Table 2.There are in slightly different values for 11  under /ℎ * =5,

k
values for T =300K are listed in the Table3.The values of advanced nonlinear  k are independent of ℎ * for the thick FGM circular cylindrical shells.The values of  k in the advanced nonlinear case with 1 c ≠ 0 are different to the linear case with 1 c = 0.The different values of  k vs.

T 2 .
=300K are shown in Fig.There are in great variant  k values under the advanced nonlinear case with 1 c ≠ 0. The compared values of * f vs. n at m= 1, 2 and 3 with /ℎ * = 5, = 0.5, * h = 2mm, T =300K are shown

Figure 2 .Figure 3 . 1 c
Figure 2. Values of  k vs. n R in nonlinear ( 1 c ≠ 0) and linear ( 1 c = 0) temperature T on the natural frequencies are investigated.The main results and new contributions of the research is more important and reasonable for the thick FGM circular cylindrical shells, especially for /ℎ * =  k 5 to consider the effect of nonlinear varied values  k and 1 c on the advanced calculation of fundamental first natural frequencies.

Table 1c .
Frequency *f with advanced nonlinear

Table 3 .
Advanced vs.  1 and n R under T= 300K  k * h (mm)