Thermal Vibration of Thick FGM Circular Cylindrical Shells by Using TSDT

: The generalized differential quadrature (GDQ) method is used to study the thermal vibration of thick functionally graded material (FGM) circular cylindrical shells. The nonlinear coefficient effect of third-order shear deformation theory (TSDT) model of displacements on the thermal equations of motion is considered. The parametric effects of FGM power law index and environment temperature on the time responses of stresses and displacements are considered. The linear varied effect of shear correction coefficients on the calculation of stiffness integration is also considered. The simply homogeneous equation is used to find the values of vibration frequency, also the dynamic GDQ discrete equation in matrix form are used to find the values of time response and transient response for the thick FGM circular cylindrical shells.


Introduction
There are some studies of shear deformation effects in the functionally graded material (FGM) shells. In 2018, Cong et al. [1] used the Reddy's third-order shear deformation theory (TSDT) for the nonlinear displacements to study the time response of displacements of double curves shallow shells, the effecting numerical solutions for honeycomb materials in geometrical parameters, material properties and damping loads are presented. In 2017, Sobhaniaragh et al. [2] used the TSDT for the displacements to study the buckling loads of FGM Carbon Nano-Tube (CNT)-reinforced shells in the environment (room temperature 300 K) without thermal strains, parametric effects on material properties and critical buckling loads are presented by using the generalized differential quadrature (GDQ) method. In 2017, Dung and Vuong [3] used an analytical method with TSDT to study the buckling of FGM shells in elastic foundation under thermal environment and external pressure. In 2016, Dai et al. [4] presented a 2000-2015 reviewing focused on coupled mechanics, e.g., thermo-mechanical responses with the first-order shear deformation theory (FSDT) models, HSDT models in widely used TSDT to study the bending, buckling, free and forced vibrations of FGM cylindrical shells by using various theoretical, analytical and numerical methods. In 2016, Fantuzzi et al. [5] used the numerical GDQ methods to study the free vibration of FGM spherical and cylindrical shells, some frequency solutions in FGM exponent number and thickness ratio are included. There are some numerical studies in the thick shells. In 2016, Kar and Panda [6] used the code of finite element method (FEM) and the TSDT displacements to obtain the numerical static bending results of deflections and stresses for the heated FGM spherical shells under thermal load and thermal environment. In 2015, Kurtaran [7] used the methods of GDQ and FSDT to obtain the numerical transient results of moderately thick laminated composite spherical and cylindrical shells. In 2012, Viola et al. [8] presented static analyses of FGM cylindrical shells under mechanical loading by using the GDQ method and a 2D unconstrained third order shear deformation theory (UTSDT), the numerical solutions for stresses without thermal effect are obtained. In 2010, Sepiani et al. [9] used the FSDT formulation to obtain the numerical free vibration and buckling results for the FGM cylindrical shells without considering the thermal effect. The advantages and disadvantages of above studies are listed as follows. The advantages are the FGM shells well studied by considering the effects of many types of shear deformation theories in displacements. The disadvantages are the FGM shells does not well studied by considering the effects of varied shear correction coefficient in the shear stresses. The importance/impact of the FGM cylindrical shells in the present study are considering the effects of varied effects of shear correction coefficient and nonlinear terms of TSDT involved into the homogeneous equation. Usually, the engineering applications towards the thermal vibration of FGM shells are used in the fields of engine, propulsion and structures for the vehicle, plane and missile.
Some GDQ computational experiences are presented in the composited FGM shells and plates. In 2017, Hong [10] presented the numerical thermal vibration results of FGM thick plates by considering the FSDT model and the varied shear correction factor effects. In 2017, Hong [11] presented the numerical thermal vibration and flutter results of a supersonic air flowed over FGM thick circular cylindrical shells. In 2017, Hong [12] presented the numerical displacement and stresses results of FGM thin laminated magnetostrictive shells by considering with velocity feedback and suitable control gain values under thermal vibration. In 2016, Hong [13] presented the thermal vibration of Terfenol-D FGM circular cylindrical shells by considering the FSDT model and the constant modified shear correction factor effects. It is interesting to investigate the thermal stresses and center displacement of GDQ computation in this nonlinear TSDT vibration approach and the varied effects of shear correction coefficient of FGM circular cylindrical shells with four edges in simply supported boundary conditions. Parametric effects of environment temperature and FGM power law index on the thermal stress and center displacement of FGM circular cylindrical shells including the effect of varied shear correction coefficient are also investigated with the vibration frequency approach of simply homogeneous equation. The main contribution and novelty of paper is to provide and investigate the numerical solutions of thermal vibrations in thick FGM circular cylindrical shells by considering the linear varied values of shear correction coefficient and nonlinear terms of TSDT.

Procedure of Formulations
For a two-material thick FGM circular cylindrical shell is used with thickness h 1 of inner layer FGM material 1 and thickness h 2 of outer layer FGM material 2, L is the axial length of FGM shells. The material properties of power-law function of FGM shells are considered with Young's modulus E fgm of FGM in standard variation form of power law index R n , the others are assumed in the simple average form by Chi and Chung, in 2006 [14]. The properties P i of individual constituent material of FGMs are functions of environment temperature T.
The time dependent of displacements u, v and w of thick FGM circular cylindrical shells are assumed in the TSDT equations including the nonlinear terms in z 3 with coefficient c 1 by Lee et al. in 2004 [15] as follows [Equations (1)-(3)]: where u 0 and v 0 are tangential displacements in the in-surface coordinates x-and θ-axes direction, respectively, w is transverse displacement in the out of surface coordinates. z-axis direction of the middle-plane of shells, ϕ x and ϕ θ are the shear rotations, R is the middle-surface radius of FGM shells, t is time. The nonlinear coefficient for c 1 = 4/[3(h * ) 2 ] is given as in TSDT approach, in which h * is the total thickness of FGM shells. For the normal stresses (σ x and σ θ ) and the shear stresses (σ xθ , σ θz and σ xz ) in the thick FGM circular cylindrical shells under temperature difference ∆T can be presented relatively to the stiffness Q ̅ ij , in-plane strains ε x , ε θ and ε xθ , not negligible shear strains ε θz and ε xz , coefficients of thermal expansion α x and α θ , coefficient of thermal shear α xθ by Lee and Reddy in 2005 [16]; by Whitney in 1987 [17]. The thermal load value of ∆T assumed in linear function of z between the thick FGM circular cylindrical shell and curing area, also satisfied the heat conduction equation in simple form for the thick FGM circular cylindrical shell by Hong in 2016 [13].
The dynamic equations of motion with TSDT for a thick FGM circular cylindrical shell can be assumed and given by Reddy in 2002 [18]. The Von Karman type of strain-displacement relations with 0 0 0 0 , and 0 ∂ are assumed and used for the strains ε x , ε θ , ε zz , ε xθ , ε θz , and ε xz as follows: By substituting the stress equations and strain-displacement relations [Equations (4)-(9)] into the dynamic equations of motion, the five numbers of dynamic equilibrium differential equations in the cylindrical coordinates with TSDT of thick FGM circular cylindrical shells in terms of partial derivatives of five unknown displacements (u 0 , v 0 and w) and shear rotations (ϕ x and ϕ θ ) subjected to partial derivatives of given external loads (f 1 ,…, f 5 ) containing thermal loads (N ̅ ,M ̅ ̃,P ̅ ), mechanical loads (p 1 ,p 2 ,q) and inertia terms can be derived and expressed in matrix forms including coefficient c 1 elements of stiffness integrals (A i s j s, B i s j s, D i s j s, E i s j s, F i s j s, H i s j s) and (A i * j *, B i * j *, D i * j *, E i * j *, F i * j *, H i * j *) by assuming that midplane strain terms where k α is the shear correction coefficient. The computed and varied values of linear k α are usually functions of total thickness of shells, FGM power law index and environment temperature presented by Hong in 2014 [19]. The stiffness Q ̅ i s j s and Q ̅ i * j * for thick FGM circular cylindrical shells with z/R terms cannot be neglected are used in the simple forms by Sepiani et al. in 2010 [9] and by Hong in 2014 [19].
The differential quadrature (DQ) method is presented firstly by Bert et al. in 1989 [20]. The GDQ method is presented and improved firstly by Shu and Du in 1997 [21]. The GDQ method is also still well used and presented in the numerical investigation for some thermal vibration of thick FGM shells. The boundary conditions in dynamic GDQ discrete equations approach are to be considered for four sides simply supported, not symmetric, orthotropic of laminated thick FGM circular cylindrical shells. For a two-dimensional function f(x,θ) at coordinates of arbitrarily typical grid point (x i , θ j ), i = 1,2,…, N and j = 1,2,…,M, in which N is the number of the total discrete grid points used in the x direction. M is the number of the total discrete grid points used in the θ direction. The dynamic GDQ discrete equations can be written into the matrix form as follows: where

Some Numerical Results and Discussions
To study the GDQ displacement results of shells layers in the stacking sequence (0° / 0°) with two constituent FGM material 1 and FGM material 2 under thermal loads as shown in Figure 1. The typical solution under four sides simply supported boundary condition is presented with no in-plane distributed forces (p 1 = p 1 = 0) and no external pressure load (q = 0). By using the simplified sinusoidal temperature πθ γ ∆ = for the thermal loads, in which γ is the frequency of applied heat flux, T̅ 1 is the amplitude of temperature, the value of γ can be calculated from the simplified heat conduction equation by Hong in 2016 [13]. Before the process of thermal vibrations of thick FGM circular cylindrical shells, it is needed to obtain the calculation values of vibration frequency ω mn in two directional vibrations with mode shape subscript numbers m and n.  Table 1. It is considering the varied effects of linear k α and ω 11 for three values of R n with the vibration frequency approach of simply homogeneous equation. The error accuracy is 8.8 × 10 −7 for the nonlinear center displacements amplitude of R n = 0.5 and L/h * = 10 cases. The N×M = 13 × 13 grid points can be treated in the very good convergence result and used further in the following GDQ computations of time responses for displacements and stresses of the thermal vibration of nonlinear TSDT thick FGM circular cylindrical shells.

Conclusions
The GDQ solutions are calculated and investigated for the displacements and stresses in the thermal vibration of thick FGM circular cylindrical shells with the vibration frequency approach of simply homogeneous equation by considering the linear varied effects of shear correction coefficient and coefficient c 1 term of TSDT. The values of center displacements amplitudes in linear case of c 1 = 0/mm2 are greater than that nonlinear case of c 1 = 0.925925/mm2 at the corresponding time for thick L/h * = 5 and 10, respectively. Thus, the values of center displacements can be modified into the more accuracy data by using the suitable c 1 terms. The values of dominated stresses σ x are all decreasing with time in the case c 1 = 0.925925/mm2 for thick L/h * = 5 and 10, respectively. The amplitude w(L/2,2π/2) of the L/h * =5 cannot withstand for higher temperature (T = 1000 K) of environment. The higher frequency of applied heat flux gets more higher amplitude