Three dimensional Free Vibration and Transient Analysis of Two Directional Functionally Graded Thick Cylindrical Panels Under Impact Loading

In this paper three dimensional free vibration and transient response of a cylindrical panel made of two directional functionally graded materials (2D-FGMs) based on three dimensional equations of elasticity and subjected to internal impact loading is considered. Material properties vary through both radial and axial directions continuously. The 3D graded finite element method (GFEM) based on Rayleigh-Ritz energy formulation and Newmark direct integration method has been applied to solve the equations in space and time domains. The fundamental normalized natural frequency, time history of displacements and stresses in three directions and velocity of radial stress wave propagation for various values of span angel of cylindrical panel and different power law exponents have been investigated. The present results show that using 2D-FGMs leads to a more flexible design than conventional 1D-FGMs. The GFEM solution have been compared with the results of an FG thick hollow cylinder and an FG curved panel, where a good agreement between them is observed.


INTRODUCTION
In recent years, the composition of several different materials is often used in structural components in order to optimize the responses of structures subjected to thermal and mechanical loads.Functionally graded materials (FGMs) are suitable to achieve this purpose.FGMs are composite materials, microscopically inhomogeneous, in which the mechanical properties vary smoothly and continuously from one surface to the other.This idea was used for the first time by Japanese researchers (Koizumi, 1993), which led to the concept of FGMs.
Cylindrical panels are of practical interest in many fields of engineering, including civil, mechanical, and aerospace, as they give rise to optimum conditions for dynamic behavior.The dynamic loadings on these structures can have serious consequences for their strength and safety.Many papers studying FGMs have been published recently.A critical litrature review of these works is presented by Birman and Byrd (2007) and Jha et al. (2013).Also, a wide range of researches has been carried out on dynamic analysis of FGM structures (Qian et al., 2004;Asemi et al., 2010;Akbarzadeh et al., 2011;Sun and Luo, 2011;Malekzadeh and Monajjemzadeh, 2013;Uymaz and Aydogdu, 2007) which mostly use the plate and shell theories.Among the FGM structures, little attention has been given to analysis of FGM cylindrical structures than those of rectangular and circular ones.For example, Shakeri et al. (2006) presented the vibration and radial wave propagation velocity in an FG thick hollow cylinder under dynamic loading by dividing the cylinder into some homogeneous sub-cylinders.Hosseini et al. (2007) studied the dynamic responses and natural frequencies of an infinite FG cylinder under internal pressure.Asgari et al. (2009) studied the dynamic behavior of a 2D-FG thick hollow cylinder with finite length under impact loading.Asemi et al. (2011) investigated the dynamic behavior and radial wave propagation of thick short length FG cylinders.Shao et al. (2004;2005) analytically obtained the elastic behavior of finite length FG hollow cylinders with simply supported end conditions under the hydrostatic pressures.In these publications, in order to satisfy the boundary conditions, the sinusoidal load along the axial direction can only be used.Bodaghi and Shakeri (2012) by using Hamilton's principle and first order shear deformation theory (FSDT) studied the free vibration and dynamic response of simply supported FG piezoelectric cylindrical panel impacted by timedependent blast pulses.Yas and Sobhani Aragh (2010) investigated the three dimensional steady state response of an FG fiber reinforced orthotropic cylindrical panel simply supported at the radial edges.Wang and Sudak (2008) presented a general method for the analysis of a thermoelastic multi-layered cylindrical panel made of an oblique pile of FG layers having orthotropic material properties.The two edges of the panel are simply-supported.Bahtui and Eslami (2007) studied the coupled thermoelastic response of an FG axisymmetric cylindrical shell.A second order shear deformation shell theory and Galerkin finite element formulation are used to formulate the problem.Davar et al. (2013) investigated the response of simply supported circular cylindrical shell made of FGM subjected to lateral impulse load.First order shear deformation theory and Love's first approximation theory are utilized in the equilibrium equations.Qu et al. (2013) studied vibrational response of FG shells of revolution with arbitrary boundary conditions.The formulation is derived by means of a modified variational principle in conjunction with a multisegment partitioning procedure on the basis of the first-order shear deformation shell theory.Ebrahimi and Najafizadeh (2014) analyzed the free vibration of a 2F-FG circular cylindrical shell.
Latin A m erican Journal of Solids and Structures 12 (2015) 205-225 The equations of motion are based on the Love's first approximation classical shell theory and the equations of motion and boundary conditions are discretized by the methods of generalized differential quadrature (GDQ) and generalized integral quadrature (GIQ).
The literature review denotes that most of the studies deal with axisymmetric FG cylindrical shells and they are based on the shear deformation theories.The application of these theories to thick structures can cause considerable errors.Therefore, for eliminating the lack of these studies, the 3D elasticity solutions not only provides realistic and accurate results but also allows further physical insights, which cannot otherwise be estimated by the other two dimensional or plate and shell theories.Also, in these studies, the material properties are assumed to have a smooth variation usually in one direction and majority of researches works into static analyses.Investigations into dynamic analysis of FG cylindrical structures are limited to one and two dimensional analyses.Also, analytical or semi-analytical solutions are available only through a number of problems with simple boundary conditions or for cylindrical panels with simply supported radial edges.Moreover, conventional functionally graded materials may also not be so effective in such design problems since all outer surface of the body will have the same composition distribution.Therefore, variation of volume fraction in two directions has a higher capability to reduce the mechanical, thermal and residual stresses and leads to a more flexible design than 1D-FGMs.Some studies have been carried out about static and dynamic analyses of structures made of 2D-FGMs (Asgari and Akhlaghi, 2010;Nemat-Allah, 2009;Zafarmand and Hassani, 2014).
Due to difficulty in obtaining analytical solutions for dynamic analysis of 2D-FG cylindrical panel based on 3D elasticity and for arbitrary boundary conditions, powerful numerical methods such as graded finite element method (GFEM) are needed.The graded finite elements, which incorporate the material property gradient at the size scale of the element, have been employed a generalized isoparametric formulation.Some works can be found in the literature on modeling nonhomogeneous structures by using GFEM (Kim and Paulino, 2009;Ashrafi et al., 2012;Zafarmand and Kadkhodayan, 2014).In these studies, it is shown that the conventional FEM formulation cause a discontinuous stress field in the direction perpendicular to the material property gradation, while the graded elements give a continuous and smooth variation.However, when the loading is parallel to the material gradation direction, the GFEM formulation estimates sharp rises in stress at the element boundaries while the conventional FEM formulation does not.
To the best knowledge of the authors, there are no studies available in the literature on dynamic analysis of 2D-FG cylindrical panel with clamped boundary conditions based on 3D elasticity and graded finite elements.The governing equations are derived based on principle of minimum potential energy and Rayleigh-Ritz method.To solve the time dependent equations, Newmark direct integration method with suitable time steps has been used.The effects of volume fraction exponents and different span angles on the time histories of displacements, stresses, velocities of radial stress wave propagation and fundamental normalized natural frequency have been investigated.

PROBLEM FORM ULATION 2.1 Volume fraction and material distribution in 2D -FG cylindrical panel
A functionally graded cylindrical panel with inner radius a , outer radius b , length L and span angle subjected to an internal impact loading is considered.The geometry and coordinate system of the cylindrical panel is shown in Figure 1.Material properties at each point can be obtained by using the linear rule of mixtures, therefore the material property ( R ) such as modulus of elasticity and mass density in the 2D-FG cylindrical panel is determined by linear combination of volume fractions and material properties of the basic materials as: Accordingly, the cylindrical panel is pure  1.It should be noted that Poisson's ratio is assumed to be a constant value of 0.3.This assumption is reasonable because of the small differences between the Poisson's ratios of basic materials.Table 1: Basic constituents of the 2D-FG cylindrical panel

Governing Equations
Neglecting body forces, the equations of motion in cylindrical coordinates are obtained as: where u, v and w are radial, circumferential and axial components of displacement, respectively and is the mass density that depends on r and z coordinates.
The stress-strain relations from the Hook's law in the matrix form are as: where the stress and strain components and the coefficients of elasticity D , are as the following relations: where denotes the Poisson's ratio and E is the Young's modulus of elasticity that depends on r and z coordinates.
The strain-displacement equations based on the theory of linear elasticity in cylindrical coordinate are: where U is the displacements vector and is a matrix containing partial differentiating equations as: The cylindrical panel is clamped on its two end edges and it is subjected to internal pressure that is a function of time.Therefore, the boundary conditions are as: The internal pressure function () pt is an arbitrary function that will be presented later.

Graded finite element modeling
In order to solve the governing equations the graded finite element method is used.In this method, in addition to displacement field, the heterogeneity of the material properties of the FGM may also be determined based on their nodal values.In this regard, shape functions similar to those of the displacement field may be used.This procedure can be used even for multidirectional functionally graded materials.Therefore, a graded finite element method (GFEM) is used to effectively trace smooth variations of the material properties at the element level.Using the graded elements for modeling of gradation of the material leads to more accurate results than dividing the solution domain into homogenous elements.The cylindrical panel is divided into a number of 8-node linear elements.For element () e , the displacements in three directions are approximated as follow: where is the matrix of linear shape functions in cylindrical coordinate and () e is the nodal displacement vector of the element that are as: To treat the material inhomogeneity by using the GFEM, it may be written: where () e  is the material property of the element.Substituting Eq. ( 16) in Eq. ( 12) gives the strain matrix of element () e as: where: Now by using Hamilton's principal and Rayleigh-Ritz energy formulation, the GFEM is imposed and finally the mass, stiffness and force matrices are obtained as following: where , T and W are potential energy, kinetic energy and virtual work done by surface tractions, respectively.These functions and their variations are: where V and A are the volume and area of the domain under consideration and P is the vector of surface traction and in the case of internal pressure is as: For each element, by imposing Eqs.(7, 16 and 20) into Eq.( 30), it can be achieved: is the variation of the nodal displacements and is arbitrary, it can be omitted from Eq. ( 31), thus this equation can be written as: MK F (32) where the characteristic matrices are defined as: As D and are not constant, for obtaining the characteristic matrices for each element, numerical integration should be applied.Now by assembling the element matrices and imposing the boundary conditions, the global equations of motion for the 2D-FG cylindrical panel can be written as: Once the finite element equations are determined, for free vibration analysis where the force matrix ( F ) is zero, by substituting it e ( is the natural frequency) into Eq.( 36), an eigen value problem would be obtained that can be solved using standard eigen value extraction procedures.Also, in the case of transient analysis, the Newmark direct integration method with suitable time steps is utilized to solve the equations.Newmark integration parameters are taken as (Zienkiewicz and Taylor, 2005): 1 2, 1 4 Newmark Newmark which lead to a constant average acceleration.This choice of parameters corresponds to a trapezoidal rule which is unconditionally stable in linear analyzes.

Validation
To validate the transient analysis of the current work, the data of an FG thick hollow cylinder under internal impact loading can be used (Asemi et al., 2011).Material distribution is assumed to be as 1D-FGM and it varies in radial direction with a power law function.In order to validate the results of this paper with the results of the present study, the span angel of the panel should  compared with (Asemi et al., 2011) Also, in order to validate the free vibration analysis of this study, the data of an FG curved panel can be used (Zahedinejad, 2010).The material properties are assumed to be varied from ceramic ( outer radius with a power law function.The boundary conditions are simply supported in four edges and the geometry is defined as: The comparison of fundamental frequency parameter ( () cc b a E ) with the published results for different power law exponents is made in Table 2.According to this table, a closed agreement between the present results and those obtained from Zahedinejad (2010)

Numerical Results
Consider a 2D-FG cylindrical panel with inner radius a = 0.2 m, outer radius b = 0.4 m and length of L = 0.5 m.The panel is clamped at 0, zL and subjected to an impact loading at its inner surface.Constituent materials are two distinct ceramics and two distinct metals described in Table 1.The Poisson's ratio is taken as a constant value of 0.3 and the loading function is assumed to be as:    The time history of radial, circumferential, shear ( r ) and axial stresses, respectively at 0.3 rm , 30 , 0.25 zm for a cylindrical panel with 3 after the unloading for different values of power law exponent are shown in Figures 9-12.It is obvious that variation of stresses is highly dependent of power law exponents.Also, as the same as circumferential displacement, the shear stress ( r ) cannot be neglected in asymmetric geometries.From the time history of the radial stress, it is possible to extract the approximate radial stress wave propagation velocity.Thus, by computing the time taken by the wave to cover the whole of the cylinder immediately after the loading, the approximate radial stress wave propagation velocities can be estimated.Table 3 shows the comparison of radial stress wave propagation velocity with analytical results (Jones, 1997) for cylindrical panels made of different homogenous materials.As it is obvious from this table, the obtained results have a good compatibility with analyti-     Results denote that the distribution of displacements and stresses and natural frequencies are strongly affected by the power law exponents and the span angle of cylindrical panel.As it can be seen from the results, the distribution of stresses has continuous variations due to using graded elements.The use of GFEM has several benefits over the use of conventional FEM in the dynamic and wave propagation analyses.In conventional FEM, the boundary of homogenous elements of an inhomogeneous material acts as discrete boundaries for the stress waves.These boundaries cause artificial wave reflections and have a cumulative effect on the magnitude and speed of propagating stress waves.In addition, these inaccuracies will be more significant in 2D-FGM problems.Therefore, by using graded finite element in which the material property is graded continuously through the elements, accuracy can be improved without refining the mesh size.

CONCLUSIONS
The main purpose of the present paper is to study the 3D transient and free vibrational behaviour of 2D-FG cylindrical panels.The 3D graded finite element method and Rayleigh-Ritz energy formulation is applied.The proposed method is validated by the result of a 1D-FG thick hollow cylinder under the impact loading and an FG curved panel which are extracted from published literature.The comparisons between the results show that the present method has a good agreement with the existing results.Different types of displacement and stress in three directions and fundamental normalized natural frequency are presented for various values of power law exponents and span angle of cylindrical panel.Also, velocity of radial stress wave propagation for different power law exponents is obtained.Results denote that the maximum stresses, stress distribution, natural frequency and velocities of radial stress wave can be controlled by the variation of material properties in two directions.Based on the achieved results, 2D-FGMs have powerful potentials for designing and optimizing structures under multi-functional requirements.

Figure 1 :
Figure 1: Geometry of the cylindrical panel

Figure
Figure 2: Volume fraction distribution of 1 m with L , respectively and the variation of desired material property depends on the volume fraction distributions of the constituents in which the volume fractions depends on r n and z n .For example, the variation of a material property such as modulus of elasticity based on the mentioned approach for the typical values of 3 rz nn is shown in Figure3.As it can be seen from this figure, the elastic modulus is equal to 1 L , respectively and varies in rz plane based on Eq. (5).The volume fractions in Eqs.(1-4) reduce to the conventional 1D-FGMs for 0 z n and in this case, the material properties vary only through the thickness direction.The basic constituents of the 2D-FG cylindrical panel are presented in Table

Figure
Figure 3: Distribution of elasticity modulus with and in the matrix form are as: the components of Φ are given in the appendix.LatinA m erican Journal ofSolids and Structures 12 (2015) 205-225 erican Journal of Solids and Structures 12 (2015) 205-225

be taken 2
to become a hollow cylinder, z n is taken zero, a = 1 m, b = 1.5 m, L = 1 m, data is shown in Figure4and a good agreement between these two results is observed.

Figure 4 :
Figure 4: Time history of the radial displacement at 1.25 rm Figure 5: Variation of internal pressure ( r p ) with time.

Figures 6
Figures6 and 7show the time history of radial displacement at

Figure 6 :Figure 7 :
Figure 6: Time history of the radial displacement at 0.3 rm , 30 , 0.25 zm with 3 for different values of

Figure 8 :
Figure 8: Time history of the circumferential displacement at 0.25 rm , 45 , 0.375 zm with 3 for different values of

Figure 9 :Figure 10 :Figure 11 :
Figure 9: Time history of the radial stress at 0.3 rm , 30 , 0.25 zm with 3 for different values of

Table 3 :
(Jones, 1997)ates the variation of radial stress wave propagation velocity at different heights of cylindrical panel for different values of power law exponents.This table denotes that velocity of radial stress wave is strongly affected by the power law exponents.Also due to the variation of material properties through the axial direction, the velocity of radial stress wave at different heights of the cylindrical panel is changing.Comparison of radial stress wave propagation velocity of homogenous cylindrical panel with analytical results(Jones, 1997)

Table 4 :
Velocity of radial stress wave propagation at different heights of cylindrical panel for different values of power law exponentsNow by introducing the normalized natural frequency as:

Table 5 :
Fundamental normalized natural frequency for different values of power law exponents with Latin A m erican Journal of Solids and Structures 12 (2015) 205-225

Table 6
illustrates the effect of span angle of the 2D-FG cylindrical panel on the fundumental normalized natural frequency with 2 It is obvious that by increasing the span angle of cylindrical panel, the fundamental natural frequency growths.
r n and 1 z n .

Table 6 :
Fundamental normalized natural frequency for different values of span angle with