NonlinearFree Vibration Response of Functionally Graded Materials Cylindrical Shell in Thermal Environment.

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NonlinearFree Vibration Response of Functionally Graded Materials Cylindrical Shell in Thermal Environment.

1P. D. Khaire, ²N. H. Ambhore, ³K. R. Jagtap

1,2 Department of Mechanical Engineering,Vishwakarma Institute of Information Technology, Kondhwa, Pune-48

3Department of Mechanical Engineering,Sinhgad Institute of Technology and Science, Narhe, Pune -41 E mail:¹[email protected], ²[email protected], ³[email protected]

Abstract- In this paper the nonlinear free vibration response of functionally graded materials (FGMs) cylindrical shell is investigated. The cylindrical shell is subjected to uniform temperature distribution with temperature independent (TID) material properties. The basic formulation is based on higher order shear deformation theory (HSDT) with Von-Karman nonlinear strain kinematics using modified C⁰ continuity. A direct iterative based nonlinear finite element method (DIFEM)developed by last two authors for the Functionally Graded Materials plate is extended for cylindrical shell. The present outlined approach has been validated with those available in literature.

Keywords: FGM, cylindrical shell, nonlinear free vibration, HSDT.

I. INTRODUCTION

Functionally Graded Material (FGM) is a composite, consisting of two or more phases, which is fabricated such that its composition varies in some spatial direction by changing the volume fraction index of constituent materials. This design is intended to take advantage of certain desirable features of each of the constituent materials. For example, if the FGM is to be used to separate regions of high and low temperature, thenat the hotter end it may consist of pure ceramic as the ceramic is having better resistance to higher temperatures. In contrast, the cooler end may be pure metal because of its better mechanical and heat transfer properties.Rapid advances in the manufacturing techniques of bulk FGMs have created exiting new possibilities of their applications in large scale structural system such as rocket heat shields, wear resistant lining in mineral processing industry, thermoelectric generators, plasma facing for nuclear reactors and electrically insulating metal/ceramic joints,thermoelectric generators, dental implantation, andbone replacement and electrically insulating metal/ceramic joints. A large number of literatures have been reportedon linear and nonlinear free vibration of plates. K. R. Jagtapet al. [1]investigated effect of random material properties on free vibration response of functionally graded materials plate with

cutouts in thermal environment by using higher order shear deformation theory with C⁰ continuity. K. R.

Jagtap et al. [2]presented stochastic nonlinear free vibration analysis of functionally graded material plate resting on elastic foundation in thermal environment by using higher order shear deformation theory with von Karman nonlinear strain kinematics with modified C⁰ continuity. Achchhe Lal et al. [3] investigated nonlinear bending response of laminated composite spherical shell panel with system randomness subjected to hygro- thermo-mechanical loading. A direct iterative based C⁰ nonlinear finite element method combined with mean centered first-order perturbation technique (FOPT) for the plate is extended for the spherical shell panel subjected to hygro-thermo-mechanical loading. Singh B.N. et al. [4] investigated composite laminate plate using finite element method. The formulation is based on higher order shear deformation theory. Randomness in the system properties are computed by using first order perturbation technique. Hiroyuki Matsunaga [5]studiedfree vibration and stability of functionally graded shallow shells according to a 2D higher-order deformation theory. Hui-Shen Shen and Hai Wang[6]investigated the large amplitude vibration behavior of a shear deformable FGM cylindrical panel resting on elastic foundations in thermal environments.

The formulation is based on a higher order shear deformation shell theory which includes shell panel- foundation interaction and the thermal effects. The material properties of FGMs are assumed to be temperature-dependent. The equations of motion are solved by a two-step perturbation technique to determine the nonlinear frequencies of the FGM cylindrical panel.

II. FORMULATION

Consider schematic diagram a FGMs cylindrical shell which consist of ceramic and metal at top and bottom layer of length a, width b, and total thickness h.

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Fig.1 Geometry of cylindrical shell

The properties of the FGMs shell are assumed to be varying through the thickness. The effective mechanical and thermal properties of the FGMs shell at an arbitrary point within the shell domain are expressed as[2],

     

   

b t b c

E z E E E V z

α z α α α V z

ρ z ρ ρ ρ V z

k z k [k k ]V z

  

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Where, t and b represents the ceramic and metal constituents, E,α,ρ and k are the effective young modulus, thermal expansion coefficient, density and thermal conductivity, VC is the volume fraction index, function of coordinate in the thickness direction (z),

 

^0.5 ^, ^,

2 2

n c

z h h

V z z

h

  

    

 

0n(2)

Where, n is volume fraction index and is always positive. For n=0, the shell is fully metal and when n=1, the composition of metal and ceramic is linear. The Poisson’s ratio  depends weakly on temperature change and is assumed tobe a constant.

2.1 Displacement field model

Higher order shear deformation theory with C⁰ continuity has been used to find displacement field model. Displacement field is given as [4]

1( ) _x 2( ) _x u u f z f z ,

1 2

(1 z) ( ) _y ( ) _y

v v f z f z

R  

    ,(3)

ww,

Where ( , , )u v w denote the displacement of a point along the (x, y, z) coordinates axe, (u, v, w) are corresponding displacements of a point on the mid plane,  and



are

respect to x and y axes, _x and _yare the slopes along x and y directions, θ =_x ^dw

dx andθ =_y ^dw dy . The function f₁(z) and f₂ (z) can be written as,

     

³

1 1 2

f z = C z - C z ; andf2

 

z = -C4

 

z ³ With

2

1 2 4

C = 1, C = C =4h . 3

The displacement vector for the modified model can be written as,

 

^{ }^{[u v w}y

xy

x]^T(4) 2.2 Strain Displacement Relations

The strain vector consisting of strains in terms of mid- plane deformation, rotations of normal and higher order terms associated with displacement for isotropic layer is,

     

^ ^ ^^l ^ ^^nl ^

 

^^t ⁽⁵⁾

Where

 

l ,

 

nl and

 

^^t are the linear and nonlinear strain vectors (Von-Karman sense), thermal strain vector respectively. The nonlinear strain vector can be written as,

1

 

[ ]

nl 2 Anl

   (6)

Where

, ,

0 1 0

2 0 0

0 0

x y

nl x y

A

w w w w

 

 

  

 

 

and ^,

, x y

w

 w 

 

 

 

The thermal strain vector

 

^^t is represented as,

 

1 2 12

0 0

x y t xy

yz zx

T

 

  



   

   

    

   

   

 

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Where₁, ₂and ₁₂are the coefficient of thermal expansion in the x, y and z directions respectively which can be obtained from the thermal coefficient in the longitudinal ₁ and transverse ₂directions of the ceramic and metal using the transformation matrix and

T is the uniform and nonuniform temperature change.

The temperature field for nonuniform temperature

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T=T z

 

T0(8)

Where ^{T z}

 

is expressed as,

 

b



t b

  

T z  T T T  z

Where ^{T z}

 

is the temperature distribution along z direction, T_t^andT_b, are temperature of top and bottom surface, and parameter ^

 

^z is defined as,

1

2 1 3 1

2 3

4 1 5 1

4 5

( ) 1 0.5 0.5

( 1)

0.5 0.5

(2 1) (3 1)

0.5 0.5

(2 1) (2 1)

n tb

b

n n

tb tb

b b

n n

tb tb

b b

k

z z

z c h n k h

k z k z

h h

n k n k

k z k z

h h

n k n k

 ^

 

   

       

   

         

    

          

With k_tb  k_t k_b and k is defined as thermal conductivity. The uniform temperature change Eq. can be written as,

( ) 0 ( _t _b) T z  T TT (9) Where,T₀is initial temperature.

2.3 Stress strain relation

The stress strain relation accounting thermal effect can be written as,

     

x y xy yz xz

Q or



  



 

 

  

 

  (10)

     

11 12

21 22

66 44

55

0 0 0

0 0 0 0

l nl t

Q Q

Q

  

 

 

   

 

 

Where

 

Qij ,

 

^ ^and

 

^ are the transformed stiffness matrix, stress and strain vectors for isotropic shell respectively.

2.4 Strain energy of the shell

The strain energy of the FGM shell is given by,

   

1 2

T v

U 



  dv⁽¹¹⁾

Above equation can be expanded as,

     

^{  }

 

^{   }

      

3

4

5

1 1

[ ] [ ]

2 2

1 [ ]

2

1 [ ]

2

T T

l l l

A A

T T

A l

T T

A

U D dA D A dA

D A dA

A D A dA

   

 

 

 

 

 



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Where [ ]D ,[D₃],[D₄]and [D₅]are shell stiffness matrices and

 

^^l is the linear mid-plane vector. The strain energy function calculated for each element above can be summed to get the total strain energy.

 

1

{ } [ ( )]

NE e e

T

l nl

U U

q K K q q



 



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Where [K_l^{], [}K_nl] and {q} are defined as global linear, nonlinear stiffness matrix and displacement vector respectively.

2.5 Work done

Because of uniform and nonuniform temperature change, pre-buckling stresses in FGM shell are generated the in-plane pre-buckling stress resultant per unit length are reason for buckling. The work done (W) by in-plane stress resultants in producing out of plane displacements ‘w’ can be expressed as,

2 2

, ,

2 2

, ,

1 [ ( ) ( )

2

2 ( ) ( ) ] 1

2

x x y y

A

xy x y

T

x x xy x

A y xy y y

W N w N w

N w w dA

w N N w

w N N w dA

 



     

   

     

   

     



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Where

N

_x,N_y and N_xy are thermal in plane, thermal compressive stress resultant per unit length.

Using finite element method and summing over the entire element above equation can be written as,

( ) ( ) ( ) ( )

1 1

{ } [ ]{ } { } [ ]{ }

NE NE

e e T e e T

T g T g

e e

W W  K  q K q

 









   ⁽¹⁵⁾

Where _T^{and [}Kg^]are defined as critical thermal buckling temperature and global geometric stiffness matrix.

2.6 Kinetic energy of FGM shell

The kinetic energy (T) of the vibrating FGM shell can be expressed as

( )^k{ } { }ˆ^T ˆ T



V u^ u dV^ ⁽¹⁶⁾

Where  and { }uˆ _{_u_ _v_ _w_} are the density and velocity vector of the shell respectively, above equation can be expressed as,

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( )

1 1

{ } [ ]{ } { }[ ]{ }

NE NE

e T

e e

T T m

q M q

 

   



 

^ ^

 

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Where, [M] is the global mass matrix.

III. EQUATION OF MOTION AND ITS SOLUTION

The governing equation for thermally induced nonlinear free vibration of the shell analysis can be derived using Lagrange’s equation of motion.

2

1

( ) 0

t

U W T dT





   ⁽¹⁸⁾

Substituting the values and obtaining in the form of nonlinear generalized eigenvalue problem as,

[ ]{ } [K q  M q]{ } 0  (19)

Where,[ ] {[K  K_l] [ K_nl( )]q _T[K_g]}

The above equation is nonlinear free vibration equation which can be solved as a linear eigenvalue problem assuming that the shell is vibrating in its principal made in each iteration, the above equation can be expressed as generalized eigenvalue problem as,

[[ ]K [M]]{ } 0q  ⁽²⁰⁾

Where   ² with  is natural frequency of the shell.

The nonlinear eigenvalue problem is solved by employing a direct iterative based C⁰nonlinear finite element method in conjunction with perturbation technique.

IV. RESULT AND DISCUSSION

A nine noded Lagrange isoparametric element with 63 DOFs per element for the present HSDT model has been used for discretizing, (4 × 4) mesh has been used for the study.The results are compared with those in literatures.

The dimensionless nonlinear fundamental frequencyof the FGMs cylindrical shell is,



b²^/h



m^/Em

   ^{, (21)}

Table 1The following temperature independent material properties are used

Types of material

E (N/m²)

α (1/C)

K (W/mK)

 (Kg m/ 3) ZrO2 151e+9 18.591e-6

204 2707

Ti-6Al-

4V 70e+9 6.941e-6 2.09 3000

Boundary Condition:

0, 0, ;

0 0,

y y

x x

v w at x a

u w at y b

 

    

4.1 Validation of fundamental frequency and parametric study

Table 2 shows the dimensionlessnonlinear fundamental frequency of FGM(Si3N4/SUS304)simply supported cylindrical shell in thermal environments with temperature independent material properties and a/b=b/R=1, a/h=20. Clearly, it is observed that the present results using C⁰ DIFEM are in good agreement with the available semi analytical method published results[6].

Table 2 validation of mean dimensionless fundamental frequency of cylindrical shell subjected uniform temperature distributionwith W_max/h=1, a/h=20.

T (K) n Present Shen H S [6]

Tc=400K Tm=400K

1 14.8713 15.9915 5 11.9025 12.8445 Table 3 showsthe effect of temperature change (∆T), thickness ratio, volume fraction index (n), amplitude ratio(W_max/h)and uniform temperature distribution on the dimensionless nonlinear fundamental frequency of SSSS supported FGM(ZrO2/ Ti-6Al-4V) cylindrical shell with R/a=10, ∆T=30K.

Table 3The effect of temperature change (∆T), thickness ratio, volume fraction index (n), amplitude ratio(Wmax/h) and uniform temperature distribution on the dimensionless nonlinear fundamental frequency of SSSS supported FGM (ZrO2/ Ti-6Al-4V) cylindrical shell with R/a=10, ∆T=30K.

a/h n W_max/h



_nl

10

0

0.3 6.8151

0.6 8.1318

0.9 9.8770

1 10.4496

l 6.2423

1

0.3 5.5676

0.6 6.6761

0.9 8.1393

1 8.6132

l 5.0831

20

0

0.3 6.8300

0.6 8.2435

0.9 10.0708

1 10.7317

l 6.1694

1

0.3 5.5736

0.6 6.7692

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1 8.8617

l 5.0113

V. CONCLUSION

A C⁰nonlinear finite element method based on direct iterative procedure [DIFEM] is used to compute the nonlinear fundamental frequency simply supported FGM cylindrical shell in thermal environment. Higher order shear deformation theory with von-Karman nonlinearity with modified C⁰ continuity is used for basic formulation. The nonlinear fundamental frequency of vibration increases with increase in the amplitude ratio. It is also observed that for same thickness ratio and the amplitude ratio, the fundamental frequency of vibration increases. It is also observed that for same thickness ratio and amplitude ratio, the nonlinear natural frequency decreases with increase in volume fraction index because of decrease in the stiffness.

REFERENCES

[1] Jagtap K. R., Achchhe Lal,Effect of random material properties on free vibration response of functionally graded materials plate with cutouts in thermal environment. International Conference on Modern Trends in Industrial Engineering, 2011.

[2] Jagtap K. R., Achchhe Lal, Singh B. N., stochastic nonlinear free vibration analysis of elastically supported functionally graded materials plate with system randomness in thermal environment , composite structures 2011;93:3185-3199.

[3] Achchhe Lal, Singh B. N., Soham Anand, Nonlinear bending response of laminated composite spherical shell panel with system randomness subjected to hygro-thermo- mechanical loading, International Journal of Mechanical sciences 2011;53:855-866.

[4] Singh B. N., Yadav D., Iyengar N.G.R., AC⁰ element for free vibrationof composite plates withuncertain material properties, advanced composite material 2003;11:331-350.

[5] Hiroyuki Matsunaga, free vibration and stability of functionally graded shallow shells according to a 2D higher-order deformation theory, composite structures 2008; 84:132-146.

[6] Hui-Shen Shen, Hai Wang, Nonlinear vibration of shear deformable FGM cylindrical panels restingon elastic foundations in thermal environments, Composites 2014; 60:167–177.

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