Nonlinear thermoelasticity, vibration, and stress wave propagation analyses of thick FGM cylinders with temperature-dependent material properties

European Journal of Mechanics A/Solids 29 (2010) 378–391

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Nonlinear thermoelasticity, vibration, and stress wave propagation analyses of thick FGM cylinders with temperature-dependent material properties M. Shariyat*, M. Khaghani, S.M.H. Lavasani Faculty of Mechanical Engineering, K.N. Toosi University of Technology, Tehran, Iran

a r t i c l e i n f o

a b s t r a c t

Article history: Received 13 September 2008 Accepted 21 October 2009 Available online 28 October 2009

In the present paper, nonlinear thermoelasticity, vibration, and stress wave propagation analyses of thick-walled cylinders made of functionally graded materials with temperature-dependent properties are performed. In contrast to researches accomplished so far, a third-order Hermitian finite element formulation is employed to guarantee both radial displacement and normal stress continuities, improve the accuracy, and prevent virtual wave source formations at the mutual boundaries of the elements. Stress wave propagation, reflection, and interference under impulsive mechanical loads in thermal environments are also studied. In contrast to the common procedure, the cylinder is not divided into isotropic sub-cylinders. Therefore, artificial wave reflections from the hard interfaces are avoided. Time variations of the temperatures, displacements, and stresses due to the dynamic loads are determined by solving the resulted highly nonlinear governing equations using an updating iterative time integration scheme and over-relaxation and under-relaxation techniques. A comprehensive sensitivity analysis includes effects of the volume fraction indices, dimensions, and temperature-dependency of the material properties is performed. Results reveal the significant effect of the temperature-dependency of the material properties on the transient stress distribution and elastic wave propagation and reflection phenomena. Interesting phenomena are noticed; among them the oblique wave formations during the wave propagation. Since examples of the present field are rare in literature, the extracted results may serve as reference results for future comparisons. Ó 2009 Elsevier Masson SAS. All rights reserved.

Keywords: Nonlinear analysis Thermal stresses Wave propagation Functionally graded material Temperature-dependency Thick-walled cylinder

1. Introduction Functionally graded materials (FGMs) are mainly employed to withstand elevated temperatures and severe thermal gradients. Low thermal conductivity, low coefficients of thermal expansion, core ductility, and smooth stress distribution have enabled the functionally graded materials to withstand higher thermal and mechanical shocks (Noda, 1991; Tanigawa, 1995). Among the FGM structures, the cylindrical FGM components have been of special interest. Numerous researchers have investigated thick FGM cylinders subjected to uniform heating (Zimmerman and Lutz, 1999; Chen and Ye, 2002) or steady state heat transfer conditions (Obata and Noda, 1994; El-abbasi and Meguid, 2000; Jabbari et al., 2002, 2003; Liew et al., 2003; Shen, 2004; Argeso and Eraslan, 2008). Some researchers have simplified the study by dividing the FGM cylinder into isotropic sub-cylinders (Liew et al., 2003).

* Corresponding author. Tel.: þ98 9122727199; fax: þ98 21 88674748. E-mail addresses: [email protected], [email protected] (M. Shariyat). 0997-7538/$ – see front matter Ó 2009 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.euromechsol.2009.10.007

Researches presented so far in the transient heat transfer analysis of the FGM cylinders are very limited and almost all of them have ignored the temperature-dependency of the material properties. Reddy and Chin (1998) and Praveen et al. (1999) have developed Lagrangian finite element formulations to study the pseudo-dynamic thermoelastic responses of the functionally graded cylinders. Obata et al. (1999) analyzed the two-dimensional unsteady thermal stresses in the FGM hollow circular cylinders by the Laplace transformation method and the perturbation technique. Using the finite difference method, Awaji and Sivakumar (2001) analyzed steady state and transient temperature responses of an FGM cylinder. Based on the multi-layered cylinder approximation, Kim and Noda (2002) studied the axisymmetric twodimensional transient thermoelasticity of the infinite hollow FGM circular cylinders using Green’s function approach. Using the numerical Laplace inversion method, Sladek et al. (2003) studied the transient heat conduction in the FGM cylinders. Wang et al. (2004) and Wang and Mai (2005) used the first-order finite element method for the spatial and the finite difference method for the time variables to study the one dimensional transient heat conduction. Hosseini et al. (2007a), and Shao et al. (2007) employed

M. Shariyat et al. / European Journal of Mechanics A/Solids 29 (2010) 378–391

analytical methods to study the transient conduction heat transfer in FGM cylinders with material properties that follow an exponential law. Recently, thermomechanical analysis of an FGM hollow circular cylinder subjected to a linearly increasing boundary temperature is developed by Shao and Ma (2008). Thermomechanical properties of the functionally graded materials were assumed to be temperature-independent. The Laplace transform technique and a series method were employed to solve ordinary differential equations of FGM cylinders with material properties that follow an exponential law. Santos et al. (2009) carried out a thermoelastic analysis for functionally graded cylindrical shells subjected to transient thermal shocks using a semi-analytical axisymmetric finite element model based on the three-dimensional linear elasticity. The material properties were assumed temperature-independent. Poultangari et al. (2008) and Tokovy and Ma (2008) have studied thermoelastic behaviors of nonaxisymmetric FGM cylinders subjected to steady state heat transfer by analytical methods, ignoring the temperature-dependency of the material properties. Almost in all of the above-mentioned researches, the temperature-dependency of the material properties was neglected. Recently transient thermal analyses taking the temperaturedependency of the material properties into account are introduced by the author (Shariyat, 2008a,b; Shariyat, 2009a). Iterative time integration and updating methods were used. Heyliger and Jilania (1992) adopted a variational method and a Ritz approximation to study the frequency responses of the inhomogeneous cylinders and spheres. Steinberg (1995) formulated the inverse spectral problem to determine properties of a cylinder with inhomogeneous materials. Han et al. (2001) presented a hybrid numerical method for analysis of transient waves in an FGM cylinder. The displacement responses were determined by employing the Fourier transformations together with the modal analysis. El-Raheb (2005) studied effects of the circumferential and the radial inhomogeneities on the transient waves of a plane-strain hollow cylinder. The cylinder was divided into isotropic subcylinders. The static-dynamic superposition method was employed to determine the transient response. Shakeri et al. (2006) and Hosseini et al. (2007b) studied vibration and transient behaviors of FGM thick hollow cylinders subjected to axisymmetric dynamic loads using a first-order Galerkin finite element method. The functionally graded cylinder was divided into isotropic sub-cylinders. In each interface between two layers, stress and displacement continuity conditions were satisfied through altering the generalized Hook’s law. Ponnusamy (2007) studied the wave propagation in a generalized thermoelastic isotropic solid cylinder with arbitrary cross-section. Almost all of the finite element formulations developed so far for thermoelasticity and stress wave propagation analyses, were presented based on the Lagrangian shape functions (Praveen et al., 1999; He et al., 2002; Wang et al., 2004; Wang and Mai, 2005; Tian et al., 2006; Shakeri et al., 2006; Hosseini et al., 2007b; Bagri and Eslami, 2007a; Santos, 2009). Since the mentioned shape functions only guarantee a C0 continuity for the displacement components, the radial stress and strain components experience jumps at the mutual boundaries of the elements especially when first-order elements are employed (Shariyat and Eslami, 1996; Shariyat, 2009a; Zienkiewicz and Taylor, 2005; Reddy, 2005). Therefore, virtual sources of wave propagation and reflection will form at the mutual boundaries of the elements that in turn may lead to unreliable results. Although the idea of dividing the FGM cylinder into some sub-cylinders has been commonly used, it may induce successive reflections and may affect the results. Bruck (2000) and Samadhiya et al. (2006) have proved that in FGM components with continuously varying properties, the pulse

379

shape is distorted with time, the wave speed is not constant, and there is no sharp interface that would cause wave reflection. The elastic wave propagation phenomenon in discretely layered FGMs is somewhat different. The transmitted wave and the reflected waves (perhaps there will be multiple reflections) from each sharp interface between the discrete layers, may affect the stress distribution. In the present paper, nonlinear transient thermoelasticity, vibration, and wave propagation analyses of functionally graded thick cylinders with temperature-dependent material properties subjected to dynamic thermomechanical loading conditions are presented. Elastic wave propagation, reflection, and interference in thermal environments are also studied. Since impulsive thermal loads are not applied in the present research, a coupled thermoelasticity analysis (Bagri and Eslami, 2007b) is not employed. A third-order Hermitian finite element formulation is developed to improve the accuracy of the results and to ensure that both radial displacement and stress components are continuous at the mutual boundaries of the elements. An iterative updating time integration solution procedure is proposed to extract the results from the highly nonlinear governing equations. Over-relaxation and underrelaxation techniques are used to accelerate the solution procedure for systems with gradual time variations and enhance the numerical stability for sensitive systems, respectively. Finally, some displacement and stress results based on the assumptions of temperature-dependency (TD) and temperature-independency (TID) of the material properties are derived and effects of various parameters on the thermoelasticity, vibration, and wave propagation characteristics are studied. 2. The governing equations The geometric as well as the thermomechanical boundary parameters of the thick-walled FGM cylinder are shown in Fig. 1. It is assumed that the FGM cylinder is constructed from a mixture of two constituent materials so that one of the boundary layers is metal-rich whereas the other boundary layer is ceramic-rich. If the representative material property (e.g. the elasticity modulus) of the inner layer is denoted by Pi while the representative material property of the outer surface is denoted by Po, the material properties of the FGM cylinder at any arbitrary point through the thickness may be expressed as:

  ro  r n P ¼ Po þ ðPi  Po Þ ro  ri

(1)

Fig. 1. Geometric and thermomechanical boundary parameters of the considered thick-walled FGM cylinder.

380

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where n is the so-called volume fraction index. Variation of the material properties in terms of the temperature may be expressed as follows (Touloukian, 1976):

  P ¼ P0 1 þ P1 T þ P2 T 2 þ P3 T 3

(2)

where P0,P1,P2 and P3 are some material constants. Combining Eqs. (1) and (2) leads to the following result:

boundary conditions. Based on the governing equation (5) and its natural (Neumann) boundary conditions (Eq. (6)), it may be easily verified that the associated functional may be written as:

#  2 Z " Z 1 vT h _ P ¼  þrrCv T T dU  kr ðT  TN Þ2 :rdG 2 vr 2 







P ¼ P0i 1 þ P1i T þ P2i T 2 þ P3i T 3 Vi þ P0o 1 þ P1o T þ P2o T 2  þ P3o T 3 Vo

Vi ¼

ro  r ro  ri

(3)

Vo ¼ 1  Vi

(4)

(5)

where r(r,T),k(r,T), and Cv(r,T) are the mass density, the thermal conductivity, and the specific heat, respectively. In the present formulation, various thermal boundary conditions are considered. For example, for an FGM cylinder with a specified heat flux (e.g. due to passing of hot gasses) at the innermost layer and a convection heat transfer at the outermost layer one has:

8 > > vT > < k þ hðT  TN Þ ¼ 0 at r ¼ ro vr > > kvT ¼ qo at r ¼ ri > : vr

(6)

It is assumed that all particles of the cylinder are initially at the ambient temperature:

Tðr; 0Þ ¼ T0

(7)

Using a Kantorovich-type approximation, variations of the temperature may be interpreted as:

Tðr; tÞ ¼ NðrÞTðtÞðeÞ

(8)

T(t)(e) is the nodal temperatures vector and N is the shape function matrix. The following relation exists between derivatives of the shape function matrix in the global and natural coordinates

N;r ¼ N;x x;r ¼ N;x

2nðeÞ 2 ¼ N;x Dr ro  ri

(10)

NT RdU ¼ 0

(11)

U

The governing equation of the transient axisymmetric heat transfer problem is

" # 1 vT vðkrÞ v2 T vT þ kr 2  rCv ¼ 0 r vr vr vt vr

r:q0 TdG

where U, Gi and Go are the solution domain, the internal boundary, and the external boundary Among the weighted residual methods, the Galerkin method is the most powerful one. Galerkin-type governing equations of the element may be derived from the following equation:

Z

n ;

Z

Gi

where Vi and Vo are the volume fractions of the constituent materials of the inner and outer boundary layers, respectively:



G0

U

(9)

x is the natural coordinate, n(e) is the number of the elements, Dr is the element length of the equally discretized thickness of the FGM cylinder, and the symbol ‘,’ stands for the partial derivative. It is evident that Eq. (5) cannot be solved by direct substitution of the approximate solution appeared in Eq. (8). This substitution leads to a single equation with many unknowns. In the finite element procedure, the suitable governing equation of the temperature distribution may be obtained through special techniques such as the semi-inverse or the weighted residual methods. In the semi-inverse methods, the governing equations of the element are determined based on extremizing the relevant functional of the problem and its

where R is the first side of the differential governing equation (Eq. (5)). For an arbitrary nodal temperature increment vector (as the variational methods), ðdTðr; tÞ ¼ NðrÞdTðtÞðeÞ Þ, the equivalent form of Eq. (11) may be written as:

Z

dT T RdU ¼ 00

U

Z

dTðeÞT NT RdU ¼ 0

(12)

U

Since dT(e)T is an arbitrary vector, Eq. (12) leads to Eq. (11) which is the standard form of the Galerkin Method. Since the governing equation of the displacements of the element are developed based on the principle of virtual work whose final form is similar to Eq. (12) with the exception of the variable nature (increment of the displacement vector should be used instead of increment of the temperature in the principle of virtual work), Eq. (11) is used to derive the governing equations of the temperature distribution of the element. However, since Eq. (5) is a second-order equation, results of Eqs. (10) and (11) (the weak formulation) are identical. According to Eqs. (5) and (9) and the following equation

r ¼ ri þ ½2ðeÞ þ x  1

Dr

(13)

2

where e is the element number and ri is the radius corresponding to the internal layer of the element, one has

h i ðeÞ k R ¼ rCv NT_  1r N;r vðvrrÞ þ krN;rr TðeÞ  2 # h 2 vðkrÞ 2 ðeÞ 1 _ TðeÞ ¼ rCv NT  r þ½2ðeÞþx1Dr N;x þ krN;xx i Dr vr Dr 2

(14)

Therefore, the thermal governing equation of the element takes the following form:

CðeÞ T_

ðeÞ

þ KðeÞ TðeÞ ¼ qðeÞ

where the damping matrix, C

CðeÞ ¼

Z

rCv NT NdU

(15) (e)

is

(16)

U

Matrices K and q have different elements for different boundary conditions. For an FGM cylinder with a specified heat flux at the innermost layer and a convection heat transfer at the outermost layer (Eq. (6)) one may write

M. Shariyat et al. / European Journal of Mechanics A/Solids 29 (2010) 378–391

KðeÞ

Z

Z   1 vðkrÞ NT;r k þ NT k;r N;r dU ¼  NT N;r dU þ r vr U U Z T G þ N hNd o

qðeÞ ¼

Go

Z

Z

vT NT k n r d G ¼ vr

G

NT qo dGi þ

Gi

Z

The governing equations of the transient thermoelasticity of the FGM cylinder may be derived based on the principle of virtual work as follows

ð17Þ

K

Z

Go

Z   vðkrÞ NT;r k þ NT k;r N;r dU ¼  N N;r dU þ r vr U U Z Z T þ N hNdGo þ NT kN;r dGi

qðeÞ ¼

Z

T1

Go

vT NT k n r d G ¼ vr

G

ð18Þ

Gi

Z

NT hTN dGo

¼ NðrÞTðtÞ

TðtÞ

ðeÞT

NðrÞ NðrÞTðtÞ

duT po dG 

Z

duT pi dG

Z

(27)

U

dUðeÞT HT pi dG ¼ 0

As before, symbols U and G denote the solution domain and the boundary, respectively. po and pi are the external and the internal pressures, respectively. Since dU(e)T is an arbitrary vector, one may conclude from Eq. (27) that the governing equation of the displacements of the element has the following form

b ðeÞ UðeÞ ¼ f ðeÞ þK

(28)

where

MðeÞ ¼

Z

r½TðtÞ; xHT HdU

(29)

BT D½TðtÞ; xBdU

(30)

U

b ðeÞ ¼ K

Z



v vr 1 r



(21)

and H and U(e) are the Hermitian shape functions matrix and the nodal displacements vector, respectively:



1 1 1 2 2 2 ð1  xÞ ð2 þ xÞ ð1  xÞ ð1 þ xÞ ð1 þ xÞ 4 4 4  1 2 ð2  xÞ ð1 þ xÞ ðx  1Þ (22) 4

H ¼ ½H1 H1 H2 H2  ¼

U10

U2

U20

Z

HT pi dG 

Gi

HT po dG þ

Go

Z

BT D½TðtÞ; x3T dU

(31)

U

Typically, the domain integrals may be calculated in the following manner

Z

3T ¼ h 3r 3q i; d ¼

Z

(20)

where

U1

Z

Gi

f ðeÞ ¼

ðeÞ

3 ¼ du ¼ dHUðeÞ ¼ BUðeÞ

UðeÞT ¼

Go

(19)

T

The strain vector may be expressed in terms of the radial displacement (u) as



rduT u€ dU þ

U

T

ðeÞ

Z

dUðeÞT HT po dG 

ðtÞ

T 2 ¼ TðtÞðeÞ NðrÞT NðrÞTðtÞðeÞ T3

d3T sdU þ

ðeÞ

where boundaries Gi and Go are the inner and the outer surfaces of the cylinder, respectively (Fig. 1). Integrals appeared in Eqs. (17) and (18) may be calculated numerically (using Gauss–Legendre method) considering Eqs. (9) and (13). The material properties and their derivatives may be calculated using the following identities:

T ¼ NðrÞT

þ

ZU

€ MðeÞ U

Go

ðeÞ

Z

dP ¼

GZ Gi U o Z U   ðeÞT T ðeÞ ðeÞT T € ðeÞ ¼ dU B D BU  3T dU þ rdU H HU dU

NT hTN dGo

and for a specified temperature at the innermost layer and a convection heat transfer at the outermost layer, one has ðeÞ

381



(23)

Lðr; xÞdU ¼

Z

2prLðr; xÞdr ¼ 2p

U

Z1

rLðr; xÞ

Dr dx Dx

1

¼ p

Z1

rLðr; xÞDrdx

(32)

1

If the matrices of the elements are assembled, the system of the governing equations of thermoelasticity of the whole FGM cylinder is obtained as follows:

_ C½TðtÞTðtÞ þ K½TðtÞTðtÞ ¼ Q ðtÞ

(33)

€ þ K½TðtÞU b M½TðtÞU ¼ FðtÞ

(34)

Therefore

2

H1;r B ¼ 4 H1 r

H1;r H1 r

H2;r H2 r

3. The proposed numerical solution procedure

3 H2;r H2 5 r

(24)

The stress vector may be determined as follows

s ¼ Dð3  3T Þ;

sT ¼ h sr sq i

(25)

where the elasticity coefficients matrix D and the thermal strain vector 3T may be expressed as

3T ¼ h aDT aDT i;

D ¼



EðrÞ 1  nðrÞ

2

1 nðrÞ

nðrÞ 1



(26)

In a thermoelastic analysis, usually a two step solution procedure is employed where in the first stage, a thermal analysis is performed and in the second stage, a stress analysis is accomplished based on the results obtained in the first stage. A Crank–Nicolson time integration method in conjunction with a Picard-type iterative scheme (Reddy, 2005) is employed to solve the resulted highly nonlinear heat transfer equations. Based on Eq. (33), one may write:

Ci T_ i þ Ki Ti ¼ Q i ;

Ciþ1 T_ iþ1 þ Kiþ1 Tiþ1 ¼ Q iþ1

(35)

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Table 1 Material properties of the constituent materials of the FGM cylinder (Reddy and Chin, 1998). Material

Property

P0

P1

P2

P3

Si3N4

E (Pa)

348.43e9 0.24 5.8723e-6 13.723 2370 555.11

3.70e-4 0 9.095e-4 0 0 1.016e-3

2.160e-7 0 0 0 0 2.92e-7

8.946e-11 0 0 0 0 1.67e-10

201.04e9 0.3262 12.330e-6 15.379 8166 496.56

3.079e-4 2.002e-4 8.086e-4 0 0 1.151e-3

6.534e-7 3.797e-7 0 0 0 1.636e-6

0 0 0 0 0 5.863e-10

n a(K1) k(W/mK) r(kg/m3) Cv(J/kg K) SUS304

E (Pa)

n a(K1) k(W/mK) r(kg/m3) Cv(J/kg K)

2  € _ þ ðDtÞ ð1  b ÞU € þb U Uiþ1 ¼ Ui þ Dt U 1 i  1 iþ1 2 i _ € _ € U iþ1 ¼ Ui þ Dt ð1  b2 ÞUi þ b2 Uiþ1

Tiþ1 ¼ Ti þ

2

Ti þ T_ iþ1



b1 ¼ b2 ¼ 0:5

(36)

Therefore, combining Eqs. (35) and (36) leads to the following equation:

 Tiþ1 ¼ K1 iþ1 Ki T i þ Q i;iþ1

(41)

where

where i is the time step counter. According to the Crank–Nicolson time integration scheme (Bathe, 2007) for the first-order systems, one has:

Dt _

vector may be used as a first estimation for the nodal temperature vector of the end of the next time step. According to the second-order Runge–Kutta time integration method for the second-order systems, the displacement and the velocity vectors of the end of the ith time step may be determined through the following equations (Bathe, 2007):

(42)

To incorporate effects of the higher natural frequencies in the response, it is recommended to choose the integration time step based on the following criterion (Gerald and Wheatley, 2003; Bathe, 2007):

Dt 

s 20

(43)

where s is the fundamental natural period time of the FGM cylinder. Therefore, based on Eq. (41), one may write;

(37)

where

Kiþ1 ¼ Ciþ1 þ aDtKiþ1 ;

Ki ¼ Ciþ1 ½I  ð1  aÞDtC1 i Ki 

Q i;iþ1 ¼ Dt½ð1  aÞCiþ1 C1 i Q i þ aQ iþ1 ;

a ¼ 0:5

ð38Þ

The iterative solution begins with an estimated nodal temperature victor, T. Values of the material properties are determined based on this vector. After each iteration, a modified nodal temperature victor is obtained and is used to update the material properties that have to be used in the next iteration. So that Eq. (37) takes the following form in the iterative solution:

Tiþ1;k ¼



Kiþ1 ðTiþ1 Þk1

1

 Ki Ti þ Q i;iþ1

(39)

where k is the iteration counter. The iterative solution is continued till convergence is achieved. In this regard, a convergence criterion like the following one may be used (Shariyat, 2009b, 2009c, 2010):

jTl;iþ1  Tl;i j max d jTl;iþ1 j

(40)

where Tl is a representative nodal temperature and d is a sufficiently small number (e.g. 0.001). The obtained nodal temperature

Table 2 The fundamental natural frequencies of the considered cylinders. Cylinder type

Metallic Ceramic FGM (metallic internal layer)

Fundamental natural frequency (Hz)

5949

n ¼ 0.5 n ¼ 1 n ¼ 5 13,150

6252

FGM (Ceramic internal layer) n ¼ 0.5 n ¼ 1 n ¼ 5

8106 10,841 11,142 8159 6631

Fig. 2. A comparison among the: (a) temperature distributions, and (b) hoop stresses of FGM cylinders with temperature-dependent (TD) and temperature-independent (TID) material properties.

M. Shariyat et al. / European Journal of Mechanics A/Solids 29 (2010) 378–391

)1 ( ðDtÞ2 _ b K Miþ1 Ui þ Dt U ¼ Miþ1 þ i 2 1 iþ1 ! ) 2 ðDtÞ2 € þ ðDtÞ F þ ð1  b1 ÞU i 2 2 iþ1 (

Uiþ1

€ U iþ1 ¼

"

2

b1 ðDtÞ

2

2

_  ðDtÞ ð1  b ÞU € Uiþ1  Ui  Dt U 1 i i 2

_ _ € € U iþ1 ¼ Ui þ Dtð1  b2 ÞUi þ b2 Dt Uiþ1

U0 ¼ U0 ;

(44)

_ ¼ V ; U 0 0

(45)

(46)

Eqs. (44)–(46) should be solved based on the prescribed initial conditions:

(47)

Although the displacement boundary conditions may be imposed directly, incorporation of the stress boundary conditions:

½sr ðtÞr¼ri ¼ pi ðtÞ;

#

€ ¼ M1 ðF  K U Þ U 0 0 0 0 0

383

½sr ðtÞr¼ro ¼ po ðtÞ

(48)

requires an additional effort. A stress boundary condition may be incorporated in two ways: (i) expressing the radial stresses appeared in Eq. (48) in terms of the nodal displacement vector using Eqs. (25), (26) and (20) and replacing the row of the corresponding node of the global matrices by the resulted equation; (ii) using the penalty method (Chandrupatla and Belegundu, 2002) to incorporate the mentioned resulted equation.

Fig. 3. Time variations of the: (a) temperatures, (b) radial displacements, (c) radial stresses, (d) hoop stresses, and (e) axial stresses, of all points of the thickness of the FGM cylinder for n ¼ 1.

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To accelerate the numerical solution to reduce the total computational time or to prevent numerical instability occurrence for sensitive systems, one may choose the following modified solution after each iteration within a specified integration time interval:

Tiþ1 ¼ Ti þ 2ðTiþ1  Ti Þ

(49)

where 2 is less than unity for the first purpose and is greater than unity for the second purpose.

4. Results and discussions In the present section, some examples relevant to transient thermoelasticity, free and forced vibrations, and wave propagation analyses of thick-walled FGM cylinders subjected to thermomechanical loads are presented. With the exception of example 1, that is a validation example, in the all remaining examples, thick cylinders with the following geometric specifications:

Fig. 4. Effect of the temperature-dependency of the material properties on time variations of the: (a) temperatures, (b) radial displacements, (c) radial stresses, (d) hoop stresses, and (e) axial stresses of the middle point of the thickness of the FGM cylinder, for n ¼ 1.

M. Shariyat et al. / European Journal of Mechanics A/Solids 29 (2010) 378–391

Fig. 5. Time variations of the: (a) temperatures and (b) radial displacements of the inner, middle, and outer points of the thickness of the FGM cylinder. In these figures, curves shown with dotted, solid, and dashed lines correspon to n ¼ 0.1, n ¼ 1, and n ¼ 5, respectively.

ri ¼ 0:1ðmÞ;

ro ¼ 0:2ðmÞ

and material properties given in Table 1 are investigated for the following four distinct types: Type Type Type Type

1: 2: 3: 4:

a pure metallic cylinder, a pure ceramic cylinder, an FGM cylinder with metallic internal layer, an FGM cylinder with ceramic internal layer.

As stated before, the FGM cylinder is assumed to be made of two constituent materials or more precisely, it is a Si3N4/SUS304 FGM cylinder. The time integration step is chosen based on Eq. (43). The fundamental natural frequencies are computed base on the transient behavior of the cylinder, as it is explained in example 4, and given in Table 2. Based on the material properties given in Table 1, it is evident that a cylinder with a higher ceramic volume fraction has a higher rigidity and subsequently has a greater fundamental natural frequency. Number of the elements is chosen based on a sensitivity analysis. Example 1: To validate results of the present approach, an FGM hollow cylinder previously studied by Wang and Mai (2005), whose

385

inner and outer radii are 50 and 150 (mm), respectively is reexamined. The material phases inside the FGM cylinder change from pure ZrO2 at the outermost layer linearly to Ti–6Al–4V at the innermost layer. The inner surface of the cylinder is suddenly heated to T i¼ 1000(K); which is maintained thereafter. Temperature of the outer surface of the cylinder is kept zero. The material properties of the ZrO2 and Ti–6Al–4V materials are chosen similar to those used by Wang and Mai (2005). In Fig. 2(a), the temperature distributions predicted by Wang and Mai (2005) are compared with the present distributions, for both temperature-dependent (TD) and temperature-independent (TID) material properties. According to Fig. 2(a), there is a good agreement between the results. The greater discrepancies are noticed for FGM cylinders with temperature-dependent materials. Results of Wang and Mai (2005) were derived based on linear elements. Iterative or updating procedures were not used by Wang and Mai (2005). Although the results show a good agreement, using linear elements to predict nonlinear variations (especially in a stress analysis) may lead to abrupt changes in the derivatives of the nodal values at the boundaries of the elements and may lead to remarkable errors in the results (Zienkiewicz and Taylor, 2005; Reddy, 2005; Shariyat, 2009a). However, the greater discrepancies are noticed in the earlier times of the heat transfer where the transient effects are more pronounced. For FGM cylinders with temperature-dependent material properties, the present temperature distribution curves have inflection points. Besides, it is evident that the influence of the temperature-dependency of the material properties grows with the temperature. The transient hoop stress distributions are shown in Fig. 2(b). Results of Wang and Mai (2005) were derived based on static governing equations of the isotropic cylinders. Therefore, the inertia effects were ignored. It is known that results obtained through modeling the nonlinear systems by means of Lagrangian elements may have noticeable errors in the neighborhoods of the boundaries (Zienkiewicz and Taylor, 2005; Reddy, 2005) as it may be seen in Fig. 2(b). Furthermore, in contrast to the present Hermitian finite element formulations, Lagrangian finite element formulations increase the compliance of the system (Zienkiewicz and Taylor, 2005; Reddy, 2004, 2007). For this reason, magnitudes of the stress values obtained by Wang and Mai (2005) are somewhat lower. Example 2: To further evaluate the influence of the temperaturedependency of the material properties on the thermoelastic behavior of the thick FGM cylinders, a type 4 cylinder subjected to a convection heat transfer with a hot fluid with temperature 600 (K) at the inner boundary while its external boundary is subjected to a convection heat transfer with the ambient (300 K) is considered. It is assumed that all points of the cylinder are initially at the ambient temperature and the initial displacements are zero. The thermomechanical parameters of the present problem are considered as:

  hi ¼ ho ¼ 1000 W=m2 K ; TiN ¼ 600ðKÞ;

Tðr; 0Þ ¼ 300ðKÞ;

ToN ¼ 300ðKÞ;

pi ¼ po ¼ 0

Time variations of the temperatures, radial displacements, radial stresses, hoop stresses, and axial stresses are plotted for all points of the thickness of the FGM cylinder in Fig. 3 for n ¼ 1. Fig. 4 shows effect of the temperature-dependency of the material properties on the time histories of the temperature distributions, radial displacements, radial stresses, hoop stresses, and axial stresses of the middle point of the thickness for n ¼ 1. Although the boundary conditions of the present problem are so chosen that influence of the temperature-dependency on the temperature results is small, even in this case, effects of the temperaturedependency of the material properties on the displacement and

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stress results are remarkable. This effect is much more remarkable in higher differences between temperatures of the internal and the external boundaries. In this special case, employing a yield criterion such as Tresca criterion reveals that the cylinder with temperatureindependent material propertied fails earlier. Therefore, the temperature-independency assumption leads to an overdesigned cylinder in the present particular case. Since the thermal diffusivity of the ceramic material is higher than that of the metallic material, the temperature gradients and temperature increases are higher in the ceramic region. Fig. 5 illustrates time variations of the temperatures and the radial displacements of the inner, middle, and outer points of the thickness of the FGM cylinder for various volume fraction indices. Comparison of Figs. 3(a) and 5(a) confirms this evidence. Comparison of Figs. 3(b) and 5(b) reveals that since the ceramic layer serves as a stiff boundary, when the cylinder is heated (through the internal or external boundaries), the internal regions of the type 3 FGM cylinder contract (move inward to attain the required thermal expansion) whereas all regions of the type 4 FGM cylinder expand. Example 3: forced vibration of a type 3 FGM cylinder subjected to an exponentially growing pressure shock at the external boundary is investigated in the present example. It is assumed that all the particles are initially at the ambient temperature (300 K). The loading data are as follows:

Tðr; tÞ ¼ 300ðKÞ;

  po ¼ 100 1  e2000t ðMPaÞ;

pi ¼ 0

Time variations of the radial displacements, radial stresses, hoop stresses, and axial stresses of the inner, middle, and outer points of the thickness are depicted in Fig. 6. As Fig. 6 shows, transient oscillations that are results of the superimposed effects of the modes of the vibration (especially the fundamental mode) have been occurred due to the rapid changes of the loading (the pressure shock). In Fig. 7, the mentioned distributions are plotted for all particles of the cylinder for n ¼ 1 for a better visualization. A comparison among the magnitudes of the mechanical stresses of the present example and the thermal stresses of the previous example implies that stresses caused by relatively low or moderate temperature gradients in the thick cylinders may be more considerable than the mechanical stresses caused by high external or internal pressures. Example 4: In the previous example, characteristics of the forced vibration behavior of the FGM cylinder are investigated. In the present example, the free vibration response as well as the fundamental natural frequency of the FGM cylinder are determined. For this purpose, a type 4 FGM cylinder is considered. It is assumed that the cylinder is subjected to an internal pressure that is increased gradually and linearly to a specified value where

Fig. 6. Time variations of the: (a) radial displacements, (b) radial stresses, (c) hoop stresses, and (d) axial stresses of the inner, middle, and outer points of the thickness of the FGM cylinder, under the applied pressure shock. In these figures, curves shown with dotted, solid, and dashed lines correspond to n ¼ 0.1, n ¼ 1, and n ¼ 5, respectively.

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Fig. 7. Time variations of the: (a) radial displacements, (b) radial stresses, (c) hoop stresses, and (d) axial stresses of the inner point of the thickness of the FGM cylinder, under the applied pressure shock (n ¼ 1).

thereafter the cylinder is unloaded abruptly. Therefore, the time history of the internal pressure is:

pi ¼



2  105 tðMPaÞ; 0;

0  t < 0:0005ðsÞ t > 0:0005ðsÞ

Other boundary and initial conditions are:

Tðr; tÞ ¼ 300ðKÞ;

uðr; 0Þ ¼ 0;

po ¼ 0

Fig. 8. Time variations of the radial displacements of the inner, middle, and outer points of the thickness of the FGM cylinder, due to the abrupt unloading. In this figure, curves shown with dotted, solid, and dashed lines correspond to n ¼ 0.1, n ¼ 1, and n ¼ 5, respectively.

Displacement responses of the inner, middle, and outer points of the thickness of the FGM cylinder are depicted in Fig. 8 for n ¼ 0.2, 1, and 5. Fig. 9 illustrates time variations of the radial displacements of all points of the thickness of the mentioned cylinder. The fundamental natural frequency may be determined based on measuring the period of the oscillations. Although the fundamental natural frequency may be determined by reducing the problem to an eigenvalue one, the method outlined here is more consistent with the presented governing equations. This method is especially useful for determining fundamental natural

Fig. 9. Time variations of the radial displacements of all points of the thickness of the FGM cylinder, due to the unloading shock (n ¼ 1).

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Table 3 The fundamental natural frequencies of the considered cylinder for various values of the volume fraction indices. n fn1(Hz)

0.2 10,866

1 8259

5 6630

with time is assumed to be a rectangular function of magnitude 100 (MPa) and Dt ¼ s=100 time duration where s is the fundamental natural period of the cylinder. Radial displacements are depicted for T ¼ 300 (K) in Fig. 10 for types 3 and 4 cylinders

frequencies of the nonlinear systems where the problem cannot be reduced to an eigenvalue one and the fundamental natural frequency depends on the magnitude of the applied loads, initial conditions, etc. The fundamental frequencies of the present cylinders are given in Table 3. Results show that as the ceramic volume fraction increases, the rigidity of the FGM cylinder increases and subsequently the fundamental natural frequency increases and the radial oscillations decrease. Example 5: Finally, wave propagation, reflection and interference phenomena are investigated for a thick cylinder subjected to an initial impulsive internal pressure and a uniform temperature rise. Variation of the impulsive internal pressure

Fig. 10. Radial displacements of a thick-walled cylinder, subjected to an impulsive internal pressure, immediately after the loading, for: (a) type 3 and (b) type 4 cylinders.

Fig. 11. Radial stresses of a type 3 FGM cylinder, subjected to an impulsive internal pressure, immediately after the loading, for: (a) n ¼ 0.2; (b) n ¼ 1; (c) n ¼ 5.

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(n ¼ 1) for equal time intervals. In Fig. 10, only early short times of the wave propagation are considered to present clear figures. As time elapses, more reflections and interferences occur. The radial stress variations of the type 3 cylinder are illustrated in Fig. 11 for different values of the volume fraction index n. As it is noticed form Figs. 10 and 11, in very short times after the impact, the radial displacements and stresses in points far from the internal boundary are zero. As the elastic wave propagates through the thickness of the cylinder, displacements and stresses of the mentioned points become nonzero. Figs. 10 and 11 imply that the speed of the elastic wave is higher in the ceramic regions of the cylinder. Fig. 12 shows the radial displacements and stresses for a uniformly preheated type 3 FGM cylinder (T ¼ 600 (K), n ¼ 1). Due to the degradation occurred in the stiffness of the preheated cylinder, its displacement components are greater. Furthermore, since the rigidities of the boundaries are lower, stress waves with less amplitude are reflected. To present a better visualization of the wave propagations, reflections and interferences, 3D plots of the results are illustrated for a uniformly preheated type 3 FGM

cylinder (T ¼ 600 (K)) in Figs. 13 and 14 for n ¼ 0.2, 1, and 5 for time intervals equal to one fundamental natural period time. As it may be noticed from Fig. 13, interference numbers grow with time. Furthermore, especially in the radial displacement response, the local vibrations are parts of a more general vibration with a grater period time. Formation of the oblique waves due to the repeated reflections is apparent in Figs. 13 and 14. Furthermore, since the boundary particles have more freedom to move, the displacements are greater at the boundary surfaces. As the volume fraction index (n) increases in type 3 FGM cylinder, volume fraction of the ceramic material increases and subsequently, amplitudes of the radial displacements and stresses decrease.

Fig. 12. Radial displacements and stresses of a uniformly preheated thick-walled type 3 cylinder (n ¼ 1).

Fig. 13. Wave propagation in a uniformly preheated type 3 thick FGM cylinder subjected to an impulsive load, for: (a) n ¼ 0.2, (b) n ¼ 1, and (c) n ¼ 5.

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more accurate results due to using a third-order Hermitian formulation instead of the common first-order Lagrangian formulations, (4) proposing an iterative updating numerical procedure to solve the resulted highly nonlinear governing equations, (5) investigating the transient behaviors of the thick-walled FGM cylinders from many aspects, and (6) employing over-relaxation and under-relaxation techniques to reduce the computational time and increase the numerical stability. Results show that the temperature-dependency of the material properties may have significant influence on the results of the displacements and stresses. Since the ceramic layer may have more rigidity, expansion of the inner parts of the type 3 cylinder may accomplish through inward movements of the particles. Furthermore, elastic wave propagations, reflections and interferences may lead to oblique waves formation. As results show, in a wave propagation process, the local oscillations may be parts of a more general oscillation with a greater period time. Temperature-dependency of the material properties may also affect various parameters of the wave propagation phenomenon in the thermal environments.

References

Fig. 14. Stress wave propagation in a uniformly preheated type 3 thick FGM cylinder subjected to an impulsive load, for: (a) n ¼ 0.2, (b) n ¼ 1, and (c) n ¼ 5.

5. Conclusions In the present paper, nonlinear thermoelasticity, free and forced vibrations, and wave propagation analyses are performed for thickwalled FGM cylinders subjected to dynamic thermomechanical loads. Some of the novelties and findings of the present paper are: (1) incorporating the temperature-dependency of the material properties in the analysis, (2) using Hermitian elements instead of Lagrangian ones that have been used in all of the previous researches, to avoid discontinuity of the radial stresses at the mutual boundaries of the elements and subsequently, prevent formation of artificial wave sources at the mentioned boundaries, (3) presenting

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