Thermal buckling behavior of functionally graded plates based on neutral surface

Accepted Manuscript Thermal buckling behavior of functionally graded plates based on neutral surface Young-Hoon Lee, Seok-In Bae, Ji-Hwan Kim PII: DOI: Reference:

S0263-8223(15)01013-2 http://dx.doi.org/10.1016/j.compstruct.2015.11.023 COST 6978

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Composite Structures

Please cite this article as: Lee, Y-H., Bae, S-I., Kim, J-H., Thermal buckling behavior of functionally graded plates based on neutral surface, Composite Structures (2015), doi: http://dx.doi.org/10.1016/j.compstruct.2015.11.023

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Thermal buckling behavior of functionally graded plates based on neutral surface Young-Hoon Lee*, Seok-In Bae*, Ji-Hwan Kim†

*

Department of Mechanical and Aerospace Engineering, College of Engineering, Seoul National University, Seoul, 151-744,South Korea [email protected], [email protected],

† Institute

of Advanced Aerospace Technology, Department of Mechanical and Aerospace

Engineering, College of Engineering, Seoul National University, Seoul, 151-744,South Korea [email protected]

Key words : Functionally graded materials, Thermal buckling behavior, Neutral surface

ABSTRACT

This work concerns with the thermal buckling behavior of Functionally Graded Materials (FGMs) based on neutral surface of a structures. Especially, the materials have nonhomogeneous properties with varying gradually from one surface to the other. In this regard, the neutral surface of the models is not the same as the mid-plane of structures due to asymmetry of material in the thickness direction, and thus it is necessary to select the neutral surface as the reference plane. Further, the material properties are temperature dependent, the location of the neutral surface depends also on the temperature change. For the crucial discussion, thermal buckling behaviors of FGMs are investigated based on the neutral surface concept. In the formulation, linear theory of plate is adopted based on the first-order shear deformation theory, and the steady state thermal conduction is considered as the one dimensional heat transfer. Numerical results show that the present works have little lower estimation of the buckling temperatures than previous data, and the shift of neutral planes are also observed by considering the static equilibrium of FGMs.

1. Introduction

Functionally graded materials (FGMs) have been emerged from manufacturing a composite material in high temperature status. Specifically, FGMs are made up of continuous mixture of ceramic and metal, and then the material properties vary smooth and continuous pattern from one surface to the other as compared with conventional composite. Furthermore, ceramic has high compressive strength and heat resistance with low fracture toughness, but metal exhibits better mechanical strength while cannot endure at high thermal environment. These attractive merits have increased the roles of FGMs in various engineering fields. Up to now, numerous research works on the materials have been performed for about three decades. Pradhan and Chakraverty [1] investigated the vibration of Euler and Timoshenko beam models using Rayleigh-Ritz method. Kapuria et al. [2] reported an experimental validation with theoretical model for the bending and vibration of the layered beams. Su and Banerjee [3] pointed out the development of dynamic stiffness method for free vibration of the Timoshenko beams. While, Na and Kim [4] suggested the optimization of volume fraction of the composite panels for the stress reductions and critical temperatures. Further, Sohn and Kim [5] revealed the structural stability of panels under aero-thermal loads. Shariat and Eslami [6] indicated the buckling of the thick plates under mechanical and thermal loads. Bouazza et al. [7] researched the thermo-elastic buckling of the models using first-order shear deformation plate theory. Ferreira et al. [8] analyzed an original hyperbolic sine shear deformation theory for the bending and free vibration analysis of plates. Additionally, Chen et al. [9] studied the vibration and stability of plates based on higher-order deformation theory. Morimoto et al. [10] discussed the buckling of rectangular plates subjected to partial heating. For equilibrium of structure, neutral surface has been used as a reference plane due to the asymmetry of material properties in the thickness direction of FGMs as compared with the previous works. Zhang [11] considered a higher-order shear deformation theory of the plate models for thermal buckling behavior using the neutral concept. Also, Zhang and Zhou [12] investigated a theoretical analysis of the thin plates based on physical neutral surface. Yaghoobi and Fereidoon [13] indicated the influence of neutral surface position on deflection. In this work, thermal buckling behavior of FGMs models with temperature dependent materials are analyzed using finite element method. Also, the aim of present study is to use the neutral surface concept and compares the numerical results with the previous data based on conventional reference plane. In order to discuss the present works, plate models are considered with and without heat conduction effects in the thickness direction. Additional

results are presented for the variation of critical temperatures according to the volume fractions for the models.

2. Formulations

FGMs with temperature dependent material properties are presented using the model with length a , width b and thickness h . In this work, the formulation is based on the first-order shear deformation theory of plate with and without considering heat transfer effects.

2.1. Functionally graded materials

A mixture of ceramic and metal is studied, and the mixture ratio is varying continuously and smoothly in the thickness direction of the model. Thus, the volume fraction of the material is defined as simple power-law:

 z 1 Vc ( z ) =  +  h 2

k

(0 ≤ k < ∞),

Vc ( z ) + Vm ( z ) = 1

(1)

where Vc , superscript k , subscripts c and m , superscript h represent the volume fraction of ceramic, the volume fraction index, ceramic and metal, thickness of plate model, respectively. Generally, the temperature dependent material properties P(T ) can be written as in Ref. [14].

P (T ) = P0 (

P−1 2 3 ) + 1 + PT 1 + P2T + PT 3 T

(2)

where T stands for the temperature, and P0, P-1, P1, P2 and P3 are constants in the cubic fit of the material properties. Considering z for the location in the thickness direction, then the rule of mixture for material properties of FGMs can be written typically as:

Peff ( z , T ) = Pm (T )Vm ( z ) + Pc (T )Vc ( z )

 z 1 = Pm (T ) + ( Pc (T ) − Pm (T ))  +  h 2

k

(3)

where subscript eff means effective all the material properties such as Young’s modulus, thermal expansion coefficient, thermal conductivity and Poisson's ratio, respectively.

2.2. Physical neutral surface

Fig. 1 shows FGMs model with the middle and the neutral surfaces, and precisely the material properties in the thickness direction are asymmetric. In this study, the neutral surface is chosen as a reference plane for deformation based on the force equilibrium of the model. Including the thermal effects on the FGMs plate, integration in the thickness direction for the first moment of elasticity modulus E ( z , T ) equals to zero in order to determine the neutral surface of the model.

h 2 h − 2

∫

E ( z , T )( z − z0 (T ))dz = 0

(4)

Thus, the position of neutral surface z0(T) can be obtained as : h 2 h − 2 h 2 h − 2

∫ z0 (T ) =

∫

E ( z , T ) zdz

(5) E ( z , T )dz

Especially, the non-dimensional neutral surface locations based on simple power-law FGMs model are obtained as:

z0 (T ) k ( Ec (T ) − Em (T )) = h 2(k + 2)(kEm (T ) + Ec (T ))

(6)

2.3. Temperature rise condition

FGMs have two kinds of temperature rise conditions such as uniform and heat conduction in the thickness direction of the model. In case of uniform temperature condition, initial temperature is to be uniformly raised to a final value Tl . Then, a uniform temperature variation across the plate thickness can be written as [15,16].

∆T = Tl − Ti

(7)

While, the model includes heat transfer effects, boundary condition is obtained by solving the heat conduction equation in the thickness of the plate as [15]. d  dT   h h K ( z, T ) = 0, T  −  = Tm , T   = Tc   dz  dz   2 2

(8)

where K ( z , T ) is a function of heat conduction through the thickness such as

 z 1 K ( z, T ) = K m + ( K c − K m )  +   h 2

k

(9)

Substitution of Eq.(9) into Eq.(8) result in a differential equation for temperature, and the solution for temperature distribution across the plate thickness is obtained using polynomial series as :

∆T T ( z ) = Tm + C

n nk +1  1  K cm   z 1   n (−1)    +   ∑ nk + 1  K m   h 2   n= 0    ∞

(10)

with ∞  ( K − Km )n  C = ∑ (−1) n c  (nk + 1) K mn  n =0 

where ∆T = Tc − Tm .

(11)

3. Governing equations

In this work, FGMs model is used based on neutral surface as a reference plane. Considering the first-order shear deformation theory of plate, and introduce the physical neutral surface. Then, the displacement fields including deviation of neutral surface z0 (T )

are u ( x, y, z , T ) = u0 ( x, y ) + ( z − z0 (T ))φx ( x, y ) v ( x, y, z , T ) = v0 ( x, y ) + ( z − z0 (T ))φ y ( x, y )

(12)

w( x, y, z , T ) = w0 ( x, y )

where u, v and w are the displacements in the x , y and z direction, while ϕ x and ϕ y are the rotations of the normal in the xz and yz planes, respectively. Further, constitutive equations of FGMs plate are

 Nb (T )   A(T ) B (T )  ε 0   N ∆T   =     −   M b (T )   B(T ) D (T )  κ   M ∆T 

E ( z, T ) 1 0 − h /2 2(1 + v( z , T )) 0 1  

[Q(T )] = [S (T )]γ , [S (T )] = k p ∫

h /2

(13.a)

(13.b)

where N b (T ) , M b (T ) and Q(T ) denote the in-plane force resultant, the moment resultant and the transverse shear force resultant vectors, respectively. Further, ε 0 , κ and γ are strain vectors based on mid-plane, curvature and transverse shear, respectively. While, k p is the shear correction factor of the plate considering the variation of volume for ceramic and metal as Ref. [17].

kp =

5 6 − (ν cVc + ν mVm )

where ν c and ν m denote the Poisson's ratio of ceramic and metal, respectively.

(14)

Further, N ∆T (T ) and M ∆T (T ) are the thermal in-plane force resultant and the thermal moment resultant vectors as :

 N ∆Tx  ( N ∆T (T ), M ∆T (T ) ) =  N ∆Ty N  ∆Txy

M ∆Tx   M ∆Ty  M ∆Txy 

(15)

 α ( z, T )  h/ 2   = ∫ (1, ( z − z0 (T )))[ E ]  α ( z , T )  ∆T ( z )dz − h /2 0   

where the temperature dependent elastic matrix is   1 v( z, T ) 0   E ( z, T )  v ( z, T ) 1 0  [ E] = 2  1 − v( z , T )  1 − v( z , T )  0 0    2

(16)

while A(T ) , B(T ) , D(T ) and S (T ) are the in-plane, in-plane bending coupling stiffness, bending stiffness and transverse shear stiffness matrices as

( A(T ), B (T ), D(T )) = ∫

h /2

− h /2

[ E ](1,( z − z0 (T )), ( z − z0 (T )) 2 ) dz

(17)

In this work, the bucking behaviors of FGMs based on the neutral surface are the interesting issues, thus critical temperatures using mid-plane are compared with the neutral surface model. In this regard, the results can be obtained by solving the following standard eigenvalue problem as [18].

(18)

[ K e + λ K T ]q D = 0 where λ and q

D

e

are the eigenvalue and eigenvector, respectively. In addition, K is the

global bending stiffness matrix obtained by assembling the A(T ) , B(T ) , D(T ) matrix and

K T is the global thermal stiffness matrix in terms of N ∆T .

Then, the lowest eigenvalue of the critical temperature Tcr is obtained as :

Tcr = Tref + ∆Tcr = Tref + λ∆T

(19)

where cr and ref denote critical and reference temperatures, respectively.

4. Numerical results and discussions

To investigate the thermal buckling behavior of the plate based on neutral surface, materials for metal and ceramic are selected as Al2O3 / Al , Si3 N 4 / SUS 304, ZrO2 / Ti . Table 1 lists elasticity modulus, coefficient of thermal expansion and thermal conductivity for constituent materials, and Poisson’s ratio is assumed to be 0.3 as in Ref. [14]. Also, the reference and the initial temperatures are equal to the same as T0 = Ti = 300 K . For the finite element model, 7x7 elements with nine-node elements are used, and the results for simplysupported and clamped boundary conditions are considered in this study. And, the relative effects on the thermal buckling behavior based on the neutral surface concept and mid-plane of the model are compared in detail.

4.1. Code verification

Before the detailed discussion for the investigation of the present work, numerical results of three cases are compared with previous data. The constituents are chosen as Alumina( Al2O3 ) and Aluminum( Al ) for temperature independent material. The properties are the Young’s modules of Al2O3 ( Ec ) and Al ( Em ) are 380(GPa ) and 70(GPa ) , respectively. Also, the thermal expansion coefficients α c , α m and the thermal conduction coefficients K c , K m are 7.4e − 6(1/ K ) 23e − 6(1/ K ) and 10.4 (W / mK ) , 204 (W / mK ) , , respectively. Firstly, Fig. 2 depicts the amount of non-dimensional shift of neutral surface from the mid-plane( z0 / h )to verify neutral surface shift for the temperature independent materials. Figure shows the neutral surface shift with respect to volume fraction, and then the maximum shift is 15.8% of thickness direction from mid-plane of the model. The results agree well with the previous work in Ref.[19].

Using Eq. (6), the maximum value ( k max (T ) ) of z0 (T ) / h is obtained as :

k max (T ) = 2

Ec (T ) Em (T )

(20)

In this work, k max (T ) is 3.3 for reference temperature and is reasonable with the result in Ref.[13]. Next, in order to verify the result of eigenvalue problem, the thermal buckling behavior of FGM plates under uniform temperature condition are examined and compared with the results of Ref [20] as in Table 2, and the present work and previous data is in good agreement. Finally, Fig. 3 presents the comparison of present results with the data in Ref [21] and are plotted with the aspect ratios ( a / b ) for different volume fraction ratios. And then, the numerical results are almost equal to the critical temperature in the reference.

4.2. Neutral surface

Fig.4 shows the non-dimensional shift( z0 / h ) of neutral surface according to volume fraction ratio is obtained for three kind of materials. It presents the effect on the ratio of position for neutral surface. As the fraction is increased, then FGMs model has richer metal portion. And thus, the shift of Al2O3 / Al is increased until volume fraction k = 3.3 and reached maximum value as 15.8% of thickness direction. And the maximum value of the shift is changed according to the ratio of metal and ceramic. Similarly, the maximum shift for Si3 N 4 / SUS 304 and approach 3.878% of thickness direction for k = 1.761 and 0.522% of the direction for k = 1.488 , respectively. Also, Al2 O3 / Al has the largest shift amount from midplane than the other materials. While, the shifts for ZrO2 / Ti are little amount change as compared with the increased volume fraction due to the small difference of Young's modulus. As a result, difference of the modulus is increased, volume fraction of maximum neutral surface shift is also increased. From now on, numerical results are discussed for Si3 N 4 / SUS 304 with temperature dependant material properties. Fig. 5 shows the effect of temperature variations according to the position of neutral surface, and interesting thing is that k increases, then the difference of the deviation amount also increased. In other words, the temperature increases, the maximum value of shift is also increased. When the reference temperatures are T = 600 K and T = 900 K , the percentage of maximum neutral surface shift

are 4.155% and 5.924% of thickness directions from mid-plane for k = 1.786 and k = 1.965 , respectively. Due to the temperature increases, the elasticity modulus difference is increased, and deviation of the neutral as well as the mid-surfaces is also increased. Further, the k max (T ) is increased due to the increase of reference temperature. Thus, the selection of reference temperature as mid-plane or neutral surface affects the evaluation of critical temperature.

4.3 Thermal buckling analysis In here, neutral surface of the FGM model is selected as the reference for the analysis of thermal buckling behavior. Fig. 6 compares the critical temperatures of Al2 O3 / Al by using neutral surface and the reference plane for maximum volume fraction k = 3.3 which stands for the maximum neutral surface shift. Fig. 6(a) shows critical temperature with and without heat transfer effects according to aspect ratios. Fig. 6(b) depicts critical temperatures for the thickness ratios, and group 'A' has lower value of the temperatures than group 'B' because of the thermal effects on the model. Thermal stiffness effects due to temperature rise condition has smaller amount than uniform temperature case. Also, stiffness due to A(T ) , B (T ) , D (T ) matrix has more effects than thermal stiffness N ∆T matrix for eigenvalue problems. As shown in the Fig.6, the effects of neutral surface on the critical temperature are smaller than mid-plane. In case, reference plane is considered as neutral surface, then the effect of stiffness is reduced. As increasing the thickness ratio of the model, then the structure becomes thinner, thus the critical temperature decreases based on neutral surface. To investigate the effect of temperature dependent material properties such as Si3 N 4 / SUS 304 , simply-supported and clamped boundary conditions of the models are discussed. Fig. 7(a) shows critical temperatures according to various aspect ratios with simply-supported boundary conditions based on mid-plane and neutral surface. Each group has slight differences of the temperatures based on mid-plane and neutral surface. While, Fig. 7(b) reveals almost the same results of the temperature for all clamped boundary conditions of models. Up to now, major interesting topics in the research works for plate model have been investigated for the clamped boundary conditions. However, even if thin rectangular plate model, the structure has 4 edges, and each edge is supported with clamped, simplysupported or free boundary conditions. Therefore, 34 set of classical boundary conditions are possible, and thus mid-plane concept should be carefully checked in engineering structures. This point was already reported in Ref.[22] for beam vibration problem. Fig. 8 presents

critical temperature of Si3 N 4 / SUS 304 for different thickness ratios. Fig. 8(a) shows little difference of critical temperature as similar to the results in Fig. 7(a), and Fig. 8(b) depicts the almost same results regardless of reference surfaces. Therefore, clamped edge conditions may not result in deviation due to the choice of reference plane. In this regard, clamped boundary conditions may be more stable than simply-supported edges. In order to know the influence on reference temperature, critical temperatures are shown in the Table 3. For

T = 300 K and T = 900 K as reference temperatures, the maximum value of neutral shift are 3.878% and 5.924% of thickness direction from mid-plane for k = 1.76 and k = 1.965 , respectively. Also, the results at reference temperature T = 900 K are varied more amount than for T = 300 K . Table 3 shows the relative error defined as :

error (%) =

TM − TN ×100% TM

(21)

where TM and TN are the temperatures at mid-plane and neutral surface as the reference planes, respectively. As indicated earlier, critical temperatures are decreased as the change of reference from middle surface to the neutral surface of the model because the force balance is exactly satisfied based on the neutral surface. The relative errors for mid-plane and neutral surface at reference temperature T = 300 K is 1.215%, and T = 900 K is 2.715%, respectively. As a result, critical temperature based on neutral surface under high temperature condition is decreased more amount than the low temperature condition. Thus, critical temperatures for thermal buckling based on the neutral surface are also affected by reference temperature.

5. Conclusions Thermal buckling analysis of Functionally Graded Materials considering temperature dependent material properties with heat transfer effects are performed based on the focus of neutral surface concept, and the first-order shear deformation theory of plate structure is used. The neutral surface locations are investigated with various conditions such as volume fractions, reference temperatures and material properties. When the Young's modulus of ceramic is larger than metal, neutral surface is away from mid-plane and locates in ceramic

part. As the neutral surface shift to ceramic part, the critical temperatures are decreased due to the reduction of elastic stiffness effect. Therefore, models with large difference of Young's modulus should consider the neutral surface concept, and the difference affects the critical temperature for thermal buckling as well as thermally related behavior for FGMs model. Due to the thermal stiffness of the plates considering heat conduction is larger than the effect of without heat conduction, the critical temperature is increased as the heat conduction. Only the neutral surface shift increases as the reference temperature is increased, and then also the critical temperature based on neutral surface is affected by reference temperature.

Acknowledgement This work was supported by the Brain Korea 21 Plus Project and Engineering Research and Institute at College of Engineering in Seoul National University during 2015.

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[11] Zhang DG. Modeling and analysis of FGM rectangular plates based on physical neutral surface and high order shear deformation theory. Int J Mech Sci 2013;68:92– 104. [12] Zhang DG, Zhou YH. A theoretical analysis of FGM thin plates based on physical neutral surface. Comput Mater Sci 2008;44:716-20. [13] Yaghoobi H, Fereidoon A. Influence of neutral position on deflection of functionally graded beam under uniformly distributed. World Appl Sci J 2010;10:337-41. [14] Reddy JN, Chin CD. Thermomechanical analysis of functionally graded cylinders and plates. J Therm Stress 2007; 21: 593–626. [15] Javaheri R, Eslami MR. Thermal buckling of functionally graded plates. AIAA J 2002; 40:162–9. [16] Mohamed BB, Mohammed SAH, Abdelouahed T. Thermal buckling of functionally graded plates according to a four-variable refined plate theory. J Therm Stress 2012;35:677-94. [17] Efraim E, Eisenberger M. Exact vibration analysis of variable thickness annular isotropic and FGM plates. J Sound Vib 2007; 299:720-38. [18] Talha M, Singh BN. Thermo-mechanical buckling analysis of finite element modeled functionally graded ceramic-metal plates. Int J Appl Mech 2011;3: 867–80. [19] Prakash T, Singha MK, Ganapathi M. Influence of neutral surface position on the nonlinear stability behavior of functionally graded plates. Comput Mech 2008;43:341– 50. [20] Javaheri R, Eslami MR. Thermal buckling of functionally graded plates based on higher order theory. J Therm Stress 2002;25:603-25. [21] Kiani Y, Bagherizadeh E, Eslami MR. Thermal buckling of clamped thin rectangular FGM plates resting on Pasternak elastic foundation (Three approximate analytical solution). J Appl Math Mech 2011;91:581-93. [22] Simsek M. Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories. Nucl Eng Des 2010;240:697-705.

List of Tables Table 1. Temperature-dependent material properties of metal and ceramic.

Table 2. Comparison of critical temperatures under uniform temperature condition. Table 3. Comparison of critical temperatures based on reference planes. List of Figures Fig. 1 Neutral surface and Mid-plane of FGMs. Fig. 2 Non-dimensional neutral surface shift from mid-plane. Fig. 3 Critical temperature variation according to aspect ratio. Fig. 4 Neutral surface shift according to volume fractions of materials. Fig. 5 Neutral surface shift due to volume index and temperatures. Fig. 6 Critical temperature of the models based on mid-plane and neutral surface: (a) Aspect ratio; (b) Thickness ratio. Fig. 7 Critical temperature of the model according to aspect ratios: (a) Simply-supported; (b) Clamped-clamped. Fig. 8 Critical temperature of the model according to thickness ratios: (a) Simply-supported; (b) Clamped-clamped.

Table 1 Temperature-dependents material properties of metal and ceramic.

E ( Pa )

0

P0 348.43e9

α (1/ K )

0

5.8723e-6

9.095e-4

0

0

k (W / mK )

0

13.723

-1.032e-3

5.466e-7

-7.876e-11

E ( Pa )

0

201.04e9

3.079e-4

-6.534e-7

0

α (1/ K )

0

12.33e-6

8.086e-4

0

0

k (W / mK )

0

15.379

-1.264e-3

2.092e-6

E ( Pa )

0

132.2e9

-3.805e-4

-6.127e-14

0

α (1/ K )

0

13.3e-6

-1.421e-3

9.549e-7

0

k (W / mK )

0

1.710

1.228e-4

6.884e-8

0

E ( Pa )

0

122.7e9

-4.605e-4

0

0

α (1/ K )

0

7.43e-6

0

0

0

k (W / mK )

0

1.100

1.545e-2

0

0

Properties

Si3 N 4

SUS 304

ZrO2

Ti − 6 Al

P−1

P1 -3.070e-4

P2 2.160e-7

P3 -8.946e-11

-7.223e-10

Table 2 Comparison of critical temperature under uniform temperature condition. a/h

80 100

Results

Volume fraction index (n) 0

1

5

10

Present

26.707

12.395

11.321

11.55

Ref. [20]

26.717

12.412

11.352

11.67

Present

17.093

7.939

7.261

7.467

Ref. [20]

17.099

7.943

7.265

7.469

Table 3 Comparison of critical Temperature considered based on reference planes.

T 300K (k=1.76) 900K (k=1.96)

Reference plane Mid Neutral Mid Neutral

Thickness Ratio a/h=20 a/h=30 468.26 209.77 462.64 207.24 1993.45 893.10 1940.03 868.98

a/h=40 a/h=50 a/h=60 a/h=80 118.33 75.83 52.70 29.66 116.89 74.91 52.06 29.30 503.78 322.84 224.36 126.29 490.14 314.09 218.27 122.86

Average Error 1.215% 2.715%

Fig. 1. Neutral surface and Mid-plane of FGMs.

18 Present Ref.[14]

Non-dimensional thickness (%)

16 14 12 10 8 6 4 2 0

0

10

20

30

40 50 60 Volume fraction,k

70

80

90

100

Fig. 2. Non-dimensional neutral surface shift from mid-plane.

16

k=0.5

Present Ref.[21]

14

Critical temperature, Tcr

k=1 12

k=5

10

8

6

4 0.5

0.6

0.7

0.8

0.9 1 1.1 Aspect ratio, a/b

1.2

1.3

1.4

1.5

Fig. 3. Critical temperature variation according to aspect ratio.

18 Al2O3/Al

Non-dimensional thickness (%)

16

Si3N4/SUS304

14

ZrO2/Ti-6Al-4V

12 10 8 6 4 2 0

0

5

10

15

20 25 30 Volume fraction,k

35

40

45

50

Fig. 4. Neutral surface shift according to volume fractions of materials.

Non-dimesional neutral surface (%)

12 10 8 6 4 2 0 0

1200 1000

10

800

20 30

400 40

Volume fraction, k

200

600 Reference Temperature,K

Fig. 5. Neutral surface shift due to volume index and temperatures.

30

25

Critical temperature

B : Non linear

Mid-plane Neutral surface

20

15

A : Uniform 10

5

0 0.5

0.6

0.7

0.8

0.9 1 1.1 Aspect ratio

1.2

1.3

1.4

1.5

(a) 200 180

Mid-plane Neutral surface

B : Non linear

Critical temperature, Tcr

160 140 120 100 80

A : Uniform

60 40 20 0 30

35

40

45

50 55 60 Thickness ratio, a/b

65

70

75

80

(b) Fig. 6. Critical temperature of the model based on mid-plane and neutral surface: (a) Aspect ratio; (b) Thickness ratio.

35 Mid-plane Neutral surface

Critical temperature, Tcr

30

B : Non linear

25

20

15

A : Uniform

10

5 0.5

0.6

0.7

0.8

0.9 1 1.1 Aspect ratio, a/b

1.2

1.3

1.4

1.5

(a) 100

B : Non linear Mid-plane Neutral surface

90

Critical temperature, Tcr

80 70 60

A : Uniform

50 40 30 20 0.5

0.6

0.7

0.8

0.9 1 1.1 Aspect ratio, a/b

1.2

1.3

1.4

1.5

(b)

Fig.7. Critical temperature of the model according to aspect ratio: (a) Simply-supported; (b) Clamped-clamped.

250

Critical temperature, Tcr

200

Mid-plane Neutral surface

B : Non linear

150

100

A : Uniform

50

0 30

35

40

45

50 55 60 Thickness ratio, a/h

65

70

75

80

(a)

700

Critical temperature, Tcr

600

Mid-plane Neutral surface

B : Non linear

500

400

300

A : Uniform

200

100

0 30

35

40

45

50 55 60 Thickness ratio, a/h

65

70

75

80

(b) Fig.8. Critical temperature of the model according to thickness ratio: (a) Simply-supported model; (b) Clamped model.