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Thermal-structural post-bucking and limit-cycle oscillation of Functionally Graded Materials
Accepted Manuscript Thermal-structural Post-bucking and Limit-Cycle Oscillation of Functionally Graded Materials Young-Hoon Lee, Ji-Hwan Kim PII: DOI: Reference:
S0263-8223(16)32087-6 http://dx.doi.org/10.1016/j.compstruct.2016.12.079 COST 8146
To appear in:
Composite Structures
Received Date: Revised Date: Accepted Date:
7 October 2016 15 December 2016 27 December 2016
Please cite this article as: Lee, Y-H., Kim, J-H., Thermal-structural Post-bucking and Limit-Cycle Oscillation of Functionally Graded Materials, Composite Structures (2017), doi: http://dx.doi.org/10.1016/j.compstruct. 2016.12.079
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Thermal-structural Post-bucking and Limit-Cycle Oscillation of Functionally Graded Materials
Young-Hoon Lee*, Ji-Hwan Kim†
* Department
of Mechanical and Aerospace Engineering, College of Engineering, Seoul National University, Seoul, 08826,South Korea [email protected],
† Institute
of Advanced Aerospace Technology, Department of Mechanical and Aerospace
Engineering, College of Engineering, Seoul National University, Seoul, 08826,South Korea [email protected]
Key words: Functionally Graded Materials; Thermal post-buckling; Limit-cycle oscillation; Neutral surface
ABSTRACT
Thermal post-buckling and limit-cycle oscillation characteristics of Functionally Graded Material (FGM) structures are investigated based on the neutral surface concept. In particular, the material properties are non-homogeneous and vary gradually from one surface to the other. Furthermore, the properties are to be considered as temperature-dependent characteristics, and the neutral surface concept is adopted instead of the mid-plane to consider the reference plane due to the asymmetric properties in the thickness direction of model. In the formulation, the First-order Shear Deformation Theory (FSDT) of plate is used, and the geometric nonlinearity is accounted for by the von Karman strain-displacement relations. Also, steady state thermal conduction effects are assumed as aone
dimensional heat transfer on the surface of the structure. For the numerical analysis, the NewtonRaphson method is applied to solve the thermal post-buckling behavior, while Newmark's time integration method is employed to resolve the limit-cycle oscillation. In order to validate the analysis results, the results of this paper based on the neutral surface are compared with the data from previous papers using the conventional approach for FGMs model. Finally, effects of the neutral surface on the non-linear thermo-mechanic behavior of structure are discussed in detail.
1. Introduction FGMs are generated by manufacturing composite materials in a high temperature state. Specifically, the materials are made up of the continuous mixture of ceramic and metal, and then the properties form a smooth and continuous pattern from one surface to the other. And, ceramic has a high compressive strength and heat resistance with low fracture toughness, while metal exhibits better mechanical strength but has lower heat resistance. These attractive merits have increased the roles of FGMs in various high quality engineering fields. For this reason, numerous research works have been performed on thermal buckling and post-buckling analysis for FGMs. Li et al. [1] presented the post-buckling of
Timoshenko
beam models under the non-uniform temperature increase. Liew et al. [2] reported the buckling and post-buckling behavior of laminated rectangular plates with uniform temperature change. Shen [3] discussed the post-buckling behavior of model under thermal loading. While, Lee and Kim [4] studied the supersonic aero-thermo post-buckling behaviors and limit-cycle oscillations of panels using Newmark's time integration method. Prakash et al. [5] pointed out the post-buckling characteristic of the skew plates under thermal load using finite element approach. Na and Kim [6] researched the thermal postbuckling behaviors of functionally graded plates using 3-D finite element method. Further, Lee and Kim [7] performed the thermo-mechanical analysis of panels in hypersonic airflows. Also, Prakash and Ganapathi [8] investigated the influence of thermal environment on the structures for supersonic flutter using finite element formulation. Bateni et al. [9] analyzed the instability of rectangular plate models under the thermal as well as mechanical loadings. On the other hand, Ibrahim et al. [10] studied the non-linear flutter and thermal buckling of panels considering not only elevated temperature but also aerodynamic loading. Furthermore, Haddapour et al. [11] revealed the nonlinear aero-elastic performance of plates in supersonic flow. For the analysis of FGMs,the neutral surface must be used as a reference plane due to the asymmetric material properties in the thickness direction. Zhang and Zhou [12] investigated a theoretical analysis of the thin plates based on the physical neutral surface. Furthermore, Prakash and Singha [13] studied nonlinear characteristics of skew plates under in-plane load. In this paper, thermal post-buckling and limit-cycle oscillation characteristics of FGM plates are analyzed based on the neutral surface of the plate. The structural models of plates are considered as temperature-dependent materials and the models are based on the FSDT of plate. Furthermore, the material properties of the model are continuously varied in the
thickness direction of the model. In order to compare the present results with the previous data based on the mid-plane of the model as the reference plane, the non-linear thermalstructural post-buckling and limit-cycle oscillation behaviors are discussed in detail. Also, the amount of neutral surface shift depends on the temperature and the volume fractions of the constituent materials. Additionally, in order to discuss the present works, the cases of models are considered with and without heat conduction effects. The shifts are discussed for the cases of Power-law (P-), Sigmoid (S-), and Exponential (E-) FGMs.
2. Material properties and 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 with respect to the middle plane. Temperature-dependent material properties are presented for FGMs with length a , width b and thickness h . Then, in this work, the FSDT of plate is adopted in this study and heat conduction effect is considered as well.
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. Then, 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. While, the temperature-dependent material properties P(T ) can be written as in Ref.
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 material point 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
As shown in Fig.1, the model is based on the continuous distribution of metal to ceramic in the thickness direction. First of all, the mass distribution is asymmetric in the thickness direction, and thus the neutral surface concept should be used to account for this. In this study, the neutral surface is chosen as a reference plane for deformation based on the force equilibrium of FGM the models. Considering the thermal effects on the FGMs, integration in the thickness direction determines the location of neutral surface in the model. Thus, the first-moment of elasticity modulus E ( z, T ) equals to zero as in Ref.[15]:
h 2 h − 2
∫
E ( z, T )( z − z0 (T ))dz = 0
(4)
Therefore, the location 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
Using this formula, temperature-dependent neutral surface shifts z0 (T ) are calculated for P-, S-, and E-type FGMs considering the specific Young's modulus.
2.3. Temperature rise condition
In this study, two cases are analyzed with uniform temperature rise and heat transfer in the thickness direction of the model [16]. Firstly, initial temperature is uniformly raised to a final value Tl with uniform temperature distribution. Then, the temperature variation across the plate thickness can be written as:
∆T = Tl − Ti
(6)
While, the model includes heat transfer effects, the heat conduction equation in the thickness of the model as [14] is to be solved:. d dT h h K ( z, T ) = 0, T − = Tm , T = Tc dz dz 2 2
(7)
where K ( z, T ) is a function of heat conduction through the thickness as,
z 1 K ( z, T ) = K m + ( K c − K m ) + h 2
k
(8)
Thus, temperature distribution across the plate thickness becomes [17] ~
T = Tc − (Tc − Tm
∫ ) ∫
z
− h /2 h /2 − h /2
3. Governing equations
(1/ K ( z , T ))dz (9)
(1/ K ( z , T ))dz
The FGMs model is to be analyzed with consideration of the FSDT of plate with heat conduction effects as well and accounting for the physical neutral surface of the structures with the finite element method.
3.1. Equations of motion
Displacement fields based on the neutral surface z0 (T ) can be written as [15] :
u ( x , y , z , T ) = u0 ( x , y ) + ( z − z0 (T ))φ x ( x , y ) v ( x , y , z , T ) = v0 ( x , y ) + ( z − z0 (T ))φ y ( x , y )
(10)
w( x, y , z , T ) = w0 ( x, y )
where u, v and w are the displacements in the x , y and z directions, while ϕ x and ϕ y stand for 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 (T ) = − M b (T ) B(T ) D (T ) κ M ∆T (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
(11.a)
(11.b)
where N b (T ) , M b (T ) and Q(T ) denote the temperature-dependent in-plane force, the moment and the transverse shear force resultant vectors, respectively. Also, ε 0 , κ and γ are strain vectors based on mid-plane, curvature and transverse shear, respectively. Furthermore, k p is the shear correction factor of the plate structure considering the variation of volume
fractions for ceramic and metal as Ref. [18].
kp =
5 6 − (ν cVc + ν mVm )
where ν c and ν m denote the Poisson's ratio of ceramic and metal, respectively.
(12)
Further, N ∆T (T ) and M ∆T (T ) are the thermal in-plane force resultant and the thermal moment resultant vectors with the expressions as:
N ∆Tx ( N ∆T (T ), M ∆T (T ) ) = N∆Ty N ∆Txy
M ∆Tx M ∆Ty M ∆Txy α ( z, T ) h /2 ~ = ∫ (1, ( z − z0 (T )))[ E ] α ( z , T ) ∆ T ( z )dz − h/ 2 0
(13)
inhere 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
(14)
while A(T ) , B(T ) , D(T ) and S (T ) are the in-plane, in-plane bending coupling , bending and transverse shear stiffness matrixes as
( A(T ), B (T ), D(T )) = ∫
h/ 2
− h /2
[ E ](1, ( z − z0 (T )), ( z − z0 (T )) 2 ) dz
(15)
To derive the equations of motion of the FGMs model, principle of virtual work is applied:
δ W = δ Wint − δ Wext = 0
(16)
now δ Wint and δWext represent the internal and external virtual works, respectively. At first,
δ Wint = ∫ δ eT dV = ∫ [δε T N b (T ) + δκ T M b (T ) + δγ T Q(T )]dA V
(17)
A
= δ d T [ K e − K ∆T
1 1 + N1(T ) + N 2(T )]d 2 3
In here, d = [u , v, w, φx , φ y ]T is the displacement vector. In addition, K e 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 (T ) .
Also, N1(T ) , N 2(T ) and P∆T are matrices
denoting the first-order and second-order non-linear stiffness, respectively. On the other hand,
δ Wext = δ d T f = ∫ ( P∆T )δ wdA = −δ d T P∆T
(18)
A
where the P∆T is thermal load vector. From now on, the notations of temperature-dependency in nonlinear terms are omitted for simplicity in the formula. Then, the governing equations of the FGMs plates in matrix form is expressed as:
ii
M d + ( K − K ∆T +
1 1 N 1 + N 2) d = P∆T 2 3
(19)
where the P∆T is thermal load vector.
3.2. Iterative procedures
In order to solve the time-dependent non-linear equations as Eq.(1920),The solution of Eq. (20) the solution is assumed as d = d s + ∆dt with d s and dt representing the time-independent and time-dependent solution parts, respectively. Then, two sets of the coupled governing equations are obtained as:
1 1 ( K − K ∆T + N1S + N 2S )d s = P∆T 2 3
(20.a)
and ii
M d t + ( K − K ∆T +
1 1 1 1 N1s + N 2 s + N 2 st + N 1t + N 2t )d t = 0 2 3 2 3
(20.b)
where the subscripts s and t denote the static and dynamic states, respectively. That is to say,
the Eq. (20.a) represents nonlinear static problem analysis such as thermal post-buckling analysis behaviors. On the other hand, the Eq. (20.b) stands for the nonlinear dynamic problem in the thermal environment for vibration and flutter behaviors, and also the static equation should be solved preliminarily due to the governing equations being coupled. In this section, a solution procedure is presented briefly. At the beginning, in order to analyze the thermal post-buckling behavior, the incremental form of Eq. (20.a) is obtained by using the Newton-Raphson iterative method:
1 1 ( K − K ∆T + N1S + N 2S )i ∆d si+1 = ∆fi 2 3
(21)
where the incremental force and updated displacement vectors are :
1 1 ∆f f = P∆T − ( K − K∆T + N1s + N 2s )i d si 2 3 and
d si +1 = d si + ∆d si +1
The post-buckling behaviors are analyzed repeatedly until it converges to incremental displacement. On the other hand, Eq.(20.b) should be fully solved to present the time-response with geometric nonlinearity resulting from the thermo-elastic behavior of the structure. In this work, the limit-cycle oscillation characteristics are directly obtained using Newmark's method in Ref. [20].
4. Numerical results and discussions
Numerical results of thermal post-buckling and limit-cycle oscillation characteristics of FGMs are summarized for Si3 N 4 / SUS 304 using the neutral surface of the model. Table 1 lists the elasticity modulus, coefficient of thermal expansion and thermal conductivity of constituent materials as in Ref. [14]. To obtain the results, 7x7 elements with nine-node quadratic models are used with a reduced integration method [19] to prevent the transverse
shear locking effect, and the data for simply-supported and clamped boundary conditions are discussed.
4.1. Code verification For the code verification, results are compared with the previous data. Firstly, the nondimensional shift ( z0 / h ) of neutral surface from the mid-plane is obtained as in Fig. 2 to verify the shift amount for the temperature-independent materials. The Figure shows the volume index along neutral surface, and then the maximum of the neutral surface shift amount is 3.8% from mid-plane of FGMs model. And also, the results agree well with the previous work. Also, the maximum value of shift ( k max (T ) ) of z0 (T ) / h is obtained
kmax (T ) = 2
Ec (T ) Em (T )
(24)
At the maximum value of the shift, k is 1.76, and is reasonable with the result in Ref. [21]. Next, Fig.3 represents thermal post-buckling behavior of FGMs according to mid-planes for simply-supported and clamped boundary conditions with a / h = 1 /100 . In the figure, group “A” and "B" confirm correctly with the results in Ref. [22]. 4.2 Neutral surface shifts
Neutral surface shifts are discussed for the temperature-dependent material properties. First of all, there are 3-types of neutral surface shift for FGMs: Power-law (P-), Sigmoid (S-), and Exponential (E-) FGMs: For the simple power-law model:
z0 (T ) k ( Ec (T ) − Em (T )) = h 2(k + 2)(kEm (T ) + Ec (T ))
Next, the sigmoid model:
(25.a)
z0 (T ) k (k + 3)( Ec (T ) − Em (T )) = h 4(k + 1)(k + 2)( Ec (T ) + Em (T ))
(25.b)
Finally, the exponential model:
z0 (T ) = h
1 1 Ec ( ln Ec − Em ) + ln Ec + Em 2 2 ( Ec − Em )(ln Ec )
(25.c)
An interesting thing to note is that the neutral surface location of exponential FGM is a special case for P-FGM with volume index k = e . Fig.4 shows the non-dimensional neutral surface shifts of P- and S-FGMs models. The shift amount of P-FGM increases with the increase of volume index up to maximum value, and the curve drops down asymptotically. However, the neutral surface shift of S-FGMs is increased to limited value 5.32% from mid-plane. Through this observation, the results of P-FGMs are expected to present various effects due to the change of neutral surfaces, and thus the PFGMs model is selected for further discussions from this point in this work. Fig. 5 depicts the effect of temperature variations as a result of the position of neutral surface, and one of the interesting things is that as k increases, then the difference of the deviation amount also increased. In this regard, neutral surface should be used in the analysis especially due to asymmetric of material properties.
4.3. Thermal post-buckling
In this part, thermal post-buckling behavior of FGMs plate is investigated with and without including the heat transfer effects. Also, the temperature is increased continuously by ∆ T from 300K to 300 K + ∆T . Without considering the heat conduction effect, Fig.6(a) and (b) present the center deflections due to thermal post-buckling behavior of simply-supported and clamped boundary conditions of the structures, respectively. Fig.6(a) compares the behavior the deflection based on neutral surface and mid-plane. As temperature increases, the center of the structure in this work moves to down-side more slowly than the results based on the mid-plane. The distribution shows the two equilibrium points for volume fraction k = 1 such as A and B at ∆T = 30 K , and the plate is deformed upward as the temperature decreasing from equilibrium
point. As the temperature decreases, the point B moves to point C, that is to say, a snapthrough could occur because the plate model needs to return to primary equilibrium point, and then the point C jumps to the point D immediately. While the previous snap-through starts at ∆T = 16 K , the points occur early at ∆T = 20 K from equilibrium point based on neutral surface concept. On the other hands, Fig. 6 (b) reveals almost the same results of the deflection for all clamped boundary conditions of model as compared the results in Fig.6(a). This means that clamped boundary conditions of the model is to be more stable than simplysupported edges in the thermal post-buckling behavior. In the next case, the heat transfer effects are considered for the analysis and then the temperature of bottom side with pure metal is expressed Tm and the top side consisting pure ceramic is represented Tc , respectively. In this regards, thermal conduction gradually occurs as temperature change from metal to ceramic in the thickness direction of the structure. Also, neutral surface of the model is selected as the reference plane for the thermal post-buckling analysis. Fig. 7 summarizes the deflection for two kinds of boundary conditions. Fig. 7 (a) represents the model moving to only up-side direction, and this denotes only one equilibrium point on the upper-side at low temperature levels. In the downward direction, the snapthrough appears to jump upwards, and the results are also delayed regardless of the volume fraction. Furthermore, the jumping point occurs later than the results in Fig. 6 (a) without considering the heat transfer effect. Without heat conduction, the result using neutral surface concept has a lower value than the data shown in Fig. 6 (a) while the results have a higher value than the existing data in Fig. 7 (a) considering heat conductions. In Fig. 7 (b), the plates have the same bifurcation points regardless of reference plane. This means that the clamped models are more stable than the simply-supported case.
4.4 Limit-cycle oscillations
The phenomena of Limit-cycle Oscillations (LCO) for FGMs may occur due to the geometric non-linearity of structures in thermal environment. In here, LCO are used estimate the boundary conditions and the reference plane of the structural model. Also, the volume fraction of the material is chosen for the mixture ratio k value of 1 in this analysis. Fig. 8 represents LCO of the model according to reference plane without heat conduction. Fig. 8 (a) shows the amplitudes of the model are increased a little bit for the simply-supported boundary condition case as the reference planes are shifted from mid-planes to the neutral
surfaces. Furthermore, Table 2 shows that the upward amplitudes are increased from 0.0693 to 0.0717, while the downward portions are almost the same. On the other hand, the clamped edge models represent almost the same results regardless of the location of the reference planes shown in Fig. 8 (b). This means that the clamped model is more stable than the simply-supported model for thermal post-buckling analysis. Additionally, the amplitudes of the clamped model are almost the same as those in Table. 2. Fig.9 describes the results that the LCO amplitudes vary with the heat transfer effects. As shown in Fig 9(b), the starting point of LCO including the heat conduction appears to be opposite, and the amplitudes of lower-side are decreased from -0.0739 to -0.0717 as in Table. 2. However, the clamped model has the complete opposite amplitude distribution according to reference plane as shown in Fig. 9 (b).
5. Conclusions In this work, the effects of neutral surface in Functionally Graded Materials (FGMs) plate models are investigated for the thermal post-buckling and Limit-cycle oscillations. Based on the First-order Shear Deformation Theory (FSDT), the matrix form of the governing equations including the geometric non-linear terms is established. To verify the present results, the numerical data is compared with previous literatures. Then, the neutral surface positions are revealed considering the volume fraction and temperatures. Furthermore, the mid-plane and the neutral surface are compared to the reference planes for the temperature-dependent material properties. With and without considering the heat transfer effects, the snap-through points appear early based on neutral surface for the thermal postbuckling analysis. In particular, the thermal deformations are almost the same for the clamped cases, while the simply-supported plates show the deviations in the deflection of the model. Including the heat transfer effects, the snap-through happens earlier than without considering the effects. Also, the bifurcation can occur slower than when using the mid-plane as the reference plane regardless of the volume fraction with clamped boundary conditions. For the LCO behaviors of the plate, the amplitude for simply-supported conditions is increased due to mid-plane as compared with the cases based on neutral surface.
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 2016.
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[14] Reddy JN, Chin CD. Thermomechanical analysis of functionally graded cylinders and plates. J Therm Stress 2007; 21: 593–626. [15] Lee YJ, Bae SI, Kim JH. Thermal buckling behavior of functionally graded plates based on neutral surface. Compost Struct 2016;137:208-14. [16] Javaheri R, Eslami MR. Thermal buckling of functionally graded plates. AIAA J 2002; 40:162–9. [17] 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. [18] Efraim E, Eisenberger M. Exact vibration analysis of variable thickness annular isotropic and FGM plates. J Sound Vib 2007; 299:720-38. [19] Zienkiewicz OC, Taylor RL, Too JM. Reduced integration technique in general analysis of plates and shells. Int J Numer Method Eng 1971;3:275-90. [20] Bathe KJ. Finite element procedures, Prentice-Hall, Englewood Cliffs, NJ, 1996, pp.780-781. [21] 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. [22] Sohn KJ and Kim JH. Structural stability of functionally graded panels subjected to aero-thermal loads. Compos Struct 2008;82:10-8.
List of Tables Table 1. Temperature-dependent material properties of metal and ceramic.
Table 2. Limit-cycle amplitudes according to reference plane. List of Figures
Fig. 1 Neutral surface and mid-plane of FGMs. Fig. 2 Non-dimensional neutral surface shift from mid-plane. Fig. 3 Non-dimensional center deflections for boundary conditions Fig.4 Neutral surface shift of P-FGMs and S-FGMs Fig. 5 Neutral surface shift due to volume index and temperatures. Fig. 6 Center deflection according to reference plane ( T = 300 K + ∆T ) (a) Simply-supported ; (b) Clamped Fig. 7 Center deflection according to reference plane considering heat conduction ( T = 300 K + ∆T ) (a) Simply-supported ; (b) Clamped Fig. 8 LCO of plates according to reference plane without heat transfer effect (a) Simply-supported ; (b) Clamped Fig. 9 LCO of plates according to reference plane with heat transfer effect (a) Simply-supported ; (b) Clamped
Table 1 Temperature-dependents material properties of metal and ceramic.
P−1
Properties
Si3 N 4
E( Pa) α (1/ K ) k (W / mK )
SUS 304
ν E( Pa) α (1/ K ) k (W / mK )
Al2O3
ν E( Pa) α (1/ K ) k (W / mK )
Ni
ν E( Pa) α (1/ K ) k (W / mK )
ν
P0
P1
P2
P3
0
348.43e9
-3.070e-4
2.160e-7
-8.946e-11
0 0
5.8723e-6 13.723
9.095e-4 -1.032e-3
0 5.466e-7
0 -7.876e-11
0
0.24
0
0
0
0 0 0 0
201.04e9 12.33e-6 15.379 0.3262
3.079e-4 8.086e-4 -1.264e-3 0
-6.534e-7 0 2.092e-6 0
0 0 -7.223e-10 0
0 0 -1123.6 0
349.55e9 6.8269e-6 -14.087 0.31
-3.853e-4 1.838e-4 -6.227e-3 0
4.027e-7 0 0 0
-1.673e-10 0 0 0
0
233.95e9
-2.794e-4
3.998e-9
0
0 0 0
9.9209e-6 187.66 0.26
8.705e-4 -2.869e-3 0
0 4.005e-6 0
0 -1.983e-9 0
Table 2 Limit-cycle amplitudes according to reference plane Limit-cycle amplitudes (w/h) Direction
Up-side Down-side
Boundary Condition Simply-supported
Clamped
Mid-plane
Neutral
Mid-plane
Neutral
0.0693 -0.0709
0.0717 -0.0707
0.0612 -0.0572
0.0612 -0.0573
z0 (T )
Neutral surface Mid-plane
Fig. 1. Neutral surface and mid-plane of FGMs.
4 Si3N4/SUS304
3.5
Ref. [16] 3
z0/h
2.5 2 1.5 1 0.5 0
0
10
20
30
40 50 60 Volume index k
70
80
90
100
Fig. 2. Non-dimensional neutral surface shift from mid-plane.
Fig. 3. Non-dimensional center deflection for boundary conditions
6
5 P-FGM S-FGM
z0/h
4
3
2
1
0
0
5
10
15
20 25 30 Volume index k
35
40
45
Fig. 4. Neutral surface shift of P-FGMs and S-FGMs
50
Non-dimesional neutral surface (%)
12 10 8 6 4 2 0 0 20 40 Volume fraction, k
200
400
600
800
1000
1200
Reference Temperature,K
Fig. 5. Neutral surface shift due to volume index and temperatures.
2 k=0 k=1 Neutral surface
Nondimensional Deflection, w/h
1.5
C
1 0.5
E
D
0
A
-0.5
F
-1
B
-1.5 -2
0
5
10
15
20
25 • T, K
30
35
40
45
50
30
35
40
45
50
(a)
2 k=0 k=1 Neutral surface
Nondimensional Deflection, w/h
1.5 1 0.5 0 -0.5 -1 -1.5 -2
0
5
10
15
20
25 • T, K
(b) Fig. 6. Non-dimensional center deflection according to reference plane without heat conduction ( ∆ T = 300 K + ∆ T ) (a) Simply-supported ; (b) Clamped
2 k=0 k=1 Neutral surface
Nondimensional Deflection, w/h
1.5 1 0.5 0 -0.5 -1 -1.5 -2
0
50
100
150
100
150
• T, K
(a) 2.5 k=0 k=1 Neutral surface
2
Nondimensional Deflection, w/h
1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2.5
0
50
• T, K
(b) Fig. 7. Non-dimensional center deflection according to reference plane considering heat conduction( ∆ T = 300 K + ∆ T ) (a) Simply-supported ; (b) Clamped
0.08 Mid-plane Neutral surface
0.06 0.04
w/h
0.02 0 -0.02 -0.04 -0.06 -0.08
0
0.005
0.01
0.015
0.02 0.025 Time (sec)
0.03
0.035
0.04
(a) 0.08 Mid-plane Neutral surface
0.06
0.04
w/h
0.02
0
-0.02
-0.04
-0.06
0
0.005
0.01
0.015
0.02 0.025 Time (sec)
0.03
0.035
0.04
(b)
Fig.8. LCO of plates according to reference plane without heat transfer effect (a) Simply-supported ; (b) Clamped
0.08 Mid-plane Neutral surface
0.06 0.04
w/h
0.02 0 -0.02 -0.04 -0.06 -0.08
0
0.005
0.01
0.015
0.02 0.025 Time (sec)
0.03
0.035
0.04
(a)
0.08 Mid-plane Neutral surface
0.06 0.04
w/h
0.02 0 -0.02 -0.04 -0.06 -0.08
0
0.005
0.01
0.015
0.02 0.025 Time (sec)
0.03
0.035
0.04
(b) Fig.9 LCO of plates according to reference plane with heat transfer effect (a) Simply-supported ; (b) Clamped
S0263-8223(16)32087-6 http://dx.doi.org/10.1016/j.compstruct.2016.12.079 COST 8146
To appear in:
Composite Structures
Received Date: Revised Date: Accepted Date:
7 October 2016 15 December 2016 27 December 2016
Please cite this article as: Lee, Y-H., Kim, J-H., Thermal-structural Post-bucking and Limit-Cycle Oscillation of Functionally Graded Materials, Composite Structures (2017), doi: http://dx.doi.org/10.1016/j.compstruct. 2016.12.079
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Thermal-structural Post-bucking and Limit-Cycle Oscillation of Functionally Graded Materials
Young-Hoon Lee*, Ji-Hwan Kim†
* Department
of Mechanical and Aerospace Engineering, College of Engineering, Seoul National University, Seoul, 08826,South Korea [email protected],
† Institute
of Advanced Aerospace Technology, Department of Mechanical and Aerospace
Engineering, College of Engineering, Seoul National University, Seoul, 08826,South Korea [email protected]
Key words: Functionally Graded Materials; Thermal post-buckling; Limit-cycle oscillation; Neutral surface
ABSTRACT
Thermal post-buckling and limit-cycle oscillation characteristics of Functionally Graded Material (FGM) structures are investigated based on the neutral surface concept. In particular, the material properties are non-homogeneous and vary gradually from one surface to the other. Furthermore, the properties are to be considered as temperature-dependent characteristics, and the neutral surface concept is adopted instead of the mid-plane to consider the reference plane due to the asymmetric properties in the thickness direction of model. In the formulation, the First-order Shear Deformation Theory (FSDT) of plate is used, and the geometric nonlinearity is accounted for by the von Karman strain-displacement relations. Also, steady state thermal conduction effects are assumed as aone
dimensional heat transfer on the surface of the structure. For the numerical analysis, the NewtonRaphson method is applied to solve the thermal post-buckling behavior, while Newmark's time integration method is employed to resolve the limit-cycle oscillation. In order to validate the analysis results, the results of this paper based on the neutral surface are compared with the data from previous papers using the conventional approach for FGMs model. Finally, effects of the neutral surface on the non-linear thermo-mechanic behavior of structure are discussed in detail.
1. Introduction FGMs are generated by manufacturing composite materials in a high temperature state. Specifically, the materials are made up of the continuous mixture of ceramic and metal, and then the properties form a smooth and continuous pattern from one surface to the other. And, ceramic has a high compressive strength and heat resistance with low fracture toughness, while metal exhibits better mechanical strength but has lower heat resistance. These attractive merits have increased the roles of FGMs in various high quality engineering fields. For this reason, numerous research works have been performed on thermal buckling and post-buckling analysis for FGMs. Li et al. [1] presented the post-buckling of
Timoshenko
beam models under the non-uniform temperature increase. Liew et al. [2] reported the buckling and post-buckling behavior of laminated rectangular plates with uniform temperature change. Shen [3] discussed the post-buckling behavior of model under thermal loading. While, Lee and Kim [4] studied the supersonic aero-thermo post-buckling behaviors and limit-cycle oscillations of panels using Newmark's time integration method. Prakash et al. [5] pointed out the post-buckling characteristic of the skew plates under thermal load using finite element approach. Na and Kim [6] researched the thermal postbuckling behaviors of functionally graded plates using 3-D finite element method. Further, Lee and Kim [7] performed the thermo-mechanical analysis of panels in hypersonic airflows. Also, Prakash and Ganapathi [8] investigated the influence of thermal environment on the structures for supersonic flutter using finite element formulation. Bateni et al. [9] analyzed the instability of rectangular plate models under the thermal as well as mechanical loadings. On the other hand, Ibrahim et al. [10] studied the non-linear flutter and thermal buckling of panels considering not only elevated temperature but also aerodynamic loading. Furthermore, Haddapour et al. [11] revealed the nonlinear aero-elastic performance of plates in supersonic flow. For the analysis of FGMs,the neutral surface must be used as a reference plane due to the asymmetric material properties in the thickness direction. Zhang and Zhou [12] investigated a theoretical analysis of the thin plates based on the physical neutral surface. Furthermore, Prakash and Singha [13] studied nonlinear characteristics of skew plates under in-plane load. In this paper, thermal post-buckling and limit-cycle oscillation characteristics of FGM plates are analyzed based on the neutral surface of the plate. The structural models of plates are considered as temperature-dependent materials and the models are based on the FSDT of plate. Furthermore, the material properties of the model are continuously varied in the
thickness direction of the model. In order to compare the present results with the previous data based on the mid-plane of the model as the reference plane, the non-linear thermalstructural post-buckling and limit-cycle oscillation behaviors are discussed in detail. Also, the amount of neutral surface shift depends on the temperature and the volume fractions of the constituent materials. Additionally, in order to discuss the present works, the cases of models are considered with and without heat conduction effects. The shifts are discussed for the cases of Power-law (P-), Sigmoid (S-), and Exponential (E-) FGMs.
2. Material properties and 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 with respect to the middle plane. Temperature-dependent material properties are presented for FGMs with length a , width b and thickness h . Then, in this work, the FSDT of plate is adopted in this study and heat conduction effect is considered as well.
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. Then, 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. While, the temperature-dependent material properties P(T ) can be written as in Ref.
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 material point 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
As shown in Fig.1, the model is based on the continuous distribution of metal to ceramic in the thickness direction. First of all, the mass distribution is asymmetric in the thickness direction, and thus the neutral surface concept should be used to account for this. In this study, the neutral surface is chosen as a reference plane for deformation based on the force equilibrium of FGM the models. Considering the thermal effects on the FGMs, integration in the thickness direction determines the location of neutral surface in the model. Thus, the first-moment of elasticity modulus E ( z, T ) equals to zero as in Ref.[15]:
h 2 h − 2
∫
E ( z, T )( z − z0 (T ))dz = 0
(4)
Therefore, the location 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
Using this formula, temperature-dependent neutral surface shifts z0 (T ) are calculated for P-, S-, and E-type FGMs considering the specific Young's modulus.
2.3. Temperature rise condition
In this study, two cases are analyzed with uniform temperature rise and heat transfer in the thickness direction of the model [16]. Firstly, initial temperature is uniformly raised to a final value Tl with uniform temperature distribution. Then, the temperature variation across the plate thickness can be written as:
∆T = Tl − Ti
(6)
While, the model includes heat transfer effects, the heat conduction equation in the thickness of the model as [14] is to be solved:. d dT h h K ( z, T ) = 0, T − = Tm , T = Tc dz dz 2 2
(7)
where K ( z, T ) is a function of heat conduction through the thickness as,
z 1 K ( z, T ) = K m + ( K c − K m ) + h 2
k
(8)
Thus, temperature distribution across the plate thickness becomes [17] ~
T = Tc − (Tc − Tm
∫ ) ∫
z
− h /2 h /2 − h /2
3. Governing equations
(1/ K ( z , T ))dz (9)
(1/ K ( z , T ))dz
The FGMs model is to be analyzed with consideration of the FSDT of plate with heat conduction effects as well and accounting for the physical neutral surface of the structures with the finite element method.
3.1. Equations of motion
Displacement fields based on the neutral surface z0 (T ) can be written as [15] :
u ( x , y , z , T ) = u0 ( x , y ) + ( z − z0 (T ))φ x ( x , y ) v ( x , y , z , T ) = v0 ( x , y ) + ( z − z0 (T ))φ y ( x , y )
(10)
w( x, y , z , T ) = w0 ( x, y )
where u, v and w are the displacements in the x , y and z directions, while ϕ x and ϕ y stand for 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 (T ) = − M b (T ) B(T ) D (T ) κ M ∆T (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
(11.a)
(11.b)
where N b (T ) , M b (T ) and Q(T ) denote the temperature-dependent in-plane force, the moment and the transverse shear force resultant vectors, respectively. Also, ε 0 , κ and γ are strain vectors based on mid-plane, curvature and transverse shear, respectively. Furthermore, k p is the shear correction factor of the plate structure considering the variation of volume
fractions for ceramic and metal as Ref. [18].
kp =
5 6 − (ν cVc + ν mVm )
where ν c and ν m denote the Poisson's ratio of ceramic and metal, respectively.
(12)
Further, N ∆T (T ) and M ∆T (T ) are the thermal in-plane force resultant and the thermal moment resultant vectors with the expressions as:
N ∆Tx ( N ∆T (T ), M ∆T (T ) ) = N∆Ty N ∆Txy
M ∆Tx M ∆Ty M ∆Txy α ( z, T ) h /2 ~ = ∫ (1, ( z − z0 (T )))[ E ] α ( z , T ) ∆ T ( z )dz − h/ 2 0
(13)
inhere 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
(14)
while A(T ) , B(T ) , D(T ) and S (T ) are the in-plane, in-plane bending coupling , bending and transverse shear stiffness matrixes as
( A(T ), B (T ), D(T )) = ∫
h/ 2
− h /2
[ E ](1, ( z − z0 (T )), ( z − z0 (T )) 2 ) dz
(15)
To derive the equations of motion of the FGMs model, principle of virtual work is applied:
δ W = δ Wint − δ Wext = 0
(16)
now δ Wint and δWext represent the internal and external virtual works, respectively. At first,
δ Wint = ∫ δ eT dV = ∫ [δε T N b (T ) + δκ T M b (T ) + δγ T Q(T )]dA V
(17)
A
= δ d T [ K e − K ∆T
1 1 + N1(T ) + N 2(T )]d 2 3
In here, d = [u , v, w, φx , φ y ]T is the displacement vector. In addition, K e 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 (T ) .
Also, N1(T ) , N 2(T ) and P∆T are matrices
denoting the first-order and second-order non-linear stiffness, respectively. On the other hand,
δ Wext = δ d T f = ∫ ( P∆T )δ wdA = −δ d T P∆T
(18)
A
where the P∆T is thermal load vector. From now on, the notations of temperature-dependency in nonlinear terms are omitted for simplicity in the formula. Then, the governing equations of the FGMs plates in matrix form is expressed as:
ii
M d + ( K − K ∆T +
1 1 N 1 + N 2) d = P∆T 2 3
(19)
where the P∆T is thermal load vector.
3.2. Iterative procedures
In order to solve the time-dependent non-linear equations as Eq.(1920),The solution of Eq. (20) the solution is assumed as d = d s + ∆dt with d s and dt representing the time-independent and time-dependent solution parts, respectively. Then, two sets of the coupled governing equations are obtained as:
1 1 ( K − K ∆T + N1S + N 2S )d s = P∆T 2 3
(20.a)
and ii
M d t + ( K − K ∆T +
1 1 1 1 N1s + N 2 s + N 2 st + N 1t + N 2t )d t = 0 2 3 2 3
(20.b)
where the subscripts s and t denote the static and dynamic states, respectively. That is to say,
the Eq. (20.a) represents nonlinear static problem analysis such as thermal post-buckling analysis behaviors. On the other hand, the Eq. (20.b) stands for the nonlinear dynamic problem in the thermal environment for vibration and flutter behaviors, and also the static equation should be solved preliminarily due to the governing equations being coupled. In this section, a solution procedure is presented briefly. At the beginning, in order to analyze the thermal post-buckling behavior, the incremental form of Eq. (20.a) is obtained by using the Newton-Raphson iterative method:
1 1 ( K − K ∆T + N1S + N 2S )i ∆d si+1 = ∆fi 2 3
(21)
where the incremental force and updated displacement vectors are :
1 1 ∆f f = P∆T − ( K − K∆T + N1s + N 2s )i d si 2 3 and
d si +1 = d si + ∆d si +1
The post-buckling behaviors are analyzed repeatedly until it converges to incremental displacement. On the other hand, Eq.(20.b) should be fully solved to present the time-response with geometric nonlinearity resulting from the thermo-elastic behavior of the structure. In this work, the limit-cycle oscillation characteristics are directly obtained using Newmark's method in Ref. [20].
4. Numerical results and discussions
Numerical results of thermal post-buckling and limit-cycle oscillation characteristics of FGMs are summarized for Si3 N 4 / SUS 304 using the neutral surface of the model. Table 1 lists the elasticity modulus, coefficient of thermal expansion and thermal conductivity of constituent materials as in Ref. [14]. To obtain the results, 7x7 elements with nine-node quadratic models are used with a reduced integration method [19] to prevent the transverse
shear locking effect, and the data for simply-supported and clamped boundary conditions are discussed.
4.1. Code verification For the code verification, results are compared with the previous data. Firstly, the nondimensional shift ( z0 / h ) of neutral surface from the mid-plane is obtained as in Fig. 2 to verify the shift amount for the temperature-independent materials. The Figure shows the volume index along neutral surface, and then the maximum of the neutral surface shift amount is 3.8% from mid-plane of FGMs model. And also, the results agree well with the previous work. Also, the maximum value of shift ( k max (T ) ) of z0 (T ) / h is obtained
kmax (T ) = 2
Ec (T ) Em (T )
(24)
At the maximum value of the shift, k is 1.76, and is reasonable with the result in Ref. [21]. Next, Fig.3 represents thermal post-buckling behavior of FGMs according to mid-planes for simply-supported and clamped boundary conditions with a / h = 1 /100 . In the figure, group “A” and "B" confirm correctly with the results in Ref. [22]. 4.2 Neutral surface shifts
Neutral surface shifts are discussed for the temperature-dependent material properties. First of all, there are 3-types of neutral surface shift for FGMs: Power-law (P-), Sigmoid (S-), and Exponential (E-) FGMs: For the simple power-law model:
z0 (T ) k ( Ec (T ) − Em (T )) = h 2(k + 2)(kEm (T ) + Ec (T ))
Next, the sigmoid model:
(25.a)
z0 (T ) k (k + 3)( Ec (T ) − Em (T )) = h 4(k + 1)(k + 2)( Ec (T ) + Em (T ))
(25.b)
Finally, the exponential model:
z0 (T ) = h
1 1 Ec ( ln Ec − Em ) + ln Ec + Em 2 2 ( Ec − Em )(ln Ec )
(25.c)
An interesting thing to note is that the neutral surface location of exponential FGM is a special case for P-FGM with volume index k = e . Fig.4 shows the non-dimensional neutral surface shifts of P- and S-FGMs models. The shift amount of P-FGM increases with the increase of volume index up to maximum value, and the curve drops down asymptotically. However, the neutral surface shift of S-FGMs is increased to limited value 5.32% from mid-plane. Through this observation, the results of P-FGMs are expected to present various effects due to the change of neutral surfaces, and thus the PFGMs model is selected for further discussions from this point in this work. Fig. 5 depicts the effect of temperature variations as a result of the position of neutral surface, and one of the interesting things is that as k increases, then the difference of the deviation amount also increased. In this regard, neutral surface should be used in the analysis especially due to asymmetric of material properties.
4.3. Thermal post-buckling
In this part, thermal post-buckling behavior of FGMs plate is investigated with and without including the heat transfer effects. Also, the temperature is increased continuously by ∆ T from 300K to 300 K + ∆T . Without considering the heat conduction effect, Fig.6(a) and (b) present the center deflections due to thermal post-buckling behavior of simply-supported and clamped boundary conditions of the structures, respectively. Fig.6(a) compares the behavior the deflection based on neutral surface and mid-plane. As temperature increases, the center of the structure in this work moves to down-side more slowly than the results based on the mid-plane. The distribution shows the two equilibrium points for volume fraction k = 1 such as A and B at ∆T = 30 K , and the plate is deformed upward as the temperature decreasing from equilibrium
point. As the temperature decreases, the point B moves to point C, that is to say, a snapthrough could occur because the plate model needs to return to primary equilibrium point, and then the point C jumps to the point D immediately. While the previous snap-through starts at ∆T = 16 K , the points occur early at ∆T = 20 K from equilibrium point based on neutral surface concept. On the other hands, Fig. 6 (b) reveals almost the same results of the deflection for all clamped boundary conditions of model as compared the results in Fig.6(a). This means that clamped boundary conditions of the model is to be more stable than simplysupported edges in the thermal post-buckling behavior. In the next case, the heat transfer effects are considered for the analysis and then the temperature of bottom side with pure metal is expressed Tm and the top side consisting pure ceramic is represented Tc , respectively. In this regards, thermal conduction gradually occurs as temperature change from metal to ceramic in the thickness direction of the structure. Also, neutral surface of the model is selected as the reference plane for the thermal post-buckling analysis. Fig. 7 summarizes the deflection for two kinds of boundary conditions. Fig. 7 (a) represents the model moving to only up-side direction, and this denotes only one equilibrium point on the upper-side at low temperature levels. In the downward direction, the snapthrough appears to jump upwards, and the results are also delayed regardless of the volume fraction. Furthermore, the jumping point occurs later than the results in Fig. 6 (a) without considering the heat transfer effect. Without heat conduction, the result using neutral surface concept has a lower value than the data shown in Fig. 6 (a) while the results have a higher value than the existing data in Fig. 7 (a) considering heat conductions. In Fig. 7 (b), the plates have the same bifurcation points regardless of reference plane. This means that the clamped models are more stable than the simply-supported case.
4.4 Limit-cycle oscillations
The phenomena of Limit-cycle Oscillations (LCO) for FGMs may occur due to the geometric non-linearity of structures in thermal environment. In here, LCO are used estimate the boundary conditions and the reference plane of the structural model. Also, the volume fraction of the material is chosen for the mixture ratio k value of 1 in this analysis. Fig. 8 represents LCO of the model according to reference plane without heat conduction. Fig. 8 (a) shows the amplitudes of the model are increased a little bit for the simply-supported boundary condition case as the reference planes are shifted from mid-planes to the neutral
surfaces. Furthermore, Table 2 shows that the upward amplitudes are increased from 0.0693 to 0.0717, while the downward portions are almost the same. On the other hand, the clamped edge models represent almost the same results regardless of the location of the reference planes shown in Fig. 8 (b). This means that the clamped model is more stable than the simply-supported model for thermal post-buckling analysis. Additionally, the amplitudes of the clamped model are almost the same as those in Table. 2. Fig.9 describes the results that the LCO amplitudes vary with the heat transfer effects. As shown in Fig 9(b), the starting point of LCO including the heat conduction appears to be opposite, and the amplitudes of lower-side are decreased from -0.0739 to -0.0717 as in Table. 2. However, the clamped model has the complete opposite amplitude distribution according to reference plane as shown in Fig. 9 (b).
5. Conclusions In this work, the effects of neutral surface in Functionally Graded Materials (FGMs) plate models are investigated for the thermal post-buckling and Limit-cycle oscillations. Based on the First-order Shear Deformation Theory (FSDT), the matrix form of the governing equations including the geometric non-linear terms is established. To verify the present results, the numerical data is compared with previous literatures. Then, the neutral surface positions are revealed considering the volume fraction and temperatures. Furthermore, the mid-plane and the neutral surface are compared to the reference planes for the temperature-dependent material properties. With and without considering the heat transfer effects, the snap-through points appear early based on neutral surface for the thermal postbuckling analysis. In particular, the thermal deformations are almost the same for the clamped cases, while the simply-supported plates show the deviations in the deflection of the model. Including the heat transfer effects, the snap-through happens earlier than without considering the effects. Also, the bifurcation can occur slower than when using the mid-plane as the reference plane regardless of the volume fraction with clamped boundary conditions. For the LCO behaviors of the plate, the amplitude for simply-supported conditions is increased due to mid-plane as compared with the cases based on neutral surface.
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 2016.
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List of Tables Table 1. Temperature-dependent material properties of metal and ceramic.
Table 2. Limit-cycle amplitudes according to reference plane. List of Figures
Fig. 1 Neutral surface and mid-plane of FGMs. Fig. 2 Non-dimensional neutral surface shift from mid-plane. Fig. 3 Non-dimensional center deflections for boundary conditions Fig.4 Neutral surface shift of P-FGMs and S-FGMs Fig. 5 Neutral surface shift due to volume index and temperatures. Fig. 6 Center deflection according to reference plane ( T = 300 K + ∆T ) (a) Simply-supported ; (b) Clamped Fig. 7 Center deflection according to reference plane considering heat conduction ( T = 300 K + ∆T ) (a) Simply-supported ; (b) Clamped Fig. 8 LCO of plates according to reference plane without heat transfer effect (a) Simply-supported ; (b) Clamped Fig. 9 LCO of plates according to reference plane with heat transfer effect (a) Simply-supported ; (b) Clamped
Table 1 Temperature-dependents material properties of metal and ceramic.
P−1
Properties
Si3 N 4
E( Pa) α (1/ K ) k (W / mK )
SUS 304
ν E( Pa) α (1/ K ) k (W / mK )
Al2O3
ν E( Pa) α (1/ K ) k (W / mK )
Ni
ν E( Pa) α (1/ K ) k (W / mK )
ν
P0
P1
P2
P3
0
348.43e9
-3.070e-4
2.160e-7
-8.946e-11
0 0
5.8723e-6 13.723
9.095e-4 -1.032e-3
0 5.466e-7
0 -7.876e-11
0
0.24
0
0
0
0 0 0 0
201.04e9 12.33e-6 15.379 0.3262
3.079e-4 8.086e-4 -1.264e-3 0
-6.534e-7 0 2.092e-6 0
0 0 -7.223e-10 0
0 0 -1123.6 0
349.55e9 6.8269e-6 -14.087 0.31
-3.853e-4 1.838e-4 -6.227e-3 0
4.027e-7 0 0 0
-1.673e-10 0 0 0
0
233.95e9
-2.794e-4
3.998e-9
0
0 0 0
9.9209e-6 187.66 0.26
8.705e-4 -2.869e-3 0
0 4.005e-6 0
0 -1.983e-9 0
Table 2 Limit-cycle amplitudes according to reference plane Limit-cycle amplitudes (w/h) Direction
Up-side Down-side
Boundary Condition Simply-supported
Clamped
Mid-plane
Neutral
Mid-plane
Neutral
0.0693 -0.0709
0.0717 -0.0707
0.0612 -0.0572
0.0612 -0.0573
z0 (T )
Neutral surface Mid-plane
Fig. 1. Neutral surface and mid-plane of FGMs.
4 Si3N4/SUS304
3.5
Ref. [16] 3
z0/h
2.5 2 1.5 1 0.5 0
0
10
20
30
40 50 60 Volume index k
70
80
90
100
Fig. 2. Non-dimensional neutral surface shift from mid-plane.
Fig. 3. Non-dimensional center deflection for boundary conditions
6
5 P-FGM S-FGM
z0/h
4
3
2
1
0
0
5
10
15
20 25 30 Volume index k
35
40
45
Fig. 4. Neutral surface shift of P-FGMs and S-FGMs
50
Non-dimesional neutral surface (%)
12 10 8 6 4 2 0 0 20 40 Volume fraction, k
200
400
600
800
1000
1200
Reference Temperature,K
Fig. 5. Neutral surface shift due to volume index and temperatures.
2 k=0 k=1 Neutral surface
Nondimensional Deflection, w/h
1.5
C
1 0.5
E
D
0
A
-0.5
F
-1
B
-1.5 -2
0
5
10
15
20
25 • T, K
30
35
40
45
50
30
35
40
45
50
(a)
2 k=0 k=1 Neutral surface
Nondimensional Deflection, w/h
1.5 1 0.5 0 -0.5 -1 -1.5 -2
0
5
10
15
20
25 • T, K
(b) Fig. 6. Non-dimensional center deflection according to reference plane without heat conduction ( ∆ T = 300 K + ∆ T ) (a) Simply-supported ; (b) Clamped
2 k=0 k=1 Neutral surface
Nondimensional Deflection, w/h
1.5 1 0.5 0 -0.5 -1 -1.5 -2
0
50
100
150
100
150
• T, K
(a) 2.5 k=0 k=1 Neutral surface
2
Nondimensional Deflection, w/h
1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2.5
0
50
• T, K
(b) Fig. 7. Non-dimensional center deflection according to reference plane considering heat conduction( ∆ T = 300 K + ∆ T ) (a) Simply-supported ; (b) Clamped
0.08 Mid-plane Neutral surface
0.06 0.04
w/h
0.02 0 -0.02 -0.04 -0.06 -0.08
0
0.005
0.01
0.015
0.02 0.025 Time (sec)
0.03
0.035
0.04
(a) 0.08 Mid-plane Neutral surface
0.06
0.04
w/h
0.02
0
-0.02
-0.04
-0.06
0
0.005
0.01
0.015
0.02 0.025 Time (sec)
0.03
0.035
0.04
(b)
Fig.8. LCO of plates according to reference plane without heat transfer effect (a) Simply-supported ; (b) Clamped
0.08 Mid-plane Neutral surface
0.06 0.04
w/h
0.02 0 -0.02 -0.04 -0.06 -0.08
0
0.005
0.01
0.015
0.02 0.025 Time (sec)
0.03
0.035
0.04
(a)
0.08 Mid-plane Neutral surface
0.06 0.04
w/h
0.02 0 -0.02 -0.04 -0.06 -0.08
0
0.005
0.01
0.015
0.02 0.025 Time (sec)
0.03
0.035
0.04
(b) Fig.9 LCO of plates according to reference plane with heat transfer effect (a) Simply-supported ; (b) Clamped