Geometrically nonlinear rapid surface heating of temperature-dependent FGM arches

Aerospace Science and Technology 90 (2019) 264–274

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Geometrically nonlinear rapid surface heating of temperature-dependent FGM arches M. Javani a , Y. Kiani b , M.R. Eslami a,∗ a b

Mechanical Engineering Department, Amirkabir University of Technology, Tehran, Iran Faculty of Engineering, Shahrekord University, Shahrekord, Iran

a r t i c l e

i n f o

Article history: Received 3 October 2018 Received in revised form 26 April 2019 Accepted 26 April 2019 Available online 3 May 2019 Keywords: Thermally induced vibrations FGM Shallow arch von Kármán non-linearity Generalized differential quadrature method Hybrid Picard-Newmark method

a b s t r a c t Based on the nonlinear dynamic analysis, thermally induced vibrations of the FGM shallow arches subjected to different sudden thermal loads are studied. Temperature and position dependence of the material properties are taken into account. Based on the uncoupled thermoelasticity assumptions, The non-linear one-dimensional transient heat conduction equation is solved numerically by a hybrid iterative GDQ method and Crank-Nicolson time marching scheme. A first order shear deformation arch theory (FSDT) is also combined with the von Kármán type of geometrical non-linearity and the Donnell kinematic assumption to obtain the equations of motion employing the Hamilton principle. Discretization of the highly coupled non-linear equations of motion is done by using the GDQ method in the arch domain. The solution of the system of the ordinary differential equations is established by means of a hybrid iterative Picard-Newmark scheme. Comparison is also made with the existing results for the case of isotropic homogeneous shallow arches, where good agreement is obtained. Also, parametric studies are proposed to show the effects of temperature dependency, geometrical non-linearity, arch thickness, power law index, and the type of thermal-mechanical boundary conditions upon the arch deflection. © 2019 Elsevier Masson SAS. All rights reserved.

1. Introduction Functionally graded materials (FGMs) as a novel class of materials have shown increasing attention in the last two decades. Especially when thermal and mechanical loads are linked together, interesting features are depicted using FGM solid structures [1–6]. For the first time, thermally induced vibration phenomenon was investigated by Boley [7]. Considering the beam-like structure with simply supported edges, he obtained the dynamic response of the beam under rapid heating. In his problem, a variable called the inertia parameter was defined as follows

B=



t T /t M

where t T and t M are the thermal and mechanical time specifications which depend on geometry and thermo-mechanical properties. His results show that the inertia parameter determines the divergence of dynamic and quasi-static responses. The dynamic response is obtained where the temperature profile is inserted into the equations of motion. On the other hand, the quasi-static re-

*

Corresponding author. E-mail addresses: [email protected] (M. Javani), [email protected] (Y. Kiani), [email protected] (M.R. Eslami). https://doi.org/10.1016/j.ast.2019.04.049 1270-9638/© 2019 Elsevier Masson SAS. All rights reserved.

sponse is obtained when the temperature profile is inserted into the equilibrium equations. In the other words, when the inertia parameter is small enough, thermally induced vibration occurs, contrariwise, quasi-static response occurs when the inertia parameter is large. As a conclusion, for thick beams, plates, and shells, dynamic and quasi-static responses are overlapped. Dynamic response caused by rapid heating was not carried out only for beams, other researchers exhibited the thermally induced vibrations for different structures with several thermal shocks. Boley and Barber [8], based on classical plate theory, analyzed a rectangular plate which is subjected to thermal shock on the upper surface when the bottom one is thermally insulated. They verified that thermally induced dynamic response takes place when the inertia parameter is small. Also, Kraus [9] conducted an axisymmetric vibration analysis for non-shallow spherical shell under rapid heating. Results of his research approved Boley’s works. Using superposition method, Venkataramana and Jana [10] obtained the solution as the sum of static solution and dynamic one of a beam subjected to a harmonically varying temperature. Total deflection tends to oscillate about the static response. Bruch et al. [11] introduced a control strategy for a beam under thermal shock and showed how deflection and velocity of a beam is sharply reduced by using the active control. Manolis and Beskos [12] carefully examined the dynamic behavior of beam-like structure utiliz-

M. Javani et al. / Aerospace Science and Technology 90 (2019) 264–274

ing numerical Laplace transform in time-domain with considering the in-plane loading and damping effects. Thermally induced deflection is conducted with emphasize on the superposition method for an orthotropic plate with various boundary conditions by Tauchert [13]. Das [14] performed the vibration analysis of a plate with an arbitrary polygonal shape which is rapidly heated. Complex variables method is utilized to obtain the dynamic deflection of the plate. Based on a higher order plate theory, to take into account the effect of transverse shear and normal deformation and 2-D finite element method, Chang [15] analyzed a thin laminated plate which is imposed to heat flux on the top surface. Stroud and Mayers [16] used a modified method based on the Reissner variational energy principle to obtain the stresses and deflections of rectangular plates which are subjected to an arbitrary temperature distribution history. Material properties are considered as temperature-dependent. As concluded, omission of temperature dependency causes errors. Alipour et al. [17] analyzed the rapid heating response of the FGM moderately thick plates with different boundary conditions, where thermo-elastic properties are assumed to be temperature-dependent. The 2-D Generalized differential quadrature method is utilized to discrete the equations of motion. These equations are traced in time by means of the well-known Newmark time-marching scheme. Separation of variables procedure as an analytical method, finite element analysis as a numerical method, and an experimental study to analyze the axisymmetric thermally induced response of circular plate are performed by Nakajo and Hayashi [18]. By considering the results of dynamic response of circular plates with clamped support, which is affected strongly by the in-plane force, it is shown that geometrically linear theory is not correct. Based on the conventional Ritz formulation and von Kármán nonlinear theory, Kiani and Eslami [19] investigated the axisymmetric thermal loading and dynamic response of a solid circular FGM plate. Also Javani et al. [20] applied the GDQ method to analyze the thermally induced vibrations in FGM annular plates. Ghiasian et al. [21] performed an investigation on the large amplitude thermally induced vibration of shear deformable beams using the Ritz formulation and hybrid Newton-Raphson-Newmark method. They showed that beams and plates which have not initial curvature with clamped edges are capable of compensating the additional moments to keep the straight state until a prescribed time in which dynamic instability occurs and structure vibrates. Response of plates and shallow shells made of viscoelastic materials based on geometrically nonlinear deflection subjected to rapid heating are analyzed by Hill and Mazumdar [22]. They used an alternative method which is basically originated from the other works of the researchers for the case of linear strain-displacement relations [23–25]. A convenient Navier method for simply supported boundary conditions is developed by Huang and Tauchert [26] to study the thermally induced vibration of composite doubly curved shells. Huang and Tauchert [27] modified their previous work by using the finite element method and considering the nonlinear strain-displacement relations. They showed that the quasistatic response of laminated shallow shells diverges when linear theory (small deflection) is used. Also, dynamic snap-through phenomenon occurs when using the non-linear strain-displacement relations. thermally induced vibration and dynamic snap-through of shallow arches was analyzed by Keibolahi et al. [28,29]. In this research the authors showed that, by definition of non-dimensional geometric parameter λ, dynamic buckling occurs when λ is large enough. Khdeir [30,31] investigated the thermally induced vibration of cross-ply laminated shallow shells and arches under thermal shock, respectively. An analytical method is proposed to obtain the dynamic response. Chang and Shyong [32] employed the finite element method based on the coupled thermoelasticity to obtain the dynamic deflection of circular cylindrical laminated

265

Fig. 1. Schematic and geometric characteristic of a shallow arch.

shells. Raja et al. [33] analyzed the piezo-hygro-thermo-elastic response of composite plates and shells. Kumar et al. [34] studied the thermally induced deflection and velocity control of cylindrical shells utilizing the piezoelectric sensors and actuators. Dynamic response of the FGM sandwich plates and shell panels subjected to rapid heating is investigated by Pandey and Pradyumna [35]. They presented the finite element method based on a higher order layerwise theory to obtain the deflection of shell panels, where thermo-mechanical properties are considered as temperature dependent. In the present research, a large amplitude thermally induced vibration analysis is investigated for shallow FGM arches under different cases of rapid heating on the top and bottom surfaces. The thermo-mechanical properties of the arch are considered to be functions of temperature and thickness coordinate. To acquire the thermally induced forces and moments acting on the arch, the one-dimensional heat conduction equation through the arch thickness is established and solved iteratively according to a hybrid generalized differential quadrature (GDQ)-Crank-Nicolson method. Based on the first order shear deformation arch theory, the von Kármán type of geometrical nonlinearity, generalized differential quadrature method, and hybrid Picard-Newmark method, the dynamic and quasi-static responses of shallow arches with arbitrary boundary conditions at edges are obtained. Finally, parametric studies show that, all of the thermo-elastic parameters such as temperature-dependency, aspect ratio, power law index, boundary conditions, and geometrically nonlinear strain-displacement relations affect the temporal evolution characteristics of the arch. 2. Fundamental equations of FG shallow arch Consider a shallow arch made of functionally graded materials with cross section b × h, radius of curvature R, and opening angle θ0 referred to the polar coordinates system (z, θ) with its origin located at the mid-surface end of the arch, as shown in Fig. 1. Here, θ and z represent the circumferential and through-the-thickness directions, respectively. Thermo-mechanical properties of an FGM circular arch should be defined according to a proper homogenization method. The Voigt rule is commonly used for this reason [36–40]. According to this rule, the mechanical and thermal properties of the FGM arch, such as Young’s modulus E, Poisson’s ratio ν , thermal expansion coefficient α , mass density ρ , specific heat C v , and thermal conductivity K are assumed as the linear function of the volume fractions of the ceramic V c and metal V m . Thus, as a function of thickness direction, a non-homogeneous property of the arch, P , may be expressed in the form

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M. Javani et al. / Aerospace Science and Technology 90 (2019) 264–274

P ( z, T ) = P m ( T ) + V c ( z) P cm ( T ), P cm ( T ) = P c ( T ) − P m ( T )

(1)

where the subscripts m and c represent the properties of metal and ceramic constituents, respectively. Temperature dependency of the material properties are assumed to follow the Touloukian model as [37]

P ( T ) = P 0 ( P −1 T −1 + 1 + P 1 T + P 2 T 2 + P 3 T 3 )

Vc =

1 2

+

z

ζ

h

Vm = 1 − Vc

,

Based on the FSDT, the stress resultants are related to the stress components through the following equations +0.5h

(Nθ θ , Mθ θ , Q θ z ) =

(2)

In which T is the temperature measured in Kelvin and P i ’s are constants and unique to each constituent. A power law distribution of the constituents across the beam thickness may be used to represent the ceramic volume fraction V c and metal volume fraction V m such as



6. Stress resultants

(7)

Substituting Eq. (6) into Eq. (7) with the aid of Eqs. (4) and (5), one obtains the stress resultants in terms of the mid-plane displacements as

⎧ ⎫ ⎨ Nθ θ ⎬

⎡

A 11 M θ θ = ⎣ B 11 ⎩ ⎭ 0 Q θz

(3)

Here, ζ is a non-negative constant called the power law index and dictates the property dispersion profile. Obviously, ceramic rich surface is at the inner surface of the arch (z = +h/2) and metal rich surface is at the outer surface of the arch (z = −h/2).

(σθ θ , zσθ θ , τθ z )dz

−0.5h

−

⎧ T ⎫ ⎨N ⎬

B 11 D 11 0

⎤⎧

0 ⎨ 0 ⎦ ⎩ A 44

1 ( v ,θ R

− w) + 1 R

1 R

ϕ,θ

1 2R 2

w ,θ + ϕ

MT ⎭ 0

⎩

⎫

w 2,θ ⎬

⎭ (8)

In the above equations, the constant coefficients A 11 , B 11 , and D 11 indicate the stretching, bending-stretching, and bending stiffness, respectively, which are calculated by

3. Kinematic assumptions

+0.5h

In this research, it is assumed that the shallow arch only vibrates in the θ − z plane. Also, displacement field is expressed based on the first order shear deformation theory (FSDT) consistent with the Timoshenko assumptions. The displacement components of the arch may be written as

( A 11 , B 11 , D 11 ) = −0.5h +0.5h

A 44 =

¯ ( z, θ, t ) = w (θ, t ) w

(4)

In the above equations v and w indicate the displacements at the mid-surface of the shallow FG arch in the θ− and z−directions, respectively. Besides, ϕ is the cross-section rotation of arch in the plane of its curvature.

E ( z, T ) 2(1 + ν ( z, T ))

−0.5h

v¯ ( z, θ, t ) = v (θ, t ) + zϕ (θ, t )

E ( z, T )(1, z, z2 )dz

dz

(9)

Besides, N T and M T are the thermal force and thermal moment resultants which are given by +0.5h T

T

(N , M ) =

(1, z) E ( z, T )α ( z, T )( T − T 0 )dz

(10)

−0.5h

7. Equations of motion 4. Non-linear strain-displacement equations The von Kármán type of geometrical non-linearity, consistent with the small strains, moderate rotations, and large displacements in the polar coordinates takes the form

εθ θ =

1 R



¯ + v¯ ,θ − w

γθ z = v¯ ,z +

1 R

1 2R 2

t2 (δ T − δ V − δ U )dt = 0

¯ 2,θ w

¯ ,θ w

The equations of motion of FGM shallow arch based on the uncoupled thermoelasticity may be derived by applying the Hamilton principle

(11)

t1

(5)

where the total virtual strain energy of the arch δ U can be written as

where εθθ represents the circumferential normal strain and γθ z denotes the shear strain component. Here and in the rest, a comma indicates the partial derivative with respect to its afterwards.

θ0 +0.5h δU = R (σθ θ δ εθ θ + τθ z δ γθ z ) dzdθ

5. Constitutive equations

Here, δ V is the virtual potential energy of the external applied loads which is absent for this work. Also, the kinetic energy δ T is given by

(12)

0 −0.5h

If stress-strain relations is assumed based on linear thermoelasticity, the constitutive law for the FGM shallow arch exposed to thermal load is

θ0 +0.5h

σθ θ = E (z, T )εθ θ − E (z, T )α (z, T ) T (z) τθ z =

E ( z, T ) 2(1 + ν ( z, T ))

γθ z

where T indicates temperature difference.





ρ (z, T ) v˙¯ δ v˙¯ + w˙¯ δ w˙¯ dzdθ =

δT = R 0 −0.5h

(6)

θ0 ¨ δ w + ( I 2 v¨ + I 3 ϕ¨ ) δ ϕ } dθ {( I 1 v¨ + I 2 ϕ¨ ) δ v + I 1 w

−R 0

(13)

M. Javani et al. / Aerospace Science and Technology 90 (2019) 264–274

where a ( ˙ ) indicates a derivative with respect to time and the inertia terms I 1 , I 2 , and I 3 are defined by +0.5h

ρ (z, T )(1, z, z2 )dz

(I1, I2, I3) =

(14)

−0.5h

Substituting Eqs. (4), (5), and (7) into Eq. (12) and by means of the suitable mathematical simplifications, the expressions for the equations of motion for the FGM circular shallow arch are obtained as

1

N θ θ,θ = I 1 v¨ + I 2 ϕ¨ R  1  1 1 ¨ N θ θ w ,θ ,θ + Q θ z,θ + N θ θ = I 1 w R R R2 1 M θ θ,θ − Q θ = I 2 v¨ + I 3 ϕ¨ R

:

δv δw

:

δϕ

:

(15)

The equations of motion in terms of the displacement components of an FGM shallow arch may be obtained using Eqs. (9) and (15). The resulting equations are

1

A 11

R 1

R A 11

( A 11 ( v ,θ θ − w ,θ ) + B 11 ϕ,θ θ + 2

R3

( A 11 ( v ,θ θ − w ,θ ) + B 11 ϕ,θ θ + 1

R A 11

1 R



1 2

−

1 2



cos

N θ θ w ,θ )δ w = 0

(i − 1)π Nθ − 1

a0 =

M−S

: :

IM − F

(16)

(17)

i = 1, 2, . . . , N θ

(18)

M−F

:

IM − C M−C

v = Q θz +

1 R

Nθ θ = Q θ z +

: :

v = w = Mθ θ = 0 Nθ θ = w = Mθ θ = 0

:

N θ θ w ,θ = M θ θ = 0 1

N θ θ w ,θ = M θ θ = 0 R v =w =ϕ=0

Nθ θ = w = ϕ = 0

(21)

j , j +1

1

β t 2

a1 =

,

1

β t

,

a2 =

1 − 2β 2β

(23)

Once the solution {X} is known at t j +1 = ( j + 1) t, the first and second derivatives of {X} at t j +1 can be computed from

where N θ is the number of grid points along the circumferential direction of the arch. Using the boundary conditions given in Eq. (17), several set of boundary conditions suitable for thermally induced vibration analysis are obtained. Accordingly, edges of the arch may take one of following boundary conditions

IM − S

It should be noted that due to presence of the von Kármán type of geometrical nonlinearity, the generalized stiffness matrix is a function of unknown time-dependent nodal vector {X}. To complete the approximation, one should approximate the time derivatives in Eq. (20). Here, the Newmark direct integration scheme based on the constant average acceleration method (α = 0.5, β = 0.25) is applied. Implementation of the Newmark method to Eq. (20) yields [41]

and



,

(20)

   K(T,X) = [K(T,X)] + a0 [M(T)]         ˙ + a2 X¨  F(T) = {F(T)} j +1 + [M(T)] a0 {X} j + a1 X (22) j j

To discrete the above equations along the arch domain, the GDQ method is imposed. Distribution of nodal points is proposed utilizing the Chebyshev-Gauss-Lobatto distribution, which reads

θi = θ0

 

¨ + [K(T,X)] {X} = {F(T)} [M(T)] X

where

w ,θ w ,θ θ ) w ,θ

The complete set of the boundary conditions are showed by means of the process of virtual displacement relieving. For θ = 0 and θ = θ0 , the boundary conditions are extracted as

Nθ θ δ v = Mθ θ δϕ = ( Q θ z +

In the above equations (I M − S), (M − S), (I M − F ), (M − F ), (I M − C ), and (M − C ) are brief of immovable-simply support, movable-simply support, immovable-free, movable-free, immovable-clamped, and movable-clamped, respectively. Similar to the motion equations, boundary conditions should be discretized using the GDQ method. Discretization of the equations of motion into nodal points in the arch domain are given in the Appendix A. Finally, the discretized equations (16) based on generalized differential quadrature method, the equations of motion take the following matrix form

     K(T,X) {X} j +1 =  F(T)

w ,θ w ,θ θ ) = I 1 v¨ + I 2 ϕ¨

( A 11 ( v ,θ − w ) + B 11 ϕ,θ + w 2 − R N T ) w ,θ θ 2R ,θ R3 A 44 1 1 + ( w ,θ θ + ϕ,θ ) + 2 ( A 11 ( v ,θ − w ) + B 11 ϕ,θ R R R A 11 2 ¨ + w − R N T ) = I1 w 2R ,θ 1 B 11 ( B 11 ( v ,θ θ − w ,θ ) + D 11 ϕ,θ θ + w ,θ w ,θ θ ) R R2 1 − A 44 ( w ,θ + ϕ ) = I 2 v¨ + I 3 ϕ¨ R +

267

        ¨ X = a0 {X} j +1 − {X} j − a1 X˙ j − a2 X¨ j j +1         ˙ X = X˙ + a3 X¨ + a4 X¨ j +1

j

j +1

(24)

and

a3 = (1 − α ) t ,

a4 = α t

(25)

The resulting equations are solved at each time step using the intime formation known from the preceding     step solution. At time ˙ , and X¨ are known or obtained t = 0, the initial values of {X}, X by solving Eq. (20) at time t = 0 and are used to initiate the time marching procedure.   Since the arch is initially at rest, the initial ˙ are assumed to be zero. An iterative scheme values {X} and X should be applied to extract the displacement vector of the shallow arch under thermal shock. Iterative Picard method is used to obtain the response [41]. 8. Temperature profile In this section, temporal evolution of temperature profile is obtained for a shallow arch subjected to rapid heating. It is assumed that the temperature profile varies only through the thickness, which is suitable with the design requirements of the FGM media. Finally, assuming Rz  1, the transient heat conduction equation through-the-thickness in the absence of heat generation takes the form [42]

 (19)

j



K ( z, T ) T ,z ,z = ρ ( z, T )C v ( z, T ) T˙

(26)

Since prior to loading, arch is at reference temperature, the initial condition is given by

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M. Javani et al. / Aerospace Science and Technology 90 (2019) 264–274

Fig. 2. Comparison of thermally induced vibration response of an arch with those of Keibolahi et al. [29].

T ( z , 0) = T 0

(27)

To solve the heat conduction Eq. (25), various types of boundary conditions may be assumed on the top and bottom surfaces of the FGM arch. Here, it is assumed that the inner surface of the arch which is ceramic rich is subjected to a time-dependent sudden temperature, whereas the outer surface which is metal rich may undergo thermally insulated boundary condition or the temperature specified time-dependent boundary condition (rapid heating). Two different types of thermal boundary conditions may be defined which are

Case 1 : T (+0.5h, t ) = T c (t ),

T (−0.5h, t ) = T m (t )

Case 2 : T (+0.5h, t ) = T c (t ),

T ,z (−0.5h, t ) = 0

(28)

Upon solution of Eq. (26) with consideration of Eqs. (27) and (28), temperature profile across the arch thickness is achieved. Under the temperature-dependent material properties assumption, thermal conductivity is a function of temperature and the heat conduction equation becomes nonlinear. The heat conduction equation is solved by means of the GDQ method, analogous to the equations of motion. According to the GDQ method, distribution of nodal points across the arch thickness is dictated by

zi = −

h 2



cos

(i − 1)π Nz − 1



,

i = 1, 2, . . . , N z

(29)

Upon applying the GDQ method to the heat conduction Eq. (28) and imposing the boundary conditions Eq. (30) to the resulting system of equations, the matrix representation of the heat conduction equation may be written as

 

[CT (T)] T˙ + [KT (T)] {T} = {FT (T)}

(30)

Since the material properties are temperature dependent, in Eq. (30) damping matrix [CT (T)], stiffness matrix [KT (T)], and even force vector {FT (T)} are functions of the nodal temperatures. Consequently, at each time step, an iterative procedure should be applied to extract the temperature profile of the shallow arch under the assumption of temperature dependent material properties. In this section, the Picard method is utilized and at each time step thermal properties are evaluated at reference temperature T 0 . Material properties are then evaluated at the obtained nodal temperatures {T} and Eq. (30) is solved again. This procedure is repeated until the temperature profile converges at the current time step. The Crank-Nicolson procedure is adopted to solve Eq. (30) along with the initial conditions (27). Details on the proceeding of Crank-Nicolson method are available in [41]. 9. Results and discussion Procedure and formulation developed in the previous sections may be used in the rest to analyze the thermally induced large

amplitude forced vibrations in a shallow arch made of FGMs subjected to rapid surface heating. In this section, first a comparison study is provided. Afterwards, novel numerical results are given to explore the effects of different parameters. In the subsequent results, unless otherwise stated, the midspan deflection is denoted θ by W while the arch rise is shown by f = R (1 − cos( 20 )). Two parameters are used to define the arch geometry which are λ = √ R θ02 S θ0 S = and μ = . Here, S is the arch span and r x = h/ 12. 4r x 4r x rx In the solution of heat conduction equation number of grid points is set equal to N z = 51. Also in the solution of equations of motion the number of grid points is set equal o N θ = 17. These values are obtained after examination of convergence and accuracy of the obtained numerical results. 9.1. Comparison study A comparison study is developed in this section to assure the validity and accuracy of the developed formulation. In this example, a thin arch made of an isotropic homogeneous material is considered. The case of an arch with both edges simply supported and immovable boundary conditions are considered. For the sake of comparison, thickness is selected as h = 1 mm. Other parameters of the arch are λ = 1.3 and μ = 100. Since arch is made of an isotropic homogeneous material, power law index is set equal to ζ = 0. The lower surface of the arch (z = +h/2) is subjected to T c = 410 K whereas the upper surface (z = −h/2) is subjected to T m = 293 K. Material properties of the arch are E = 207.7877 GPa, ρ = 8166 kg/m3 , α = 15.321 × 10−6 1/K, K = 12.14291 W/m K, C v = 390.3507 J/kg K, and ν = 0.3. Comparison is performed in Fig. 2. Comparison is done with the results of Keibolahi et al. [29] which is developed by means of the thin arch theory. It is seen that our results match well with those of Keibolahi et al. [29]. The small divergence between our results and those of Keibolahi et al. [29] belongs to different arch theory and different solution method. 9.2. Parametric studies After validating the developed solution method and numerical results, novel numerical results are given for the case of FGM shallow arches subjected to thermal shock. For the numerical results of this section, SUS304/Si3 N4 FGM arch is assumed for development of the numerical results. The coefficients for these constituents are given in Table 1. 9.3. Example 1: influence of temperature dependency The first study in this section is devoted to analyze the effect of temperature dependency of the constituents on the response of circular shallow FGM arches under the action of rapid heating. For this example, an arch with both edges simply supported is considered. Both ends of the arch are immovable. Other characteristics

M. Javani et al. / Aerospace Science and Technology 90 (2019) 264–274

269

Table 1 Temperature dependent coefficients for SUS304 and Si3 N4 [39]. Material

Property

P −1

P0

P1

P2

P3

SUS304

α [1/K]

0 0 0 0 0 0

12.33e − 6 201.04e + 9 15.379 8166 0.3262 496.56

8.086e − 4 3.079e − 4 −1.264e − 3 0 −2.002e − 4 −1.151e − 3

0

0 0

0 3.797e − 7 1.636e − 6

0 0

0 0 0 0 0 0

5.8723e − 6 348.43e + 9 13.723 2370 0.24 555.11

9.095e − 4 −3.07e − 4 −1.032e − 3 0 0 1.016e − 3

0 2.16e − 7 5.466e − 7 0 0 2.92e − 7

0

E [Pa] K [W/m K] ρ [kg/m3 ]

ν C v [J/kg K] Si3 N4

α [1/K] E [Pa] K [W/m K] ρ [kg/m3 ]

ν C v [J/kg K]

−6.534e − 7 2.092e − 6

−7.223e − 10 −5.863e − 10 −8.946e − 11 −7.876e − 11 0 0

−1.67e − 10

Fig. 3. Effect of temperature dependence on temperature profile, lateral deflection, thermally induced force, and moment of the FGM shallow arch with both edges immovable and simply supported. Characteristics of the arch are ζ = 0, h = 8 mm, λ = 1.3, μ = 50, T c = 700 K, and T m = 300 K.

270

M. Javani et al. / Aerospace Science and Technology 90 (2019) 264–274

Fig. 4. Effect of temperature dependence on temperature profile, lateral deflection, thermally induced force, and moment of the FGM shallow arch with both edges immovable and simply supported. Characteristics of the arch are ζ = 2, h = 8 mm, λ = 1.3, μ = 50, T c = 700 K, and T m = 300 K.

of the arch are h = 8 mm, λ = 1.3 mm, μ = 50 mm, T c = 700 K, and T m = 300 K. Two power law indices are considered, which are ζ = 0 and ζ = 2. Illustrations from this example are provided in Figs. 3 and 4. As seen from the results of these figures, when temperature dependence for the constituents is taken into account, temperature profile is underestimated which is more accurate compared to the case of temperature independent material properties. Also, deflections estimated by temperature dependent material properties are higher than those with constant material properties. This is expected, since with temperature dependent material properties, as the temperature elevates, arch looses stiffness resulting in higher lateral deflection. As expected, due to the change in material properties such as elasticity modulus, thermal conductivity, and thermal expansion coefficient, the thermally induced force and moment resultants are also dependent on temperature where accurate results are obtained under temperature dependent material properties. It is again seen that when constant

material properties are used, thermally induced bending moment and axial force are underestimated. Since temperature dependence is an important factor in accurate estimation of the arch behavior under thermal shock, in the subsequent results the temperature dependency of the constituents is considered. 9.4. Example 2: influence of inertia In this example, the effect of inertia terms on the response of FGM arches subjected to rapid surface heating is investigated. For this example a class of shallow arches which are simply supported on both ends are considered. Also, the case of immovable arches is analyzed. Different power law indices are considered which are ζ = 0, 0.5, 1 and 5. Geometrical features of the arch are: h = 1 mm, λ = 1.3, and μ = 200. The metal rich surface is kept at reference temperature while the ceramic rich surface is exposed to a sud-

M. Javani et al. / Aerospace Science and Technology 90 (2019) 264–274

271

Fig. 5. The effect of power law index on the thermally induced vibrations of shallow FGM arches.

den temperature elevation at T c = 328 K. Results of this study are illustrated in Fig. 5. In the results of this figure, for each power law index the quasi-static response, the dynamic response, and the steady-state response are provided. As seen from the results of this figure, the dynamic response oscillates around the quasi-static response. This is an important conclusion which shows the occurrence of thermally induced vibrations. It is seen that the power law index has a significant influence on the magnitude and frequency of thermally induced vibrations.

while the metal rich surface is kept at reference temperature. For the other case, the metal rich surface is thermally insulated while the ceramic rich surface is subjected to T c = 337.75 K. Results of this study are illustrated in Fig. 6. It is observed that thermally induced vibrations indeed exist especially when the boundary conditions are of the temperature specified type. The deflections induced by rapid heating and the frequency of thermally induced vibrations are highly dependent to the type of in-plane boundary conditions, as seen from the results of Fig. 6.

9.5. Example 3: influence of in-plane edge supports

9.6. Example 4: influence of out-of-plane edge supports

The next comparison study analyzes the influence of in-plane edge supports on the thermally induced vibrations. In this example, an FGM arch with geometrical properties h = 1 mm, λ = 1.3, and μ = 200 is considered. The power law index is set equal to ζ = 2. Two different types of thermal loads are analyzed. In the first case, the ceramic rich surface is subjected to T c = 337.75 K

The thermally induced vibrations in a shallow arch made of FGMs with different boundary conditions is demonstrated in Figs. 7 and 8. Results of Figs. 7 and 8 are associated to, respectively, immovable free and immovable clamped arches. In Fig. 7, deflections at different positions are shown. It is seen that the nature and amplitude of deflections are different in different posi-

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M. Javani et al. / Aerospace Science and Technology 90 (2019) 264–274

Fig. 6. The effect of in-plane edge supports on the thermally induced vibrations of shallow FGM arches.

Fig. 7. Thermally induced vibration example in an immovable free arch at different positions.

Fig. 8. Thermally induced vibration example in an immovable clamped arch.

tions. Observations of Fig. 8 show that the oscillation induced by thermal shock are small in comparison with other boundary conditions due to the local flexural rigidity of the clamped edge. 9.7. Example 5: influence of geometrical non-linearity The next study examines the effect of geometrical non-linearity on the thermally induced vibrations of a shallow arch. Two different arches are considered. The case of immovable and movable arches. For this example, one end of the arch is assumed to be clamped while the other one is simply supported. Linear distribution of material properties is assumed. The characteristics of the

arch are h = 1 mm, λ = 1.3, and μ = 250. It is seen that for arches with immovable edge supports, geometrical nonlinearity changes the nature of vibrations and also amplitude and period of vibrations. However, for movable arches difference between the linear and nonlinear results is small (see Fig. 9). 10. Conclusion In this research, the vibrations induced by rapid surface heating in a shallow FGM arch is investigated. For this reason, the kinematics of a shear deformable shallow arch and the Donnell type of kinematic assumptions compatible with the von Kármán assump-

M. Javani et al. / Aerospace Science and Technology 90 (2019) 264–274

273

Fig. 9. Importance of geometrical nonlinearity on the thermally induced vibration of arches.

tions are used. Material properties of the arch are assumed to be temperature dependent. The heat conduction equation of the arch is solved using the one dimensional model. The obtained temperature profile is inserted into the equations of motion of the arch. These equations are discretized using the GDQ method and solved by means of the Newton-Raphson and the Newmark method. Different examples are shown to discuss the effects of various parameters. It is shown that in thin arches thermally induced vibrations indeed exist. Power law index, thermal and mechanical boundary conditions, temperature dependence, and geometrical non-linearity are the main factors which affect the thermally induced deflections and frequencies.

+

+

+

Nθ 1 

2R

(

j =1

A 44 1

(

R 1 2R

Nθ 

¯ i j w j )( A

R

Nθ 

j =1

¯ i j w j )( A

¯ ijϕ j − R N T ) A

Nθ  ¯ ij v j − wi ( A ( A 11 2

1

¯ ijϕ j) + A

R

j =1

j =1 Nθ 

Nθ 

B¯ i j w j + Nθ 

Nθ  j =1

k =1 Nθ 

(

+ B 11

¯ ik w k )) + B 11 A

j =1

¯ ik w k )) A

k =1

¯ ij ϕ j − R N T ) = I1 w ¨i A

(A.2)

j =1

Declaration of Competing Interest

Nθ Nθ Nθ Nθ    1  ¯ ¯ ¯ ( B ( B v − A w + ( A w )( B¯ ik w k )) 11 ij j ij j ij j 2

1 R

There is no competing interest. Appendix A

+ D 11

The governing equations of motion in the arch using the GDQ method may be expressed as Nθ Nθ Nθ Nθ    1  ¯ ¯ ¯ ( A ( B v − A w + ( A w )( B¯ ik w k )) 11 ij j ij j ij j 2

j =1

Nθ 

B¯ i j ϕ j ) − A 44 (

j =1

j =1

+ B 11

Nθ 

R

j =1

j =1

v s = 0 or

k =1

B¯ i j ϕ j ) = I 1 v¨ i + I 2 ϕ¨ i

(A.1)

Nθ Nθ Nθ   1  ¯ il w l )( A 11 ( ¯ ij w j ¯ ij v j − ( A B A 3

+

+

j =1

l =1

Nθ 1 

R

(

1 R3

Nθ 

¯ i j w j )( A

j =1 Nθ 

(

l =1

Nθ 

B¯ il w l )( A 11 (

j =1

k =1

¯ i j w j + ϕi ) = I 2 v¨ i + I 3 ϕ¨ i A

Nθ 

¯ ij v j − wi A

B¯ i j ϕ j )

(A.3)

j =1

R

j =1

1 2R

Nθ 

(

Nθ 

¯ sj w j )( A

j =1 Nθ 

¯ sk w k )) A

k =1

¯ sj ϕ j − R N T ) = 0, A

s = 1, N θ

j =1

j =1

k =1

j =1

Nθ  ¯ sj v j − w s ( A 11 ( A

+ B 11

j =1

B¯ ik w k )) + B 11

1

+

j =1

R

1 R

Nθ 

where i = 2, 3, . . . , N θ − 1. Also, boundary condition should be discretized using GDQ method as follows

1 R

R

j =1

ws = 0

or

Nθ 1

R

+

¯ sj w j + ϕs + A

j =1

1 2R

Nθ 

(

j =1

Nθ  ¯ sj v j − w s ( A ( A 11 2

1 R

Nθ 

¯ sj w j )( A

k =1

j =1

¯ sk w k )) A

(A.4)

274

M. Javani et al. / Aerospace Science and Technology 90 (2019) 264–274

Nθ 

+ B 11

Nθ 

¯ sj ϕ j − R N T )( A

j =1

¯ sl w l ) = 0, A

s=1, N θ

l =1

(A.5)

ϕs = 0 or

1 R

Nθ 

( B 11 (

¯ sj v j − w s A

j =1

+

Nθ 1 

2R

(

+ D 11

Nθ 

¯ sj w j )( A

j =1 Nθ 

¯ sk w k )) A

k =1

¯ sj ϕ j − R M T ) = 0, A

s = 1, N θ

(A.6)

j =1

¯ i j and B¯ i j are weighting coefficients of the first and Where A second order derivative, respectively, and are defined as follows

⎧ ϒ(θi ) ⎪ ⎪ ⎨ (θi −θ j )ϒ(θ j ) ¯ ij = Nθ A  ⎪ ¯ ik A ⎪ ⎩−

when

i = j

when

i= j

k=1,k=i

i , j = 1, 2, ..., N θ

(A.7)

in which

ϒ(θi ) =

Nθ 

(θi − θk )

(A.8)

k=1,k=i

and





¯

¯ ii A¯ i j − A i j B¯ i j = 2 A (θi −θ j ) i , j = 1, 2, ..., N θ

⎧ N ⎪ ⎨ B¯ = − θ B¯ ii ik k=1,k=i ⎪ ⎩ i = 1, 2, ..., N θ

 when

when

i = j

i = j.

(A.9)

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