Nonlinear thermomechanical stability of shear deformable FGM shallow spherical shells resting on elastic foundations with temperature dependent properties

Accepted Manuscript Nonlinear thermomechanical stability of shear deformable FGM shallow spherical shells resting on elastic foundations with temperature dependent properties Hoang Van Tung PII: DOI: Reference:

S0263-8223(14)00171-8 http://dx.doi.org/10.1016/j.compstruct.2014.04.004 COST 5657

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Composite Structures

Please cite this article as: Tung, H.V., Nonlinear thermomechanical stability of shear deformable FGM shallow spherical shells resting on elastic foundations with temperature dependent properties, Composite Structures (2014), doi: http://dx.doi.org/10.1016/j.compstruct.2014.04.004

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Nonlinear thermomechanical stability of shear deformable FGM shallow spherical shells resting on elastic foundations with temperature dependent properties Hoang Van Tung Faculty of Civil Engineering, Hanoi Architectural University, Ha Noi, Viet Nam E-mail: [email protected] Tel: +84-906193585 Abstract: This paper presents an analytical approach to investigate the nonlinear stability of clamped functionally graded material (FGM) shallow spherical (SS) shells and circular plates resting on elastic foundations, subjected to uniform external pressure and exposed to thermal environments. Material properties are assumed to be temperature dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents. Formulations for axisymmetrically deformed SS shells are based on the first order shear deformation theory taking geometrical nonlinearity, initial geometrical imperfection and interaction of Pasternak type elastic foundations into consideration. Approximate solutions are assumed to satisfy clamped immovable boundary conditions and Galerkin method is applied to derive expressions of buckling loads and load-deflection curves for FGM SS shells. Specialization of these expressions gives corresponding relations of FGM circular plates, and an iterative algorithm is adopted to obtain buckling temperatures and postbuckling temperature-deflection curves for thermally loaded FGM circular plates. The effects of material, geometry and foundation parameters, imperfection and temperature dependence of material properties on the nonlinear response of FGM SS shells and circular plates are analyzed and discussed in detail.

Keywords: Functionally graded material; Nonlinear stability; Shallow spherical shell; Circular plate; Elastic foundation; Temperature dependent properties.

1

1. Introduction Spherical shells are important components widely used as major load carrying portions in the structures of aircraft, missile and aerospace vehicle. They also find many applications in various industries such as shipbuilding, underground structures and building constructions. Since these shell structures are frequently exposed to severe mechanical and thermal loading conditions, their static and dynamic response are important problems and have received considerable attention. Stability and vibration problems of isotropic SS shells have been considered in works [1-6]. Uemura [1] analyzed the effects of initial deformation on the axisymmetrical buckling of SS shells under external pressure. Nath and Alwar [2] used Chebyshev series expansion to study the nonlinear static and transient behavior of spherical shells under simply supported and clamped boundary edge conditions. Nonlinear free vibration of clamped isotropic SS shells has been investigated by Sathyamoorthy [3] by using an analytical method. Eslami and his collaborators [4,5] applied adjacent equilibrium criterion and analytical solutions to study linear thermal and mechanical buckling of simply supported thin isotropic SS shells with and without imperfection. An analysis on nonlinear buckling of SS shells under uniform external pressure has been given by Li et al. [6] using the first order shear deformation theory and a semi-analytical approach. Nonlinear analysis on free and forced vibrations, static and dynamic stability of thin and moderately thick orthotropic SS shells have been addressed in works [7-10]. Also, axisymmetric buckling and postbuckling behavior of symmetrically laminated moderately thick SS shells under uniform external pressure and simply supported and clamped boundary conditions with movable or immovable edge constraints have been analyzed by Xu [11] using a FourierBessel series solution, and by Nath and Sandeep [12] making use of an iterative Chebyshev series solution technique. The effects of elastic foundations on the static and dynamic response of isotropic and orthotropic SS shells under external pressure have been studied in works [13-16]. Functionally graded materials (FGMs) are novel composites usually composed of ceramic and metal constituents. Due to smooth and gradual variation of material constituents, FGMs are capable of reducing or eliminating disadvantageous problems of conventional composites such as debonding and huge stress concentration. By virtue of high stiffness and performance temperature resistance capacity, FGMs can withstand severe 2

thermal environments and are suitable for applications in temperature shielding components such as aircraft, missile and aerospace structures. Therefore, stability and vibration of FGM plates and shallow shells in general, and FGM circular plates and SS shells in particular have received much attention. Eslami and his co-workers [17,18] made use of approximate analytical solutions, the classical shell theory and adjacent equilibrium criterion [29] to investigate linear buckling of simply supported thin FGM shallow and deep spherical shells without elastic foundations and subjected to external pressure and thermal loadings. Recently, Boroujerdy and Eslami [19,20] studied thermal and thermomechanical stability of simply supported thin piezo-FGM SS shells by utilizing a similar approach. Bich and Tung [21] presented an investigation on the nonlinear axisymmetrical static response of clamped thin FGM SS shells subjected to external pressure and thermal loads by making use of the classical shell theory and analytical solutions. Also, nonlinear unsymmetrical static and dynamic buckling behavior of thin FGM SS shells have been analyzed by Bich et al. [22] basing on an analytical approach and approximate solutions. Investigations on stability and vibration of FGM circular plates have been performed in works [23-27]. Based on the classical plate theory and shooting method, Ma and Wang [23] and Li et al. [24] dealt with the axisymmetric large deflection bending, thermal and thermomechanical postbuckling behavior of thin FGM circular plates without and with initial imperfection. Najafizadeh and Heydari [25] employed a variational method and adjacent equilibrium criterion based on the higher order shear deformation theory to investigate the linear buckling of clamped FGM circular plates under thermal loads and axisymmetric deformation. Alternatively, thermal buckling of clamped perfect FGM circular plates have been analyzed by Tran et al. [26] utilizing an isogeometric finite element formulation. It is evident from the open literature that very little attention has been addressed the nonlinear thermomechanical response of FGM SS shells and thermal postbuckling behavior of FGM circular plates resting on elastic foundations accounting for initial imperfection and temperature dependence of material properties. This paper presents an analytical investigation on the nonlinear thermomechanical response of axisymmetrically deformed FGM SS shells and circular plates resting on elastic foundations and subjected to combined action of pressure and thermal loads. Material properties are assumed to be temperature dependent and graded in the thickness direction 3

according to a simple power law distribution in terms of the volume fractions of constituents. Formulations are based on the first order shear deformation shell theory accounting for geometrical nonlinearity, initial geometrical imperfection and interaction of Pasternak type elastic foundations. Approximate solutions are assumed to satisfy immovable clamped boundary conditions and Galerkin method is applied to derive loaddeflection relations. An iterative procedure is adopted to obtain thermal buckling loads and postbuckling curves of FGM circular plates. The effects played by material, geometrical and foundation stiffness parameters, imperfection and temperature dependence of material properties on the nonlinear thermomechanical stability of FGM SS shells and circular plates are analyzed and discussed in detail.

2. FGM shallow spherical shell on an elastic foundation Consider a functionally graded shallow spherical shell of radius of curvature R , base radius a , thickness h and rise of shell H . The shell is immovably clamped at boundary edge and rested on a Pasternak elastic foundation as shown in Fig.1. The shell is made from a mixture of ceramics and metals, and is defined in a coordinate system ( ϕ ,θ , z ) whose origin is located on the middle surface of the shell, ϕ and θ are in the meridional and circumferential directions, respectively, of the shell and z is perpendicular to the middle surface and points inwards ( −h / 2 ≤ z ≤ h / 2 ). Material composition of the shell is smoothly varied along the thickness direction in such a way that the inner surface is metal-rich and the outer surface is ceramic-rich by following a simple power law in terms of the volume fractions of the constituents as n

⎛ 2z + h ⎞ Vm ( z ) = ⎜ ⎟ , Vc ( z ) = 1 − Vm ( z ) ⎝ 2h ⎠

(1)

where Vm and Vc are the volume fractions of metal and ceramic constituents, respectively, and nonnegative number n ≥ 0 is volume fraction index that defines the distribution of constituents in FGM. Practically, FGMs are most commonly used in high temperature environments and significant changes in material properties are inherent. Usually, the elasticity modulus decreases and the thermal expansion coefficient increases at elevated temperatures. 4

Therefore, it is essential to account for this temperature dependence for accurate and reliable prediction of the response of FGM structures. Effective properties Preff of the FGM SS shells can be determined by the linear rule of mixture as

Preff ( z , T ) = Prm (T )Vm ( z ) + Prc (T )Vc ( z )

(2)

where Pr denotes a specific material property assumed to be temperature-dependent in the present study, and subscripts m and c represent the metal and ceramic constituents, respectively. From Eqs. (1) and (2) the effective elasticity modulus E and thermal expansion coefficient α of the FGM SS shells can be written in the form ⎛ 2z + h ⎞ E ( z , T ) = Ec (T ) + Emc (T ) ⎜ ⎟ ⎝ 2h ⎠ ⎛ 2z + h ⎞ ⎟ ⎝ 2h ⎠

α ( z , T ) = α c (T ) + α mc (T ) ⎜

n

n

(3)

where

Emc (T ) = Em (T ) − Ec (T ) , α mc (T ) = α m (T ) − α c (T ) and

(4)

Poisson’s ratio ν is assumed to be constant. It is evident that the effective

properties of an FGM SS shell are both position and temperature dependent, and E = Em ,

α = α m at z = h / 2 and E = Ec , α = α c at z = −h / 2 . Furthermore, specialization of Eqs. (3) for n = 0 gives corresponding properties of isotropic metal shells, and percentage of ceramic constituent in the FGM shell is enhanced as n index increases. As n tends to infinity, Eqs. (3) turn to corresponding properties of pure ceramic spherical shell.

3. Formulation In this study, the first order shear deformation theory is used to derive basic equations and Galerkin method is applied to obtain expressions of buckling loads and load-deflection relations for moderately thick FGM SS shells supported by elastic foundations, exposed to thermal environments and subjected to uniform external pressure.

5

The imperfect FGM SS shell is assumed to be under axisymmetric deformation and displacement components u , v , w in ϕ , θ , z directions, respectively, at a distance z from the middle surface are represented as [11]

u (r , z ) = u (r ) + zψ (r ) v (r , z ) = 0

(5)

w(r , z ) = w(r ) + w* (r ) in which r = R sin ϕ , u is displacement in the meridional direction at the middle surface, w is the deflection of the shell, and ψ is the rotation of a normal to the middle surface. Also, w* (r ) is an imperfection function representing initial small deviation of shell surface from perfectly spherical configuration. Due to shallowness of the shell, it is approximately assumed that cos ϕ = 1 , Rdϕ = dr and R = a 2 /(2 H ) . The non-zero strain components of the SS shell are defined as

ε r = ε r 0 + z χ r , εθ = εθ 0 + z χθ , ε rz = ψ + w,r + w,*r

(6)

where a comma denotes differentiation with respect to the corresponding variable, i.e. ( ),r = d ( ) / dr , and the strains at the middle surface ε r 0 , ε θ 0 and curvatures χ r , χθ are related to the displacements and rotation in the form

ε r 0 = u,r − w / R + w,2r / 2 + w,r w,*r , ε r 0 =

u w − r R

(7)

χ r = ψ ,r , χθ = ψ / r . Based on Hooke law, stress-strain relations for an FGM SS shell including temperature effects are

σr =

E ( z, T ) [ε r + νεθ − (1 + ν )α ( z, T )ΔT ] 1 −ν 2

σθ =

E ( z, T ) [εθ + νε r − (1 + ν )α ( z, T )ΔT ] 1 −ν 2

σ rz =

E ( z, T ) ε rz 2(1 + ν )

(8)

where ΔT denotes the change of environment temperature from thermal stress free initial state.

6

The force and moment resultants are expressed in terms of the stress components through the thickness as h/2

h/2

−h / 2

−h / 2

( N r , Nθ ) = ∫ (σ r ,σ θ ) dz , ( M r , M θ ) = ∫ (σ r ,σ θ ) zdz

h/2

, Qr = K S

∫

σ rz dz

(9)

−h / 2

where K S is shear correction coefficient. Substitution of Eqs. (6) into Eqs. (8) and putting the result into Eqs. (9), the force and moment resultants are written in the form Nr =

Φ E1 E ε + νε θ 0 ) + 2 2 ( χ r + νχθ ) − 0 2 ( r0 1 −ν 1 −ν 1 −ν

Nθ =

Φ E1 E ε + νε r 0 ) + 2 2 ( χθ + νχ r ) − 0 2 ( θ0 1 −ν 1 −ν 1 −ν

Qr =

K S E1 (ψ + w,r + w,*r ) 2(1 + ν )

Mr =

E E2 Φ ε + νε θ 0 ) + 3 2 ( χ r + νχθ ) − 1 2 ( r0 1 −ν 1 −ν 1 −ν

Mθ =

E E2 Φ ε + νε r 0 ) + 3 2 ( χθ + νχ r ) − 1 2 ( θ0 1 −ν 1 −ν 1 −ν

(10)

(11)

where E1 = Ec (T )h + Emc (T )h /(n + 1) , E2 = Emc (T )h 2 [1/(n + 2) − 1/(2n + 2) ] , E3 = Ec (T )h3 /12 + Emc (T )h3 [1/(n + 3) − 1/(n + 2) + 1/(4n + 4) ] ,

(Φ 0 , Φ1 ) =

(12)

n n ⎡ ⎛ 2z + h ⎞ ⎤ ⎡ ⎛ 2z + h ⎞ ⎤ α α + + E T E T T T ( ) ( ) ( ) ( ) ⎢ c mc mc ⎜ ⎟ ⎥⎢ c ⎜ ⎟ ⎥ ΔT (1, z ) dz . ∫ h h 2 2 ⎝ ⎠ ⎝ ⎠ ⎥⎦ ⎢ ⎥ ⎢ −h / 2 ⎣ ⎦⎣ h/2

The nonlinear equilibrium equations of an imperfect SS shell resting on an elastic foundation based on the first order shear deformation theory are [6,11]

( rN r ),r − Nθ

=0

(13a)

( rM r ),r − M θ − rQr = 0

(13b)

( rQr ),r + ( N r + Nθ ) + ( rN r ( w,r + w,*r )),r + r ( q − q f ) = 0

(13c)

r R

7

where q is external pressure uniformly distributed on the outer surface of the shell, and the shell-foundation interaction q f is represented by Pasternak model [14,16]

q f = k1w − k2 ( w,rr + w,r / r )

(14)

in which k1 is Winkler foundation modulus and k2 is the shear layer foundation stiffness of Pasternak model. Introduction of Eqs. (7) into Eqs. (10), (11) and substituting the resulting expressions and Eq. (14) into Eqs. (13a) - (13c), system of equilibrium equations of imperfect FGM SS shells are written in terms of displacement components and the rotation as L1 ≡

E1 1 −ν 2

w,r 1 ⎡ u 2 * ⎢ ru,rr + u,r − r − (1 + ν )r R + 2 (1 −ν ) w,r + (1 −ν ) w,r w,r ⎣

E ⎛ ψ⎞ + r ( w,r w,rr + w,rr w,*r + w,r w,*rr )⎤⎦ + 2 2 ⎜ rψ ,rr + ψ ,r − ⎟ = 0 r⎠ 1 −ν ⎝ L2 ≡

E2 1 −ν 2

w,r 1 ⎡ u 2 * ⎢ ru,rr + u,r − r − (1 + ν )r R + 2 (1 −ν ) w,r + (1 −ν ) w,r w,r ⎣

+ r ( w,r w,rr + w,rr w,*r + w,r w,*rr )⎤⎦ + L3 ≡

(15b)

E3 ⎛ ψ ⎞ K S E1 rψ ,rr + ψ ,r − ⎟ − ( rψ + rw,r + rw,*r ) = 0 2 ⎜ r ⎠ 2 (1 + ν ) 1 −ν ⎝

K S E1 ⎡ψ + rψ ,r + w,r + w,*r + r ( w,rr + w,*rr )⎤ + E2 ( rψ ,r + ψ ) ⎦ R (1 −ν ) 2 (1 + ν ) ⎣ +

+

(15a)

E1 ⎛ 2 r 2 * ⎞ ⎜ ru,r + u − rw + w,r + rw,r w,r ⎟ R (1 −ν ) ⎝ R 2 ⎠

E1 ⎡ (1 + ν ) 1 + ν ) u,r ( w,r + w,*r ) − w + rw,r ) ( w,r + w,*r ) ( 2 ⎢( 1 −ν ⎣ R r − (1 + ν ) w ( w,rr + w,*rr ) + ru,rr ( w,r + w,*r ) + ( ru,r + ν u ) ( w,rr + w,*rr ) R 1 3 ⎤ + w,2r ( w,r + w,*r ) + rw,2r ( w,rr + w,*rr ) + w,2r w,*r + 3rw,r w,rr w,*r ⎥ (15c) 2 2 ⎦ E + 2 2 ⎡⎣( w,r + w,*r ) ( (1 + ν )ψ ,r + rψ ,rr ) + ( w,rr + w,*rr ) ( rψ ,r + νψ )⎤⎦ 1 −ν 2r Φ 0 Φ + rq − k1rw + k2 ( rw,rr + w,r ) − − 0 ⎡⎣ w,r + w,*r + r ( w,rr + w,*rr )⎤⎦ = 0 R (1 −ν ) 1 −ν

The FGM SS shell is assumed to be clamped and immovable in the meridional direction at the boundary edge, and under axisymmetrical deformation. The symmetry 8

condition at the center r = 0 and boundary conditions at r = a are expressed in the form [3,11,12]

To satisfy

ψ =0

at r = 0

w=0 , ψ =0 , u=0

at r = a

(16)

symmetry and boundary conditions (16), the following approximate

solutions for the displacement components and the rotation are assumed [3] r (a2 − r 2 ) a2 − r 2 ) a2 − r 2 ) ( ( r (a − r ) * u =U , ψ =Ψ , w =W , w = μh a2 a4 a4 a3 2

2

(17)

where U , Ψ are coefficients to be determined, W is the amplitude of the deflection and

μ is a small coefficient representing imperfection size. In this study, imperfection function is assumed to be in the form of the deflection for sake of simplicity. Subsequently, solutions (17) are substituted into the equilibrium equations (15a) - (15c) and Galerkin method is applied to the resulting equations. Specifically, the following integrations are performed a

∫ L1′r ( a − r ) dr = 0 , 0

a

2 2 ∫ L2′ r ( a − r ) dr = 0 , 0

a

∫ L′ ( a 3

0

2

− r 2 ) dr = 0 2

(18)

where L1′, L2′ , L3′ are the resulting expressions obtained by substituting the solutions (17) into L1 , L2 , L3 at the left sides of equations (15a) – (15c), respectively. Performing the integrations (18) yields the following equations U+

4 E2 4 2 Ψ − (1 + ν )λ2W − (35 − 29ν )W (W + 2μ ) = 0 3E1 5 105λ1

(19a)

⎡ 64 E3 16 ⎤ 256 + E2U + ⎢ (1 −ν ) K S E1λ12 ⎥ Ψ − (1 + ν ) E2λ2W 315 315 ⎣ 45 ⎦ − q=

64 512 (1 −ν ) K S E1λ1 (W + μ ) − (1 −ν ) E2W (W + 2μ ) = 0 315 945λ1

⎡ 5 K S E1 16 (1 + 2ν ) E2 20 K S E1 20 E2λ2 ⎤ W + μ)− ⎢ + Ψ+ Ψ (W + μ ) 2 ( 2⎥ 7 (1 + ν ) λ1 7 (1 −ν 2 ) λ13 ⎣ 7 (1 + ν ) λ1 7 (1 −ν ) λ1 ⎦

+

⎤ 2 E1λ2 ⎡ 64λ2 17 128 W− U− W (W + 2μ )⎥ 2 ⎢ 16 231λ1 (1 −ν ) λ1 ⎣ 21 ⎦ 9

(19b)

+

⎡ 5 1024 2 W (W + 3μ ) (30 + 79ν )U (W + μ ) + ⎢ 273λ1 (1 −ν ) λ ⎣112 E1

2

3 1

−

+

256 (1 + ν ) λ2W (W + μ )⎤⎥ 33 ⎦

⎤ 4Φ 0 ⎡ 2 E1 10 ⎛2 ⎞ K1 + 5K 2 ⎟W + W + μ )⎥ ( ⎢λ2 − 2 4 ⎜ 21(1 −ν ) λ1 ⎝ 3 (1 −ν ) λ1 ⎣ 7λ1 ⎠ ⎦

(19c)

where

E1 = E1 / h , E2 = E2 / h 2 , E3 = E3 / h3 , U = U / h , W = W / h , Φ 0 = Φ 0 / h ,

λ1 = a / h , λ2 = H / a , K1 =

12 (1 −ν 2 ) a 4 E1h 2

k1 , K 2 =

12 (1 −ν 2 ) a 2 E1h 2

k2 .

(20)

From Eqs. (19a) and (19b), U and Ψ can be determined in terms of material and geometry parameters and W as

Ψ = e10 + e11W + e12W 2

(21a)

U = e20 + e21W + e22W 2

(21b)

where e10 =

64 (1 −ν ) K S E12λ1μ , e

e11 =

4 64 2μ E1E2 (10ν − 118 ) , (1 + ν ) E1E2λ2 + (1 −ν ) K S E12λ1 + e e 3λ1e

e12 =

4E e E1E2 (10ν − 118) , e20 = − 2 10 , 3E1 3λ1e

e21 =

4 4μ 4E e (1 + ν ) λ2 + (35 − 29ν ) − 2 11 , 5 105λ1 3E1

e22 =

2 (35 − 29ν ) 4 E2e12 − , e = 16 (1 −ν ) K S E12λ12 + 448 E1E3 − 420 E22 . 105λ1 3E1

(22)

This study only considers case of uniform temperature rise. Specifically, environment temperature is assumed to be uniformly raised from initial value T0 at which the FGM SS shell is thermal stress free to final one T and temperature change ΔT = T − T0 is independent of thickness variable z . By virtue of Eqs. (12), thermal parameter Φ 0 can be expressed as 10

Φ 0 = PΔTh

(23a)

where P = Ec (T )α c (T ) +

Ec (T )α mc (T ) + Emc (T )α c (T ) Emc (T )α mc (T ) . + n +1 2n + 1

(23b)

Introduction of Ψ and U from Eqs. (21a), (21b) and Φ 0 from Eqs. (23a), (23b) into Eq. (19c) gives the following nonlinear relation between external pressure and nondimensional maximum deflection W including the effect of elevated temperature. ⎡ 4λ 40 (W + μ ) ⎤ 2 q = e30 + e31W + e32W 2 + e33W 3 + ⎢ PΔT − 2 ⎥ − − (1 ν ) λ 7(1 ν ) λ ⎥ 1 1 ⎦ ⎣⎢

(24)

where e30 =

20 K S E1μ 17 E1λ2e20 − β e10 − 2 7 (1 + ν ) λ1 8 (1 −ν ) λ12 +

5 E1 (30 + 79ν ) μ e20 112 (1 −ν

2

)λ

3 1

+

16 (1 + 2ν ) E2 7 (1 −ν 2 ) λ13

μ e10 ,

20 K S E1 128E1λ22 17 E1λ2e21 − e11β + − e31 = 2 2 7 (1 + ν ) λ1 21(1 −ν ) λ1 8 (1 −ν ) λ12 − +

5 E (30 + 79ν ) 512 E1λ2 μ 256 E1λ2 μ + 1 + − μ e e ( ) 21 20 3 231(1 −ν ) λ1 112 (1 −ν 2 ) λ13 33 (1 −ν ) λ13 16 (1 + 2ν ) E2 7 (1 −ν

2

)λ

3 1

(e10 + μ e11 ) +

2 E1 ⎛2 ⎞ K1 + 5 K 2 ⎟ , 2 4 ⎜ 21(1 −ν ) λ1 ⎝ 3 ⎠

e32 = − β e12 −

5 E (30 + 79ν ) 17 E1λ2e22 256 E1λ2 − + 1 (e21 + μ e22 ) 2 3 8 (1 −ν ) λ1 231(1 −ν ) λ1 112 (1 −ν 2 ) λ13

+

16 (1 + 2ν ) E2 1024 E1μ 256 E1λ2 − + (e11 + μ e12 ) , 3 2 4 91(1 −ν ) λ1 33 (1 −ν ) λ1 7 (1 −ν 2 ) λ13

e33 =

β=

5 E1 (30 + 79ν ) 112 (1 −ν 2 ) λ13

e22 +

16 (1 + 2ν ) E2 1024 E1 + e12 , 273 (1 −ν 2 ) λ14 7 (1 −ν 2 ) λ13

5 K S E1 20 E2λ2 + . 7 (1 + ν ) λ1 7 (1 −ν ) λ12

11

(25)

Eq. (24) is explicit expression of load-deflection curves for clamped immovable FGM SS shells resting on Pasternak elastic foundations and subjected to combined pressure and thermal loadings. As environment temperature is maintained at initial value T = T0 , i.e. ΔT = 0 , pressure-loaded FGM SS shells may exhibit an extremum type buckling behavior and extremum points of pressure-deflection curves can be determined from condition dq = e31 + 2e32W + 3e33W 2 = 0 dW

(26)

which gives W1,2 =

2 −e32 ∓ e32 − 3e31e33 3e33

(27)

and critical buckling pressures of FGM SS shells are obtained as

qcr = q (W1 ) = e30 + e31W1 + e32W12 + e33W13

(28)

providing material and shell geometry parameters satisfy condition 2 e32 − 3e31e33 ≥ 0

(29)

Conversely, in case of ΔT ≠ 0 and due to temperature dependence of material properties, the effects of elevated temperature are included in all terms at right hand side of Eq. (24), and thermomechanically loaded FGM SS shells may experience a bifurcation type buckling behavior and corresponding critical buckling pressures are predicted as

⎛ 10 μ ⎞ 4 P qcrΔT = e30 + ⎜ λ2 − ΔT . ⎟ 7λ1 ⎠ (1 −ν )λ1 ⎝

(30)

Subsequently, specialization of Eq. (24) for case of clamped immovable FGM circular plates resting on elastic foundations and exposed to thermal environments, i.e. q = 0 and

λ2 = H / a = 0 , gives the following relation ΔT =

7 (1 −ν ) λ12

(e′ 40 P (W + μ )

30

′ W + e32 ′ W 2 + e33 ′ W3) + e31

12

(31)

′ , e31 ′ , e32 ′ , e33 ′ are where λ1 = a / h is radius to thickness ratio of circular plate, and e30 received by specialization of e30 , e31 , e32 , e33 , respectively, for case of λ2 = 0 . For geometrically perfect FGM circular plates, i.e. μ = 0 , Eq. (31) is reduced to the form 7 (1 −ν ) λ12 ′ + e32 ′ W + e33 ′ W 2 ). e31 ΔT = ( 40 P

(32)

Obviously, a bifurcation type buckling occurs for thermally loaded FGM circular plates and critical buckling temperature change can be predicted as 1 −ν ) E1λ12 ⎡ 20 K S ( ΔTcr = ⎢ 40 (1 + ν ) P ⎣ λ12

−

320 2 ⎛2 ⎞⎤ K K + (33) 5 (1 −ν ) K S2 E12 + 1 2 ⎟⎥ . ⎜ e 3 (1 −ν ) λ14 ⎝ 3 ⎠⎦

In case of temperature independent material properties, Eqs. (33) and (31) are closedform expressions of thermal buckling loads and postbuckling curves, respectively. In contrast, as properties of constituent materials in FGM are temperature dependent, Eqs. (31) and (33) are implicit expressions of temperature-deflection relation and critical buckling temperature change, respectively, and an iterative process is adopted to obtain critical buckling temperature and the postbuckling equilibrium curves of thermally loaded FGM circular plates. To predict the critical buckling temperatures, iterative process includes the following steps (1.1) Begin with solving the critical buckling temperature ΔTcr from Eq. (33) using temperature independent material properties calculated at reference temperature T0 . (1.2) Update the right hand side of Eq. (33) using the property values at T = T0 + ΔTcr to obtain a new critical buckling temperature. (1.3) Repeat step (1.2) until the critical buckling temperature converge to a prescribed error tolerance. Specifically, the iteration process is converged as the relative difference between the two consecutive solutions is smaller than a prescribed tolerance. Similarly, the temperature-deflection curve is traced by the following steps

13

(2.1) Begin with the dimensionless deflection W = W / h = 0 . (2.2) Use the iterative process (1.1) – (1.3) for Eq. (31) or (32). (2.3) Specify the new value of W / h and repeat step (2.2) until the postbuckling temperature converges to a prescribed error tolerance. (2.4) Repeat steps (2.2), (2.3) to obtain the thermal postbuckling curve.

4. Results and discussion 4.1. Comparison study

To validate proposed approach, consider a clamped immovable isotropic SS shell subjected to uniform external pressure in the absence of elastic foundations and thermal loadings. Critical buckling pressures of a clamped isotropic SS shell are calculated by using explicit expression (28) and compared in Fig. 2 with those reported by Nath and Sandeep [12] utilizing an iterative Chebyshev series solution technique. It is evident that an excellent agreement is obtained in this comparison. As second example for verification, thermal buckling of a clamped FGM circular plate under uniform temperature rise and without elastic foundations is considered. The combination of materials consists of aluminum ( Al ) and alumina ( Al2O3 ). Temperature independent elasticity modulus and thermal expansion coefficient are Em = 70 GPa ,

α m = 23 × 10−6 oC −1 for aluminum and Ec = 380 GPa , α c = 7.4 × 10−6 oC −1 for alumina, whereas Poisson’s ratio is a constant ν = 0.3 for both materials. This problem was also studied by Najafizadeh and Heydari [25] using the variational method and adjacent equilibrium criterion and by Tran et al. [26] basing on an isogeometric finite element approach within the framework of the higher order shear deformation plate theory. Critical buckling temperature changes are calculated by closed-form relation (33) and compared in Fig. 3 with results obtained in works [25,26] in which n* is volume fraction index in case of

Vm and Vc are interchanged in Eq. (1), i.e. Vc ( z ) = ( (2 z + h) /(2h) ) . Again, a good n*

correlation is achieved. The remainder of this section presents numerical results for FGM SS shells composed of silicon nitride ( Si3 N 4 ) and stainless steel ( SUS 304 ). The material properties Pr , such as 14

elasticity modulus E and thermal expansion coefficient α , can be expressed as a nonlinear function of temperature [27] 2 3 Pr = P0 ( P−1T −1 + 1 + PT ) 1 + P2T + PT 3

(34)

in which T = T0 + ΔT and T0 = 300 K (room temperature), P0 , P−1 , P1 , P2 and P3 are the coefficients of temperature T ( K ) and are unique to the constituent materials. Specific values of these coefficients for E (in Pa ) and α (in 1/ K ) of silicon nitride and stainless steel are given by Reddy and Chin [28] and are listed in Table 1. Poisson’s ratio is assumed to be a constant ν = 0.3 . In addition, temperature-dependent and temperature-independent material properties will be written as T-D and T-ID properties, respectively, for sake of brevity. The T-ID are material properties calculated at room temperature T0 = 300 K . Table 1: Temperature-dependent thermo-elastic coefficients for silicon nitride and stainless steel (Reddy and Chin [28]). Materials

Properties

P0

P−1

P1

P2

P3

Silicon nitride

E ( Pa )

348.43e+9

0

-3.070e-4

2.160e-7

-8.946e-11

α (1/ K )

5.8723e-6

0

9.095e-4

0

0

E ( Pa )

201.04e+9

0

3.079e-4

-6.534e-7

0

α (1/ K )

12.330e-6

0

8.086e-4

0

0

Stainless steel

4.2. FGM shallow spherical shells In what follows, the nonlinear response of axisymmetrically deformed FGM SS shells will be analyzed. The shells are assumed to be clamped and immovable along boundary edge. Unless otherwise specified, the environment temperature is maintained at reference value T0 , i.e. ΔT = 0 , and spherical shell is free from elastic foundation interaction, i.e. K1 = K 2 = 0 . In characterizing the behavior of the SS shells, deformations in which the central region of a shell moves toward the plane that contains the periphery of the shell are referred to as inward deflections (positive deflections). Deformations in the opposite direction are referred to as outward deflections (negative deflections). Similarly, positive or

15

negative imperfections produce perturbations in the spherical shell geometry that move the central region of a shell inward or outward, respectively. Table 2 shows the effects of volume fraction index n , geometry ratio H / a , nondimensional stiffness parameters ( K1 , K 2 ) of elastic foundation and imperfection size μ on the critical buckling pressures qcr ( MPa ) for FGM SS shells at room temperature. It is evident that critical pressure loads are pronouncedly enhanced as n index, H / a ratio and foundation stiffness are increased. Moreover, negative initial imperfection also has beneficial influence on buckling resistance capacity of FGM SS shells. It seems that effective curvature of spherical shell to be higher for smaller values of μ . Furthermore, the critical pressures of FGM SS shells are increased and, especially, more shallow spherical shells may avoid buckling due to the presence of elastic foundations. Table 2: Critical buckling pressures qcr ( MPa ) of FGM SS shells at room temperature ( a / h = 20 , T0 = 300 K , ΔT = 0 ). ( K1 , K 2 )

(0,0)

(50,10)

(30,15)

H / a = 0.1

n

H / a = 0.2

μ = −0.1

0

0.1

μ = −0.1

0

0.1

0

16.34

15.59

14.92

83.05

77.50

72.09

2

21.78

20.64

19.60

112.62

104.94

97.47

10

23.98

22.75

21.64

123.77

115.38

107.20

0

-

-

-

100.23

94.45

88.78

2

-

-

-

136.00

127.90

120.05

10

-

-

-

149.41

140.65

132.07

0

-

-

-

102.82

97.01

91.32

2

-

-

-

139.41

131.37

123.48

10

-

-

-

153.26

144.46

135.85

(-) no buckling The effects of foundation stiffness and imperfection parameters on the critical buckling loads and postbuckling load-deflection curves for FGM SS shells subjected to uniform 16

external pressure are graphically illustrated in Figs. 4 and 5. Fig. 4 indicates that the critical buckling pressures are enhanced due to increase of stiffness parameters of elastic foundations, especially stiffness of shear layer K 2 of Pasternak foundation model. In contrast, Fig. 5 shows that critical buckling pressures are decreased as imperfection parameter μ is increased from −0.1 to 0.1 . Fig. 6 shows the effects of the volume fraction index n and elastic foundation on the nonlinear response of FGM SS shells under uniform external pressure. As can be seen, ceramic-rich shells have both higher critical loads and more severe snap-through instability. The pressure-deflection curves become higher and more stable, i.e. very benign snapthrough phenomenon, as the FGM SS shell is supported by an elastic foundation. The effects of shell rise to base radius ratio H / a on the nonlinear stability of FGM SS shells are analyzed in Fig. 7. It is clear that the nonlinear response of spherical shells is very sensitive to variation of H / a ratio. Specifically, an increase in H / a ratio gives higher buckling loads followed by a more intense snap-through phenomenon. Fig. 8 plotted as counterparts of Fig. 7 for case of K1 = 50 , K 2 = 20 , and Fig. 9 depicted with various values of non-dimensional stiffness parameters K1 , K 2 show pronounced effects of the support of elastic foundations on the nonlinear response of pressure-loaded FGM SS shells. As can be observed, pressure-deflection curves are higher and more stable in the presence of elastic foundations. In addition, parameter K 2 of Pasternak foundation model has more sensitive effects on the loading carrying capacity of FGM SS shells. It is evident from Table 2 and Figs. 8 and 9 that elastic foundations have pronounced and beneficial influences on the stability of FGM SS shells with two respects are that enhancement of loading capacity and reduction or even elimination of snap-through instability. The effects of thermal environments and temperature dependence of material properties on the nonlinear thermomechanical response of FGM SS shells are illustrated in Figs. 10 and 11. Due to the presence of thermal environment, pressure-loaded spherical shells exhibit a bifurcation-type buckling behavior with bifurcation point pressure is increased as temperature change to be higher. It can be explained that pre-existent thermal loading makes spherical shell furface deflected outwards (negative deflection) and curvature of the 17

shell developed prior to application of external pressure. As temperature dependence of material properties is incorporated, the FGM SS shells have higher buckling pressures and lower loading capacity in deep region of postbuckling behavior. Fig. 11, plotted as counterparts of Fig. 10 for case of K1 = 50 , K 2 = 10 , again indicates very useful effects of elastic foundations on the stability of FGM SS shells subjected to combined thermomechanical loads. 4.3. FGM circular plates Table 3 shows the effects of material, geometrical and elastic foundation parameters and temperature dependence of material properties on critical buckling temperatures for clamped FGM circular plates exposed to elevated temperature environment. Table 3: Critical buckling temperatures Tcr = T0 + ΔTcr ( K ) of perfect FGM circular plates ( T0 = 300 K ). ( K1 , K 2 )

n

a / h = 15

a / h = 20

a / h = 30

a / h = 40

(0,0)

0

522 a (556 b)

433 (445)

362 (365)

336 (337)

1

589 (647)

475 (496)

383 (388)

348 (350)

5

645 (733)

512 (545)

402 (410)

359 (362)

∞

704 (824)

550 (597)

422 (433)

371 (375)

0

813 (890)

619 (689)

456 (473)

392 (398)

1

954 (1245)

714 (833)

507 (538)

423 (434)

5

1075 (1506)

800 (981)

555 (603)

453 (471)

∞

1165 (1714)

864 (1098)

592 (655)

476 (500)

0

849 (1052)

644 (724)

469 (489)

399 (406)

1

1000 (1331)

745 (881)

523 (559)

433 (445)

5

1127 (1617)

836 (1043)

575 (631)

466 (486)

∞

1221 (1841)

902 (1169)

614 (686)

490 (518)

(100,10)

(50,20)

a

T-D properties, b T-ID properties

18

It is clear that critical buckling temperatures are considerably enhanced as FGM plate is ceramic-rich and/or supported by elastic foundations. The temperature dependence of material properties has deteriorative influences on thermal buckling resistance capacity of FGM circular plates. Furthermore, the difference of critical buckling temperatures for two cases of T-D and T-ID material properties is more pronounced as FGM circular plates are thicker, ceramic-rich or supported by more stiff foundations. In contrast, the effects of temperature dependent material properties on buckling temperatures are negligibly small for relatively thin and/or metal-rich FGM circular plates. Fig. 12, plotted by using Eq. (24) for case of ΔT = 0 and H / a = 0 , shows pressuredeflection curves versus various values of n index and foundation stiffness ( K1 , K 2 ) for FGM circular plates under uniform external pressure. It is obvious that equilibrium paths are stable with no snap-through phenomenon and loading capacity of FGM circular plates is remarkably improved due to the support of elastic foundations. Finally, the effects of material distribution, elastic foundations and temperature dependent material properties on the thermal postbuckling of geometrically perfect FGM circular plates are considered in Figs. 13 and 14. As expected, ceramic-rich FGM plates have higher buckling temperature and postbuckling strength. These figures also indicate that deteriorative influences of temperature dependent material properties on reduction of thermal loading carrying capability of FGM circular plates become more pronounced as FGM plates are ceramic-rich and/or supported by foundations with higher values of stiffness parameters.

5. Concluding remarks From above analysis, the following remarks are suggested: (a) Material and geometry parameters have sensitive influences on the nonlinear response of clamped FGM SS shells and an appropriate distribution of constituents in FGM is needed to enhance loading capacity and avoid fracture resulting from a severe snapthrough instability.

19

(b) Elastic foundations, especially shear layer of Pasternak type foundations, have very beneficial influences on both loading carrying capability and reduction of intensity of snapthrough instability. Furthermore, extremum-type buckling of FGM SS shells may be prevented and pressure-deflection paths are stable as stiffness parameters of elastic foundations become higher. (c) Effective curvature of SS shells are changed and nonlinear response of pressureloaded FGM SS shells is affected due to initial geometrical imperfection. Specifically, both critical buckling pressure and severity of snap-through instability are reduced as imperfection parameter μ increases. (d) Temperature dependence of material properties has deteriorative effects on the thermal buckling and postbuckling behavior of FGM circular plates, especially for thicker and ceramic-rich FGM circular plates or plates supported by foundations with higher stiffness parameters.

Acknowledgements This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.01-2012.02.

20

References [1] Uemura M. Axisymmetrical buckling of an initially deformed shallow spherical shell under external pressure. Int J Nonlinear Mech 1971; 6: 177-92. [2] Nath Y, Alwar RS. Non-linear static and dynamic response of spherical shells. Int J Nonlinear Mech 1978; 13: 157-70. [3] Sathyamoorthy M. Vibrations of moderately thick shallow spherical shells at large amplitudes. J Sound Vib 1994; 172(1): 63-70. [4] Eslami MR, Ghorbani HR, Shakeri M. Thermoelastic buckling of thin spherical shells. J Therm Stress 2001; 24: 1177-98. [5] Shahsiah R, Eslami MR. Thermal and mechanical instability of an imperfect shallow spherical cap. J Therm Stress 2003; 26(7): 723-37. [6] Li QS, Liu J, Tang J. Buckling of shallow spherical shells including the effects of transverse shear deformation. Int J Mech Sci 2003; 45: 1519-29. [7] Varadan TK, Pandalai KAV. Nonlinear flexural oscillations of orthotropic shallow spherical shells. Comput Struct 1978; 9: 417-25. [8] Chao CC, Lin IS. Static and dynamic snap-through of orthotropic spherical caps. Compos Struct 1990; 14: 281-301. [9] Sathyamoorthy M. Nonlinear vibrations of moderately thick orthotropic shallow spherical shells. Comput Struct 1995; 57(1): 59-65. [10] Dube GP, Joshi S, Dumir PC. Nonlinear analysis of thick shallow spherical and conical orthotropic caps using Galerkin’s method. Appl Math Modelling 2001; 25: 755-73. [11] Xu CS. Buckling and postbuckling of symmetrically laminated moderately thick spherical caps. Int J Solids Struct 1991; 28(9): 1171-84. [12] Nath Y, Sandeep K. Postbuckling of symmetrically laminated, moderately thick, axisymmetric shallow spherical shells. Int J Mech Sci 1993; 35(1): 965-75. [13] Dumir PC. Nonlinear axisymmetric response of orthotropic thin spherical caps on elastic foundations. Int J Mech Sci 1985; 27: 751-60.

21

[14] Nath Y, Jain RK. Non-linear studies of orthotropic shallow spherical shells on elastic foundations. Int J Nonlinear Mech 1986;21:447-58. [15] Nie GH. Asymptotic buckling analysis of imperfect shallow spherical shells on nonlinear elastic foundation. Int J Mech Sci 2001; 43: 543-55. [16] Civalek O. Geometrically nonlinear dynamic and static analysis of shallow spherical shell resting on two-parameters elastic foundations. Int J Pres Vess Pip 2014;113:1-9. [17] Shahsiah R, Eslami MR, Naj R. Thermal instability of functionally graded shallow spherical shell. J Therm Stresses 2006; 29(8): 771-90. [18] Shahsiah R, Eslami MR, Boroujerdy MS. Thermal instability of functionally graded deep spherical shell. Arch Appl Mech 2011;81(10):1455-71. [19] Boroujerdy MS, Eslami MR. Thermal buckling of piezo-FGM shallow spherical shells. Meccanica 2013;48(4):887-99. [20] Boroujerdy MS, Eslami MR. Nonlinear axisymmetric thermomechanical response of piezo-FGM shallow spherical shells. Arch Appl Mech 2013;83(12):1681-93. [21] Bich DH, Tung HV. Non-linear axisymmetric response of functionally graded shallow spherical shells under uniform external pressure including temperature effects. Int J Nonlinear Mech 2011; 46: 1195-204. [22] Bich DH, Dung DV, Hoa LK. Nonlinear static and dynamic buckling analysis of functionally graded shallow spherical shells including temperature effects. Compos Struct 2012;94: 2952-60. [23] Ma LS, Wang TJ. Nonlinear bending and post-buckling of a functionally graded circular plate under mechanical and thermal loadings. Int J Solids Struct 2003; 40: 3311-30. [24] Li SR, Zhang JH, Zhao YG. Nonlinear thermomechanical post-buckling of circular FGM plate with geometric imperfection. Thin Wall Struct 2007; 45: 528-36. [25] Najafizadeh MM, Heydari HR. Thermal buckling of functionally graded circular plates based on higher order shear deformation plate theory. European J Mech A-Solids 2004; 23: 1085-100.

22

[26] Tran LV, Thai CH, Nguyen XH. An isogeometric finite element formulation for thermal buckling analysis of functionally graded plates. Finite Elem Anal Des 2013;73:6576. [27] Touloukian YS. Thermophysical properties of high temperature solid materials. New York: Macmillan,1967. [28] Reddy JN, Chin CD. Thermoelastic analysis of functionally graded cylinders and plates. J Therm Stresses 1998; 21: 593-626. [29] Brush DO, Almroth BO. Buckling of bars,plates and shells. New York: McGrawHill;1975.

23

h

ψ ϕ,u

k2

H

w k1 r

R

a z Fig. 1. Configuration and coordinates of a shallow spherical shell on an elastic foundation.

4

4

qcra /Eh 200 180 160

Present Nath and Sandeep [12] isotropic spherical shell, ν = 0.3, a/h = 20

140 120 100 80 60 40 20 0 0.1

0.15

0.2 H/a

0.25

Fig. 2. Comparison of nondimensional buckling loads for clamped immovable isotropic spherical shells.

24

o

ΔT cr ( C)

350

Najafizadeh and Heydari [25] Tran et al. [26] Present

300

* 250 1: n = 0 *

200

2: n = 0.5 * 3: n = 100

1

150

2

100

3

50 0 0.01

0.02

0.03 0.04 0.05 h/a Fig. 3. Comparison of critical buckling temperature change for clamped ( Al / Al2O3 ) FGM circular plates under uniform temperature rise.

qcr (GPa) 0.4 n = 2, a/h = 20, μ = 0, T 0 = 300 K, ΔT = 0 0.35 1: K1 = 0, K2 = 0 0.3 2: K = 100, K = 0 1

0.25

2

4

3: K1 = 100, K2 = 10

3

4: K1 = 100, K2 = 20

2 1

0.2 0.15 0.1 0.05 0 0.1

0.15

0.2 0.25 H/a Fig. 4. Effects of stiffness parameters of elastic foundations on the critical buckling pressure of FGM shallow spherical shells.

25

q (GPa) 0.06 n = 2, a/h = 20, H/a = 0.15

1 2 3 4 5

0.05

T0 = 300 K, Δ T = 0 K1 = 0, K2 = 0

0.04 0.03

1: μ = - 0.1 2: μ = - 0.05

0.02

3: μ = 0

0.01

4: μ = 0.05 5: μ = 0.1

5 4 3 2 1

0 -0.01

0

0.5

1

1.5

2

2.5

3

3.5

W/h

Fig. 5. Effects of imperfection on the nonlinear response of FGM shallow spherical shells under uniform external pressure.

q (GPa) 0.09 a/h = 20, H/a = 0.15, μ = 0, T 0 = 300 K, ΔT = 0 0.08 3

0.07 2

0.06 0.05 0.04

1: n = 0 2: n = 1 3: n = 100

2

0.03

1

0.02

K1 = 0, K2 = 0 K1 = 50, K2 = 10

0.01 0 0

1

3

0.5

1

1.5

2

2.5

3

3.5

W/h Fig. 6. Effects of n index and elastic foundation on the nonlinear response of FGM shallow spherical shells under external pressure.

26

q (GPa) 0.1 H/a = 0.2

0.09 0.08

H/a = 0.175

0.07 0.06 0.05

H/a = 0.15

0.04 H/a = 0.125

0.03

H/a = 0.1

0.02 0.01 0

n = 1, a/h = 20, K1 = 0, K2 = 0, T0 = 300 K, Δ T = 0, μ = 0 0

0.5

1

1.5

2

2.5

3

3.5

W/h

Fig. 7. Effects of H / a ratio on the nonlinear response of perfect FGM shallow spherical shells under external pressure. q (GPa) 0.18 n = 1, a/h = 20, K1 = 50, K2 = 20, μ = 0 0.16 T0 = 300 K, ΔT = 0 0.14

H/a = 0.2

0.12 H/a = 0.175

0.1

H/a = 0.15

0.08 0.06

H/a = 0.125

0.04

H/a = 0.1

0.02 0

0

0.5

1

1.5

2

2.5

3

3.5

W/h

Fig. 8. Counterparts of Fig. 7 for case of K1 = 50 , K 2 = 20 .

27

q (GPa) 0.1 n = 1, a/h = 20, H/a = 0.15, T0 = 300 K, Δ T = 0, μ = 0 0.09 5

0.08

4

0.07 0.06

3

0.05 0.04

1: K1= 0, K2 = 0

0.03

2: K1= 50, K2 = 0 3: K1= 50, K2 = 10 4: K1= 100, K2 = 10 5: K1= 50, K2 = 20

0.02 0.01 0

2

0

0.5

1

1.5

1

2

2.5

3

3.5

W/h

Fig. 9. Effects of nondimensional stiffness parameters K1 , K 2 of elastic foundations on the nonlinear response of FGM shallow spherical shells. q (GPa) 0.05 3

T-ID T-D

n = 2, a/h = 20, H/a = 0.1

0.04

T0 = 300 K, μ = 0 K1 = 0, K2 = 0

2

0.03

1 0.02 1 0.01

1: Δ T = 0

2

2: Δ T = 300 K

0

3: Δ T = 500 K 3

-0.01 -0.02

0

0.5

1

1.5 W/h

2

2.5

3

Fig. 10. Effects of thermal environments on the nonlinear thermomechanical response of FGM shallow spherical shells.

28

q (GPa) 0.1 T-ID T-D

0.09

n = 2, a/h = 20, H/a = 0.1, T0 = 300 K, μ = 0 K1 = 50, K2 = 10

0.08 0.07

1

0.06 0.05

3 2

2

0.04 0.03

1: Δ T = 0

1

0.02

3: Δ T = 500 K

0.01 0

3

2: Δ T = 300 K

0

0.5

1

1.5 W/h

2

2.5

3

Fig. 11. Counterparts of Fig. 10 for case of K1 = 50 , K 2 = 10 . q (GPa) 0.16 (K1,K2) = (0,0) (K1,K2) = (50,10)

0.14

a/h = 20, T0 = 300 K, Δ T = 0, μ = 0

0.12

3

1: metal

0.1

2: n = 2 3: ceramic

0.08

3

0.06

2 1

0.04

2 1

0.02 0

0

0.5

1 W/h

1.5

2

Fig. 12. Effects of material distribution and foundation stiffness on the nonlinear response of FGM circular plates under uniform external pressure.

29

T (K) 1100 T-ID T-D

1000

a/h = 20, K1 = 0, K2 = 0, T0 = 300 K, μ = 0 900

1: metal (n = 0) 2: n = 2

800

3: ceramic

700 600

3

2

1

500 400

0

0.5

1

1.5

W/h

Fig. 13. Effects of material distribution and temperature dependent properties on the thermal postbuckling of FGM circular plates. T (K) 1600 1: K1 = 0, K2 = 0

T-ID T-D

2: K1 = 50, K2 = 10

1400

3: K1 = 100, K2 = 20 1200 3 1000

2

800

1 600 n = 5, a/h = 20, T0 = 300 K, μ = 0 400

0

0.5

1

1.5

W/h Fig. 14. Effects of elastic foundations and temperature dependent properties on the thermal postbuckling of perfect FGM circular plates. 30