Axisymmetric post-buckling behavior of saturated porous circular plates

Thin-Walled Structures 112 (2017) 149–158

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Axisymmetric post-buckling behavior of saturated porous circular plates M.R. Feyzi, A.R. Khorshidvand

⁎

MARK

Department of Mechanical Engineering, Islamic Azad University, South Tehran Branch, 11365/4435 Tehran, Iran

A R T I C L E I N F O

A B S T R A C T

Keywords: Post-buckling Porous material Circular plate Shooting method

This study aimed to investigate axisymmetric post-buckling behavior of a circular plate made of porous material under uniformly distributed radial compression with simply supported and clamped boundary conditions. Pores are saturated with fluid and plate properties vary continuously in the thickness direction. Governing equations are obtained based on classical plate theory and applying Sanders nonlinear strain-displacement relation. Shooting numerical method is used to solve the governing equations of problem. Effects of porosity coefficient, pore distribution, pore fluid compressibility, thickness change and boundary conditions on the post-buckling behavior of the plate are investigated. The results obtained for post-bucking of homogeneous/isotropic plates and critical buckling load of porous plates are compared with the results of other researchers.

1. Introduction Porous materials are composed of two components; a solid matrix and fluid within matrix pores that can be liquid or gas. Porous materials exist in nature such as stone, wood and bone and may be made artificially such as metal, ceramic and polymer foams and they are used as structural components in various industries such as aerospace, transportation, building, etc. Biot [1] who is the pioneer in developing the theory of poroelasticity, studied buckling of a fluid-saturated porous slab under axial compression and showed that critical buckling load is proportional to pore compressibility. Magnucki and Stasiewicz [2] investigated buckling of a simply supported porous beam and showed porosity effect on the strength and buckling load of the beam. Buckling and bending of a rectangular porous plate with varying properties across the thickness and under in-plane compression and transverse pressure were studied by Magnucki et al. [3]. Magnucka-Blandzi [4] examined buckling of a circular porous plate and showed that the critical load linearly decrease with increase porosity of the plate; he also studied dynamic stability of a circular plate made of metal foam and showed porosity effect on critical loads with numerical results [5]. He continued his research in this field and investigated rectangular sandwich plate with metal foam core and simply supported boundary condition. In this study, he considered the middle plane of the plate as symmetry plane and by numerical method obtained critical buckling loads for a set of sandwich plates [6]. Jasion et al. [7] obtained global and local buckling for sandwich beam and plate with metal foam core by experimental, numerical and analytical methods, and compared the obtained results.

⁎

Wen [8] obtained an analytical solution for saturated porous plate with an incompressible fluid and showed that there is a significant interaction between the solid and flow. Closed-form solution for buckling of porous circular plate saturated with fluid and with three different types of pore distribution in thickness direction including nonlinear nonsymmetric, nonlinear symmetric and monotonous and based on classical plate theory (CPT) under mechanical and thermal loads was obtained by Jabbari et al. [9,10]. Buckling of porous circular plates integrated with piezoelectric layers was investigated under mechanical and thermal loads and based on CPT by [11–13] and under thermal load and based on first-order shear deformation plate theory by [14]. Buckling analysis of porous plates with functional properties is similar to functionally graded material (FGM) plates to some extent. Woo and Meguid [15] obtained an analytical solution in terms of Fourier series for the coupled large deflection of FG plates and shallow shells under transverse mechanical loads and a temperature field. Closed-form solution for the critical buckling temperature of a rectangular FG plate was obtained under different types of thermal loads and based on classical and higher-order shear deformation plate theories by Javaheri and Eslami [16,17]. They showed that classical plate theory overestimates buckling temperatures. They also examined Buckling of FG Plates under in-plane Compression and based on CPT [18]. Najafizadeh and Eslami [19] presented buckling of a circular plate with functional properties under uniform radial compression. Ma and Wang [20] investigated axisymmetric post-buckling of an FG circular plate under uniformly distributed radial compressive load. They also [21] studied bending and thermal post-buckling of an FG circular plate based on classical nonlinear von Karman plate theory. The governing

Corresponding author. E-mail address: [email protected] (A.R. Khorshidvand).

http://dx.doi.org/10.1016/j.tws.2016.11.026 Received 10 June 2016; Received in revised form 24 November 2016; Accepted 30 November 2016 0263-8231/ © 2016 Elsevier Ltd. All rights reserved.

Thin-Walled Structures 112 (2017) 149–158

M.R. Feyzi, A.R. Khorshidvand

equations were solved with the shooting method. Effects of material constant and boundary conditions on temperature distribution, nonlinear bending, critical buckling temperature and thermal post-buckling behavior were indicated. They [22] studied the relationships between axisymmetric bending and buckling of FG circular plates based on third-order plate theory (TPT) and CPT. They compared TPT solutions with first-order plate theory (FPT) and CPT solutions and showed that TPT solutions are almost the same as FPT solutions and FPT is sufficient to consider shear deformation effect on the axisymmetric bending and buckling of FG plate. Post-buckling of FG circular plate with geometric imperfection under transverse mechanical load and transversely nonuniform temperature rise was studied by Li et al. [23]. Effects of geometric imperfections on buckling and post-buckling of FG plates were studied by many researchers such as [24–26]. Post-buckling of an FG circular plate under asymmetric transverse and in-plane loadings was studied by Fallah et al. [27]. To the authors' knowledge, there is no study in previous works regarding post-buckling of porous circular Plates. Therefore in this study, mechanical post-buckling of a saturated porous circular plate is investigated. It is assumed that mechanical properties vary continuously in the thickness direction. Based on classical plate theory and Sanders assumption, governing equations of the problem are obtained as a system of differential equations and shooting method is used to solve them. Both clamped and simply supported boundary conditions are considered. The effects of geometric parameters, poroelastic material parameters and boundary conditions on the post-buckling behavior of plate are investigated.

Fig. 1. Variation of shear modulus through the dimensionless thickness for nonlinear nonsymmetric pore distribution for different values of e1.

2. Governing equations 2.1. Mechanical properties of poroelastic plates In this study, a circular plate with radius b and thickness h is considered, which is made of porous material and its pores are saturated with fluid. Cylindrical coordinate axes are located on the mid-plane of the plate and z axis is in the thickness direction. Plate properties vary continuously along the thickness. For pore distribution in the thickness direction, three different cases are considered [2,9,10]. At first case, pore distribution is nonlinear symmetric and the middle plane of plate is its symmetry plane and moduli of elasticity, which depend on pore distribution, are as follows

⎡ ⎡ ⎛ z ⎞⎤ ⎛ z ⎞⎤ E G E (z ) = E0⎢1−e1cos⎜π ⎟⎥G(z ) = G0⎢1−e1cos⎜π ⎟⎥e1=1− 1 =1− 1 ⎝ h ⎠⎦ ⎝ h ⎠⎦ E0 G0 ⎣ ⎣

Fig. 2. Variation of shear modulus through the dimensionless thickness for nonlinear symmetric pore distribution for different values of e1.

extensional strains, but in the symmetrical and monotonous cases, such couplings are neglected [14]. Variations of shear modulus with porous distribution in the thickness direction are shown in Fig. 1 for nonlinear nonsymmetric distribution, in Fig. 2 for nonlinear symmetric distribution and in Fig. 3 for monotonous distribution.

(1)

where e1 is the porosity coefficient of the plate (0 < e1<1); E1 and E0 are Young's moduli at the middle plane (z = 0) and the upper and lower surfaces of the plate (z = ± h /2), respectively; G1 and G0 are shear moduli at (z = 0) and (z = ± h /2), respectively. The relationship between elastic modulus and shear modulus is Ej =2Gj (1 + ν ),j = 0,1 and Poisson's ratio (ν ) is assumed to be constant across the plate thickness. At second case, pore distribution is nonlinear nonsymmetric and moduli of elasticity are expressed as follows

⎧ ⎧ ⎡⎛ π ⎞⎛ ⎡⎛ π ⎞⎛ h ⎞⎤⎫ h ⎞⎤⎫ E (z ) = E0⎨1−e1cos⎢⎜ ⎟⎜z + ⎟⎥⎬G(z ) = G0⎨1−e1cos⎢⎜ ⎟⎜z + ⎟⎥⎬ 2 ⎠⎦⎭ 2 ⎠⎦⎭ ⎣⎝ 2h ⎠⎝ ⎣ ⎝ 2h ⎠ ⎝ ⎩ ⎩ ⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

(2) in which E1 and E0 are Young's moduli at the lower surface (z = − h /2) and the upper surface (z = + h /2) of the plate, respectively; G1 and G0 are shear moduli at the lower and upper surfaces, respectively. At third case, pore distribution is monotonous and moduli of elasticity in this case, are:

E (z ) = E0(1−e1)G (z ) = G0(1−e1)

(3)

In the asymmetrical case, there is mechanical coupling between extensional forces and curvatures and between bending moments and

Fig. 3. Variation of shear modulus through the dimensionless thickness for monotonous pore distribution for different values of e1.

150

Thin-Walled Structures 112 (2017) 149–158

M.R. Feyzi, A.R. Khorshidvand

2.2. Stress-strain constitutive relation

Nr , r +

Biot [1] theory of linear poroelasticity has two features: (1) an increase of pore pressure induces a dilation of pore, and (2) compression of the pore causes a rise of pore pressure. Stress-strain relation for poroelastic materials is expressed as follow [28]:

σij =2Gεii +

2Gνu εkk δij − 1 − 2νu

αpδij

M=

h /2

(Nr , Nθ ) =

νu =

2G(νu − ν ) (6)

(A2 , A3 , A4 ) = (7)

σr = A1εr + B1εθ σθ = A1εθ + B1εr

(8)

p = M (−αεkk )

(9)

h /2

∫−h/2 A1(1,z, z2)dz(B2, B3, B4) = ∫−h/2 B1(1,z, z2)dz

(16)

By substituting Eqs. (12) and (15) in Eqs. (13), the governing equilibrium equations of the problem in terms of displacement components are obtained 2⎞ ⎛ 2 dU U d 2W dW 1 dU 1 ⎛ dW ⎞ ⎟ A2 ⎜⎜ 2 + − 2 + + ⎟⎟ ⎜ 2 r dr 2r ⎝ dr ⎠ ⎠ r dr dr ⎝ dr ⎛ ⎛ d 3W ⎞2 ⎞ ⎛ 1 d 2W 1 dW ⎞ ⎜ − 1 ⎜ dW ⎟ ⎟=0 ⎟ B + A3 ⎜ − 3 − + + 2 ⎟ ⎜ r dr 2 ⎝ dr r 2 dr ⎠ ⎝ 2r ⎝ dr ⎠ ⎠

(17)

2⎤ A ⎛ U ⎞⎛ 1 dW ⎞ B ⎛ U ⎞⎛ d 2W ⎞ A2 ⎡⎢ dU 1 ⎛ dW ⎞ ⎥⎛ d 2W ⎞ + ⎜ ⎟ ⎟ ⎜ 2 ⎟ + 2 ⎜ ⎟⎜ 2 ⎟ + 2 ⎜ ⎟⎜ 2 ⎝ dr ⎠ ⎥⎦⎝ dr ⎠ A4 ⎝ r ⎠⎝ r dr ⎠ A4 ⎝ r ⎠⎝ dr ⎠ A4 ⎢⎣ dr

where the constants A1 and B1 in terms of the constants C1 and C2 are:

⎡ ⎤ νu νu − ν C1 = 2⎢1 + + ⎥G (z ) 1 − 2νu ⎣ (1 − 2νu )(1 − 2ν ) ⎦ ⎡ (ν − ν )(1+νu ) ⎛ C ⎞⎤ 2 C2 = C1 − 2G (z )A1=( )⎢1 + νu + u ⎜1 − 2 ⎟⎥G ( z ) 2 ⎢ C1 ⎠⎥⎦ 1 − 2ν ⎝ 1 − νu ⎣ ⎡ (ν − ν )(1+νu ) ⎛ C ⎞⎤ 2 )⎢(1+νu )νu + u ⎜ 1 − 2 ⎟⎥G (z ) 2 ⎢ 1 − 2ν C1 ⎠⎥⎦ ⎝ 1 − νu ⎣

(15)

where h /2

αB(1 − 2ν ) ν+ 3 αB(1 − 2ν ) 1− 3

(14)

⎧ Nr ⎫ ⎡ A2 B2 ⎤⎧ εr ⎫ ⎡ A3 B3 ⎤⎧ kr ⎫⎧ Mr ⎫ ⎡ A3 B3 ⎤⎧ εr ⎫ ⎨ ⎬=⎢ ⎥⎨ ⎬ ⎥⎨ ⎬⎨ ⎬ = ⎢ ⎥⎨ ⎬ + ⎢ ⎩ Nθ ⎭ ⎣ B2 A2 ⎦⎩ εθ ⎭ ⎣ B3 A3 ⎦⎩ kθ ⎭⎩ Mθ ⎭ ⎣ B3 A3 ⎦⎩ εθ ⎭ ⎡ A B ⎤⎧ k ⎫ + ⎢ 4 4 ⎥⎨ r ⎬ ⎣ B4 A4 ⎦⎩ kθ ⎭

where p is pore fluid pressure; M is Biot's modulus; νu is undrained Poisson's ratio (0 < ν<νu<0.5); α is Biot coefficient of effective stress (0 < α < 1); B is Skempton pore pressure coefficient (marker of pore fluid properties); ξ is variation of fluid volume content; and εkk is volumetric strain. By simplifying Eq. (4) to plane-stress condition in cylindrical coordinates and under undrained condition (ξ = 0), following relations are obtained [10]

B1=(

h /2

∫−h/2 (σr, σθ )dz(Mr , Mθ ) = ∫−h/2 (σr, σθ )zdz

Substitution of Eqs. (8) and (11) into Eqs. (14), yields

(4)

(5)

α 2(1 − 2νu )(1 − 2ν )

(13)

where N and M are force and moment components, respectively and are defined as follows

where

p = M (ξ − αεkk )

(Nr − Nθ ) =0(rNr W , r − Mθ ) + (rMr ), rr =0 ,r r

+

2⎤ B ⎛ 3 d 2W dW ⎞ B2 ⎡⎢ dU 1 ⎛ dW ⎞ ⎥⎛ 1 dW ⎞ ⎟ + ⎜ ⎟ + 3 ⎜− ⎟ ⎜ 2 ⎝ dr ⎠ ⎥⎦⎝ r dr ⎠ A4 ⎢⎣ dr A4 ⎝ r dr 2 dr ⎠

+

⎡ 2 ⎛ d 2W ⎞2 ⎤ B2A3 ⎛ 1 d 2W dW ⎞ A3 ⎢ 1 ⎛ dW ⎞ ⎜ ⎟ − 2⎜ ⎟ −⎜ 2⎟⎥+ A4 ⎢⎣ r ⎝ dr ⎠ A4 A2 ⎝ r dr 2 dr ⎠ ⎝ dr ⎠ ⎥⎦

⎛ A3 A3 ⎞⎛ d 4W 2 d 3W 1 d 2W 1 dW ⎞ ⎟ = ⎜1− − 2 + 3 ⎟⎜ 4 + 3 2 A A r ⎝ dr r dr r dr ⎠ 4 2 ⎠⎝ dr

(10)

(18)

The continuity and symmetry conditions at the center of plate (r = 0) are 2.3. Strain-displacement relations

U =0,

Based on classical plate theory and according to [29], strain components at distance z from the middle plane are determined by following relations

εr =

εr + zkrεθ =

εθ + zkθ

dW =0, dr

(11)

dW =0, dr

W =0,

1 (W , r )2 εθ = 2

U 1 kr = −W , rrkθ =− W , r r r

(19)

Nr =−P

(20)

and boundary conditions at the edge of the plate (r = b ) for simply supported case, are

W =0, U ,r +

r→0⎝

The plate edge in r direction is movable and in z direction is clamped or simply supported. Boundary conditions at the edge of the plate (r = b ) for clamped case, are

εθ are engineering strain components in the middle plane where εr , kθ are curvatures. These strains and curvatures, in terms of and kr , displacement components and according to the Sanders assumption are [30] εr =

⎛ d 3W 1 d 2W ⎞ ⎟=0 lim⎜ 3 + r dr 2 ⎠ dr

W =Finite,

Mr =0,

Nr =−P

(21)

For making the governing equations and boundary conditions dimensionless, following dimensionless parameters are defined [10,20]

(12)

where W and U are displacement components of a point on the midplane of the plate in z and r directions, respectively and Comma in subscript denotes differentiation. 2.4. Governing equilibrium equations Governing equilibrium equations of a thin circular plate under radial uniform compressive load and with polar symmetry condition utilizing principle of minimum total potential energy are presented as follows [10]

r , b

f3 =

A3 B h2 f4 = 2 , A2 h A4

f5 =

B3h , A4

f6 =

f8 =

B3 b 2P λ= 2 , A2 h h A2

P* =

P , G0b

Mr* =

w=

W , h

u=

Ub , h2

A3 h , A4

x=

f1 =

f2 =

A2 h 2 , A4

B2 B ,f = 4 A2 7 A4 b 2Mr hA4

,

(22)

Dimensionless governing equations and boundary conditions are derived as follows: 151

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M.R. Feyzi, A.R. Khorshidvand

⎧ ⎛ ⎛1 ⎞ ⎛1 ⎞ 2 1 1 1 ⎞ y4 + 2 y3 − 3 y2 + ⎨f4 ⎜y6 + y22⎟y3 + f2 ⎜ y5y3⎟ + f4 ⎜ y5y2⎟ ⎝x ⎠ ⎝x ⎠ ⎩ ⎝ x 2 ⎠ x x ⎛1 ⎞⎫ ⎛ ⎛1 ⎞ 1 ⎞⎛ 1 ⎞ + f2 ⎜y6 + y22⎟⎜ y2⎟ + (f2 f3 − 3f5 )⎜ y3y2⎟ − f1 ⎜ 2 y22 + y32⎟⎬ ⎝ ⎝x ⎠ ⎝x ⎠⎭ 2 ⎠⎝ x ⎠

2 ⎛ d 3w d 2u u d 2w dw 1 du 1 ⎛ dw ⎞ 1 d 2w 1 dw ⎞ ⎟ + − 2 + + − 2 ⎜ ⎟ − f3 ⎜ 3 + 2 2 x dx x dx 2 2x ⎝ dx ⎠ ⎝ dx dx x dx dx x dx ⎠ ⎛ ⎛ ⎞2 ⎞ 1 dw − f6 ⎜⎜ ⎜ ⎟ ⎟⎟=0 2 ⎝ x ⎝ dx ⎠ ⎠ (23)

ω=−

/ (1 − f1 f3 )

2⎤ ⎡ ⎛ u ⎞⎛ 1 dw ⎞ ⎛ u ⎞⎛ d 2w ⎞ du 1 ⎛ dw ⎞ ⎛ d 2w ⎞ + ⎜ ⎟ ⎥⎜ 2 ⎟ + f4 ⎜ ⎟⎜ 2 ⎟ + f2 ⎜ ⎟⎜ f2 ⎢ ⎟ ⎢⎣ dx ⎝ x ⎠⎝ x dx ⎠ ⎝ x ⎠⎝ dx ⎠ 2 ⎝ dx ⎠ ⎥⎦⎝ dx ⎠

⎛1 ⎞ ⎛ 1 1 1 2 1 1 ⎞ y2 + f3 ⎜y4 + y3 − 2 y2⎟ + f6 ⎜ y22⎟ μ=− y6 + 2 y5 − y3y2 − ⎝ 2x ⎠ ⎠ ⎝ x 2 x x x x

⎡ ⎛ ⎞2 ⎛ 2 ⎞2 ⎤ 2⎤ ⎡ du 1 ⎛ dw ⎞ ⎛ 1 dw ⎞ 1 dw dw + f4 ⎢ + ⎜ ⎟ ⎥⎜ ⎟ − f1 ⎢ 2 ⎜ ⎟ + ⎜ 2 ⎟ ⎥ ⎢ x ⎝ dx ⎠ ⎢⎣ dx 2 ⎝ dx ⎠ ⎥⎦⎝ x dx ⎠ ⎝ dx ⎠ ⎥⎦ ⎣ ⎛ 1 d 2w dw ⎞ ⎛ d 4w 2 d 3w 1 d 2w ⎟ = (1−f1 f3 )⎜ 4 + + (f2 f3 −3f5 )⎜ − 2 2 3 2 x dx ⎝ x dx dx ⎠ ⎝ dx x dx ⎞ 1 dw ⎟ + 3 x dx ⎠

u=0,

dw =0, dx

Nr =−λ,atx =1 Nr =−λ , at x =1

w = δ,

⎡0 1 0 0 0 0⎤ ⎢ ⎥ 1 B0 = ⎢ 0 0 Δx 1 0 0 ⎥ , ⎢⎣ 0 0 0 0 1 0 ⎥⎦

⎡1 ⎢ B1 = ⎢ 0 ⎢⎣ 0

(24)

⎡1 ⎢ B1 = ⎢ 0 ⎢ ⎢0 ⎣

Mr*=0,

(Simply supported)

(25)

where δ is a dimensionless parameter to determine deflection at the center of the plate and Nr , Mr* are 2 ⎛ d 2w ⎞ ⎛ 1 dw ⎞ ⎛u⎞ du 1 ⎛ dw ⎞ Nr = + ⎜ ⎟ − f3 ⎜ 2 ⎟ + f6 ⎜ ⎟ − f8 ⎜ ⎟M * ⎝ x dx ⎠ r ⎝x⎠ dx 2 ⎝ dx ⎠ ⎝ dx ⎠ 2⎞ ⎛ ⎛ 1 dw ⎞ ⎛u⎞ du d 2w 1 ⎛ dw ⎞ = f1 ⎜⎜ + ⎜ ⎟ ⎟⎟ − + f5 ⎜ ⎟ − f7 ⎜ ⎟ 2 ⎝ x dx ⎠ ⎠ ⎠ ⎝ ⎝ dx dx x 2 dx ⎠ ⎝

⎡1 ⎢0 B0 = ⎢ ⎢0 ⎢⎣ 0

(26)

3. Numerical method

B0 Y (∆x ) = b0 , B1Y (1) = b1

⎧ y2 ⎫ ⎧ dw / dx ⎫ ⎪ y ⎪ ⎪ d 2w / dx 2 ⎪ ⎪ ⎪ 3 ⎪ 3⎪ ⎪y ⎪ ⎪ ⎪ 3 H (x, Y ) = ⎨ 4 ⎬ = ⎨ d 4w / dx 4 ⎬ ⎪ ω ⎪ ⎪ d w / dx ⎪ ⎪ y6 ⎪ ⎪ du / dx ⎪ ⎪ μ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ 2 d u / dx 2 ⎭

2

−

f8

−f3

x

0 f1 y2 2 y2 2

− −

0 0 0⎤ ⎧0⎪ ⎫ 0 0 0 ⎥, b = ⎪ ⎨ 0 ⎬,(Clamped) ⎥ 1 ⎪ ⎪ f ⎩− λ ⎭ 0 6 1 ⎥⎦ x

0 0 0⎤ ⎥ ⎧ 0⎫ f ⎪ ⎪ −1 0 5 f1⎥ , b = ⎨ ,(Simply Supported) 0⎬ x 1 ⎪ ⎪ ⎥ f6 ⎩ ⎭ − λ −f3 0 1 ⎥⎦ x

(34)

0

f7 x f8 x

0 0 0 0 0⎤ 1 0 0 0 0⎥ ⎥, 0 1 1 0 0⎥ Δx ⎥ 0 0 0 1 0⎦

⎧δ ⎫ ⎪0⎪ b0 = ⎨ ⎬ ⎪0⎪ ⎩0⎭

(35)

(36)

⎧ ⎫ ⎡ ⎤ B1 = ⎢ 1 0 0 0 0 0 ⎥ , b1 = ⎨ 0 ⎬,(Clamped) ⎣0 1 0 0 0 0⎦ ⎩0⎭

(37)

⎡1 B1 = ⎢ ⎢⎣ 0

(38)

0 f1 y2 2

−

0 0 0⎤ ⎧ ⎫ ⎥ , b = ⎨ 0 ⎬,(Simply Supported) f −1 0 5 f1⎥⎦ 1 ⎩ 0 ⎭ x 0

f7 x

dZ = H (x , Z ) dx

(39) T

Z(Δx ) = { z1 z2 z3 z 4 z5 z 6 } = I

(40)

T

I (D) = { d1 0 d 2 −d 2 /Δx 0 d3} , (λGiven )I (δ,D) = { δ 0 d1 −d1/Δx 0 d 2 }T , (δGiven )

(41)

where I is initial values vector and D is unknown initial values vector and is defined as

(27)

T

(λGiven )D = { d1 d 2 }T , (δGiven )

D = { d1 d 2 d3 } ,

(28)

(42)

The aim is to find values for D so that Eq. (43) be satisfied, these values are indicated with D*.

where

⎧ y1 ⎫ ⎧ w ⎫ ⎪ y ⎪ ⎪ dw / dx ⎪ ⎪ 2⎪ ⎪ 2 2⎪ ⎪ ⎪y ⎪ ⎪ Y = ⎨ y3 ⎬ = ⎨ d 3w / dx 3 ⎬ ⎪ 4 ⎪ ⎪ d w / dx ⎪ ⎪ y5 ⎪ ⎪ u ⎪ ⎪ ⎪y ⎪ ⎪ ⎩ 6 ⎭ ⎩ du / dx ⎭

y2

0 0

(33)

Initial value problem (IVP) related to the boundary value problem is expressed as

In order to solve the governing differential equations, as a boundary value problem (BVP), shooting method [20,21,31,32] is used. The governing equilibrium equations of the problem as a system of two coupled differential equations for numerical solution, are written as a system of six first-order differential equations. In order to avoid the singularities at x = 0 , the small quantity ∆x (∆x > 0) which is close to zero is considered, thus the problem is solved on the interval [∆x, 1]. Boundary value problem is rewritten in the following standard form using the vector/matrix notations

,

0 1

⎧ 0⎫ ⎪ ⎪ b0 = ⎨ 0 ⎬ ⎪ ⎪ ⎩0⎭

for given deflection case

This problem can be expressed in two different ways; in the first case, deflection of the plate center (δ ) is given and Nr =−λ is removed from boundary conditions in Eqs. (25). In the second case, applied load on the edge of the plate (λ ) is given and w = δ is removed in Eqs. (25).

dY = H (x , Y ) dx

(32)

for given load case

⎛ d 3w 1 d 2w ⎞ dw ⎟=0, at x =0 w=0, =0, lim ⎜ + x →0⎝ dx 3 dx x dx 2 ⎠

(Clamped)w=0,

(31)

B1Z(1; D*) − b1 = 0 ,

(λGiven )B1Z(1; δ, D*) − b1 = 0 , (δGiven )

(43)

D* is assumed as the root of Eq. (43) and a Newton-Raphson numerical method is used to obtain it. In order to solve the IVP (Eqs. (39) and (40)), the fourth-order Runge-Kutta method is used. In this case, the solution of IVP with initial values I in x = ∆x is the same as the solution of BVP (Eqs. (27) and (28)), since such solution satisfies boundary conditions in the edge of the plate (x = 1), Thus

(29)

Y (x ) = Z(x; D*) ,

(λGiven )Y (x; δ ) = Z(x; δ, D*) ,

(δGiven )

(44)

In order to achieve the post-buckling equilibrium paths of plate, in given deflection case, a small value is considered for δ and after solving BVP, applied load proportional to the deflection of the plate center, in

(30) 152

Thin-Walled Structures 112 (2017) 149–158

M.R. Feyzi, A.R. Khorshidvand

Table 1 Comparisons of the results of the present numerical method with results obtained from analytical method by [10], for critical buckling load (P* × 10 5 ) . (ν = 0.3 , B = 0 ). Materials

e1

Homogenous & isotropic

0

Porous/ nonlinear, nonsymmetrical distribution

0.3

0.5

0.7

h /b

0.01 0.02 0.03 0.01 0.02 0.03 0.01 0.02 0.03 0.01 0.02 0.03

Clamped plate

Simply supported plate

Reference

Present

Reference

Present

0.3496 2.7971 9.4404 0.2852 2.2815 7.6999 0.2364 1.8916 6.3842 0.1794 1.4353 4.8442

0.3496 2.7965 9.4381 0.2851 2.2808 7.6978 0.2364 1.8910 6.3822 0.1793 1.4346 4.8425

0.1 0.8005 2.7018

0.0999 0.7996 2.6986

Fig. 4. Post-buckling paths of a saturated porous plate for clamped boundary condition and for different values of e1, and comparison of homogeneous/isotropic case (e1=0 ) with

equilibrium state, is obtained. Then, with continuation method δ is increased step by step and results of each step are used as an initial guess for next step.

results of [20]. (B = 0, ν = 0. 3 ).

4. Numerical results and discussions In this paper, post-buckling behavior of a saturated porous circular plate was studied, and effects of geometric parameters, poroelastic material parameters and boundary conditions on the post-buckling behavior were investigated. Comparisons of obtained results from the present numerical method with obtained results from the analytical method [10] for critical buckling load are shown in Tables 1, 2. In Table 1, critical buckling load for a saturated porous plate under clamped boundary condition and a homogeneous/isotropic plate under both clamped and simply supported boundary conditions is shown for different values of porosity coefficients and thickness ratios. In Table 2, critical buckling loads for a porous plate under clamped boundary condition and for different values of Skempton coefficients are presented. Also, obtained results for the homogeneous/isotropic plate are compared, in Figs. 4 and 5 for post-buckling paths and in Fig. 8 for post-buckling configurations, with results of [20]. There are good agreements between obtained results from the present numerical method with existing results in the literature that confirms validity of the present numerical method.

Fig. 5. Post-buckling paths of a saturated porous plate for simply supported boundary condition and for different values of e1, and comparison of homogeneous/isotropic case (e1=0 ) with results of [20]. (B = 0, ν = 0. 3 ).

conditions, are shown. It is observed that the post-buckling strength of the homogeneous/isotropic plate (e1=0 ), is more than the porous plate. Increasing the porosity coefficient decreases the post-buckling strength. The greater the value of deflection of the plate, the more considerable the effect of porosity coefficient on plate strength reduction. Also, with comparison of two figures it can be found that for each specified value of dimensionless central deflection (W (0)/ h ), the corresponding post-buckling load for the clamed case in comparison with the simply supported case is more than double. Fig. 6 is composed of four magnified views of Fig. 4. As it can be seen, the homogeneous/isotropic plate has a strictly increasing postbuckling path, but in the porous plate, the post-buckling load does not increase monotonically with increasing the central deflection and postbuckling loads for small deflections are lower than critical buckling load. In other words, in the porous plate after the buckling, applied load decreases and then increases with deflection increasing. From Fig. 6, it can be concluded that post-buckling load reduction value (in small deflections) decreases with decreasing of porosity coefficient and in homogeneous/isotropic plate tends to zero. Fig. 7 is a magnified view of Fig. 5. According to this figure, it can be found that in the porous plate with the simply supported boundary condition, unlike the clamped case, buckling does not occur but from the beginning of loading the plate starts bending. Buckling behavior of

4.1. Porosity In this section, the effect of porosity coefficient on post-buckling behavior of a saturated porous plate with nonlinear nonsymmetric pore distribution is investigated. In Figs. 4 and 5 post-buckling equilibrium paths for different porosity coefficients and clamped and simply supported boundary Table 2 Comparisons of the results of the present numerical method with results obtained from analytical method by [10], for critical buckling load (P* × 107 ) and for clamped boundary conditions. (ν = 0. 3,e1=0. 5, h /b = 0. 01) . Materials

B

Reference

Present

Porous/nonlinear, symmetric distribution

0.3 0.5 0.7 0.3 0.5 0.7 0.3 0.5 0.7

29.80428 30.52959 31.21832 24.65218 25.28168 25.88022 18.20553 18.65789 19.08871

29.3065 29.7422 30.168 24.6483 25.2777 25.8762 18.7322 19.521 20.2751

Porous/nonlinear, nonsymmetrical distribution

Porous/monotonous distribution

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Fig. 6. Zoomed views of Fig. 4. Fig. 8. Post-buckling configurations of a saturated porous plate for clamped boundary condition and for different values of e1, and comparison of homogeneous/isotropic case (e1=0 ) with results of [20]. (B = 0, ν = 0. 3 ).

Fig. 7. Zoomed view of Fig. 5.

homogeneous/isotropic plate under simply supported boundary condition is the same as clamped boundary condition. Therefore, in the circular plate with simply supported edge, buckling occurs only for the case e1=0 (i.e. homogeneous/isotropic plate) and for the case e1≠0 (i.e. porous plate) bending occurs. In Figs. 8 and 9, for each of loads λ = 20,100 , the post-buckling configurations of the saturated porous plate are shown for different porosity coefficients and for clamped and simply supported boundary conditions, respectively. As it can be seen, deflection of porous plate is more than homogeneous/isotropic plate, and increasing the porosity coefficient increases the plate deflection. Also, deflection of simply supported plate is more than that of clamped plate. In addition, as shown in these figures, with increase of applied load, curvatures of the post-buckling configurations curves decrease near the plate center and increase near the plate edge. Variations of the bending moments at the center of the plate versus applied load on the edge of the plate for different values of porosity coefficient are studied in Figs. 10 and 11 for clamped and simply supported boundary conditions, respectively. It can be observed that, as the plate starts buckling, bending moment at the plate center increases rapidly and then decreases slowly with increase of the applied load. This decline occurs because of the decline of curvature near the plate center (as mentioned above).

Fig. 9. Post-buckling configurations of a saturated porous plate for simply supported boundary condition and for different values of e1. (B = 0, ν = 0. 3 ).

Fig. 12 is a magnified view of Fig. 10 for the small values of applied load. This figure shows that in the porous plate with clamped boundary condition, bending moment at the center of the plate has a negative value in the beginning of buckling, and after the buckling when the post-bucking load reduces to its minimum value, central bending moment tends to zero, then both of them (i.e. post-buckling load and central bending moment) increase with increasing the deflection; But homogeneous/isotropic plate has a strictly increasing post-buckling path and load reduction does not occur after the buckling, therefore central bending moment does not have a negative value. Magnified view of Fig. 11 is plotted for the small values of applied load in Fig. 13. As it can be seen, in the porous plate with simply supported edge from beginning of loading, central bending moment increases slowly and then increases rapidly with increasing the postbuckling load. Homogeneous/isotropic plate with simply supported boundary condition behaves like clamped boundary condition thus 154

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Fig. 13. Zoomed view of Fig. 11.

Fig. 10. Central bending moments Vs. applied load for clamped plate and for different values of e1. (B = 0, ν = 0. 3 ).

Fig. 14. Variations of applied load (P*) Vs. thickness to radial ratio for clamped plate and for different values of e1 for each of two cases W (0 )/h = 3, 7 . (B = 0, ν = 0. 3 ).

Fig. 11. Central bending moments Vs. applied load for simply supported plate and for different values of e1. (B = 0 , ν = 0. 3 ).

Fig. 15. Variations of applied load (P*) Vs. thickness to radial ratio for simply supported plate and for different values of e1 for each of two cases W (0 )/h = 3, 7 . (B = 0, ν = 0. 3 ).

Fig. 12. Zoomed view of Fig. 10.

starts buckling in a certain critical load, then it's bending moment increases.

4.2. Thickness In Figs. 14 and 15 for each of the dimensionless central deflections W (0)/ h=3,7, curves of post-buckling load versus thickness to radial ratio 155

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Fig. 18. Variation of applied load ( λ ) with coefficient of porosity (e1) for a plate with Fig. 16. Post-buckling configurations of a saturated porous plate for clamped boundary condition and for different pore distributions with e1=0. 3 and comparison with

clamped boundary condition (W (0 )/h = 7 , B = 0, ν = 0. 3 ).

and

for

different

cases

of

pore

distributions.

homogeneous/isotropic plate. ( λ = 100, B = 0, ν = 0. 3 ).

are shown for a saturated porous plate with nonlinear nonsymmetric pore distribution and for different porosity coefficients and for clamped and simply supported boundary conditions, respectively. According to the results presented in these figures, increasing the thickness increases the post-buckling strength; in other words, to create a certain deflection, by increasing the plate thickness, more load must be applied on the edge of the plate. It is also clear that, increasing the porosity coefficient, decreases the post-buckling strength of the plate. 4.3. Pore distribution In Figs. 16 and 17, the effect of pore distribution on post-buckling behavior of a saturated porous plate is investigated for clamped and simply supported boundary conditions and is compared with a homogeneous/isotropic plate. The plates are subjected to the post-buckling load λ = 100 . These figures show that the porous plate with any pore distribution has lower post-buckling strength than the homogeneous/ isotropic plate. Also, it can be seen that the post-buckling strength of the porous plate is highest in the nonlinear symmetric distribution case, because near the upper and lower surfaces of the plate, porosity tends to zero. For example, this phenomenon is also observed in wide-flange beams (W-beams) that a large proportion of cross section is in farthest distance from the neutral axis and near the upper and lower surfaces.

Fig. 19. Variation of applied load ( λ ) with coefficient of porosity (e1) for a plate with simply supported boundary condition and for different cases of pore distributions. (W (0 )/h = 7 , B = 0, ν = 0. 3 ).

Effect of porous change on the post-buckling behavior of the porous plate for different pore distributions and for clamped and simply supported boundary conditions is shown in Figs. 18 and 19, respectively; It can be found that with increasing porosity coefficient, the post-buckling strength of the porous plate with any pore distributions decreases almost linearly. 4.4. Pore fluid properties Under the undrained condition (ξ=0), that fluid within porous solid cannot escape, Skempton pore pressure coefficient depends on pore fluid compressibility and lies in the interval (0,1). So that, if pore fluid compressibility increases (B → 0), saturated porous plate behaves like a porous plate without fluid under the drained condition, and if pore fluid compressibility decreases (B → 1), plate behavior will be close to a rigid body [10]. In order to investigate the effect of pore fluid compressibility on post-buckling behavior of a saturated porous plate with different pore distributions, Figs. 20 and 21 are studied. It can be observed that by decreasing the pore fluid compressibility, the post-buckling strength of plate increases. With the aim of increasing the strength of the porous plate versus post-buckling load, appropriate distribution of pores in the thickness direction, is more effective than decreasing the pore fluid compressibility.

Fig. 17. Post-buckling configurations of a saturated porous plate for simply supported boundary condition and for different pore distributions with e1=0. 3 and comparison with homogeneous/isotropic plate. ( λ = 100, B = 0, ν = 0. 3 ).

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5)

6) 7)

8)

9) Fig. 20. Variation of applied load ( λ ) with Skempton coefficient ( B ) for a saturated porous plate for clamped boundary condition and different pore distributions. (e1=0. 5, W (0 )/h = 7 , ν = 0. 3 ).

10) 11) 12)

in which porosity value decreases near the top and bottom surfaces, shows more post-buckling strength compared to other pore distributions. In a porous plate with clamped boundary condition, after buckling, applied load decreases in small deflections and then increases with increasing the deflection. Load reduction after buckling is directly proportional to the porosity coefficient. In a porous plate with the simply supported boundary condition buckling does not occur, unlike the clamped boundary condition, but from the beginning of loading the plate starts bending so deflection increases slowly and then increases rapidly with increasing the post-buckling load. Bending moment at the center of plate increases rapidly in beginning of buckling and then decreases slowly with increase of post-buckling load. In order to form a specified deflection in the plate, the required post-buckling load for the clamped case in comparison with the simply supported case is more than double. By increasing plate thickness, post-buckling strength increases nonlinearly. Increasing pore fluid compressibility reduces post-buckling strength. In order to increase post-buckling strength of plate, appropriate distribution of pores in the thickness direction is more effective than fluid compressibility decline.

Acknowledgement This work was supported by the Islamic Azad University, South Tehran Branch, Tehran, Iran. References [1] M.A. Biot, Theory of buckling of a porous slab and Its thermoelastic analogy, J. Appl. Mech. 31 (1964) 194–198. [2] K. Magnucki, P. Stasiewicz, Elastic buckling of a porous beam, J. Theor. Appl. Mech. 42 (2004) 859–868. [3] K. Magnucki, M. Malinowski, J. Kasprzak, Bending and buckling of a rectangular porous plate, Steel Compos. Struct. 6 (2006) 319–333. [4] E. Magnucka-Blandzi, Axi-symmetrical deflection and buckling of circular porouscellular plate, Thin-Walled Struct. 46 (2008) 333–337. [5] E. Magnucka-Blandzi, Dynamic stability of a metal foam circular plate, J. Theor. Appl. Mech. 47 (2009) 421–433. [6] E. Magnucka-Blandzi, Mathematical modelling of a rectangular sandwich plate with a metal foam core, J. Theor. Appl. Mech. 49 (2011) 439–455. [7] P. Jasion, E. Magnucka-Blandzi, W. Szyc, K. Magnucki, Global and local buckling of sandwich circular and beam-rectangular plates with metal foam core, Thin-Walled Struct. 61 (2012) 154–161. [8] P.H. Wen, The analytical solutions of incompressible saturated poroelastic circular Mindlin's plate, J. Appl. Mech. 79 (2012) (051009-051009). [9] M. Jabbari, M. Hashemitaheri, A. Mojahedin, M.R. Eslami, Thermal buckling analysis of functionally graded thin circular plate made of saturated porous materials, J. Therm. Stress. 37 (2014) 202–220. [10] M. Jabbari, A. Mojahedin, A. Khorshidvand, M. Eslami, Buckling analysis of a functionally graded thin circular plate made of saturated porous materials, J. Eng. Mech. 140 (2013) 287–295. [11] M. Jabbari, E.F. Joubaneh, A. Khorshidvand, M. Eslami, Buckling analysis of porous circular plate with piezoelectric actuator layers under uniform radial compression, Int. J. Mech. Sci. 70 (2013) 50–56. [12] M. Jabbari, A. Mojahedin, E.F. Joubaneh, Thermal buckling analysis of circular plates made of piezoelectric and saturated porous functionally graded material layers, J. Eng. Mech. 141 (2014) 04014148. [13] A. Khorshidvand, E.F. Joubaneh, M. Jabbari, M. Eslami, Buckling analysis of a porous circular plate with piezoelectric sensor–actuator layers under uniform radial compression, Acta Mech. 225 (2014) 179–193. [14] M. Jabbari, E.F. Joubaneh, A. Mojahedin, Thermal buckling analysis of porous circular plate with piezoelectric actuators based on first order shear deformation theory, Int. J. Mech. Sci. 83 (2014) 57–64. [15] J. Woo, S.A. Meguid, Nonlinear analysis of functionally graded plates and shallow shells, Int. J. Solids Struct. 38 (2001) 7409–7421. [16] R. Javaheri, M.R. Eslami, Thermal buckling of functionally graded plates, AIAA J. 40 (2002) 162–169. [17] R. Javaheri, M.R. Eslami, Thermal buckling Of functionally graded plates based On higher order theory, J. Therm. Stress. 25 (2002) 603–625.

Fig. 21. Variation of applied load ( λ ) with Skempton coefficient ( B ) for a saturated porous plate for simply supported boundary condition and different pore distributions. (e1=0. 5, W (0 )/h = 7 , ν = 0. 3 ).

5. Conclusions Post-buckling of a saturated porous circular plate was studied under uniform radial compression. It was assumed that mechanical properties vary continuously in thickness direction. Pore distribution in the thickness direction was considered by three different functions. Based on classical plate theory and Sanders assumption, governing equilibrium equations of the problem in terms of the displacement components and as a system of coupled nonlinear differential equations were obtained. Boundary conditions were considered clamped and simply supported. In order to solve this boundary value problem, a shooting method in conjunction with a Newton-Raphson method was employed. The effects of boundary conditions, geometrical parameters and poroelastic material parameters on post-buckling of the plate were expressed. The conclusions of this study are presented as follows: 1) By decreasing the porosity coefficient, the post-buckling behavior of porous plate will be close to the homogeneous/isotropic plate. 2) By increasing the porosity coefficient, the post-buckling strength of plate with different pore distributions decreases almost linearly. 3) Increasing deflection of the plate increases the effect of porosity coefficient on the post-buckling strength reduction. 4) A plate with symmetric pore distribution in the thickness direction, 157

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