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Investigating post-buckling of geometrically imperfect metal foam nanobeams with symmetric and asymmetric porosity distributions
Accepted Manuscript Investigating post-buckling of geometrically imperfect metal foam nanobeams with symmetric and asymmetric porosity distributions Mohammad Reza Barati, Ashraf M. Zenkour PII: DOI: Reference:
S0263-8223(17)32217-1 http://dx.doi.org/10.1016/j.compstruct.2017.09.008 COST 8871
To appear in:
Composite Structures
Received Date: Revised Date: Accepted Date:
16 July 2017 19 August 2017 14 September 2017
Please cite this article as: Barati, M.R., Zenkour, A.M., Investigating post-buckling of geometrically imperfect metal foam nanobeams with symmetric and asymmetric porosity distributions, Composite Structures (2017), doi: http:// dx.doi.org/10.1016/j.compstruct.2017.09.008
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Investigating post-buckling of geometrically imperfect metal foam nanobeams with symmetric and asymmetric porosity distributions Mohammad Reza Barati1, Ashraf M. Zenkour2,3 1
2
Aerospace Engineering Department & Center of Excellence in Computational Aerospace, AmirKabir University of Technology, Tehran 15875-4413, Iran
Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia 3 Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafr El-Sheikh 33516, Egypt * Corresponding author email: [email protected], [email protected]
Abstract In this research, analysis of post-buckling behavior of porous metal foam nanobeams is performed based on a nonlocal nonlinear refined shear deformation beam model with geometric nonlinearity and imperfection. In the metal foam nanobeam, porosities are dispersed by uniform, symmetric and asymmetric models. The present nanobeam model satisfies the shear deformation effect needless of any shear correction factor. The post-buckling load-deflection relation is obtained by solving the governing equations having cubic nonlinearity applying Galerkin’s method needless of any iteration process. New results show the importance of porosity coefficient, porosity distribution, geometrical imperfection, nonlocal parameter, foundation parameters and slenderness ratio on nonlinear buckling behavior of porous nanoscale beams. Specially, porosities have a great impact on post-buckling configuration of both ideal and imperfect nanobeams. Keywords: Post-buckling, Refined beam theory, Porous nanobeam, Nonlocal elasticity, Porosities
1. Introduction Porous materials, such as metal foams, are an important category of lightweight materials with application to aerospace engineering, automotive industry and civil constructions owning to supreme multi-functionality offered by low specific weight, efficient capacity of energy dissipation and enhanced machinability. Usually, the variation of porosity through the thickness
of porous plates causes a smooth change in mechanical properties. Therefore, this type of materials has received broad interest by some researchers. Jabbari et al. [1, 2] examined porosity distribution effect on buckling characteristics of saturated porous plates. Chen et al. [3] studied static bending and buckling of metal foam porous beams with functionally graded porosities using a shear deformation beam model. In another work, Chen et al. [4, 5] explored linear and nonlinear vibration behavior of metal foam beams with different porosity distributions. They stated that uniform and non-uniform porosity distributions have a significant influence on vibration frequencies of the plates due to the reduction in their stiffness. Also, Rezaei and Saidi [6] studied the influence of non-uniform (asymmetric) porosities on vibration frequencies of thick porous plates using a unified higher order plate model. Different from metal foam structures, porosities occur in the functionally graded metal-ceramic structures during their construction. Interested readers on this issue are referred to previous investigations [7-11]. It is clear that all of the above-mentioned studies on the mechanical behavior of metal foam structures using higher order theories are conducted with ignorance of small size influences. Recent experimental results indicate that when the size of the structures reduces to nano scale, the influences of small scale play a notable role in mechanical responses of such nanostructures. The defect of the classical continuum theory is that it does not take into account the size effects in micro/nano scale structures. So, Eringen’s nonlocal elasticity theory [12] is proposed to overcome this problem which includes small scale effects with good accuracy to model micro/nano scale devices and systems. Based on the nonlocal constitutive relations of Eringen, a number of studies have been carried out to predict the mechanical responses of nanostructures [13-23]. The post-buckling of nanoscale beams via nonlocal elasticity has been a topic of investigation in recent years. Mohammadi et al. [24] examined post-buckling behavior of homogenous nanobeams with geometric imperfection based on classical beam model. Liu et al. [25] explored post-buckling behavior of perfect piezoelectric nanobeams based on nonlocal elasticity theory. Li and Hu [26] performed post-buckling analysis of functionally graded nanobeams accounting for nonlocal and strain gradient effects. Also, Thongyothee and Chucheepsakul [27] performed postbuckling analysis of perfect nanobeams accounting for nonlocal and surface stress effects based on classical beam theory.
Searching the literature reveals that there is no published paper on buckling of metal foam nanobeams with the effect of porosity distribution. However, there are some published papers on analysis of mechanical behaviors of porous FG nanostructures based on modified power-law function. Mechab et al. [28] examined vibrational characteristics of porous FG nanoplates resting on elastic medium using a higher order refined plate theory. Barati [29] explored forced vibration behavior of FG nanobeams with porosities under dynamic loads and resting on an elastic foundation. In another study, Ebrahimi and Barati [30] presented vibration analysis of porous FG nanobeams under magneto-electric field. Most recently, Barati [31] explored wave propagation in porous double-layered FG nanobeams in thermal environments. This research is concerned with the analysis of post-buckling behavior of nonlocal porous nanobeams made of metal foam having geometrical imperfection. A nonlinear refined beam model is developed for modeling of nanobeam. Three types of porosity dispersion are adopted. The post-buckling load-deflection relation is obtained by solving the governing equations having cubic nonlinearity applying Galerkin’s method. It is shown that the post-buckling loads of porous nanobeams are significantly affected by porosities coefficient, porosity distribution, geometrical imperfection, nonlocality, elastic foundation constants and slenderness ratio. 2. Porous nanobeam model with different porosity distributions Assume a porous nanobeam with thickness h as illustrated in Figs.1 and 2. Different types of porosity distribution have been considered: 1: uniform distribution, 2: non-uniform distribution 1 (symmetric), 3: non-uniform distribution 1 (asymmetric). In the case of non-uniform distribution 1, the lowest values of elasticity moduli and mass density occur at the mid-plane of the nanobeam due to the largest size of nano-pores; while the highest values of elasticity moduli and mass density occur at the top and bottom sides. In the case of non-uniform distribution 2, the elasticity moduli and mass density change gradually from their highest values at the top surface to a lowest value at bottom surface. The mechanical properties of a porous nanobeam with different types of porosity distributions can be expressed by [5]: •
Uniform porosity distribution
E = E1 (1 − e0 χ )
(1a)
G = G1 (1 − e0 χ )
(1b)
ρ = ρ1 (1 − e0 χ ) •
(1c)
Non-uniform distribution 1
πz E ( z ) = E1 (1 − e0 cos ) h
(2a)
πz G( z) = G1 (1 − e0 cos ) h
(2b)
πz ) h
(2c)
ρ ( z ) = ρ1 (1 − em cos
•
Non-uniform distribution 2
πz π E( z) = E1 (1 − e0 cos + ) 2h 4
(3a)
πz π G( z) = G1 (1 − e0 cos + ) 2h 4
(3b)
πz π + ) 2h 4
(3c)
ρ ( z) = ρ1 (1 − em cos
where E1, G1 and
ρ1
are the maximum values of elasticity moduli, shear moduli and mass
density; e0 and em are the coefficients of porosity and mass density, respectively defined by
E G e0 = 1 − 2 = 1 − 2 E1 G1
(4a)
ρ2 ρ1
(4b)
em = 1 −
Also, em can be determined based on the typical mechanical properties of an open-cell metal foam as:
E2 ρ2 = E1 ρ1
(5a)
2
(5b)
em =1− 1− e0
In the case of uniform porosity distribution, the material properties are constant through the thickness direction and they are only dependent on porosity coefficient e0. Then, the coefficient
χ is expressed by: 1 12 2 1 − e0 − + 1 χ= − π e0 e0 π
2
(6)
3. Theoretical formulation Among many developed higher order shear deformation theories [32-, this study uses a refined beam theory due to the fact that the procedure of obtaining governing equations is very similar to classical beam model: u1 ( x, z ) = u ( x ) − ( z − z * )
∂wb ∂w − [ f ( z ) − z ** ] s ∂x ∂x
(7a) (7b)
u3 (x, z) = w( x) = wb (x) + ws ( x) With the following shear strain function:
z 5z 3 f (z) = − + 2 4 3h
(8)
and h/ 2
z* =
∫ ∫
− h /2 h/ 2
E ( z ) zdz
− h /2
E ( z) dz
** , z =
∫
h /2
E ( z ) f ( z)dz
− h /2 h /2
∫
− h /2
(9)
E ( z) dz
Also, u is displacement component of the mid-surface and
wb and ws denote the bending and
shear transverse displacement, respectively. Nonzero strains of the refined beam model are expressed as follows:
2 2 ∂u 1 ∂w 2 * ∂ wb ** ∂ w s εx = + ( ) − (z − z ) − [ f ( z) − z ] ∂x 2 ∂x ∂x 2 ∂x 2 ∂w γ xz = g ( z ) s ∂x
(10)
Also, extended Hamilton’s principle expresses that: t
∫ δ (U − V ) dt = 0
(11)
0
here, U is strain energy and V is work done by external forces. The first variation of the strain energy can be calculated as:
δ U = ∫ σ ijδ εij dV =∫ (σ xδ ε x + σ xzδ γ xz ) dV V
(12)
V
σij are the components of the stress tensor, εij are the components of the strain tensor. Substituting Eq. (10) into Eq.(12) yields: ∂ (wb + w s ) ∂δ (wb + ws ) ∂δ u + ]− M ∫0 ∂x ∂x ∂x ∂ 2δ w s ∂δ ws + Q xz ]d x ∂x 2 ∂x L
δU = −M
s x
[ N x[
b x
∂ 2δ w b ∂x 2
(13)
in which the variables introduced in arriving at the last expression are defined as follows:
( Nx , M xb , M xs ) = ∫
h /2
− h/2
Qxz = ∫
h /2
−h /2
(1, z − z* , f − z** )σ x dz, (14)
g ( z)σ xz dz
The first variation of the work done by non-conservative forces can be written as: L
δ V = ∫ [ − kL + k p 0
L ∂ 2 ( wb + ws ) − k NL ( wb + ws ) 3 ]δ ( wb + ws ) dx − Pδ u 2 0 ∂x
(15)
where kL, kp, and kNL are linear, shear and nonlinear coefficients of elastic substrate. The governing equations can be derived from Hamilton’s principle by setting the coefficients of δ u , δ w b and δ w s to zero as:
∂N x =0 ∂x
(16)
∂ 2 M xb = − N + k L ( wb + w s ) − k P ∇ 2 ( w b + w s ) + k N L ( wb + w s ) 3 ∂x 2
(17)
∂ 2 M xs ∂ Q xz + + N − k L ( w b + w s ) + k P ∇ 2 ( w b + w s ) − k N L ( w b + w s ) 3 = 0 2 ∂x ∂x
(18)
in which ∂ ∂ ( wb + w s ) (N x ) N = ∂x ∂x
(19)
From Eq.(16), it can be deduced that Nx = cte . Thus, it is possible to represent Eq.(19) as:
∂ 2 ( wb + ws ) N = Nx ∂x 2
(20)
3.1. Nonlocal governing equations According to Eringen nonlocal elasticity model which contains wide range interactions between points in a continuum solid, the stress state at a point inside a body is regarded to be a function of all neighbor points’ strains. There are two forms are nonlocal elasticity which are differential and integral models (Zhu and Li 2017). These two models are not equivalent in many cases. Accordingly, the differential form of nonlocal elasticity is an approximate model. Hence, in the present work in order to capture the small size impacts nonlocal elasticity theory is implemented in which a linear differential framework of constitutive equations is expressed as:
(1− (e0a)2∇2 )σij = Cijklε kl
(21)
in which ∇ 2 denotes the Laplacian operator; Cijkl and ε kl are the elastic constants and strain components. Therefore, the scale length e a 0
considers the influences of small size on the
response of nano-scale structures. Thus, the constitutive relations of nonlocal theory for a higher order porous nanobeam can be stated as:
∂2σxx = E(z)ε xx ∂x2 ∂ 2σ xz − ( e0 a ) 2 = G ( z )γ xz ∂x 2
σ xx − (e0a)2
(22)
σ xz
(23)
Integrating Eqs. (25) and (26) over the nanobeam thickness, one can obtain the force-strain and the moment-strain of the nonlocal refined beams as follows:
∂u 1 ∂(wb + ws ) 2 1 ∂(wb* + ws* ) 2 ) − ( )] N x = A[ + ( 2 ∂x 2 ∂x ∂x
(24)
∂ 2 wb ∂ 2 wb* ∂ 2 ws ∂ 2 ws* ∂ 2 ( wb + w s ) 2 − ) − E ( − ) + ( e a ) ( − N 0 x ∂x 2 ∂x 2 ∂x 2 ∂x 2 ∂x 2 + k L ( wb + w s ) − ( k P ) ∇ 2 ( wb + w s ) + k NL ( wb + ws ) 3 ) M xb =
−D(
∂ 2 wb ∂ 2 wb* ∂ 2 ws ∂ 2 w*s ∂ 2 ( wb + ws ) ∂Qxz 2 − ) − F ( − ) + ( e a ) ( − N − 0 x ∂x 2 ∂x ∂x 2 ∂x 2 ∂x 2 ∂x 2 2 3 + k L ( wb + ws ) − (k P )∇ ( wb + ws ) + k NL ( wb + ws ) )
M xs = − E (
2 2 (1 − ( e0 a ) ∇ ) Q xz
= As
∂ws ∂x
(25)
(26)
(27)
in which: A=∫
h/ 2
F=∫
h/ 2
− h/ 2
− h/ 2
E ( z ) dz , D = ∫
h/2
−h/2
E ( z )( f − z ** ) 2 dz ,
E ( z )( z − z * ) 2 dz , E = ∫
h/2
−h/2
E ( z )( z − z * )( f − z ** ) dz
E (z) 2 g dz − h / 2 2(1 + v )
As = ∫
h/2
(28)
The nonlinear governing equations based on the nonlinear refined beam theory in terms of the displacement can be derived by substituting Eqs. (24)-(27), into Eqs. (16)-(18) as follows: A(
∂ ( w b + w s ) ∂ 2 ( w b + w s ) ∂ ( w b* + w *s ) ∂ 2 ( w b* + w s* ) ∂ 2u ) + A ( + )=0 ∂x 2 ∂x ∂x 2 ∂x ∂x 2
∂ 4 wb ∂ 4 wb* ∂ 4 ws ∂ 4 w*s ∂w ∂w ∂ ∂3 ) ) − ) − E ( − ) + ( N − µ (N x x 4 4 4 4 3 ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x + (1 − (e0 a )2 ∇ 2 )( − k L ( wb + ws ) + k P ∇ 2 ( wb + ws ) − k NL ( wb + ws )3 ) = 0
(29 )
− D(
(30
)
∂ 4 wb ∂ 4 wb* ∂ 4 ws ∂ 4 w s* ∂ 2 ws ∂ 2 ws* ∂ ∂w ∂3 ∂w ) ( ) ( ( ( ) − − F − + N ) − µ N ) + A − x x 44 ∂x 4 ∂x 4 ∂x 4 ∂x 4 ∂x ∂x ∂x 3 ∂x ∂x 2 ∂x 2 + (1 − (e0 a ) 2 ∇ 2 )( − k L ( wb + ws ) + k P ∇ 2 ( wb + ws ) − k N L ( wb + w s ) 3 ) = 0 −E (
(31 )
Now, it is possible to represent Eq.(32) as:
∂ ∂u 1 ∂ ( wb + ws ) 2 1 ∂ ( wb* + w*s ) 2 + ( ) − ( ) =0 ∂x ∂x 2 ∂x 2 ∂x
(32)
It means that ∂u 1 ∂ ( wb + w s ) 2 1 ∂ ( wb* + w s* ) 2 =− ( ) + ( ) + c1 2 2 ∂x ∂x ∂x
(33)
Now, integrating Eq.(36) yields:
u=−
1 x ∂ ( wb + ws ) 2 1 x ∂ ( wb* + ws* ) 2 ( ) dx + ( ) dx + c1 x + c2 2 ∫0 2 ∫0 ∂x ∂x
(34)
Then, by applying boundary conditions [36]: u(0)=0, u(L)=-PL/A One can obtain
c2 = 0 c1 =
P 1 L ∂ ( wb + ws ) 2 1 L ∂ ( wb* + ws* ) 2 − ( ) dx ( ) dx − ∫ ∫ A 2L 0 ∂x 2L 0 ∂x
(35)
Inserting Eq.(33) into Eqs.(30) and (31) yields the following system of equations which are decoupled from the axial displacement:
∂ 4 wb ∂ 4 wb* ∂ 4 w s ∂ 4 w s* − ) − E( − ) 4 4 ∂x ∂x ∂x 4 ∂x 4 1 L ∂ ( wb + ws ) 2 1 L ∂ ( wb* + ws* ) 2 P ∂ 2 ( wb + w s ) + A[ + ( ) dx − ( ) dx − ] 2 L ∫0 ∂x 2 L ∫0 ∂x A ∂x 2 1 L ∂ ( wb + w s ) 2 1 L ∂ ( wb* + w *s ) 2 P ∂ 4 ( wb + ws ) − µ A[ + ( ) dx − ( ) dx − ] 2 L ∫0 ∂x 2 L ∫0 ∂x A ∂x 4 + (1 − ( e0 a ) 2 ∇ 2 )( − k L ( wb + w s ) + k P ∇ 2 ( wb + w s ) − k NL ( wb + ws ) 3 ) = 0 −D(
(36)
∂ 4 wb ∂ 4 wb* ∂ 4 w s ∂ 4 w s* ∂ 2 w s ∂ 2 w s* − ) − F ( − ) + A ( − ) s ∂x 4 ∂x 4 ∂x 4 ∂x 4 ∂x 2 ∂x 2 1 L ∂ ( wb + w s ) 2 1 L ∂ ( wb* + w s* ) 2 P ∂ 2 ( wb + w s ) + A[ + ( ) dx − ( ) dx − ] 2 L ∫0 ∂x 2 L ∫0 ∂x A ∂x 2 1 L ∂ ( wb + w s ) 2 1 L ∂ ( wb* + w s* ) 2 P ∂ 4 ( wb + w s ) − µ A[ + ( ) dx − ( ) d x − ] 2 L ∫0 ∂x 2 L ∫0 ∂x A ∂x 4 2 2 ∂ 2 ( wb + w s ) 2 ∂ wb 2 ∂ ws I ( ) I ( ) − k L ( wb + w s ) + (1 − ( e0 a ) 2 ∇ 2 )( − I 0 + ∇ + ∇ 4 5 ∂t 2 ∂t 2 ∂t 2 + k P ∇ 2 ( wb + w s ) − k NL ( wb + w s ) 3 ) = 0 −E(
(37)
4. Solution procedure A hybrid analytical-numerical procedure of obtaining post-buckling loads of a porous metal foam nanobeam based on Galerkin’s method is presented. To this end, the displacements are adopted as: ∞
wb = ∑Wbm Xm ( x)
(38a)
m=1 ∞
ws = ∑Wsm X m ( x)
(38b)
m =1
where ( Wbm , Wsm) are the unknown coefficients and the functions
X m are the test functions to
satisfy the following boundary conditions: wb = ws = 0,
∂ 2 wb ∂ 2 ws = =0 ∂x 2 ∂x 2
wb = ws = 0,
∂wb ∂ws = =0 ∂x ∂x
for simply supported edges for clamped edges
(39a) (39b)
The initial configuration of nanobeam due to geometrical imperfection can be defined as [37]:
wb* = W b*Φ ( x ) = W b* sin π ws* = W s*Φ ( x ) = W s* sin π
x L for S-S boundary conditions x L
x wb* = Wb*Φ ( x ) = 0.5Wb* 1 − cos 2π L for C-C boundary conditions x ws* = W s*Φ ( x ) = 0.5W s* 1 − cos 2π L
(40a)
(40b)
in which W * is the mid-span initial rise. Inserting Eq. (38) into Eqs. (36)–(37) and implementing Galerkin’s method gives:
k1,1Wbm + k1,2Wsm + G*W 3 +Ψ1,1Wb* +Ψ1,2Ws* = 0
(41a)
* 3 +Ψ2,1Wb* +Ψ2,2Ws* = 0 k2,1Wbm + k2,2Wsm + GW
(41b)
in which W = Wbm + Wsm and k1,1 = − DΓ40 − k L (Γ00 − µ0 Γ 20 ) + kP (Γ 20 − µ0 Γ 40 ) − PΓ 20 + µ PΓ 40 −
A Wb* 2L
L
2
( )∫
A Wb* 2L
0
2
( )∫
(Φ′) 2 dxΓ 20 + µ
L
0
(Φ′) 2 dxΓ 40
(42a)
k1,2 = − EΓ40 − kL (Γ00 − µ0 Γ20 ) + kP (Γ 20 − µ0Γ 40 ) − PΓ20 + µ PΓ40 −
A Wb* 2L
2
( )∫
L
0
(Φ′) 2 dxΓ20 + µ
A Wb* 2L
2
( )∫
L
0
(Φ′)2 dxΓ40
(42b)
k2,2 = − F Γ 40 − kL (Γ00 − µ0 Γ 20 ) + k P (Γ 20 − µ 0 Γ 40 ) + As Γ 20 − PΓ 20 + µ PΓ 40 −
A Ws* 2L
2
( )∫
G * = A(
L
0
(Φ′) 2 dxΓ 20 + µ
A Ws* 2L
2
( )∫
L
0
(Φ′) 2 dxΓ 40
1 1 Γ11Γ 20 ) − µ A( Γ11Γ 40 ) − K NL (Γ 0000 − µ (6Γ1100 + 3Γ 2000 )) 2L 2L L
Ψ 1,1 = D ∫ Φ (4) X m dx
(42c)
(42d) (42e)
0
L
Ψ 1,2 = Ψ 2,1 = E ∫ Φ (4) X m dx
(42f)
0
L
L
Ψ 2,2 = F ∫ Φ ( 4 ) X m dx − As ∫ Φ ( 2 ) X m dx 0
0
(42g)
and L
{Γ00 , Γ20 , Γ40 , Γ11} = ∫ {X m X m , X m'' X m , X m'''' X m , X m' X m' }dx 0
L
{Γ0000 , Γ1100 , Γ2000} = ∫ {X m X m X m X m , X m' X m' X m X m , X m'' X m X m X m }dx 0
'
in which X m are the derivatives of function X m with respect to x. The function
X m for
different boundary conditions is defined by:
X = sin
S-S:
X = 0.51 − cos
C-C:
(43)
2 (44)
Also, non-dimensional parameters are adopted as: KL = kL
e a L4 L2 L4 , Kp = kp , K NL = k NL , µ = 0 D D A L
(45)
5. Results and discussions In this research, analysis of post-buckling behavior of porous metal foam nanobeams is performed based on a nonlocal nonlinear refined shear deformation beam model with geometric nonlinearity and imperfection. In the metal foam nanobeam, porosities are dispersed by uniform, symmetric and asymmetric models. First of all, the critical buckling load of the nanobeam is validated with those of classical nanobeam model obtained by Reddy (2007). These results are tabulated in Table 1 for fully simply-supported edge conditions and a good agreement is observed. Also, the ratio of nonlinear buckling load to linear buckling load of a beam is validated with those of Fallah and Aghdam (2011) for different dimensionless amplitudes, as shown in Table2. In the present study, the material properties of metal foam nanobeam are considered as:
•
= 200 GPa, = 7850 / , = 0.33,
Fig. 3 shows the influence of porosity coefficient on the post-buckling load of geometrically perfect and imperfect porous nanobeams at L/h=10, W * / h = 0.1 and KL=Kp=0 for uniform porosity distribution. Various values of porosity coefficient are considered (e0=0.2, 0.4 and 0.6). For an ideal (perfect) nanobeam, the starting point ( W / h = 0 ) is critical buckling load. But, for an imperfect nanobeam ( W * / h ≠ 0 ), there is no critical buckling load, since the nanobeam is at its initial configuration. It is well-known that the nonlinear buckling load gets larger with the increase of dimensionless amplitude. This is due to the intrinsic stiffening effect. Also, increase of porosity coefficient results in smaller buckling loads for both ideal and imperfect nanobeams. This is due to a significant reduction in stiffness of nanobeams in the presence of porosities inside the material structure. Porosity distribution effect on the post-buckling characteristics of porous nanobeams is presented in Fig. 4 at µ=0.2, KL=0 and Kp=0. Obtained results show that the nanobeam with porosity distribution 1 has the highest nonlinear buckling load whereas the results of the nanobeams with uniform porosity distribution and graded porosity distribution 2 are quite close. This indicates that the nanobeam with symmetrically distributed porosity can achieve the highest beam stiffness hence the best mechanical performance. Therefore, porosity distribution has a major role on the buckling behavior and should be considered in stability analysis of nanobeams. As stated, the material properties of porous nanobeam are constant thorough the thickness for uniform porosity distribution. While, the material properties are maximum at upper and lower surfaces for nonuniform porosity distribution 1. These observations are valid for both S-S and C-C boundary conditions. Fig.5 shows the post-buckling load of perfect and imperfect porous nanobeam versus dimensionless deflection for different nonlocal parameters when L/h=10, KL=KP=KNL=0 and
e0=0.5. Uniform porosity distribution is considered in this figure. Post-buckling loads of a macro scale beam are obtained by setting µ =0. One can see that nonlocal coefficient provides a stiffness reduction mechanism leading to smaller post-buckling loads for both ideal and imperfect nanobeams. So, nonlocal modeling of nanobeams yields lower post-buckling loads compared with local model. It is clearly observable that an imperfect nonlocal nanobeam has no critical buckling load. Because, the nanobeam is at the initial state and nonlinear buckling
analysis can be performed at dimensionless amplitudes ( W / h ) larger than the initial amplitude ( W * / h ).
Effect of slenderness ratio (L/h) on post-buckling behavior of nonlocal porous nanobeams with S-S boundary conditions is presented in Fig.6 at µ =0.2, e0=0.5. Uniform porosity distribution is considered in this figure. Both geometrically ideal (perfect) and imperfect nanobeams are also considered. It is evident that nanobeams are more flexible at larger slenderness ratios. Therefore, obtained post-buckling loads become smaller with increase of slenderness ratio at a fixed normalized amplitude ( W / h ). However, obtained post-buckling loads for various values of slenderness ratio depend on the magnitude of normalized amplitude. For smaller slenderness ratios, post-buckling load increases with a higher rate with respect to normalized amplitude than higher slenderness ratios or thinner nanobeams. This is because the nanobeam is stiffer at small slenderness ratios. Geometrical imperfection ( W * / h ) effect on post-buckling behavior of porous nanobeams with uniform porosity distribution is plotted in Fig.7 when µ =0.2, e0=0.5. One can see that the initial configuration of nanobeam has a great influence on the post-buckling load-deflection curves. As stated, the critical buckling load vanished with the consideration of initial geometrical imperfection or in the region of the small bending. Actually, in the case of perfect configuration ( W * / h = 0 ), the nanobeam is first critically buckled. Then, nanobeam buckling strength increases
with the rise of dimensionless amplitude. But, in the case of imperfect configuration ( W * / h ≠ 0 ), there is no buckling strength before the initial state of the nanobeam. So, the buckling load is zero at the starting point for an imperfect nanobeams. After that, higher amplitudes of nanobeams need larger compressive loads. Finally, it can be deduced that pot-buckling curves of perfect and imperfect nanobeams become closer to each other at large dimensionless amplitudes. In Fig.8, post-buckling characteristics of a metal foam nanobeam for various elastic substrate coefficients are plotted at L/h=10, µ =0.2. Winkler coefficient has less important effect on the post-buckling path than Pasternak coefficient by providing a discontinuous interaction of substrate-nanobeam system. Increasing in substrate coefficients yields larger post-buckling loads. However, nonlinear foundation coefficient effect depends on the magnitude of dimensionless amplitude. Actually, effect of nonlinear foundation coefficient becomes more sensible at larger amplitudes.
6. Conclusions This research was concerned with the analysis of post-buckling behavior of nonlocal porous nanobeams made of metal foam having geometrical imperfection. A nonlinear refined beam model was developed for modeling of nanobeam. Three types of porosity dispersion were adopted. It was reported that post-buckling path of metal foam nanobeams is dependent on the value of nonlocal coefficient. Since, structural stiffness reduces with the increase of nonlocal coefficient. Another important factor on post-buckling behavior of metal foam nanobeams was the type of porosity dispersion in the material structure. The smallest post-buckling load was obtained in the case of uniform porosity dispersion. Considering geometrical imperfection, the post-buckling path was significantly different from ideal metal foam nanobeams. Conflict of Interest: The authors declare that they have no conflict of interest. References 1.
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Table 1. Comparison of buckling loads of nanobeams for various nonlocal parameters. µ
0 0.5 1 1.5 2
L/h=20 Reddy (2007)
present
L/h=100 Reddy(2007)
present
9.8696 9.4055 8.9830 8.5969 8.2426
9.8696 9.4055 8.9830 8.5969 8.2426
9.8696 9.4055 8.9830 8.5969 8.2426
9.8696 9.4055 8.9830 8.5969 8.2426
Table 2. Comparison of nonlinear buckling load ratio of beams for various dimensionless amplitudes.
W / h
Fallah and Aghdam (2011)
present
0 0.5 1 1.5
1.000 1.018 1.085 1.199
1.000 1.018 1.085 1.199
(a) Non-uniform porosity distribution 1
(b) Non-uniform porosity distribution 2
(c) Uniform porosity distribution
Fig. 1. Porosity distributions in the thickness direction.
Fig. 2. Configuration and coordinates of nanobeam.
(a) S-S
(b) C-C
Fig.3. Post-buckling load of perfect and imperfect porous nanobeam versus dimensionless deflection for different uniform porosity coefficients (L/h=10, KL=KP=KNL=0, µ =0.2).
(a) S-S
(b) C-C
Fig.4. Post-buckling load of perfect and imperfect porous nanobeam versus dimensionless deflection for different porosity distributions (L/h=10, KL=KP=KNL=0, µ =0.2, e0=0.5).
(a) S-S
(b) C-C
Fig.5. Post-buckling load of perfect and imperfect porous nanobeam versus dimensionless deflection for different nonlocal parameters (L/h=10, KL=KP=KNL=0, e0=0.5).
Fig.6. Post-buckling load of perfect and imperfect porous nanobeam versus dimensionless deflection for different slenderness ratios (KL=KP=KNL=0, µ =0.2, e0=0.5).
Fig.7. Post-buckling load of porous nanobeam versus dimensionless deflection for different imperfection parameters (L/h=10, KL=KP=KNL=0, µ =0.2, e0=0.5).
Fig.8. Post-buckling load of porous nanobeam versus dimensionless deflection for different foundation parameters (L/h=10, KL=KP=KNL=0, µ =0.2, e0=0.5).
S0263-8223(17)32217-1 http://dx.doi.org/10.1016/j.compstruct.2017.09.008 COST 8871
To appear in:
Composite Structures
Received Date: Revised Date: Accepted Date:
16 July 2017 19 August 2017 14 September 2017
Please cite this article as: Barati, M.R., Zenkour, A.M., Investigating post-buckling of geometrically imperfect metal foam nanobeams with symmetric and asymmetric porosity distributions, Composite Structures (2017), doi: http:// dx.doi.org/10.1016/j.compstruct.2017.09.008
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Investigating post-buckling of geometrically imperfect metal foam nanobeams with symmetric and asymmetric porosity distributions Mohammad Reza Barati1, Ashraf M. Zenkour2,3 1
2
Aerospace Engineering Department & Center of Excellence in Computational Aerospace, AmirKabir University of Technology, Tehran 15875-4413, Iran
Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia 3 Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafr El-Sheikh 33516, Egypt * Corresponding author email: [email protected], [email protected]
Abstract In this research, analysis of post-buckling behavior of porous metal foam nanobeams is performed based on a nonlocal nonlinear refined shear deformation beam model with geometric nonlinearity and imperfection. In the metal foam nanobeam, porosities are dispersed by uniform, symmetric and asymmetric models. The present nanobeam model satisfies the shear deformation effect needless of any shear correction factor. The post-buckling load-deflection relation is obtained by solving the governing equations having cubic nonlinearity applying Galerkin’s method needless of any iteration process. New results show the importance of porosity coefficient, porosity distribution, geometrical imperfection, nonlocal parameter, foundation parameters and slenderness ratio on nonlinear buckling behavior of porous nanoscale beams. Specially, porosities have a great impact on post-buckling configuration of both ideal and imperfect nanobeams. Keywords: Post-buckling, Refined beam theory, Porous nanobeam, Nonlocal elasticity, Porosities
1. Introduction Porous materials, such as metal foams, are an important category of lightweight materials with application to aerospace engineering, automotive industry and civil constructions owning to supreme multi-functionality offered by low specific weight, efficient capacity of energy dissipation and enhanced machinability. Usually, the variation of porosity through the thickness
of porous plates causes a smooth change in mechanical properties. Therefore, this type of materials has received broad interest by some researchers. Jabbari et al. [1, 2] examined porosity distribution effect on buckling characteristics of saturated porous plates. Chen et al. [3] studied static bending and buckling of metal foam porous beams with functionally graded porosities using a shear deformation beam model. In another work, Chen et al. [4, 5] explored linear and nonlinear vibration behavior of metal foam beams with different porosity distributions. They stated that uniform and non-uniform porosity distributions have a significant influence on vibration frequencies of the plates due to the reduction in their stiffness. Also, Rezaei and Saidi [6] studied the influence of non-uniform (asymmetric) porosities on vibration frequencies of thick porous plates using a unified higher order plate model. Different from metal foam structures, porosities occur in the functionally graded metal-ceramic structures during their construction. Interested readers on this issue are referred to previous investigations [7-11]. It is clear that all of the above-mentioned studies on the mechanical behavior of metal foam structures using higher order theories are conducted with ignorance of small size influences. Recent experimental results indicate that when the size of the structures reduces to nano scale, the influences of small scale play a notable role in mechanical responses of such nanostructures. The defect of the classical continuum theory is that it does not take into account the size effects in micro/nano scale structures. So, Eringen’s nonlocal elasticity theory [12] is proposed to overcome this problem which includes small scale effects with good accuracy to model micro/nano scale devices and systems. Based on the nonlocal constitutive relations of Eringen, a number of studies have been carried out to predict the mechanical responses of nanostructures [13-23]. The post-buckling of nanoscale beams via nonlocal elasticity has been a topic of investigation in recent years. Mohammadi et al. [24] examined post-buckling behavior of homogenous nanobeams with geometric imperfection based on classical beam model. Liu et al. [25] explored post-buckling behavior of perfect piezoelectric nanobeams based on nonlocal elasticity theory. Li and Hu [26] performed post-buckling analysis of functionally graded nanobeams accounting for nonlocal and strain gradient effects. Also, Thongyothee and Chucheepsakul [27] performed postbuckling analysis of perfect nanobeams accounting for nonlocal and surface stress effects based on classical beam theory.
Searching the literature reveals that there is no published paper on buckling of metal foam nanobeams with the effect of porosity distribution. However, there are some published papers on analysis of mechanical behaviors of porous FG nanostructures based on modified power-law function. Mechab et al. [28] examined vibrational characteristics of porous FG nanoplates resting on elastic medium using a higher order refined plate theory. Barati [29] explored forced vibration behavior of FG nanobeams with porosities under dynamic loads and resting on an elastic foundation. In another study, Ebrahimi and Barati [30] presented vibration analysis of porous FG nanobeams under magneto-electric field. Most recently, Barati [31] explored wave propagation in porous double-layered FG nanobeams in thermal environments. This research is concerned with the analysis of post-buckling behavior of nonlocal porous nanobeams made of metal foam having geometrical imperfection. A nonlinear refined beam model is developed for modeling of nanobeam. Three types of porosity dispersion are adopted. The post-buckling load-deflection relation is obtained by solving the governing equations having cubic nonlinearity applying Galerkin’s method. It is shown that the post-buckling loads of porous nanobeams are significantly affected by porosities coefficient, porosity distribution, geometrical imperfection, nonlocality, elastic foundation constants and slenderness ratio. 2. Porous nanobeam model with different porosity distributions Assume a porous nanobeam with thickness h as illustrated in Figs.1 and 2. Different types of porosity distribution have been considered: 1: uniform distribution, 2: non-uniform distribution 1 (symmetric), 3: non-uniform distribution 1 (asymmetric). In the case of non-uniform distribution 1, the lowest values of elasticity moduli and mass density occur at the mid-plane of the nanobeam due to the largest size of nano-pores; while the highest values of elasticity moduli and mass density occur at the top and bottom sides. In the case of non-uniform distribution 2, the elasticity moduli and mass density change gradually from their highest values at the top surface to a lowest value at bottom surface. The mechanical properties of a porous nanobeam with different types of porosity distributions can be expressed by [5]: •
Uniform porosity distribution
E = E1 (1 − e0 χ )
(1a)
G = G1 (1 − e0 χ )
(1b)
ρ = ρ1 (1 − e0 χ ) •
(1c)
Non-uniform distribution 1
πz E ( z ) = E1 (1 − e0 cos ) h
(2a)
πz G( z) = G1 (1 − e0 cos ) h
(2b)
πz ) h
(2c)
ρ ( z ) = ρ1 (1 − em cos
•
Non-uniform distribution 2
πz π E( z) = E1 (1 − e0 cos + ) 2h 4
(3a)
πz π G( z) = G1 (1 − e0 cos + ) 2h 4
(3b)
πz π + ) 2h 4
(3c)
ρ ( z) = ρ1 (1 − em cos
where E1, G1 and
ρ1
are the maximum values of elasticity moduli, shear moduli and mass
density; e0 and em are the coefficients of porosity and mass density, respectively defined by
E G e0 = 1 − 2 = 1 − 2 E1 G1
(4a)
ρ2 ρ1
(4b)
em = 1 −
Also, em can be determined based on the typical mechanical properties of an open-cell metal foam as:
E2 ρ2 = E1 ρ1
(5a)
2
(5b)
em =1− 1− e0
In the case of uniform porosity distribution, the material properties are constant through the thickness direction and they are only dependent on porosity coefficient e0. Then, the coefficient
χ is expressed by: 1 12 2 1 − e0 − + 1 χ= − π e0 e0 π
2
(6)
3. Theoretical formulation Among many developed higher order shear deformation theories [32-, this study uses a refined beam theory due to the fact that the procedure of obtaining governing equations is very similar to classical beam model: u1 ( x, z ) = u ( x ) − ( z − z * )
∂wb ∂w − [ f ( z ) − z ** ] s ∂x ∂x
(7a) (7b)
u3 (x, z) = w( x) = wb (x) + ws ( x) With the following shear strain function:
z 5z 3 f (z) = − + 2 4 3h
(8)
and h/ 2
z* =
∫ ∫
− h /2 h/ 2
E ( z ) zdz
− h /2
E ( z) dz
** , z =
∫
h /2
E ( z ) f ( z)dz
− h /2 h /2
∫
− h /2
(9)
E ( z) dz
Also, u is displacement component of the mid-surface and
wb and ws denote the bending and
shear transverse displacement, respectively. Nonzero strains of the refined beam model are expressed as follows:
2 2 ∂u 1 ∂w 2 * ∂ wb ** ∂ w s εx = + ( ) − (z − z ) − [ f ( z) − z ] ∂x 2 ∂x ∂x 2 ∂x 2 ∂w γ xz = g ( z ) s ∂x
(10)
Also, extended Hamilton’s principle expresses that: t
∫ δ (U − V ) dt = 0
(11)
0
here, U is strain energy and V is work done by external forces. The first variation of the strain energy can be calculated as:
δ U = ∫ σ ijδ εij dV =∫ (σ xδ ε x + σ xzδ γ xz ) dV V
(12)
V
σij are the components of the stress tensor, εij are the components of the strain tensor. Substituting Eq. (10) into Eq.(12) yields: ∂ (wb + w s ) ∂δ (wb + ws ) ∂δ u + ]− M ∫0 ∂x ∂x ∂x ∂ 2δ w s ∂δ ws + Q xz ]d x ∂x 2 ∂x L
δU = −M
s x
[ N x[
b x
∂ 2δ w b ∂x 2
(13)
in which the variables introduced in arriving at the last expression are defined as follows:
( Nx , M xb , M xs ) = ∫
h /2
− h/2
Qxz = ∫
h /2
−h /2
(1, z − z* , f − z** )σ x dz, (14)
g ( z)σ xz dz
The first variation of the work done by non-conservative forces can be written as: L
δ V = ∫ [ − kL + k p 0
L ∂ 2 ( wb + ws ) − k NL ( wb + ws ) 3 ]δ ( wb + ws ) dx − Pδ u 2 0 ∂x
(15)
where kL, kp, and kNL are linear, shear and nonlinear coefficients of elastic substrate. The governing equations can be derived from Hamilton’s principle by setting the coefficients of δ u , δ w b and δ w s to zero as:
∂N x =0 ∂x
(16)
∂ 2 M xb = − N + k L ( wb + w s ) − k P ∇ 2 ( w b + w s ) + k N L ( wb + w s ) 3 ∂x 2
(17)
∂ 2 M xs ∂ Q xz + + N − k L ( w b + w s ) + k P ∇ 2 ( w b + w s ) − k N L ( w b + w s ) 3 = 0 2 ∂x ∂x
(18)
in which ∂ ∂ ( wb + w s ) (N x ) N = ∂x ∂x
(19)
From Eq.(16), it can be deduced that Nx = cte . Thus, it is possible to represent Eq.(19) as:
∂ 2 ( wb + ws ) N = Nx ∂x 2
(20)
3.1. Nonlocal governing equations According to Eringen nonlocal elasticity model which contains wide range interactions between points in a continuum solid, the stress state at a point inside a body is regarded to be a function of all neighbor points’ strains. There are two forms are nonlocal elasticity which are differential and integral models (Zhu and Li 2017). These two models are not equivalent in many cases. Accordingly, the differential form of nonlocal elasticity is an approximate model. Hence, in the present work in order to capture the small size impacts nonlocal elasticity theory is implemented in which a linear differential framework of constitutive equations is expressed as:
(1− (e0a)2∇2 )σij = Cijklε kl
(21)
in which ∇ 2 denotes the Laplacian operator; Cijkl and ε kl are the elastic constants and strain components. Therefore, the scale length e a 0
considers the influences of small size on the
response of nano-scale structures. Thus, the constitutive relations of nonlocal theory for a higher order porous nanobeam can be stated as:
∂2σxx = E(z)ε xx ∂x2 ∂ 2σ xz − ( e0 a ) 2 = G ( z )γ xz ∂x 2
σ xx − (e0a)2
(22)
σ xz
(23)
Integrating Eqs. (25) and (26) over the nanobeam thickness, one can obtain the force-strain and the moment-strain of the nonlocal refined beams as follows:
∂u 1 ∂(wb + ws ) 2 1 ∂(wb* + ws* ) 2 ) − ( )] N x = A[ + ( 2 ∂x 2 ∂x ∂x
(24)
∂ 2 wb ∂ 2 wb* ∂ 2 ws ∂ 2 ws* ∂ 2 ( wb + w s ) 2 − ) − E ( − ) + ( e a ) ( − N 0 x ∂x 2 ∂x 2 ∂x 2 ∂x 2 ∂x 2 + k L ( wb + w s ) − ( k P ) ∇ 2 ( wb + w s ) + k NL ( wb + ws ) 3 ) M xb =
−D(
∂ 2 wb ∂ 2 wb* ∂ 2 ws ∂ 2 w*s ∂ 2 ( wb + ws ) ∂Qxz 2 − ) − F ( − ) + ( e a ) ( − N − 0 x ∂x 2 ∂x ∂x 2 ∂x 2 ∂x 2 ∂x 2 2 3 + k L ( wb + ws ) − (k P )∇ ( wb + ws ) + k NL ( wb + ws ) )
M xs = − E (
2 2 (1 − ( e0 a ) ∇ ) Q xz
= As
∂ws ∂x
(25)
(26)
(27)
in which: A=∫
h/ 2
F=∫
h/ 2
− h/ 2
− h/ 2
E ( z ) dz , D = ∫
h/2
−h/2
E ( z )( f − z ** ) 2 dz ,
E ( z )( z − z * ) 2 dz , E = ∫
h/2
−h/2
E ( z )( z − z * )( f − z ** ) dz
E (z) 2 g dz − h / 2 2(1 + v )
As = ∫
h/2
(28)
The nonlinear governing equations based on the nonlinear refined beam theory in terms of the displacement can be derived by substituting Eqs. (24)-(27), into Eqs. (16)-(18) as follows: A(
∂ ( w b + w s ) ∂ 2 ( w b + w s ) ∂ ( w b* + w *s ) ∂ 2 ( w b* + w s* ) ∂ 2u ) + A ( + )=0 ∂x 2 ∂x ∂x 2 ∂x ∂x 2
∂ 4 wb ∂ 4 wb* ∂ 4 ws ∂ 4 w*s ∂w ∂w ∂ ∂3 ) ) − ) − E ( − ) + ( N − µ (N x x 4 4 4 4 3 ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x + (1 − (e0 a )2 ∇ 2 )( − k L ( wb + ws ) + k P ∇ 2 ( wb + ws ) − k NL ( wb + ws )3 ) = 0
(29 )
− D(
(30
)
∂ 4 wb ∂ 4 wb* ∂ 4 ws ∂ 4 w s* ∂ 2 ws ∂ 2 ws* ∂ ∂w ∂3 ∂w ) ( ) ( ( ( ) − − F − + N ) − µ N ) + A − x x 44 ∂x 4 ∂x 4 ∂x 4 ∂x 4 ∂x ∂x ∂x 3 ∂x ∂x 2 ∂x 2 + (1 − (e0 a ) 2 ∇ 2 )( − k L ( wb + ws ) + k P ∇ 2 ( wb + ws ) − k N L ( wb + w s ) 3 ) = 0 −E (
(31 )
Now, it is possible to represent Eq.(32) as:
∂ ∂u 1 ∂ ( wb + ws ) 2 1 ∂ ( wb* + w*s ) 2 + ( ) − ( ) =0 ∂x ∂x 2 ∂x 2 ∂x
(32)
It means that ∂u 1 ∂ ( wb + w s ) 2 1 ∂ ( wb* + w s* ) 2 =− ( ) + ( ) + c1 2 2 ∂x ∂x ∂x
(33)
Now, integrating Eq.(36) yields:
u=−
1 x ∂ ( wb + ws ) 2 1 x ∂ ( wb* + ws* ) 2 ( ) dx + ( ) dx + c1 x + c2 2 ∫0 2 ∫0 ∂x ∂x
(34)
Then, by applying boundary conditions [36]: u(0)=0, u(L)=-PL/A One can obtain
c2 = 0 c1 =
P 1 L ∂ ( wb + ws ) 2 1 L ∂ ( wb* + ws* ) 2 − ( ) dx ( ) dx − ∫ ∫ A 2L 0 ∂x 2L 0 ∂x
(35)
Inserting Eq.(33) into Eqs.(30) and (31) yields the following system of equations which are decoupled from the axial displacement:
∂ 4 wb ∂ 4 wb* ∂ 4 w s ∂ 4 w s* − ) − E( − ) 4 4 ∂x ∂x ∂x 4 ∂x 4 1 L ∂ ( wb + ws ) 2 1 L ∂ ( wb* + ws* ) 2 P ∂ 2 ( wb + w s ) + A[ + ( ) dx − ( ) dx − ] 2 L ∫0 ∂x 2 L ∫0 ∂x A ∂x 2 1 L ∂ ( wb + w s ) 2 1 L ∂ ( wb* + w *s ) 2 P ∂ 4 ( wb + ws ) − µ A[ + ( ) dx − ( ) dx − ] 2 L ∫0 ∂x 2 L ∫0 ∂x A ∂x 4 + (1 − ( e0 a ) 2 ∇ 2 )( − k L ( wb + w s ) + k P ∇ 2 ( wb + w s ) − k NL ( wb + ws ) 3 ) = 0 −D(
(36)
∂ 4 wb ∂ 4 wb* ∂ 4 w s ∂ 4 w s* ∂ 2 w s ∂ 2 w s* − ) − F ( − ) + A ( − ) s ∂x 4 ∂x 4 ∂x 4 ∂x 4 ∂x 2 ∂x 2 1 L ∂ ( wb + w s ) 2 1 L ∂ ( wb* + w s* ) 2 P ∂ 2 ( wb + w s ) + A[ + ( ) dx − ( ) dx − ] 2 L ∫0 ∂x 2 L ∫0 ∂x A ∂x 2 1 L ∂ ( wb + w s ) 2 1 L ∂ ( wb* + w s* ) 2 P ∂ 4 ( wb + w s ) − µ A[ + ( ) dx − ( ) d x − ] 2 L ∫0 ∂x 2 L ∫0 ∂x A ∂x 4 2 2 ∂ 2 ( wb + w s ) 2 ∂ wb 2 ∂ ws I ( ) I ( ) − k L ( wb + w s ) + (1 − ( e0 a ) 2 ∇ 2 )( − I 0 + ∇ + ∇ 4 5 ∂t 2 ∂t 2 ∂t 2 + k P ∇ 2 ( wb + w s ) − k NL ( wb + w s ) 3 ) = 0 −E(
(37)
4. Solution procedure A hybrid analytical-numerical procedure of obtaining post-buckling loads of a porous metal foam nanobeam based on Galerkin’s method is presented. To this end, the displacements are adopted as: ∞
wb = ∑Wbm Xm ( x)
(38a)
m=1 ∞
ws = ∑Wsm X m ( x)
(38b)
m =1
where ( Wbm , Wsm) are the unknown coefficients and the functions
X m are the test functions to
satisfy the following boundary conditions: wb = ws = 0,
∂ 2 wb ∂ 2 ws = =0 ∂x 2 ∂x 2
wb = ws = 0,
∂wb ∂ws = =0 ∂x ∂x
for simply supported edges for clamped edges
(39a) (39b)
The initial configuration of nanobeam due to geometrical imperfection can be defined as [37]:
wb* = W b*Φ ( x ) = W b* sin π ws* = W s*Φ ( x ) = W s* sin π
x L for S-S boundary conditions x L
x wb* = Wb*Φ ( x ) = 0.5Wb* 1 − cos 2π L for C-C boundary conditions x ws* = W s*Φ ( x ) = 0.5W s* 1 − cos 2π L
(40a)
(40b)
in which W * is the mid-span initial rise. Inserting Eq. (38) into Eqs. (36)–(37) and implementing Galerkin’s method gives:
k1,1Wbm + k1,2Wsm + G*W 3 +Ψ1,1Wb* +Ψ1,2Ws* = 0
(41a)
* 3 +Ψ2,1Wb* +Ψ2,2Ws* = 0 k2,1Wbm + k2,2Wsm + GW
(41b)
in which W = Wbm + Wsm and k1,1 = − DΓ40 − k L (Γ00 − µ0 Γ 20 ) + kP (Γ 20 − µ0 Γ 40 ) − PΓ 20 + µ PΓ 40 −
A Wb* 2L
L
2
( )∫
A Wb* 2L
0
2
( )∫
(Φ′) 2 dxΓ 20 + µ
L
0
(Φ′) 2 dxΓ 40
(42a)
k1,2 = − EΓ40 − kL (Γ00 − µ0 Γ20 ) + kP (Γ 20 − µ0Γ 40 ) − PΓ20 + µ PΓ40 −
A Wb* 2L
2
( )∫
L
0
(Φ′) 2 dxΓ20 + µ
A Wb* 2L
2
( )∫
L
0
(Φ′)2 dxΓ40
(42b)
k2,2 = − F Γ 40 − kL (Γ00 − µ0 Γ 20 ) + k P (Γ 20 − µ 0 Γ 40 ) + As Γ 20 − PΓ 20 + µ PΓ 40 −
A Ws* 2L
2
( )∫
G * = A(
L
0
(Φ′) 2 dxΓ 20 + µ
A Ws* 2L
2
( )∫
L
0
(Φ′) 2 dxΓ 40
1 1 Γ11Γ 20 ) − µ A( Γ11Γ 40 ) − K NL (Γ 0000 − µ (6Γ1100 + 3Γ 2000 )) 2L 2L L
Ψ 1,1 = D ∫ Φ (4) X m dx
(42c)
(42d) (42e)
0
L
Ψ 1,2 = Ψ 2,1 = E ∫ Φ (4) X m dx
(42f)
0
L
L
Ψ 2,2 = F ∫ Φ ( 4 ) X m dx − As ∫ Φ ( 2 ) X m dx 0
0
(42g)
and L
{Γ00 , Γ20 , Γ40 , Γ11} = ∫ {X m X m , X m'' X m , X m'''' X m , X m' X m' }dx 0
L
{Γ0000 , Γ1100 , Γ2000} = ∫ {X m X m X m X m , X m' X m' X m X m , X m'' X m X m X m }dx 0
'
in which X m are the derivatives of function X m with respect to x. The function
X m for
different boundary conditions is defined by:
X = sin
S-S:
X = 0.51 − cos
C-C:
(43)
2 (44)
Also, non-dimensional parameters are adopted as: KL = kL
e a L4 L2 L4 , Kp = kp , K NL = k NL , µ = 0 D D A L
(45)
5. Results and discussions In this research, analysis of post-buckling behavior of porous metal foam nanobeams is performed based on a nonlocal nonlinear refined shear deformation beam model with geometric nonlinearity and imperfection. In the metal foam nanobeam, porosities are dispersed by uniform, symmetric and asymmetric models. First of all, the critical buckling load of the nanobeam is validated with those of classical nanobeam model obtained by Reddy (2007). These results are tabulated in Table 1 for fully simply-supported edge conditions and a good agreement is observed. Also, the ratio of nonlinear buckling load to linear buckling load of a beam is validated with those of Fallah and Aghdam (2011) for different dimensionless amplitudes, as shown in Table2. In the present study, the material properties of metal foam nanobeam are considered as:
•
= 200 GPa, = 7850 / , = 0.33,
Fig. 3 shows the influence of porosity coefficient on the post-buckling load of geometrically perfect and imperfect porous nanobeams at L/h=10, W * / h = 0.1 and KL=Kp=0 for uniform porosity distribution. Various values of porosity coefficient are considered (e0=0.2, 0.4 and 0.6). For an ideal (perfect) nanobeam, the starting point ( W / h = 0 ) is critical buckling load. But, for an imperfect nanobeam ( W * / h ≠ 0 ), there is no critical buckling load, since the nanobeam is at its initial configuration. It is well-known that the nonlinear buckling load gets larger with the increase of dimensionless amplitude. This is due to the intrinsic stiffening effect. Also, increase of porosity coefficient results in smaller buckling loads for both ideal and imperfect nanobeams. This is due to a significant reduction in stiffness of nanobeams in the presence of porosities inside the material structure. Porosity distribution effect on the post-buckling characteristics of porous nanobeams is presented in Fig. 4 at µ=0.2, KL=0 and Kp=0. Obtained results show that the nanobeam with porosity distribution 1 has the highest nonlinear buckling load whereas the results of the nanobeams with uniform porosity distribution and graded porosity distribution 2 are quite close. This indicates that the nanobeam with symmetrically distributed porosity can achieve the highest beam stiffness hence the best mechanical performance. Therefore, porosity distribution has a major role on the buckling behavior and should be considered in stability analysis of nanobeams. As stated, the material properties of porous nanobeam are constant thorough the thickness for uniform porosity distribution. While, the material properties are maximum at upper and lower surfaces for nonuniform porosity distribution 1. These observations are valid for both S-S and C-C boundary conditions. Fig.5 shows the post-buckling load of perfect and imperfect porous nanobeam versus dimensionless deflection for different nonlocal parameters when L/h=10, KL=KP=KNL=0 and
e0=0.5. Uniform porosity distribution is considered in this figure. Post-buckling loads of a macro scale beam are obtained by setting µ =0. One can see that nonlocal coefficient provides a stiffness reduction mechanism leading to smaller post-buckling loads for both ideal and imperfect nanobeams. So, nonlocal modeling of nanobeams yields lower post-buckling loads compared with local model. It is clearly observable that an imperfect nonlocal nanobeam has no critical buckling load. Because, the nanobeam is at the initial state and nonlinear buckling
analysis can be performed at dimensionless amplitudes ( W / h ) larger than the initial amplitude ( W * / h ).
Effect of slenderness ratio (L/h) on post-buckling behavior of nonlocal porous nanobeams with S-S boundary conditions is presented in Fig.6 at µ =0.2, e0=0.5. Uniform porosity distribution is considered in this figure. Both geometrically ideal (perfect) and imperfect nanobeams are also considered. It is evident that nanobeams are more flexible at larger slenderness ratios. Therefore, obtained post-buckling loads become smaller with increase of slenderness ratio at a fixed normalized amplitude ( W / h ). However, obtained post-buckling loads for various values of slenderness ratio depend on the magnitude of normalized amplitude. For smaller slenderness ratios, post-buckling load increases with a higher rate with respect to normalized amplitude than higher slenderness ratios or thinner nanobeams. This is because the nanobeam is stiffer at small slenderness ratios. Geometrical imperfection ( W * / h ) effect on post-buckling behavior of porous nanobeams with uniform porosity distribution is plotted in Fig.7 when µ =0.2, e0=0.5. One can see that the initial configuration of nanobeam has a great influence on the post-buckling load-deflection curves. As stated, the critical buckling load vanished with the consideration of initial geometrical imperfection or in the region of the small bending. Actually, in the case of perfect configuration ( W * / h = 0 ), the nanobeam is first critically buckled. Then, nanobeam buckling strength increases
with the rise of dimensionless amplitude. But, in the case of imperfect configuration ( W * / h ≠ 0 ), there is no buckling strength before the initial state of the nanobeam. So, the buckling load is zero at the starting point for an imperfect nanobeams. After that, higher amplitudes of nanobeams need larger compressive loads. Finally, it can be deduced that pot-buckling curves of perfect and imperfect nanobeams become closer to each other at large dimensionless amplitudes. In Fig.8, post-buckling characteristics of a metal foam nanobeam for various elastic substrate coefficients are plotted at L/h=10, µ =0.2. Winkler coefficient has less important effect on the post-buckling path than Pasternak coefficient by providing a discontinuous interaction of substrate-nanobeam system. Increasing in substrate coefficients yields larger post-buckling loads. However, nonlinear foundation coefficient effect depends on the magnitude of dimensionless amplitude. Actually, effect of nonlinear foundation coefficient becomes more sensible at larger amplitudes.
6. Conclusions This research was concerned with the analysis of post-buckling behavior of nonlocal porous nanobeams made of metal foam having geometrical imperfection. A nonlinear refined beam model was developed for modeling of nanobeam. Three types of porosity dispersion were adopted. It was reported that post-buckling path of metal foam nanobeams is dependent on the value of nonlocal coefficient. Since, structural stiffness reduces with the increase of nonlocal coefficient. Another important factor on post-buckling behavior of metal foam nanobeams was the type of porosity dispersion in the material structure. The smallest post-buckling load was obtained in the case of uniform porosity dispersion. Considering geometrical imperfection, the post-buckling path was significantly different from ideal metal foam nanobeams. Conflict of Interest: The authors declare that they have no conflict of interest. References 1.
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Table 1. Comparison of buckling loads of nanobeams for various nonlocal parameters. µ
0 0.5 1 1.5 2
L/h=20 Reddy (2007)
present
L/h=100 Reddy(2007)
present
9.8696 9.4055 8.9830 8.5969 8.2426
9.8696 9.4055 8.9830 8.5969 8.2426
9.8696 9.4055 8.9830 8.5969 8.2426
9.8696 9.4055 8.9830 8.5969 8.2426
Table 2. Comparison of nonlinear buckling load ratio of beams for various dimensionless amplitudes.
W / h
Fallah and Aghdam (2011)
present
0 0.5 1 1.5
1.000 1.018 1.085 1.199
1.000 1.018 1.085 1.199
(a) Non-uniform porosity distribution 1
(b) Non-uniform porosity distribution 2
(c) Uniform porosity distribution
Fig. 1. Porosity distributions in the thickness direction.
Fig. 2. Configuration and coordinates of nanobeam.
(a) S-S
(b) C-C
Fig.3. Post-buckling load of perfect and imperfect porous nanobeam versus dimensionless deflection for different uniform porosity coefficients (L/h=10, KL=KP=KNL=0, µ =0.2).
(a) S-S
(b) C-C
Fig.4. Post-buckling load of perfect and imperfect porous nanobeam versus dimensionless deflection for different porosity distributions (L/h=10, KL=KP=KNL=0, µ =0.2, e0=0.5).
(a) S-S
(b) C-C
Fig.5. Post-buckling load of perfect and imperfect porous nanobeam versus dimensionless deflection for different nonlocal parameters (L/h=10, KL=KP=KNL=0, e0=0.5).
Fig.6. Post-buckling load of perfect and imperfect porous nanobeam versus dimensionless deflection for different slenderness ratios (KL=KP=KNL=0, µ =0.2, e0=0.5).
Fig.7. Post-buckling load of porous nanobeam versus dimensionless deflection for different imperfection parameters (L/h=10, KL=KP=KNL=0, µ =0.2, e0=0.5).
Fig.8. Post-buckling load of porous nanobeam versus dimensionless deflection for different foundation parameters (L/h=10, KL=KP=KNL=0, µ =0.2, e0=0.5).