CC BY
5
VOJNOTEHNICKI GLASNIK / MILITARY TECHNICAL COURIER, 2024, Vol. 72, Issue 3
e
Analytical investigation on the buckling and free vibration of porous laminated FG-CNTRC plates
Tahir Ghazoula, Mohamed Atif Benattab,
Abdelwahhab Khatir0, Youcef Beldjelilid,
Baghdad Kroure, Mohamed Bachir Bouiadjraf
a University of Djillali Liabes, Structures and Advanced Materials in Civil Engineering and Public Works Laboratory,
Sidi Bel Abbes, People's Democratic Republic of Algeria, e-mail: [email protected], corresponding author,
ORCID Ю: https://orcid.org/0009-0006-9869-4339
b University of Djillali Liabes, Structures and Advanced Materials in Civil Engineering and Public Works Laboratory,
Sidi Bel Abbes, People's Democratic Republic of Algeria, e-mail: [email protected],
ORCID iD: https://orcid.org/0009-0007-5854-9054
c Polytechnic University of Marche, Structural Section DICEA,
Ancona, Italian Republic, e-mail: [email protected],
ORCID iD: https://orcid.org/0000-0003-4920-5165
d University of Djillali Liabes, Structures and Advanced Materials in Civil Engineering and Public Works Laboratory,
Sidi Bel Abbes, People's Democratic Republic of Algeria, e-mail: [email protected],
ORCID iD: https://orcid.org/0000-0003-3877-9665
e University of Djillali Liabes, Structures and Advanced Materials in Civil Engineering and Public Works Laboratory,
Sidi Bel Abbes, People's Democratic Republic of Algeria, e-mail: [email protected],
ORCID iD: https://orcid.org/0000-0002-8265-9807
f University of Djillali Liabes, Structures and Advanced Materials in Civil Engineering and Public Works Laboratory,
Sidi Bel Abbes, People's Democratic Republic of Algeria;
Thematic Agency for Research in Science and Technology,
Algiers, People's Democratic Republic of Algeria, e-mail: [email protected],
ORCID iD: https://orcid.org/0009-0008-4814-6187
doi https://doi.Org/10.5937/vojtehg72-50469
FIELD: mechanics, materials ARTICLE TYPE: original scientific paper
Abstract:
Introduction/purpose: The aim of this study is to examine the buckling and free vibration behavior of laminated composite plates reinforced with carbon
nanotubes when various sources of uncertainty are taken into account with the main focus being the existence of porosity.
Methods: A porous laminated plate model is developed using a high order shear deformation theory. Different configurations of functionally graded aligned single-walled carbon nanotubes throughout the thickness of each layer are being investigated. The effective properties of materials are evaluated through the extended rule of mixture while considering an upper bound for the effect of porosity. The governing equations are derived and solved using the virtual work principle and Navier's approach. The validity of the current formulation is confirmed by comparing the results with the existing data from literature sources. The impact of numerous parameters such as porosity, carbon nanotube volume fraction, reinforcement distribution types, lamination scheme, and the number of layers on the buckling and free vibration responses is investigated in detail.
Results: A key finding of this study is the significant reduction in buckling resistance of laminated FG-CNTRC plates due to porosity, contrasting with the minor impact on the free vibration response.
Conclusion: The results of this paper emphasize the critical role of porosity in structural integrity and provide novel insights into the behaviour of advanced composite materials.
Key words: buckling, free vibration, laminated composite plate, porosity, functionally graded material, carbon nanotubes.
Introduction
Carbon nanotubes (CNTs) are recognized as excellent reinforcements for advanced composites owing to their superior properties and low density. In contemporary industries such as aeronautical, mechanical and civil engineering, these advanced composite materials integrated in the form of shells, plates or beams as structural components have found significant applications. As required, the accurate evaluation of the mechanical reactions of structures fabricated from CNTs reinforced composite materials (CNTRC) becomes crucial for engineering design and manufacture.
Over recent years, significant attention has been devoted by researchers to functionally graded CNTRC materials (FG-CNTRC) featuring spatially varying characteristics based on a specific non-uniform distribution of the reinforcement phase. The discovery of this interesting feature led to different studies on the mechanics of CNTRC structures. Shen (2009) has introduced the first study on FG-CNTRC. His findings suggested that functionally graded reinforcement might increase the bending moment. According to Kwon et al. (2011), FG-CNTRC can be
Ghazoul, T. et al, Analytical investigation on the buckling and free vibration of porous laminated FG-CNTRC plates, pp.1242-1271
VOJNOTEHNICKI GLASNIK / MILITARY TECHNICAL COURIER, 2024, Vol. 72, Issue 3
e
achieved utilizing a powder metallurgy manufacturing method with a nonuniform dispersion of CNTs throughout the layer. The bending and free vibration of different kinds of FG-CNTRC plates were investigated by Zhu et al. (2012). They discover that reinforcements distributed near the bottom and top are more effective than those dispersed close to the mid-plane in enhancing the stiffness of CNTRC plates. Lei et al. (2013) used the element-free kp-Ritz method to study the buckling of FG-CNTRC plates. They found that for all different distributions of CNTs, those closer to the bottom and top exhibit larger buckling load values contrasted with other types of reinforcement arrangements. Liew et al. (2015) conducted a comprehensible review of the existing literature encompassing static, buckling, dynamic and non-linear analyses of FG-CNTRC.
On the other hand, researchers assert that multi-layered composite structures offer superior mechanical performance compared to singlelayer structures. Laminated composite plates are highly attractive for structural applications due to their exceptional weight-to-stiffness ratio and the capacity to tailor the lamination scheme to meet specific design requirements. In conjunction with the expansion application of laminated composites in engineering structures, numerous deformation plate theories have been developed to precisely forecast their behavior. Kirchhoff (1850) developed the simplest theory, known as the classical plate theory (CLPT); however, it is not applicable to thick plates as it ignores completely the effects of shear deformation. Mindlin (1951) proposed the first-order shear deformation plate theory (FSDT) which considers shear deformation effects and is applicable to both thin and moderately thick plates. Nevertheless, the FSDT requires the use of a shear correction factor to meet zero shear conditions on the surfaces of the plate. Determining the correct value of this factor is challenging, which is considered a shortcoming of the FSDT. Later, higher-order shear deformation theories (HSDT) were developed (Reddy, 1984; Shimpi et al, 2003; Mantari et al, 2012) to overcome the limitations of the FSDT. These theories avoid the need for shear correction factors by assuming nonlinear stress variation through the thickness. Subsequently, researchers arrived at a class of the refined plate theory (RPT) by disassembling transverse displacements into bending and shear components (Thai & Choi, 2011; Thai & Vo, 2013). The HSDT is often desirable for its superior accuracy over the CLPT and the FSDT. However, HSDTs with five or six variables are even more accurate for analyzing laminated composite plates than RPTs. Sayyad & Ghugal (2015) provided a synthesis of recent research conducted on multi-layered composite plates that utilized various shear deformation theories.
Stimulated by the concept of FG-CNTRC and the benefits of multilayered composite structures, researchers have been inspired to study FG-CNTRC multi-layered structures. This approach has the advantage of arranging the CNTs where the reinforcement is most effective and the ability to orient them in such a way as to obtain the highest mechanical properties from the laminated composite structures. Malekzadeh & Shojaee (2013) examined the buckling of CNTRC quadrilateral laminated plates using the FSDT. Zhang & Selim (2017) incorporated Reddy’s HSDT for free vibration analyses of FG-CNTRC thick laminated plates. Lei et al. (2018) used the CLPT combined with the element-free method to analyse the vibration behavior of matrix-cracked hybrid (FG-CNT/conventional fiber) laminated composite plates. Alimoradzadeh et al. (2023) analysed the thermo-mechanical buckling of FG-Fiber laminated composite beams using the Euler-Bernoulli beam theory. Chiker et al. (2023) investigated how the uncertainty caused by the non-ideal aligned distribution of FG-CNTs nano-fillers affects the free vibration characteristics of laminated FG-CNTRC plates. Fu et al. (2019) introduced a model based on the nth-order shear deformation theory to conduct static analysis of FG-CNTRC laminated plates supported by elastic foundations under thermal conditions. Arani et al. (2021) used Reddy's shear deformation theory to analyze the forced and free vibrations of FG-CNTRC laminated cylindrical panels. Tran et al. (2020) devised a novel four-variable RPT to perform static analysis on smart FG-CNTRC laminated plates incorporating a piezoelectric actuator and subjected to electro-mechanical loads. Daikh et al. (2023) analyzed the static bending response of composite laminated beams reinforced with randomly oriented FG-CNTs and fiber reinforcements on an elastic foundation, employing the finite element method. An analytical model for examining the free vibration of thick laminated plates reinforced with FG-graphene composites was introduced by Ma & Jin (2023).
However, perfection remains elusive, and fiber composite materials are no exception. During their manufacturing, there are many imperfections that are created. These imperfections exert a considerable influence on the mechanical characteristics of composite materials, often causing them to deviate from expected values. Thoughtfully planned and tightly controlled manufacturing processes can decrease the occurrence of defects, thereby enhancing material properties, albeit typically resulting in increased costs (Ciriscioli et al, 1991). This would change if one could evaluate and quantify the effect of defects and, based on that, allow a certain level of defects in composite materials parts as a means to estimate the quality of a composite. In this way, the desired properties and safety
Ghazoul, T. et al, Analytical investigation on the buckling and free vibration of porous laminated FG-CNTRC plates, pp.1242-1271
VOJNOTEHNICKI GLASNIK / MILITARY TECHNICAL COURIER, 2024, Vol. 72, Issue 3
e
factors can be achieved at a lower cost and with reduced manufacturing energy.
As a result of its importance in fiber composite materials, porosity stands out as the most extensively studied manufacturing defect. In modern composite materials, removing pores has become increasingly challenging, due to a higher complexity of parts and the heightened viscosity of modified polymers (Lee et al, 2006). Mehdikhani et al. (2019) conducted a review on porosity in fiber composite materials, including its formation, characteristics, and effects.
The literature contains a considerable amount of research on the impact of porosity on the inter-laminar shear strength of Fiber/epoxy composites (Hernandez et al, 2011; Stamopoulos et al, 2016; Hayashi & Takahashi, 2017). Others are keen on assessing how porosity affects the physical and mechanical characteristics of unidirectional fiber plant composites (Madsen & Lilholt, 2003; Madsen et al, 2009). However, studies focusing on the effect of porosity on CNTRCs are rare, particularly concerning the analysis of their mechanical responses. Guessas et al.
(2018) analytically studied the impact of matrix porosity on the buckling response of a CNTRC porous plate via FSDT considering a semi-empirical approach established by Phani & Niyogi (1987) describing the relationship between porosity and Young’s modulus in brittle solids. Medani et al.
(2019) employed identical assumptions to study the static and dynamic behavior of an FG-CNTRC porous sandwich plate. In their studies, it was found that porosity has an undeniable effect on the mechanical responses of CNTRC structures.
From the aforementioned review, it is clear that a limited number of studies have been conducted to analyze the structural responses of CNTRCs with porosities. The literature currently available also indicates that multi-layered FG-CNTRC structures have captured the interest of numerous researchers, not only because of their intricate construction, but also because they pose challenges.
The current study focuses on developing an analytical model to analyze the buckling and free vibration performance of laminated FG-CNTRC plates while accounting for the presence of porosity. The extended rule of mixture is formulated to assess the effective material properties of the resulting nanocomposites in the presence of porosity, taking into account an upper limit for its influence. Four different distribution types of CNTs along the thickness of the layers are being examined, encompassing uniform distribution as well as three other functionally graded distributions. The adopted shear deformation theory ensures the appropriate distribution of transverse shear strains throughout the
thickness and imposes tangential stress-free boundary conditions on the surfaces. The governing equations are derived through the virtual work principle and solved using Navier’s solutions. The results obtained by the present method are compared with the results from the existing literature. Detailed parametric analyses are performed to investigate the influences of porosity, CNTs distributions, CNTs volume fraction, the number of layers, CNTs fiber orientation, stacking sequence and aspect ratio on the buckling and free vibration characteristics of porous laminated FG-CNTRC plates.
Theoretical formulations
Effective material properties
Consider a rectangular plate with the total thickness h , the length a and the width b , composed of N perfectly bonded layers, with reference to the coordinate system depicted in Figure 1.
The component layers are presumed to consist of a blend of singlewalled CNTs (SWCNTs) and an isotropic matrix. In consideration, there are four different types of CNT distribution throughout the layer thickness: uniform distribution (UD) and functionally graded distributions denoted by FG-O, FG-V and FG-X.
Figure 1 - Different configurations of CNTs through the layer thickness
As a result, the volume fraction of CNTs can be expressed based on their configuration throughout the thickness of each individual layer as:
Ghazoul, T. et al, Analytical investigation on the buckling and free vibration of porous laminated FG-CNTRC plates, pp.1242-1271
VOJNOTEHNICKI GLASNIK / MILITARY TECHNICAL COURIER, 2024, Vol. 72, Issue 3
e
UD
V = <
y CNT >
V
y CNT
1 + — V
FG - V
1-
2| z
Vcnt FG - O
2 z
where
V =
y CNT
V
' Г7
W
FG - X
CNT m C WCNT + (P 1 P )-(P
(2)
WCNT is the mass fraction of CNTs, pCNT and p
/ Pm )WcnT
are the densities of the CNTs and the matrix, respectively. VCNT and v*nt are the volume fractions of CNTs for UD and FG-CNTRC, respectively.
It is assumed that the mass volume of CNTs in the UD-CNTRC layer and the FG-CNTRC layers is the same, meaning: vcnt = vCnt .
The extended rule of mixture, as a straightforward and convenient micromechanics model, is used to determine the effective material properties of the CNTRC layer (Shen, 2009):
+ VmEm (3a)
E1 = WcntEi -
VC,
12
e2 ес
Vm
(3b)
ъ_
G12
V V
cnt + m
GCNT Gm
(3c)
where е™т and ecnt are the Young’s moduli of CNTs in longitudinal and transverse directions, respectively, and g12 is the shear modulus. g^ is the
shear modulus of CNTs, Em and Gm are the Young’s modulus and shear modulus of the matrix. i1 ,i2 andi3 are the efficiency parameters which serve to account for load transfer between CNTs and the polymeric matrix. VCNT and vm are CNTs and matrix volume fractions which satisfy the relation:
VCNT + Vm =1 (4)
In the previous study, Hagstrand et al. (2005) theoretically and experimentally investigated how void content affects the structural flexural performance of unidirectional glass fiber reinforced polypropylene composite. Consequently, they set an upper threshold for the impact of void content on the composite elastic modulus using the relationship:
E (Vvoid ) = E0(1- Vvoid ) (5)
where E(vvoid) and E0 represent the elastic modulus in the presence and absence of voids, respectively. vvoid represents the void volume fraction.
Similarly, the effective elastic moduli of the CNTRC in equation (3) as a function of porosity can be derived as follows:
EP = Ei (1- P ),i = 1,2,12 (6)
where P represents the volume fraction of porosity and Ep is the effective elastic modulus in the presence of porosity.
As Poisson’s ratio exhibits weak dependence on the position, it is assumed thatv12 remains constant across the thickness of CNTRC plates:
V„ = VCNTvT + Vmvm (7)
where vCNT and vm are the Poisson’s ratios of CNTs and the matrix, respectively.
Density p as a function of porosity is given by:
P= V CNT РСШ + (Vm - P ) Pm (8)
where pCNT and pm are the density of CNTs and the matrix, respectively.
Kinematics equations
According to the theory of material point situated at (x, y,t) in the plate domain, the displacements field can be formulated as follows:
(x, y, z, t) = и0 (x, y, t) - z.w x + f (z)фх (9a)
’(x. y, z,t) = v0 (x. y,t) - zWy + f ( My (9b)
w (x, y, z, t )= w0 (x, y, t) (9c)
Here U, vand w represent the displacements in the x, y and z directions, respectively. и 0, v0 and w0 denote mid-plane displacements, while yx and фу denote shear rotations. All generalized displacements are the functions
ofx, y and time t. f (z) denotes the shape function governing the
Ghazoul, T. et al, Analytical investigation on the buckling and free vibration of porous laminated FG-CNTRC plates, pp.1242-1271
VOJNOTEHNICKI GLASNIK / MILITARY TECHNICAL COURIER, 2024, Vol. 72, Issue 3
e
transverse shear strain and stress distribution across the thickness. In terms of HSDTs, it is defined as (Sayyad & Ghugal, 2015):
4
f (z H
z
(10)
It is crucial to note that, depending on the selected shape function, the displacement field in (9) can be readily adjusted to fit other plate theories. For instance, the CLPT is achieved by using the zero shape function while the FSDT is obtained with f (z) = z . From the theory of
small deformations, the non-zero strain components related to the displacement field in (9) are defined as follows:
s x uc, 0,x -w0 0,лх
\ s y = V0,y + z< -w0 0, yy •+ f (z) Ф r y,y
7xy„ u0, y + V -2w0 0,xy J Ф +Ф x, y y,x
with
)
df (z)
dz
(11)
(12)
Constitutive relations
Given that the plate consists of multiple orthotropic layers, the stress state within each layer is expressed as:
Ox Q12 Q16 0 0 sx
°y Q12 Q22 Q26 0 0 sy
О = Q16 Q26 Q66 0 0 ■ 7xy
°yz 0 0 0 Q44 Q45 Tyz
0 xz _ 0 0 0 Q45 Q55 _ Jxz J
where [of; {s} and [Q. ]k are the stress vector, the strain vector and the transformed stiffness matrix, respectively.
The components of the transformed stiffness matrix [Q. ]k are defined as follows:
Qk = Qkcosd + 2 ( + 2Qk66 )cos20sin2 в + Qksin4 в;
Q\ = ( + Qn -4Qk66)cos20sin2 в + Qkn (sin4в + cos4в);
Q^2 = Qkn sin4 0 + 2( + 2Qi)cos2 0sin2 в + Qk21 cos4 в;
Qk =(Qiki - Q^ - 2Qk6 )cos3 в sin в + (Qjk2 - Q2k2 + 2Q6k6 )sin30cose;
Ql =(Qk -Q\, -2Qk66)cosesin3e + (Qk -Q2k2 + 2Q66)sin^cos30; (14)
Q6k6 = (( + Qk2 - 2Q,k2 - 2Q6k6) cos2 в sin2 в + Q6k6 (sin4e + cos4^);
Q44 = Q4k4cos2 в + Q5k sin2 в;
Q45 = ( - Q4k4) cos в sin в;
Qk = Qk cos2 в + Q4k4sin2 в;
where в is the angle formed by the global plate coordinate and the individual layer's local material coordinate. Qk are the plane stress-reduced
stiffnesses expressed in terms of the engineering constants along the material axes of the layer as follows:
Qki =
E11
l- V12V21
ek _ V12 E22
12 _ 1
1 - V12V21
;Q2k2 =
1 - V12V21
;Q6k = GQ = G23;Qk = G13 (15)
Equations of motion
The governing equations are derived here using the virtual work principle. The principle can be expressed analytically as:
+ V - T)dt = 0 (16)
The strain energy of the plate is computed by:
SUP = j [cxSex + ay8sy + ах/уху + v^y^ + vjy^ ] dxdydz (17)
SUP =j
NxSU0,x + NySV0,y + (SU0,y + SV0,x ) -
MwxSWo^ -M;Sw0 yy - 2MWySw0 xy + Mf Sfx x + My S(f>y, y + Mf (Sfxy + Sfyx) +
QyzSfy + QxzSfx
dxdy
where
N
iNi, Щ , Mf ) = SjZt (1, z’f (z))akdz,i = X У tv
. . » zk_i
(18)
(19a)
Ghazoul, T. et al, Analytical investigation on the buckling and free vibration of porous laminated FG-CNTRC plates, pp.1242-1271
VOJNOTEHNICKI GLASNIK / MILITARY TECHNICAL COURIER, 2024, Vol. 72, Issue 3
e
N
Qi=Zj ? f (z )akdz’
i = xz, yz
(19b)
By substituting equation (13) into equation (19) and integrating across the thickness of the plate, the stress resultants can be linked to the strain through the following relations:
Nx
Ny
Nxy
M;
i m; \ =
My Mf
Mfy
Mxy
Aii Au A16
AAA
Ml 22 26
,A16 A26 A66
[ j
[C ]
[*. ] [j
[ D ] [ j [j [ F ]
i v,
0,y
u0, y + v0,x
-w0
-w,
0,yy
-2w,
0,xy
+ Ф
x ,y y,
; i, j = 1,2,6 (20a)
where
H H
44 45
H45 H55
(20b)
N _
(, ,D,,Cj,E,,F,)=) Q(z,z2,f ((zf (z),f2(z))j=1,2,6 (21a)
k=1
N
H , =EJZ4k(f'(z)) <*;i,j = 4,5
(21b)
by:
The virtual potential energy of the external applied load is computed
8V = f[ n W0, x8w0,x + N; W;, y8w;,y ] dxdydz
(22)
where №x = yxNcr and №y = yyNcr are the in-plan compressive forces.
The expression for the kinetic energy of the mass system is as follows:
8T = jp[u8u + v8v + W8W ] dxdydz (23)
ST — j
I1 (uoSuo + voSvo + WoSWo ) +
I2 (W0,xSU0 + W0,ySV0 + U0SW0,x + V0SW0,y ) + 73 (W0,xSW0,x + W0,ySW0,y) +
14 (SU0 + (x + XyS~V0 + V0Sy ) -75 (SW0,x + W0,xSXt + Фу^О,y + W0,ySy ) + h ((x(x +<j>yS<iy )
dxdy
(24)
where
N
(Il, 12,13,14,I5,1б )=!jV ((, f (z ), z/ (z ), /2 (z ))
?,_1 * k-l
(25)
Substituting (18), (22) and (24) into (16) gives the equations of motion for the plate:
Su0 : Nx, x + Ny, y = О ***4 II -12 W0, x +14Xx
Sv0 : Nxy, x + Ny, y = = I1V0 -12 W0, y +14фy
Sw0 : Mw + 2MW ' + Mw + N0 w„ + № w0
q x,xx xy, xy y, yy x 0, xx y 0, yy
II > + 2 (,x + vQ, y )- -13 ( W0,xx + W0,yy )+ )5 (x,
6фх\ :Myx + К, y - Qxz 14U0 15 W0, x +1A
8фу : My, y + Mfy, ,x - Qyz — 14V0 - _I5 W0, y + G\
(26)
Analytical solutions
In the current study, the rectangular plates are taken to be simply supported at all edges. The exact solution of (26) can be derived analytically by applying the following boundary conditions:
- Mf — tJul л _ „ и
(27)
v0 — w0 —фу — Nx — MW — Mf — 0 at x — 0, a
U — Wq —Фх — Ny — MyW — My — 0 at y — 0,b
The following displacement expansions are provided to derive the closed-form solutions of (26) and fulfill the simply supported boundary conditions given in (27), following Navier's approach:
Ghazoul, T. et al, Analytical investigation on the buckling and free vibration of porous laminated FG-CNTRC plates, pp.1242-1271
VOJNOTEHNICKI GLASNIK / MILITARY TECHNICAL COURIER, 2024, Vol. 72, Issue 3
e
u0 (x,y, t) = YTFmneim‘cos(ax)sin(Ay)
m=1 n=1
CO CO
v0 (x, y,t) = ^ТХтпеш sin(ax)cos(Py)
m=1 n=1
CO CO
w0 ( y, t) = YTWmn^* sin(ax)sm (py) (28)
m=1 n=1
CO CO
Ф.x (x,y, t) = ХХФxmneM cos(ax)sin(Py)
m=1 n=1
CO CO
Фу ( У, t ) = Цф m eiOt sin(ax)cos(Py)
m=1 n=1
where и ,V , w , Ф and Ф™ are arbitrary parameters that need to be
mn m mn xm n
determined. o represents the eigen frequency linked to the (m,n) eigen mode. a = mn I a and p = nnl b .
By substituting equations (20) and (28) into equation (26), we can then derive the analytical solutions from the following equations
s11 S12 S13 S14 S15 mn m12 m13 m14 m15 0
S12 S22 S23 S24 S25 m12 m22 m23 m24 m25 0
S13 S23 S33 S34 S35 — aO m13 m23 m33 m34 m35 = ■ 0
S14 S24 S34 S44 S45 m14 m24 m34 m44 m45 0
V _ S15 S25 S35 S45 S55 _ m15 m25 m35 m45 m55 _ J 0^
where
S11 =_Ana —A66P ;S12 =~aP{yA\2 + A66)зS13 =Вца +RaaP +2B66aP ; S14 =~Cua —СббР,s15 =—iC(p(C[2 +C66),S’22 =—Ar22P — A66a ,
S23 = B\2a P+ B22P + 2B66Pa ; S24 = S15; S25 = —C22P — C66a ;
S33 =-Dna4 —2(( +2D66 ) —A2P +^a2 +^p2; s34 = Ellai + EaP1 + 2E66aP2; s35 = El2a2p+E22f? + 2E66Pa2;
S44 =-^j1a2 —F66p —H55; S45 =-^((2 + F66 ); ) = —F22P — Fa" — H44
m,1 = m22 = —Л; ml2 = ml5 = m24 = m45 = 0; m^ = I2a;
^4 = m25 = —/4;m22 = I2P;m22 = —/ — I3 (a2 + P); (30b)
m34 = I5a; m35 = I5P; m44 = m55 =—4
The numerical results in this study are presented using the subsequent dimensionless parameters:
N_ = N~a
n2 D0
, D0 =-
Emh3
12
i-(* )2
(31)
Results and the discussion
The material properties of the CNTRC layers utilized throughout this work are provided in Table 1. (10,10) armchair SWCNTs are selected as reinforcements, and Poly{(mphenylenevinylene)-co-[(2,5-dioctoxy-p-phenylene) vinylene]}, referred to as PmPV, as the matrix. g ™t and g™t are assumed to be equal to gc2nt .
Table 1 - Characteristics of the CNTs and the matrix
Ej (GPa) E 2 (GPa) Gj2 (GPa) P (kg/m3) V12
CNTs (10,10) armchair SWCNT 5.6466x103 7.08x103 1.9445x103 1400 0.175
Matrix PmPV 2.1 2.1 0.7358 1150 0.34
The efficiency parameters v, for the CNTs are defined in Table 2. v3 is assumed to be equal to v2 ( Zhu et al, 2012).
Table 2 - Efficiency parameters of the CNTs
Vя y CNT Vi V 2
0.11 0.149 0.934
0.14 0.150 0.941
0.17 0.149 1.381
Buckling analysis
It is worth mentioning that the FG-V distribution type is excluded from consideration in this section. This is due to the presence of stretchingbending coupling attribute in the FG-V layer, provoked by its asymmetry, causing deflections and bending moments when the plate experiences compressive loading.
Ghazoul, T. et al, Analytical investigation on the buckling and free vibration of porous laminated FG-CNTRC plates, pp.1242-1271
VOJNOTEHNICKI GLASNIK / MILITARY TECHNICAL COURIER, 2024, Vol. 72, Issue 3
e
As a first step, it is crucial to verify the accuracy and efficiency of the mathematical formulation presented in earlier sections for forecasting the buckling behavior of porous laminated FG-CNTRC plates. For this purpose, a comparison is conducted between the dimensionless critical buckling load Ncr of an FG-CNTRC plate obtained by the present method and those reported by Zhu et al. (2012) (HSDT), Wattanasakulpong & Chaikittiratana (2015) (TSDPT, SSDPT) and Guessas et al. (2018) (FSDT) based on various theories, as shown in Table 3. The results confirm the excellent agreement between the current results and those of previous studies.
Table 3 - Comparison of the dimensionless critical buckling load of FG-CNTRC square
plates
b/h V * CNT Source Uniaxial compressive load (r,=-i; у, = о) Biaxial compressive load (r, =-1; r, = -i)
UD FG-O FG-X UD FG-O FG-X
10 0.11 HSDT 20.6814 14.4990 24.2864 10.3407 7.2495 12.1432
TSDPT 20.6814 14.4990 24.2864 10.3407 7.2495 12.1432
SSDPT 20.7286 14.4515 24.3943 10.3643 7.2257 12.1972
FSDT 20.5412 14.9792 23.9594 10.2706 7.4896 11.9797
present 20.6814 14.4990 24.2864 10.3407 7.2495 12.1432
100 0.17 TSDPT 65.0043 35.1143 94.7137 32.5021 17.5572 47.3569
SSDPT 65.0053 35.1126 94.7163 32.5026 17.5563 47.3581
present 65.0043 35.1143 94.7137 32.5021 17.5572 47.3569
Table 4 presents the dimensionless critical buckling load of symmetric cross-ply laminated FG-CNTRC plate with and without porosity (P) under uniaxial and biaxial compressive loading. The results reveal that porosity exerts a significant influence on the critical buckling load, which decreases considerably with an increase in porosity. This decrease is attributed to the negative effect of porosity on the rigidity of the plate. Furthermore, It can be observed that the critical buckling load is high for the FG-X distribution type and low for the FG-O distribution type. In agreement with Lei et al. (2013), it is concluded that CNTs distributed near the top and bottom surfaces of each layer outperform those distributed closer to the mid-plane in enhancing the stiffness of the laminated FG-CNTRC plate. Additionally, regarding the effect of the number of layers on the critical buckling load, it is observed that increasing the number of layers leads to an increase in the critical buckling load.
Table 4 - Effect of porosity on the dimensionless critical buckling load of the porous cross-ply laminated FG-CNTRC square plate (b /h=10; y'CNT = 0.11)
Lamination scheme P Uniaxial compressive load ( Yx = -1; 7y = 0 ) Biaxial compressive load (Yx =-1; Yy = -1)
UD FG-O FG-X UD FG-O FG-X
[0°/90°/0°] 0 21.7292 20.1795 23.3611 10.8646 10.0898 11.6806
0.1 19.5563 18.1615 21.0250 9.7781 9.0808 10.5125
0.2 17.3833 16.1436 18.6889 8.6917 8.0718 9.3445
[0°/90°/ 90°/0°] 0 23.6661 22.7649 24.6746 11.8331 11.3825 12.3373
0.1 21.2995 20.4884 22.2072 10.6500 10.2442 11.1036
0.2 18.9329 18.2119 19.7397 9.4665 9.1060 9.8699
[079070°/ 90°/0°] 0 24.9562 24.4626 25.5669 12.4781 12.2313 12.7835
0.1 22.4606 22.0164 23.0102 11.2303 11.0082 11.5051
0.2 19.9649 19.5701 20.4535 9.9825 9.7850 10.2268
[079070° /90707 90°/0°] 0 25.8034 25.5837 26.1409 12.9017 12.7918 13.0705
0.1 23.2231 23.0253 23.5268 11.6115 11.5127 11.7634
0.2 20.6427 20.4669 20.9127 10.3214 10.2335 10.4564
[0790707 90707907 079070°] 0 26.1425 26.0335 26.3688 13.0712 13.0168 13.1844
0.1 23.5282 23.4302 23.7320 11.7641 11.7151 11.8660
0.2 20.9140 20.8268 21.0951 10.4570 10.4134 10.5475
Table 5 - Dimensionless critical buckling load of the porous cross-ply laminated FG-CNTRC square plate for different CNTs volume fractions (b /h=10)
V * v CNT P Uniaxial compressive load ( Yx =-1; Yy = 0 ) Biaxial compressive load (Yx=-1; Yy =-1)
UD FG-O FG-X UD FG-O FG-X
0.11 0 24.9562 24.4626 25.5669 12.4781 12.2313 12.7835
0.1 22.4606 22.0164 23.0102 11.2303 11.0082 11.5051
0.2 19.9649 19.5701 20.4535 9.9825 9.7850 10.2268
0.14 0 28.9296 28.3851 29.7241 14.4648 14.1925 14.8620
0.1 26.0366 25.5466 26.7517 13.0183 12.7733 13.3758
0.2 23.1437 22.7081 23.7793 11.5718 11.3540 11.8896
0.17 0 38.8356 38.2169 39.9517 19.4178 19.1084 19.9758
0.1 34.9521 34.3952 35.9565 17.4760 17.1976 17.9783
0.2 31.0685 30.5735 31.9613 15.5343 15.2868 15.9807
Ghazoul, T. et al, Analytical investigation on the buckling and free vibration of porous laminated FG-CNTRC plates, pp.1242-1271
VOJNOTEHNICKI GLASNIK / MILITARY TECHNICAL COURIER, 2024, Vol. 72, Issue 3
e
The dimensionless critical buckling load of porous symmetric cross-ply [07907079070°] laminated FG-CNTRC plates for various CNTs volume fractions and different porosity volume fraction are presented in Table 5. The findings indicate that that as the volume fraction of CNTs increases, the critical buckling load rises accordingly. This is attributed to the fact that the stiffness of the laminated FG-CNTRC plate further increases by adding an extra amount of CNTs volume fraction. However, this increase becomes less significant in the presence of porosity.
Figure 2 provides the variation of the dimensionless critical buckling load of porous symmetric angle-ply laminated FG-CNTRC plates [в°/-в °/ в °/-в °/в °] in relation to the variation of the CNTs orientation angle в for different porosity volume fractions. Upon inspection of Figure 2, it is evident that the critical buckling load increases as в changes from 0° to 45°, then decreases as в changes from 45° to 90°, with the critical buckling loads being symmetric to в = 45°. The latter presents the largest values of critical buckling load for all reinforcement types and in both uniaxial and biaxial compressive loading. Consistent with previous findings, plates with porosity exhibit low resistance against buckling compared to plates without porosity.
Figure 2 - Dimensionless critical buckling load of the porous angle-ply laminated FG-CNTRC square plate (b/h=10; у"сят = 0.11)
Free vibration analysis
For free vibration analysis, numerical validation is also performed. The present method is compared with that of Huang et al. (2017) for the free vibration of anti-symmetrically laminated FG-CNTRC plates in the absence of porosity, and the results show good agreement, as illustrated in Table 6. Table 6 lists the dimensionless fundamental frequency о for varying patterns of the CNTs distribution, the number of layers, the CNTs volume fraction V*CNT and the width-to-thickness ratio b / h. The small variation in the lowerb / h ratios is attributed to the model proposed by Huang et al. (2017), which is a four-variable FSDT.
Table 6 - Comparison of dimensionless fundamental frequencies of FG-CNTRC antisymmetric cross-ply laminated square plates
Source b h [079070790°] [0790707907079070790°]
V* y CNT
0.11 0.14 0.17 0.11 0.14 0.17
UD Present 10 14.405 15.546 17.840 15.161 16.344 18.779
20 16.776 18.500 20.707 17.866 19.711 22.049
50 17.696 19.701 21.810 18.943 21.122 23.340
Huang et al. (2017) 10 14.640 15.832 18.124 15.338 16.555 18.993
20 16.872 18.624 20.820 17.944 19.811 22.142
50 17.714 19.726 21.831 18.958 21.142 23.358
FG-V Present 10 14.299 15.444 17.737 15.150 16.348 18.799
20 16.621 18.330 20.529 17.838 19.687 22.032
50 17.518 19.498 21.597 18.906 21.081 23.303
Huang et al. (2017) 10 14.518 15.717 17.993 15.302 16.534 18.975
20 16.683 18.424 20.596 17.881 19.754 22.078
50 17.495 19.484 21.565 18.883 21.065 23.271
FG-O Present 10 14.166 15.293 17.574 15.116 16.308 18.757
20 16.496 18.190 20.375 17.808 19.652 21.995
50 17.396 19.362 21.448 18.878 21.050 23.269
Huang et al. (2017) 10 14.450 15.649 17.909 15.288 16.519 18.957
20 16.581 18.314 20.470 17.859 19.729 22.049
50 17.378 19.354 21.421 18.856 21.036 23.238
FG-X Present 10 14.668 15.852 18.202 15.238 16.445 18.910
20 17.071 18.839 21.092 17.944 19.806 22.165
50 18.003 20.054 22.204 19.019 21.211 23.445
Huang et al. (2017) 10 14.794 16.007 18.344 15.363 16.596 19.052
20 17.099 18.888 21.120 17.977 19.859 22.197
50 17.975 20.032 22.165 18.995 21.193 23.411
Ghazoul, T. et al, Analytical investigation on the buckling and free vibration of porous laminated FG-CNTRC plates, pp.1242-1271
VOJNOTEHNICKI GLASNIK / MILITARY TECHNICAL COURIER, 2024, Vol. 72, Issue 3
e
Table 7 presents the three lowest dimensionless frequencies for symmetric cross-ply porous laminated FG-CNTRC plates with different porosity volume fractions. Upon comparing the results, it can be observed that the frequencies show weak dependence on porosity, with the frequency decreasing very slightly with the increase of porosity. The same observation can be made from Figures 4 and 5. This can be attributed to the fact that the effect of porosity on the density is as important as its effect on the rigidity of the plate. Additionally, the FG-X distribution type exhibits the largest frequencies compared to the UD, FG-V and FG-O for every mode. It is also evident that the effect of the number of layers is manifested in the increase of the frequency as the number of layers increases.
Table 7 - Effect of porosity on the dimensionless frequency of the porous cross-ply laminated FG-CNTRC square plate (b / h = 10; Vl*NT = 0.11)
p [079070°] [07907079070°]
UD FG-V FG-O FG-X UD FG-V FG-O FG-X
1st Mode 0 13.8958 13.6781 13.3887 14.4106 14.9041 14.9219 14.7553 15.0858
0.1 13.8779 13.6604 13.3713 14.3919 14.8848 14.9026 14.7362 15.0662
0.2 13.8555 13.6383 13.3498 14.3687 14.8608 14.8786 14.7124 15.0419
Mode 0 20.8993 20.0286 19.2837 22.2633 26.8626 26.9065 26.5834 27.2181
0.1 20.8722 20.0027 19.2588 22.2344 26.8278 26.8716 26.5490 27.1828
0.2 20.8386 19.9704 19.2277 22.1985 26.7845 26.8283 26.5062 27.1390
3rd Mode 0 33.2531 32.8548 32.0150 34.4913 33.7417 33.8055 33.3006 34.2971
0.1 33.2100 32.8123 31.9736 34.4466 33.6980 33.7617 33.2575 34.2527
0.2 33.1564 32.7594 31.9220 34.3909 33.6436 33.7072 33.2038 34.1974
[0790707907079070° ] [079070790707907079070°]
1st Mode 0 15.1576 15.1748 15.0925 15.2564 15.2578 15.2748 15.2257 15.3237
0.1 15.1379 15.1552 15.0729 15.2367 15.2381 15.2551 15.2060 15.3039
0.2 15.1135 15.1307 15.0486 15.2121 15.2135 15.2304 15.1814 15.2792
Mode 0 28.5479 28.5952 28.4286 28.7590 29.4004 29.4495 29.3453 29.5527
0.1 28.5109 28.5581 28.3918 28.7217 29.3624 29.4114 29.3073 29.5144
0.2 28.4649 28.5121 28.3460 28.6754 29.3150 29.3639 29.2600 29.4668
3rd Mode 0 33.6851 33.7463 33.4966 33.9928 33.4966 33.5563 33.4098 33.7017
0.1 33.6415 33.7026 33.4532 33.9488 33.4533 33.5128 33.3665 33.6580
0.2 33.5872 33.6482 33.3992 33.8940 33.3993 33.4588 33.3127 33.6037
Figure 3 depicts the effect of porosity on the amplitude of a four layered unidirectional [0707070°] porous laminated FG-CNTRC plate in
different modes of vibration. It is clear that the results are almost identical in terms of values for all modes of vibration.
Thus, we can state that porosity has a negligible effect on the amplitude of the plate.
This result further confirms the weakening effect of porosity on the free vibration response of the laminated FG-CNTRC plate.
Figure 3 - Amplitude of the unidirectional porous laminated FG-CNTRC square plate
(b / h = 10 ; Vcm = 0.11)
The variation in the dimensionless fundamental frequency of porous anti-symmetric cross-ply laminated FG-CNTRC plates [079070790°] with respect to the variation of the CNTs volume fraction and the porosity volume fraction is depicted in Figure 4.
It is noticeable that the increase in the CNTs volume fraction leads to a significant increase in the fundamental frequency for all CNTs distribution types.
Ghazoul, T. et al, Analytical investigation on the buckling and free vibration of porous laminated FG-CNTRC plates, pp.1242-1271
VOJNOTEHNICKI GLASNIK / MILITARY TECHNICAL COURIER, 2024, Vol. 72, Issue 3
e
Figure 4 - Variation of the dimensionless frequency of the four layered porous antisymmetric cross-ply laminated FG-CNTRC square plate in the function of the variation of the CNTs volume fraction (b / h = 10 )
Figure 5 presents the variation of the dimensionless fundamental frequency of the anti-symmetric angle-ply laminated FG-CNTRC plate [в°/-в°/в 7-0 °] in relation to the variation of the CNTs orientation angle (в) for different porosity volume fractions.
The fundamental frequency increases as в changes from 0° to 45°, then decreases as в changes from 45° to 90°, in which the fundamental frequencies are symmetric to в = 45°. This last mentioned value (в =45°) presents the largest values of fundamental frequency for all cases.
According to the above investigations, it can be noticed that porosity has a low impact on the frequency of laminated FG-CNTRC plates compared to the effects of the other parameters.
Figure 5 - Dimensionless frequency of the porous angle-ply laminated FG-CNTRC square
plate (b / h = 10 ; VlCNT = 0.11)
The minimal influence of porosity on the plate's amplitude and frequency across various vibration modes suggests the resilient nature of the laminated FG-CNTRC plate structure to porosity effects. While porosity typically alters material stiffness, the specific arrangement of composite layers in this case seems to counteract such effects. Consistently uniform results imply effective mitigation of porosity-induced changes in amplitude and frequency by the plate's design and material composition. This highlights the robustness of the laminated FG-CNTRC plate against porosity-induced variations in vibrational characteristics, further confirming its weakened free vibration response due to porosity.
Conclusion
In this paper, the buckling and free vibration of laminated FG-CNTRC plates in the presence of porosity are analytically examined. A mathematical model based on a five-unknown HSDT is developed. The assumption is made that the laminates consist of layers that are perfectly bonded, with carbon nanotubes (CNTs) distributed throughout the thickness of each layer, accounting for different types of FG distributions.
Ghazoul, T. et al, Analytical investigation on the buckling and free vibration of porous laminated FG-CNTRC plates, pp.1242-1271
VOJNOTEHNICKI GLASNIK / MILITARY TECHNICAL COURIER, 2024, Vol. 72, Issue 3
e
The effective material properties are determined using the extended rule of mixture considering an upper limit of the impact of porosity. The accuracy of the present formulation has been validated, and detailed parametric studies have been conducted to examine the effects of several parameters on the critical buckling load and natural frequency of porous laminated FG-CNTRC plates. Some typical conclusions are noted:
- In terms of porosity, it is found that porosity has a considerable negative effect on the critical buckling load.
- Porosity has a weak effect on the free vibration characteristics.
- Increasing the CNTs volume fraction enhances plate rigidity.The enhancement becomes less significant in the presence of porosity.
- For the CNTs distribution type, it is derived that reinforcements distributed near the top and bottom surfaces of the layer outperform those distributed closer to the mid-plane in enhancing the stiffness of the plate.
- The number of layers significantly impacts the responses to free vibration and buckling. The critical buckling load and natural frequency rise in proportion to the increase in the number of layers.
- The lamination angle significantly influences both the critical buckling load and natural frequency. Because of the axial symmetry of the laminated square plate orientations at 45°, any changes in these properties are also symmetrical.
In future works, it would be beneficial to explore the influence of environmental conditions on laminated FG-CNTRC plates' mechanical properties and durability. Additionally, investigating dynamic responses under various loading conditions and conducting experimental validation could enhance reliability. Exploring advanced manufacturing techniques and novel materials could also improve performance and expand application possibilities.
References
Alimoradzadeh, M., Heidari, H., Tornabene, F. & Dimitri, R. 2023. ThermoMechanical Buckling and Non-Linear Free Oscillation of Functionally Graded Fiber-Reinforced Composite Laminated (FG-FRCL) Beams. Applied Sciences, 13(8), art.number:4904. Available at: https://doi.org/10.3390/app13084904.
Arani, A.G., Kiani, F. & Afshari, H. 2021. Free and forced vibration analysis of laminated functionally graded CNT-reinforced composite cylindrical panels. Journal of Sandwich Structures & Materials, 23(1), pp.255-278. Available at: https://doi.org/10.1177/1099636219830787.
Chiker, Y., Bachene, M., Attaf, B., Hafaifa, A. & Guemana, M. 2023. Uncertainty influence of nanofiller dispersibilities on the free vibration behavior of multi-layered functionally graded carbon nanotube-reinforced composite laminated plates. Acta Mechanics, 234(4), pp. 1687-1711. Available at: https://doi.org/10.1007/s00707-022-03438-6.
Ciriscioli, P.R., Springer, G.S. & Lee, W.I. 1991. An Expert System for Autoclave Curing of Composites. Journal of Composite Materials, 25(12), pp.1542-1587. Available at: https://doi.org/10.1177/002199839102501201.
Daikh, A.A., Belarbi, M.-O., Salami, S.J., Ladmek, M., Belkacem, A., Houari, M.S.A., Ahmed, H.M. & Eltaher, M.A. 2023. A three-unknown refined shear beam model for the bending of randomly oriented FG-CNT/fiber-reinforced composite laminated beams rested on a new variable elastic foundation. Acta Mechanica, 234(10), pp.5171-5186. Available at: https://doi.org/10.1007/s00707-023-03657-5.
Fu, T., Chen, Z., Yu, H., Wang, Z. & Liu, X. 2019. Mechanical behavior of laminated functionally graded carbon nanotube reinforced composite plates resting on elastic foundations in thermal environments. Journal of Composite Materials, 53(9), pp. 1159-1179. Available at:
https://doi.org/10.1177/0021998318796170.
Guessas, H., Zidour, M., Meradjah, M. & Tounsi, A. 2018. The critical buckling load of reinforced nanocomposite porous plates. Structural Engineering and Mechanics, 67(2), pp. 115-123. Available at:
https://doi.org/10.12989/sem.2018.67.2.115.
Hagstrand, P.-O., Bonjour, F. & Manson, J.-A.E. 2005. The influence of void content on the structural flexural performance of unidirectional glass fibre reinforced polypropylene composites. Composites Part A: Applied Science and Manufacturing, 36(5), pp.705-714. Available at:
https://doi.org/10.1016/j.compositesa.2004.03.007.
Hayashi, T. & Takahashi, J. 2017. Influence of void content on the flexural fracture behaviour of carbon fiber reinforced polypropylene. Journal of Composite Materials, 51(29), pp.4067-4078. Available at:
https://doi.org/10.1177/0021998317698215.
Hernandez, S., Sket, F., Molina-Aldaregui'a, J.M., Gonzalez, C. & LLorca, J. 2011. Effect of curing cycle on void distribution and interlaminar shear strength in polymer-matrix composites. Composites Science and Technology, 71(10), pp.1331-1341. Available at: https://doi.org/10.1177/0021998317698215.
Huang, B., Guo, Y., Wang, J., Du, J., Qian, Z., Ma, T. & Yi, L. 2017. Bending and free vibration analyses of antisymmetrically laminated carbon nanotube-reinforced functionally graded plates. Journal of Composite Materials, 51(22), pp.3111-3125. Available at: https://doi.org/10.1177/0021998316685165.
Kirchhoff, G. 1850. Uber das Gleichgewicht und die Bewegung einer elastischen Scheibe. Journal fur die reine und angewandte Mathematik (Crelles Journal), 1850(40), pp.51-88. Available
at: https://doi.org/10.1515/crll. 1850.40.51.
Ghazoul, T. et al, Analytical investigation on the buckling and free vibration of porous laminated FG-CNTRC plates, pp.1242-1271
VOJNOTEHNICKI GLASNIK / MILITARY TECHNICAL COURIER, 2024, Vol. 72, Issue 3
e
Kwon, H., Bradbury, C.R. & Leparoux, M. 2011. Fabrication of Functionally Graded Carbon Nanotube-Reinforced Aluminum Matrix Composite. Advanced Engineering Materials, 13(4), pp.325-329. Available at:
https://doi.org/10.1002/adem.201000251.
Lee, J., Kim, J. & Hyeon, T. 2006. Recent Progress in the Synthesis of Porous Carbon Materials. Advanced Materials, 18(16), pp.2073-2094. Available at: https://doi.org/10.1002/adma.200501576.
Lei, Z.X., Liew, K.M. & Yu, J.L. 2013. Buckling analysis of functionally graded carbon nanotube-reinforced composite plates using the element-free kp-Ritz method. Composite Structures, 98, pp.160-168. Available at:
https://doi.org/10.1016/j.compstruct.2012.11.006.
Lei, Z.X., Zhang, L.W. & Liew, K.M. 2018. Modeling large amplitude vibration of matrix cracked hybrid laminated plates containing CNTR-FG layers. Applied Mathematical Modelling, 55, pp.33-48. Available at:
https://doi.org/10.1016/j.apm.2017.10.032.
Liew, K.M., Lei, Z.X. & Zhang, L.W. 2015. Mechanical analysis of functionally graded carbon nanotube reinforced composites: A review. Composite Structures, 120, pp.90-97. Available at: https://doi.org/10.1016/j.compstruct.2014.09.041.
Ma, R. & Jin, Q. 2023. Free Vibration Analysis of Functionally Graded Graphene-Reinforced Composite-Laminated Plates. Journal of Aerospace Engineering, 36(3), art. ID: 04023016. Available at:
https://doi.org/10.1061/JAEEEZ.ASENG-4657.
Madsen, B. & Lilholt, H. 2003. Physical and mechanical properties of unidirectional plant fibre composites—an evaluation of the influence of porosity. Eco-Composites, 63(9), pp. 1265-1272. Available at:
https://doi.org/10.1016/S0266-3538(03)00097-6.
Madsen, B., Thygesen, A. & Lilholt, H. 2009. Plant fibre composites -porosity and stiffness. Composites Science and Technology, 69(7), pp.10571069. Available at: https://doi.org/10.1016/j.compscitech.2009.01.016.
Malekzadeh, P. & Shojaee, M. 2013. Buckling analysis of quadrilateral laminated plates with carbon nanotubes reinforced composite layers. Thin-Walled Structures, 71, pp.108-118. Available at:
https://doi.org/10.1016/j.tws.2013.05.008.
Mantari, J.L., Oktem, A.S. & Guedes Soares, C. 2012. A new trigonometric shear deformation theory for isotropic, laminated composite and sandwich plates. International Journal of Solids and Structures, 49(1), pp.43-53. Available at: https://doi.org/10.1016/j.ijsolstr.2011.09.008.
Medani, M., Benahmed, A., Zidour, M., Heireche, H. & Tounsi, A. 2019. Static and dynamic behavior of (FG-CNT) reinforced porous sandwich plate using energy principle. Steel and Composite Structures, 32(5), pp.595-610. Available at: https://doi.org/10.12989/scs.2019.32.5.595.
Mehdikhani, M., Gorbatikh, L., Verpoest, I. & Lomov, S.V. 2019. Voids in fiber-reinforced polymer composites: A review on their formation, characteristics, and effects on mechanical performance. Journal of Composite Materials, 53(12), pp.1579-1669. Available at: https://doi.org/10.1177/0021998318772152.
Mindlin, R.D. 1951. Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. Journal of Applied Mechanics, 18(1), pp.31-38. Available at: https://doi.org/10.1115/1.4010217.
Phani, K.K. & Niyogi, S.K. 1987. Young's modulus of porous brittle solids. Journal of Materials Science, 22(1), pp.257-263. Available at:
https://doi.org/10.1007/BF01160581.
Reddy, J.N. 1984. A Simple Higher-Order Theory for Laminated Composite Plates. Journal of Applied Mechanics, 51(4), pp. 745-752. Available at: https://doi.org/10.1115/1.3167719.
Sayyad, A.S. & Ghugal, Y.M. 2015. On the free vibration analysis of laminated composite and sandwich plates: A review of recent literature with some numerical results. Composite Structures, 129, pp. 177-201. Available at: https://doi.org/10.1016/j.compstruct.2015.04.007.
Shen, H.S. 2009. Nonlinear bending of functionally graded carbon nanotube-reinforced composite plates in thermal environments. Composite Structures, 91(1), pp.9-19. Available at: https://doi.org/10.1016/j.compstruct.2009.04.026.
Shimpi, R.P., Arya, H. & Naik, N.K. 2003. A Higher Order Displacement Model for the Plate Analysis. Journal of Reinforced Plastics and Composites, 22(18), pp.1667-1688. Available at: https://doi.org/10.1177/073168403027618.
Stamopoulos, A.G., Tserpes, K.I., Prucha, P. & Vavrfk, D. 2016. Evaluation of porosity effects on the mechanical properties of carbon fiber-reinforced plastic unidirectional laminates by X-ray computed tomography and mechanical testing. Journal of Composite Materials, 50, pp.2087-2098. Available at:
https://doi.org/10.1077/0021998315602049.
Thai, H.-T. & Choi, D.-H. 2011. A refined plate theory for functionally graded plates resting on elastic foundation. Composites Science and Technology, 71(16), pp.1850-1858. Available at: https://doi.org/10.1016/j.compscitech.2011.08.016.
Thai, H.-T. & Vo, T.P. 2013. A new sinusoidal shear deformation theory for bending, buckling, and vibration of functionally graded plates. Applied Mathematical Modelling, 37(5), pp. 3269-3281. Available at:
https://doi.org/10.1016/j.apm.2012.08.008.
Tran, H.Q., Vu, V.T., Tran, M.T. & Nguyen-Tri, P. 2020. A new four-variable refined plate theory for static analysis of smart laminated functionally graded carbon nanotube reinforced composite plates. Mechanics of Materials, 142, art. number: 103294. Available at:
https://doi.org/10.1016/j.mechmat.2019.103294.
Wattanasakulpong, N. & Chaikittiratana, A. 2015. Exact solutions for static and dynamic analyses of carbon nanotube-reinforced composite plates with Pasternak elastic foundation. Applied Mathematical Modelling, 39(18), pp.54595472. Available at: https://doi.org/10.1016/j.apm.2014.12.058.
Zhang, L.W. & Selim, B.A. 2017. Vibration analysis of CNT-reinforced thick laminated composite plates based on Reddy's higher-order shear deformation theory. Composite Structures, 160, pp.689-705. Available at:
https://doi.org/10.1016/j.compstruct.2016.10.102.
Ghazoul, T. et al, Analytical investigation on the buckling and free vibration of porous laminated FG-CNTRC plates, pp.1242-1271
VOJNOTEHNICKI GLASNIK / MILITARY TECHNICAL COURIER, 2024, Vol. 72, Issue 3
e
Zhu, P., Lei, Z.X. & Liew, K.M. 2012. Static and free vibration analyses of carbon nanotube-reinforced composite plates using finite element method with first order shear deformation plate theory. Composite Structures, 94(4), pp.14501460. Available at: https://doi.org/10.1016/j.compstruct.2011.11.010.
Investigacion analftica sobre el pandeo y la vibracion libre de placas laminadas porosas FG-CNTRC
Tahir Ghazoula, autor de correspondencia, Mohamed Atif Benattaa, Abdelwahhab Khatirb, Youcef Beldjelilia,
Baghdad Kroura, Mohamed Bachir Bouiadjraa
a Universidad de Djillali Liabes, Laboratorio de Estructuras y Materiales Avanzados en Ingenierla Civil y Obras Publicas,
Sidi Bel Abbes, Republica Argelina Democratica y Popular, b Universidad Politecnica de Marche, Seccion Estructural DICEA,
Ancona, Republica Italiana
CAMPO: mecanica, materiales
TIPO DE ARTICULO: articulo cientifico original
Resumen:
Introduccion/objetivo: El objetivo de este estudio es examinar el comportamiento de pandeo y vibracion libre de las placas compuestas laminadas, reforzadas con nanotubos de carbono cuando se tienen en cuenta varias fuentes de incertidumbre, siendo el enfoque principal la existencia de porosidad.
Metodos: Se desarrolla un modelo de placa laminada porosa utilizando la teoria de la deformacion por corte de alto orden. Se investigan diferentes configuraciones de nanotubos de carbono de pared simple alineados y funcionalmente graduados en todo el espesor de cada capa. Las propiedades efectivas de los materiales se evaluan a traves de la regla extendida de mezcla mientras se considera un limite superior para el efecto de la porosidad. Las ecuaciones que rigen se derivan y resuelven utilizando el principio de trabajo virtual y el enfoque de Navier. La validez de la formulacion actual se confirma comparando nuestros resultados con los datos existentes de fuentes bibliograficas. Se investiga en detalle el impacto de numerosos parametros como la porosidad, la fraccion de volumen de nanotubos de carbono, los tipos de distribucion de refuerzo, el esquema de laminacion y el numero de capas en las respuestas de pandeo y vibracion libre.
Resultados: Un hallazgo clave de este estudio es la reduccion significativa en la resistencia al pandeo de las placas laminadas FG-CNTRC debido a la porosidad, en contraste con el impacto menor en la respuesta de vibracion libre.
Conclusion: Los resultados de este artlculo enfatizan el papel crltico de la porosidad en la integridad estructural y brindan nuevos conocimientos sobre el comportamiento de los materiales compuestos avanzados.
Palabras claves: pandeo, vibracion libre, placa compuesta laminada, porosidad, material funcionalmente graduado, nanotubos de carbono.
Аналитическое исследование изгиба и свободной вибрации пористых слоистых пластин FG-CNTRC
Тахир Газула, корреспондент, Мухаммед Атиф Бената3,
Абделвахаб Катирб, Йусуф Белджелили3,
Багдад Кроура, Мухаммед Башир Бу]ажераа
а Университет Джиллали Лиабес, Лаборатория конструкций и современных материалов в гражданском строительстве и общественных работах,
г. Сиди-Бель-Аббес, Алжирская Народно-Демократическая Республика б Политехнический университет Марке, структурное подразделение DICEA, г. Анкона, Итальянская Республика
РУБРИКА ГРНТИ: 30.19.00 Механика деформируемого твердого тела,
81.09.00 Материаловедение ВИД СТАТЬИ: оригинальная научная статья
Резюме:
Введение/цель: Целью данного исследования является изучение изгиба и свободной вибрации многослойных композитных пластин, армированных углеродными нанотрубками, при учете различных источников неопределенности с акцентом на наличие пористости.
Методы: Модель многослойной пористой пластины разработана с применением теории сдвиговой деформации высокого порядка. Исследовались различные конфигурации функционально упорядоченных выровненных одностенных углеродных нанотрубок по толщине каждого слоя. Эффективные свойства материалов оцениваются с помощью расширенного правила смешивания с учетом верхнего предела пористости. Управляющие уравнения получены и решены с помощью принципа виртуальной работы и подхода Навье. Обоснованность этой формулировки подтверждается сравнением результатов с данными из существующих научных источников. Подробно исследовано влияние многочисленных параметров, таких как пористость, объемная доля углеродных нанотрубок, виды распределения армирования, схема ламинирования и количество слоев на изгибе и реакция на свободную вибрацию.
Результаты: Ключевым выводом данного исследования
является значительное снижение сопротивления
Ghazoul, T. et al, Analytical investigation on the buckling and free vibration of porous laminated FG-CNTRC plates, pp.1242-1271
VOJNOTEHNICKI GLASNIK / MILITARY TECHNICAL COURIER, 2024, Vol. 72, Issue 3
e
ламинированных пластин FG-CNTRC на изгиб из-за их пористости, в отличии от незначительного влияния на отклик на свободную вибрацию.
Выводы: Результаты данной статьи подчеркивают
критическую роль пористости в целостности структуры и дают новое представление о поведении современных композитных материалов.
Ключевые слова: изгиб, свободная вибрация, многослойная композитная пластина, пористость, функционально
распределенный материал, углеродные нанотрубки.
Аналитичко испитива^е порозних ламинираних плоча од функционално градираних композита о]ачаних угъеничним наноцевима (FG-CNTRC) на изви]а^е и слободне вибраци]е
Тахир Газула, аутор за преписку, Мухамед Атиф Бенатаа,
Абделвахаб Катир* б, Jусуф Белуелилиа,
Багдад Кроура, Мухамед Башир Бу)ажераа
a Универзитет Ъилали Лиабес, Лаборатори)а за напредне конструкц^е и материале у гра^евинарству и (авним радовима, Сиди Бел Абес, Народна Демократска Република Алжир
б Политехнички универзитет Марке, Структурни одсек DICEA,
Анкона, Република Итали]а
ОБЛАСТ: механика, матери)али
КАТЕГОРША (ТИП) ЧЛАНКА: оригинални научни рад
Сажетак:
Увод/циъ: Циъ ове студи]е]есте да испита понашаъе ламинираних композитних плоча о]ачаних угъеничним наноцевима при извцаъу и слободним вибраци'ама када се узима]у у обзир различити извори несигурности и када ]е фокус на посто]аъу порозности.
Методе: Модел порозне ламиниране плоче развуен ]е помоПу смица]не деформационе теори'е вишег реда. Испитане су различите конфигураций функционално градираних угъеничних наноцеви с ]едноструким зидом, поре^аних целом дебъином сваког сло]а. Ефективна сво]ства материала процежена су кроз проширено правило о смешама, узима]уПи у обзир горъу границу ефекта порозности. ВодеПе]едначине изведене су и решене помоПу принципа виртуалног рада и Нави'еровог приступа. Валидност наведене формулаци'е потвр^ена ]е поре^еъем доби'ених резултата са подацима из посто]еПе литературе. Детаъно ]е испитан утица] бро]них параметара попут порозности, запреминског удела угъеничних наноцеви, типова дистрибуци'е
о]ача^а, шеме ламинацие, као и 6poja сло]ева на извуаъе и одговора на слободне вибраци'е.
Резултати: Къучни налаз ове студне ]есте да }е знатно сманена отпорност на извуа^е ламинираних FG-CNTRC плоча услед порозности, за разлику од минорног утица а на одговор на слободне вибраци е.
Закъучак: Резултати овог рада истичу критичну улогу порозности у интегритету структуре и пружа}у нове увиде у понашаъе напредних композитних матери ала.
К^учне речи: извуаъе, слободна вибраци'а, ламинирана композитна плоча, порозност, функционално градиран
материал, уг^еничне наноцеви.
Paper received on: 15.04.2024.
Manuscript corrections submitted on: 24.09.2024.
Paper accepted for publishing on: 25.09.2024.
© 2024 The Authors. Published by Vojnotehnicki glasnik / Military Technical Courier (www.vtg.mod.gov.rs, втг.мо.упр.срб). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/rs/).
Ghazoul, T. et al, Analytical investigation on the buckling and free vibration of porous laminated FG-CNTRC plates, pp.1242-1271
