Static and free vibration analysis of functionally graded carbon nanotube reinforced skew plates Enrique García-Macías a,⇑ , Rafael Castro-Triguero b , Erick I. Saavedra Flores c , Michael I. Friswell d , Rafael Gallego e a Department of Continuum Mechanics and Structural Analysis, School of Engineering, University of Seville, Camino de los Descubrimientos s/n, E-41092-Seville, Spain b Department of Mechanics, University of Cordoba, Campus de Rabanales, Cordoba, CP 14071, Spain c Departamento de Ingeniería en Obras Civiles, Universidad de Santiago de Chile, Av. Ecuador 3659, Santiago, Chile d Zienkiewicz Centre for Computational Engineering, College of Engineering, Swansea University, Singleton Park, SA2 8PP, United Kingdom e Dept. Structural Mechanics and Hydraulic Engineering, University of Granada, 18071 Granada, Spain article info Article history: Available online 6 January 2016 Keywords: Vibration analysis Skew shells Hu–Washizu functional Shell finite elements Uniaxially aligned CNT reinforcements Functionally graded material abstract The remarkable mechanical and sensing properties of carbon nanotubes (CNTs) suggest that they are ideal candidates for high performance and self-sensing composites. However, the study of CNT-based composites is still under development. This paper provides results of static and dynamic numerical sim- ulations of thin and moderately thick functionally graded (FG-CNTRC) skew plates with uniaxially aligned reinforcements. The shell element is formulated in oblique coordinates and based on the first-order shear deformation plate theory. The theoretical development rests upon the Hu–Washizu principle. Independent approximations of displacements (bilinear), strains and stresses (piecewise constant subre- gions) provide a consistent mechanism to formulate an efficient four-noded skew element with a total of twenty degrees of freedom. An invariant definition of the elastic transversely isotropic tensor is employed based on the representation theorem. The FG-CNTRC skew plates are studied for a uniform and three different distributions (two symmetric and one asymmetric) of CNTs. Detailed parametric studies have been carried out to investigate the influences of skew angle, CNT volume fraction, thickness-to-width ratio, aspect ratio and boundary conditions. In addition, the effects of fiber orientation are also examined. The obtained results are compared to the FE commercial package ANSYS and the limited existing bibliography with good agreement. Ó 2015 Elsevier Ltd. All rights reserved. 1. Introduction Since the discovery of carbon nanotubes (CNTs) by Ijima [1] in 1991, many researchers have investigated their unique capabilities as reinforcements in composite materials. Due to their remarkable mechanical, electrical and thermal properties, carbon nanotubes are considered ideal reinforcing fibers for advanced high strength materials and smart materials with self sensing capabilities [2,3]. In actual structural applications, it is important to develop theoret- ical models in order to predict the response of structural elements made of carbon nanotube-reinforced composites (CNTRC). In par- ticular, skew plates are widely employed in civil and aeronautical engineering applications such as panels in skew bridges, construc- tion of wings, tails and fins of swept-wing aircraft, etc. However, due to the mathematical difficulties involved in their formulation, works on the static and dynamic analysis of CNTRC skew elements are scarce in the literature [4]. The number of publications dealing with static and dynamic analysis of CNTRC structural elements have increased considerably in recent years. Wuite and Adali [5] studied the bending behavior of classical symmetric cross-ply and angle-ply laminated beams reinforced by aligned CNTs and isotropic beams reinforced by ran- domly oriented CNTs. By using a micromechanical constitutive model based on the Mori–Tanaka method, they highlighted that small percentages of CNT reinforcement lead to significant improvement in beam stiffness. Vodenitcharova and Zhang [6] developed a continuum model for the uniform bending and bending-induced buckling of a straight nanocomposite beam with circular cross section reinforced by a single-walled carbon nan- otube (SWNT). The results showed that although the addition of a matrix to a SWNT increases the load carrying capacity, the thicker matrix layers the SWNT buckles locally at smaller bending angles and greater flattening ratios. Formica et al. [7] studied the vibrational properties of cantilevered CNTRC plates with an http://dx.doi.org/10.1016/j.compstruct.2015.12.044 0263-8223/Ó 2015 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. E-mail address: [email protected](E. García-Macías). Composite Structures 140 (2016) 473–490 Contents lists available at ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct
18
Embed
Static and free vibration analysis of functionally graded ...michael.friswell.com/PDF_Files/J289.pdf · Static and free vibration analysis of functionally graded carbon nanotube reinforced
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Static and free vibration analysis of functionally graded carbon nanotubereinforced skew plates
http://dx.doi.org/10.1016/j.compstruct.2015.12.0440263-8223/� 2015 Elsevier Ltd. All rights reserved.
⇑ Corresponding author.E-mail address: [email protected] (E. García-Macías).
Enrique García-Macías a,⇑, Rafael Castro-Triguero b, Erick I. Saavedra Flores c, Michael I. Friswell d,Rafael Gallego e
aDepartment of Continuum Mechanics and Structural Analysis, School of Engineering, University of Seville, Camino de los Descubrimientos s/n, E-41092-Seville, SpainbDepartment of Mechanics, University of Cordoba, Campus de Rabanales, Cordoba, CP 14071, SpaincDepartamento de Ingeniería en Obras Civiles, Universidad de Santiago de Chile, Av. Ecuador 3659, Santiago, Chiled Zienkiewicz Centre for Computational Engineering, College of Engineering, Swansea University, Singleton Park, SA2 8PP, United KingdomeDept. Structural Mechanics and Hydraulic Engineering, University of Granada, 18071 Granada, Spain
The remarkable mechanical and sensing properties of carbon nanotubes (CNTs) suggest that they areideal candidates for high performance and self-sensing composites. However, the study of CNT-basedcomposites is still under development. This paper provides results of static and dynamic numerical sim-ulations of thin and moderately thick functionally graded (FG-CNTRC) skew plates with uniaxially alignedreinforcements. The shell element is formulated in oblique coordinates and based on the first-order sheardeformation plate theory. The theoretical development rests upon the Hu–Washizu principle.Independent approximations of displacements (bilinear), strains and stresses (piecewise constant subre-gions) provide a consistent mechanism to formulate an efficient four-noded skew element with a total oftwenty degrees of freedom. An invariant definition of the elastic transversely isotropic tensor is employedbased on the representation theorem. The FG-CNTRC skew plates are studied for a uniform and threedifferent distributions (two symmetric and one asymmetric) of CNTs. Detailed parametric studies havebeen carried out to investigate the influences of skew angle, CNT volume fraction, thickness-to-widthratio, aspect ratio and boundary conditions. In addition, the effects of fiber orientation are also examined.The obtained results are compared to the FE commercial package ANSYS and the limited existingbibliography with good agreement.
� 2015 Elsevier Ltd. All rights reserved.
1. Introduction works on the static and dynamic analysis of CNTRC skew elements
Since the discovery of carbon nanotubes (CNTs) by Ijima [1] in1991, many researchers have investigated their unique capabilitiesas reinforcements in composite materials. Due to their remarkablemechanical, electrical and thermal properties, carbon nanotubesare considered ideal reinforcing fibers for advanced high strengthmaterials and smart materials with self sensing capabilities [2,3].In actual structural applications, it is important to develop theoret-ical models in order to predict the response of structural elementsmade of carbon nanotube-reinforced composites (CNTRC). In par-ticular, skew plates are widely employed in civil and aeronauticalengineering applications such as panels in skew bridges, construc-tion of wings, tails and fins of swept-wing aircraft, etc. However,due to the mathematical difficulties involved in their formulation,
are scarce in the literature [4].The number of publications dealing with static and dynamic
analysis of CNTRC structural elements have increased considerablyin recent years. Wuite and Adali [5] studied the bending behaviorof classical symmetric cross-ply and angle-ply laminated beamsreinforced by aligned CNTs and isotropic beams reinforced by ran-domly oriented CNTs. By using a micromechanical constitutivemodel based on the Mori–Tanaka method, they highlighted thatsmall percentages of CNT reinforcement lead to significantimprovement in beam stiffness. Vodenitcharova and Zhang [6]developed a continuum model for the uniform bending andbending-induced buckling of a straight nanocomposite beam withcircular cross section reinforced by a single-walled carbon nan-otube (SWNT). The results showed that although the addition ofa matrix to a SWNT increases the load carrying capacity, thethicker matrix layers the SWNT buckles locally at smaller bendingangles and greater flattening ratios. Formica et al. [7] studied thevibrational properties of cantilevered CNTRC plates with an
474 E. García-Macías et al. / Composite Structures 140 (2016) 473–490
Eshelby–Mori–Tanaka approach and finite element modeling. Theresults demonstrated the ability of CNTs to tune the vibrationalproperties of composites and increase the fundamental frequenciesup to 500%. These exceptional properties have motivated manyresearchers to optimize the contribution of CNTs. According to thisprinciple, Arani et al. [8] investigated analytically and numericallythe buckling behavior of CNTRC rectangular plates. Based on clas-sical laminate plate theory and the third-order shear deformationtheory for moderately thick plates, they optimized the orientationof CNTs to achieve the highest critical load. Another example ofthis interest is the research carried out by Rokni et al. [9]. By divid-ing a beam along its longitudinal and thickness direction with theinclusion proportion as the design variable, they proposed a newtwo-dimensional optimum distribution of reinforcements of apolymer composite micro-beams to maximize the fundamentalnatural frequency given a weight percentage of CNTs.
Functionally graded materials (FGMs) belong to a branch ofadvanced materials characterized by spatially varying properties.This concept has promoted the development of a wide range ofapplications of functionally graded composite materials since itsorigin in 1984 (see e.g. [10]). Inspired by this idea, Shen [11] pro-posed non-uniform distributions of CNTs within an isotropicmatrix. In this work, nonlinear vibration of functionally gradedCNT-reinforced composite (FG-CNTRC) plates in thermal environ-ments was presented. Researchers have employed many differentmethodologies to model FG-CNTRCs andmost of them are recordedin a recent review by Liew et al. [12]. Zhu et al. [13] carried outbending and free vibration analysis of FG-CNTRC plates by using afinite element model based on the first-order shear deformationplate theory (FSDT). Ke et al. [14] presented nonlinear free vibrationanalysis of FG-CNTRC beams within the framework of Timoshenkobeam theory and Ritzmethod solved by a direct iterative technique.They concluded that symmetrical distributions of CNTs providehigher linear and nonlinear natural frequencies for FG-CNTRCbeams than with uniform or unsymmetrical distribution of CNTs.Zhang et al. [15] proposed a meshless local Petrov–Galerkinapproach based on the moving Kriging interpolation technique toanalyze the geometrically nonlinear thermoelastic behavior offunctionally graded plates in thermal environments. Shen andZhang [16] analyzed the thermal buckling and postbuckling behav-ior of uniform and symmetric FG-CNTRC plates under in-plane tem-perature variation. These results showed that the bucklingtemperature as well as thermal postbuckling strength of the platecan be increased with functionally graded reinforcement. However,in some cases the plate with intermediate nanotube volume frac-tionmay not present intermediate buckling temperature and initialthermal postbuckling strength. Aragh et al. [17] proposed an Eshelby–Mori–Tanaka approach and a 2-D generalized differentialquadrature method (GDQM) to investigate the vibrational behaviorof rectangular plates resting on elastic foundations. Yas and Hesh-mati used Timoshenko beam theory to analyze the vibration ofstraight uniform [18] and non-uniform [19] FG-CNTRC beams sub-jected to moving loads. Alibeigloo and Liew [20] studied the bend-ing behavior of FG-CNTRC plates with simply supported edgessubjected to thermo-mechanical loading conditions by threedimensional elasticity theory and using the Fourier series expan-sion and state-spacemethod. This work was extended by Alibeiglooand Emtehani [21] for various boundary conditions by using the dif-ferential quadrature method. Zhang et al. [22] proposed a state-space Levy method for the vibration analysis of FG-CNT compositeplates subjected to in-plane loads based on higher-order sheardeformation theory. This research analyzed three different sym-metric distributions of the reinforcements along the thickness,namely UD, FG-X and FG-O. They concluded that FG-X providesthe largest frequency and critical buckling in-plane load. Whereas,the frequency for the FGO-CNT plate was the lowest. Wu and Li [23]
used a unified formulation of Reissner’s mixed variational theorem(RMVT) based finite prism methods (FPMs) to study the three-dimensional free vibration behavior of FG-CNTRC plates. Free vibra-tion analyses of quadrilateral laminated plates were carried out byMalekzadeh and Zarei [24] using first shear deformation theory anddiscretization of the spatial derivatives by the differential quadra-ture method (DQM). Furthermore, mesh-free methods, employedin many different fields such as elastodynamic problems [25] andwave equations [26], have also been widely employed in the simu-lation of FG-CNTRCs. Zhang et al. [27] employed a local Krigingmesheless method to evaluate the mechanical and thermal buck-ling behaviors of ceramic–metal functionally graded plates (FGPs).Lei et al. [28] presented parametric studies of the dynamic stabilityof CNTRC-FG cylindrical panels under static and periodic axial forceusing themesh-free-kp-Ritz method and the Eshelby–Mori–Tanakahomogenization framework. Lei et al. [29] employed thismethodol-ogy to carry out vibration analysis of thin-to-moderately thick lam-inated FG-CNT rectangular plates. Zhang and Liew [30] presenteddetailed parametric studies of the large deflection behaviors ofquadrilateral FG-CNT for different types of CNT distributions. Theyshowed that the geometric parameters such as side angle,thickness-to-width ratio or plate aspect ratios are more significantthanmaterial parameters such as CNT distribution and CNT volumefraction. Zhang et al. [31] employed the ILMS-Ritz method to assessthe postbuckling behavior of FG-CNT plates with edges elasticallyrestrained against translation and rotation under axial compres-sion. Some other results can be found in the literature dealing withthe buckling analysis of FG-CNTRC thick plates resting on Winkler[32] and Pasternak foundations [33], free vibration analysis of trian-gular plates [34], cylindrical panels [35,36], three-dimensional freevibration analysis of FG-CNTRC plates [37], vibration of thick func-tionally graded carbon nanotube-reinforced composite plates rest-ing on elastic Winkler foundations [38], vibration analysis offunctionally graded carbon nanotube reinforced thick plates withelastically restrained edges [30], etc.
In the case of skew plates, the verification of their mathematicalformulation is difficult because of the lack of exact solutions, andthose available in literature are based on approximate methods.Over the past four decades, a lot of research has been carried outon the study of isotropic skew plates [39–42]. In contrast, researchwork dealing with the analysis of anisotropic skew plates is ratherscant, and even more so for FG-CNT composite materials. However,Zhang et al. [30] obtained the buckling solution of FG-CNT rein-forced composite moderately thick skew plates using theelement-free IMLS-Ritz method and first-order shear deformationtheory (FSDT). The same authors [4] also provided approximatesolutions for the free vibration of uniform and a symmetric distri-bution of the volume fraction of CNT in moderately thick FG-CNTskew plates. This methodology was also employed by Lei et al.[43] to perform buckling analysis of thick FG-CNT skew plates rest-ing on Pasternak foundations. Geometrically nonlinear large defor-mation analysis of FG-CNT skew plates resting on Pasternakfoundations was carried out by Zhang and Liew [44].
In this paper, we develop an efficient finite element formulationbased on the Hu–Washizu principle to obtain approximatesolutions for static and free vibration of various types ofFG-CNTRC skew plates with moderate thickness. The shell theoryis formulated in oblique coordinates and includes the effects oftransverse shear strains by first-order shear deformation theory(FSDT). An invariant definition of the elastic transversely isotropictensor based on the representation theorem is also defined in obli-que coordinates. Independent approximations of displacements(bilinear), strains and stresses (piecewise constant within subre-gions) provide a consistent mechanism to formulate four-nodedskew elements with a total of twenty degrees of freedom. A setof eigenvalue equations for the FG-CNTRC skew plate vibration is
E. García-Macías et al. / Composite Structures 140 (2016) 473–490 475
derived, from which the natural frequencies and mode shapes canbe obtained. Detailed parametric studies have been carried out toinvestigate the influences of skew angle, carbon nanotube volumefraction, plate thickness-to-width ratio, plate aspect ratio, bound-ary conditions and the distribution profile of reinforcements (uni-form and three non-uniform distributions) on the static anddynamic response of the FG-CNTRC skew plates. The results arecompared to commercial codes and the limited existing bibliogra-phy with very good agreement.
2. Functionally graded CNTRC plates
Fig. 1 shows the four types of FG-CNTRC skew plates consideredin this paper, with length a, width b, thickness t and fiber orienta-tion angle u. UD-CNTRC represents the uniform distribution andFG-V, FG-O and FG-X CNTRC are the functionally graded distribu-tions of carbon nanotubes in the thickness direction of the compos-ite skew plates. The effective material properties of the two-phasenanocomposites mixture of uniaxially aligned CNTs reinforce-ments and a polymeric matrix, can be estimated according to theMori–Tanaka scheme [45] or the rule of mixtures [3,46]. Theaccuracy of the extended rule of mixtures (EROM) has been widelydiscussed and a remarkable synergism with the Mori–Tanakascheme for functionally graded ceramic–metal beams is reportedin [17]. Due to the simplicity and convenience, in the presentstudy, the extended rule of mixture was employed by introducingthe CNT efficiency parameters and the effective material propertiesof CNTRC skew plates can thus be written as [11]
E11 ¼ g1VCNTECNT11 þ VmE
m ð1aÞg2
E22¼ VCNT
ECNT22
þ Vm
Em ð1bÞ
g3
G12¼ VCNT
GCNT12
þ Vm
Gm ð1cÞ
where ECNT11 ECNT
22 and GCNT12 indicate the Young’s moduli and shear
modulus of SWCNTs, respectively, and Em and Gm represent
Fig. 1. Geometry and configurations of the functionally graded carbon nanotube-reinforced (FG-CNTRC) skew plates.
corresponding properties of the isotropic matrix. To account forthe scale-dependent material properties, the CNT efficiencyparameters, gj (j = 1,2,3), were introduced and can be calculated bymatching the effective properties of the CNTRC obtained from amolecular dynamics (MD) or multi-scale simulations with thosefrom the rule of mixtures. VCNT and Vm are the volume fractions ofthe carbon nanotubes and matrix, respectively, and the sum of thevolume fractions of the two constituents should equal unity.Similarly, the thermal expansion coefficients, a11 and a22, in thelongitudinal and transverse directions respectively, Poisson’s ratiom12 and the densityq of the nanocomposite plates can be determinedin the same way as
where mCNT12 and mm are Poisson’s ratios, and aCNT11 ;aCNT
22 and am are thethermal expansion coefficients of the CNT and matrix, respectively.Note that m12 is considered as constant over the thickness of thefunctionally graded CNTRC skew plates.
And the other effective mechanical properties are
E33 ¼ E22; G13 ¼ G12; G23 ¼ 12
E22
1þ m23;
m13 ¼ m12; m31 ¼ m21; m32 ¼ m23 ¼ m21;
m21 ¼ m12E22
E11
ð3Þ
The uniform and three types of functionally gradeddistributions of the carbon nanotubes along the thickness directionof the nanocomposite skew plates shown in Fig. 1 are assumed tobe
VCNT ¼ V�CNT ðUD CNTRCÞ
VCNT ¼ 4 zj jt V�
CNT ðFG-X CNTRCÞVCNT ¼ 1þ 2z
t
� �V�
CNT ðFG-V CNTRCÞVCNT ¼ 2 1� 2 zj j
t
� �V�
CNT ðFG-O CNTRCÞ
ð4Þ
3. Finite element formulation
3.1. Parametrization of the geometry
Consider CNTRC skew plate of length a, width b, thickness t andskew angle a as shown in Fig. 1. The midsurface of the shell to beconsidered in this paper is given in terms of skew coordinatesðh1; h2Þ, hence the change of coordinates is given by
x ¼ h1 þ h2 cosa
y ¼ h2 sina
z ¼ h3ð5Þ
This parametrization leads a covariant basis ar defined by Eq. (6)
~a1 ¼100
8><>:
9>=>;; ~a2 ¼
cosasina0
8><>:
9>=>; and ~a3 ¼
001
8><>:
9>=>; ð6Þ
The covariant metric tensor is noted by a has a value of sin2 aand leads a contravariant basis ar defined by Eq. (7)
~a1 ¼1
� tan�1 a0
8><>:
9>=>;; ~a2 ¼
0csca0
8><>:
9>=>; and ~a3 ¼ ~a3 ð7Þ
476 E. García-Macías et al. / Composite Structures 140 (2016) 473–490
The theoretical formulation is derived by a variational formula-tion. Denoting byUðcÞ the strain energy and by c and r the vectorscontaining the strain and stress components, respectively, amodified potential of Hu–Washizu assumes the form [47]
PHW ½v;c;r� ¼ZV½UðcÞ�rT ðc�DvÞ�Pb�dV�
ZSv̂
ðv� v̂ÞrndS�ZSt
Pt dS
ð8ÞIn Eq. (8), v and the index b represent the displacement vector
and the body forces, respectively, whereas v̂ are prescribed dis-placements on the part of the boundary in which displacementsare prescribed ðSv̂Þ.
The displacement field is constructed by first-order shear defor-mation. Hence the in-plane deformation cab is expressed in termsof the extensional ð0cabÞ and flexural ð1cabÞ components of theCauchy-Green strain tensor as
cab ¼ 0cab þ h3 1cab: ð9ÞDenoting by Va and V3 the tangential displacements of the mid-
surface in the ha and h3 directions, and by /a the rotations aboutthe ha lines, the strains in terms of the aforementioned displace-ments and rotations have the form
Extensional strains : 0cab ¼12
Va jj b þ Vb jja� �
; ð10aÞ
Flexural strains : 1cab ¼12
ffiffiffia
peal /
ljj b þ
ffiffiffia
peb l /
ljj a
� �; ð10bÞ
Shear strains : 2ca 3 ¼ V3; a þffiffiffia
peal /
l ð10cÞIn Eqs. (10) ea b denote the permutation tensor associated with
the undeformed surface and a double bar ð:Þjj signifies covariant dif-ferentiation with respect to the undeformed surface. In vectorialform
0c ¼0c110c2220c12
8><>:
9>=>;; 1c ¼
1c111c2221c12
8><>:
9>=>; and cS ¼
c13c23
� �ð11Þ
The thin body assumption is considered in the z-direction, andthus it is often possible to neglect the transverse normal stress s33.The stress–strain relationships are defined by
sab ¼ @U@cab
¼ Cabcd ccd
sa3 ¼ 2Ea3b3 cb3s33 ¼ 0
ð12Þ
And the free-energy density takes the form
U ¼ 12Cabcd cab ccd þ 2Ea3b3 ca3 cb3 ð13Þ
3.3. Linearly elastic transversely isotropic constitutive matrix in non-orthogonal coordinates
The definition of non-orthogonal coordinates requires a coher-ent definition of the stress–strain relationships. On the basis ofthe representation theorems of transversely isotropic tensorsdeveloped by Spencer [48], Lumbarda and Chen [49] obtainedthe constitutive tensor of linear elastic transversely isotropic mate-rials in a general coordinates system as
Cijkl ¼X6r¼1
crIrijkl ð14Þ
The Ir are a set of linearly independent fourth order tensors thatform a basis of an algebra of order 6 and the cr are six elasticparameters. In component form, the fourth-order tensors Ir aredefined by
� � ð15eÞI6ijkl ¼ ninjnknl ð15fÞwhere ni are the rectangular components of an unit vector parallelto the axis of the transverse isotropy, defined in the mid-plane ofthe skew plate as ~n¼ ðcosu; sinu;0Þ (see Fig. 1), and aij are thecomponents of the contravariant basis ar defined in Eq. (7). Thematerial parameters, cr , are defined as
c1 ¼ 2l; c2 ¼ k; c3 ¼ c4 ¼ a; c5 ¼ 2ðlo � lÞ; c6 ¼ b ð16ÞThe material parameters cr depend on five elastic constants: l
and k, shear modulus within the plane of isotropy and the Laméconstant, the out-of-plane elastic shear modulus l0;a and b. Inmatrix notation the 4th order elasticity tensor of transversely iso-tropic material for a preferred x direction in a Cartesian coordinatesystem gives
C ¼
2aþ bþ k� 2lþ 4l0 aþ k aþ k 0 0 0aþ k kþ 2l k 0 0 0aþ k k kþ 2l 0 0 00 0 0 l 0 00 0 0 0 l0 00 0 0 0 0 l0
2666666664
3777777775
ð17ÞThe relation between elastic invariant constants and the engi-
neering constants can be found by comparing Eq. (17) with theclassical transversely isotropic stiffness tensor. This comparisonleads to
Once the constitutive tensor is obtained, the plane stress stiff-ness matrix can be obtained numerically by deleting the rowsand columns associated with the z-direction in the compliancematrix. By inverting the resulting compliance matrix, the constitu-tive equations are written in Voigt’s notation in the form
s11s22s12s23s13
26666664
37777775¼
Q11ðzÞ Q12ðzÞ 0 0 0Q12ðzÞ Q22ðzÞ 0 0 0
0 0 Q66ðzÞ 0 00 0 0 Q44ðzÞ 00 0 0 0 Q55ðzÞ
26666664
37777775�
c11c22c12c23c13
26666664
37777775
ð19Þ
E. García-Macías et al. / Composite Structures 140 (2016) 473–490 477
CijE;C
ijC ;C
ijB
� �¼
Z h=2
�h=2QijðzÞ � ð1; z; z2Þdz ði; j ¼ 1;2;6Þ;
CijS ¼ 1
ks
Z h=2
�h=2Qijdz ði; j ¼ 4;5Þ
ð20Þ
Note that Qij varies with z according to the grading profile of theCNTRC along the thickness. ks denotes the transverse shear correc-tion factor for FGM, given by [50]
ks ¼ 6� ðmiV i þ mmVmÞ5
ð21Þ
3.4. Stiffness matrix of skew plate element
The strain-energy density per unit of area at the reference sur-face can be defined by
U ¼Z h=2
�h=2Udz ð22Þ
From Eq. (9) and Eq. (13), the strain-energy density can beexpressed as
U ¼Z h=2
�h=2
12Cabcd
0cab þ h3 1cab� �
0ccd þ h3 1ccd� �
þ 2Ea3b3 ca3 cb3
� �dz ð23Þ
Expression (23) for the strain energy can be represented as thesum of the extensional ðUEÞ, bending ðUBÞ, coupling ðUCÞ and trans-verse shear ðUSÞ strain energy as
3.4.1. DiscretizationThe shell element derived in the present study is a four-noded
skewed isoparametric finite element (see Fig. 2) with five degreesof freedom at each node: three physical components of thedisplacements u1;u2;u3 and two components of the rotationsu1;u2 Eq. (25). Bilinear shape functions Nk are chosen for the phys-ical components of the displacements and rotations in the follow-ing way
ui ¼X4k¼1
uki Nk and ua ¼
X4k¼1
u ka Nk; ð25Þ
Nk ¼ 14ð1þ nk nÞ ð1þ gk gÞ; i ¼ 1;2;3 and a ¼ 1;2: ð26Þ
As mentioned before, the use of the Hu–Washizu principle andthe independent approximation of strain and stress yields a series
Fig. 2. Four node skew quadrilateral shell finite element.
of desirable features important for the reliability, convergencebehavior, and efficiency of the elemental formulation such as theavoidance of superfluous energy and zero energy modes. Further-more, the discrete approximation is drawn in a consistent mannerfrom the general theory of the continuum and the mechanicalbehavior of the finite element, without resorting to special manip-ulations or computational procedures. In addition, it has beenshown [47,51,52] that essential prerequisites for the achievementof these goals are: the identification of constant and higher-orderdeformational modes which are contained in the displacement/rotation assumptions, the realization that the constant terms arenecessary for convergence, and that higher-order terms reappearin different strain components. Therefore, our approximation doesnot need to retain the higher-order terms in two different straincomponents (they are needed only to inhibit a mode).
For instance, the following assumptions for the extensionalstrains have been shown to serve the aforementioned goals
c11 ¼ �c11 þ ��c11g;c22 ¼ �c22 þ ��c22 n
c12 ¼ �c12 þ �̂c11 nþ �̂c22g:ð27Þ
Note that, according to the above ideas, the underlined terms inEqs. (27) are not considered. The elimination of such terms allowsthe reduction of excessive internal energy and to improve conver-gence. Furthermore, the replacement of the linear variation of thestrains and stresses by piecewise constant approximations leads tocomputational advantages that are most important in repetitivecomputations. The piecewise constant approximations can beimproved by introducing four subdomains over the finite element(see Fig. 3). For example, Fig. 4 illustrates the piecewise approxi-mation of c11 and c22 over two subdomains. The membrane shearstrain c12 is approximated by a constant.
Considering the piecewise approximations through the foursubdomains and expressing strains in physical components(e;j; c), the extensional, bending and shear strain over every sub-domain are defined as
Extensional strains ðe11; e22; e12Þ
e11 ¼ e A11 in AI þ AII
e B11 in AIII þ AIV
(;
e22 ¼ e C22 in AI þ AIV
eD22 in AII þ AIII
(and e12 ¼ �e12 in A
ð28Þ
Fig. 3. Subdomain areas throughout the finite element.
Fig. 4. Schematic representation of the piecewise constant extensional strain approximation.
478 E. García-Macías et al. / Composite Structures 140 (2016) 473–490
Bending strains ðj11;j22;j12Þ
j11 ¼ j A11 in AI þ AII
j B11 in AIII þ AIV
(;
j22 ¼ j C22 in AI þ AIV
jD22 in AII þ AIII
(and j12 ¼ �j12 in A
ð29Þ
Shear strains ðc13; c23Þ
c1 ¼ cA1 in AI þ AII
c B1 in AIII þ AIV
(and c2 ¼ c C
2 in AI þ AIV
cD2 in AII þ AIII
(
ð30ÞAs a consequence of this approximation, the strain energy term inthe Hu–Washizu variational principle takes the form ofZAUdA ¼
ZAI
UI dAþ . . .þZAIV
UIV dA ¼XIVi¼I
ZAi
Ui dA
¼ 12�e T �DE �eþ 1
2�j T �DB �jþ 1
2�e T �DC �jþ 1
2�j T �DC �eþ 1
2�c T �DS �c
ð31Þwhere the vectors �e; �j and �c are defined by
�e ¼
eA11eB11eC22eD222e12
8>>>><>>>>:
9>>>>=>>>>;; �j ¼
jA11
jB11
jC22
jD22
2j12
8>>>><>>>>:
9>>>>=>>>>;; �c ¼
cA1cB1cC2cD2
8>><>>:
9>>=>>; ð32Þ
The matrices �DE; �DB; �DC and �DS are the discretized elasticitymatrices that depend on the geometry of the surface —i.e., on thecontravariant ðaabÞ and covariant ðaabÞ components of the metrictensors— and can be represented as follows
�DE ¼
RA1þA2
DE ð1;1ÞdA 0RA1DE ð1;2ÞdA
RA2DE ð1;2ÞdA
RA1þA2
DE ð1;3ÞdARA3þA4
DE ð1;1ÞdARA4DE ð1;2ÞdA
RA3DE ð1;2ÞdA
RA3þA4
DE ð1;3ÞdARA1þA4
DE ð2;2ÞdA 0RA1þA4
DE ð2;3ÞdARA2þA3
DE ð2;2ÞdARA2þA3
DE ð2;3ÞdAsym
RA DE ð3;3ÞdA
2666664
3777775;
ð33Þ
�DS ¼
RA1þA2
DS ð1;1ÞdA 0RA1DS ð1;2ÞdA
RA2DS ð1;2ÞdAR
A3þA4DS ð1;1ÞdA
RA4DS ð1;2ÞdA
RA3DS ð1;2ÞdAR
A1þA4DS ð2;2ÞdA 0
symRA2þA3
DS ð2;2ÞdA
26664
37775
ð34Þ
Furthermore, the parameters for the stress resultants areexpressed by the following vector forms
NT ¼ NA11 NB
11 NC22 ND
22 N12
�;
MT ¼ MA11 MB
11 MC22 MD
22 M12
�and
Q T ¼ Q A11 QB
11 QC22 QD
22 Q12
�:
ð35Þ
In addition, by introducing the matrices
AN ¼ AM ¼
AI þ AII 0 0 0 00 AIII þ AIV 0 0 00 0 AI þ AIV 0 00 0 0 AII þ AIII 00 0 0 0 A
along with the discretized strain–displacement relationships, thebilinear approximations for the displacements and rotations, andalso the discrete parameters for the strains and stresses, the discreteform of the generalized variational principle of Hu–Washizu isgiven by
PHW ¼ 12�e T �DE �eþ 1
2�j T �DB �jþ 1
2�e T �DC �jþ 1
2�j T �DC �e
þ 12�c T �DS �c� 1
2N T AN �eþ �e T ANN
� �� 12
M T AM �jþ �j T AMM� �
� 12
Q T AQ �cþ �c T AQ Q� �
þ 12
N T EDþ D T EN� �
þ 12
M T BDþ D T BM� �
þ 12
Q T GDþ D T GQ� �
ð37Þ
The Hu–Washizu variational principle establishes that ifthe variation is taken with respect to nodal displacements androtations (D), strains, and stresses, then all field equations ofelasticity and all boundary conditions appear as Euler–Lagrange
E. García-Macías et al. / Composite Structures 140 (2016) 473–490 479
equations. In particular, the stationary condition for the functional,dPHW ¼ 0, enforces the following governing discretized fieldequation
(a) Variation of the stress resultants leads to the discrete strain–displacement relationships
ED� AN �e ¼ 0 ) �e ¼ A�1N ED;
BD� AM �j ¼ 0 ) �j ¼ A�1M BD and
GD� AQ �c ¼ 0 ) �c ¼ A�1Q GD:
ð39Þ
(b) Variation of the strain parameters yields the discrete consti-tutive equations
�DE �eþ �DC �j� ANN ¼ 0 ) N ¼ A�1N
�DE �eþ A�1N
�DC �j
�DF �jþ �DC �e� AMM ¼ 0 ) M ¼ A�1M
�DF �jþ A�1M
�DC �e
�DS �c� AQ Q ¼ 0 ) Q ¼ A�1Q
�DS �c
ð40Þ(c) Variation of the nodal displacements/rotations leads to thediscrete form of the equilibrium equations
E T Nþ B T Mþ G T Q � p ¼ 0: ð41Þ
By introducing Eqs. (39) in Eqs. (40), the parameters for the stressresultants can be expressed in terms of nodal displacements as
N ¼ A�1N
�DEA�1N EDþ A�1
N�DCA
�1M BD;
M ¼ A�1M
�DFA�1M BDþ A�1
M�DCA
�1N ED
Q ¼ A�1Q
�DSA�1Q GD:
ð42Þ
In a compact way, the introduction of expressions (42) into Eq.(41) yields the discrete equilibrium expressed in terms of nodaldisplacements and rotations as
KExtension þ KBending þ KCoupling þ KShear
�D ¼ p ð43Þ
Therefore, the stiffness matrix, K20�20, is defined by the sum ofthe following four terms
KExtension ¼ A�1N
�DEA�1N E; ð44Þ
KBending ¼ B T A�1M
�DFA�1M B; ð45Þ
KCoupling ¼ B T A�1M
�DCA�1N Eþ E T A�1
N�DCA
�1M B; ð46Þ
KShear ¼ G T A�1Q
�DSA�1Q G: ð47Þ
3.5. The governing eigenvalue equation
The eigenvalue problem for the undamped free vibration prob-lem takes the well-known form
Ku ¼ x2Mu; ð48Þwhere K is the stiffness matrix of the system, u represents theeigenvectors, x is the natural frequency in rad/s and M is the massmatrix of the structure. The consistent element mass matrix isderived by discretizing the kinetic energy
dUK ¼ 12
ZVq2v d€vdV ; ð49Þ
and by employing the displacement field defined by first-ordershear deformation, the integral (49) assumes the form
dUK ¼ZAq d€u1 d€u2 d€u3 d €u1 d €u2½ �
I1 I1A 0 I2 I2A
I1A I1 0 I2A I20 0 I1 0 0I2 I2A 0 I3 I3A
I2A I2 0 I3A I3
26666664
37777775
u1
u2
u3
u1
u2
26666664
37777775dA;
ð50Þwhere the terms I1; I2; I3 and A (the contravariant components rela-tionship) are defined by
I1 ¼Z h=2
�h=2qðzÞdz; I2 ¼
Z h=2
�h=2qðzÞzdz; I3 ¼
Z h=2
�h=2qðzÞz2dz ð51Þ
A ¼ a12ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia11 � a22
p ¼ � cosðaÞ ð52Þ
In addition, by the definition of the displacements and rotationsthrough the shape functions, nodal displacements and nodal rota-tions in Eq. (25), the consistent mass matrix can be represented by
M ¼
M11 M12 M13 M14
M22 M23 M24
M33 M34
sym M44
26664
37775
20�20
ð53Þ
Every Mij term of the mass matrix, where i and j represent therow and the column respectively, assumes the following form
Mij ¼
RA I1NiNj dA
RAAI1NiNj dA 0
RA I2NiNj dA
RAAI2NiNj dAR
A I1NiNj dA 0RAAI2NiNj dA
RA I2NiNj dAR
A I1NiNj dA 0 0RA I3NiNj dA
RAAI3NiNj dA
symRA I3NiNj dA
26666664
37777775
5�5
ð54Þ
Finally, we remark that all the aspects of numerical implemen-tation associated with the above expressions are carried out bymeans of the commercial software package MATHEMATICA [53], whichis particularly useful for the treatment of symbolic and algebraiccomputations.
4. Numerical results
In this section, a set of static and free vibration analyses are pre-sented to demonstrate the applicability of the proposed finite ele-ment formulation to FG-CNTRC thin and moderately thick skewplates. Firstly, some results are compared to the limited existingbibliography, for isotropic and FG-CNTRC skew plates. Then, newbending and free vibration analyses are presented to broadenknowledge about mechanical characteristics of FG-CNTRC skewplates by taking into consideration not previously consideredaspects such as asymmetric reinforcement distributions and orien-tation of CNTs.
4.1. Comparison studies
In order to show the validity of the proposed finite element for-mulation, convergence analyses are carried out in order to checkthe stability of the solution. Also, the free vibration resultsobtained by Liew et al. [54] and Zhang et al. [4] for isotropic skewplates are compared to the ones obtained by the proposed method.Then, the free vibration results for FG-CNTRC skew plates pre-sented by Zhang et al. [4] are also verified.
4.1.1. Convergence and comparison of free vibration analysis ofisotropic skew plates
In these first tests, comparison studies of free vibrationanalysis are carried out for isotropic skew plates with skew anglesa ¼ 90�; 60�; 45� and 30�, thickness-to-width ratios of t=b ¼ 0:001
480 E. García-Macías et al. / Composite Structures 140 (2016) 473–490
(thin plate) and 0:2 (moderately thick plate) and with fourdifferent kinds of boundary conditions, namely all edges simplysupported (SSSS) or clamped (CCCC), and two opposite edgessimply supported and the other two clamped (SCSC) or free (SFSF).The boundary conditions at any edge can be defined as follows
us ¼ uz ¼ cs ¼ 0 ( Simply supported edge ðSÞun ¼ us ¼ uz ¼ cn ¼ cs ¼ 0 ( Clamped edge ðCÞ
(ð55Þ
where the subscripts n and s denote the normal and tangentialdirections, respectively. The non-dimensional frequency parameterfor vibration analysis is defined by
�x ¼ xb2
p2
ffiffiffiffiffiffiqtD
rð56Þ
wherex is the angular frequency of the CNTRC plates, q is the platedensity per unit volume and D ¼ Et3=12ð1� m2Þ is the plate flexural
Fig. 5. Corner stress singularities (Von Mises) of fully clamped (CCCC) isotropic skew plat(a ¼ 90�; 60�; 45� and 30�).
Fig. 6. First frequency parameter �x1 convergence analysis for SSSS and SFSF isotropi
rigidity. A value of m ¼ 0:3 for Poisson’s ratio is used for this analy-sis. Skew plates are characterized by the presence of stress singular-ities at the shell corners. Because of the simplifying assumptionscommonly adopted, these problems worsen with increasing skewangle and can lead to divergent solutions. Fig. 5 shows the VonMises stress field of fully clamped isotropic skew plates withincreasing skew angles. The existence of stress concentrations atobtuse corners is highlighted. The effect of presence of thesesingularities on the dynamic characteristics of skew plates is welldocumented. For example McGee et al. [42] and Huang et al. [55]analyzed the influence of the bending stress singularities by usingthe Ritz method. By the implementation of comparison functionsor so called corner functions the authors studied different boundaryconditions and achieved great improvements in the convergence ofthe solution. The presence of these singularities requires the devel-opment of a convergence analysis of the dynamic characteristics inorder to prove the stability of the solution. In Fig. 6, the solutions interms of the first frequency parameter �x1 are represented for four
es subjected to transverse uniform loading (qo ¼ �0:1 MPa) and varying skew angle
c skew plates in terms of mesh size N � N (a=b ¼ 1; t=b ¼ 0:2; a ¼ 45� , u ¼ 0�).
Table 1Comparison study of frequency parameters ðx2=p2Þ= ffiffiffiffiffiffiffiffiffiffiffi
qt=Dp
of isotropic skew plates with CCCC boundary conditions (a=b ¼ 1, t=b ¼ 0:001, u ¼ 0�).
Table 2Material properties of Poly (methyl methacrylate) (PMMA) at room temperature of300 K and ð10;10Þ single walled carbon nanotubes (SWCNT).
ð10;10Þ SWCNT [16] PMMA ðT ¼ 300 KÞ
ECNT11 ¼ 5:6466 TPa Em ¼ 2:5 GPa
ECNT22 ¼ 7:0800 TPa mm ¼ 0:34
GCNT12 ¼ 1:9445 TPa am ¼ 45� 10�6=K
mCNT12 ¼ 0:175
E. García-Macías et al. / Composite Structures 140 (2016) 473–490 481
sets of mesh sizes (8� 8; 16� 16; 32� 32 and 40� 40) and for askew plate with a=b ¼ 1; a ¼ 45� and t=b ¼ 0:2 having CCCC andSFSF boundary conditions. As expected, fewer elements are neededfor the lower modes to reach an acceptable convergence as
Table 3Comparison of Young’s moduli for PMMA=CNT composites reinforced by ð10;10Þ SWCNT
Fig. 7. First frequency parameter �x1 convergence analysis for SSSS and SV�
CNT ¼ 0:12; a=b ¼ 1; t=b ¼ 0:001; a ¼ 30� , u ¼ 0�).
compared to the higher modes. These studies show that 24� 24elements are sufficient to reach accurate vibration results. There-fore, for the subsequent calculations, this mesh size is adopted.
The first eight frequency parameters for CCCC boundary condi-tions are presented in Table 1 together with the published resultsin references [54,4]. It can be seen that the present frequencyparameters match very well for all cases. It is remarkable thatthe stiffening effect of increasing skew angles in this type of struc-tural element is seen in all posterior results.
4.1.2. Convergence and comparison of free vibration analysis of FG-CNTRC skew plates
The next comparison analysis refers to free vibration ofFG-CNTRC skew plates. A new convergence analysis is performedin order to check the suitability of the discretization for accurate
at T = 300 K with MD simulation [16].
f mixtures
Pa) g1 E22 (GPa) g2
0.137 2.9 1.0220.142 4.9 1.6260.141 5.5 1.585
FSF FG-CNTRC skew plates in terms of mesh size N � N (UD-CNTRC,
482 E. García-Macías et al. / Composite Structures 140 (2016) 473–490
predictions of this new scenario with transversely isotropicmaterials. The matrix Poly (methyl methacrylate), referredto as PMMA, is selected and the material properties areassumed to be mm ¼ 0:34; am ¼ 45 � ð1þ 0:0005 �DtÞ � 10�6=K andEm ¼ ð3:52� 0:0034 � TÞ GPa. The armchair (10,10) SWCNTs areselected as reinforcements with properties taken from the MDsimulation carried out by Shen and Zhang [16]. The material prop-erties of these two phases are summarized in Table 2. In this study,
Table 4Comparison study of frequency parameters �x for a skew plate with CCCC boundary cond
it is assumed that the effective material properties are independentof the geometry of the CNTRC plates. The detailed material proper-ties of PMMA=CNT for the FG-CNTRC skew plates are selected fromthe MD results reported by Han and Elliot [56]. The CNT efficiencyparameters are taken from [16] and are presented in Table 3.
In this section, the nondimensional frequency parameter �x isdefined for composites by using the matrix’s material propertiesas follows
E. García-Macías et al. / Composite Structures 140 (2016) 473–490 483
�x ¼ xb2
p2
ffiffiffiffiffiffiffiffiffiqmtD
r; D ¼ Emt3=12ð1� mm2Þ ð57Þ
In Fig. 7, the solutions in terms of the first frequency parameter�x1 for the mesh sizes (8� 8; 16� 16; 32� 32 and 40� 40) arerepresented for a skew plate with a=b ¼ 1; a ¼ 30�, UD-CNTRC,V�
CNT ¼ 0:12 and t=b ¼ 0:001 having CCCC and SFSF boundary
Table 6Effects of CNT volume fraction VCNT and width-to-thickness ratio ðb=tÞ on the non-dimensioq0 ¼ �0:1 MPa with SSSS boundary conditions (a=b ¼ 1, u ¼ 0�).
Table 7Effects of CNT volume fraction VCNT and width-to-thickness ratio ðb=tÞ on the non-dimensioq0 ¼ �0:1 MPa with CCCC boundary conditions (a=b ¼ 1, u ¼ 0�).
conditions. As in the previous case, a mesh pattern of 24� 24 isconsidered sufficient for convergence, and, henceforth this meshsize is adopted.
The results obtained by Zhang et al. [4] are compared to theones obtained by the proposed method. Comparison of the firsteight frequency parameters of the fully clamped CNTRC skew
nal central deflection wo=t for CNTRC skew plates under a uniformly distributed load
FG� V FG� O
NSYS Present ANSYS Present ANSYS
.271E�03 2.983E�03 2.980E�03 4.341E�03 4.337E�03
.094E�03 2.812E�03 2.678E�03 3.824E�03 3.719E�03
.721E�03 2.279E�03 2.117E�03 2.856E�03 2.741E�03
.917E�04 1.200E�03 1.141E�03 1.364E�03 1.335E�03
.733E�01 1.232E+00 1.232E+00 2.105E+00 2.105E+00
.329E�01 1.125E+00 1.120E+00 1.797E+00 1.794E+00
.135E�01 8.890E�01 8.824E�01 1.282E+00 1.278E+00
.443E�01 4.586E�01 4.528E�01 5.721E�01 5.719E�01
.448E�03 1.935E�03 1.933E�03 2.866E�03 2.863E�03
.320E�03 1.789E�03 1.715E�03 2.480E�03 2.423E�03
.065E�03 1.417E�03 1.331E�03 1.813E�03 1.752E�03
.956E�04 7.248E�04 6.960E�04 8.440E�04 8.308E�04
.275E�01 8.404E�01 8.403E�01 1.436E+00 1.436E+00
.914E�01 7.533E�01 7.509E�01 1.204E+00 1.202E+00
.995E�01 5.796E�01 5.758E�01 8.390E�01 8.370E�01
.139E�01 2.873E�01 2.838E�01 3.639E�01 3.642E�01
.045E�03 1.359E�03 1.357E�03 1.988E�03 1.986E�03
.641E�04 1.305E�03 1.233E�03 1.820E�03 1.756E�03
.941E�04 1.080E�03 9.916E�04 1.425E�03 1.349E�03
.600E�04 5.824E�04 5.487E�04 7.218E�04 6.945E�04
.280E�01 5.278E�01 5.276E�01 9.280E�01 9.279E�01
.126E�01 4.905E�01 4.883E�01 8.245E�01 8.226E�01
.641E�01 3.980E�01 3.947E�01 6.212E�01 6.188E�01
.506E�01 2.134E�01 2.106E�01 2.965E�01 2.957E�01
nal central deflection wo=t for CNTRC skew plates under a uniformly distributed load
FG� V FG� O
NSYS Present ANSYS Present ANSYS
.260E�03 1.516E�03 1.513E�03 1.673E�03 1.669E�03
.135E�03 1.415E�03 1.336E�03 1.537E�03 1.464E�03
.952E�04 1.069E�03 1.022E�03 1.145E�03 1.107E�03
.931E�04 5.016E�04 5.357E�04 5.282E�04 5.691E�04
.687E�01 3.420E�01 3.417E�01 4.470E�01 4.467E�01
.671E�01 3.266E�01 3.215E�01 4.164E�01 4.126E�01
.527E�01 2.717E�01 2.647E�01 3.335E�01 3.293E�01
.861E�02 1.437E�01 1.401E�01 1.662E�01 1.656E�01
.628E�04 9.405E�04 9.384E�04 1.050E�03 1.048E�03
.830E�04 8.656E�04 8.222E�04 9.538E�04 9.130E�04
.338E�04 6.453E�04 6.218E�04 7.031E�04 6.839E�04
.909E�04 2.986E�04 3.211E�04 3.205E�04 3.464E�04
.137E�01 2.342E�01 2.340E�01 3.063E�01 3.060E�01
.117E�01 2.197E�01 2.167E�01 2.809E�01 2.788E�01
.985E�02 1.776E�01 1.737E�01 2.197E�01 2.176E�01
.148E�02 8.982E�02 8.789E�02 1.058E�01 1.056E�01
.118E�04 7.297E�04 7.278E�04 8.044E�04 8.028E�04
.483E�04 6.911E�04 6.480E�04 7.648E�04 7.187E�04
.299E�04 5.295E�04 5.013E�04 5.911E�04 5.587E�04
.368E�04 2.516E�04 2.670E�04 2.819E�04 2.959E�04
.310E�02 1.486E�01 1.484E�01 1.938E�01 1.935E�01
.249E�02 1.445E�01 1.421E�01 1.866E�01 1.844E�01
.659E�02 1.238E�01 1.206E�01 1.577E�01 1.550E�01
.347E�02 6.866E�02 6.711E�02 8.566E�02 8.484E�02
Fig. 8. Variation of the non-dimensional central deflection wo=t under uniformly distributed load q0 ¼ �0:1 MPa with SSSS and CCCC boundary conditions(a=b ¼ 1; VCNT ¼ 17%; b=t ¼ 50, u ¼ 0�).
Fig. 9. Non-dimensional central axial stress �rxx ¼ r�t2qoj j�a2 in CNTRC skew plates under a uniform load q0 ¼ �0:1 MPa and various boundary conditions
(VCNT ¼ 17%; a=b ¼ 1; b=t ¼ 50; a ¼ 30�; u ¼ 0�).
484 E. García-Macías et al. / Composite Structures 140 (2016) 473–490
Table 8Comparison study of frequency parameter k1 of skew plates with SSSS boundary conditions (a=b ¼ 1; u ¼ 0�).
486 E. García-Macías et al. / Composite Structures 140 (2016) 473–490
E. García-Macías et al. / Composite Structures 140 (2016) 473–490 487
plates with uniform and FG-X distributions of reinforcement isgiven in Table 4. As previously noted, decreasing skew angles resultin higher natural frequencies. Moreover, results provided by Zhanget al. [4] about the influence of the aspect ratio t=b on the non-dimensional frequency parameter are also compared with the pro-posed approach in Table 5. Here it is also noticeable that increasingthe a/b ratio decreases the frequency parameters. It can be seenthat the results obtained by the proposed methodology match verywell with the cited reference in both analyses. These close agree-ments, in combination with the convergence analyses, serve to ver-ify the present approach and establish the foundation for itsapplication to FG-CNTRC skew plates.
4.2. Results for FG-CNTRC skew plates
The results obtained by the proposed methodology have beenshown to be stable and similar to those provided in the literature.Some new results are now presented. Here we analyze the static
Fig. 10. Effect of CNT volume fraction on the firs
response of functionally graded PMMA/CNT skew plates under uni-form transverse loads ðqoÞ, the free vibration analysis of FG skewplates with symmetrical and unsymmetrical reinforcement distri-butions, and finally we show the advantages of the invariant defini-tion of the constitutive relationships from the analysis of theinfluence of fiber direction on the natural frequencies. The variousnon-dimensional parameters usedwithin this section are defined as
Non� dimensional frequency parameter : k ¼ xb2
t
ffiffiffiffiffiffiffiqm
Em
r; ð58aÞ
Central deflection : �w ¼ uz
tð58bÞ
Central axial stress : �r ¼ r � t2qoj j � a2 ð58cÞ
where wo is the vertical deflection at the central point. Note thenon-dimensional frequency parameter is slightly different fromthe one employed earlier.
t frequency parameter k1 (a=b ¼ 1, u ¼ 0�).
488 E. García-Macías et al. / Composite Structures 140 (2016) 473–490
4.2.1. Bending of FG-CNTRC skew platesSeveral numerical examples are provided to investigate the
bending analysis of FG-CNTRC skew plates under uniformtransverse loading qo ¼ �0:1 MPa. Four types of FG skew plates,UD-CNTRC, FG-V, FG-O and FG-X CNTRC are considered with sev-eral boundary conditions. In order to demonstrate the accuracyof the FE model used in the present study, results given by ANSYS(SHELL181, four-noded element with six degrees of freedom ateach node) for the same mesh density are provided. Tables 6 and7 shows the non-dimensional central deflection �w for the fourtypes of FG CNTRC skew plates subjected to a uniform transverseload qo with different values of width-to-thickness ratio(b=t ¼ 10, 50) and varying skew angles for SSSS and CCCC bound-ary conditions. It is noticeable that the volume fraction of the CNTshas so much influence on the central deflection of the plates. Forinstance, for uniform distributions only 6% increase in the volumefraction of CNT may lead to more than 60% decrease in the centraldeflection. Likewise, the values of non-dimensional deflectionsdecrease as the skew angle increases. It is also notable that thecentral deflections of FG-V and FG-O CNTRC plates are larger thanthe deflections of UD-CNTRC plates while those of the FG-X CNTRCplates are smaller. This is because the profile of the reinforcementdistribution affects the stiffness of the plates. This phenomenonhighlights the advantage of FG materials, in which a desiredstiffness can be achieved by adjusting the distribution of CNTs
Fig. 11. First eight mode shapes of a fully clamped UD-CNTRC skew plate for skew
along the thickness direction of the plates. It is concluded that rein-forcements distributed close to the top and bottom induce higherstiffness values of plates. Fig. 8 shows the non-dimensional centraldeflections of skew plates with a=b ¼ 1; VCNT ¼ 17%; b=t ¼ 50, forSSSS and CCCC boundary conditions. The stiffening effect of highervalues of skew angle a can be seen clearly. Similar conclusions canbe extracted from stress analysis. Fig. 9 shows the non-dimensional stress �rxx distribution along the thickness for CNTRCskew plates with a skew angle of a ¼ 30�, subjected to a uniformtransverse load qo with the volume fraction VCNT ¼ 17%. Due tothe symmetric distribution (with respect to the mid-plane) of rein-forcements for UD, FG-O and FG-X CNTRC skew plates, the centralaxial stress distributions is anti-symmetric. In the case of FG-VCNTRC and FG-O CNTRC skew plates, the axial stress is close to zeroat the bottom and top respectively. This is because the concentra-tion of CNTs vanishes at these points for these two distributions.
4.2.2. Free vibration analysis FG-CNTRC skew platesThe free vibration analyses of fully clamped and simply sup-
ported FG-CNTRC skew plates are given in Tables 8 and 9. The pre-sented four types of CNTRC are considered with the CNT volumefractions of 12%; 17% and 28%. The plate geometry is defined bythe following parameters, a=b=1 and b=t ¼ 10, 50. Here we presentthe first free vibration analysis of asymmetric FG-CNTRC skewplates, and the results are compared to those from the commercial
angles a ¼ 90� ; 60�; 45� and 30�; V�CNT ¼ 12%; a=b ¼ 1; t=b ¼ 0:02 and u ¼ 0� .
Fig. 12. Effect of fiber angle u on the first frequency parameter k1 of a CNTRC skew plate (UD-CNTRC, VCNT ¼ 12%; a=b ¼ 1; b=t ¼ 50).
E. García-Macías et al. / Composite Structures 140 (2016) 473–490 489
code ANSYS. Fig. 10 shows the evolution of the frequency parame-ters with varying skew angles for each reinforcement distributionseparately. As expected from the previous analysis, higher skewangles give stiffener behaviors and therefore higher frequencyparameters. This can be explained in terms of the plate area andthe perpendicular distance between the non-skew edges. Withhigher skew angles, the distance between the non-skew edgesdecreases which increases the frequency values. Furthermore, lar-ger volume fractions of CNTs lead to higher values of frequencyparameters, due to an increase in the stiffness of the CNTRC platewhen the CNT volume fraction is higher. Moreover, as could beseen in the bending simulations, we observe that the FG-X plateslead to the stiffest solutions and possess the highest frequencyparameters. The explanation of this phenomenon is the same asmentioned before; reinforcements distributed closer to theextremes result in stiffener plates than those distributed nearerto the mid-plane.
Fig. 11 shows the vibration mode shapes of fully clampedUD-CNTRC plates (V�
CNT ¼ 12%; a=b ¼ 1 and t=b ¼ 0:02) for skewangles a ¼ 90�; 60�; 45� and 30�. It is observed from these figuresthat mode crossing occurs as the skew angle increases.
4.2.3. Effect of direction of CNTs in FG-CNTRC skew plates on naturalfrequencies
Taking advantage of the invariant definition of the constitutivetensor for transversely isotropic materials, characterized by an unitvector parallel to the axis of the transverse isotropy,~n ¼ ðcosu; sinu;0Þ, we analyze the influence of the angle u onthe frequency parameters k. Fig. 12 shows the variation of the firstfrequency parameter for several values of skew angle a and twoboundary conditions, CCCC and SSSS. As in all the previousanalyses, the frequency parameters increase for higher values ofskew angle. Moreover, the results for skew angles of a ¼ 90� areperfectly symmetric around u ¼ 90�. In contrast, the curves for
higher skew angles present increasing levels of asymmetry. This isdue to the increment of stiffness provided by fibers coupled withthe stiffening effect of the skew angle. The curves can be separatedinto two sets divided aroundu ¼ 90�. For the SSSS boundary condi-tion set, it is clear that the increased stiffness is associated withfibers aligning the direction of the longest diagonal. Otherwise,the frequency values for the second set decreases in all cases forfiber angles aboveu ¼ 90�. In the case of the CCCC boundary condi-tion set this behavior is repeated although the maximum values areapproximately obtained with fibers aligned in the horizontal Carte-sian direction. This result shows the importance of taking into con-sideration the direction of the CNTs in order to optimize themechanical response of the FG-CNTRC skew plates. For example,with a skew angle of a ¼ 30� and SSSS boundary conditions, thevariation of the u may increase the first frequency parameter byup to 11:7% and decrease it by up to 18:6%.
5. Conclusions
In this paper, static and free vibration analyses of moderatelythick FG-CNTRC skew plates are presented. An efficient finite ele-ment formulation based on the Hu–Washizu principle is pre-sented. The shell theory is formulated in oblique coordinatesand includes the effects of transverse shear strains by first-ordershear deformation theory (FSDT). An invariant definition of theelastic transversely isotropic tensor based on the representationtheorem is defined in oblique coordinates. Independent approxi-mations of displacements (bilinear), strains and stresses (piece-wise constant within subregions) provide a consistentmechanism to formulate four-noded skew elements with a totalnumber of twenty degrees of freedom. A set of eigenvalueequations for the FG-CNTRC skew plate vibration is derived, fromwhich the natural frequencies and mode shapes can be obtained.Detailed parametric studies have been carried out to investigate
490 E. García-Macías et al. / Composite Structures 140 (2016) 473–490
the influences of skew angle, carbon nanotube volume fraction,plate thickness-to-width ratio, plate aspect ratio, boundary condi-tion and distribution profile of reinforcements (uniform and threenon-uniform distributions) on the static and free vibration charac-teristics of the FG-CNTRC skew plates. The results are compared tocommercial code ANSYS and limited existing bibliography withvery good agreement.
Acknowledgement
This research was supported by Spanish ministry of economyand competitively under the Project Ref: DPI2014-53947-R. E. G-M was also supported by a FPU contract-fellowship from the Span-ish Ministry of Education Ref: FPU13/04892.
References
[1] Iijima S. Helical microtubules of graphitic carbon. Nature (London, UnitedKingdom) 1991;354:56–8.
[2] Gibson RF. A review of recent research on mechanics of multifunctionalcomposite materials and structures. Compos Struct 2010;92:2793–810.
[3] Esawi AM, Farag MM. Carbon nanotube reinforced composites: potential andcurrent challenges. Mater Des 2007;28:2394–401.
[4] Zhang L, Lei Z, Liew K. Vibration characteristic of moderately thick functionallygraded carbon nanotube reinforced composite skew plates. Compos Struct2015;122:172–83.
[5] Wuite J, Adali S. Deflection and stress behaviour of nanocomposite reinforcedbeams using a multiscale analysis. Compos Struct 2005;71:388–96.
[6] Vodenitcharova T, Zhang L. Bending and local buckling of a nanocompositebeam reinforced by a single-walled carbon nanotube. Int J Solids Struct2006;43:3006–24.
[7] Formica G, Lacarbonara W, Alessi R. Vibrations of carbon nanotube-reinforcedcomposites. J Sound Vib 2010;329:1875–89.
[8] Arani AG, Maghamikia S, Mohammadimehr M, Arefmanesh A. Bucklinganalysis of laminated composite rectangular plates reinforced by SWCNTSusing analytical and finite element methods. J Mech Sci Technol2011;25:809–20.
[9] Rokni H, Milani AS, Seethaler RJ. 2Doptimum distribution of carbon nanotubesto maximize fundamental natural frequency of polymer composite micro-beams. Compos Part B: Eng 2012;43:779–85.
[10] Udupa G, Rao SS, Gangadharan K. Functionally graded composite materials: Anoverview. Proc Mater Sci 2014;5:1291–9.
[12] Liew K, Lei Z, Zhang L. Mechanical analysis of functionally graded carbonnanotube reinforced composites: a review. Compos Struct 2015;120:90–7.
[13] Zhu P, Lei Z, Liew KM. Static and free vibration analyses of carbon nanotube-reinforced composite plates using finite element method with first order sheardeformation plate theory. Compos Struct 2012;94:1450–60.
[14] Ke L-L, Yang J, Kitipornchai S. Nonlinear free vibration of functionally gradedcarbon nanotube-reinforced composite beams. Compos Struct2010;92:676–83.
[15] Zhu P, Zhang L, Liew K. Geometrically nonlinear thermomechanical analysis ofmoderately thick functionally graded plates using a local petrovgalerkinapproach with moving kriging interpolation. Compos Struct2014;107:298–314.
[16] Shen H-S, Zhang C-L. Thermal buckling and postbuckling behavior offunctionally graded carbon nanotube-reinforced composite plates. Mater Des2010;31:3403–11.
[17] Sobhani Aragh B, Nasrollah Barati A, Hedayati H. Eshelby–Mori–Tanakaapproach for vibrational behavior of continuously graded carbon nanotube-reinforced cylindrical panels. Compos Part B: Eng 2012;43:1943–54.
[18] Yas M, Heshmati M. Dynamic analysis of functionally graded nanocompositebeams reinforced by randomly oriented carbon nanotube under the action ofmoving load. Appl Math Modell 2012;36:1371–94.
[19] Heshmati M, Yas M. Vibrations of non-uniform functionally graded MWCNTS-polystyrene nanocomposite beams under action of moving load. Mater Des2013;46:206–18.
[20] Alibeigloo A, Liew K. Thermoelastic analysis of functionally graded carbonnanotube-reinforced composite plate using theory of elasticity. Compos Struct2013;106:873–81.
[21] Alibeigloo A, Emtehani A. Static and free vibration analyses of carbonnanotube-reinforced composite plate using differential quadrature method.Meccanica 2015;50:61–76.
[22] Zhang L, Song Z, Liew K. State-space levy method for vibration analysis of fg-cnt composite plates subjected to in-plane loads based on higher-order sheardeformation theory. Compos Struct 2015;134:989–1003.
[23] Wu C-P, Li H-Y. Three-dimensional free vibration analysis of functionallygraded carbon nanotube-reinforced composite plates with various boundaryconditions. Journal Vib Control 2014:362–70.
[24] Malekzadeh P, Zarei A. Free vibration of quadrilateral laminated plates withcarbon nanotube reinforced composite layers. Thin-Walled Struct2014;82:221–32.
[25] Zhang L, Li D, Liew K. An element-free computational framework forelastodynamic problems based on the IMLS-Ritz method. Eng Anal BoundElem 2015;54:39–46.
[26] Zhang L, Huang D, Liew K. An element-free IMLS-Ritz method for numericalsolution of three-dimensional wave equations. Comput Methods Appl MechEng 2015;297:116–39.
[27] Zhang L, Zhu P, Liew K. Thermal buckling of functionally graded plates using alocal kriging meshless method. Compos Struct 2014;108:472–92.
[28] Lei Z, Zhang L, Liew K, Yu J. Dynamic stability analysis of carbon nanotube-reinforced functionally graded cylindrical panels using the element-free kp-Ritz method. Compos Struct 2014;113:328–38.
[29] Lei Z, Zhang L, Liew K. Free vibration analysis of laminated FG-CNT reinforcedcomposite rectangular plates using the kp-Ritz method. Compos Struct2015;127:245–59.
[30] Zhang L, Lei Z, Liew K. Buckling analysis of FG-CNT reinforced composite thickskew plates using an element-free approach. Compos Part B: Eng2015;75:36–46.
[31] Zhang L, Liew K, Reddy J. Postbuckling of carbon nanotube reinforcedfunctionally graded plates with edges elastically restrained againsttranslation and rotation under axial compression. Comput Methods ApplMech Eng 2016;298:1–28.
[32] Zhang L, Lei Z, Liew K. An element-free IMLS-Ritz framework for bucklinganalysis of FG-CNT reinforced composite thick plates resting on winklerfoundations. Eng Anal Bound Elem 2015;58:7–17.
[33] Zhang L, Song Z, Liew K. Nonlinear bending analysis of FG-CNT reinforcedcomposite thick plates resting on Pasternak foundations using the element-free IMLS-Ritz method. Compos Struct 2015;128:165–75.
[34] Zhang L, Lei Z, Liew K. Free vibration analysis of functionally graded carbonnanotube-reinforced composite triangular plates using the FSDT and element-free IMLS-Ritz method. Compos Struct 2015;120:189–99.
[35] Zhang L, Lei Z, Liew K, Yu J. Static and dynamic of carbon nanotube reinforcedfunctionally graded cylindrical panels. Compos Struct 2014;111:205–12.
[36] Lei Z, Zhang L, Liew K. Vibration analysis of CNT-reinforced functionally gradedrotating cylindrical panels using the element-free kp-Ritz method. ComposPart B: Eng 2015;77:291–303.
[37] Shahrbabaki EA, Alibeigloo A. Three-dimensional free vibration of carbonnanotube-reinforced composite plates with various boundary conditions usingRitz method. Compos Struct 2014;111:362–70.
[38] Zhang L, Lei Z, Liew K. Computation of vibration solution for functionallygraded carbon nanotube-reinforced composite thick plates resting on elasticfoundations using the element-free IMLS-Ritz method. Appl Math Comput2015;256:488–504.
[39] Leissa A. Plate vibration research, 1976–1980: classical theory; 1981.[40] Leissa A. Recent studies in plate vibrations, 1981–1985, part i: classical theory;
1987.[41] Liew K, Xiang Y, Kitipornchai S. Research on thick plate vibration: a literature
survey. J Sound Vib 1995;180:163–76.[42] McGee O, Kim J, Kim Y. Corner stress singularity effects on the vibration of
rhombic plates with combinations of clamped and simply supported edges. JSound Vib 1996;193:555–80.
[43] Lei Z, Zhang L, Liew K. Buckling of FG-CNT reinforced composite thick skewplates resting on pasternak foundations based on an element-free approach.Appl Math Comput 2015;266:773–91.
[44] Zhang L, Liew K. Large deflection analysis of FG-CNT reinforced compositeskew plates resting on pasternak foundations using an element-free approach.Compos Struct 2015;132:974–83.
[45] Shi D-L, Feng X-Q, Huang YY, Hwang K-C, Gao H. The effect of nanotubewaviness and agglomeration on the elastic property of carbon nanotube-reinforced composites. J Eng Mater Technol 2004;126:250–7.
[46] Fidelus J, Wiesel E, Gojny F, Schulte K, Wagner H. Thermo-mechanicalproperties of randomly oriented carbon/epoxy nanocomposites. Compos PartA: Appl Sci Manuf 2005;36:1555–61.
[47] Wempner G, Talaslidis D. Mechanics of solids and shells. CRC Press; 2003.[48] Spencer A. The formulation of constitutive equation for anisotropic solids. In:
Mechanical behavior of anisotropic solids/Comportment Mchanique desSolides Anisotropes. Springer; 1982. p. 3–26.
[49] Lubarda V, Chen M. On the elastic moduli and compliances of transverselyisotropic and orthotropic materials. J Mech Mater Struct 2008;3:153–71.
[50] Efraim E, Eisenberger M. Exact vibration analysis of variable thickness thickannular isotropic and FGM plates. J Sound Vib 2007;299:720–38.
[51] Talaslidis D, Wempner G. The linear isoparametric triangular element: theoryand application. Comput Methods Appl Mech Eng 1993;103:375–97.
[52] Wempner G, Talaslidis A, Hwang C. A simple and efficient approximation ofshells via quadrilateral elements. J Appl Mech 1982;49:115–20.
[53] Wolfram S. Mathematica. Wolfram Media; 1999.[54] Liew K, Xiang Y, Kitipornchai S, Wang C. Vibration of thick skew plates based
on Mindlin shear deformation plate theory. J Sound Vib 1993;168:39–69.[55] Huang C, Leissa A, Chang M. Vibrations of skewed cantilevered triangular,
trapezoidal and parallelogram Mindlin plates with considering corner stresssingularities. Int J Numer Methods Eng 2005;62:1789–806.
[56] Han Y, Elliott J. Molecular dynamics simulations of the elastic properties ofpolymer/carbon nanotube composites. Comput Mater Sci 2007;39:315–23.