Static and free vibration analyses of functionally graded porous variable-thickness plates using an edge-based smoothed finite element method

Trung Thnh Trn ,Quoc-Ho Phm ,Trung Nguyen-Thoi

a Department of Mechanics,Le Quy Don Technical University,Hanoi,Viet Nam

b Division of Computational Mathematics and Engineering,Institute for Computational Science,Ton Duc Thang University,Ho Chi Minh City,Viet Nam

c Faculty of Civil Engineering,Ton Duc Thang University,Ho Chi Minh City,Viet Nam

Keywords:Functionally graded porous(FGP)plates Edge-based smoothed finite element method(ES-FEM)Mixed interpolation of tensorial components(MITC)Static bending Free vibration

ABSTRACT The main purpose of this paper is to present numerical results of static bending and free vibration of functionally graded porous(FGP)variable-thickness plates by using an edge-based smoothed finite element method(ES-FEM)associate with the mixed interpolation of tensorial components technique for the three-node triangular element(MITC3),so-called ES-MITC3.This ES-MITC3 element is performed to eliminate the shear locking problem and to enhance the accuracy of the existing MITC3 element.In the ES-MITC3 element,the stiffness matrices are obtained by using the strain smoothing technique over the smoothing domains formed by two adjacent MITC3 triangular elements sharing an edge.Materials of the plate are FGP with a power-law index(k)and maximum porosity distributions(Ω)in the forms of cosine functions.The influences of some geometric parameters,material properties on static bending,and natural frequency of the FGP variable-thickness plates are examined in detail.

1.Introduction

In recent years,FGP material has attracted great interest from researchers over the world due to porosity appeared in materials during the manufacturing process or intentionally created.Porosities inside materials can be distributed with many different types.They can be distributed uniform,non-uniform,or graded function.Basically,porosity reduces the stiffness of the structure,however with engineering properties such as lightweight,excellent energyabsorbing capability,great thermal resistant properties and so on,they still have been widely applied in various fields including aerospace,automotive industry,and civil engineering.A lot of analytical and numerical studies on the FGP structures have been performed and some of typical work can be summarized as follows.Kim et al.[1]investigated the static bending,free vibration,and buckling of FGP micro-plates using modified couples stress base on the analytical method(AM).Barati and Zenkour[2]analyzed the vibration of the FGP cylindrical shells reinforced by the graphene platelet(GPL)using first-order shear deformation theory(FSDT)and Galerkin’s method.Zenkour[3]developed a new Quasi-3D to calculate the bending of the FGP plates.Zenkour and Barati[4]considered electro-thermoelastic vibration of FGP plates integrated with piezoelectric layers by using AM and they also investigated post-buckling of FG beams reinforced by GPL with geometrical imperfection[5].Daikh and Zenkour analyzed the influence of porosity on the bending of FG sandwich plates[6]and calculated free vibration and buckling of FG sandwich plates in Ref.[7].Sobhy et al.[8]considered the effect of porosity distribution to buckling and free vibration of FG nanoplate using quasi-3D refined theory.Mashat and his co-workers[9]used a quasi 3-D higher-order deformation theory(HSDT)to analyze the bending of FGP plates resting on elastic foundations(EF)under hygro-thermo-mechanical loads.Nguyen et al.developed the polygonal finite element method(PFEM)combined with HSDT to calculate nonlinear static and dynamic responses of FGP plates[10],static bending and free vibration of FGP plates reinforced by GPL[11],and active-controlled vibration of FGP plate reinforced by GPL[12].In addition,Nguyen et al.[13]controlled of geometrically nonlinear responses of smart FGP plates reinforced GPL based on B′ezier extraction of the Non-Uniform Rational B-Spline(NURBS).Rezaei[14,15]based on AM to examine the free vibration of rectangular and porous-cellular plates.Zhao et al.[16]investigated the free vibration of FGP shallow shells using an improved Fourier method,and then analyzed the dynamics of the FGP doubly-curved panels and shells[17].Li et al.[18]analyzed nonlinear vibration and dynamic buckling of the sandwich FGP plate reinforced GPL on the elastic foundation(EF).With the nonlinear problem,Sahmani et al.[19]used the nonlocal method to analyze nonlinear large-amplitude vibrations of FGP micro/nano-plates with GPL reinforce.Wu et al.[20]considered the dynamic of FGP structures by using FEM.

Variable-thickness structures are extensively used in many types of high-performance surfaces like aircraft,civil engineering,and other engineering fields.Using these structures will help adjust the weight of structural,and hence help maximize the capacity of the material.Gagnon and Gosselin[21]studied the static bending of variable thickness homogeneous plates using the finite strip method(FSM).Sakiyama et al.[22]analyzed the free vibration homogeneous plates using the function approximation method(FAM).Singh and Saxena[23]considered a rectangular variablethickness plate using the Rayleigh-Ritz method based on basis functions satisfying essential boundary conditions(BC).Nerantzaki and Katsikades[24]used an analog equation solution to analyze the free vibration and dynamic behaviors of the variable-thickness plates.Mikami and Yoshimura[25]have applied the collocation method with orthogonal polynomials to calculate the natural frequencies for linear variable-thickness plates based on Reissner-Mindlin plate theory.Al-Kaabi and his colleagues have shown a method based on a variational principle in conjunction with finite difference technique for analysis of the Reissner-Mindlin plate of linearly[26]and parabolically[27]varying thickness.Based on the FSDT,Mizusawa et al.[28]have developed the spline strip method to study the natural frequencies for the tapered rectangular plates.All of these papers only investigate plates with two opposite simply supported(SS)edges perpendicular to the direction of thickness variation.Cheung et al.[29]calculated the free vibration of Reissner-Mindlin variable-thickness plates using the Rayleigh-Ritz method with different BC.Manh and Nguyen[30]combined FSDT with isogeometric analysis(IGA)to study the static bending and buckling of the composite variable-thickness plates.Lieu et al.[31]based on the IGA to compute static bending and free vibration of bi-directional FGM variable-thickness plates.Gupta et al.[32]examined the influence of crack location on the vibration of non-uniform thickness FGM micro-plate and in Ref.[33]Dhurvey simulated the buckling of the composite variable-thickness plates using ANSYS software.Thang et al.[34]investigated the effects of variable thickness on buckling and post-buckling of imperfect sigmoid FGM plates on elastic medium(EM)subjected to compressive loading.Thien et al.[35]developed the IGA to analyze buckling analysis of non-uniform thickness nanoplates in an EM.In addition,Zenkour[36]presented hygrothermal mechanical bending of variable-thickness plates using AM.In Ref.[37],Allam and his colleagues investigated thermoelastic stresses in FG variablethickness rotating annular disks using infinitesimal theory.

To improve the convergence and accuracy of the plate and shell structural analyses,the origin MITC3 element[38]is combined with the ES-FEM[39]to give the so-called ES-MITC3 element[40-45].In the formulation of the ES-MITC3,the system stiffness matrix is employed using strains smoothed over the smoothing domains associated with the edges of the triangular elements.The numerical results of the present study demonstrated that the ESMITC3 has the following superior properties[40]:(1)the ESMITC3 can avoid transverse shear locking phenomenon even with the ratio of the thickness to the length of the structures reach 10-8;(2)the ES-MITC3 has higher accuracy than the existing triangular elements such as MITC3[38],DSG3[46]and CS-DSG3[47];and is a good competitor with the quadrilateral element MITC4 element[48].

According to the best of authors’knowledge,static bending and free vibration analyses of FGP variable-thickness plates using ESMITC3 have not yet been published,especially with the variable thickness of FGP plates in both directions with any BC.Therefore,this paper aims to fill in this gap by developing the ES-MITC3 method for static bending and free vibration analyses of the FGP variable-thickness plates.The formulation is based on the FSDT due to its simplicity and computational efficiency.The accuracy and reliability of the present approach are verified by comparing with those of other available numerical results.Moreover,the numerical and graphical results illustrate the effect of maximum porosity value,power-law index,and the law of variable thickness on the static bending and free vibration of FGP plates.

2.The FGP variable-thickness plates

Consider an FGP variable-thickness plate as depicted in Fig.1 which has laws of variable thickness are shown in Fig.2.Type 1:the plate has a constant thickness as shown in Fig.2(a);Type 2:the thickness varies linearly in thex-axis as shown in Fig.2(b);Type 3:the thickness varies linearly in thex-axis andy-axis as shown in Fig.2(c).

The FGP with the variation of two constituents and three different distributions of porosity through thickness are presented as[1]:

in whichΩis the maximum porosity value.The typical material properties of FGP in the thickness direction of the plate are given by the rule:

Fig.1.The FGP plate model with varying thickness.

Fig.2.The laws of variable thickness of FGP plates.

wherePtandPbare the typical material properties at the top and the bottom surfaces,respectively andkis the power-law index.Fig.3(a)shows the normalized distribution of porosity throughthickness.Fig.3(b),(c),(d)show the normalized distributions of three different cases of porosity in whichΩ=0.5,k=0.5,1,10,andPt/Pb=10.The porosity distribution of Case 1 is symmetric with respect to the mid-plane of plates and a center enhanced distribution.Case 2 and Case 3 are bottom and top surface-enhanced distributions,respectively.From Fig.3(a-c),we can see that all cases of distribution porosity increase the hardness of the upper surface of the plate,however,case 3 increases the stiffness of the plate through thickness as not strongly as the other two cases.

3.The weak form and FEM formulation for FGP plates

3.1.The first order shear deformation theory for FGP plates

The displacement fields of FGP plate in present work based on FSDT model can be expressed as[49]:

in whichu0,v0,w0denotes displacement variables of mid-surface of the plate(z=0)andθx,θyare the rotations of a transverse normal about they-axis andx-axis,respectively.

For the bending plate,the strain components can be expressed as follows:

with the membrane strain

and the bending strain

and the transverse shear strain is expressed by:

From Hooke’s law,the linear stress-strain relations of the FGP plates can be expressed as:

Fig.3.Distribution of porosity and typical material property.

where

in whichE(z),υ(z)are the effective Young’s modulus,Poisson’s ratio which is calculated by Eq.(2),respectively.

3.2.Weak form equations

To obtain the governing equations of the FGP plates,the Hamilton’s principle is applied in the following form[49]:

where U,V and K are the strain energy,the work done by external loads and the kinetic energy of plate,respectively.

The strain energy is expressed as:

in whichε=[εmκ]Tand

with A,B,D,and Dscan be given by

The work done by external transverse loads are expressed by

The kinetic energy is given by

As this plate structure has variation thickness,all matrices in Eq.(13),Eq.(14)and Eq.(17)depend on the law of varying thickness.The limits of integration depend on the position of the point on the plate.

Substituting Eqs.(11),(15)and(16)into Eq.(10),the weak formulation for static and free vibration of FGP plates,respectively,is finally obtained as

4.Formulation of an ES-MITC3 method for FGP plates

4.1.Formulation of finite element based on the MITC3

The bounded domainΨis discretized intoneMITC3 elements withnnnodes such thatThen the generalized displacements at any pointfor elements of the FGP plate can be approximated as:

wherenneis the number of nodes of the plate element;N(x)andis the shape function and the nodal degrees of freedom(DOF)of ueassociated with thejth node of the element,respectively.

The linear membrane and the bending strains of the MICT3 element can be expressed in matrix forms as follows

where

To eliminate the shear locking phenomenon as the thickness of the plate becomes very small,the MITC3 element based on FSDT is proposed by Lee et al.[38].In their study,the transverse shear strains of the classical triangular element are independent interpolated by computing at the middle of triangular element edges,named typing points.The transverse shear strain field associated to typing points with 5 DOFs per node can be written as:

in which

Fig.4.Three-node triangular element in the local coordinates.

with

where

in whicha=x2-x1,b=y2-y1,c=y3-y1andd=x3-x1are pointed out in andAeis the area of the three-node triangular element shown in Fig.4.

Substituting the discrete displacement field into Eqs.(18)and(19),we obtained the discrete system equations for static and free vibration analysis of the FGP plate using MITC3 based on FSDT formulation,respectively as

where K is the stiffness matrix of FGP plate and F represents the load vector.

with

whereωis the natural frequency and M is the mass matrix

4.2.Formulation of an ES-MITC3 method for FGP plates

In the ES-FEM,a domainΨis divided intonksmoothing domainsΨkbased on edges of elements,such asandfori≠j.An edge-based smoothing domainΨkassociated with the inner edgekis formed by connecting two endnodes of the edge to centroids of adjacent MITC3 elements as shown in Fig.5.

Now,by applying the ES-FEM[39],the smoothed strain~εk,a smoothed shear strain~γkover the smoothing domainΨkcan be created by computing the integration of the compatible strains,the strainεand the shear strainγ，respectively,in Eqs.(18)and(19)such as:

Fig.5.The smoothing domainΨk is formed by triangular elements.

whereΦk(x)is a given smoothing function that satisfies at least unity property∫

ΨkΦk(x)dΨ=1.In this study,we use the constant smoothing function.

in whichAkis the area of the smoothing domainΨkand is given by

wherenekis the number of the adjacent MITC3 elements in the smoothing domainΨk;andAiis the area of theith MITC3 element attached to the edgek.

By substituting Eqs.(21),(22)and(29)into Eqs.(44)and(45)then,the approximation of the smoothed strains on the smoothing domainΨkcan be expressed by:

The global stiffness matrix of FGP plate using the ES-MITC3 is assembled by

where~Kkis the ES-MITC3 stiffness matrix of the smoothing domain Ψkand given by

in which

The advantage of the present ES-MITC3 as follows:

Using three-node triangular elements that are much easily generated automatically even for complicated geometry domains.

The ES-MITC3 element only uses the 3-node element with the same number of DOF as the MITC3 element,not increasing DOF to improve accuracy.

5.Accuracy of the proposed method

Firstly,we consider simple support(SSSS)and fully clamped(CCCC)FGP plates with three cases of distributions of porosity with parameter geometrya=b=1 m,the thickness of the plate is constanth=a/50.Material properties are shown in Table 4 with power-law indexk=1 and maximum porosity valueΩ=0.5.The maximum displacement of plate with different BC are presented in Table 1 and Fig.6.The natural frequencies of the plate are shown in Table 2 and Fig.7.

Secondly,the authors investigate a rectangular isotropic plate that has linear thickness variations in thex-directions andy-directions,lengtha=0.5 m,b=2a,h0=0.3 m,h1=h2=0.2 m,Young’s moduliE=2 GPa.The plate is one short-edge clamped,subjected to a uniform loadP=2000 Pa.Fig.8 shows the

Table 1Maximum displacements of the plate with different meshes.

Fig.6.The convergence of displacement of the plate with different meshes.a)The SSSS plate;b)The CCCC plate.

Table 2Natural frequencies of the plate with different meshes.

6.Numerical results and discussions

In this section,we use the material properties of the FGM plate from the study of Kim et al.[1].The moduli and mass densities of two constituents are shown in Table 4.

6.1.Static bending of the FGP plates

In this subsection,we consider an SSSS FGP plate with dimensiona=b=1 m,h0=a/50,the thickness is varied linearly inxaxis andy-axis.The material properties of the FGP plate is shown in Table 2 with porosity distribution of case 1 in whichΩ=0.5 and power-indexk=1.The plate subjected to uniform load with intensityp=-1 alongz-axis.From Fig.9(a)for plates with constant thicknessh0=h1=h2=a/50,it can be seen that the deformation field is symmetric,and the maximum of deflection is at the center of the plate.However,the deflection of the plate with variable thickness is not symmetric,the maximum of deflection will be near the thinner thickness and far away from thicker thickness as shown in Fig.9(b and c).

Fig.7.The convergence of natural frequency of the plate with different meshes.a)The SSSS plate;b)the CCCC plate.

Fig.8.Displacement at some points of the plate.

Table 3Nondimensional natural frequenciesω*.

Table 4Material properties of the FGP plate.

Fig.9.The visualization of the deformation of an SSSS FGP plate(top view).

Fig.10.Displacement at some points of the plate and stress at the point of plate that has maximum deflection along with z-axis.

Table 5Maximum of deflection of the FGP plate.

In Fig.10(a),the deflection of some points alongx-aix withy=0.5 m(type 1,type 2)andy=7/12(type 3)are presented.Fig.10(b-d)show the stress at the point of plate that has maximum deflection.Maximum of deflection and stress are shown in Table 5.This phenomenon shows that the variation of plate thickness significant effects on the static bending response of the FGP plate.

In order to investigate the effect of power-indexkto static bending of the FGP plate.We change thekfrom 0 to 10 for all cases of porosity distribution.From Fig.11 and Table 6,it can be observed that with the sameΩvalue,the maximum deflection of FGP plate in porosity distribution of case 3 is greatest and the smallest result is for the porosity distribution of case 1.Whenkincreases,the volume of metal increases,the stiffness of the FGP plate will be reduced and hence the deflection increases.

Next,the authors investigate the effect of maximum porosityvalueΩon bending of the FGP plate.Maximum porosity values are chosen asΩ=0,0.2,0.4,0.6,0.8,1 for all cases of porosity distribution.From to Fig.12 and Table 7,whenk=0(the material of the plate is ceramic),it can be seen that the deflection of FGP plate in porosity distribution of case 1 is the smallest,and the deflection of FGP plate with porosity distribution of case 2 and case 3 are the same.These results are appropriate because,with the homogeneous FGP plate,the porosity distributions of cases 2 and 3 are symmetric,and the values of stiffness obtained are not different.Withknot equal to zero,the deflection of FGP plate in porosity distribution of case 1 is the smallest,and the largest is for the porosity distribution of case 3.This also proves that when the porosity is more distributed on the upper surface of the FGP plate according to thickness,stiffness of the plate is reduced(case 3).

Table 6The maximum deflection of FGP plate(type 3)with different values of k(μm).

Fig.11.The maximum deflection of FGP plate(type 3)with different of values of k.

Fig.12.The maximum deflection of FGP plate(type 3)with different values of k.

Table 7The maximum deflection of FGP plate(type 3)with different values ofΩ(μm).

Table 8Natural frequencies of the FGP plate.

6.2.Free vibration of FGP plates

In this subsection,an SSSS FGP plate is considered with geometric parameters:a=b=1 m,h0=a/50,the thickness is varied linearly inx-axis andy-axis(type 3).The material properties of the FGP plate are similar to the static bending problem with porosity distributionΩ=0.5 and power-indexk=1.The first six natural frequencies of the FGP plate are shown in Table 8 and the first six mode shape are presented in Fig.13.In these figures,the mode shape of vibration of the variable-thickness FGP plate is not symmetric because the thickness at each position on the plate is different.The maximum values of the mode shape are traveled toward a smaller thickness.

Fig.13.The first six mode shapes of the FGP plate(type 3).

Fig.14.The first natural frequencies of FGP plate with different values of k.

Table 9The first natural frequencies of the FGP plate with different values of k(Hz).

Now,we investigate the influence of power-law indexkon free vibration of the FGP plate.Power-law index gets valuesk=0,2,4,6,8,10 for all cases of porosity distribution.In Fig.14 and Table 9,it can be seen that with the sameΩvalue,the first natural frequency of the FGP plate in porosity distribution of case 1 is greatest,and the smallest is for the porosity distribution of case 3.WithΩ=0.25,0.5,0.75 whenkincreases from 0 to 1,the first natural frequencies of the FGP plate will be reduced,butkincreases from 1 to 10 when the first natural frequencies of the FGP plate will increase.However,withΩ=1 the first natural frequencies of the FGP plate with porosity distribution of case 1 and case 2 increase whenkchanges from 2 to 10.In case 3 the first natural frequencies of the FGP plate increase from 4 to 10.

Finally,the effect of maximum porosity distribution on the free vibration of the FGP plate(type 3)is considered.The value of maximum porosityΩchanges from 0 to 1 for all cases of porosity distribution.From Fig.15 to Table 10,in can be found that whenk=0(the material of plate is ceramic),the first natural frequencies of FGP plate in porosity distribution of case 1 increase,the first natural frequencies in case 2 and case 3 are similar.Withkvaries from zero,the first natural frequency of FGP plate in porosity distribution of case 1 is the largest,and the smallest is for the porosity distribution of case 3.

7.Conclusions

In this paper,the static and free vibration analyses of the FGP variable-thickness plates are studied using the ES-MITC3.Numerical results of static bending and free vibration obtained by the present approach are compared to other available solutions.From the present formulation and the numerical results,we can withdraw some following points:

-For static and free vibration analyses of the FGP variablethickness plates,the ES-MITC3 element which can eliminate“the shear locking”phenomenon will give the more accurate results than the standard triangular elements and the original MITC3 element.

-The law of variable thickness significant effects on displacement,stress,and free vibration of the plates.The material parameters also change the stiffness and mass of the plates.Specifically,with the same geometry and BC,when the power-indexkincreases,the“stiffness”of the plate will decrease and when the maximum porosity distributionsΩincrease,the“stiffness”of plate will decrease,respectively.

Fig.15.Natural frequencies of the FGP plate with different values ofΩ.

Table 10The natural frequencies of the FGP plate with different values ofΩ(Hz).

-Numerical results in present work are useful for calculation,design,and testing of material parameters in engineering and technologies.

-The present approach can be developed for investigating the plates with laws of variable thickness and also for analysing the FGP plates subjected to other loads.

Data availability

The data used to support the findings of this study are included within the article.

Declaration of competing interest

The authors declare that they have no conflicts of interest.

AcknowledgmentsThis research is funded by Vietnam National Foundation for Science and Technology Development(NAFOSTED)under Grant number 107.02-2019.330.