Pham Van Vinh ,Le Quang Huy

a Department of Solid Mechanics,Le Quy Don Technical University,236 Hoang Quoc Viet,Hanoi,Viet Nam

b Institute of Techniques for Special Engineering,Le Quy Don Technical University,236 Hoang Quoc Viet,Hanoi,Viet Nam

Keywords:Functionally graded sandwich plates Porous plates Hyperbolic shear deformation theory Bending analysis Free vibration analysis Buckling analysis

ABSTRACT This study focusses on establishing the finite element model based on a new hyperbolic sheareformation theory to investigate the static bending,free vibration,and buckling of the functionally graded sandwich plates with porosity.The novel sandwich plate consists of one homogenous ceramic core and two different functionally graded face sheets which can be widely applied in many fields of engineering and defence technology.The discrete governing equations of motion are carried out via Hamilton’s principle and finite element method.The computation program is coded in MATLAB software and used to study the mechanical behavior of the functionally graded sandwich plate with porosity.The present finite element algorithm can be employed to study the plates with arbitrary shape and boundary conditions.The obtained results are compared with available results in the literature to con firm the reliability of the present algorithm.Also,a comprehensive investigation of the effects of several parameters on the bending,free vibration,and buckling response of functionally graded sandwich plates is presented.The numerical results shows that the distribution of porosity plays signi ficant role on the mechanical behavior of the functionally graded sandwich plates.

1.Introduction

In many fields of engineering and industry,traditional materials such as wood and metal are widely used for a long time ago.However,the mechanical properties of these materials do not meet the special requirements in many fields such as aerospace engineering,submarine engineering,defence engineering and nuclear power plant.In 1984,a group of material scientists in Japan proposed the functionally graded materials(FGMs)which are mixtures of two or more individual ingredients with a smooth and continuous varying of volume fractions and mechanical properties through the thickness of the plates and beams[1].After that,the application of these materials is increased quickly.Therefore,a lot of scientists paid their attention to investigate the mechanical and thermal behaviors of these structures[2-6].

On the other hand,FGMs have been applied to multi-layered structures such as laminated or sandwich structures because of the gradual variation of the material properties at the interfaces between the face layers.These structures are usually used in hightemperature environments[7],so it is necessary to have an excellent understanding of the static and dynamic response of these structures.Nguyen et al.[8]applied first-order shear deformation theory(FSDT)to analyze the vibration and buckling behavior of functionally graded sandwich plates(FGSPs),in which a new improvement of the transverse shear stiffness has been employed to improve the accuracy and ef ficiency of FSDT.Nguyen et al.[9]developed a new re fined simple FSDT for static bending and free vibration analysis of advanced composite plates.Thai et al.[10]analyzed the mechanical behavior of FGSP via a new FSDT,where the transverse displacement was divided into bending and shear parts.Also,it has been applied to analyze the static bending behavior of FGSP by Mantari et al.[11].However,the shear stress of the FSDT does not equal to zeros at the surface of the plates,so it needs a shear correction factor which depends on the material,geometry as well as boundary conditions(BCs),so it is dif ficult to predict the exact value of the shear correction factor.It has prompted scientists to develop new theories that are more suitable to analyze beams,plates and shells.Zenkour[12,13]developed third-order shear deformation theory(TSDT)and sinusoidal shear deformation theory(SSDT)to investigate the de flections,stresses,free vibration and buckling behavior of FGSPs.Tounsi and his coworkers[14-20]developed many simple and ef ficient HSDTs with non-polynomial shape functions to study the static and dynamic response of FGSPs.Vinh et al.[21,22]modi fied single variable shear deformation theory for static bending and free vibration analysis of FGM plates and FGM nanoplates.The thermomechanical bending of FGSP has been investigated by Li et al.[23]using a four-variable re fined plate theory.In the work[24],Nguyen et al.developed a new HSDT with inverse trigonometric shape function to research the bending,free vibration and buckling of FGSP.Daikh[25,26]used HSDT with fifth-order polynomial shape function to investigate the effects of porosity on the bending,free vibration and buckling behavior of power-law and sigmoid FGSPs.Daikh et al.[27]used a hyperbolic shear deformation theory to analyze the static bending of multilayer nonlocal strain gradient nanobeams reenforced by carbon nanotubes.Sobhy[28]developed a four-variable shear deformation theory for hygro-thermal buckling of porous FGM sandwich microplates and microbeams.Taj et al.[29]analyzed the FG skew sandwich plates using HSDT in combination with finite element method(FEM).Xuan et al.[30]used isogeometric finite element analysis(IGA)based HSDT to analyze composite sandwich plates.Although the HSDT satis fies the stress-free conditions at two surfaces of the plates and does not need any shear correction factors,these theories neglect the effects of the thickness stretching on the behaviors of FGSPs,which are very important in the cases of thick plates.

To take into account the thickness stretching effects on the thick plates,various quasi-3D theories have been developed.Daikh et al.[31]established a quasi-3D theory in combination with nonlocal strain gradient for bending analysis of sigmoid FG sandwich nanoplates.Neves et al.[32,33]developed quasi-3D theories to investigate the static and dynamic response of the FGSP using the meshless method and radial basis functions method.Sobhy et al.[34]established a new quasi-3D theory to analyze free vibration and buckling behavior of the FGM nanoplates.Akavci[35]developed a new HSDT and quasi-3D theory to study the behavior of FGSP resting on elastic foundations.Bessaim et al.[36]established a new HSDT and normal(quasi-3D)deformation theory to research the bending and free vibration of FGSP with isotropic face sheets.The bending analysis of FGSP had been investigated by Zenkour[37]via a simple four-unknown shear and normal deformation(quasi-3D)theory.Furthermore,the FSDT and HSDT have been modi fied to Zig-Zag theory to study FGSP by Iurlaro et al.[38],Neves et al.[39],Dorduncu[40]and Garg et al.[41]to analyze the static and dynamic behavior of FGSPs.Liu et al.[42]used IGA in cooperation with higher-order layer-wise theory to analyze laminated composite and FG sandwich plates.Pandey et al.[43]used the layer-wise theory to analyze the free vibration of FGSP in the thermal environment.Burlayenko et al.[44]used threedimensional finite elements to investigate the static bending and free vibration behavior of the FGSP with the material properties are calculated via Mori-Tanaka homogenization method.

Fig.1.The geometry and structure of the functionally graded sandwich plate with porosity.

The use of FGSPs in the fact shows that these structures usually contact to different loads and environments such as static loads,dynamic loads,blast loads and high-temperature environments[45].On the other hand,porosity is usually appeared in materials during the fabrication process or intentionally created.By including porosity,the stiffness of the structures is reduced,but it also reduces the mass of the structures.Besides,the optimization of the material distribution,as well as the porous distribution through the thickness of these structures,can improve the strength of the structures or avoid the stress concentration phenomenon at the surfaces.So,the sandwich structures can be made of many different types of FGM layers to maintain these features.Hence,FGSP with porosity has been widely applied in many fields of engineering including defence technology.For example,the FGSP with porosity can be used to make the tank armor that can withstand nuclear explosions.The cover of military aircraft or special military equipment can be made from FGSP with porosity to reduce their weight.Besides,the outer skin and fuel tanks of missiles are made of special FGSP with porosity to reduce the total weight and increase heat resistance.A lot of scientists have been focused on the investigation of the static and dynamic response of the isotropic and sandwich FG plates with porosity.Shahsavari et al.[46]developed a new quasi-3D hyperbolic theory for free vibration of FG plates with porosities resting on elastic foundations.Zenkour[47]analyzed the mechanical behavior of FG single-layered and sandwich plates with porosities.Barati et al.[48,49]analyzed vibration and post-buckling of porous graphene platelet reinforced beams and cylindrical shells with different porosity distributions.Sobhy et al.[50]considered the effects of porosity on the buckling and vibration of double-FGM nanoplates via a quasi-3Dre fined theory.Zenkour et al.[51]studied the effects of porosity on the thermal buckling behavior of actuated FG piezoelectric nanoplates.The displacement and stresses of FG porous plates are investigated by Zenkour[52]via a quasi-3D re fined theory.Mashat et al.[53]developed a new quasi-3D higher-order plate theory for bending analysis of porous FG plates resting on elastic foundations under hygro-thermomechanical loads.

In this study,a novel functionally graded sandwich plate(FGSP)with three types of porous distribution is introduced and investigated with static bending,free vibration and buckling problems.The outlines of the paper are as follows:the basic formulation of the problem is given in section 2,including the construction of FGSP with three types of porosity,the formulation of the new hyperbolic shear deformation theory and finite element formulations.Section 3 gives the convergency and veri fication study as well as the benchmark numerical results of the static bending,free vibration and buckling behavior of the FGSP with porosity with many useful discussions in each subsection.Section 4 gives some important summaries and good ideas for future works on the investigation of these structures.

2.Problem formulation

2.1.Functionally graded sandwich plates with porosity

The mechanical behaviors of a novel sandwich plate with porosity are investigated in this study.The sandwich plate consists of one homogenous ceramic core and two different functionally graded face sheets.The dimension of the sandwich plates is a in x-direction,b in y-direction and the total thickness is h as shown in Fig.1.A group of three numbers as“i-j-k”is used to denote the ratio of the thicknesses of the bottom-core-top layers.It means the thickness of the bottom layer is h.i/（i+j+k）,that of core layer is h.j/（i+j+k）and that of top layer is h.k/（i+j+k）.

2.1.1.The FGSP model with even porous face sheets(porosity I)

The variation of the effective material properties through the thickness of the FGSP with porosity I are obtained by the following formulae

where P，Pand Pare the material properties of the materials at bottom surface,top surface and core layer of the sandwich plates,andξis the coef ficient of porosity（ξ?1）.

2.1.2.The FGSP model with linear-uneven porous face sheets(porosity II)

For the FGSP with one perfect core and two linear-uneven porous face sheets,the effective material properties are calculated by the following formula

2.1.3.The FGSP model with linear-uneven porous core(porosity III)

The effective material properties of FGSP with one linearuneven porous core and two perfect face sheets are described by the following formula

The perfect FGSP are obtained easily by setting the porous coef ficientξ=0 in Eqs.(1)-(3).

2.2.Finite element formulation

2.2.1.Displacement field and strains

The higher-order shear deformation theory is adopted in this study to describe the displacement of the sandwich plate as follows

where u，v，w，β，β，θ，θare seven unknown displacement functions at the middle surface of the plate and f（z）,g（z）are the shape function.Numerous shape functions have been introduced in the literature.In this study,the novel hyperbolic shape functions of f（z）and g（z）are given as follows

The strain fields of the plate are

The lower comma is used to denote the derivation respect to the following variable.It can be seen that f=0 and g≠0 at z=±h/2,soγ（±h/2）has to equal to zeros to satisfy the shear stress free conditions at the top and bottom surfaces of the plate.This condition of the shear strain will be treated at element level in finite element formulation later.As a consequence,the shear strain vector can be obtained viaγas follows

2.2.2.Constitutive relations

The linear constitutive relations of the plates are

where E（z），νare Young’s modulus and Poisson’s ratio,respectively.It is noticed that the Poisson’s ratio is assumed constant and equal to each material.

In this study,Hamilton’s principle is adopted to obtain the equation of motion of the plates

whereδΠ，δW，δV，δT are respectively the variations of strain energy,work done by external force,work done by in-plane compressive load and kinetic energy of the plate.

2.2.3.Strain energy

The expression of the variation of the strain energy is obtained as follows

After rearranging into matrix form and integrating through the thickness of the plate,the variation of strain energy of the plate can be calculated as follows

2.2.4.Work done by external force

The variation of work done by external transverse load is obtained by

2.2.5.Work done by in-plane compressive loads

The variation of work done by in-plane compressive loads is calculated by

By including the displacement field Eq.(4)into Eq.(23),the variation of work done by in-plane compressive loads is obtained as

where.?=｛?/?x，?/?y｝.,q=｛u，v，w｝.The matrix Sare given by

2.2.6.Kinetic energy

The variation of the kinetic energy of the plate is obtained as

2.2.7.Finite element formulations

A four-node quadrilateral plate element with seven degrees of freedom is employed to investigate the FGSP with porosity.The nodal displacement vector of the i-th node is

The coordinates and displacement variables at any points of the element are approximated via the shape functions as following formulae

where Nare the linear shape functions and Nis a quadratic shape function which are given by

Eq.(39)is inserted into Eq.(32),it leads to

2.2.7.1Introducing Eq.(41)into expression ofδε,one gets

The variation of the axial strain vector can be rewritten in short form as

The variation of the shear strain vector can be expressed in the matrix form as

Inserting Eqs.48-52 into(15),and using the trivial manner of classical FEM,one gets the finite element equations of static bending,free vibration and buckling problem of the plates.

For the bending problem

For the free vibration problem

In which,K，M，K，f，U are respectively the global stiffness matrix,the global mass matrix,the global geometric stiffness matrix,the global nodal force vector and global displacement vector of the plate.These matrices and vectors are assembled by the element stiffness matrix K，the element mass matrix M，the element geometric stiffness matrix Kand the element nodal force vector f.They are computed by the following formulae

3.Numerical results and discussions

3.1.Convergency and veri fication study

To verify the convergent rate and the accuracy of the present algorithm,some comparisons between the results of the present procedure with published data will be considered in this subsection.

Firstly,a square FGM sandwich plates with one homogenous ceramic core of ZrOand two FGM face sheets of Al/ZrOis examined.Young’s modulus of ZrOand Al are E=151 GPaand E=70 GPa,respectively;while the Poisson’s ratio is constant ν=0.3,and the side-to-thickness ratio of the plate is a/h=10.The plate is simply supported at all edges and subjected to sinusoidal load.The non-dimensional center de flections,axial and shear stresses are computed as the following formulae（E=1 GPa）

Table 1 presents the comparison between the present numerical results and those of Zenkour[12]of an FG sandwich plate without porosity.It can see clearly that the numerical results converge at the mesh of 32×32.The results in Tables 2 and 3 are calculated using the mesh of 32×32.According to Tables 1-3,the present results at the mesh of 32×32 are in good agreement with the results of Zenkour[12]using TSDT and SSDT.

Table 1 The convergency and comparison of the non-dimensional center de flection of FG sandwich plates.

Table 2 The comparison of the non-dimensional axial stress of FG sandwich plates.

Table 3 The comparison of the non-dimensional shear stress of FG sandwich plates.

Table 4 The convergency and comparison of the non-dimensional center de flection of FGSP with porosity.

Table 5 The comparison of the non-dimensional axial stress of FGSP with porosity.

Table 6 The comparison of the non-dimensional shear stress of FGSP with porosity.

Next,the authors examine the static bending of a square FGSP of Al/ZrOwith different values of porosity coef ficients.The geometry and materials properties are similar to those of previous comparison,the volume fraction index is p=2.It is noticed that this sandwich plate is achieved easily from FGSP of porosity I by setting two metal ingredients at the bottom and top layer with similar material properties.The present results are compared with those of Daikh and Zenkour[25]using an analytical solution.The comparison the non-dimensional center de flections,axial stress and transverse shear stress are presented in Tables 4-6.It is obvious that the numerical results converge at the mesh of 32×32,and close to the results of Daikh and Zenkour[25]using the analytical solution.

Table 7 The convergency and comparison of the non-dimensional frequency of FG sandwich plates.

Secondly,the non-dimensional frequency and critical buckling load of a square FG sandwich of Al/AlOusing the present algorithm are compared to those of Zenkour[13]using TSDT and SSDT.In this examination,the square FG sandwich plate is made of one homogenous ceramic core of AlO,two similar FGMface sheets of Al/AlO,and the side-to-thickness ratio of a/h=10.The material properties of AlOare E=380 GPa,ρ=3800 kg/m,ν=0.3,and those of Al are E=70 GPa,ρ=2707 kg/m,ν=0.3.The following non-dimensional parameters are used

The numerical results of the present algorithm and those of Zenkour[13]are exhibited in Tables 7 and 8.It is obvious that the frequency and critical buckling load of the sandwich plate converge at the mesh of 32×32 and those are very closed to the results of Zenkour[13].

Continuously,the non-dimensional frequency and critical buckling load of FGsandwich plates of Al/AlOwith porosity using the present algorithm are compared to those of Daikh and Zenkour[26]using an analytical method.In this examination,the square FGSP is made of one homogenous ceramic core of AlO,two similar FGM face sheets of Al/AlO.The geometry and materials properties are similar to those of previous comparison,the volume fraction index is p=2.The numerical results of the present algorithm and those of Daikh and Zenkour[26]are performed in Tables 9 and 10.It is obvious that the frequency and critical buckling load of the sandwich plate converge at the mesh of 32×32 and those are very closed to the results of Daikh and Zenkour[26].

Table 8 The comparison of the non-dimensional critical buckling load of FG sandwich plates.

Table 9 The convergency and comparison of the non-dimensional frequency of FGSP with porosity.

Table 10 The comparison of the non-dimensional critical buckling load of FGSP with porosity.

Table 11 The non-dimensional center de flections of FGSP with porosity.

Table 12 The non-dimensional axial stress of FGSP with porosity.

Table 13 The non-dimensional shear stress of FGSP with porosity.

Table 14 The non-dimensional fundamental frequency of FGSP with porosity(SSSS).

According to several comparison studies,the numerical results of the present algorithm converge at the mesh of 32×32,and are in good agreement with published results.Hence,in the rest of the paper,the mesh of 32×32 is used to investigate the mechanical behavior of the FGSP with porosity.

3.2.Parameter study and discussions

In the recent work,the FGSP contains one homogenous ceramic core of AlO,one bottom face sheet of Al/AlOand one top face sheet of SUS304/AlO.The material properties of AlOare E=380 GPa,ρ=3800 kg/m,ν=0.3,those of Al are E=70 GPa,ρ=2707 kg/m,ν=0.3,and those of SUS304 are E=207 GPa,ρ=8166 kg/m,ν=0.3.The effective Young’s modulus and mass density through the thickness of the(1-1-1)perfect FGSP are demonstrated in Fig.2.Figs.3-5 presents the effective Young’s modulus and mass density through the thickness of the(1-1-1)FGSP with porosity.In two cases of porosity I and porosity II,the porosity reduces the effective Young’s modulus and mass density of the materials at two face sheets,while the porosity III reduces the effective Young’s modulus and mass density of the material at the ceramic core of the sandwich plates.

Fig.2.The effective Young’s modulus and mass density of perfect FGSP.

Fig.3.The effective Young’s modulus and mass density of FGSP with porosity I（ξ=0.2）.

Four types of boundary conditions of the plate are considered,which are fully clamped at all edges(CCCC),fully simply supported at all edges(SSSS),clamped at two opposite edges and simply supported at two opposite edges(SCSC),and clamped at two continuous edges and simply supported at next two edges(SSCC).The following formulations are used to estimate the nondimensional de flections,stresses,natural frequencies and critical buckling loads of the FGSP with porosity

Fig.4.The effective Young’s modulus and mass density of FGSP with porosity II（ξ=0.2）.

Fig.5.The effective Young’s modulus and mass density of FGSP with porosity III（ξ=0.2）.

Fig.6.The distribution of the axial stress through the thickness of FGSP with porosity(SSSS).

Fig.7.The distribution of the shear stress through the thickness of FGSP with porosity(SSSS).

Fig.8.The effects of several parameters on the center de flection of the(1-1-1)FGSP with porosity.

3.2.1.Static bending analysis of FGSP with porosity

For the static bending analysis,the FGSP is subjected to a sinusoidal distribution load with the maximum value of q=1.The non-dimensional center de flections,axial stress and transverse shear stress of the fully simple supported at all edges FGSP with b/a=1，a/h=10 are presented in Tables 11-13 for several values of power-law index and coef ficient of porosity.It can be seen that the de flections and stresses of the plates do not depend on the schemes of the sandwich plates whenξ=0，p=0.In the general,the inclusion of the porosity effects leads to the rise of the deflections and stresses of the plates.The porosity III does not have effects on the behavior of(1-0-1)FGSP.The reason is that the(1-0-1)FGSP does not consist of the core layer.

Figs.6 and 7 demonstrate the distribution of the axial and shear stresses through the thickness of the FGSP with different scheme and values of coef ficient of porosity.It can see clearly that,although the scheme and distribution of porosity of the sandwich plates are symmetric,the distribution of the axial and shear stresses through the thickness of the FGSP are still asymmetric.It is due to the fact that the ingredients of two face sheets of the FGSP are different.These figures show that the scheme of the sandwich plate and the porosity affects strongly on the distribution of the axial and shear stresses.Figs.6(b)and 7(b)show that the maximum values of the axial and shear stresses of the FGSP with porosity I are highest in comparison with other ones.

Fig.8 demonstrates the effects of some parameters on the center de flections of the(1-1-1)FGSP with porosity.From Fig.8(a),it is obvious that when the aspect ratio b/a increases,the center deflections of the plates increase.The increase rate of SSSS plates is greatest while the growing speed of CCCC ones is the smallest.It also sees that the center de flections of the CCCC plates and SCSC ones are similar when the aspect ratio greater than 2.The in fluence of the side-to-thickness ratio on the center de flections of the FGSP with porosity is exhibited in Fig.8(b).The center de flections of the plates increase as the increase of the side-to-thickness ratio.Once again,the growing speed of the SSSS plates is greatest while the speed of the growing of CCCC ones is smallest.The in fluence of the power-law index p on the de flections of the FGSP is demonstrated in Fig.8(c)while the effects of the porosity coef ficientξon the center displacement of the FGSP are shown in Fig.8(d).The deflections of the plates increase as the increase of the power-law index p and porosity coef ficientξ.From these two demonstrations,it is obvious that the effects of porosity III are much weaker than porosity I and porosity II,and the porosity I have signi ficant effects on the behavior of the FGSP.In the case of porosity I,the center displacement of the(1-1-1)FGSP withξ=0.2 is approximately 1.3 times those without porosity.

3.2.2.Free vibration analysis of FGSP with porosity

Continuously,this subsection focusses on the analysis of free vibration of the square FGSP with porosity,and the side-tothickness of a/h=10.The non-dimensional fundamental frequency of the fully simple supported FGSP with porosity is given in Table 14.The non-dimensional first six frequencies of the FGSP with porosity subjected to different boundary conditions are demonstrated in Table 15.

Table 15 The non-dimensional first six frequencies of square FGSP with porosity（a/h=10，p=2，ξ=0.2）.

Next,a(1-1-1)FGSP with porosity is considered here for the parameter study.The effects of the aspect ratio b/a and the side-tothickness ratio a/h are demonstrated in Fig.9(a and b)for four cases of the boundary conditions.The frequencies of the fully clamped sandwich plates are greatest while those of fully simple supported ones are smallest.When the aspect ratio b/a and the side-tothickness ratio a/h rise,the frequencies of the plates decrease.Fig.9(c)demonstrates the effects of the power-law index p on the frequencies of the FGSP with porosity.When the power-law index increase,the frequencies decrease.The effects of porosity on the free vibration of the FGSP are demonstrated in Fig.9(d).From this figure,when the porous coef ficientξincrease,the frequencies of the FGSP with porosity I and porosity II decrease rapidly,while the frequencies of the FGSP with porosity III increase slowly.According to Fig.9(c and d),it can be concluded that the porosity III has weak effects on the frequencies of the FGSP while the porosity I and II have strong effects on the frequencies of that ones.

Fig.10 illustrates the first nine mode shapes of the FGSP with porosity II subjected to SSCC boundary condition.Because the boundary condition is asymmetric,the mode shapes of the plate are asymmetric.

3.2.3.Buckling analysis of FGSP with porosity

The buckling behavior of the FGSP with porosity is investigated in this subsection.The plate is subjected to biaxial compressive load.The non-dimensional critical buckling load of the fully simple supported(SSSS)FGSP with porosity and b/a=1，a/h=10 is given in Table 16.It can see clearly that the power-law index and the porosity have signi ficant effects on the buckling behavior of the plates.On the other hand,the effects of the boundary conditions on the critical buckling loads of the sandwich plates with porosity are presented in Table 17,where b/a=1，a/h=10 and p=2.The critical buckling loads of the CCCC plates are greater than other ones,and those of SSSS plates are smallest.

Table 16 The non-dimensional critical buckling load of FGSP with porosity.

Table 17 The non-dimensional critical buckling loads of FGSP with porosity with different BCs（p=2）.

Fig.9.The effects of several parameters on the center de flection of the(1-1-1)FGSP with porosity.

Fig.10.The first nine mode shapes of the SSCC(1-1-1)FGSP with porosity II（b/a=1，a/h=10，p=2，ξ=0.2）.

Continuously,a(1-1-1)FGSP with porosity is examined in this subsection.The effects of the aspect ratio b/a on the critical buckling loads of the FGSP with porosity II are illustrated in Fig.11(a).The critical buckling loads decrease as the increase of b/a.Fig.11(b)demonstrates the effects of the side-to-thickness ratio a/h on the critical buckling loads of the FGSP with porosity.It can be seen that the side-to-thickness ratio have signi ficant effects on the critical buckling loads of the FGSP.When the ratio a/h increases,the critical

buckling loads of the sandwich plates decrease rapidly.For SSSS plates,the critical buckling loads of the plate with a/h=50 is approximately 20 times smaller than those of the plate with a/h=5.The critical buckling loads of the CCCC plates with a/h=5 is approximately 40 times greater than those of the plate with a/h=50.Continuously,the in fluence of the power-law index p on the critical buckling loads of the FGSP with porosity is demonstrated in Fig.11(c).The critical buckling loads of the plates decrease when p increases.The speeds of the decrease when the power-law index increases from 0 to 2 is greater than those of the plate when the power-law index increases from 2 to 10.Besides,it can see clearly that the effects of porosity III are very small in comparison with the effects of porosity I and porosity II.Fig.11(d)presents the dependence of the critical buckling loads on the varying of the porous coef ficient.In the general,the critical buckling loads of the porous plates are smaller than those of the perfect ones.The critical buckling loads of the plates with porosity I and porosity II decrease very fast when the porous coef ficientξincreases.However,the critical buckling loads of the plates with porosity III decrease slowly when the coef ficientξincreases.In general,the distribution of the porosities through the thickness of the plates plays a signi ficant role on the buckling behavior of the FGSP with porosity.

Fig.11.The effects of several parameters on the center de flection of the(1-1-1)FGSP with porosity.

4.Conclusions

In the conclusion of this study,a comprehensive study on the bending,free vibration and buckling analysis of the FGSP with porosity has been carried out.A finite element procedure based on a novel hyperbolic shear deformation theory has been established to predict the static and dynamic response of the FGSP with porosity.The accuracy and ef ficiency of the numerical results of the present algorithm are provided by comparing the present results and available results in some special cases.The present finite element algorithm can be applied to analyze the plates with arbitrary shape and boundary conditions.Some useful conclusions can be achieved as follows.

·The bending,free vibration and buckling behaviors of the FGSP are completely different from the conventional FGSP,especially the distribution of the stresses through the thickness of the plates.

·The inclusion of the porous effect leads to an increase of the de flections and critical buckling loads.However,the trend of the change in natural frequencies depends on the type of porous distribution.

·The location and distribution of the porosity affect strongly on the behavior of the FGSP.The porosity located at two face sheets has signi ficant effects and the porosity located at core layer has small effects on the mechanical behavior of the FGSP.

The novel FGSP with porosity has a signi ficant potential application in many fields of the aerospace,nuclear energy or marine engineering.So,it is necessary to have more works on the behavior of FGSP subjected to many types of loads such as thermal load,hygro-thermal load or blast pressure.

This research did not receive any speci fic grant from funding agencies in the public,commercial,or not-for-pro fit sectors.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to in fluence the work reported in this paper.