• <tr id="yyy80"></tr>
  • <sup id="yyy80"></sup>
  • <tfoot id="yyy80"><noscript id="yyy80"></noscript></tfoot>
  • 99热精品在线国产_美女午夜性视频免费_国产精品国产高清国产av_av欧美777_自拍偷自拍亚洲精品老妇_亚洲熟女精品中文字幕_www日本黄色视频网_国产精品野战在线观看 ?

    Finite element analysis of functionally graded sandwich plates with porosity via a new hyperbolic shear deformation theory

    2022-03-29 07:08:32PhamVanVinhLeQuangHuy
    Defence Technology 2022年3期

    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.

    一个人看视频在线观看www免费| 成人美女网站在线观看视频| 国产亚洲av片在线观看秒播厂| 嫩草影院入口| 男女啪啪激烈高潮av片| 观看av在线不卡| 久久久国产一区二区| 久久99精品国语久久久| 黄色日韩在线| 男的添女的下面高潮视频| 精品亚洲成国产av| 51国产日韩欧美| 国产精品一区www在线观看| 我要看黄色一级片免费的| 我要看黄色一级片免费的| 五月天丁香电影| 免费在线观看成人毛片| 久久毛片免费看一区二区三区| 在线看a的网站| 国产精品一及| av不卡在线播放| 国产乱人偷精品视频| 国产在线免费精品| 亚洲欧洲国产日韩| 能在线免费看毛片的网站| 国产精品久久久久久久久免| 少妇高潮的动态图| 久久精品久久久久久噜噜老黄| 欧美另类一区| 国产视频内射| 欧美老熟妇乱子伦牲交| 亚洲精品久久久久久婷婷小说| 国产男女内射视频| 午夜激情福利司机影院| 欧美+日韩+精品| 午夜福利在线在线| 欧美一区二区亚洲| 国产精品伦人一区二区| 精品少妇黑人巨大在线播放| 日韩中字成人| 国精品久久久久久国模美| 免费看日本二区| 久久久欧美国产精品| 成人毛片a级毛片在线播放| 日韩中文字幕视频在线看片 | 人妻 亚洲 视频| 两个人的视频大全免费| 中国国产av一级| 久久精品夜色国产| 青春草视频在线免费观看| 黄色配什么色好看| 亚洲国产日韩一区二区| 丰满乱子伦码专区| av播播在线观看一区| 最近最新中文字幕免费大全7| 国产一区二区三区av在线| 只有这里有精品99| 七月丁香在线播放| 少妇人妻久久综合中文| 男女边吃奶边做爰视频| 黄色怎么调成土黄色| 春色校园在线视频观看| 国产成人aa在线观看| 热re99久久精品国产66热6| 亚洲国产色片| 亚洲人与动物交配视频| 少妇熟女欧美另类| 欧美成人午夜免费资源| 国产在线一区二区三区精| 国产一区有黄有色的免费视频| 成年美女黄网站色视频大全免费 | 国产精品爽爽va在线观看网站| 欧美性感艳星| 久久97久久精品| 少妇人妻精品综合一区二区| 亚洲第一av免费看| 国产精品一区二区在线观看99| 国产一区二区三区av在线| 我的女老师完整版在线观看| 丰满迷人的少妇在线观看| 国产成人免费无遮挡视频| av不卡在线播放| 久久久久久久久久久丰满| 日本欧美视频一区| 亚洲人成网站在线观看播放| 女性生殖器流出的白浆| 永久网站在线| 国产亚洲5aaaaa淫片| 少妇被粗大猛烈的视频| 天堂8中文在线网| 久久综合国产亚洲精品| 亚洲国产精品成人久久小说| 美女高潮的动态| 久久99热6这里只有精品| 国产v大片淫在线免费观看| 99热这里只有是精品50| 99re6热这里在线精品视频| 亚洲欧美日韩东京热| 成人亚洲欧美一区二区av| 22中文网久久字幕| 在线观看一区二区三区| 中国三级夫妇交换| 九色成人免费人妻av| 日日撸夜夜添| 久久久午夜欧美精品| 日韩制服骚丝袜av| 2018国产大陆天天弄谢| 亚洲精品久久午夜乱码| 亚洲精品日本国产第一区| 97在线人人人人妻| 另类亚洲欧美激情| 日本色播在线视频| 欧美成人a在线观看| av在线app专区| 色网站视频免费| 婷婷色综合大香蕉| 一区二区av电影网| 精品久久久久久电影网| 97在线视频观看| videossex国产| 丝袜脚勾引网站| 亚洲欧美日韩另类电影网站 | 91狼人影院| 亚洲av欧美aⅴ国产| 久久人人爽人人爽人人片va| 国产一级毛片在线| 国产白丝娇喘喷水9色精品| 日本vs欧美在线观看视频 | 欧美一级a爱片免费观看看| 天天躁夜夜躁狠狠久久av| 一级毛片久久久久久久久女| 国产色爽女视频免费观看| 两个人的视频大全免费| 色视频在线一区二区三区| 欧美极品一区二区三区四区| 欧美精品一区二区大全| 亚洲精品久久午夜乱码| 成人综合一区亚洲| 久久国产乱子免费精品| 久久国内精品自在自线图片| 在线精品无人区一区二区三 | 中文字幕久久专区| 免费观看a级毛片全部| 欧美成人a在线观看| 免费少妇av软件| 日韩av不卡免费在线播放| 成人漫画全彩无遮挡| 欧美 日韩 精品 国产| av福利片在线观看| 青春草亚洲视频在线观看| 国产v大片淫在线免费观看| 成人亚洲欧美一区二区av| 大片电影免费在线观看免费| 国产精品久久久久久久久免| 日韩欧美 国产精品| 舔av片在线| 久久久久久久大尺度免费视频| 久久久久久久久久人人人人人人| 激情五月婷婷亚洲| 久久精品国产a三级三级三级| 国产免费一级a男人的天堂| 伦理电影免费视频| 如何舔出高潮| 亚洲av日韩在线播放| 成人影院久久| 亚洲国产毛片av蜜桃av| 欧美日韩视频精品一区| 一本一本综合久久| 黄色配什么色好看| 国产亚洲欧美精品永久| 中文字幕制服av| 寂寞人妻少妇视频99o| a级毛片免费高清观看在线播放| 婷婷色av中文字幕| 我的老师免费观看完整版| 亚洲av电影在线观看一区二区三区| 亚洲不卡免费看| 午夜福利网站1000一区二区三区| 日日摸夜夜添夜夜爱| 一个人免费看片子| 亚洲真实伦在线观看| 一二三四中文在线观看免费高清| 久久久精品免费免费高清| 黄片wwwwww| 中文乱码字字幕精品一区二区三区| 在线播放无遮挡| 久久久成人免费电影| 狂野欧美激情性bbbbbb| 黑丝袜美女国产一区| 男女国产视频网站| 蜜桃亚洲精品一区二区三区| 欧美高清成人免费视频www| 日韩成人av中文字幕在线观看| 七月丁香在线播放| 青春草亚洲视频在线观看| 九九在线视频观看精品| 我的老师免费观看完整版| 哪个播放器可以免费观看大片| 国产亚洲最大av| 国产中年淑女户外野战色| 大又大粗又爽又黄少妇毛片口| 夜夜看夜夜爽夜夜摸| 亚洲一级一片aⅴ在线观看| 一本久久精品| 涩涩av久久男人的天堂| 一个人看的www免费观看视频| 欧美bdsm另类| 女人十人毛片免费观看3o分钟| 干丝袜人妻中文字幕| 久久久久久久久久成人| av免费观看日本| 美女福利国产在线 | 精品人妻偷拍中文字幕| 久久青草综合色| 国产成人a区在线观看| 成人黄色视频免费在线看| 美女cb高潮喷水在线观看| 国产精品不卡视频一区二区| 五月玫瑰六月丁香| 亚洲av不卡在线观看| 人妻夜夜爽99麻豆av| 亚洲,一卡二卡三卡| 国产伦精品一区二区三区四那| 3wmmmm亚洲av在线观看| 水蜜桃什么品种好| 夜夜看夜夜爽夜夜摸| 黄色怎么调成土黄色| 精品一区二区三卡| 中国三级夫妇交换| 晚上一个人看的免费电影| 永久免费av网站大全| 91精品国产国语对白视频| 久久这里有精品视频免费| 国产伦精品一区二区三区视频9| 国产中年淑女户外野战色| 精品国产露脸久久av麻豆| 久久综合国产亚洲精品| 精品一品国产午夜福利视频| 黑人高潮一二区| 男人狂女人下面高潮的视频| 纵有疾风起免费观看全集完整版| 性色avwww在线观看| 欧美老熟妇乱子伦牲交| 欧美精品国产亚洲| 免费黄频网站在线观看国产| 精品国产乱码久久久久久小说| a 毛片基地| 黄色日韩在线| 久久国产亚洲av麻豆专区| av专区在线播放| 亚洲国产精品专区欧美| 国产v大片淫在线免费观看| 天天躁夜夜躁狠狠久久av| 亚洲国产最新在线播放| 成人免费观看视频高清| 91在线精品国自产拍蜜月| 久久精品熟女亚洲av麻豆精品| 99热国产这里只有精品6| 一级黄片播放器| 国产有黄有色有爽视频| 男男h啪啪无遮挡| 亚洲无线观看免费| 噜噜噜噜噜久久久久久91| 国产一区有黄有色的免费视频| 午夜日本视频在线| 日韩av免费高清视频| 日韩欧美 国产精品| 久久久久视频综合| 国产精品嫩草影院av在线观看| 日韩电影二区| 91精品一卡2卡3卡4卡| 亚洲欧美精品自产自拍| 七月丁香在线播放| 亚洲,一卡二卡三卡| 久久精品国产a三级三级三级| 国产精品久久久久久久电影| 免费久久久久久久精品成人欧美视频 | 国产伦精品一区二区三区四那| 婷婷色麻豆天堂久久| 偷拍熟女少妇极品色| 亚洲av综合色区一区| 五月开心婷婷网| 大码成人一级视频| 亚洲国产成人一精品久久久| 国产真实伦视频高清在线观看| 国产淫片久久久久久久久| 亚洲成色77777| 综合色丁香网| 小蜜桃在线观看免费完整版高清| 亚洲内射少妇av| 又爽又黄a免费视频| 美女主播在线视频| 在线免费十八禁| 久久国产精品大桥未久av | 如何舔出高潮| 成人黄色视频免费在线看| 久久久久国产精品人妻一区二区| 国产精品一区二区性色av| 日韩欧美精品免费久久| 三级经典国产精品| 天天躁日日操中文字幕| 女人久久www免费人成看片| 亚洲av电影在线观看一区二区三区| 成人一区二区视频在线观看| 在线观看三级黄色| 亚洲国产毛片av蜜桃av| 各种免费的搞黄视频| 国产欧美日韩一区二区三区在线 | 久久热精品热| 国产美女午夜福利| 成人毛片a级毛片在线播放| 免费观看的影片在线观看| 国产白丝娇喘喷水9色精品| 黄色怎么调成土黄色| 亚洲精品一二三| 九色成人免费人妻av| 国产成人精品一,二区| 国产成人91sexporn| 国产爽快片一区二区三区| 成人黄色视频免费在线看| 建设人人有责人人尽责人人享有的 | 精品午夜福利在线看| 亚洲精品国产av蜜桃| 91aial.com中文字幕在线观看| 久久人人爽人人片av| 22中文网久久字幕| 久久精品夜色国产| 亚洲精品自拍成人| 亚洲国产精品成人久久小说| 日本免费在线观看一区| 国内精品宾馆在线| 欧美xxⅹ黑人| 国产精品熟女久久久久浪| 夜夜看夜夜爽夜夜摸| av国产久精品久网站免费入址| 女人久久www免费人成看片| 最近中文字幕高清免费大全6| 色5月婷婷丁香| 欧美另类一区| 亚洲欧美成人精品一区二区| 成人影院久久| 欧美老熟妇乱子伦牲交| 在线观看美女被高潮喷水网站| 91精品一卡2卡3卡4卡| 成人特级av手机在线观看| 国产伦精品一区二区三区四那| 国产成人精品婷婷| 日本黄大片高清| 日本一二三区视频观看| 亚洲欧美一区二区三区黑人 | 成人综合一区亚洲| 免费久久久久久久精品成人欧美视频 | 性高湖久久久久久久久免费观看| 六月丁香七月| 久久久久久久久久人人人人人人| 一本久久精品| 人妻夜夜爽99麻豆av| 老司机影院成人| 欧美三级亚洲精品| 99久国产av精品国产电影| 成年美女黄网站色视频大全免费 | 欧美成人a在线观看| 久久久久久久久久成人| 亚洲,一卡二卡三卡| 久久久久国产网址| 亚洲成人手机| 精品国产一区二区三区久久久樱花 | 直男gayav资源| 久久久久精品久久久久真实原创| 久久久久久人妻| 精品国产乱码久久久久久小说| 午夜精品国产一区二区电影| 国产精品av视频在线免费观看| 狠狠精品人妻久久久久久综合| 日本午夜av视频| 美女视频免费永久观看网站| 久久久久精品久久久久真实原创| av黄色大香蕉| 中文精品一卡2卡3卡4更新| 一级爰片在线观看| 2018国产大陆天天弄谢| 久久久欧美国产精品| 我要看日韩黄色一级片| 国产精品99久久久久久久久| 国产成人免费无遮挡视频| 国产白丝娇喘喷水9色精品| 嫩草影院入口| 日本-黄色视频高清免费观看| 夜夜骑夜夜射夜夜干| 插逼视频在线观看| 国产精品熟女久久久久浪| 纵有疾风起免费观看全集完整版| 少妇熟女欧美另类| 日本午夜av视频| 精品99又大又爽又粗少妇毛片| 亚洲欧美成人综合另类久久久| 亚洲人成网站在线播| 最新中文字幕久久久久| 国产亚洲av片在线观看秒播厂| 爱豆传媒免费全集在线观看| 大又大粗又爽又黄少妇毛片口| 国产精品免费大片| 国产成人午夜福利电影在线观看| 欧美一区二区亚洲| 女人久久www免费人成看片| 18禁在线无遮挡免费观看视频| 欧美区成人在线视频| 毛片女人毛片| 亚洲精品亚洲一区二区| 久久久久国产精品人妻一区二区| 国产真实伦视频高清在线观看| 晚上一个人看的免费电影| 视频区图区小说| 久久精品国产a三级三级三级| 99久久中文字幕三级久久日本| 免费黄频网站在线观看国产| 国产成人精品婷婷| 欧美日本视频| 啦啦啦中文免费视频观看日本| 不卡视频在线观看欧美| 毛片一级片免费看久久久久| 欧美日韩一区二区视频在线观看视频在线| 国产亚洲精品久久久com| 国产精品一区二区性色av| 最近的中文字幕免费完整| 亚洲精品久久久久久婷婷小说| 成人二区视频| 菩萨蛮人人尽说江南好唐韦庄| 久久久久久久亚洲中文字幕| 国产亚洲午夜精品一区二区久久| 欧美老熟妇乱子伦牲交| 少妇猛男粗大的猛烈进出视频| 插逼视频在线观看| 欧美xxxx性猛交bbbb| 国产精品国产三级专区第一集| 美女高潮的动态| 最新中文字幕久久久久| 在线观看一区二区三区激情| 欧美国产精品一级二级三级 | 亚洲成人中文字幕在线播放| 涩涩av久久男人的天堂| 男女边吃奶边做爰视频| 不卡视频在线观看欧美| 一区二区三区乱码不卡18| 1000部很黄的大片| 欧美丝袜亚洲另类| 欧美bdsm另类| 在线观看一区二区三区激情| 舔av片在线| 国产成人精品福利久久| 午夜福利影视在线免费观看| 日韩av在线免费看完整版不卡| 久久99热6这里只有精品| av播播在线观看一区| 久久久久性生活片| 亚洲精品亚洲一区二区| 高清欧美精品videossex| 久久久久久久久久久免费av| 99热网站在线观看| 中文乱码字字幕精品一区二区三区| 免费av不卡在线播放| 观看美女的网站| 久久久欧美国产精品| 国产男女内射视频| 亚洲av成人精品一区久久| 搡女人真爽免费视频火全软件| 香蕉精品网在线| 91aial.com中文字幕在线观看| 在线免费观看不下载黄p国产| 成人毛片a级毛片在线播放| 在现免费观看毛片| 日本免费在线观看一区| 成人无遮挡网站| 伦理电影大哥的女人| 嫩草影院新地址| 欧美人与善性xxx| 久久人人爽av亚洲精品天堂 | 少妇的逼好多水| 国产成人精品一,二区| 亚洲精品色激情综合| 久久久a久久爽久久v久久| 日本vs欧美在线观看视频 | 最近的中文字幕免费完整| 国产精品久久久久久久电影| 观看美女的网站| 80岁老熟妇乱子伦牲交| 一级毛片久久久久久久久女| 精品国产一区二区三区久久久樱花 | 国产成人精品一,二区| 国产高清国产精品国产三级 | 丝袜脚勾引网站| 亚洲熟女精品中文字幕| 国产精品麻豆人妻色哟哟久久| 欧美另类一区| 网址你懂的国产日韩在线| 毛片女人毛片| 欧美精品一区二区免费开放| 老女人水多毛片| 欧美另类一区| 网址你懂的国产日韩在线| 亚洲国产精品成人久久小说| 美女主播在线视频| 国产色婷婷99| 综合色丁香网| 制服丝袜香蕉在线| 国产一级毛片在线| 国产男人的电影天堂91| 国产伦在线观看视频一区| 日韩欧美精品免费久久| 国产精品国产三级专区第一集| 老熟女久久久| 亚洲精品国产成人久久av| 久久久久久久久久久免费av| 国产精品99久久99久久久不卡 | 亚洲美女黄色视频免费看| 伊人久久国产一区二区| 国产精品爽爽va在线观看网站| 国产黄片美女视频| 久久精品夜色国产| 久久久成人免费电影| 一级黄片播放器| 五月伊人婷婷丁香| 成人美女网站在线观看视频| 日韩成人av中文字幕在线观看| av.在线天堂| 人体艺术视频欧美日本| 免费观看a级毛片全部| 色视频www国产| 免费观看在线日韩| 99久久精品国产国产毛片| 亚洲中文av在线| 在线观看美女被高潮喷水网站| av视频免费观看在线观看| 久久久精品94久久精品| 80岁老熟妇乱子伦牲交| 久久久久视频综合| 最近最新中文字幕大全电影3| 精品国产三级普通话版| 日日啪夜夜爽| 日韩三级伦理在线观看| 男女无遮挡免费网站观看| 国产成人精品婷婷| 少妇的逼水好多| 中国国产av一级| 一个人看视频在线观看www免费| 国产成人免费无遮挡视频| 国产高清不卡午夜福利| 美女cb高潮喷水在线观看| 身体一侧抽搐| 狂野欧美白嫩少妇大欣赏| 国产精品偷伦视频观看了| 妹子高潮喷水视频| 26uuu在线亚洲综合色| 直男gayav资源| av又黄又爽大尺度在线免费看| 日韩欧美一区视频在线观看 | 中文精品一卡2卡3卡4更新| 欧美变态另类bdsm刘玥| 深爱激情五月婷婷| 国产精品一区二区三区四区免费观看| 在线 av 中文字幕| 日韩中字成人| av视频免费观看在线观看| 能在线免费看毛片的网站| 国产精品成人在线| 亚洲人成网站高清观看| 国精品久久久久久国模美| 中文字幕av成人在线电影| 久久精品国产亚洲av天美| a级毛片免费高清观看在线播放| 老司机影院毛片| 麻豆精品久久久久久蜜桃| 色吧在线观看| 久久鲁丝午夜福利片| 国产美女午夜福利| 七月丁香在线播放| 伊人久久精品亚洲午夜| 国产美女午夜福利| 色吧在线观看| 亚洲国产日韩一区二区| 最近的中文字幕免费完整| 色吧在线观看| 看免费成人av毛片| 在线免费观看不下载黄p国产| 最近最新中文字幕免费大全7| 伊人久久精品亚洲午夜| 国产成人aa在线观看| 七月丁香在线播放| 免费看光身美女| 最近2019中文字幕mv第一页| 国产精品一及| 十八禁网站网址无遮挡 | av专区在线播放| 欧美日韩国产mv在线观看视频 | 乱码一卡2卡4卡精品| 天美传媒精品一区二区| 国产乱人视频| 丝袜喷水一区| 亚洲色图综合在线观看| 久久久午夜欧美精品| av在线老鸭窝| 在线观看免费高清a一片| 自拍偷自拍亚洲精品老妇| 美女中出高潮动态图| 亚洲国产最新在线播放| 亚洲熟女精品中文字幕| 午夜视频国产福利| 午夜老司机福利剧场| 欧美日韩视频精品一区| 国产av一区二区精品久久 | av福利片在线观看| 国产精品久久久久久精品古装| 免费av中文字幕在线| 久久人妻熟女aⅴ| 国产久久久一区二区三区|