Are“Higher-Order”and “Layer-wise Zig-Zag” Plate&Shell Theories Necessary for Functionally Graded Materials and Structures?

Yaping Zhang,Qifeng Fan,Leiting Dong,3and Satya N.Atluri

Are“Higher-Order”and “Layer-wise Zig-Zag” Plate&Shell Theories Necessary for Functionally Graded Materials and Structures?

Yaping Zhang1,Qifeng Fan2,Leiting Dong2,3and Satya N.Atluri4

Similar to the very vast prior literature on analyzing laminated composite structures,“higher-order”and “l(fā)ayer-wise higher-order”plate and shell theories for functionally-graded(FG)materials and structures are also widely popularized in the literature of the past two decades.However,such higher-order theories involve(1)postulating very complex assumptions for plate/shell kinematics in the thickness direction,(2)defining generalized variables of displacements,strains,and stresses,and(3)developing very complex governing equilibrium,compatibility,and constitutive equations in terms of newly-defined generalized kinematic and generalized kinetic variables.Their industrial applications are thus hindered by their inherent complexity,and the fact that it is difficult for end-users(front-line structural engineers)to completely understand all the newly-defined generalized DOFs for FEM in the higher-order and layer-wise theories.In an entirely different way,very simple 20-node and 27-node 3-D continuum solid-shell elements are developed in this paper,based on the simple theory of 3D solid mechanics,for static and dynamic analyses of functionally-graded plates and shells.A simple Over-Integration(a 4-point Gauss integration in the thickness direction)is used to evaluate the stiffness matrices of each element,while only a single element is used in the thickness direction without increasing the number of degrees of freedom.A stress-recovery approach is used to compute the distribution of transverse stresses by considering the equations of 3D elasticity in Cartesian as well as cylindrical polar coordinates.Comprehensive numerical results are presented for static and dynamic analyses of FG plates and shells,which agree well,either with the existing solutions in the published literature,or with the computationally very expensive solutions obtained by using simple 3D isoparametric elements(with standard Gauss Quadrature)available in NASTRAN(wherein many 3D elements are used in the thickness direction to capture the varying material properties).The effects of the material gradient index,the span-to-thickness ratio,the aspect ratio and the boundary conditions are also studied in the solutions of FG structures.Because the proposed methodology merely involves:(2)standard displacement DOFs at each node,(2)involves a simple 4-point Gaussian over-integration in the thickness direction,(3)relies only on the simple theory of solid mechanics,and(4)is capable of accurately and efficiently predicting the static and dynamical behavior of FG structures in a very simple and cost-effective manner,it is thus believed by the authors that the painstaking and cumbersome development of“higher-order”or“l(fā)ayer-wise higher-order”theories is not entirely necessary for the analyses of FG plates and shells.

functionally graded plates and shells,20-node hexahedral element,27-node hexahedral element,over-integration,higher order theory,layer-wise theory.

1 Introduction

Functionally graded materials(FGM)were proposed as heat-shielding structural materials by Japanese material scientists in 1984[Koizumi(1997)].Typically,FGMs are mixtures of ceramics and metals with material properties varying smoothly from one structural surface to another.In this way,thermal-stress concentrations developed at material interfaces of conventional structural components can be avoided.This excellent feature has promising applications for aircrafts,space vehicles,automobiles and other engineering structures.Thus,it is important to develop a simple and accurate tool for analyzing static and dynamic behaviors of FGM structures.

Similar to the very vast literature on laminated composite structures,specialized plate and shell theories for functionally-graded(FG)materials and structures were also extensively studied in the literature of the past two decades.These theories involve expanding the displacements using first-order,or higher-order power-series,or other types of functions,in the thickness direction of plates/shells.For example,the Kirchhoff and Reissner(or Mindlin)theories are the most widely-used plate/shell theories,see[Timoshenko and Woinowsky(1959),Reissner(1945)and Mindlin(1951)],which are embedded in almost every standard FEM packages such as Ansys,Abaqus,Nastran,etc.These and other first-order theories are also applied for static and dynamic analyses of FGM plates/shells,see[Zenkour(2006);Cheng and Batra(2000);Batra and Jin(2005)].However,for relatively thick FGM structures,Kirchhoff and Reissner assumptions usually underestimate the deflections and overestimates natural frequencies,as transverse shear strains are neglected(Kirchhoff)or assumed to be constant(Reissner)in such theories.

To overcome the aforementioned limitations,many higher-order shear deformation theories(HSDT)were later proposed,see[Lo,Christensen and Wu(1977);Reddy(1984);Reddy and Robbins(1994)],wherein the variations ofin-plane displacements in the thickness direction are mostly approximated using third-order polynomials.These and other higher-order theories were used to analyze FGM plates and shells also,see[Reddy(2000);Qian,Batra and Chen(2004);Ferreira,Batra,Roque,Qian and Martins(2005)].In[Carrera,Brischetto,Cinefra and Soave(2011)],it is also concluded that the normal deflection should also be expanded in the thickness direction to include the effects of thickness-stretching.Moreover,layer-wise theories,which expand displacements in each of many artificial layers of FGM plates/shells,were also used to improve the accuracy of the solution,see[Ramirez,Heyliger and Pan(2006);Carrera,Brischetto and Robaldo(2008)].

In order to derive higher-order or layer-wise theories of plates and shells,kinematic assumptions are substituted into the principle of potential energy of 3D elasticity.By exploring the stationarity conditions,very complex governing differential equations in terms of newly defined generalized displacements,strains and stresses can be derived,see[Reddy(2004)]for example.However,such complex differential equations cannot be directly solved.One usually goes back to derive the weak-forms of these governing differential equations,and thus develop the corresponding the finite element models to solve the problem numerically.In this sense,defining the many generalized displacements,strains,stresses,and deriving the complex higherorder or layer-wise theories and differential equations seems unnecessary.One can directly use the variational principle of 3D elasticity to develop finite elements for the modeling of plates and shells.Moreover,it is difficult for end-users to completely understand all the newly-defined generalized DOFs in higher-order theories(or their FEM counterparts),which have ambiguous physical meanings.Furthermore,it becomes very problematic when boundary conditions have to be enforced correctly for these generalized degrees of freedom,by the end-users.

In an entirely different way,[Dong,El-Gizawy,Juhany,Atluri(2014a,b)]directly developed 2D quadrilateral 4-node elements(for beam-type structures),and 3D hexahedral 8-node finite elements(for plate and shell structures),for FG as well as laminated structures,based on the simple theories of 2D and 3D solid mechanics,respectively.Because the traditional displacement-based lowest order elements suffer from shear locking,a technique of locking-alleviation was used in Dong,El-Gizawy,Juhany,Atluri(2014a,b),by independently assuming locking-free element strains.Without using mixed variational principles,and thereby bypassing the troublesome LBB conditions of stability,a simple collocation method was used to satisfy the compatibility between the independently assumed strains and those de-rived from the independently assumed displacement fields,at carefully selected required number of points inside the elements.Over-integration was also used in the thickness direction to accurately evaluate the stiffness matrix of FG and laminated elements.However,for very thick laminated structures with only a few layers,the computational accuracy is slightly compromised if only a single lowest-order finite element is used in the thickness direction.Therefore displacement-based higherorder solid(continuum)elements were also developed in[Fan,Zhang,Dong,Li,Atluri(2015)]for analyzing very thick laminates.Similarly,for FGM plates and shells,if the span-to-thickness ratio is very large,or if the material properties are varying in the thickness direction with very large gradients,it will be beneficial to have very simple higher-order displacements-based solid elements,with the standard displacement degrees of freedom,to improve the accuracy of solutions.

Inthisstudy,standard displacement-based3Dhexahedral20-nodeelement(DPH20)and a 3D 27-node element(DPH27),with over-integration in the thickness direction,are developed to carry out static and dynamic analyses of functionally graded plates and shells.Comprehensive numerical results are presented,which agree very well with existing solutions in the published literature,or with the computationally very-expensive solutions obtained by using standard 3D isoparametric element methodologies(with standard Gauss quadrature)available in commercial FEM codes.Because the proposed methodology:(1)merely involves the standard displacement DOFs at each node,(2)relies only on the simple theory of 3D solid mechanics,(3)simple 4-point Gauss quadrature in the thickness direction,and(4)is capable of accurately and efficiently predicting the static and dynamical behavior of FGM structures in a very simple and cost-effective manner,it is thus believed by the authors that the painstaking development of specialized“higherorder”or“l(fā)ayer-wise”theories is not entirely necessary for the analyses of FGM plates and shells.In the following sections,details of the proposed methodology are described and many numerical examples of various laminated plates/shells with different loads and boundary conditions are provided.

2 Formulation

2.1 Primal FEM with over-integration

As illustrated in Fig.1,nodal shape functions of the 20-node and 27-node hexahedral elements can be defined in terms of local isoperimetric coordinates.We firstly define the following function for nodei:

Figure 1:Numbering of nodes for 20-node/27-node hexahedral elements in ξ,η,ζcoordinates.

where ξi,ηiand ζiare the natural coordinates of nodei,andg(ξi,ξ),g(ηi,η)andg(ζi,ζ)are defined as:

Shape functions are therefore defined as follows.

For vertex nodesi=1,2,...,8:

whereGefor each vertex node is the sum of theGivalues of nodes on the three adjacent edges.

For mid-edge nodesi=9,10,...,20:

Nodal shape functions for the 27-node hexagonal element is slightly different,which can be found in most books of finite elements[Atluri(2005),Zienkiewicz and Taylor(1997)].

Thus the displacement fields within the element are interpolated by using nodal shape functions:

or in equivalent matrix-vector notations:

where?uerepresents nodal displacements of the element.

Strain fields within the element are obtained by differentiating Eq.(7)with respect to Cartesian coordinates:

whereLis a linear differential operator.

By using the Galerkin Weak-Form or equivalent variational principles[Atluri(2005);Dong,Alotaibi,Mohiuddine and Atluri(2014c)],the element stiffness matrix and mass matrix are computed by:

As discussed in[Dong,EI-Gizawy,Juhany and Atluri(2014b,c)],a scheme of overintegration in the thickness direction is used to evaluate the element stiffness and mass matrices for FGM.We know that for homogeneous materials,a 3×3×3 Gauss integration is mostly used for evaluating Eq.(9).However,for FGM materials,the material properties are varying in the thickness direction,therefore the conventional 3×3×3 Gauss integration will be insufficient.Thus,an increased-order integration,which is a 4-point Gauss integration in the thickness direction,is used to capture to varying material properties in the thickness direction.

The transverse normal and shear stresses are computed by using a stress-recovery approach considering the equilibrium equations of 3D linear elasticity.For the laminated plates,the distribution of transverse stresses can be obtained by numerically evaluating:

wherez=z0denote the lower surface of the plate.

For cylindrical shells,the distribution of transverse stresses can also be evaluated,by numerically solving the following 3 differential equations:

In Eq.(11),the left hand-side involves stress components to be recovered,and the right-hand side are directly evaluated from the solutions of DPH20orDPH27.Each equation is a first-order single-variable ODE,which can be solved with a variety of computational methods,see[Dong,Alotaibi,Mohiuddine and Atluri(2014c)].In this study,simple collocation of Eq.(11)is implemented at a variety of points along the thickness direction. Combined with the traction free condition at the inner surface of the cylindrical shell,stress components σrθ,σrz,σrrcan be efficiently recovered from the computed in-plane normal and shear stresses.

2.2 Functionally graded material properties

Figure 2:Geometry and the reference coordinate system for the FG plate.

We firstly consider a FG plate of lengtha,widthb,and thicknessh,as shown in Fig.2.Thex-,y-,andz-coordinates are along the length,width,and height directions of the plate,respectively.z=0 is placed at the mid-surface of the plate.

The FG plate is mostly made by a mixture of two material phases,for example,a metal and a ceramic.The material properties of the FG plate,such as Young’s modulusE,mass density ρ and Poisson’s ratio μ,are assumed to be varying continuously throughout the thickness of plate according to the power law distribution of volume fraction of constituents.According to the rule of mixtures,the effective material properties can be expressed as:

wherePm,Pc,VmandVcare defined as the material properties and volume fractions of metal and ceramic,respectively.

In this paper,the volume fraction of ceramic is assumed to be subjected to the following power-law distribution along the thickness direction:

wherepis the material gradient index which should only be positive.Then the effective material properties of the FG plate can be expressed as:

where the subscriptscandmrepresent ceramic and metal,respectively.The material properties of metals and ceramics used in this study are listed in Table 1.

Table 1:Material properties of metals and ceramics used in this study.

As an example,the variation of Young’s modulusEin the thickness direction of a FGAl/Al2O3plate with various values of gradient indexpisshownFig.3.Forp=0 and p=∞,the FG plate is purely ceramic(Al2O3)or metallic(Al),respectively.For other positive values ofp,the material properties vary smoothly from metalrich-surface to ceramic-rich surface with different gradients.

For functionally graded shells,material properties also vary smoothly from one surface to the other in the thickness direction,which is similar to the FGM plate.

Figure 3:Variation of Young’s modulus E through the dimensionless thickness(z/h)of Al/Al2O3plate.

3 Numerical Examples and Results

3.1 Static analysis

3.1.1A simply-supported Al/Al2O3FG plate subjected to a uniformly distributed lateral load

The first example studies a simply-supported thick-section Al/Al2O3FG square plate subjected to a uniformly distributed lateral load:q=1GPa.The plate is square witha=b=100 mm,and the plate thickness ish=10mm.Material parameters used for the FG plate are listed in Table 1.Three different gradient indexes(p=0.2,2,10)are used.

Figure 4:Finite element model for the FG plate(a/h=10)by(a)NASTRAN and(b)the present DPH20/DPH27 elements.

We solve these problems using a uniform 10×10 mesh with DPH20 and DPH27 elements respectively,as well as using NASTRAN.We can see the difference of meshes between NASTRAN and the present DPH20/DPH27 model in Fig.4.When modeled by Nastran,it takes about half an hour to obtain the numerical results using 250,000 elements.In contrast,the present DPH20/DPH27 model requires only 100 elements and about 15 seconds of computational time on a regular PC with i7 CPU.Computed in-plane and transverse shear stress are shown in Figs.5-10,from which we can see that NASTRAN and the present method give similar results,though the computation time differs by two orders of magnitudes.Moreover,the DPH20 solution takes slightly less computational time than the DPH27 solution,as it contains a smaller number of DOFs.In fact,in all the following numerical examples,DPH20 always gives almost the same computational results as compared to DPH27,thus only results for DPH20 are demonstrated in the following subsections.

Figure 5:Computed in-plane normal stress σxxa tx=y=50mm for the thicksection Al/Al2O3FG plate with gradient index p=0.2.

Figure 6:Computed transverse shear stress σxzat x=y=10mm for the thicksection Al/Al2O3FG plate with gradient index p=0.2.

Figure 7:Computed in-plane normal stress σxxat x=y=50mm for the thicksection Al/Al2O3FG plate with gradient index p=2.

Figure 8:Computed transverse shear stress σxzat x=y=10mm for the thicksection Al/Al2O3FG plate with gradient index p=2.

Figure 9:Computed in-plane normal stress σxxat x=y=50mm for the thicksection Al/Al2O3FG plate with gradient index p=10.

Figure 10:Computed transverse shear stress σxzat x=y=10mm for the thicksection Al/Al2O3FG plate with gradient index p=10.

3.1.2A simply-supported Al/Al2O3FG plate subjected to a uniform distributed lateral load

Figure 11:Geometry and coordinate system of a FG plate.

In this subsection,we consider a simply-supported thick-section Al/Al2O3FG shell subjected to a uniformly distributed outer-pressure:q=1GPa.Material parameters used in the FG plate are listed in Table 1.The inner radius and outer radius of the cylindrical shell arerin=10mmandrout=11mmrespectively.The spans of the cylindrical shell inzdirection and in θ direction arel=10mmand ? = π respectively.The FG shell is simply supported at θ =0,π andz=0,10.

We solve this problem using a uniform 60×20 mesh with DPH20 elements,as well as using NASTRAN(with 1.2 million DOFs),as shown in Fig.12.Computed distributions of stresses are shown in Fig.13.It can be clearly seen that the results by the present DPH20 elements agree well with those by NASTRAN even though DPH20 requires significantly less DOFs and computational time.

Figure 14:Mode shapes of simply-supported Al/Al2O3square FG plates with a/h=5 and p=1.

3.2 Free vibration analysis

3.2.1Modal analysis of functionally graded plates

Figure 12:Finite element model for the FG shell by(a)NASTRAN and(b)the present DPH20 elements.

Simply-supported Al/Al2O3square FG plates with various gradient indexes are analyzed in this subsection.Material parameters used in the FG plates are listed in Table 1.The span-to-thickness ratioa/h=10 is adopted in the example.The non-dimensional frequency parametersare used to compare results obtained in this paper to the results in the literatures.We solve this problem using a uniform 20×20 mesh with DPH20 elements.The present results are shown in Table 2 along with the exact solution by[Jin,Su,Shi,Ye and Gao(2014)]and the solution by a variational Ritz method[Huang,McGee and Wang(2013)].It is seen that the DPH20 solution agrees well with the exact solution of 3D elasticity to a satisfactory precision.

Figure 13:Computed σθθ,σrr,σzzatπ,z=5mm for the thick-section Al/Al2O3FG plate.

We then consider extremely thick simply-supported Al/Al2O3square FG plates with span-to-thickness ratioa/h=5.Two different kinds of meshes with 20×20×1 and 20×20×2 DPH20 elements(one or two layers of elements in the thickness direction)are used.The first five non-dimensional frequency parametersare presented in Table 3. Although both models give acceptable solutions,more accurate results are obtained by using two layers of elements(3.33%and 0.12%maximum errors for 20×20×1 and 20×20×2 meshes respectively).

Therefore,a uniform 20×20×2 mesh with DPH20 elements will be used if the FG plate is extremely thick(i.e.a/h≤5)in the following numerical examples.

Simply-supported Al/Al2O3square FG plates with span-to-thickness ratio varied from 5 to 20 and gradient index varied from 0 to 10 are analyzed to verify the accuracy and ef?cient of the present method.The non-dimensional frequency parametersobtained using the DPH20 element are compared with HSDT solutions[Thai,Park and Choi(2013);Hosseini-Hashemi,Fadaee and Atashipour(2011a);Matsunaga(2008)],FSDT solutions[Hosseini-Hashemi,Fadaee

and Atashipour(2011b);Zuo,Yang,Chen,Xie and Zhang(2014)]and CPT solutions[Thai,Park and Choi(2013)]in Table 4.Obviously,the solutions given by the proposed DPH20 element are in excellent agreement with the HSDT solutions[Thai,Park and Choi(2013);Hosseini-Hashemi,Fadaee and Atashipour(2011a);Matsunaga(2008)]and FSDT solutions[Hosseini-Hashemi,Fadaee and Atashipour(2011b);Zuo,Yang,Chen,Xie and Zhang(2014)].The results also indicate that the CPT over-predicts the natural frequency of FG plates,especially for the thick plate at higher modes of vibration.Moreover,it is found that the nondimensional frequency parameter decreases as the gradient index increases.This is because larger gradient index leads to decrease of stiffness.The mode shapes of simply-supported Al/Al2O3square FG plates witha/h=5 and 10 are depicted in Figs.14 and 15,respectively.

Table 2:The non-dimensional frequency parametersof the simply-supported Al/Al2O3 square FG plates with a/h=10.

Table 2:The non-dimensional frequency parametersof the simply-supported Al/Al2O3 square FG plates with a/h=10.

p Method ωn 1 2 3 4 5 Jin et al.(2014)5.779 13.810 13.810 19.480 19.480 0 Huang et al.(2013)5.777 13.810 13.810 19.480 19.480 DPH20 5.795 13.898 13.898 19.483 19.483 Jin et al.(2014)4.428 10.630 10.630 16.200 16.200 1 Huang et al.(2013)4.426 10.630 10.630 16.200 16.200 DPH20 4.434 10.668 10.668 16.202 16.202 Jin et al.(2014)3.774 8.931 8.931 12.640 12.640 5 Huang et al.(2013)3.772 8.927 8.927 12.640 12.640 DPH20 3.795 9.047 9.047 12.642 12.642

Table 3:The non-dimensional frequency parameters of simply-supported Al/Al2O3 square FG plates with a/h=5.

Table 3:The non-dimensional frequency parameters of simply-supported Al/Al2O3 square FG plates with a/h=5.

p Method ωn 1 2 3 4 5 Jin et al.(2014)5.304 9.742 9.742 11.650 11.650 Huang et al.(2013)5.304 9.742 9.742 11.650 11.650 0 20×20×1 elements 5.356 9.742 9.742 11.849 11.849 20×20×2 elements 5.307 9.742 9.742 11.664 11.664 Jin et al.(2014)4.100 8.089 8.089 9.108 9.108 Huang et al.(2013)4.099 8.089 8.089 9.107 1 9.107 20×20×1 elements 4.123 8.090 8.090 9.201 9.201 20×20×2 elements 4.101 8.089 8.089 9.118 9.118 Jin et al.(2014)3.405 6.296 6.296 7.344 7.344 Huang et al.(2013)3.405 6.296 6.296 7.343 5 7.343 20×20×1 elements 3.470 6.304 6.304 7.589 7.589 20×20×2 elements 3.406 6.296 6.296 7.352 7.352

Figure 14:Mode shapes of simply-supported Al/Al2O3 square FG plates with a/h=5 and p=1.

Figure 15:Mode shapes of simply-supported Al/Al2O3 square FG plates with a/h=10 and p=1.

?

Figure 16:First six mode shapes of a SSSS Al/ZrO2 square FG plates with a/h=10 and p=1.

Figure 17:First six mode shapes of a SCSC Al/ZrO2 square FG plates with a/h=10 and p=1.

Figure 18:First six mode shapes of a SCSF Al/ZrO2 square FG plates with a/h=10 and p=1.

In Table 5, the non-dimensional fundamental frequency parametersof Al/ZrO2rectangular FG plates with different boundary conditions are presented.There are five different sets of boundary conditions,namely,SSSS(simply supported at all edges),SCSC(clamped atx=0,aand simply supported aty=0,b),SCSF(clamped atx=0,simply supported aty=0,band free atx=a),SSSC(clamped atx=0 and simply supported atx=a,y=0,b)and SSSF(simply supported atx=0,y=0,band free atx=a).The results obtained by the present method with various values of aspect ratio(b/a=1 and 2),span-to-thickness ratio(a/h=5a rev c)ompared with exact solutions reported in[Jin,Su,Shi,Ye and Gao(2014)].Very good agreement is observed for all the computations,and the difference of the frequency parameters does not exceed 0.91%for the worst case.The first six 3D mode shapes of SSSS,SCSC and SCSF Al/ZrO2square FG plates are shown in Figs.16–18.

3.2.2Modal analysis of functionally graded shells

In this subsection,a clamped FG cylindrical shell is studied with geometric parametersa/h=10,a/R=0.1,a=band with different gradient indexes from 0 to∞.The constituents of the FG shell considered in this example are Si3N4and SUS304,whose material properties are given in Table 1.The non-dimensional frequency parameter adopted for comparison isin whichDm=Emh3/12(1?v2m).We solve these problems using a uniform 20×20×1 mesh with DPH20 elements.The first four non-dimensional frequency parameters are presented in Table 6 along with HSDT solutions[Neves,Ferreira,Carrera,Cinefra,Roque,Jorge and Soares(2013);Pradyumna and Bandyopadhyay(2008);Yang and Shen(2003)].From Table 6,it is found that the solutions given by the proposed DPH20 element are in excellent agreement with the HSDT solutions.

Figure 19:Finite element model for the FG cylindrical shell by(a)NASTRAN and(b)the present DPH20 elements.

We also consider FG cylindrical shells with different boundary conditions.The constituents of the FG shell are Si3N4and SUS304,whose material properties are given in Table 1.The gradient indexp=2 is used in this example.The inner radius and outer radius of the cylindrical shell arerin=60mmandrout=70mmrespectively.The spans of the cylindrical shell in z direction and in θ direction arel=100mmand φ = π/2 respectively.

Figure 20:First six non-dimensional frequency parameters and their corresponding mode shapes of a CFFF FG cylindrical shell by(a)NASTRAN and(b)the present DPH20 elements.

Figure 21:First six non-dimensional frequency parameters and their corresponding mode shapes of a SSSS FG cylindrical shell by(a)NASTRAN and(b)the present DPH20 elements.

Figure 22:First six non-dimensional frequency parameters and their corresponding mode shapes of a CFFF FG cylindrical shell by(a)NASTRAN and(b)the present DPH20 elements.

Table 6:The non-dimensional frequency parameterclamped FG cylindrical shells.

Table 6:The non-dimensional frequency parameterclamped FG cylindrical shells.

Mode Method p=0 p=0.2 p=2 p=10 p=∞Pradyumna et al.(2008)72.9613 60.0269 39.1457 33.3666 32.0274 Yang et al.(2003)74.5180 57.4790 40.7500 35.8520 32.7610 1 Neves et al.(2013)a 74.2634 60.0061 40.5259 35.1663 32.6108 Neves et al.(2013)b 74.5821 60.3431 40.8262 35.4229 32.8593 present 75.5192 61.7080 41.3946 35.8943 33.4029 Pradyumna et al.(2008)138.5552 113.8806 74.2915 63.2869 60.5546 Yang et al.(2003)144.6630 111.7170 78.8170 69.0750 63.3140 2 Neves et al.(2013)a 141.6779 114.3788 76.9725 66.6482 61.9329 Neves et al.(2013)b 142.4281 115.2134 77.6639 67.1883 62.4886 present 144.9942 118.3963 79.1736 68.5137 63.8204 Pradyumna et al.(2008)138.5552 114.0266 74.3868 63.3668 60.6302 Yang et al.(2003)145.7400 112.5310 79.4070 69.6090 63.8060 3 Neves et al.(2013)a 141.8485 114.5495 77.0818 66.7332 62.0082 Neves et al.(2013)b 142.6024 115.3665 77.7541 67.2689 62.5668 present 145.1461 118.5338 79.2588 68.5822 63.8848 Pradyumna et al.(2008)195.5366 160.6235 104.7687 89.1970 85.1788 Yang et al.(2003)206.9920 159.8550 112.4570 98.3860 90.3700 4 Neves et al.(2013)a 199.1566 160.7355 107.9484 93.3350 86.8160 Neves et al.(2013)b 200.3158 162.0337 108.9677 94.0923 87.6341 present 204.4336 166.8808 111.3657 96.2167 89.6707

Three different boundary conditions are studied which are CFFF,SSSS and CSSF.We solve these problems using a uniform 20×20×1 mesh with DPH20 elements,as well as using NASTRAN.The comparison between the meshes by NASTRAN(with 0.33 million DOFs)and the present method is shown in Fig.19.The nondimensional frequency parameter used for comparison iswhichDm=Emh3/12(1?v2m).The first six mode shapes along with the corresponding non-dimensional frequency parameters are shown in Figs.20–22.Very good agreement is obtained for each of the different cases.

4 Conclusion

Through extensive numerical results of static and dynamic analyses of functionallygraded plates and shells,it is demonstrated that the proposed DPH20 and DPH27 elements are entirely capable of accurately and ef?ciently predicting the static and dynamical behaviors of FG structures in a very simple and cost-effective manner.Because higher-order and layer-wise plate and shell theories involve(1)postulating very complex assumptions of plate/shell kinematics in the thickness direction,(2)defining generalized variables of displacements,strains,and stresses,and(3)developing very complex governing equilibrium,compatibility,and constitutive equations in terms of newly-defined generalized variables,while the currently proposed DPH20 and DPH27 elements merely involve displacement DOFs at each node,and rely only on the simple theory of solid mechanics,it is thus concluded by the authors that the development of higher-order or layer-wise theories are not entirely necessary for analyses of FG structures.

Acknowledgement:This research is supported by the Mechanics Section,Vehicle Technology Division,of the US Army Research Labs.The support of National Natural Science Foundation of China(grant No.11502069)and Natural Science Foundation of Jiangsu Province(grant No.BK20140838)is also thankfully acknowledged.

在GeoIPAS軟件平臺(tái)支持下，以地球化學(xué)圖為依據(jù)，結(jié)合區(qū)內(nèi)各元素分布分配特征和地質(zhì)構(gòu)造條件，確定單元素異常下限并圈定單元素異常。異常下限計(jì)算采用對(duì)數(shù)計(jì)算公式：

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1Taizhou Polytechnic College,China.

2School of Aeronautic Science and Engineering,Beihang University,China.

3Corresponding Author,Email:dong.leiting@gmail.com

4Department of Mechanical Engineering,Texas Tech University,USA.