Buckling Characteristics and Influencing Factors of Composite Grid Sandwich Plates with Soft Core

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（College of Naval Architecture and Ocean,Naval University of Engineering,Wuhan 430033,China）

Abstract: The buckling characteristics and influencing factors of a composite grid sandwich plate with soft core materials are investigated based on the high-order shear deformation theory. Firstly, the equivalent elastic parameters of the soft core with composite grid reinforced are obtained by using the method of orthotropic homogenization. Furthermore, the buckling governing equations are derived based on the high-order shear displacement field assumption and energy method, and the theoretical solution of the buckling of a simply-supported grid sandwich plate under in-plane compressive loads is given. Finally, the parameters’influence on typical grid sandwich plates is discussed in the exam⁃ple, such as the span-thickness ratio, surface layer to total thickness ratio, grid spacing and grid com⁃posite layup angle.

Key words:grid;sandwich plate;higher-order shear deformation theory;buckling;energy method

0 Introduction

The sandwich composite structures are composed of stiffer surface layers and softer core mate⁃rials,which have the advantages of high rigidity and functional-structure integration,and thus have been increasingly used in aerospace and marine structures[1-3].

The grid sandwich plates containing composite grids inside the core can not only fully utilize the high rigidity of the sandwich structure, but also effectively overcome the disadvantages of the conventional sandwich plates, such as the large shear deformation, easy extension of local damage and difficult selection of core materials which need to have both excellent mechanical and function⁃ality properties. The research on grid sandwich structures and other grid structures (single-layer spatial grid structures and grid stiffened structure) has been the target of intense research in the field of new composite material structures[4-11].

In the existing research, some work on grid sandwich plate with the core materials through the methods of simulation and experiment has been carried out.Sun[7-8]studied the compressive proper⁃ties of composite sandwich structures with grid reinforced honeycomb core, through the in-plane compression tests and finite element analysis, it was indicated that the combined core sandwich specimens provided increased stiffness, specific stiffness, energy absorption and critical load,which were higher than the sum of honeycomb core sandwich specimens and grid core sandwich specimens. Sharaf and Fam[9-10]studied the effects of foam core density and grid ribbed form on the performance of small-scale composite grid sandwich panels by finite element simulation, and ob⁃tained the optimized grid spacing under its material system, and later carried out experiments and simulation studies on the stiffness and strength of large-scale shell plates under uniform load and concentrated load. Lu[11]used ANSYS finite element software to simulate the vibration attenuation performance and its influence parameters of the grid sandwich plate, and verified it through experi⁃ments. Through the query of the current literature, the theoretical research on the buckling of grid sandwich panels with soft core materials is still rare.

In order to obtain the accurate predictions of buckling characteristics, a number of plate theo⁃ries have been derived to analyze the composite sandwich plates, including CLPT (classical plate theories), FSDT (first-order shear deformation theories) and HSDT (higher-order shear deformation theories). The CLPT yields acceptable results only for the analysis of thin plates, neglecting the ef⁃fect of transverse shearing on deformation. In order to take into account the effects of transverse shearing deformation, the FSDT have been developed. However, the accuracy of FSDT depends on the shear correction factor which may be difficult to compute. The HSDT provide better accuracy for transverse shear stresses without the need of a shear correction factor.

1 Equivalent elastic parameters of the soft core with composite grid reinforced

Fig.1 Schematic diagram of grid sandwich plate with soft core

In order to consider the grid’s effect on the elastic parameters of the core materials,the orthogonal anisotropic equivalent elastic parameters of the soft core with grid reinforced in this paper are carried out based on the energy method in Ref.[12]and the elastic parameters are solved. Different from Ref.[12], where the grid’s material is isotropic, the grids in this paper are made of orthotropic composite laminate. The distribution of grids in the sandwich plate and selection of typical repre⁃sentative units are shown in Fig.2.

Fig.2 Selection of typical representative units

wheretbis the thickness of grid，lcyis 1/2 of the width of the core unit inydirection，andlcxis 1/2 of the width of the core unit inxdirection，the unit of all these is mm.

To consider the effect of composite layup angles, it is assumed that the Young’s modulus along the length of the grids’laminate isEbx. By deriving, the calculation equations of the orthotro⁃py equivalent parameters (such asEc1,Ec2,μc12,Gc13,Gc23andGc12) of the soft core with composite grids can be concluded as follows:

whereEbxrepresents Young’s modulus in the length direction of laminate in the grid,Gbxzrepre⁃sents in-plane shear modulus of laminate in grid,μbxyrepresentsxy-direction Poisson’s ratio of laminate in grid,Eprepresents Young’s modulus of core,μprepresents Poisson’s ratio of core,Gprepresents shear modulus of core, andDrepresents bending stiffness of the laminated grid, which can be calculated by the following bending theory of laminated composite beams:

and for the symmetric laminates,B11= 0,thenD=D11.

By Eq.(2), the orthotropic equivalent elastic parameters of the soft core with grid reinforced can be obtained, and the influence of the following parameters on the equivalent parameters can be considered, such as the elastic parameters and layup angle of the grids’laminate, the thickness of the grid, the grids’spacing and the core’s elastic parameters. As Eq.(2) takes all the relevant de⁃sign parameters of the soft core with reinforced grids into account, it can meet the engineering de⁃sign needs.

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2 Higher order shear theory solution for buckling characteristics of grid sandwich plates

In order to inverstigate the buckling char⁃acteristics of the above grid sandwich plate and consider the influence of shear deformation on the deflection, high-order shear theory is used to solve this problem, making the soft core with grid as one layer of the structure, which is equivalent to orthotropic material. The coordi⁃nate system and thez-direction coordinates of each layer of the structure are shown in Fig.3.

2.1 Displacement field hypothesis

The displacement components in the plate are expressed as follows[13]：

Fig.3 Coordinate system of the sandwich plate and coordinates of each layer

whereu0、v0、w0、θxandθyare five unknown parameters.

According to the geometric relationship equation between strain and displacement,and substituting Eq.(4)into Eq.(3),the following can be derived:

2.2 Buckling differential equation

2.3 Calculation equation of internal force and moment

It is assumed that the internal forces and moments are represented by the following vector:

2.4 Solution of buckling load of sandwich plates with four simply-supported sides

In this paper,the Navier method is used to solve the buckling load in single direction (x-direc⁃tion)of a rectangular sandwich plate with four sides simply supported and symmetric cross-ply lam⁃inated surface.It is assumed that the rectangular plate has a length ofaand a width ofb.

The simply-supported boundary conditions are as follows:

whereα=mπ/a,β=nπ/b. Through the above equation, the unknowns in the displacement field shown in Eq.(3)are converted intoUmn,Vmn,Wmn,XmnandYmn.

For a (0°/90°) symmetric cross-ply laminated sandwich panel structure, it can be concluded that:

Using the Matlab programming, the buckling characteristics of the grid sandwich plates with soft core and the influence of related parameters can be calculated and analyzed.

3 Numerical results and discussion

Example 1: To verify the correctness of the equations and procedures written in this paper,the example deals with (0°/90°/0°/90°/0°/core/0°/90°/0°/90°/0°) symmetrically simply-supported conventional sandwich plate with orthotropic-honeycomb core loaded by uniformly-distributed nor⁃mal pressureNx0on the edgesx= 0 andx=a[14].

Cross-ply laminated face sheets are each of thicknesstf,whilehis the total thickness of plate.The sandwich plate is made of following materials[14].

Face sheets(Orthotropic):

Tab.1 Comparison of calculation results of buckling load of conventional sandwich panels

It can be seen from the data in Tab.1 that the results obtained by the method in this paper are very close to those in Ref.[14], and the maximum error is 6.6% (whiletf/h= 0.15 anda/h= 10).The comparison proves the correctness of the relevant theoretical equation and Matlab program in this paper.

Example 2:The influence of composite grid and its design parameters on the buckling load of grid sandwich plates with soft core are stud⁃ied. The materials’parameters of the sand⁃wich panel are as follows: the face sheets are made of glass fiber reinforced composite material, the core is made of soft functional material,and the grids are made of carbon fi⁃ber reinforced composite material. The elas⁃tic parameters of each material in this exam⁃ple are shown in Tab.2.

(1)Influence of composite grids on the buckling load

To study the effect of grids on the buckling load of soft core sandwich panels, the method in this paper is used to calculate the dimensionless buckling load of the conventional sandwich plate(without grid) and grid-enhanced sandwich plate in thexdirection:Nˉcr0andNˉcr1, respectively for different span-thickness ratio (a/h) and ratio of surface to total thickness (tf/h). For conventional sandwich panels, the data of soft core in Tab.1 are used directly as the core’s elastic parameters. For grid-reinforced sandwich pan⁃els, to illustrate the effect of the grids on the buckling characteristics, a sparse grid is used: grid spacingsx=sy=a/25, grid thicknesstb=a/500, the laying angle of the composite material in gridsθ= 45°.The spanthickness ratios and ratio of face sheet to total thick⁃ness change within the following range:tf/h= 1/20-1/10,a/h= 5-100. Through calculation, the curves of ratioγ=Nˉcr1/Nˉcr0for differenttf/handa/hare shown in Fig.4.

Tab.2 Elastic parameters of the sandwich panel material system

Fig.4 Effect of grid on buckling load of sandwich plate

It can be seen that as the ratioa/hincreases,the influence of grids on the buckling load of the sandwich plate is gradually reduced (the reason should be that the effect of shear deformation is re⁃duced). When the span-thickness ratio of the sandwich panel is sufficiently large (i.e., the thin plate),the effect of the grid will be substantially negligible,for example,for a sandwich panel wheretf/h= 1/5,the effect of the grid on the buckling load of the sandwich panel will be less than 5%.

At the same time, the influence of grids on the buckling load of the sandwich panel reduces with the increase of the ratioa/h. For example, for the sandwich panel wherea/h= 20, whentf/h=1/5,the ratioγ=Nˉcr1/Nˉcr0= 1.228,however whentf/h= 1/20,the ratioγ=Nˉcr1/Nˉcr0= 5.096.

(2)Influence of laminate’s laying angle of grids on the buckling loadNˉcr1

In order to study the influence of the laying angle of composite used in grids on the buckling load of grid sandwich plate,the following grid sandwich plates are taken as an example to calculate the dimensionless buckling loadNˉcr1for different layup angles.

The thicknesstbof the grid isa/400,the grid spacingsx=sy=a/20,the ratio of surface to total thicknesstf/h= 1/7, the span-thickness ratioa/h= 20. The material parameters used are as shown in Tab.2. The varia⁃tion of the buckling loadNˉcr1of the sandwich plate with the grid layup angle changing from 0° to 45° is shown in Fig.5.It can be seen that for the material system in this paper, the optimized ply angle of composite used in the grids is 45°,in this case, the buckling loadNˉcr1of the structure is the larg⁃est.

(3)Influence of grids’spacing on the buckling loadNˉcr1

In order to study the influence of grids’spacing on the buckling load of the grid sandwich panels with soft core,the buckling loadNˉcr1is calulated as the grid vertical and horizontal grid spacings(sx=sy=s)changes within range ofa/100-a/5.In this example,the grid’s layup angleθ=45°,the ratio of surface to to⁃tal thicknesstf/h= 1/7,the span-thickness ratioa/h= 20,and the thicknesstb=a/400.

Fig.5 Variation of the buckling load of the sandwich panel with the lay⁃ing angle of the grid composite

Through calculation, the variation law of the buckling loadNˉcr1of the sandwich plate with the grid spacingsis shown in Fig.6. It can be seen that the buckling load of the sandwich plate tends to decrease nonlinearly with the increase of the grid spacing.

Fig.6 Variation of buckling load of sandwich plate with grid spacing

4 Conclusions

To account for the effect of transverse shearing strains of the soft core and the influence of the grids,the buckling behavior of a grid sandwich plate with soft core and the influence of the design parameters of grids are studied theoretically based on the HSDT（high-order shear deformation theory）in this paper, the HSDT provide a better accuracy for calculation of transverse shear deformation than the CLPT, and without the need of a shear correc⁃tion factor in FSDT,which may be difficult to compute.The following conclusion can been drawn:

(1)The enhancement effect of the grids (i.e.the ratioγ=Nˉcr1/Nˉcr0)decreases with the increase of span-thickness ratio (a/h) while the enhancement effect increases as the ratio of surface to total thickness(tf/h)decreases.

(2) With the other parameters remaining unchanged, the buckling loadNˉcr1of the grid sand⁃wich plates tends to decrease nonlinearly with the increase of the grid spacing (sxandsy).

(3) For the material system in this paper, the optimum ply angle of the grid laminate is 45°, in the case of which,the buckling loadNˉcr1of the sandwich plate is the largest.