A strategy resisting wrinkling of sandwich structures reinforced using functionally-graded carbon nanotubes

Xiohui REN, Senlin ZHANG, Zhen WU,*

a School of Mechanical Engineering, Xi’an Aeronautical University, Xi’an 710065, China

b School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, China

KEYWORDS

Abstract Sandwich structures have been widely applied in the wing and the horizontal tail of the aircraft,so face sheets of such structure might occur wrinkling deformation in the process of service,which will largely decrease capability of sustaining loads.As a result,this paper aims at proposing a reasonable strategy resisting wrinkling deformation of sandwich structures.To this end, an enhanced higher-order model has been proposed for wrinkling analysis of sandwich structures.Buckling behaviors of a five-layer sandwich plate are firstly analyzed,which is utilized to assess performance of the proposed model.Subsequently, wrinkling behaviors of four sandwich plates are further investigated by utilizing present model, which have been evaluated by using quasi threedimensional(3D)elasticity solutions,3D Finite Element Method(3D-FEM)results and experimental datum.Finally,the present model is utilized to study the buckling and the wrinkling behaviors of sandwich plates reinforced by Carbon Nano Tubes (CNTs).In addition, influence of distribution profile of CNTs on wrinkling behaviors has been analyzed, and a typical distribution profile of CNTs has been chosen to resist wrinkling deformation.Without increase of additional weight,the present strategy can effectively resist wrinkling deformation of sandwich plates, which is rarely reported in published literature.

1.Introduction

The lightening design is the eternal goal pursued by the aircraft designers, so that sandwich structures have been widely adopted in aircraft structures,such as the wing,the horizontal tail, and the rudder.For the sandwich structures subjected to the complicated loads, the wrinkling behaviors might occur earlier than other failure modes.Therefore,it is very necessary to present a reasonable strategy resisting the wrinkling deformation of sandwich structures.To achieve this goal, an accurate and efficient model is required to capture the wrinkling mechanism of sandwich structures.After the characteristics of wrinkling deformation are mastered,this work attempts to propose an approach to prevent the wrinkling deformation of sandwich structures reinforced by using functionally-graded carbon nanotubes.

Many investigators have attempted to study the buckling and the wrinkling response of the sandwich panels.Frostig1proposed a higher-order theory for the buckling and the skin wrinkling of sandwich plates,where the laminated plate theory was employed for the skins, and the assumed stress distribution was utilized for the soft core.The unified theory has been proposed by Niu and Talreja2for the buckling and wrinkling analysis of sandwich plates with isotropic skins.Hadi and Matthews3employed this model to study the wrinkling behaviors of anisotropic sandwich plates.Vonach and Rammerstorfer4presented an analytical solution for the wrinkling behaviors of sandwich plates by means of the Rayleigh-Ritz method.Dafedar et al.5utilized a Layerwise higher-order theory for buckling analysis of sandwich plates.In addition, Birman and Bert6attempted to study wrinkling behaviors of sandwich plates made up of composite skins subjected to biaxial loads.An analytical method has been developed by Aiello and Ombres7to investigate buckling behaviors of sandwich plates made up of hybrid composite skins and the flexible core.Kheirikhah et al.8constructed a higher-order model to study buckling behaviors of the soft-core sandwich plates by means of the energy method.In addition,Kheirikhah et al.9extended the higher-order model to investigate biaxial wrinkling behaviors of sandwich plates with flexural core.Gu and Asaro10studied the skin wrinkling behaviors of sandwich panels subjected to the thermal–mechanical environment.More investigations on the wrinkling behaviors of sandwich panels can be found in the articles11–13.

Shen and Zhu14have studied the buckling and the postbuckling behaviors of the sandwich plates reinforced by Carbon Nano Tubes (CNTs).Moreover, effect of the functionally graded reinforcement of face sheets on the buckling behaviors has been explored.On the basis of the firstorder shear deformation theory, Malekzadeh and Shojaee15investigated the buckling behaviors of laminated plates reinforced with the carbon nanotubes, and the influence of distributions of the CNTs on critical loads has been also explored.By means of Reissner’s mixed variational theorem, Wu and Chang16proposed a finite layer method for buckling analysis of laminated plated reinforced by the CNTs.Based on the first-order theory, Lei et al.17employed the meshless method to study buckling behaviors of laminated plates reinforced by the CNTs.Kumar and Srinivas18utilized the layerwise formulation to study free vibration,buckling and bending behaviors of laminated plates reinforced by the multi-walled carbon nanotubes.Based on the refined zig-zag model,19Di Sciuva and Sorrenti20employed the Ritz method to investigate the static and dynamic behaviors of sandwich plates reinforced by the CNTs.In terms of the Carrera’s unified formulation,Naderi21studied free vibration of annular sector sandwich plates reinforced with the CNTs.On the basis of the hybrid higher-order theory, Van Do et al.22employed the isogeometric analysis method to explore the bending and buckling behaviors of composite plates reinforced by the functionally graded carbon nanotubes.Thakur et al.23employed the nonpolynomial shear deformation theory to study dynamic behaviors of composite folded plates.Furthermore, the proposed model was extended to investigate dynamic response of flat and folded composite plates subjected to hygrothermal conditions24.By utilizing finite element method based on the modified strain gradient theory, Alimirzaei et al.25studied the nonlinear static, buckling and vibration response of the micro-composite beams reinforced by the boron nitrid nanotube.Based on the refined higher-order theory,Yogesh et al.26proposed an isoparametric quadrilateral element to investigate the size-dependent vibration behaviors of the porous graded nanostructures.By applying the isogeometric finite element formulation based on the strain gradient theory,Thanh et al.27studied the linear and nonlinear behaviors of the functionally graded material nanoplate, where the Newton-Raphson iteration technique was employed to obtain the nonlinear responses.To improve accuracy in predicting frequencies of the functionally graded nanoplates, the stiffness matrix of finite element has been modified and enhanced by using the machine learning technique.28In terms of the improved firstorder shear deformation theory, Vinh et al.29presented a four-node quadrilateral plate element to predict bending and buckling response of the functionally graded plates with porosity,where a good performance of the proposed model has been evaluated by comparing with the published results.More articles on finite element formulation for sandwich plates can be found in review paper presented by Irfan and Siddiqui30.

In addition, Qiu et al.31utilized the interval analysis and probabilistic approach to study buckling behaviors of composite plates.Hu and Ke32proposed a nonlinear constitutive model to analyze the failure behaviors of composite laminates subjected to uniaxial compression.Wu et al.33proposed a finite element formulation to study the influence of aperture on dynamic behaviors of sandwich plates, and these results have been verified by experiments.Based on the integral first-order beam theory, Bourada et al.34studied the dynamic and buckling behaviors of the concrete beams reinforced by the CNTs.Bendenia et al.35also employed the first-order theory to investigate the static and dynamic response of nanocomposite sandwich plates.By means of the higher-order model,Rachid et al.36researched the bending response of composite beams reinforced by the CNTs,where effects of nonlinear distribution of CNTs on mechanical behaviors of nanocomposite beams were investigated in detail.On the basis of the refined third-order model, Al-Furjan et al.37studied bending behaviors of composite disk reinforced by functionally gradedgraphene nanoplatelets.Huang et al.38attempted to study static stability of doubly curved micro-shell panels reinforced by the CNTs through using the higher-order shear deformation model with seven unknowns.Kong et al.39utilized both the first-order model and the third-order model to investigate vibrations of the electrorheological sandwich disk.

By reviewing literature, it is found that many investigators have attempted to study the wrinkling behaviors of sandwich panels.Furthermore, the carbon nano tubes have been employed to resist the buckling deformation of the laminated plates and the sandwich panels.To the best of our knowledge,however, the strategies resisting the wrinkling of sandwich structures by using functionally-graded carbon nanotubes are less investigated.Therefore, this paper focuses on presenting a reasonable strategy resisting the wrinkling deformation of sandwich structures.To this end, an accurate and efficient model considering transverse normal strain is required to capture the wrinkling mechanism of sandwich structures, as the wrinkling behavior of sandwich panels is a typical threedimensional issue.Thus, an enhanced higher-order model is proposed to study the characteristics of wrinkling deformation.Compared with the quasi three dimensional elasticity solution and the 3D-FEM results,the performance of the present model will be verified by analyzing the typical wrinkling of sandwich panels.In addition, the recent published models are also selected to assess the present model.Subsequently, the present model is utilized to study the effect of distributions of the CNTs on wrinkling deformation of sandwich panels.After the influencing mechanism of the CNTs distributions on the wrinkling behaviors of sandwich panels is mastered, a strategy resisting the wrinkling of sandwich structures can be proposed without increasing additional weight.

2.An enhanced higher-order plate model containing transverse normal strain

When the stiffness of face sheets is significantly different from that of the core, the sandwich plates subjected to in-plane loads might encounter the wrinkling deformation of face sheets, which will largely reduce the load-bearing capacity of the sandwich structures.As a result, it is necessary to propose an accurate model to predict wrinkling behaviors of sandwich structures,and a strategy will be required to oppose the occurrence of wrinkling deformation.The typical feature of wrinkling behaviors in sandwich structures is that the face sheets are subjected to bending deformation,whereas the core is subjected to the compressive deformation.To accurately capture the inconsistent deformation of the face sheets and the core,an enhanced higher-order plate model will be proposed in this work, in which the local functions will be utilized to describe the local deformation of face sheets.Thus, the starting displacements at the kth ply can be expressed as follows

It is observed that the local displacement functions uLand vLhave been introduced in the in-plane displacement field,which are utilized to improve the capability simulating the local deformation of the face sheets.The local displacement functions uLand vLcan be acquired by using the Chebyshev polynomials.With the increase of the order number of the Chebyshev polynomials, the accuracy of the model can be obviously improved, but displacement variables will be increased.Moreover, the additional displacement variables will be retained in the final displacement field.By comprehensively considering the efficiency and accuracy, the secondorder Chebyshev polynomial will be employed to construct the local displacement functions, which can be presented as follows

In the existing higher-order models,40–41the third-order polynomial according to the thickness coordinate z is generally utilized to simulate the distribution of in-plane displacements through the thickness direction.However,it is difficult to predict the wrinkling response of sandwich plates by using the third-order polynomial.To accurately describe the global deformation of sandwich plates, the fifth-order polynomial according to the thickness coordinate z is employed to obtain the global displacement components.The fifth-order global displacement polynomial can be written as follows

where,u0,v0and w0signify the displacements along x and y at any points of midplane, respectively; displacement parameters ui,viand wi(i=1–5)denote the higher-order terms in the Taylor’s series expansion across z coordinate.

The displacement polynomial in Eq.(4) can automatically fulfill the continuity at the interfaces of adjacent layers.If the local displacement functions in Eq.(2) are added into the in-plane displacement components, the in-plane displacement component can be regenerated, in which the rebuilding displacement components are unable to meet the continuity conditions at the interfaces.Therefore,the consistency of in-plane displacements at the interface of adjacent layers will be enforced to eliminate the additional local variables.Through utilizing the continuity conditions of in-plane displacement components, the following equation can be presented as follows

For a sandwich structure without any damages, transverse shear stresses are required to satisfy the coincident conditions at the interfaces of adjacent plies.The continuity conditions of transverse shear stresses at the interfaces can be expressed as follows

Employing relationships of the strains and the stresses, the transverse shear stresses at the kth layer of a sandwich plate can be expressed as follows

In addition, this work aims at the buckling and the wrinkling analysis of the sandwich structures subjected to compression loads.As a result, transverse shear stresses on the upper and the lower surfaces are required to be zero, namely the transverse shear free conditions.After all conditions are enforced, the finial displacement field of the present model can be expressed as follows

where the coefficients Φiand Ψiare the functions of material constants and thickness coordinate z.

3.Analytical formulation for buckling and wrinkling analysis

3.1.Governing equations

Taking all stress components σx, σy, σz, τxy, τxzand τyzinto account,the governing equations of a sandwich plate subjected to the in-plane stresses Sxand Sycan be acquired by using the principle of virtual displacements, where the in-plane stresses Sxand Syare assumed to remain constant during the buckling and the wrinkling procedure.The principle of virtual displacements for the buckling and the wrinkling procedures can be written as follows

in which, U= [u v w]Tdenotes the displacement vector; V denotes the volume occupied by a whole plate.Relationship between the stress vector and the strain vector can be given by

where Qijcan be found in Appendix A.

In terms of the present model, the strain components for a sandwich plate can be presented by

Implementing integration of the Eq.(10) by parts and collecting the variational displacement parameters δu0, δui, δv0,δvi, δw0and δwi(i = 1, 2, 3, 4, 5), the governing equations by means of the present model can be given by

3.2.Analytical solution for simply-supported plates

For a simply-supported rectangular plate,the boundary conditions on the boundaries x=0 and x=a can be presented by

In addition, the boundary conditions on the boundaries y = 0 and y = b can be written as follows

According to the Navier’s solution procedure,40–41the generalized displacement parameters fulfilling the simply-supported boundary conditions in Eqs.(15)and(16)can be presented by

where i = 1–5; a and b denote the length and the width of a plate, respectively.

Through substituting Eq.(17)into the Eq.(14),the governing equations can be rewritten by collecting the coefficients for the generalized displacements of any fixed values r and s.The displacement vector U-can be expressed as follows

For the buckling and the wrinkling problems, the stability equation can be written as the following eigenvalue problem:

where the matrix K represents the stiffness matrix, and the matrix S signifies the geometric stiffness matrix caused by the in-plane stresses.The parameter Λ denotes the buckling or wrinkling load.

4.Numerical results and discussion

If the minimum wrinkling load is less than the minimum buckling load of the sandwich structures, it means that the wrinkling failure is earlier than the buckling failure, which will significantly decrease the load-bearing capability of the sandwich structures.In this work, we will attempt to employ the Functionally Graded (FG) face sheets reinforced by Carbon Nano Tubes (CNTs) to replace the aluminum or composite face sheets.Such strategy can effectively resist the wrinkling deformation of sandwich structures by designing the reasonable distribution of CNTs.Thus, the FG face sheets of sandwich plates can be reinforced by the (10,10) armchair singlewalled CNTs.42Furthermore, the extended rule of mixtures17will be employed to obtain material properties of the FG face sheets.On the basis of the extended rule of mixtures given by Lei et al.,17the material properties of the FG face sheets reinforced by CNTs can be expressed as

Furthermore,the volume fractions of CNTs corresponding to the five distribution profiles through the thickness of a plate have been demonstrated by Di Sciuva and Sorrenti.20Volume fractions of the five distribution profiles have been shown in Table 1,where tksignifies the thickness of the kth ply.The chosen distribution profiles of CNTs through the thickness can be observed in Fig.1.The parametercan be expressed as follows

in which, mcrefers to a mass fraction of the CNTs; ρcand ρmrespectively signify the densities of the CNTs and the homogenous matrix.

Acronyms demonstrating the different distribution profiles of CNTs through the thickness of a plate have been shown in Table 1, where the distribution profiles UD, FG-?, FG-?,FG-◇and FG-X can be found in Fig.1.Geometry of a three-layer sandwich plate can be found in Fig.2.

5.Numerical examples

In this section, buckling behaviors of a sandwich plate composed of aluminum skins and soft core are firstly analyzed.Moreover, the quasi 3D elasticity solutions5and the results of 3D-FEM in ABAQUS have been employed to verify the performance of the proposed model.Subsequently, the wrinkling behaviors of the sandwich plates made up of aluminum or composite skins are also investigated.To demonstrate distinction between the buckling and the wrinkling behaviors of the sandwich plates, the displacement modes with respect to buckling and wrinkling behaviors acquired from the proposed model have also been shown.To resist occurrence of wrinkling behaviors in sandwich plates,the CNTs have been proposed to enhance stiffness of the face sheets.Moreover,the influence of distributions of the CNTs along the thickness of face sheets onthe buckling and the wrinkling behaviors is also studied in sandwich plates, the CNTs have been proposed to enhance stiffness of the face sheets.Moreover,the influence of distributions of the CNTs along the thickness of face sheets on the buckling and the wrinkling behaviors is also studied.

Table 1 Volume fractions of different distribution profiles of CNTs through thickness direction of a plate.

Example 1.Buckling behaviors of a five-layer sandwich plate subjected to uniaxial compression along x direction are studied.The sandwich plate is made up of the composite face skins and the soft core,and material properties of the composite face skins and the soft core5can be expressed as follows.

Firstly, buckling behaviors of a five-layer sandwich plate are analyzed by using the present model and the 3D-FEM,and all results have been presented in Table 2.5,24,44For thick sandwich plates(a/h<10),it can be observed that the present model, 3D-FEM and HYF135can produce the minimum buckling loads as the half-wave number is equal to one.Nevertheless, the models HYF23 and BHSDT24will produce the minimum buckling loads as the half-wave number is equal to two.Furthermore, the models HYF23,5HSDT44and BHSDT24largely overestimate buckling loads of sandwich plates in comparison with the quasi 3D model HYF135.

Fig.2 Geometry of a three-layer sandwich plate.

For a moderately thick plate (a/h = 10), present result agrees well the result of the quasi 3D model HYF135and the 3D-FEM results.Moreover, the minimum buckling loads of these models can be obtained as the half-wave number is equal to two.However, the minimum buckling loads of the models HYF235and BHSDT24are acquired as the half-wave number is equal to three.For the present model, 3D-FEM and BHSDT,24comparison of buckling loads with increase of half-wave number for a five-layer sandwich plate (a/h = 10) has been plotted in Fig.3.The 3D-FEM denotes the results acquired by utilizing 4 × 105C3D8R elements in ABAQUS,whereas the 3D-FEM*signifies the results obtained by using 7.84 × 105C3D8R elements in ABAQUS.It can be found that a large number of C3D8R elements have to be used to improve accuracy of buckling loads with increase of halfwave number, which will cost a large amount of computing time.

Example 2.Wrinkling behaviors of two sandwich plates subjected to uniaxial compression along x direction and made up of the aluminum or composite face sheets and the soft core are investigated.

Material properties of the aluminum or composite face skins and the soft core5can be presented by.

Aluminum face skins:Ef=70 GPa,vf=0?3,tf=0?65 mm;

Fig.1 Distribution configuration of CNTs along thickness of a plate.

Table 2 Comparison of buckling loads for a five-ply sandwich plate.

Fig.3 Variation of buckling loads with increase of half-wave number for a five-layer sandwich plate (a/h = 10).

This case will study the wrinkling behaviors of sandwich plates,and the influence of adhesive layers on wrinkling behaviors is also explored in detail.Firstly, wrinkling behavior of a square sandwich plate made up of aluminum skins and soft core (Sandwich A) is investigated, where the length and the height are 228 mm and 26.3 mm and the adhesive layers are not contained.In Table 3,3,5,45it is found that the present result is in good agreement with the 3D-FEM result, result of HYF135and analytical solutions.3,45However, result of HYF23 is less accurate.To demonstrate the difference between the buckling and the wrinkling behaviors,displacement modes of a sandwich plate with the length and the height of 100 mm and 26.3 mm are plotted in Fig.4.By observing Fig.4, it is shown that the sandwich plate is subjected to buckling deformation as the half-wave number is less than 7,where deformation of midplane is in accordance with those of the top and the bottom surfaces.Nevertheless, as the half-wave number is more than 6, the sandwich plate will subject to the wrinkling deformation.For the wrinkling behaviors,the midplane is subjected to the compressive deformation,whereas the top and the bottom surfaces will subject to the bending deformation.Fig.5 shows the variation of buckling and wrinkling loads with the increase of half-wave number for a three-layer sandwich plate(tf=0.65 mm,tc=25 mm,a=b=100 mm), and it can be seen that critical loads obtained from the present model and the 3D-FEM suddenly drop as the half-wave number is more than 6.Moreover, the minimum wrinkling load is obviously lower than the minimum buckling load, which will decrease the resistance to compression.In addition, the models BHSDT24and SHSDT46lose the capability to predict the wrinkling behaviors of a sandwich plate as an effective transverse normal strain has been neglected.In addition, wrinkling loads of a sandwich plate with incorporation of adhesive layers(Sandwich A*) are also calculated.Moreover, it is found that the present results agree well with the results of 3D-FEM and HYF13 in Table 3.Subsequently,the wrinkling behaviors of a sandwich plate composed of the composite face sheets and the soft core will be analyzed.Firstly, the sandwich plate without incorporation of adhesive layers(Sandwich B)is analyzed,and the present result agrees well with the analytical solutions3,45.Furthermore,the wrinkling behaviors of a sandwich plate with incorporation of adhesive layers (Sandwich B*) are further analyzed by using the present model.Moreover, the presentresults are compared with those of 3D-FEM, HYF13,5HYF23,5analytical solutions3,45and experimental datum.45Numerical results show that the present result is in good agreement with the analytical solutions3,45and experimental datum.45Therefore, the capability of the proposed model has been verified by using the quasi 3D model,53D-FEM and experimental datum, so that the proposed model will be extended to investigate the buckling and the wrinkling behaviors of the sandwich plates with face sheets reinforced by the CNTs.

Table 3 Comparison of wrinkling loads for three-ply sandwich plates (a = b = 228 mm, tc = 25 mm).

Fig.4 Displacement modes corresponding to buckling and wrinkling behaviors computed by using present model (tf =0.65 mm, tc = 25 mm, a = b = 100 mm).

Fig.5 Variation of buckling and wrinkling loads with increase of half-wave number for a three-layer sandwich plate (tf =0.65 mm, tc = 25 mm, a = b = 100 mm).

Example 3.Influence of the CNT distributions on wrinkling behaviors of the sandwich plates subjected to uniaxial compression along x direction is to be studied.

Firstly, effects of volume fractionof CNTs on critical loads of the sandwich plate will be studied.For a thick sandwich plate reinforced by uniform distribution of carbon nanotubes (UD-UD, a/h = 2, tc/tf= 50), the buckling and wrinkling loads with increase of half-wave number have been shown in Table 4.It is clearly shown that the sandwich plate subjects to the buckling deformation when the half-wave number is equal to 1,2 and 3.By improving volume fractionof CNTs, the minimum buckling load can be largely increased.As the half-wave number is more than 3, the sandwich plates are mainly subjected to the wrinkling deformation.With the increase of the half-wave number, the wrinkling loads show a trend from decline to rise.However, it is interesting that the critical loads with asterisk in Table 4 gradually decrease with increase of the half-wave number again, which can be clearly found in Fig.6.To explain such phenomenon,displacement modes of the half plate corresponding to buckling and wrinkling behaviors computed by using present model (UDUD, a/h = 2, tc/tf= 50,=0?28) have been plotted in Fig.7.It can be clearly observed that the sandwich plates are mainly subjected to the in-plane buckling deformation(Buckling*) as the half-wave number is equal to 10, 11 and 12, which attributes to enhancement of the bending stiffness in the sandwich plate.Furthermore,the buckling and the wrinkling loads of the moderately thick sandwich plates (a/h = 5)are also presented in Table 4.In addition, these results areclearly shown in Fig.8.With the increase of volume fractionof CNTs, it is found that the buckling and the wrinkling loads can be largely increased, whereas the minimum wrinkling load of the sandwich plates are still less than the minimum buckling load.

Table 4 Comparison of buckling loads and wrinkling loads for a three-ply sandwich plate reinforced by functionally-graded carbon nanotubes (UD-UD, tc/tf =50).Volume fractionm

Fig.6 Effect of CNTs volume fraction on critical loads of the sandwich plate reinforced by functionally-graded carbon nanotubes (UD-UD, a/h = 2, tc/tf = 50).

Fig.7 Displacement modes corresponding to buckling and wrinkling behaviors computed by using present model (UD-UD,a/h = 2, tc/tf = 50, =0?28).

Fig.8 Effect of CNTs volume fraction r on critical loads of the sandwich plate reinforced by functionally-graded carbon nanotubes (UD-UD, a/h = 5, telr = 50).

To avoid the phenomenon that the wrinkling deformation of the sandwich plates occurs earlier than the buckling deformation, this work will investigate the influence of distribution profiles of CNTs along the thickness on the buckling and wrinkling loads.For a moderately thick sandwich plate(tc/tf=50,a/h=5),the buckling and wrinkling loads of sandwich plates with different distribution profiles of CNTs have been shown in Table 5.For the CNTs volume fraction= 0?12,the minimum buckling loads with respect to the UD-UD profile, the?-?profile,the ?-?profile,the ◇-◇profile and the X-X profile are respectively 5.2906, 5.1821, 5.0938, 5.0543 and 5.5299.In addition, the minimum wrinkling loads with respect to the UD-UD profile, the ?-?profile, the ?-?profile, the ◇-◇profile and the X-X profile are respectively 3.3588,2.7704,2.7907,2.4115 and 4.1010.Numerical results show that the utilization of the X-X configuration can largely improve the capability of the sandwich plates resisting the buckling and the wrinkling deformations.

Moreover, distribution profiles of CNTs have more influences on the wrinkling loads than the buckling loads, which can be found in Fig.9.For the fixed half-wave number (r), it can also be found that wrinkling load of sandwich plate with face sheets reinforced by the X-X profile of CNTs is obviously greater than those of sandwich plate with face sheets reinforced by other profiles.With the increase of the half-wave number,wrinkling loads of sandwich plate with the X-X profile are much greater than those of sandwich plates with other profiles.Therefore, the X-X distribution profiles of CNTs should be selected to resist the wrinkling deformation.Wrinkling phenomenon of sandwich plates shows that face sheets produce the local bending deformation with the increase of compressive loads.By utilizing the X-X distribution profiles of CNTs,bending stiffness of face sheets can be largely improved,as the CNTs mainly distributed on the upper and the lower surfaces of face sheets.As a result, using the X-X profile of CNTs, capability resisting wrinkling deformation of sandwich plate can be obviously improved.However,wrinkling loads are still less than the buckling loads, which means that wrinkling deformation still occurs earlier than buckling deformation.

In Table 5, the minimum buckling loads (= 0?28) with respect to the UD-UD profile,the ?-?profile,the ?-?profile,the ◇-◇profile and the X-X profile are respectively 5.9951,5.6587,5.5775,5.4172 and 6.3241.Nevertheless, the minimum wrinkling loads with respect to the UD-UD profile, the ?-?profile, the ?-?profile, the ◇-◇profile and the X-X profile are 5.1885, 4.2484, 4.2760, 3.6843 and 6.3591, respectively.It is excited to find that the minimum wrinkling load of the sandwich plate reinforced by the X-X distribution profile of the CNTs is more than the minimum buckling load, which means that the buckling deformation occurs earlier than the wrinkling deformation.In Fig.10,it can also be found that the minimum wrinkling loads of sandwich plates reinforced by the UD-UD profile, the ?-?profile, the ?-?profile and the ◇-◇profile of the CNTs are still less than buckling loads,although the CNTs volume fraction has been increased.As a result, the wrinkling behaviors of sandwich structures can be resisted by using the face skins reinforced by the X-X distribution profile of the CNTs.Therefore, a strategy resisting the wrinkling deformation of sandwich structures can be acquired by reasonably designing the distribution profiles of CNTs through the thickness of face sheets.

Table 5 Influence of distributions of CNTs through thickness on buckling loads and wrinkling loads for a three-ply sandwich plate reinforced by functionally-graded carbon nanotubes(tc/tf =50, a/h=5).

Fig.9 Effect of distributions of CNTs along thickness on buckling and wrinkling loads for a three-ply sandwich plate reinforced by functionally-graded carbon nanotubes(tc/tf=50,a/h = 5, =0?12).

Fig.10 Effect of distributions of CNTs along thickness on buckling and wrinkling loads for a three-ply sandwich plate reinforced by functionally-graded carbon nanotubes(tc/tf=50,a/h = 5, =0?28).

6.Conclusions

Compared with the laminated composite structures, the sandwich structures under in-plane compressive loads possess the unique behaviors,namely wrinkling deformation.The winkling deformation is the typical 3D mechanical behaviors, so the models neglecting transverse normal strain will lose the capability to predict such issue.Therefore, an accurate higher-order model has been developed for the buckling and the wrinkling analysis of the sandwich structures.After the typical cases are chosen to evaluate the performance of the present model, the proposed model is utilized to study the mechanical behaviors of the sandwich plates reinforced by the CNTs.In addition,the strategy resisting the wrinkling of sandwich structures has been proposed by using functionally-graded carbon nanotubes,and the valuable conclusions have been given by.

(1) The proposed higher-order plate model can fulfill the continuity conditions of transverse shear stresses at the interfaces, and effect of transverse normal deformation is also taken into account.Thus, the proposed model can reasonably predict wrinkling behaviors of sandwich plates, whereas the exiting higher-order models discarding transverse normal strain have no capability to analyze the wrinkling deformation of sandwich plates.

(2) For a thick five-layer sandwich plate (a/h < 10), the buckling loads obtained from the present model are in excellent agreement with the 3D-FEM results and the quasi 3D elasticity solutions.However, the other models largely overestimate the buckling loads, and these buckling loads are acquired as the half-wave number is equal to 2,which is different from the 3D-FEM results and the quasi 3D elasticity solutions.For the moderately thick plate, the present model can still accurately predict buckling loads.

(3) By analyzing the wrinkling behaviors of four sandwich plates with or without incorporation of adhesive layers, it shows that the present model can accurately produce the wrinkling loads, which have been verified by using the 3D-FEM results and the testing Datum.Moreover, incorporation of adhesive layers has a significant influence on the wrinkling loads of the sandwich plates.

(4)As the face sheets are reinforced by the CNTs,the buckling and the wrinkling loads can be obviously increased.Moreover, distributions of CNTs have more influences on the wrinkling loads than the buckling loads.In addition,the wrinkling behaviors of the sandwich structures can be resisted by using the face sheets reinforced by CNTs with the X-X distribution profile and improving the volume fraction of CNTs.Thus,the X-X distribution profile is proposed to reduce the occurrence of the wrinkling behaviors in sandwich structures.

Declaration of Competing Interest

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

Acknowledgements

The work described in this paper was supported by the National Natural Sciences Foundation of China (No.12172295) and SKLLIM1902, China.

Appendix A.The transformed elastic constant matrix can be written as follows40: