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1.State Key Laboratory of Mechanics and Control of Mechanical Structures，Nanjing University of Aeronautics and Astronautics，Nanjing 210016，P.R.China；2.Beijing Institute of Space Long March Vehicle，Beijing 100048，P.R.China；3.Computational Structure Technique &Simulation Center，Aircraft Strength Research Institute of China，Xi’an 710065，P.R.China

Abstract:Intelligent structures like zero Poisson’s ratio（ZPR）cellular structures have been widely applied to the engineering fields such as morphing wings in recent decades，owing to their outstanding characteristics including light weight and low effective modulus.In-plane and out-of-plane mechanical properties of ZPR cellular structures are investigated in this paper.A theoretical method for calculating in-plane tensile modulus，in-plane shear modulus and out-of-plane bending modulus of ZPR cellular structures is proposed，and the impacts of the unit cell geometrical configurations on in-plane tensile modulus，in-plane shear modulus and out-of-plane bending modulus are studied systematically based on finite element（FE）simulation.Experimental tests validate the feasibility and effectiveness of the theoretical and FE analysis.And the results show that the in-plane and out-of-plane mechanical properties of ZPR cellular structures can be manipulated by designing cell geometrical parameters.

Key words：cellular structure；zero Poisson’s ratio（ZPR）；mechanical properties；parameter design；morphing application

0 Introduction

Morphing wings are regarded as a developing direction of aircraft，which can significantly improve the aerodynamic and aeromechanics performance，reduce the fuel consumption and expand the flight envelope，depending on flight missions and condi?tions［1-4］.The morphing wings system，as a creative concept emerging in aircraft design，is composed by a morphing skin，a uniformly distributed driving sys?tem with high power energy density and a control device［5-6］.The most important characteristics of morphing skin are low in-plane stiffness and high out-of-plane stiffness to bear the aerodynamic load［5，7］.To obtain optimal performance of morphing skin，many scholars have launched thousands of trials vari?ous mechanism model.Composite corrugated struc?tures have been proposed for morphing skin panels in trailing edge region of a wing by Thill et al.［8］.Af?ter that，the mechanical properties of a series of cor?rugated sandwich structure have been studied sys?tematically［9-10］.However，the corrugated sandwich structure has strong load bearing capability but usu?ally with heavy weight or low morphing capability.Kikuta［11］investigated the mechanical properties of a type of thermoplastic polyurethanes，copolyester elastomer，shape memory polymer，or woven mate?rials that could be used as a skin for a morphing wing.Unfortunately，those materials remain prob?lems of brittleness and thermal fatigue.

Honeycomb structures with a favorable balance of properties containing remarkable lightweight and outstanding mechanical properties are fascinating candidates for application in the fields of morphing skin［12-13］.Chen and Chang et al.［14-15］used sandwiched morphing skin consisted of cellular structure and flexible silicone rubber for morphing aircraft.In addi?tion，honeycomb configurations like SILICOMB［16］and re-entrant［17-18］showed a similar characteristics of high morphing capability and low weight.Among these previous experiments，when honeycombs with positive Poisson’s ratio and negative Poisson’s ratio are bearing the out-of-plane aerodynamic loads，the inherent anticlastic and synclastic curvature would appear，which is a key challenge for their application in morphing wings［19-20］.However，the zero Poisson’s ratio（ZPR）feature can preclude significant in?crease of effective stiffness in horizontal direction by limiting the contraction（or bulging）in vertical direc?tion［21-22］.Honeycombs with ZPR feature present no contraction in vertical direction when loaded along the wingspan，which exactly satisfies the require?ments of the application of morphing wings.

The majority of honeycomb structures pro?posed in previous studies were suitable for one-di?mensional uniaxial morphing or two-dimensional shear morphing.The in-plane properties of conven?tional hexagonal cellular structures were investigat?ed by Gibson and Ashby et al.［23］.Applying finite el?ement（FE）homogenization，Huang et al.［24］found that the thickness and the corner radius of cell wall would affect the elastic properties of two types of honeycomb with ZPR.Gong et al.［25］investigated a honeycomb with four-angle star-shaped cells that can achieve deformations along two orthogonal di?rections.Similarly，in Liu’s reports［26-27］，the inplane equivalent elastic modulus of ZPR hybrid and accordion cellular structure was deduced detail.However，the calculation methods of out-of-plane equivalent elastic modulus were not given in those reports.In fact，the camber is the most fundamental and crucial factor for wings to generate lift.Chang?ing the camber trailing edge can effectively change the airflow separation on the wing surface and signif?icantly improve the flight mobility of the aircraft，es?pecially for low-speed aircraft that are usually in low Reynolds number flight conditions and whose perfor?mance mainly depends on laminar boundary layer flow.

To meet requirements of mechanical perfor?mance of morphing wings undergoing complicated working condition，it is necessary to clarify the inplane and out-of-plane mechanical properties of hon?eycomb structures.In this work，in-plane and outof-plane mechanical properties of ZPR cellular struc?tures are investigated through a combination of theo?retical analysis and FE homogenization.The param?eters analysis is performed to describe the impacts of the unit cell geometrical configurations on the inplane and out?of-plane mechanical properties of ZPR cellular structures.Finally，a series of experiments are carried out to validate the feasibility and effec?tiveness of the theoretical and FE analysis.

1 Model and Experiment Tests

1.1 Analytical models

Fig.1（a）shows the schematic diagram of the hon?eycomb with ZPR，where the direction?1 and direc?tion-2 represent transverse and vertical direction，re?spectively.During the deformation analysis pro?cess，the unit cell of the honeycombs is regarded a homogeneous plate with effective modulus owing to the cyclic substructure［28］，as illustrated in the red marked area.Fig.1（b）shows the geometric parame?ters of the unit cell with internal anglesθ，wherelrepresents the length of inclined wall，h=2αlthe length of vertical walls，t=βlthe thickness of the hexagon，andbthe thickness of whole honeycombs perpendicular to 1-2 plane.Here，αandβrepresent the aspect ratio and cell wall thickness ratio，respec?tively.

Fig.1 Schematic diagram of the honeycomb with ZPR

Obviously，the large tensile deformation along direction-2 cannot be achieved in the proposed cellu?lar structure，so that this study considered only ar?eas regarding the in-plane tensile modulus along di?rection-1.In order to calculate the in-plane tensile modulus along direction-1，a unit cell structure is se?lected，and its analytical model is shown in Fig.2（a）.The first step for model is simplification as shown in Fig.2（b），which transforms the model into a quar?ter model owing to the biaxial symmetry.After sim?plification，the fixed boundary is set to the left end of the model，while the right is set with a concen?trated forceFand a momentM.On one hand，the deformation of the honeycombs with ZPR is mostly driven by the bending of the inclined walls when honeycombs withstand a load along direction-1［29］.One the other hand，the length of the vertical wall along direction-1 is small than that of the inclined walls.Therefore，the tensile deformation of the ver?tical honeycomb wall is ignored，and only the bend?ing and tensile deformation of the cellular structure is considered in this paper.

Fig.2 Analytical model used to calculate the in-plane tensile modulus

According to the equilibrium equations，it can be concluded that the vertical force is zero，and the momentMis

The strain energyUof a cantilever beam sub?jected to bending momentM（x）and axial loadFN（x）can be expressed as

whereE，IandAare the Young’s modulus，inertia moment and cross sectional area，respectively.

In this paper，it is assumed that bending mo?ment in the anti-clockwise direction is positive.By now the bending moment is

And the axial load is

Substituting Eqs.（3）and（4）into Eq.（2），it can be concluded that the strain energyUis

whereEsis the Young’s modulus of raw materials.

According to the Castigliano’s second theo?rem［30］，when the elastic system is enduring static load，the displacementδiof the point of force action can be calculated by the partial derivative of the strain energyUwith respect to any applied forceFi，shown as

Combining Eq.（5）and Eq.（6），it can be con?cluded that horizontal displacement of the point of force action is

According to the homogenization theory，the equivalent tensile modulusE1can be deduced by the equivalent stressσ1and strainε1that

Substituting Eqs.（7—9）into Eq.（10），it can be concluded that the homogenized and dimension?less tensile modulus can be expressed as

It is most important to research the equivalent shear modulus of the proposed cellular structures for application of flexible skin undergoing sweep morph?ing.Fig.3 shows the schematics of unit cell model used to calculate the equivalent shear modules.The fixed boundary is set to the left end of the model，while the right is set with a concentrated forceFalong direction-2 and a momentMalong direction-1.

Fig.3 Schematics of unit cell model used to calculate the equivalent shear modules

According to the equilibrium equations，it can be concluded that the momentMis zero，andFis

And the vertical deformation induced by shear force can been obtained according to Euler-Bernoulli theory［31］

It can be concluded that the shear strain is

According to the definition of shear modulus，the homogenized and dimensionless shear modulus can be expressed as

whereE2is the equivalent shear modulus of the ZPR cellular structures.

For a cellular structure，which is designed for a wing with a variable camber trailing edge，it is es?sential to understand the bending deformation.Ac?cording to the Chen’s report［32］，the calculation method for in-plane Young’s modulus of honey?comb cell was inapplicable to out-of-plane bending modulus since the moment acting on inclined walls are different for in-plane deformation and bending deformation.Fig.4（a）shows the loading scheme of the honeycomb cell.After simplification，the theo?retical model in Fig.4（a）has been transformed into an 1/4 model，as shown in Fig.4（b），owing to the biaxial symmetry.The deformation of the vertical honeycomb wall is ignored.

Fig.4 Analytical model used to calculate the out-of-plane bending modulus

The momentMis divided into two parts，namely bending momentMbendingand torsional mo?mentMtorsion，shown as

Then the bending angle around the direction-1 induced byMbendingcan be obtained

Different from the torsion of a bar with circular section，the plane assumption is no longer applica?ble to the torsion of a bar with rectangular section.In fact，after the torsion deformation occurs，the lat?eral circumferential boundary of a bar with rectangu?lar section becomes a spatial curve.For a bar with rectangular section，the normal stress caused by the constrained torsion is very small，resulting in that the constrained torsion is no different from the natu?ral torsion.A theoretical formula of the bending an?gle around the direction-1 induced byMtorsionis pro?posed that

whereqis the torsion coefficient，Gs=the shear modulus of the based materials，andKthe Polar moment of inertia of solid rectangular sec?tion［28］and can be calculated by

The torsion coefficientqof a bar with rectangu?lar section has been detailed in the Huang’s investi?gation［33］.

ReplaceEswith the Kirchhoff plate formulato calculate the out-of-plane bending modu?lus of the beam in a relatively large width and thus one can obtain the equivalent out-of-plane bending modulus of the proposed honeycomb cell as

whereνandIb=are the Poisson’s ratio and the second moment of cross sectional area of the beam，respectively.

The total angular deformation of the structure around the direction-1 can be calculated as

Apparently，combining Eqs.（17—23），the ho?mogenized and dimensionless bending modulus can be calculated by the following form

1.2 FE homogenization

In order to verify the calculation results in theo?ry，the numerical simulation with a commercial FE software ABAQUS（version 6.14）was carried out.The material selected to perform the simulation was photosensitive resin with the elasticity modulus of 2 800 MPa，and the Poisson’s ratio of 0.33.Inter?nal geometric parameter ofl=10 mm is adopted.The simulation was carried out with aspect ratioα（2，2.5 and 3），cell wall thickness ratioβ（0.1，0.15，0.2）and cell angleθranged from ?45° to 45°with even step at 5°.

Fig.5 shows the FE model used to calculate the in-plane tensile modulus of honeycombs with ZPR.In the model，a 2-node linear element B21 is used to model the honeycomb block.The boundary condi?tions with fixed surfaceAand loaded surfaceBare shown in Fig.5.For the tensile deformation，the sur?faceBwas loaded with 10 kN uniform load along di?rection-X.Accordingly，the surfaceBwas loaded with 10 kN uniform load along direction-Yfor shear deformation.

Fig.5 FE model used to calculate the in-plane mechanics of honeycombs with ZPR

As shown in Fig.6，3D model with 3×6 unit cells was employed to calculate the out-of-plane bending modulus.Mesh convergence results show that the model with a minimum element size oft/2 can obtain accurate results.In the model，an 8-node linear brick C3D8R is used to model the honeycomb block.As for the boundary conditions，surfaceAwas constrained and surfaceBwas loaded with 10 kN concentrated force along direction-Zin the form of coupling.And the rest of surfaces was set as free boundary.Then the equivalent bending modulus can be calculated by the following form

Fig.6 FE model used to calculate the out-of-plane mechan?ics of honeycomb with ZPR

wherewis the deflection of the structure along the direction-Z.

1.3 Manufacturing and experiment tests

As the honeycomb structure is complex，it is difficult to machine them by traditional technology.Thus，the state-of-the-art additive layer manufactur?ing technology is utilized to prepare the cellular structure with ZPR，which makes it possible to eval?uate the mechanical performance of structure in a convenient and cost-efficient way.All honeycomb samples employed in this investigation were fabricat?ed using a Stereolithography（SLA）machine（3DSL-360Hi，Shanghai Digital Electronics Tech?nology Co.，Ltd，China）with photosensitive resin（PS）.The manufactured precision of the test speci?men is highly dependent on the machine type and the minimum thickness dimension of the printed structure is restricted to 1 mm with 200 μm preci?sion.Firstly，the designed model of honeycomb pan?el was imported as“.stl”file and then sliced by the printing software.Then，the honeycomb panel was built layer-by-layer with a layer thickness of 50 μm through a down-top printing process.After cleaning off the uncured photosensitive resin off the surface，the printed honeycombs were put in an oven to go through drying process.The geometric average val?ue of the Young’s modulus of the PS，SZUVW8001，is 2 800 MPa.

The in-plane and out-of-plane mechanical prop?erties of the proposed ZPR cellular structures were conducted on former testing machine with a constant displacement rate of 1 mm/s.Corresponding force and displacement during tests were recorded to calcu?late the tensile modulus，shear modulus and bending modulus of ZPR cellular structures.For the test of homogenized tensile modulus，the ZPR cellular structure specimens had dimensions of 130 mm×120 mm×5 mm，and load transmitting blocks with appropriate width were designed in tensile specimens to prefer the standards ASTM D638-08［34］.A meth?od used for calculating shear modulus was different from off-axial test.Specimen with unit cell had di?mensions of 40 mm×15 mm×5 mm，as illustrated in Fig.7.Ears to fit the grip was designed，via which the shear forces could be transmitted to the corre?sponding edges of unit cells so that the effective shear modulus of unit cells could be both obtained un?der axial movement［35］.In addition，a three-point bending tests were carried out to measure the out-ofplane bending performance of the honeycomb with di?mensions of 100 mm×120 mm×5 mm.A constant displacement rate of 5 mm/min and a span length of 80 mm were used during three-point bending test.

Fig.7 Schematic of shear tested specimen

2 Results and Discussion

2.1 Parametric analysis

From Eqs.（11，16，24），one can make predic?tions and calculations of the in-plane and out-ofplane mechanics of ZPR cellular structures based on the geometric parameters includingα，βandθ.Fig.8 shows the comparison of FE results and theoretical results for the dimensionless equivalent elastic modu?lusE1/Esof the proposed ZPR cellular structures varying with the geometric parameters.The results show that the in-plane stiffness of the proposed ZPR cellular structures in this investigation resembles the Gibson-Ashby model owing to the similar hexagonal geometrical configurations［23］.However，in the Ru?bert’s analytical model［36］，the ZPR cellular struc?tures were identified as only undergoing pure bend?ing，resulting that the transverse stiffness approach infinity atθ=0°，which are not agree with practice situation.After taking axial deformation of cell walls into consideration，in-plane tensile modulus of the ZPR cellular structures no longer be infinity atθ=0°in this study.As shown in Figs.8，9，the analytical results of the dimensionless equivalent elastic modu?lus of ZPR cellular structures are consistent with the FE results，and the average relative error is around 3%.Making a general survey of the overall situation at the same time，the maximum error is 7.5% occur?ring atα=3，β=0.15 andθ=35° between analytical result and FE simulation.

Fig.8 FE and analytical results of E1/Es of the ZPR cellular structure versus cell angle

As shown in Fig.8（a），E1/Esreaches its ex?treme value atθ=0° firstly，and at this point，in?clined walls in horizontal direction undergo the pure tensile，leading to a maximumE1/Esvalue.Second，E1/Eshas a symmetric distribution centered onθ=0° whenαandβare definite values.Ifθ>0° is taken alone，E1/Esdecreases as the cell angleθincreases.Finally，E1/Esincreases with the increasing of the parameterβwhen other parameters remain con?stant.Similarly to Fig.8（a），E1/Esreaches its ex?treme value atθ=0° and has a normal distribution centered onθ=0° when parametersαandβremain constant in Fig.8（b）.Whenθ>0°，E1/Esdecreases as the cell angleθincreases.In addition，E1/Esde?creases with the increasing of the parameterαwhen other parameters remain constant owing to the in?creased cross-sectional area induced by the parame?tersα.Comparing Fig.8（a）and Fig.8（b），the varia?tions of parameterαproduce smaller fluctuations onE1/Esthan the variations of parameterβdo.The variations ofE1/Esvalues are of the same order of magnitude whenαranges from 2 to 3，while the in?crement value ofE1/Esranges from one order of magnitude to two orders of magnitude whenβranged from 0.1 to 0.2.It gives a good reference model for the designs of the in-plane elastic modulus by various parameterβ.

Fig.9 shows the comparison of FE results and theoretical results for the dimensionless equivalent shear modulusE2/Esof ZPR cellular structures vary?ing with the geometric parameters.The analytical results of the dimensionless equivalent shear modu?lusE2/Esof ZPR cellular structures are also consis?tent with the FE results.And the mean relative devi?ations ofE2/Esof ZPR cellular structures between analytical results and FE results are less than 3%，while the max deviation is 7% occurring atα=2，β=0.2 andθ=45°.Unlike the dimensionless equivalent elastic modulusE1/Es，the dimensionless equivalent shear modulusE2/Esis not symmetrical?ly distributed.E2/Esfirstly decreases，then increases with the increasing of cell angleθfrom ?45° to 45°whenαandβare definite values.In addition，the minimum values ofE2/Esare concerned with the geo?metric parameters of the unit cell.The FE results，it should be noted，are somewhat below the analytical ones，which is contributed by the difference between FE model and theoretical model［37］.In fact，the beam model applied to the FE simulation is the Timoshen?ko beam，while to theoretical analysis is the Euler-Bernoulli beam.As shown in Fig.9（a），E2/Esin?creases with the increasing of the parameterβwhen other parameters are definite values.It can be found in Fig.9（b）that the increase of the parameterαleads to a decrease of dimensionless equivalent shear mod?ulus when other parameters remain constant.

Fig.9 FE and analytical results of E2/Es of the ZPR cellular structure versus cell angle

Fig.10 shows the comparison of FE results and theoretical results for the dimensionless equivalent bending modulusEb/Esof ZPR cellular structures varying with the geometric parameters.The analyti?cal results of the dimensionless equivalent bending modulusEb/Esof ZPR cellular structures are also consistent with the FE results.And the mean rela?tive deviations ofEb/Esof ZPR cellular structures between analytical results and FE results are less than 20%，while the max deviation is 35% occur?ring atα=2，β=0.2 andθ=0°.In addition，the de?viations between analytical results and FE results decrease with the cell angleθincreases whenθ>0°.Both in Fig.10（a）and Fig.10（b），Eb/Esreaches its extreme value atθ=0°，and at this point，inclined walls in horizontal direction undergo the pure bend?ing，leading to a maximumEb/Esvalue.Whenαandβare definite values，Eb/Eshas a symmetric dis?tribution centered onθ=0°.If you takeθ>0° alone，an increasing cell angleθleads to a decrease ofEb/Es.In addition，the out-of-plane bending modulus in?creases with increasing of the parameterβ，while de?creases with increasing of the parameterα，which is similar to what observed by Huang et al.［34］.From these simulations it is apparent that the simplifying assumption for the analytical model（i.e.，neglect?ing the deformation of the vertical wall）leads to dis?crepancies against the higher fidelity FE model.The deformation of the vertical wall versus the whole cell deformation increases for increasing dimensions of the vertical wall.An increase of the cell wall as?pect ratio not only leads to a slightly decrease of the equivalent bending modulus，but also leads to a rela?tively larger discrepancy between the analytical and the FE results.

Fig.10 FE and analytical results of Eb/Es of the ZPR cellu?lar structure versus cell angle

2.2 Experiment results

To verify the correctness of previous analytical and FE results，honeycomb specimens withl=10，α=2，β=0.1 andθ=45° were used in experimental test for in-plane and out-of-plane mechanics.Fig.11 shows the tensile，shear and bending mechanical properties of the honeycomb with ZPR.Then the corresponding equivalent tensile moduli，equivalent shear moduli and equivalent bending moduli are cal?culated and listed in Table 1.In addition，the analyti?cal and FE results are also listed in Table 1 to make a comparison with experimental results.For the equivalent tensile modulusE1，the deviation be?tween experimental and analytical results is 1.02%，and the deviation between experimental and FE re?sults is 1.52%.For the equivalent shear modulusE2，the experimental results show discrepancies be?tween 5.26% and 1.05% over the analytical and FE results，respectively.In addition，for the equivalent bending modulusEb，the deviation between experi?mental and analytical results is 12.03%，and the de?viation between experimental and FE results is 15.27%.Those discrepancies between the analyti?cal，FE and experimental results can be attributed to several reasons.First，the SLA is very similar to fu?sion deposition molding，both having a layerwise de?position and an additional degree of internal porosity that do not meet the assumption of a homogeneous and isotropic honeycomb materials［38］.Second，in the tensile，shear and bending theoretical models，the deformation of the vertical honeycomb wall is ig?nored.In addition，even when the fixture produces a pure shear deformation as closely to the ideal case，there are still differences between a pure shear defor?mation and an approximate one［39］.As shown in Ta?ble 1，for whether in-plane or out-of-plane mechani?cal properties，the deviation between experimental data and numerical prediction is less than 2%.

Fig.11 Mechanical properties of the honeycomb with ZPR

Table 1 Comparison of the analytical,FEM,and experi?ment results MPa

3 Conclusions

Analytical models are established for the calcu?lation of in-plane and out-of-plane mechanical prop?erties of ZPR cellular structures in combination with the FE analysis.The experimental test validates cor?rectness and effectiveness of the theoretical and FE analysis.According to the parametric analysis，these cell geometric parameters provide different contributions to the effective mechanical properties and lead to a separate design for the in-plane and out-of-plane performances.Therefore，the superior technique could be an efficient reference to mechanic engineering such as morphing aircraft design.For example，a large cell angleθof inclined wall and a large aspect ratioαcould be employed for one di?mensional morphing.And the variations of aspect ra?tio produce smaller fluctuations on the dimension?less equivalent elastic modulus than the variations of the thickness of inclined wall.To obtain the better shear morphing，ZPR cellular structures with large aspect ratioαand lower cell angleθcould be select?ed.Aimed at some complex flight conditions，the multi-objective optimization could be performed with a comprehensive consideration of the in-plane and out-of-plane mechanics of ZPR cellular struc?tures.However，this work still has some weakness?es and drawbacks.The results，both analytical and experimental，presented so far do not consider the effects of geometric and material nonlinearities.Hence，they are only applicable for small deforma?tions and not accurate when the honeycomb cores present large local strains or material plastic behav?iors.In future work，the mechanical properties of the ZPR honeycomb structure should be explored by considering the nonlinear behavior of honeycomb un?der large deformation.