Dong-mei Yin,Bao-ming Li,Hong-cheng Xiao

National Key Laboratory of Transient Physics,Nanjing University of Science & Technology,Nanjing,210094,China

Keywords:Lightweight design Filament-wound composites Bridging model Three-dimensional elastic properties

ABSTRACT This work provides a method to predict the three-dimensional equivalent elastic properties of the filament-wound composites based on the multi-scale homogenization principle.In the meso-scale,a representative volume element(RVE)is defined and the bridging model is adopted to establish a theoretical predictive model for its three-dimensional equivalent elastic constants.The results obtained through this method for the previous experimental model are compared with the ones gained respectively by experiments and classical laminate theory to verify the reliability of this model.In addition,the effects of some winding parameters,such as winding angle,on the equivalent elastic behavior of the filament-wound composites are analyzed.The rules gained can provide a theoretical reference for the optimum design of filament-wound composites.

1.Introduction

The filament wound composites have been widely applied in many industries,such as aerospace,energy and transportation.In the meanwhile,because of their high specific stiffness,specific strength and designable ability,they are the one of ideal choices for the lightweight design of weapon equipments.The whole filament winding process can be automatic.In order to make full use of the bearing ability of fiber bundles(tow),these reinforced materials will be arranged on the direction of bearing loading.Due to the cross winding of fiber bundles,there will be fiber undulation in the local place of the filament wound composites,which is similar to the characteristic of fiber undulation in the braid composites.Its complex structure enlarges the difficulty of the investigation of its macro mechanical properties.Therefore,many researchers have developed a series of studies.Some researches are on the basis of the macro scale,which mainly adopted the classical laminate theory[1-3]or the experimental methods[4-6].

But with different winding patterns,there will be different structures in the micro scale.These different microscopic structures may lead to varied macroscopic mechanical properties and different local stress/strain concentrations under the same loading[7 and 8].The effects of the microscopic structures on the macroscopic mechanical properties can not be embodied by the laminates model,and also need huge experiments to verify.Hence,based on the principle of multi-scale analysis,some investigations of the mechanical properties of filament wound composites have been done through the micro scale analysis.In which most researches about its elastic behavior can be classified as analytical models and numerical models.In the analytical approaches,the microscopic structure in periodic distribution is selected as a representative unit cell or a representative volume element according to the characteristic of the internal microscopic structure of the filament wound composites.Sun Jiang et al.[9]divided this kind of representative unit cell into laminate fields and filament undulation fields,and used the classic laminate theory and volume average to calculate the axial modulus of the filament wound tube.Also the similar method was adopted by D.Zindel et al.[10]to forecast the elastic properties of the filament wound composites with flexible matrix material.In this model,the nonlinear behavior of the flexible matrix has been considered on the basis of experimentally obtained nonlinear lamina properties.Base on RVE and periodic arrangement of RVE,the stiffness of filament wound composite is treated as a stiffness field in Ref.[11].The function of this stiffness field is obtained by 2D fourier series,numerical integration and analytical method is applied in deduction of relation of stiffness function and stiffness matrix of fiber bundles.In Refs.[12,13],through relating the homogeneous properties in-plane strains to the volumeaveraged in-plane stresses,a three-dimensional micromechanical model for the elastic properties of filament wound composites is set up,which considering the effects of undulated laminas.The bounds on the material behavior are provided based on the isostrain or iso-stress conditions assumed at each given station along the length of the RVE.In numerical models,based on the homogenization theory,the finite element models of the RVE for the filament wound composites with considering the effect of fiber undulating zone were often established to analyze their elastic properties.Jiang Yun-peng et al.[14]adopted the shell element to establish the finite element model of the RVE.While the solid element is used to model the finite element model of the RVE in Refs.[15,16].The influences of different technological parameters,such as winding angle,fiber undulating zone and intersecting area,on the equivalent modulus are analyzed.But only the equivalent inplane properties of the filament wound composites can be obtained in Refs.[14,15].In order to improve the calculation precision and reduce the calculation time,Fu Fuchao[17]divided the RVE into 16 sub-RVEs,and different RVEs under different winding patterns can be gained by rearranging of these sub-RVEs.

The previous investigations reveal that the three dimensional(3D)elastic properties of the filament wound composites obtained by the experiments are more accurate,but it consumes too much time and cost.Based on the micromechanics,whether the analytical approaches or the finite element methods are applied to forecast its elastic properties,it is necessary to improve the efficiency of prediction on the premise of ensuring certain accuracy.At present,the main idea is to calculate the macroscopic mechanical properties of the whole composites on the basis of its basic constituent materials(fiber and matrix).The bridging model provided by Huang[18,19]has advantages in this respect.For this reason,this work is aimed at establishing a simple and accurate analytical method to calculate the 3D equivalent elastic properties of filament wound composites based on the bridging model(BM).Firstly,based on the multi-scale homogenization theory,a RVE is chosen,which characterizes all the micro-architecture details of the filament wound composites,such as fiber undulation,winding angle and lamination arrangement.Then referring to the principle of volume average in the analysis method of braided composites[20,21],the present research will establish a theoretical prediction model for the 3D equivalent elastic properties of the filament wound composites with the spiral winding.The previous experimental datas[9]and the classical laminate theory(CLT)model will be used to verify the reliability of this model.Besides,the effects of some winding parameters on the 3D elastic behavior of the filament wound composites,such as winding angle,will be investigated in this work.

2.Homogenization approach

2.1.Representative volume element for the filament wound tube

The winding products are produced with the cross winding of fiber bundles,and often adopt the winding angle±φ(φ≠0°,90°),which can homogenize the products’deformations.Therefore,this work will focus on the tubes with the spiral winding angles.There will be multiple identical rhombus patterns in the final winding tube,as displayed in Fig.1.Based on the homogenization theory,a rhombus pattern is selected as the representative volume element.It can be further divided into several lower-level RVEs,which can be considered as two kinds of regions according to whether there is undulation or not.One is the laminated structure regions(including the RVE 3 and RVE 4 with stacking sequences of±φandφ,respectively),and the other(including the RVE 1,RVE 2 and RVE 5)have the fiber bundles undulation similar in the braid composites,as illustrated in Fig.2.Because the RVE 1 and RVE 2 locate along the two helical fiber paths,one lamina traverses over or under another lamina while the other lamina is not undulated.The RVE 5 locates along a circumferential path,and its two laminas traverse over or under each other.The winding pattern is related to the corresponding winding process parameters.One of the important parameters is the number of circuits performed until the tow is deposited just next to the first circuit,Nc,which can be calculated by Refs.[5,17]:

Fig.1.The filament wound pattern for a tube.

Fig.2.The rhombus-shaped RVE.

Where Spis the equal fraction divided in the circumferential direction of the mandrel between two circuits,and Npis the number of circuits needed to cover mandrel completely,is given as:

Where D is the diameter of the tube,φis the winding angle,and the wdis the bandwidth of the tow.

If it is assumed that the thickness of a single±φlayers is 2t,the volume fractions of these five regions(1-5)in the whole RVE can be got through calculating their area fractions in the xoy plane of the whole RVE area,as shown in Fig.2,and they can be expressed as follows:

Where t is the thickness of a tow,and is assumed to be equal to the thickness of a single layer.Lu is the undulation length,as shown in Fig.3.While a and b are related to Ncandφ,as follows:

2.2.Prediction of the 3D equivalent elastic properties

In order to investigate the 3D equivalent elastic properties of the filament wound cylinder,the 3D continuum homogenization procedures in RVEs at multiple length scales is employed in this work.In the largest length scale,the model includes all layers of the filament wound cylinder.In the middle length scale,the research object is the RVE of a single±φlayers,and it is further divided into five regions(lower-level RVEs).In the smallest length scale,each lower-level RVE(Region 1-Region 5)is divided into multiple unidirectional composites,and each unidirectional composite is comprised of the fiber tows and matrix.Taking one lower-level RVE(Region 1)for example,it is displayed in Fig.3.In the figure,z-axis is located in the direction of the thickness of a single±φlayers,ξ-axis is along the undulation length,and hu is the function for the distance between the tow’s center and theξ-axis.The angleβdisplays the orientation of fiber tow’s undulation.Therefore,the fiber tow’s undulation degree can be described by the parameters of hu andβ,which should be expressed in different forms in the different regions.

Fig.3.Multiple unidirectional composites in the lower-level RVE(Region 1).

For the Region 1:

Each region is divided into n equal parts along the length direction.The length of each part is defined as:

Each part is assumed to be composed of p different unidirectional composites,and the whole RVE consists of q=5np unidirectional composites.The volume fraction of the jjth unidirectional composites in the kth region can be calculated as:

Then the volume fractions of the iith unidirectional composites in the whole RVE are given by

It is assumed that the fiber tow is a transversely isotropic linear elastic material,while the matrix is considered as an isotropic elastic material.According to the bridging model in Ref.[18and19],the relationship between the incremental stresses in the tow and matrix of the unidirectional composite is:

The elements in the bridging matrix can be expressed as:

In which Em,Gmandνmare the elastic modulus,shear modulus and Poisson’s ratio of the matrix,respectively.andare the elastic moduli in the longitudinal direction and the transverse direction of the fiber tow,respectively.Gf12andνf12are the shear modulus and Poisson’s ratio in plane for the fiber tow.The parametersαandγcan be determined by the experiment or the analytical computation.Here we set them 0.3 for the fiber reinforced resin composites[22].

According to the bridging model,the compliance matrix of each unidirectional composite in its material coordinate system(1-2-3)can be derived as:

Then the stiffness matrix of each unidirectional composite in its material coordinate system(1-2-3)can be written as［C］i=［S］-1i.Transforming this stiffness matrix into the global coordinate system(x-y-z),the stiffness matrix of each unidirectional composite in the global coordinate system will be obtained by

In which［T］iσis the transformation matrix of stress vector and superscript“T”represents the transpose of the matrix.In the global coordinate system,x and y axes follow axial and hoop directions of the cylinder,respectively,while the z-axis is located in the thickness direction of the winding layer.According to the relationship between the global coordinate system and the material coordinate system for each unidirectional composite in the different regions,as exhibited in Figs.2 and 3,the transformation matrix is expressed in different forms.

In the laminate region:

In the undulation region:

In which

Whereβis the angle which fiber tow is oriented out of the plane in the undulation region.

It is supposed that all unidirectional composites have the same strains as the whole RVE.Therefore,based on the iso-strain,the equivalent stiffness matrix of the whole RVE can be obtained through the superposition of the stiffness matrices of all unidirectional composites by adopting the volume average.It is written by

Its equivalent compliance matrix can be calculated by.According to the composite mechanics[23],the stiffness and compliance matrices of the anisotropic material are still both symmetric about their diagonals in the global coordinate system.Then the equivalent elastic constants of the RVE can be received by the elements of this compliance matrix,and on the basis of homogenization theory,they also represent the equivalent elastic constants of the filament wound cylinder with±φwinding layers.

3.Calculation and discussion

3.1.Verification of the model

The experimental models in Ref.[9]are used to verify the above analysis model.These tubes are produced by filament winding with carbon fiber bundles impregnated with epoxy resin,and its material parameters are listed in Table 1.The average fiber volume fraction over the tube is 50%.The tube’s diameter is D=26 mm,and the parameter of winding pattern Ncis 4.The winding angles of the tubes in the experiment vary from 6.9°to 59.4°.

Table 1Material parameters of carbon fiber bundle and epoxy resin.

The equivalent axial elastic modulus Exxand in-plane Poisson’s ratioνxyof these tubes with different winding angles are predicted by the analysis model(BM)provided in this work,and are compared with the ones obtained respectively by the experiment and CLT model in Figs.4 and 5.In which the parameter n is set to 100,and Lu is about 6 times of thickness of the fiber tow.According to Ref.[11],if the distribution of fiber is uniform,the volume content of fiber tow in each position has no obvious change.Therefore,it is assumed that the fiber tow volume fractions for different unidirectional composites are equal to the average fiber volume fraction over the tube.

Fig.4.The comparison of equivalent axial elastic modulus.

Fig.5.The comparison of the Poisson’s ratioνxy.

It can be observed in this figure that the variety trends of equivalent axial elastic modulus and in-plane Poisson’s ratio obtained by these three methods are consistent.For the range of winding angles from 6.9°to 45.8°adopted in the experimental models,the axial elastic moduli predicted by the BM are more aligned to the ones gained in the experiment.Because the fiber tow’s undulation can affect the in-plane stiffness of the filament wound structure[5,10 and 13],but these effects cannot be obtained by the CLT model.However,the error of the values of the in-plane Poisson’s ratio by these three methods is relatively large.It is also relevant to the reason that the data collected in the experiment is less and unstable[9].Therefore,it is concluded that the analysis model in this work is reliable to some extent.

3.2.The influence of some of winding parameters

To study the influence of winding angleφon the 3D elastic constants of the filament wound tube,the above model is still used here,and the values ofφare from 5°to 85°.The variations of some of elastic constants of the filament wound tube with winding angle φare displayed in Fig.6.The axial elastic modulus Exxbasically decreases with the increase of winding angleφ,while the hoop elastic modulus Eyygrows,as shown in Fig.6(a).The former declines obviously at the smallerφ(<45°),while the latter enlarges obviously at the largerφ(＞45°).This result agrees with the one predicted by the classic laminate theory in Ref.[24].The radial elastic modulus Ezzhas a relatively flat growth with the increase of φ.Fig.6(b)gives the changes of shear moduli of the filament wound tube with winding angleφ.It is distinct that the shear modulus Gxyrises firstly and decreases latterly with the growth ofφ,while the changes of other shear moduli are relatively less.Moreover,when the angleφis enlarging,the Poisson’s ratioνxyincreases first and then drops obviously,as displayed in Fig.6(c),while the variations of other Poisson’s ratios are relatively smaller.It can be concluded that the effects of winding angle on the elastic constants of the filament wound tube in the plane x-y presents relatively significantly.

In addition,some elements of its equivalent compliance matrix obtained in the results,which are related to the coupling coefficients of tension-shear and shear-shear,are non zero.It means that the filament wound composites with winding angle±φis not a classical orthotropic material if considering the fiber undulation.Fig.7 exhibits the variations of these elements of its equivalent compliance matrix with the winding angleφ.It can be found that the effects ofφon the elements S34,S14and S24are relatively bigger,especially for S34,and the element S56is affected less.It reveals that the shears in the plane zoy are sensitive to the change of the winding angle,especially for the shear leaded by the radial loading.The shear coupling between the plane zox and xoy is very small,even can be ignored.However,these elements of equivalent compliance matrix related to the coupling coefficients are very minor,and they will have little effects on the macroscopic mechanical properties of the filament wound tube.So it can be ignored in some researches.

Fig.7.The variations of some elements of the equivalent compliance matrix of filament wound composites with winding angleφ.

To investigate the influence of the parameter Ncon the 3D elastic constants of the filament wound tube,a series of models with Np=25 andφ=25°are set up,while Nc=1,2,3,4,6,7,8,9,11,12.The diameter and material parameters of the tube are the same to the above,and the variations of some equivalent elastic constants of the filament wound tube with Ncare displayed in Fig.8.With the growth of Nc,the radial modulus Ezzincreases significantly,while the axial modulus Exxdrops,and the hoop modulus Eyyis basically unchanged.Also the shear modulus Gxydecreases,and the variations of the other shear moduli are small.It is because that larger Ncproduces more undulated weaving in the tubes,and it will weak the stiffness in the xoy plane of the composites.This phenomenon is also found in Ref.[14,25].The calculation results also show that the Poisson’s ratiosνxyandνxzdecrease when the Ncis growing.But the change of the Poisson’s ratioνyzis opposite to them.

In additions,Fig.9 depicts some predicted elements of the equivalent compliance matrix for the tubes with different winding parameter Nc.As seen from the figure,the predicted absolute values of S34and S24are increasing with the rising of Nc,and the absolute values of other elements are quite small,which are almost unaffected by Nc.Though the values of the elements related to the coupling effects are relatively very small,the shears in the plane yOz caused by the hoop and axial loads are relatively sensitive to the winding parameter Nc.

Fig.8.The variations of some of elastic constants of the filament wound tube with Nc.

Fig.9.The variations of some elements of the equivalent compliance matrix of filament wound tube with Nc.

4.Conclusions

A multi-scale homogenization procedure for predicting the 3D equivalent elastic properties of filament-wound composite cylinders has been developed.This model adopts the bridging model to calculate the elastic properties of different unidirectional composites in the RVE,which is chosen on the basis of the filament winding pattern.Referring to the analysis method of braided composites,the volume average is applied to calculate the equivalent stiffness matrix of the filament wound cylinder.A comparison among the results predicted by this method and the ones obtained by the previous experiments and CLT model for the filamentwound composite tubes is carried out to validate the reliability of this theory model.The trends of the results gained by these three methods are in good agreement,especially for the axial elastic moduli of the tubes with the winding angles from 6.9°to 45.8°.It means that the analysis model in this investigation is reliable to some extent.

In additions,the effects of winding angles and winding parameter Ncon the elastic properties of the filament wound composite cylinder are analyzed by this model.The results indicate that if considering the fiber undulation,the filament wound composites with±φwinding angle can not be thought as a classical orthotropic material.The effects of winding angle on the elastic constants of the filament wound tube in the plane x-y are relatively obvious.The shears in the plane yoz are sensitive to the changes of the winding angle and the winding parameter Nc.These conclusions can provide a reference to the design of the mechanical properties of filament wound composites.