針刺過程中的力學行為分析

Abstract：A staged modelingapproach isproposed todividetheneedling process into three stages：deformation，insertion andsmoothness，andconstructdynamiccoupling modelsofrigid force，cuting forceandfriction forcerespectively.Based ontheBousinesqcontacttheory，theinitialtip-fabriccontactbehaviorisanalyzed，combinedwiththeelasticcuting theorytoquantifythe fiber breakage mechanism，andthe Winkler foundation modeland LuGredynamic frictionmodelareused to characterizethefrictionpropertiesoftheneedeshaft.Itisshowthat：thestiffessforce inthedeformationstageisaffectedbythegeometrical parametersofthenedletipandtheelasticmodulusof thefabric；thecutingforceintheinsertion stageiscloselyrelated tothemicrodeformationof thefiber；andthe dynamic friction behavior inthesmooth stage canbe depicted bythe LuGre model with highaccuracy.Thecomplete needling force prediction modelfinallestablished provides atheoretical basis for optimizing composite needling process parameters，reducing fiber damage and equipment design.

Keywords：fiber fabric； force model；composite materials；needling technology

1 Introduction

Needling technology，as one of the key processes in composites manufacturing，plays an important role in enhancingthe interlaminarbondingandoverallmechanical properties of fiber pre- fabricated bodystructures' es[1-2].During the needling process，the mechanical interaction between the needle body and the fiber fabricdirectly affects the degree of damagewithin the material，the uniformity of fiber distribution andthe mechanical properties of the final composite. Although some studies have initially explored the mechanics of needling，mostoftheexistingmodelsare based on simplified assumptions（e.g.，ignoring dynamic friction effects and not distinguishing between deformation and cutting stage force mechanisms），which leads to insufficient prediction accuracy in complex needle geometry and multi - scale fiber response scenarios， restricting theoptimization of the process and the enhancement of material properties.

Aiming at the above problems，this study proposes a staged modeling method，which divides the needling process systeminto three stages：deformation，insertion and stabilization， and constructs dynamic coupling models respectively. Through the coupling analysisof multi- stage mechanical behavior，a high -precision needling force prediction model is established，aiming to provide theoretical support for the parameter optimization，fiber damage control and equipment design of the needling process in composite manufacturing.

2 Insertion force analysis

During insertion，theperforated needleisinserted verticallyintothefabricata fixedrate v₀ feed.The needle is divided into two parts，the tip and the needle shaft，whenthe needleisin contact with thefabric，it first deforms it，and this stage is subject to the stiffness force；the tip of the needle enters the fabric and interactswith the fabric，and at this stage the needle is subject to two kinds of forces，one is the cutting force requiredto deformorbreakthe fiber，and theotheristhe friction forceexerted by the fabricin the tangential direction on the wall of the needle；when the tip of the needle passes through the fabric completely，the needle receivesonly the friction force fromthefabric.Therefore，it is necessary to calculate the stiffness force F_s on the fabric surface，the cutting force F_c on the needle tip and the friction force F_f onthe needle shaft，respectively，to establish a prediction model for the needling force：

F_n=F_s+F_c+F_f

The needling process is modeled in three parts ： deformation phase，insertion phase，and stable phase. Inthe deformation phase，the needling resistance is dominated by the stiffness force，in the insertion phase， theneedling force is dominated by the cutting force and friction force，and in the stable phase，the needling force is dominated by the friction force，as shown in Figure 1.

Fig.1Three phases of the needle insertion process

3 Modeling of the stiffness force

Theinteraction between the needleand the fabric in the deformation phase can be regarded as the Boussinesq problem，as shownin Figure2.Sneddon solved and derived an analytical solution by means of the Hankel transform and double integral equations[3].The generalized formulas for the needling depthand the stiffness force，f，being：

Fig.2（a）Model of stiffness force（b）Needle geometry

Where function is specified by the following definitions：taking the tip of the needle as the origin，the curved expression of the needle is y=f （r）， where r=αx（rlt;α） ，andhence f （2 is theradius of the contact circle， h is the depth of the needle tip penetration into the fabric；and E_r isthe reduced modulus determined by the following equation ：

Where E₁ ， E₂andv₁ ， v₂ are the Young's modulus andPoissonsratiooftheneedleandfabricand Poisson sratio of the fabric，respectively. Approximating the beveled needle tip as an axisymmetric conical needle， we take f ，where εx=acotα denotes the normal penetration， α is bevel angle.

Then the profile function of the needle tipf （x） can be written as

Substituting Eq.（5）and Eq.（2）into Eq. （3） gives the stiffness force as

Combining Eq.（4）and（6）gives the final stiffness force equation.

Wherein， h and represent the axial length and laterallengthof the insertionof the skewed needleinto the fabric，respectively.

4 Modeling of the cutting force

Based on the elementary cutting tool（ECT）theory^?4? ，the cutting edge of the needle is divided into consecutive infinitesimal segments as shown in Figure 3. The fiber is subjected to the joint action of the cutting force f，and the reaction force p.

Fig.3Cutting force model of infinitesimal fiber segments Since the forces on the fibers in the model are

symmetric，only half ofthemodel region needstobe considered，and the equilibrium equation for the model is[5]

Where E_f and I_f are the Young's modulus and momentof inertiaofthefiber， k isthe stiffness constant of thefiber，andisthedeflectionofthefiberinthey-direction.The cuting force can be calculated from the following equation ：

5 Modeling of the friction force

Theneedle enters the fabric from the extrusion of theneedleshaftbythefabric.Theforcedistributionon the needle shaft is approximately uniform and the direction of force distribution is perpendicular to the surface ofthe needle shaft，as shown inFigure 4.Thedistributed force acting on the needle wall can be described by the Winkler’s foundation model， namely[6] ：

Where E_n ， I_n is the stiffness coefficient and moment of inertia， E_m ， I_?m isthe stiffness coefficient and Poissons ratio of the needle， b isthe circumference of theneedle，and h isthe contact length of the needle withthefabric.

Fig.4Winkler’sfoundationmodelandLuGredynamic frictionmodel

The friction part uses the LuGre dynamic friction model，which describes the frictional behavior ofthe contact surfaces through the interaction of microscopic elastic bristles. The core equations of the model include[7-8]

Where v isthe relative velocity of the two contact surfaces， σ₀ and σ₁ are the stiffness and damping coefficients of the micro elastic bristles， σ₂ ， μ_c ，and μ_s

aretheviscous damping，normalized Coulomb，andviscous friction coefficients，and β isthe static parameter.

Assuming zero initial conditions，the general solu tion of the ordinary differential equationis

When v₀ islarge enough，the friction force can be expressed as

In summary，the complete insertion force model equation can be expressed as：

Where h_A is the position of the maximum deformation surface before the needle pierces the fabric and h_B isthe position of the needle completely through the fabric.

6 Conclusion

In thispaper，thedynamic evolution law ofme chanical behavior during needling is systematicallyrevealed through a staged modeling approach. The conclusions are as follows ：

（1）In the deformation stage，based on the stiffness force model derived from the Boussinesq contact theory，the effects of the needle tip geometrical parameters and the fabric elastic modulus on the initial resistanceare clarified.

（2）In the insertion stage，the cutting force model constructed by combining with the ECT theory quantifiesthe relationship between the fiber fracture critical conditions and the cutting efficiency of the needle tip.

（3）In the stable stage，the Winkler’s foundation model is used in conjunction with the LuGre dynamic friction model，thedynamic friction characteristicsof the needle shaft and the fabric were accurately portrayed.

References

[1]CHEN X，CHEN L，ZHANG C，et al. Three-dimensional needle -punching for composites-A review [J]. CompositesPart A：Applied science and manufacturing，2016，85：12-30.

[2]SUNY，FANW，SONGC，etal.Effects of stitch yarns on interlaminar shear behavior of three-dimensional stitched carbon fiber epoxy composites at room temperature and high temperature [J]. AdvancedCompositesand Hybrid Materials，2022，5（3）：1951- 1965.

[3]SNEDDON IN. The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitraryprofile[J]. International journal of engineering science，1965，3（1）：47-57.

[4]LiR，Shih AJ.Tool temperature in titanium drilling[J].Journal of Manufacturing Science and Engineering，20o7，129（4）：740- 749.

[5]ZHAO HUA F，COOK R D. Beam elements on two-parameter elasticfoundations[J].Journal of EngineeringMechanics，1983， 109 （6）：1390-1402.

[6]CHENGL，JIG，FEI S，et al.A mechanistic predictive model for ultrasound guided insertion process ofO.11 mm Z-pins [J].Composite Structures，2022，294：115781.

[7]KARNOPP D.Computer simulation of stick - slip friction in mechanical dynamic systems [J].Journal of dynamic systems，measurement，and control，1985，107（1）：100-103.

[8]ASADIANA，PATELRV，KERMANIMR.Dynamics of translational friction in needle-tissue interaction during needle insertion [J]．Annals of biomedical engineering，2014，42：73-85.