Wei Zhang，Gang Wei，Xin－Ke Xiao，Zi－Tao Guo

(Hypervelocity Impact Research Center，Harbin Institute of Technology，Harbin 150080，China)

1 Introduction

High velocity impact often results in not only large plastic deformation but also fracture provided that the impact velocity is high enough.Such behavior is usually beyond the prediction capacity oftheory analytical method and is often relied on numerical simulations［1－2］， for example finite element calculation.Subjected to highly localized loading such as ballistic penetration，materials are highly affected by large strains，high strain rates，temperature softening，varying stress states and loading history，which may finally lead to failure［3］.Thus，it is of great importance to take these effects in the material models，for example，strength model and fracture criterion.Regarding the fracture criterion，many experimental studies have shown that the ductility depends highly on the stress triaxiality for ductile materials. In the literatures，several fracture criteria have been proposed based on this realization［4－6］，where some criteria are more complex than others.Although many of these ductile fracture models are proposed，only few are currently available in leading commercial finite element codes.For example，the constant fracture strain and the Johnson－Cook fracture criterion have been incorporated into ABAQUS/Explicit and LS－DYNA.Meanwhile，these fracture criteria often predict the same physical process with different results，as shown in Refs.［3，7 －8］.Thus，it is of practical importance to validate suitable fracture criterion/criteria for a certain problem.

During the Taylor impact test，in addition to large plastic deformation，high strain rate and temperature rise，variousstressstateswere generated in the impacting cylinder［1，9］.All these effects make the Taylor impact test as one ideal benchmark to evaluate the prediction capacity of a fracture criterion.

In our previous work［9］，the cylinders of 7A04－T6 aluminum alloy were fired against hardened tool steel plate.Three deformation and fracture modes，i.e.，mushrooming，shear cracking and fragmentation，were found at increasing velocities.Velocity ranges enabled each deformation and fracture modes were identified.In the present paper，finite element simulations were conducted with the help of material performance tests to evaluate the prediction capacity of five different fracture criteria.

2 Material Property Modeling and Taylor Impact Fracture Modes

2.1 Material Property Modeling

The constitutive relation and fracture behavior of 7A04－T6 aluminum alloy were investigated by conducting a series of material tests.The te st results and the developed material behavior models were presented in Ref.［10］.Here，only the main material models are briefly shown.

A constitutive relation based on the Johnson－Cook(J－C)constitutive law is used to describe the isotropic hardening，strain rate and elevated temperature effects［11］in impact simulations.The Johnson－Cook(JC)constitutive law reads

where f(εeq)=A+is the flow stress law at the room temperature and a reference strain rate.Usually，the strain hardening term f(εeq)in J－C constitutive law［11］i s accurate enough.In Ref. ［ 10］，however，it was found that this term cannot well predict the loadelongation curve obtained in the uniaxial tensile test.In order to improve prediction，the strain hardening term of Eq.(1)was replaced by a piecewise function.The proposed strain hardening term yields to the Voce empirical Eq.［12］for εeq＜ εu－ A/E，i.e.，before necking point in the uniaxial tension test.The model operates with three material constants;A，A1and t1.While εeq≥εu－ A/E(at and after necking point in the uniaxial tension test)，f(εeq)reads

The critical failure criterion developed for 7A04－T6 aluminum alloy reads

Here，the stress triaxiality ratio σ*is defined as σ*= σH/σeq.In Ref. ［1］，a cut－off value for the negative stress triaxiality at σ*= －1/3 was adopted.In the present paper，the simulations with and without this cut－off idea are conducted and the prediction results are shown.

By assuming adiabatic conditions，the temperature rise due to convention of plastic work into heat can be calculated as

Material constants in Eqs.(1)，(4)and(5)can be found in Ref.［10］.

一是水權(quán)產(chǎn)權(quán)體系尚未建立，水權(quán)提供者的權(quán)利還不明確。現(xiàn)行的取水權(quán)尚未作為水資源資產(chǎn)產(chǎn)權(quán)進(jìn)行確權(quán)登記發(fā)證，其獲得的水資源使用權(quán)的權(quán)利行使與義務(wù)履行不明確。目前的取水許可證更多反映的是行政管理的內(nèi)容，不具備產(chǎn)權(quán)證的特質(zhì)；農(nóng)村集體經(jīng)濟(jì)組織的水塘和修建管理的水庫中的水資源使用權(quán)沒有確權(quán)登記制度；水資源使用權(quán)的權(quán)利保護(hù)與權(quán)利流轉(zhuǎn)等制度還不健全。

In addition，the C－L，the constant fracture strain，the maximum shear stress and the maximum principle stress fracture models have been used for trying to predict the Taylor impact fracture of7A04－T6 aluminum alloy in the paper.Based on the material tests in Ref.［10］，all related parameters of the several fracture models can be obtained.

The Cockroft－Latham fracture criterion［5］reads，

The material constant Wcrfor C－L fracture criterion is calibrated by FE simulation on the tension tests on smooth cylinderspecimens and notched cylinder specimens at reference strain rate and room temperature，similar to the method used in Ref.［7］.The material constants for the followed fracture criteria of themaximum shearstressand themaximum principle stress were obtained in the same way.It was found Wcr∈［330.0，363.0］MPa by FE calculation.

The Constant strain fracture criterion reads

In the Taylor rod，compression and shear are the predominant stress states，although the tension also comes into being sometimes，as shown in Refs.［1，9］.Thus，two parallel simulations were conducted using the Constant strain fracture criterion，i.e.，fracture strain was taken as either that at compression or that at torsion.From Ref.［10］，it was found that εf=0.305 at σ*= － 1/3 and εf=0.609 at σ*=0.

The maximum shear stress fracture criterion reads

FE Simulations shows σScr∈［330.0，363.0］MPa.

FE Simulations shows σ1cr∈［632，1150］MPa.

2.2 Taylor Impact Fracture Modes

In Ref. ［9］，three deformation and fracture modes， i.e.， mushrooming， shearcrackingand fragmentation，were found in cylinders of 7A04－T6 aluminum alloy fired against hardened tool steel plate.Fig.1 shows the fracture modes at increasing impact velocities while Fig.2 shows the fractured Taylor rods.

Fig.1 Fracture modes of 7A04-T6 aluminum alloy Taylor rods

3 Prediction Results of Five Different Fracture Criteria

A 3－D finite element model is built.The geometry of the projectiles is a cylinder of 12.62 mm×50.82 mm and the target is a cylinder of 120 mm ×25 mm，as shown in Fig.3.Other details of the FE modeling can be found in Ref.［9］，where the material model and related constants are included.

Fig.2 Deformation and fracture modes of 7A04-T6 aluminum alloy Taylor rods［9］

Fig.3 FEM modeling of the Taylor impact test［9］

3.1 Modified Johnson-Cook Fracture Criterion

Fig.4 shows the prediction results by using the modified Johnson－Cook fracture criterion(as shown in Eq.(4)).By the comparison of the results in Figs.2 and 4，it can be seen that the modified J－C fracture criterion can predict close fracture modes of 7A04－T6 aluminum alloyprovided that the cut－off idea is adopted.When the cut－off idea is not used，i.e.，the material can fail under compression，projectiles suffer from severe fracture than that happed in the tests.

Fig.4 Deformation and fracture modes predicted by two methods regarding the fracture strain at low stress triaxiality values

3.2 Cockroft-Latham Fracture Criterion

Fig.5 shows the simulation results by the C－L fracture criterion(see Eq.(6)).The simulation using Wcr=100 MPa can predict the fracture modes of 7A04－T6 aluminum alloy Taylor rods reasonably although the simulation using Wcr=80 MPa predicts more fractures.

Fig.5 Deformationandfacturemodesof 7A04-T6projectiles predicted by C-L fracture criterion

3.3 Constant Strain Fracture Criterion

The simulation results using both sets of fracture strain show that 7A04－T6 aluminum alloy Taylor rods suffer from severe fracture，as shown in Fig.6.

Fig.6 Deformation and fracture behavior predicted by constant strain fracture criterion

Fig.4 indicates that the modeling of the fracture behaviorofa materialundercompression is of extremely important for Taylor fracture prediction.Thus，parallel simulations are conducted by using cutoff idea，i.e.，by setting

The simulation results are shown in Fig.7.It is clearly seen that shear cracking and fragmentation can be predicted by using fracture strain at compression.However，the predicted striking velocity range is not in agreement with the test results(as shown in Fig.1).

3.4 Maximum Shear Stress Fracture Criterion

Taylor impact simulations show that the fracture pattern predicted by using the maximum shear stress fracture criterion and σScr∈［330.0，363.0］MPa is not in agreement with the test results.The striking projectile’s nose elements suffer from eroding layer by layer.Even by setting σScr=450 MPa，no reasonable fracture can be obtained，as shown in Fig.8.

Fig.7 Deformation and fracture behavior of 7A04-T6 projectiles predicted by constant strain fracture criterion with the stress triaxiality cut-off idea

Fig.8 Numerical simulation obtained deformation and fracture process in the section plane of the projectile at V0=301.5 m/s when σSCr=450.0 MPa

3.5 Maximum Principle Stress Fracture Criterion

The simulations，by using the maximum principle stress fracture criterion，show that projectiles suffer from mushrooming at all the tested striking velocities and no fracture can be predicted by setting σ1cr=1150 MPa.Severe brittle fracture is observed in projectiles by setting σ1cr=681.2 and 834 MPa，as shown in Fig.9.It is clearly seen that the projectiles seem to fracture in direction parallel to the projectile axle.Obviously，the fracture mode is not consistent with the experiment results.

4 Discussions and Conclusions

Based on the Taylor impact test results of our previous study，the prediction capacity of five fracture criteria in finite element calculation is evaluated.The fracture criteria constants are calibrated according to a series of material tests.

The simulation results show that the modified Johnson－Cook fracture criterion with the stress triaxiality cut－off idea and the C－L fracture criterion can predict the Taylor fracture behavior of 7A04－T6 aluminum alloy rods.Constant fracture strain criterion predicts severe fracture without adopting the stress triaxiality cut－off idea. By including the stress triaxiality cut－off idea， the constant fracture strain criterion can predict shear cracking and fragmentation.However，the predicted striking velocity range of the two fracture patterns is not in agreement with the tests.Unfortunately，both the maximum shear stress and the maximum principle stressfracture criteria predict fracture patterns far from the reality.

The evaluation proves that the C－L facture criterion has a similar prediction capacity with the modified Johnson－Cook fracture criterion regarding the complex fracture behavior of Taylor rods of 7A04－T6 aluminum alloy.However，the C－L fracture criterion has only one material constant and thus needs only one tension test on smooth cylinder specimen at room temperature and reference strain rate，while the modified Johnson－Cook fracture criterion has 6 constants to calibrate and needs a series of material tests.

Moreover，the simulation results show that fracture strain cut－off at stress triaxiality less than － 1/3 is necessary to predict reasonable Taylorfracture patterns.The C－L fracture implicitly recognizes this idea.This may be a critical reason why the only one parameter C－L fracture criterion can predict well.

Fig.9 Numerical simulation obtained projectile deformation and fracture modes when σ1cr=834.0 MPa

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