Strain Rate-Dependent Constitutive and Low Stress Triaxiality Fracture Behavior Investigation of 6005 Al Alloy

School of Traffic and Transportation Engineering, Central South University, Changsha 410075, China Key Laboratory of Traffic Safety on Track, Central South University, Ministry of Education, Changsha 410075, China National and Local Joint Engineering Research Center of Safety Technology for Rail Vehicle, Changsha 410075, China Smart Transport Key Laboratory of Hunan Province, Changsha 410075, China Hunan Industry Polytechnic, Changsha 410208, China Joint International Research Laboratory of Key Technology for Rail Traffic Safety, Changsha 410075, China


Introduction
Al-Mg-Si-Cu alloy, due to its excellent processing property and mechanical behavior with medium strength, superior corrosion resistance, and advanced welding performance, has been playing an important role in various industrial applications and scientific research [1].It is crucial to ensure impact safety, while lightweight materials are extensively used to improve train speed [2][3][4].A large number of studies [5][6][7][8] on the energy absorption components of the train show that the dynamic mechanical response of the material has a great influence on the performance of crash energy absorption.Previous studies [9][10][11] have investigated the dynamic response of different aluminum alloys.6005 Al alloy, as a common manufacturing material for high-speed train, attracted extensive attention.e effect of temperature on the micromechanics of ductile fracture and the continuous cooling precipitation, including temperature-and time-dependent precipitation behavior, were investigated in [12,13].
e influence of the age-hardening behavior and microstructural characterization of precipitates are also studied by electron microscopy [14].
To study the dynamic mechanical properties of materials, experimental studies, constitutive model, and numerical simulations have been developed under various loading conditions.Electronic testing machines and the split-Hopkinson bar (SHPB) system have been used to investigate mechanical behavior under tension, compression, or other loading conditions [15][16][17].Constitutive models including the physical model [18] and phenomenological model [19] have been developed to describe the flow stress and mechanical behavior of materials under various loading conditions.Numerical simulation, based on the finite element method (FEM), is always used to calculate the flow behavior of variety of materials under specified loading conditions [20,21].
e constitutive behavior model plays an important role in numerical simulation.e Johnson-Cook model [22], one of the most widely used phenomenological models, is used to describe the coupling factors of strain hardening, strain-rate hardening, and thermal. is model is also widely employed in numerical simulation because the five material constants can be easily acquired from the experimental data.
Ductility, considered as an important mechanical property, has been investigated on many materials [10,23,24].Stress triaxiality, defined as the ratio of the hydrostatic stress and von Mises stress to depict the stress state of materials, is found to be an important factor that influences fracture strain greatly according to previous studies [23,24].Several fracture models [25,26] have been proposed to describe the dependency of fracture on the stress triaxiality.e JC fracture model [25] is a common model widely used in the commercial finite element code.However, the JC fracture model is not very accurate especially for the low triaxiality region.Bao and Wierzbicki (BW) [27] developed a more flexible fracture locus to reflect the fracture behavior in the fracture strain-triaxiality space.Wierzbicki et al. [28] compared seven representative criteria and found that the X-W criterion was performing well in a wide triaxiality range.Even if the BW fracture locus is more practical than the JC one, it still does not consider the Lode angle effect.Recently, the importance of the two parameters, triaxiality and Lode angle, has been stressed for the ductile fracture of metals.e MMC is a modification of the original criterion whose description and derivation can be found in [29] and has been successfully used by Gilioli et al. [30] to study the ballistic impact failure of aluminum 6061-T6.Previous studies [10,11,24] mostly focus on high triaxialities, which means that the material is under axial tension loading.
e fracture behavior of 6005 Al alloy under the tension and shearing loading would be studied.
In this paper, various loading conditions are set to obtain the dynamic behavior of 6005 Al alloy.Not only the strainrate effect would be investigated through a series of experiments at different strain rates using the MTS and SHTB testing systems, but also the mechanical behavior under the combined force with tension and shearing was studied.
ere are several strain rates including quasi-static and dynamic (10 −3 s −1 , 10 −2 s −1 , 10 −1 s −1 , 700 s −1 , 1100 s −1 , and 3300 s −1 ).To investigate the fracture strain and ultimate strength of the material under the combined force with tension and shearing, tensile shearing tests under the different angles (0 °, 30 °, 45 °, 60 °, and 90 °, resp.) are performed.Furthermore, a modified Johnson-Cook constitutive model would be obtained to describe the dynamic behavior of 6005 Al alloy.Model predictions would be compared with the testing data.Finally, a fracture criterion would be proposed to describe the relationship between fracture strain and stress triaxiality at the stress state of low triaxialities.

Materials and Experiments.
e present study has been carried out on 6005 Al alloy.e chemical composition of the material is listed in Table 1.Two groups of experiments are set to study the mechanical property.ey are tensile tests at various strain rates and tensile shearing tests of double-notch specimens.ree repeated tests under the same condition are carried out to ensure the accuracy and reliability of results, especially for dynamic loading tests.

Low Strain Rate Tensile Tests.
To more reasonably study the mechanical behavior of 6005 Al alloy under the quasistatic and low strain-rate loading condition, tension tests based on four strain rates (from 10 −4 s −1 to 10 −1 s −1 ) are adopted by the Mechanical Testing and Simulation (MTS) 647 hydraulic wedge grip.e maximum axial load ability of the machine is 200 kN.e tension specimens are used with the dimensions of 100 mm in the gauge length L 0 and 20 mm in the width of the cross section and 8 mm in the thickness of the cross section, as shown in Figure 1(a).
e engineering stress σ E and engineering strain ε E can be expressed as follows: where A 0 is the cross section area of the specimens.F and ΔL are the tensile load and the elongation of the specimens, respectively.Furthermore, the true stress σ T and true strain ε T are obtained by the following equations [25]: Note that (2) is inapplicable after the onset of necking.Figure 1(b) shows the specimen fixed in the MTS.ree repeated tests are conducted to ensure the repeatability.
2.1.2.SHTB Tests.SHTB test system is used to study the dynamic behavior of 6005 Al alloy.e striker of SHTB is 20 mm in diameter and 500 mm in length.e incident bar is 20 mm in diameter and 3300 mm in length, while the transmitted bar is 20 mm in diameter and 1500 mm in length.To obtain a good signal of the strain and signal-to-interference ratio, both bars are made of LY12 Al alloy with ultrahigh hardness.e bullet, promoted by the launching system with high-pressure gas, impacts the flange of the incident bar.en, a tensile loading wave generated through the incident bar.A strain gauge, affixed to the surface of the incident bar, records the incident signal and the reflected signal, while another one affixed to the surface of the transmitted bar records the transmission signal.All of signals simplified through a high dynamic strain meter are stored and recorded by an oscilloscope.Finally, a computer deals with the strain signals.Figure 2 shows the experimental device of the SHTB system.
Based on the hypothesis of the one-dimensional stress wave and strain and stress axial uniformity, the expression of 2 Advances in Materials Science and Engineering engineering stress, engineering strain, and strain rate is calculated by the following equations: where E is the elastic modulus of the incident bar.A and A S are the cross section areas of the incident bar and specimens, respectively.C 0 is the stress wave speed and L is the gauge length of the specimens.ε t (t) and ε r (t) are the signals of the transmission wave and re ection wave._ ε(t) is the strain rate.e true stress and true strain of the dynamic tensile experiment can also be obtained by (2).
In this study, three di erent strain rates, that is, 700 s −1 , 1100 s −1 , and 3300 s −1 , are conducted to research the dynamic behavior at high strain rates.ree repeat measurements are performed in each test in order to improve the accuracy of results.e specimen is xed in the SHTB as shown in Figure 2(c).

Tensile Shearing
Tests.Tensile shearing tests have been carried out to investigate the relationship between stress triaxiality, double-notch specimens with di erent stress triaxialities are machined.e specimens are similarly used with the dimensions of 60 mm in overall length, 5 mm in gauge length, 3 mm in notch length, 20 mm in width, and 7 mm in thickness.e angles between loading direction and the bearing cross section between two notches, denoted as α, are designed as 0 °, 30 °, 45 °, 60 °, and 90 °, respectively.Figure 3 shows the specimens of shearing tests.e tensile shearing tests of 0 °specimens can be regarded as pure shear tests, while other tests are under the combined action of tension and shearing.For these tests, the normal stress and shearing stress can be obtained through the following equations:

Advances in Materials Science and Engineering
where A min is the bearing area of the notched specimens given as follows: where t and l are the thickness and length of the bearing cross section between two notches.For the plane stress state, principal stresses can be calculated with the following equation: e stress triaxiality is de ned as   Advances in Materials Science and Engineering where σ h is the hydrostatic stress, σ m is the von Mises equivalent stress, and σ 1 , σ 2 , and σ 3 represent three principal stresses.According to ( 4)-( 7), the stress triaxiality can be rewritten as e initial stress triaxialities of di erent specimens are listed in Table 2.It can be found that the stress triaxiality increases from 0 to 0.33 with the increase of α.
ese tests are conducted by MTS, and the loading speed is 1 mm/min.

Constitutive Model.
For tensile loading, to re ect the relationship between strain, stress, strain-rate hardening, and ambient temperature, the Johnson-Cook model [22], one of the widely used constitutive models, is commonly applied to describe the ow behavior, which is shown as follows: where σ and ε are the stress and equivalent plastic strain, respectively.A is the yield stress of the material.B and n are the parameters of strain hardening.C is the strain-rate hardening parameter, determined by the material._ ε * _ ε/_ ε 0 is the dimensionless plastic strain rate, where _ ε is the strain rate and _ ε 0 is the reference strain rate.T * (T − T r )/ (T m − T r ) is the dimensionless temperature indicating the thermal softening e ect, where T r is the room temperature and T m is the melting point of the material.During high strain-rate loading, specimens are reasonable to be assumed in adiabatic conditions.In this case, the rate of temperature change is calculated through where ρ and C p and are the density and heat capacity of the material.χ is the conversion factor of work into heat.e temperature rise can be estimated using the following equation: where ε e is the strain corresponding to the maximum stress.ρ and C p are assumed as constants, while χ is taken as 0.9 for metals [31].e related parameters are given in Table 3. e equation can be rewritten as All tensile tests in this study are conducted at room temperature.Furthermore, strain hardening and strain-rate hardening are only considered without the e ect of temperature for low strain-rate tests.erefore, the JC model can be simpli ed as the following equation: In this study, 10 −4 s −1 is speci ed as the reference strain rate, that is, _ ε 0 10 −4 s −1 .e constitutive behavior model can be rewritten as en, take logarithm on both sides of the equation, and the equation can be rewritten as Unknown parameters A, B, and n can be determined with the least squares method.In B and n represent the intercept of the straight line and the slope of the straight line, respectively.
Next, to obtain the value of C, the equation can be transformed as follows: where σ 1 A + Bε n , σ 1 represents the stress of the material at 10 −4 s −1 strain rate, and σ 2 (_ ε) represents the yield stress at a strain rate of _ ε.

Flow Stress at Di erent Strain Rates.
e true stressstrain curves under di erent tensile strain rates before the onset of necking are plotted in Figure 4. e tension process can be generally divided into two stages: elastic stage and plastic stage.One thing to note about the di erence of Young's modulus between low and high strain rates at the elastic stage is the fact that, during Hopkinson tensile tests, the stress wave in the specimen needs two or three re ections to achieve uniformity, resulting in the inhomogeneity of stress in the specimen.erefore, the stress-strain relation at high strain rates is inaccurate.For the plastic stage, the strain hardening e ect can be seen under each condition.A stronger strain hardening e ect is observed at higher strain rates (700 s −1 -3300 s −1 ) in the plastic stage.In addition, a clear strain-rate hardening e ect is observed when comparing curves of di erent strain rates.e yield strength and ultimate strength for 6005 Al alloy under di erent strain rates are all listed in Table 3.
From Table 4, it can be seen that the yield stress and ultimate stress gradually increase when the strain rate increases.e yield stress experiences an increase of 35.7% from quasi-static to 3300 s −1 , while the ultimate stress increases by 50.3%.Evidently, 6005 Al alloy is a material with great dependence on the strain rate.

Results of Tensile Shearing Tests.
For this group of tests, the central cross section of specimens is under tension-shear combined loading, which is a complex stress state.e true stress and true strain cannot be obtained directly due to the di erent deformation mode for each test.e experimental results are thus exhibited with the curves of force and displacement, as shown in Figure 5.As the comparison shows, the maximum tensile force for di erent specimens di ers a lot.As the notch angle increases from 0 °to 90 °, the maximum tensile force increases monotonously from 7718 N to 16301 N experiencing an increase of 111.21%.e fracture displacements scatter from 0.76 mm to 1.9 mm, and no clear correlation to notch angle is observed.

Material Parameter Identi cation.
e experimental data of the quasi-static tensile are used to determine the three material constants A, B, and n, and the yield stress data at the strain rate range from quasi-static to 0.1 s −1 to determine C of the simpli ed Johnson-Cook model.
e temperature soft parameter m can be obtained by comparing the results from experiments and the simpli ed JC model.e four parameters for 6005 Al alloy at di erent strain rates can be obtained with the method mentioned in Section 2.2 as shown in Table 5. e relationship between ln (σ − A) and ln ε is shown in Figure 6.
To compare the predicted results from the JC constitutive model and experimented results, both of the two data groups are plotted in Figure 7.However, the values of yield stress at high strain rates are underestimated.Considering the temperature rise caused by the adiabatic process at high strain rates is limited (30 K calculated by (9d)), the temperature e ect is neglected.It could be observed that the prediction agrees well with the experiments at the quasistatic strain rate, and with the increase of strain rate, the prediction deviates from experimented results increasingly.is deviation may be caused by neglecting the variations of the strain-rate coe cients.at is, the parameter C in the JC model is supposed to be a variable instead of a constant.Apparently, the original JC constitutive model cannot depict the ow stress behavior at dynamic loading adequately.erefore, it is necessary to modify the original JC constitutive model.

e Modi cation of the JC Constitutive Model.
For the simpli ed JC constitutive model neglecting the temperature e ect, ow stress is a function of strain and strain rate.
e relationship between ow stress and ln _ ε * is considered as linear in the original JC model, which is an ideal case.Actually, the strain-rate coe cient may also correlate with strain.In other words, for a material under a certain strainrate loading, the strain-rate coe cient may vary with the    8 exhibits the relationship between the strain-rate coe cient and strain.On one hand, a nonlinear relationship can be found between C and ε.On the other hand, for di erent strain rates, the relationship is di erent.A nonlinear relationship can be observed as well when it comes to C and ln _ ε * , as shown in Figure 9. erefore, it can be assumed that C is a function of strains and strain rates.e relationship is given as follows: From Figures 8 and 9, considering the nonlinear relationship between C and the two variables, C can be described as a binary quadratic polynomial of ε and In _ ε * independently.In addition, the interaction e ect between the two variables should be taken into consideration.A quadratic polynomial regression model can thus be used to describe the relationship between C and ε and between C and In _ ε * shown as follows: where C 0 -C 5 are coe cients needed to be solved.e relationship between C and the two independent variables can be described with a surface in the three-dimensional space coordinate system.e six coe cients can be obtained directly by nonlinear surface tting (shown in Figure 10).e blue surface represents experimental data, and the red one is the tting surface.e parameters of the modi ed JC constitutive model of 6005 Al alloy are listed in Table 6.Comparison between the predicted and experimented results is shown in Figure 11.Obviously, the modi ed JC model predicts the ow stress more accurately compared with the original JC model.It is indicated that the modi cation using the C function is effective to depict the ow behavior of the investigated material.

FEA Modeling of Tensile Shearing
Tests.In this paper, numerical simulations are used to obtain the actual initial stress triaxiality and equivalent plastic strain to fracture.ABAQUS/Standard, as a common FEA code, is employed to simulate tensile shearing tests.Five tensile shearing tests of di erent angle specimens are simulated.e material property is de ned as elastic-plastic and isotropic hardening.e stressstrain data of the plastic stage from the quasi-static tensile test are input.e element type is the three-dimensional hexahedral element with reduced integration.e 3D nite element model meshes of specimens are all shown in Figure 12.
e minimum size of elements is 0.

Plastic strain
Quasi-static-JC 0.001 s -1 -JC 0.01s -1 -JC 0.1s -1 -JC 700 s -1 -JC 1100 s -1 -JC 3300 s -1 -JC Quasi-static-Exp.0.001 s -1 -Exp.0.01 s -1 -Exp.0.1s -1 -Exp.700 s -1 -Exp.1100s -1 -Exp.3300s -1 -Exp.Advances in Materials Science and Engineering boundary conditions are imposed on the ones on the other end.It is worth stressing that fracture criterion is not applied in modeling for the reason that the numerical simulations are performed to validate the numerical analysis results.Results obtained from the numerical simulations and experiments are compared in Figure 13.e two groups of force-displacement curves are in good agreement before the onset of fracture, as can be seen in Figure 13.Because no failure criterion is employed in the simulations, the simulations do not re ect the fracture behavior.However, due to the good agreement before the onset of fracture, it is still reliable to obtain the equivalent plastic strain to fracture and stress triaxiality from numerical simulations.
e FEA model can thus be used to formulate the fracture strain.
3.6.Fracture Criterion for 6005 Al Alloy.In a certain triaxiality region, the coupling of the triaxiality and fracture strain constitutes a fracture criterion.e model built in Section 3.5 is used to obtain the equivalent plastic strain to fracture.By comparing the force-displacement curves of simulations and experiments, the fracture time step can be determined.e equivalent plastic strain at this time step is the fracture strain.It should be noted that the object to study is the critical elements where the fracture begins.e point is to determine the location where the fracture begins.It is observed from the nite element models that the maximum equivalent plastic strain at the fracture time step locates in the root of notches, which means that the fracture begins there.Figure 14 shows the distribution of equivalent plastic strain at the fracture time step in a typical numerical simulation.
In Section 2.1.3,a theoretical calculation method based on the assumption of the plane stress state for the initial stress triaxiality is proposed.However, specimens under tensile loading are actually in the three-dimensional stress state.erefore, it is necessary to acquire the actual initial stress triaxiality from the numerical simulations.e two sets of initial stress triaxialities and corresponding fracture strains are listed in Table 7.
According to Table 7, the actual initial stress triaxialities, within the range from 0.04 to 0.741, are obviously greater than the theoretical values.It may be caused due to the threedimensional stress instead of the plane stress state in which the specimen is.e actual initial stress triaxialities coupled with equivalent plastic strains to fracture are plotted in Figure 15.Results indicate that the equivalent plastic strain to fracture presents nonmonotonic dependence on the initial stress triaxiality.In the stress triaxiality range from 0.04 to 0.369, which is a low triaxiality level, when σ * increased, the fracture strain increases as well from 0.767 to 1.423.A drop shows up as the stress triaxiality exceeds 0.369, a relatively high triaxiality level.According to the study by Bao and Wierzbicki [23], for low triaxialities (0.04-0.369), a parabolic t is found to be satisfactory.e tting result is shown as (12).A plot of the equation is shown in Figure 16.

Discussion
4.1.Evaluation of the Modi ed JC Model.In Section 3.2, a modi ed JC constitutive model has been proposed in which C is considered to be relative to strain and strain rate.
Previous studies [32,33] have modi ed JC constitutive model for high-strength alloys by changing the form of the strain hardening factor and for AA6005-T6 by multiplying the strain-rate factor with a true stress.e method used in the present study has been successfully employed to modify the JC model for 7050-T745 aluminum alloy in [34].
To accurately evaluate the forecasting error of two constitutive models of 6005 aluminum alloy, three evaluating indices are investigated.ey are the correlation coe cient, normalized mean bias value, and average absolute relative error denoted as R, NMBE, and AARE, respectively, which are expressed as follows:   Advances in Materials Science and Engineering    Advances in Materials Science and Engineering R where E i and P i are the experimental and predicted values, while E and P are the mean values of them, respectively.N is the total number of data used in this study.e correlation coe cient is used to characterize the degree of a linear correlation relationship between two variables.
e NMBE represents the mean bias value of predicted values, and a positive NMBE value means overprediction, while negative one means underprediction from a model.AARE is an unbiased statistic that estimates the accuracy of predictions by comparing the relative error term by term.8. e average absolute relative errors of the original JC model and modi ed JC model are 3.71% and 1.34%, while the normalized mean bias errors are 6.8% and −0.049%, respectively.
ese results show that the modi ed JC model features higher prediction accuracy than the original JC model.
In addition, the absolute relative error is employed to analyze the performance of the two models.e absolute relative error can be calculated with the following equation:

Absolute relative error
It can be obtained that the absolute relative error given by the original JC model varies from 0.0031% to 8.14%, while the value is in the range of 0.05% to 3.26% for the modi ed JC model.e distribution of the absolute relative errors is plotted in Figure 18.For almost 95% data sets, the relative error of the modi ed JC model is concentrated in the range of 0 to 2%, while the relative error scatters from 0 to 9% erratically for the original JC model.In conclusion, the modi ed Johnson-Cook model can describe the ow behavior of 6005 aluminum alloy more accurately than the original Johnson-Cook model.

e Fracture Behavior under Low Stress Triaxiality.
In this paper, a parabolic equation is established to determine the fracture strain, in which stress triaxiality is the only independent variable.Several studies [10,23,24] have investigated the fracture behavior of aluminum alloy, while most of them focus on the high stress triaxiality.e Johnson-Cook fracture model is the most commonly used failure criterion.Bao and Wierzbicki [23] have studied the fracture ductility of AA2024-T351 in the entire range of stress triaxialities (−1/3 to 0.95).ree expressions were given for negative (−1/3 to 0), low (0-0.4),and high (0.4-0.95) triaxialities, respectively.A similar tendency of fracture strain is observed for AA6005 in the range from 0.04 to 0.741.
With an elongation beyond 10%, 6005 Al alloy can be regarded as a ductile material.To further investigate the fracture mode of 6005 Al alloy during tensile, pure shear, and tensile-shear combined tests, some magni ed fracture surfaces are collected by a scanning electron microscope (SEM).Figure 19 shows the typical fracture surfaces of tensile shearing specimens with the notch degree from 0 °to 90 °. Figure 19(a) shows the pure shearing fracture surface, where there are typical "ripple waves" and the direction of the shear fracture surface is parallel to that of the maximum shear stress.It indicates that the fracture mechanism is shear fracture.During the pure shear test, the stress triaxiality is about zero.
e shear band appears in the center of the specimen.Cracks are produced within the shear band and then coalesced.When the equivalent plastic strain reaches the critical value, the fracture takes place.Figure 19(e) shows the fracture surface of 90 °specimen, where there exist a number of dimples with di erent size.During the tensile test on 90 °specimen, there is a high stress triaxiality in the root of the notch.e specimen fractures after experiencing the process of nucleation, growth, and coalescence of voids.For tensile shearing tests, dimple and shear combination of di erent degrees can be seen in Figures 19(b)-(19d).In the

Conclusion
In this study, the dynamic constitutive behavior of 6005 Al alloy is investigated by means of experiments and numerical simulations.Tensile tests under di erent strain rates indicate that the ow stress of the investigated material is rather sensitive to strain rate.What is more, a clear e ect of strain hardening is observed.Based on the tensile test data, a simpli ed Johnson-Cook model was established to describe the ow stress of 6005 Al alloy at various strain rates.Furthermore, the original JC model was modi ed using the   A set of tensile shearing tests were carried out to study the fracture behavior under low stress triaxialities.Numerical simulations based on FEA were used to obtain the actual initial stress triaxialities and equivalent plastic strain to fracture.e tendency of the fracture strain is nonmonotonic in the triaxiality range of 0.04 to 0.74.A parabolic fracture criterion was thus proposed in the triaxiality range of 0.04 to 0.369.Finally, the fractography analysis for 6005 Al alloy indicates that the material has a typical ductile fracture mechanism including the shear fracture under pure shear and the dimple fracture under uniaxial tensile.

Figure 1 :Figure 2 :
Figure 1: (a) Specimen for the conventional tension test; (b) MTS 647 hydraulic wedge grip used for the conventional tension test.

Figure 4 :
Figure 4: True strain-stress curves of 6005 Al alloy at di erent strain rates.

Figure 7 :Figure 6 :Figure 8 :
Figure 7: Comparison between predicted results of the JC model and experimental results at di erent strain rates.

Figure 11 :
Figure 11: Comparison between the predicted results and experimented results: (a) low strain rates and (b) high strain rates.

Figure 13 :
Figure 13: Comparison between numerical simulations and experiments.

Figure 14 :
Figure 14:e distribution of equivalent plastic strain at the fracture time step in a typical numerical simulation.

Figure 15 :
Figure 15: Actual initial stress triaxialities coupled with equivalent plastic strains to fracture.

Figure 17 :
Figure 17: Correlation analysis between experimental values and predicted values: (a) the original JC model and (b) the modi ed JC model.

Figure 18 :
Figure 18: e distribution of the absolute relative errors of (a) the original JC model and (b) modi ed JC model.

Table 1 :
e chemical composition of 6005 Al alloy.

Table 3 :
Values of material parameters for 6005 Al alloy.

Table 4 :
e yield and ultimate stresses under di erent strain rates for 6005 Al alloy.

Table 5 :
Parameters of the Johnson-Cook model of 6005 Al alloy.
2. e nodes on one end of the 3D nite element model are xed, while displacement

Table 6 :
e material constants of the modi ed JC constitutive model of 6005 Al alloy.

Table 7 :
e calculated and actual initial stress triaxialities and corresponding fracture strains.

Table 8 :
e performance of the two models.
Advances in Materials Science and Engineering range of low stress triaxialities, fracture is caused by a combined function of shear and void growth.