Yao Di,Cai Lixun,Bao Chen

School of Mechanics and Engineering,Southwest Jiaotong University,Chengdu 610031,China

Anewapproachonneckingconstitutiverelationships of ductile materials at elevated temperatures

Yao Di,Cai Lixun*,Bao Chen*

School of Mechanics and Engineering,Southwest Jiaotong University,Chengdu 610031,China

A new method is presented to determine the full-range,uniaxial constitutive relationship of materials by tensile tests on funnel specimens with small curvature radius and finite element analysis(FEA).An iteration method using FEA APDL(ANSYS parametric design language)programming has been developed to approach the necking constitutive relationship of materials.Test results from SAE 304 stainless steel at room temperature show that simulated load vs displacement curve,diameter at funnel root vs displacement curve,and funnel deformation contours are close to modeled results.Due to this new method,full-range constitutive relationships and true stress and effective true strain at failure are found for 316L stainless steel,TA17 titanium alloy and A508-III stainless steel at room temperature,and 316L stainless steel at various elevated temperatures.

1.Introduction

The true stress-strain curves of ductile materials before necking initiation can be easily obtained by conventional uniaxial tensile testing using a standard round bar.

where σTis true stress,εTis true strain,σEis engineering stress,and εEis engineering strain.However,after the necking deformation occurs,the acquisition of uniaxial true stress and true strain becomes difficult due to the rapid reduction in local cross section of the straight round bar.Bridgeman1made an assumption that the contour of the cross section in the necking deformation zone was circular,and the equivalent strain was uniformly distributed on this section.Therefore,the corrected stress in the necking deformation zone of a round bar could be found as follows:

where S is nominal stress and,as shown in Fig.1,d is the minimum diameter of the cross section in the necking zone,and R is the radius of the necking section.

The Bridgeman correction of stress leads to material curves affected by an error that can be greater than 10%and requires a significant amount of experimental work in order to measure the evolving curvature radii of necking profiles at different stages of each tensile test.2

Fig.1 Necking shape.

To calculate the necking deformation in the work zone of a round bar,Chen3,Needleman et al.4,5,and Saje6achieved necking simulation of a round bar by finite element analysis(FEA).To induce necking deformation,Chen3made an artif icial taper on the round bar,Needleman et al.4,5used bifurcation criterion and Saje6set a rigid restriction at the end of the bar.However,calculation accuracy was limited due to the low level of the technique’s use of computers and FEA,and there was no experimental verification for this method.Gurson7proposed a void growth model under an axisymmetric stress state to describe the large deformation of materials.Chu and Needleman8,and Tvergarrd9developed a GTN(Gurson-Tvergarrd-Needleman)model by improving Gurson’s model.The GTN involves the highly complex determination of nearly 10 parameters and its accuracy cannot be ensured.

Accuracy of simulation results for necking was calculated for the first time by Li.10By adjusting main stress and main strain,Norris et al.11proposed an preliminary iteration method to obtain the true stress-strain curve after necking.Thereafter,Matic et al.12–14proposed a method to evaluate the full-range constitutive relationship of ductile alloys by adjusting the power-law parameters in the constitutive model.However,many materials’constitutive relationships have nonpower-law hardening constitutive relationships after necking or even during hardening stages.Zhang et al.15,16also used the parameter searching method to acquire full-range constitutive relationships where the load-displacement curve was set to be the target of convergence.Choi17,Nayebi18,Cabezas and Celentano19,and Lee20et al.made several attempts to acquire full-range constitutive relationships of ductile materials by different methods,but the resulting strain ranges were limited,and the validity of methods was also questionable.Joun et al.21,22completely simulated the necking of a straight round bar without defects using a rigid plastic finite element method.However,the validity and the accuracy of that research still need to be further tested.In recent research reported by Xue et al.23full-range constitutive relationships were obtained by theabove-mentioned parametersearchingmethod.Yao et al.24,25proposed a method to obtain a full-range constitutive relationship by using a standard straight round bar and a funnel-shaped round bar simultaneously,but the dispersion of materials has a remarkable effect on this method that cannot be easily countered.

In this study,the finite element analysis aided testing(FAT)method has been proposed to acquire the full-range constitutive relationships of ductile materials by using a funnelshaped round bar.This method contains the determination of true stress-strain relationships both before and after necking.The application of the funnel-shaped round bar can directly simulate the necking phenomenon without artificial defects.By directly adjusting the input data of a constitutive relationship in FEA,the full-range true stress-strain curve can be determined if the experimental load-displacement records coincide with the numerical results.Additionally,an optical observation based on a digital image correlation(DIC)technique is employed to verify the validity of the FAT method by checking root diameter variation in a funnel-shaped round bar,outline of the deformed specimen,and strain distribution on the specimen.Based on the proposed FAT method,the full-range constitutive relationships are estimated for SS304(SAE 304 stainless steel),TA17 titanium alloy and A508-III steel at room temperature,and SS316L(SAE 316L stainless steel)at various elevated temperatures.The critical failure true stress and true strain are also given.Stainless steel is now widely used in the aviation and aerospace industries for things such as engine components and wing parts because of its high strength,elongation and anti-fatigue performance.TA17 titanium alloy is a typical light-weight aerospace alloy applied as the main material for air frames and engines.A508-III steel is a reaction pressure vessel material that has high strength with relatively low hardening.According to analyses of stress and strain distributions on cross sections in the necking zone,the failure mechanisms of ductile materials will been discussed in detail.

2.Research conditions

2.1.Testing system

The uniaxial tensile test system includes the universal test machine material testing system(MTS),room temperature strain extensometer MTS632.12C-21(25 mm gauge length,50%measuring range,5‰precision)and high temperature strain extensometer MTS632.68F-08(12 mm gauge length,20%measuring range,5‰precision),centering grips system and VIC-3D(video image correlation-3 dimensional)optical measuring system.The control mode of the tensile test is displacement and the test speed is 0.02 mm/s.The VIC-3D noncontact optical measurement system was used to obtain the diameter at the funnel root d and the deformation contours of the funnel specimen.The impact effect of bias-load was eliminated by the centering grips system.Elastic modulus testing results in four directions of the same specimen showed that the test error does not exceed 0.5%using the centering grips.

2.2.Materials and specimens

The materials to be tested are SS304,A508-III,TA17 and SS316L.SS304,TA17 and SSA508-III were tested at room temperature,and SS316L was tested at room temperature,300 °C and 500 °C.After solid solution strengthening,the mechanical properties of SS304 and SS316l are quite stable.

Fig.2 shows dimensions of the straight round bar specimens and funnel-shaped specimens.The radius of the funnelshaped specimen R0=13 mm is used in SS304 testing,R0=5 mm specimen is used in SS316L and TA17,and R0=10 mm specimen is used in A508-III testing.

2.3.Finite element model and analysis

Fig.3 shows the tensile deformation of the funnel specimen during the VIC-3D measurement and FEA.An axisymmetric meshing model was built to simulate the deformation behaviors of the funnel specimen.Considering the symmetry of the specimen,an axisymmetric element Plane 182 with 4 nodes and plastic ability,large deformation and large strain analyses were used in FEA.To ensure simulation accuracy,mesh refinement was applied at the root of the funnel.The boundary conditions are shown in Fig.3;the tensile test is the one fixed end and the other is applied displacement.

3.Testing results

The true strain-stress curves of materials before necking can be obtained by tests of round bar specimens to verify the authenticity of the method’s iterative results.The strain distribution on the surface of a funnel-shaped SS304 specimen was obtained by the VIC-3D testing system.

Fig.3 Finite element simulation.

3.1.Uniaxial testing results

The uniaxial tensile test results of SS304,A508-III,TA17 and SS316L at room temperature,SS316L at 300 °C and 500 °C were completed.After logarithmic processing,the results are shown in Fig.4.P-V curves of the funnel specimens are shown in Fig.5.

3.2.Testing results of VIC-3D system

After VIC-3D testing of the diameter at funnel root d vs displacement V,funnel deformation contours and the strain distribution on the surface of the funnel section were obtained.

4.Iterative method to obtain true stress-strain curve of materials by funnel-shaped specimen

Theoretically,when a true stress-strain curve is input into the commercial FEA code as the fundamental material model,output results should be consistent with experimental results.An iterative procedure to determine the full-range true stress-strain curve has been established by checking the truth of the load-displacement curve simulated for a funnel-shaped round bar produced from FEA.

4.1.Acquiring true stress-strain curve before necking using a funnel-shaped round bar

According to the tensile test on a funnel-shaped round bar,the uniaxial force P and the mean strain εmare directly obtained.As shown in Fig.6,the mean stress σmat the root cross section can be obtained by the uniaxial load P and the diameter d at the root of the funnel-shaped specimen,

where d0is the initial diameter at the root of the funnel shaped specimen shown in Fig.1.Then,a reference stress,σr,can be defined through the Bridgeman correction as

where R0is the radius of the funnel section.

Due to inhomogeneous deformation at the root of the funnel-shaped specimen,a reference strain,εr,can be defined as26

where Feis the Geometric correction coefficient,L0is the span offunnel arc,A0is the area of cross section at the root of the funnel-shaped specimen,and H and Aiare as shown in Fig.7(a).The initial mean stress-strain(σm-εm)curve and the reference strain-stress(σr-εr)curve of the SS304 funnel-shaped specimen with R=13 mm are shown in Fig.7(b).

Fig.4 True strain-stress curves of different materials before necking at different temperatures(controller mode:displacement;controller rate:0.02 mm/s).

Fig.5 Tensile testing results of different material funnel shaped specimens(controller mode:displacement;controller rate:0.02 mm/s).

Fig.6 εmand σmin funnel-shaped specimen.

Use the σr-εrcurve as the multilinear constitutive relationship model in the FEA software.The loading model in FEA is a change of displacement in the gauge section.The amount of the displacement Vfcan be obtained from the tensile tests when the force P reaches maximum.The true stress-strain curve before necking can be obtained by the following iterative method:

(a)Calculating the simulated P-V curve before necking,as shown in Fig.8,the P-V curve and the stress-strain curve consist of a series of data points Pi-Vi,σi-Vi,and εi-Vidue to the control model of displacement.

(b)The stress σnewcan be modified by the equation

where Fi,Eis the force obtained by testing,and Fi,Fis the force calculated by FEA.Thus,the stress in the σr-εrcurve is updated.The εnewcan be obtained by output from Von Mises strain at the center node of the root cross section using the correspondence relationship with the displacement Vi.

(c)Taking the updated constitutive relationship curve σnewεnewas the mutilinear constitutive relationship model in the FEA software,calculate the simulated P-V curve before necking;if the simulating curve coincides with the testing curve,stop the iterative process.If not,repeat processes(a)and(b).

With the continuous iteration,the simulated P-V curve will become more and more close to the testing P-V curve.The iterative stress-strain curves will agree well,so the iterative process can stop;the true stress-strain curve before necking can be obtained from the last iteration.Fig.8(b)shows the whole iterative procedure to estimate the true stress-strain curve before necking.

Fig.7 Method to obtain σr-εrcurve of SS304.

Fig.8 Iterative processes before necking.

After several iterations,the true stress-strain curve before necking is obtained by the funnel-shaped specimen using the iterative method.Compared to the experimental stress-strain curve before necking,as shown in Fig.9,the two curves match closely.Therefore,it can be proven that the iterative method is suitable for the estimation of true stress-strain curve before necking,and this iteration procedure ought to be further recommended to estimate full-range true stress-strain curves including necking deformation.The iterative method to obtain the true stress-strain curves of ductile materials is named the FAT method.

4.2.Acquiring true stress-strain curve after necking using a funnel-shaped round bar

The above stress-strain curve before necking can be described by using the strain hardening Chaboche model.27By regarding this model as the initial input constitutive relationship for the FEA code,the full-range true stress-strain curve,including necking deformation,can be obtained through several iterative analyses.The P-V curves and stress-strain curves are shown in Fig.10.From Fig.10 it can be seen that,compared to experimental results,a coincident numerical P-V curve is presented with only a two-time iteration.

4.3.Validity offAT method

Fig.9 Comparison of stress-strain curves between test and iterative method.

Fig.11 shows the evolution of the diameter d at the root of the funnel-shaped specimen with respect to the displacement V resulting from the VIC-3D measurement and the iterative method.Apparently,these two curves are similar to each other,and the validity of the funnel zone outlines and the strain distribution on the surface of the funnel obtained by the iterative method can be also confirmed by the VIC-3D measuring results,as shown in Fig.12.

The full-range uniaxial constitutive relationship curve of SS316L at different temperatures can also be obtained,as shown in Fig.13.

The full-range uniaxial constitutive relationship curves of SSA508-III at room temperature are shown in Fig.14.

The full-range uniaxial constitutive relationship curves of TA17 at room temperature are shown in Fig.15.As shown in the figure,the FAT method can be applied to light weight aerospace alloys,which is significant for the fracture analysis of air frames and other parts.

4.4.Acquisition of breaking strain and stress of stainless steel

The displacement Vfcan be obtained from testing.Thus,the critical fracture stress and strain are obtained from FEA when the displacement load reaches Vf.Table 1 shows the critical fracture strain and stress of SS304,SS316L,TA17 and A508-III.

4.5.Application offull-range constitutive relationship to sheet specimens with a center hole

The full-range constitutive relationships based on the proposed FAT method have been verified in many cases of different shaped specimens.An additionalexampleisshown in Fig.15.It is the application to tensile tests of sheet specimens with a circular hole in center(CHS specimen).The material to be tested is SS316L.Fig.16 shows dimensions and meshing model in FEA of the specimen.As we can see,the simulated force-gauge displacement(P-V)curve and the testing results coincide.

It is important that the full-range constitutive relationship(FFCR)of ductile materials obtained by the FAT method can be used in a relatively complex structure.The loading results are accurately predicted by the FFCR curve.Table 2 shows critical fracture strain and stress obtained for the CHS specimens.

Fig.10 Iterative processes after necking.

Fig.11 Comparison between VIC-3D result and iterative result.

Fig.12 Strain distribution on funnel-shaped specimen.

Fig.13 Full-range constitutive relationship curves of SS316L at different temperatures(1#specimen tensile testing results were used in calculations).

Fig.14 Full-range constitutive relationship curves of A508-III at room temperature.

Fig.15 Full-range constitutive relationship curves of TA17 at room temperature.

Table 1 Critical fracture strain and stress of ductile materials.

Table 2 Critical fracture strain and stress obtained for CHS specimens.

5.Conclusions

(1)To obtain full-range constitutive relationships including necking deformation,an finite element aided testing(FAT)method was proposed.Based on tensile testing of a funnel-shaped specimen,an iteration procedure was recommended.

(2)VIC-3D test results were completed to validate results of the FAT method.

(3)The FAT method was applied to obtain the constitutive relationships of SS316L at elevated temperatures.

(4)Failure stress and strain are given for different kinds of specimens based on the full-range constitutive relationship curve obtained by the FAT method,which is meaningful for analyses of large deformations and ductile fractures in structures.

Acknowledgments

This study was co-supported by the National Natural Science Foundation of China(No.11472228)and the Sichuan Youth Science and Technology Innovation Team Projects(No.2013TD0004).

Fig.16 Application offull-range constitutive relationship on CHS specimens.

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Yao Di is a Ph.D.student in School of Engineering Mechanics,Southwest Jiaotong University,major in fracture mechanics and constitutive relationship of ductile material.

Cai Lixun is a professor in School of Engineering Mechanics,Southwest Jiaotong University,major in fracture mechanics,materials and structural strength and its testing technology.

Bao Chen is an associate professor in School of Engineering Mechanics,Southwest Jiaotong University,major in theory and testing technology offracture mechanics.

11 November 2015;revised 1 April 2016;accepted 28 August 2016

Available online 21 October 2016

Ductile material;

Elevated temperatures;

Finite element aided testing(FAT)method;

Fracture stress-strain;

True stress-strain

Ⓒ2016 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.This is anopenaccessarticleundertheCCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

*Corresponding authors.Tel.:+86 28 87602706(L.Cai);+86 28 87600850(C.Bao).

E-mail addresses:Di_yaodic@163.com(D.Yao),Lix_cai@263.net(L.Cai),Bchxx@163.com(C.Bao).

Peer review under responsibility of Editorial Committee of CJA.

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http://dx.doi.org/10.1016/j.cja.2016.10.011

1000-9361Ⓒ2016 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.

This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).