A new method for acquiring true stress strain curves over a large range of strains using a tensile test and finite element method

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1 Available online at Mechanics of Materials 4 (28) A new method for acquiring true stress strain curves over a large range of strains using a tensile test and finite element method ManSoo Joun a,b, *, Jea Gun Eom c, Min Cheol Lee a a School of Mechanical and Aerospace Engineering, Gyeongsang National University, 9 Gajwa-dong, Jinju-City, GyeongNam 66-71, Republic of Korea b Research Center for Aircraft Parts Technology, Gyeongsang National University, Republic of Korea c Technology Innovation Center, Gyeongsang National University, Republic of Korea Received 1 May 27 Abstract This paper presents a method for acquiring true stress strain curves over large range of strains using engineering stress strain curves obtained from a tensile test coupled with a finite element analysis. The results from the tensile test are analyzed using a rigid-plastic finite element method combined with a perfect analysis model for a simple bar to provide the deformation information. The reference true stress strain curve, which predicts the necking point exactly, is modified iteratively to minimize the difference in the tensile force between the tensile test and the analyzed results. The validity of the approach is verified by comparing tensile test results with finite element solutions obtained using a modified true stress strain curve. Ó 27 Elsevier Ltd. All rights reserved. Keywords: Flow stress; Large strain; Stress strain curve; Tensile test 1. Introduction Metal-forming simulation techniques have become generalized in industry. As a result, material properties, including the true stress strain curves, are indispensable for process design engineers * Corresponding author. Address: School of Mechanical and Aerospace Engineering, Gyeongsang National University, 9 Gajwa-dong, Jinju-City, GyeongNam 66-71, Republic of Korea. Tel./fax: address: msjoun@gsnu.ac.kr (M. Joun). because the accuracy of a simulation depends mainly on that of the material properties used. This is especially true for true stress strain curves. A true stress strain curve is affected by the manufacturing history, metallurgical treatments, and chemical composition of the material. Therefore, metal-forming simulation engineers require true stress strain curves that reflect the special conditions of their materials. However, it is difficult to obtain the material properties from experiments and very limited information about true stress strain curves can be found in the literature. Most simulation engineers /$ - see front matter Ó 27 Elsevier Ltd. All rights reserved. doi:1.116/j.mechmat

2 M. Joun et al. / Mechanics of Materials 4 (28) use the material properties supplied by software companies, which are very limited and sometimes unproven. True stress strain curves can be obtained using tensile (Bridgman, 1952; Cabezas and Celentano, 24; Koc and Štok, 24; Komori, 22; Mirone, 24; Zhang, 1995; Zhang et al., 1999), compression (Choi et al., 1997; Gelin and Ghouati, 1995; Haggag et al., 199; Lee and Altan, 1972; Michino and Tanaka, 1996; Osakada et al., 1991), ball indentation (Cheng and Cheng, 1999; Huber and Tsakmakis, 1999a,b; Lee et al., 25; Nayebi et al., 22), punch (Campitelli et al., 24; Husain et al., 24; Isselin et al., 26), torsion (Bressan and Unfer, 26), and notch tensile (Springmann and Kuna, 25) tests. Haddadi et al. (26) and Bouvier et al. (26) studied the anisotropic behaviors of sheet metals under large plastic deformations using the simple shear test. Most of these methods obtain true stress strain relations only for strains less than.5. However, the maximum strain often exceeds 1. in bulk metal forming, such as in forging, extrusion, and rolling. Sometimes it reaches 3. in multi-stage automatic cold forging, the so-called cold-former forging used to produce fasteners. Recently, many researchers have tried to obtain true stress strain curves using finite element methods, see e.g. (Cabezas and Celentano, 24; Campitelli et al., 24; Choi et al., 1997; Husain et al., 24; Isselin et al., 26; Lee et al., 25; Mirone, 24; Nayebi et al., 22; Springmann and Kuna, 25). In a tensile test, the true strain reaches its maximum value at the smallest cross-section in the necked region, and it may exceed 1.5 just before a ductile material fractures. Therefore, one should be able to obtain the flow stress of materials at a large strain if finite element methods are used to predict the localized deformation behavior during a tensile test. A few researchers have attempted to obtain the flow stress at a large strain using simulation and experimental approaches, but these applications have been quite limited, see e.g. (Cabezas and Celentano, 24; Mirone, 24). The first step in obtaining the true stress at a large strain from a tensile test is to predict the onset of necking exactly using analytical, numerical, or experimental methods. Many researchers have applied finite element methods to predict the onset of necking (Joun et al., 27). However, all researchers who have used simple bar models between gage marks of a tensile test specimen have included various imperfections or constraints at the ends to allow necking to take place artificially. Several researchers have used a full tensile test specimen, including a grip, as the analysis model. A full specimen model causes some difficulty when matching experimental data with predictions and thereby generalizing the approach. Dumoulin et al. (23) satisfied the Considère criterion (Considère, 1885) using a full model of a sheet specimen. However, they did not discuss the accuracy of their predictions compared with experiments in a quantitative manner. Joun et al. (26) were the first to obtain accurate finite element solutions that satisfied the Considère criterion exactly in an engineering sense using a perfect tensile test analysis model, that is, a cylindrical specimen consisting of a simple bar model without any imperfections. They recently predicted the exact onset of necking using a rigidplastic finite element method (Joun et al., 27) and Hollomon s constitutive law. This paper presents a new method based on our previous research (Joun et al., 27) and an iterative error-reducing scheme to obtain the true stress strain relationship at a large strain from the localized deformation behavior in the necked region. 2. Introduction to finite element analysis of the tensile test In this chapter, we summarized the previous study (Joun et al., 27) on finite element analysis of the tensile test, on which the current study is based Description of a tensile test Fig. 1 shows a typical tensile test result selected to illustrate and apply our approach. The material is SWCH1A. The radius of the cylindrical tensile test specimen (see Fig. 2a) is mm and its distance between the two gauge marks is 25 mm. In Table 1, the experimental results of elongation versus tensile load are listed around the maximum load point and Fig. 1 shows the engineering stress strain curve obtained by the tensile test of the specimen. As seen in Fig. 1, the tensile strength is 357 MPa and the engineering strain at the necking point is.135. The universal test machine used is Instron 5592-HVL and the test speed was kept below 1. mm/min to reduce the effect of rate-dependency of the material.

3 588 M. Joun et al. / Mechanics of Materials 4 (28) Engineering stress (MPa) a Gage mark b y v y 1/6 mm/s σ = yx Measured Engineering strain Fig. 1. Experimental results of a tensile test Rigid-plastic finite element formulae A plastic flow analysis problem in tensile test is to find the velocity field v i which satisfies the following boundary value problem: The material is denoted as the domain V with the boundary S. The boundary S, as denoted in Fig. 2b, can be divided into the velocity-prescribed boundary S vi, where the velocity is given as v i ¼ v i the traction-prescribed boundary S ti, where the stress vector is given as t ðnþ i ¼ t ðnþ i It is assumed that the material is incompressible, i.e., v i,i =, isotropic and rigid-viscoplastic and obeys the Huber von Mises yield criterion and its associated flow rule, that is r ij ¼ 2r 3_e _e ij ð1þ where r ij and _e ij are deviatoric stress tensor and strain-rate tensor, respectively. The material is not rate-dependent and the flow stress r in Eq. (1) is assumed to be a function of only effective strain e, i.e.,r ¼ rðeþ. It is also assumed that the effect of acceleration and gravity on force equilibrium is negligible and the process is isothermal. When the penalty method for the incompressibility condition is employed, the weak-form of the above boundary value problem can be written as Z Z r ii x ijdv þ K _e ij x ij dv X Z t i x i ds ¼ ð2þ S ti V V Gage mark v = x σ = xy (n) t x = (n) t y = where x ij ¼ 1 2 ðx i;j þ x j;i Þ, and the weighting function x i is arbitrary except that it vanishes on S vi. K is a large positive constant called penalty constant that maintains the incompressibility condition approximately and has a meaning of K _e ii ¼ r ij =3. Previously, we defined the reference stress strain curve as follows (Joun et al., 27): r ¼ K N e n N ð3þ v = σ = Fig. 2. Finite element model of the tensile test specimen: (a) analysis domain and (b) finite element mesh system and boundary conditions. y yx x where K N is the reference strength coefficient and n N is the reference strain hardening exponent. The reference strain hardening exponent, denoted as n N,is defined as the true strain at the necking point, that is n N ¼ lnð1 þ e N e Þ ð4þ Table 1 Experimental results of the tensile test around the maximum load point Elongation (mm) Load (N) 1,929 1,928 1,946 1,951 1,947 1,929 1,935 1,928 1,926

4 M. Joun et al. / Mechanics of Materials 4 (28) where e N e is the engineering strain at the necking point. The reference strength coefficient, denoted as K N, is defined by making the flow stress curve of Eq. (3) pass through the necking point in the true stress strain curve. Therefore, the reference strength coefficient can be found from r N e K N ¼ ð1 þ en e Þ ð5þ ½lnð1 þ e N e ÞŠlnð1þeN e Þ where r N e is the engineering stress at the maximum load point, i.e., the necking point Analysis model for the tensile test simulation and the simulation results True stress (MPa) Measured and fitted Extrapolated True strain Fig. 3. stress strain curve, defined by r ¼ K N e nn. If we select the whole tensile specimen as the analysis domain, sometimes called a full analysis model, it is difficult to trace the deformation between the gauge marks on the tensile specimen, as it is not trivial to construct a structured finite element mesh system with nodes on the gauge marks. An important factor in tensile testing is the deformation between the gauge marks. The material between the gauge marks and the grips helps to maintain uniaxial loading in the material between the gauge marks. Therefore, in the computer simulation of the tensile test, it is desirable to take one half of the material between the gauge marks as the axisymmetric analysis model and to consider the velocity or displacement prescribed boundary in the axial direction, and the traction-free boundary in the radial direction at both ends as indicated in Fig. 2b. Of course, the analysis model should produce similar results to the full analysis model. The analysis model has no imperfection that purposefully triggers localization of the specimen at the desired region. Experience allows us to postulate that necking occurs at either end in the perfect analysis model used in this study. Therefore, one quarter of the longitudinal crosssection between the two gage marks is selected as a finite element analysis domain, as seen in Fig. 2a. The analysis domain is discretized into a structured finite element mesh system with linear isoparametric quadrilateral elements, as shown in Fig. 2b, by dividing the rectangle evenly by 15 and 3 divisions in the x and y directions, respectively. The finite element model in Fig. 2b does not contain any imperfections and the traction in the radial direction is free, which is nearly the same as the real tensile test. Therefore, the analysis model is ideal. With the engineering stress and strain obtained from the tensile test in Section 2.1, n N and K N are calculated as.127 and MPa from Eqs. (4) and (5), respectively. Fig. 3 shows the reference stress strain curve defined by Eq. (3). Finite element analysis was performed by a commercial forging simulator AFDEX2D (Joun and Lee (1997), Joun et al. (1998), which was developed based on the rigid-thermoviscoplastic finite element method. To prevent smoothing of the state variables, including effective stress and strain, we avoided re-meshing during the simulation. The solution was obtained after 3 solution steps using the reference stress strain curve shown in Fig. 3. In Table 2, the predicted results of elongation versus tensile load are listed around the maximum load point. The predictions and experiments of the tensile loads with respect to elongations are compared in Fig. 4. As summarized in Tables 1 and 2, the predicted and measured tensile loads at the Table 2 Simulation results of the tensile test around the maximum load point Elongation (mm) Load (N) 1,95 1,95 1,95 19,51 1,95 1,95 1,95 1,95 1,95

5 59 M. Joun et al. / Mechanics of Materials 4 (28) Load (N) Measured Elongation (mm) Fig. 4. Comparison of experiments with tensile test predictions. necking point are 1,951 N and 1,951 N, respectively, and the predicted and measured elongations at the necking point are mm and mm, respectively. The predicted and experimental results are identical at the necking point. This is because of the reference stress strain curve employed. Consequently, it has been revealed (Joun et al., 27) that the necking point can be exactly predicted with Hollomon s constitutive law, which compares well with the related theory (Considère, 1885). 3. Acquisition of the stress strain relationship after necking It was explained in the previous chapter that the reference stress strain curve in Fig. 3 must be used to predict the necking point exactly in an engineering sense. However, problems arise from the fact that the difference between predictions and experiments increases with the elongation, as shown in Fig. 4, and that the true strain of the material during cold forging sometimes exceeds the true strain at the necking point by more than a dozen times. Therefore, the reference stress strain curve cannot be used to predict the material behavior exactly after necking occurs. An appropriate scheme is thus necessary to obtain an improved true stress strain curve from the reference stress strain curve. In this paper, we predict the exact engineering stress strain curve using a finite element simulation of a tensile test by improving iteratively the true stress strain curve. After necking occurs, the non-uniformity of the true strain increases rapidly in the longitudinal direction. The maximum strain occurs at the minimum cross-section where the shear stress is free due to symmetry and the non-uniformity of the strain distribution is comparatively low. Therefore, it is relatively easy to define the representative strain at the minimum cross section. Through finite element analysis, one can trace the minimum cross-section of the tensile test specimen at a specified or sampled elongation d i. The representative strain of the minimum cross-section at elongation d i, denoted as e i R, can be calculated from finite element solutions of the tensile test. The difference between the measured load F i t and the predicted load F i e at elongation di can be reduced by modifying the true stress r i R corresponding to the representative strain e i R. In this paper, the representative strain e i R is defined using the following average area scheme: R e i R ¼ eda A i ð6þ A i where A i indicates the area of the minimum crosssection of the tensile test specimen at the sampled elongation d i. It is believed that the representative strain defined by Eq. (6) alleviates the effect of the heterogeneities of stress and strain occurred during tensile test. The current true stress r i R;old ¼ ri R at ei R; is modified to give the new true stress r i R;new by multiplying the current true stress by F i t as follows: F i e r i R;new ¼ ri R;old F i t ð7þ F i e An iterative algorithm based on the above idea is proposed to obtain the improved true stress strain curve. The reference stress strain curve is used before necking occurs. After necking, the true stress strain relationship is interpolated linearly using the sampled points (e i R ; ri R ) defined at the elongation d i, as shown in Fig. 5. The detailed procedure used to calculate the improved sampled points (e i R ; ri R ) at the sampled elongation di is as follows. In the algorithm, e i;j R and r i;j R are the j-times modified strain and stress, respectively, at the sampled elongation d i. Step 1: Calculate the reference strain hardening exponent n N and the reference strength coefficient K N from tensile test experiments using Eqs. (4) and (5). Select the sampled elongations d i (i =1,2,...,M) from the experimental data after the necking point.

6 M. Joun et al. / Mechanics of Materials 4 (28) True stress (MPa) Sampled points Measured and fitted First improved 9 1 an example, and gives the improved stress strain curve from the first iteration. This curve was improved from the reference stress strain curve using our approach. Fig. 6 compares the predicted load elongation curves obtained using the reference and improved stress strain curves with the measured load elongation curve. The load elongation curve predicted from the improved stress strain curve was considerably more accurate. Quite accurate results were obtained after a single iteration, although the maximum elongation was quite large. The maximum error was 474 N or 6.4% of the measured load. This error could be True strain Fig. 5. Modified true stress strain curve calculated after first iteration Step 2: Conduct a finite element analysis of the tensile test using the reference stress strain curve and then calculate e R i ði ¼ 1; 2;...; MÞ at the sampled elongation d i from the finite element solutions of the tensile test. Step 3: Set j = 1 and e i;j R ¼ e i R and then calculate r i;j R from r i;j R ¼ K N ðe i;j R Þ n N ði ¼ 1; 2...; MÞ. Step 4: Conduct a finite element analysis of the tensile test using both the perfect analysis model and the true stress strain curve defined by n N, K N and ðe i;j R ; r i;j R Þði ¼ l; 2;...; MÞ. Then check the convergence of the soluti on at the sampled el ongati on d i (i = l,2,...,m) by comparing the measured load F i t with the predicted load F i e.if convergence is achieved, stop the iterations. Otherwise, calculate e i R from the finite element solutions and set e i;jþ1 R ¼ e i R. Step 5: Calculate the stress r i R at e ¼ e i;jþ1 R ði ¼ 1; 2;...; MÞ by linearly interpolating the sampled points, ðe i;j R ; r i;j R Þði ¼ 1; 2;...; MÞ and calculate the improved stress r i;jþ1 R at e ¼ e i;jþ1 R r i;jþ1 R ¼ r i R F i t F i e ð8þ Step 6: Replace j with j + 1 and return to Step Application example Fig. 5 shows points corresponding to the sampled elongation d i on the reference stress strain curve as Load (N) True stress (MPa) Measured First improved Elongation (mm) Fig. 6. Comparison of the elongation tensile force curves Measured and fitted First improved Second improved Third improved Fourth improved True strain Fig. 7. Comparison of the stress strain curves.

7 592 M. Joun et al. / Mechanics of Materials 4 (28) Load (N) Elongation (mm) reduced or minimized through additional iterations. Figs. 7 and 8 show the improved stress strain curves for several iterations and their corresponding predicted load elongation curves, respectively. Table 3 lists the maximum errors of the predicted loads relative to the measured loads with the number of iterations. After four iterations, the maximum error was reduced to less than.3%, i.e., it led to the exact solution in an engineering sense. Therefore, the convergence characteristics of our scheme are quite good. 5. Concluding remarks Measured First improved Second improved Third improved Fourth improved Fig. 8. Comparison of the load elongation curves. Table 3 Reduction in error with the number of iterations Number of iterations Maximum error (%) An approach for acquiring true stress strain curves at large strains by coupling experiments with an analysis based on a tensile test and a rigid-plastic finite element method was presented. The approach uses the reference stress strain curve before necking occurs to predict the necking point exactly. An iterative scheme then minimizes the error between the measured and predicted load elongation curves after necking occurs by improving the true stress strain curve. Our approach can predict the flow stress at large strains using only the measured load elongation curve of a material and a tensile test analysis, yielding exact results from an engineering viewpoint. This is very important for simulating bulk metal forming. The approach is simple and systematic, and it can be embedded into commercial metalforming simulation software with ease. Acknowledgements This work was supported by the Program for the Training of Graduate Students in Regional Innovation which was conducted by the Ministry of Commerce Industry and Energy of the Korean Government. s Bouvier, S., Haddadi, H., Levée, P., Teodosiu, C., 26. Simple sheat tests: experimenta; techniques and characterization if the plastic anisotropy of rolled sheets at large strains. Journal of Materials Processing Technology 172, Bressan, J.D., Unfer, R.K., 26. Construction and validation tests of a torsion test machine. Journal of Materials Processing Technology 179 (1 3), Bridgman, P.W., Studies in Large Flow and Fracture. McGraw-Hill, New York. Cabezas, E.E., Celentano, D.J., 24. Experimental and numerical analysis of the tensile test using sheet specimens. Finite Elements in Analysis and Design 4 (5 6), Campitelli, E.N., Spätig, P., Bonadé, R., Hoffelner, W., Victoria, M., 24. Assessment of the constitutive properties from small ball punch test: experiment and modeling. Journal of Nuclear Materials 335 (3), Cheng, Y.T., Cheng, C.T., Can stress strain relationships be obtained from indentation curves using conical and pyramidal indenters? Journal of Materials Research 14 (9), Choi, Y., Kim, B.M., Choi, J.C., A method of determining flow stress and friction factor by the ring compression test. In: Proceedings of the Korean Society of Mechanical Engineering 1997 Spring Annual Meeting, Chonnam National University, Republic of Korea, pp Considère, M., L emploi du fer de l acier dans les constructions. Annales des Ponts et Chaussées 9, Dumoulin, S., Tabourot, L., Chappuis, C., Vacher, P., Arrieux, R., 23. Determination of the equivalent stress-equivalent strain relationship of a copper sample under tensile loading. Journal of Materials Processing Technology 133 (1 2), Gelin, J.C., Ghouati, O., The inverse approach for the determination of constitutive equations in metal forming. CIRP Annals 44 (1), Haddadi, H., Bouvier, S., Banu, M., Maier, C., Teodosiu, C., 26. Towards an accurate description of the anisotropic behavior of sheet metals under large plastic deformations: modelling, numerical analysis and identification. International Journal of Plasticity 22, Haggag, F.M., Nanstad, R.K., Hutton, J.T., Thomas, D.L., Swain, R.L., 199. Use of automated ball indentation testing to measure flow properties and estimate fracture toughness in

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