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Stress correction method for flow stress identification by tensile test using notched round bar
Accepted Manuscript Title: STRESS CORRECTION METHOD FOR FLOW STRESS IDENTIFICATION BY TENSILE TEST USING NOTCHED ROUND BAR Authors: Masanobu Murata, Yoshinori Yoshida, Takeshi Nishiwaki PII: DOI: Reference:
S0924-0136(17)30356-4 http://dx.doi.org/doi:10.1016/j.jmatprotec.2017.08.008 PROTEC 15343
To appear in:
Journal of Materials Processing Technology
Received date: Revised date: Accepted date:
16-3-2017 29-7-2017 5-8-2017
Please cite this article as: Murata, Masanobu, Yoshida, Yoshinori, Nishiwaki, Takeshi, STRESS CORRECTION METHOD FOR FLOW STRESS IDENTIFICATION BY TENSILE TEST USING NOTCHED ROUND BAR.Journal of Materials Processing Technology http://dx.doi.org/10.1016/j.jmatprotec.2017.08.008 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
STRESS CORRECTION METHOD FOR FLOW STRESS IDENTIFICATION BY TENSILE TEST USING NOTCHED ROUND BAR
Masanobu Murataa,*, Yoshinori Yoshidab, Takeshi Nishiwakic a
System Technology Dept., Nagoya Municipal Industrial Research Institute,
Nagoya, Aichi 456-0058, Japan b
Center for Advanced Die Engineering and Technology (G-CADET), Gifu
University, Gifu-City, Gifu 501-1193, Japan c
Dept. of Mechanical Engineering, Daido University, Nagoya, Aichi 457-8530,
Japan
Corresponding author. Masanobu Murata E-mail addresses: [email protected] (M. Murata) 3-4-41, Rokuban, Atsuta-ku, Nagoya-shi, Aichi, 456-0058, Japan
1
ABSTRACT In this paper, a stress correction method for flow stress identification using notched round bar tensile test is proposed. Flow stress is evaluated in uniform elongation and local elongation until final fracture in a tensile test with circumference notched round bar tensile test specimens. Tensile load and change in the shape of the notch are measured by image analysis. In order to correct the average tensile stress to the flow stress, inverse analysis is applied to the tensile test. For the validation of the inverse analysis, numerical tensile tests are performed by FEM. As a result of applying the inverse analysis for the numerical tensile tests, the corrected flow stress completely reproduces the two types of reference flow stress curves which are determined by Swift's and Voce's law. On the other hand, the flow stress corrected by Bridgman's method, which is a conventional stress correction method, overestimated these reference flow stress curves. In the case of the actual tensile test of low carbon steel SS400 (in JIS), the flow stress corrected by inverse analysis correspond to Swift’s law determined in uniform elongation. As well as numerical tensile test results, the flow stress corrected by Bridgman’s method is higher than that of obtained by the inverse
2
analysis.
Keywords: tensile test, notched specimen, image analysis, inverse analysis, flow stress, large strain.
1. Introduction In forging and plate forming, plastic working simulations by the finite element method (FEM) have been increasingly applied to shorten the time required for product development and cost reduction. The improvement of the prediction accuracy of the simulation of various factors, including machining force and product shape, is demanded. The improvement of the accuracy of the identification of flow stress curves is required because the prediction accuracy of plastic working simulation is significantly affected by flow stress curves, which represent the work hardening behavior of materials. In general, the flow stress curves of materials are obtained by a tensile test using dumbbell-shaped specimens because of its simple method. Fig. 1 shows the relationship between flow stress and average tensile stress and schematics of round bar specimens used in the tensile test. Once necking occurs in a specimen, the necking area is subjected to multiaxial stress. Therefore, the
3
average stress in the direction of the tensile axis obtained by continuous measurement (hereafter, average tensile stress), σzave = P/A, does not agree with the flow stress of the material, σflow, where P is the tensile load and A is the minimum perpendicular cross-sectional area during continuous measurement. The direct measurement of flow stress after the occurrence of necking is not possible.
Average tensile stress zave P / A
P
P
Necking
A
Stress
A
R
Flow stress
flow
2a P
Equivalent strain Uniform elongation
Post necking
P
zave flow
zave flow
Uniform elongation
Post necking
Fig. 1 Difference between the average tensile stress and the material flow stress in local elongation after necking
4
In the plastic working of actual products, a large strain exceeding the uniform elongation strain is mostly applied to products. Thus, σflow for large strains exceeding the uniform elongation strain is usually predicted by extrapolation on the basis of work hardening models, such as the Swift’s law (Swift, 1952) and Voce’s law (Voce, 1948). However, problems related to the selection of an appropriate hardening model and the prediction accuracy of parameters still remain.
As the method of identifying σflow from σzave after the occurrence of necking, Bridgman’s stress analysis (hereafter, Bridgman’s method) has been proposed. Bridgman (1952) analyzed the stress state at the necking area in a round bar used in the tensile test by elementary analysis assuming that the stress state is an axisymmetric problem. He demonstrated that σzave can be corrected to σflow by continuously measuring the radius of curvature R and the minimum cross-sectional radius a at the bottom of the necking area. A similar analysis was carried out by Davidenkov and Spiridonova (1946). Tsuchida et al. (2012) measured R and a by the stepwise tensile test, flow stress until just before fracture of various metals and alloys was evaluated using Bridgman's method.
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Yoshida et al. (2004) automated the measurement of the shape of the necking area by image analysis and succeeded in identifying σflow until fracture and the critical damage values. Cabezas and Celentano, (2004) applied Bridgman’s method to uniaxial tensile test in order to obtain σflow after necking occurs. By comparing the tensile test and its FEM analysis, the validity of the obtained σflow was evaluated. Stress correction by Bridgman’s method consists of simple equations; however, many assumptions are made during the derivation of the equations, which may lead to a low accuracy of stress correction. La Rosa et al., (2003) performed FEM analysis using the corrected flow stress curves obtained by the Bridgman’s method. The magnitude of the approximations intrinsic to the Bridgman’s method were quantified from the detailed comparison of the experimental and FEM results. Mirone, (2004) pointed out that the radius of curvature at the necking area is not always necessary for stress correction by comparing experimental results with the FEM analysis results in detail. Mirone, (2004) also developed empirical equations for stress correction with higher accuracy than that of Bridgman’s method. However, there is no guarantee that these equations can be applied to all materials.
Along with the recent significant improvement of computer performance,
6
the number of cases in which inverse analysis by FEM is applied to determine the unknown parameters of materials has been increasing. Hasegawa et al. (2009) identified the strain hardening exponent of the power law hardening model, with the aim of reproducing the relationship between elongation and load of a uniaxial tensile test by FEM. The agreement was improved by assuming the strain hardening exponent as the first and second order functions of the strain. Coppieters et al., (2011) identified the parameters of the Swift’s and Voce’s law by inverse analysis using the strain measurement data of digital image correlation (DIC). Kim et al., (2013) applied virtual fields method (VFM) and inverse analysis to the uniaxial tensile test of the sheet specimen, and identified the parameters of Swift’s and modified Voce’s law. These attempts were premised on work hardening models to express the work hardening behavior of the material after occurrence of necking. However, there is a problem that the accuracy of flow stress curves to be identified depend on the selected work hardening model when the parameters of the work hardening model are the target of identification. On the other hand, other attempts have also been made to predict σflow after occurrence of necking without using the work hardening model. Dunand and Mohr, (2010) calibrated the hardening modulus in each strain section of piecewise
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linear hardening model divided into three sections, in order to reproduce experimentally-measured force–displacement curve of low ductility aluminum plate. However, parameter calibration are considerable complicated for high ductility materials that many divided sections are required. An attempt using similar piecewise linear hardening model are also performed by Kajberg and Lindkvist, (2004). In this case, the number of section divisions is four. Joun et al., (2008) reported an attempt to increase the number of section divisions. They developed an iterative algorithm to correct the piecewise linear flow stress curves, in order to reproduce experimentally-measured force–displacement curve of uniaxial round bar tensile test. However, since the calibrated data does not include deformation information of the necking, there is no guarantee that deformation of the necking is consistent. In this study, we propose a new stress correction method to identify σflow until fracture including after the necking with high accuracy which does not depend on any work hardening model. We focus on a notched round bar tensile test which is simple in shape of specimen and can easily change stress loading path with change in the initial notch radius. Tensile load and change in the shape of the notch are measured by the tensile tests with image analysis. σflow is then identified by correcting the obtained σzave. For the determination
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of the amount of stress correction, inverse analysis by FEM is used. We conducted a study which consist of actual experiments and numerical experiments. In the actual experiment, tensile tests were carried out using notched round bar specimens which ware made of low carbon steel SS400 (in JIS) with high ductility and the proposed method was applied to identify σflow until fracture. In addition, to demonstrate the validity of this method, numerical experiments by FEM using notched round bar specimens was carried out to confirm if the reference flow stress curve σref, which is the correct curve
prepared
beforehand,
can
be
appropriately reproduced.
Stress
correction using the conventional method (Bridgman’s method) was also carried out by experiment and by numerical experiments to compare the accuracy of stress correction between the two methods. In this study, in order to confirm the effectiveness of the proposed method, all experiments and numerical calculations are limited to cold and quasi-static conditions.
2. Experimenta method 2.1 Materials and shape of specimens Specimens used in the tensile test were obtained by cutting a low carbon steel SS400 (in JIS) round bar and used in the experiment. Table 1 is a summary
9
of the chemical composition of SS400. As shown in Fig. 2, four notched round bar specimens (a)–(d) and a smooth round bar specimen (e) were used. Table 2 is a summary of the material properties of the smooth round bar specimen obtained by the tensile test and the parameters of the Swift’s law identified in the range of uniform elongation. Here, εp is the equivalent plastic strain. The necking occur at the notch on round bar specimens every time, it is easy to measure the shapes of necking specimens. In addition, by changing the initial notch radius R0, the history of the stress applied to the necking area can be changed during the test. If the obtained flow stress curves are the same as each other, regardless of various stress history, our method will be considered valid.
Table 1 Chemical composition of SS400
C
Si
Mn
P
S
0.07%
0.16%
0.6%
0.024%
0.041%
10
(a)
(b)
(C)
(D)
(E)
Fig. 2 Notched round bar specimens (a)-(d) and a smooth round bar specimen (e) (unit: mm) Table 2 Material properties of SS400
Tensile
Yield
Uniform
strength
strength
elongation
473MPa
357MPa
19%
F*
n*
ε0*
788MPa
0.19
0.002
*Approximated using σ = F(ε0 + εp)n for εp =0.1 - 0.19
2.2 Tensile test method Tensile tests were carried out using the four notched round bar specimens
11
at a tensile speed of 3 mm/min. The process of deformation at the necking area until fracture was recorded using a charge-coupled device (CCD) camera (Point
Grey
Research
Inc.,
Grasshopper
GRAS-20S4M).
The
minimum
cross-sectional radius a and the radius of curvature R at the bottom of the necking area, which is used in Bridgman’s method as explained later, were continuously determined by image analysis. Here, R was determined by least-squares approximation along the circular arc at the bottom of the necking area on the image (Yoshida et al., 2004). The following two types of curve were obtained from the results of the tensile test. One is the average tensile stress (σzave)–equivalent strain (εeq) curve used as the reference for stress correction. The other is the curve of tensile load (P) – change in the cross-sectional radius at the bottom of the necking area (a0−a), which is used as a target curve in the inverse analysis explained later. Assuming that the specimen is a round rod and subjected to isotropic deformation, σzave is given by σzave = P/(π/a2)
(1)
The equivalent strain εeq is given by εeq = 2 ln (a0/a)
(2)
where the average of the strain distributed over the cross-sectional area
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of the bottom of the necking area and elastic deformation are not taken into consideration.
3. Numerical experiment method To verify the validity of the proposed method, the tensile test was numerically experimented by FEM. Assuming the axial symmetric and mirror symmetric at the bottom of the necking area, half of the longitudinal cross section of a specimen was modeled using axisymmetric elements, as shown in Fig. 3. The size of an average element at the bottom of the necking area is 0.1 mm. It is assumed that the material is homogeneous and isotropic elasto-plastic and follows the von-Mises yield criterion and associated flow rule. LS-DYNA971 (Livermore Software Technology Corporation) was used as the FEM analysis solver. The tensile test was numerically simulated by applying a forced displacement at upper-end nodes.
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Fig. 3 Mesh in tensile specimen and boundary conditions
Two types of σref, i.e., σ(swift) and σ(voce) , following the Swift’s law given by ref ref Eq. (3) and Voce’s law given by Eq. (4), respectively, were used in the numerical simulation. σ(swift) = 830 (εp + 0.002)0.22 (MPa) ref σ(voce) = 602.7－338.0exp(-12.9εp) ref
(3) (MPa)
(4)
The parameters are adjusted so that σ(swift) and σ(voce) are in good agreement in ref ref the range of uniform elongation and show differences after uniform elongation. Similar to the experiment explained in Section 2.2, the σzave–εeq curve and
14
P–(a0−a) curve were calculated and prepared as the reference curve and numerical target curve, respectively, using the results of tensile tests. The coordinates of nodes in the vicinity of the bottom of the necking area are approximated by a circular arc, and the radius of curvature R was determined for every 0.1 mm of tensile displacement.
4. Stress correction method 4.1 Stress correction by Bridgman’s method (conventional method) According to the stress analysis by Bridgman’s method, the stress components at the bottom of the necking area in the tensile test using round bar specimens are given by a2 2aR r 2 σr σθ σflow ln 2aR
(5)
a2 2aR r 2 σz σflow 1 ln 2aR
(6)
where σr is the radial stress, σθ is the circumferential stress, σz is the stress of tensile direction, and r is the radial distance from the center of the cross section of the necking area. Because the value obtained by integrating σz over the cross section of the bottom of the necking area is equal to P,
15
σflow
1 R a 1 2 ln1 a 2R
σ zave
(7)
is derived. The term before σzave on the right-hand side of Eq. (7) is the restraint coefficient (hereafter, Bridgman’s correction coefficient) used to correct σzave to σflow and is determined from R and a. By multiplying Bridgman’s correction coefficient with the σzave–εeq curves obtained from experimental and numerical results, we obtained σflow–εeq curves.
4.2 Stress correction by inverse analysis (proposed method) (a) Parameter representation of flow stress curves In the proposed method, σzave is corrected by inverse analysis so that the P–(a0−a) curve, the numerical target curve, can be reproduced by FEM analysis. The σflow–εeq curve used for the inverse analysis is defined in Fig. 4 using the σzave–εeq curve obtained from experimental or numerical results as the reference curve. In the range of uniform elongation, σzave values measured using a smooth round bar specimen are assumed to be σflow. For strain exceeding the uniform elongation strain, σflow is expressed by piecewise linear approximation with N sections of the range of strain. Flow stress at each equivalent strain σflow
I
is given by
16
σflow
I
= xIσzave
I
(0≤xI≤1, I = 1, 2,…N)
(8)
σflow I ≤ σflow I+1 (I = 1, 2,…N)
(9)
where the average tensile stress σzaveI is multiplied by x = xI (I = 1, 2, …N). Here, x is the design variable for the optimization calculation and corresponds to Bridgman’s correction coefficient. In this study, the restraint conditions given by Eq. (9) are considered, assuming that the deformations under cold-working and quasistatic conditions, under which work softening rarely occurs, are targeted. In other words, the shaded region in Fig. 4 can be represented by the equation that defines the flow stress curve. The range of equivalent strain is divided with an interval of 0.05 for εu ≤ εeq ≤ 0.5 and 0.1 for 0.5 ≤ εeq ≤ εf. Here, εu is the uniform elongation strain and εf is the average equivalent strain at the bottom of the necking area at fracture. Denoting the radius of the bottom of the necking area at fracture as af, εf = 2ln(a0/af). For instance, Assuming that εu = 0.2 and εf = 1.2 regardless of σref and R0 in the stress correction of numerical simulation results, the number of design variables is 13 (x1, …, x13).
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Equivalent stress
Corrected flow stress curves Average tensile stress curves zave 2
flow 0 u
zave I
zave 1
flow I
zave N
flow N
u
f
u
Uniform elongation
Equivalent strain
Fig. 4 Corrected flow stress curves for inverse analysis
(b) Method of inverse analysis Tensile test analysis for inverse analysis is performed under the same FEM condition as in section 3, namely it is assumed that the material is homogeneous and isotropic elasto-plastic and follows the von-Mises yield criterion and associated flow rule. The axisymmetric element and the boundary condition of Fig. 3 are used. The parameterized piecewise linear work hardening as shown in Fig. 4 is used as the flow stress curve for inverse analysis. The experimental or numerical target curve of inverse analysis is the P–(a0−a) curve obtained until fracture, as explained previously. The P–(a0−a) curve contains much information on σflow after uniform elongation. Fig. 5 shows the error between a target tensile load Pi and the tensile load
18
calculated by inverse analysis Fi(x). Here, Fi(x) is a calculation result of tensile load, which is also the response surface of a polynomial equation of the first degree with respect to x. The objective function is the minimization of the mean-square error e between Pi and Fi(x), as defined in Eq. (10), and an optimal value of x is determined. 1 e n
n
i 1
Fi(x) Pi Max Pi
2
min
(10)
Here, n is the number of divisions of the target curve. In this study, n = 50 to 100, P–(a0−a) curve obtained from experiments or numerical experiments is divided into 50 to 100. In this study, the successive response surface method (SRSM) developed by Stander and Craig (2002), a type of iterative approximation response surface method, was used to search for the optimal value of x. SRSM is appropriate for local optimization and has shorter computation time than using global optimization such as genetic algorithm (GA). LS-OPT4.2 (Livermore Software Technology Corporation) was used as the optimization software. Fig. 6 shows the flowchart of SRSM. D-optimal design is used to select sampling points for experimental design and tensile test analysis are carried out. Subsequently, a response surface Fi(x) is created for each Pi. Function e,
19
given by Eq. (10), is optimized. To search for the minimum value of e, the adaptive simulated annealing method (ASA), was used. Until e sufficiently decreases, the region of interest for parameter x is moved and reduced during iterative calculation.
Tensile load P, F
Computed curve i 1 i 2
i3
Target curve
Fi ( x )
i
Pi
in
Change in radius a0-a Fig. 5 Error between target curve and computed curve
20
Start Sampling Perform tensile test analysis Creation of metamodels Optimization Move or/and contract the region of interest Evaluation NG
OK
End
Fig. 6 Flowchart of optimization method
5. Results and discussion 5.1 Stress correction using tensile test results of numerical experiment and discussion Fig. 7 shows the flowchart of stress correction for verifying the numerical experiment results. σflow, corrected by Bridgman’s method and inverse analysis, and σref, used as the input of the numerical simulation, were compared to evaluate the reproducibility.
21
Prepare the reference flow stress curves ( swift ) ref 830( p 0.002)0.22 (MPa )
( voce ) ref 602.7 338.0e 12.9p (MPa )
Numerical tensile tests using FEM (R0 = 3, 6, 10 and 20mm) σzave-εeq P-(a0-a)
σzave-εeq R, a
Stress correction using inverse analysis
Stress correction using Bridgman’s method
σflow-εeq
σflow-εeq Compare with each other
Fig. 7 Flowchart of the stress corrections and the validations for using the numerical tensile test results
Fig. 8 shows σzave–εeq curves obtained using Eqs. (1) and (2) and numerical experiment results, as well as a reference flow stress curve using σ(swift) . ref σzave values for any R0 values are always higher than σ(swift) ; this tendency is ref more significant with decreasing R0. This is because, with decreasing R0, the effect of the multiaxial stress state, rather than the uniaxial stress state, becomes significant.
22
Average tensile stress σzave /MPa Stress /MPa stress /MPa or Equivalent
1400 Average tensile stress curves
1200
R0 3
zave
6 10 20mm
1000 800
Reference flow stress curve
600
( swift ) ref 830( p 0.002)0.22
400 200 0 0.0
0.2
0.4 0.6 0.8 Equivalent strain εεeq Equivalent , εp eq /-
1.0
1.2
Fig. 8 Average tensile stress curves obtained from numerical tensile tests (Reference flow stress curve : σ(swift) ) ref
(a) Stress correction by Bridgman’s method Fig. 9 shows the results of the stress correction of σzave–εeq curves by Bridgman’s method in accordance with the procedure explained in Section 4.1. σflow obtained by stress correction using either σ(swift) or σ(voce) almost lies on ref ref a curve regardless of R0. However, with increasing εeq, σflow deviates from σ (swift) ref
and σ(voce) . The flow stress is overestimated by ~12 and ~16% at maximum ref
when the Swift’s law and the Voce’s law are applied respectively. Fig. 10 shows the stress distribution predicted by Bridgman’s method using Eqs. (5)–(7) at the bottom of the necking area at εf = 1.2 (R0 = 20 mm, reference
23
flow stress curve: σ(swift) ). The stress distribution predicted by Bridgman’s ref method cannot sufficiently reproduce the stress of the numerical experiment results. Therefore, the flow stress obtained by Bridgman’s method is considered to be overestimated.
1400
1000
1400
800
1200
600
1000
400
200 0
Stress /MPa
Stress /MPa
1200
800 600
Equivalent stress /MPa
1200 1000 800
( swift ) ref 830( p 0.002)0.22
Swift Voce
600
400 200 0
( voce) ref 602.7Swift 338.0e
12.9 p
Voce s method) Corrected flow stress (Bridgman'
系列10 Swift, R0 3mm 系列9R0 6mm Swift, 系列8R0 10mm Swift, 系列7R0 20mm Swift,
Voce, R0 3mm Voce, R0 6mm Voce, R0 10mm Voce, R0 20mm
系列10 0.4 0.6 0.8 系列9 1.0 系列8 Equivalent strain εεeq , 系列7 εp eq /-
0.0 0.2 1.2 400 0.0 0.2 0.4 0.6 0.8 1.0 1.2 200 Equivalent strain 2ln(a0/a) /0 Fig. 9 Stress correction results using Bridgman’s method about numerical 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Equivalent strain 2ln(a0/a) /-
tensile tests
24
1600
Numerical tensile test σz
1400
Eq. (6) σz
Stress /MPa
1200 Eq. (7) σflow
1000 800
z θ
r
Numerical tensile test σflow
600
Numerical tensile testσr =σθ
400
200
Eq. (5) σr =σθ
0 0.0
0.5
1.0
1.5 2.0 r /mm Distance from center of neck bottom r /mm
2.5
Fig. 10 Distribution of stress components on neck at εf =1.2 (R0 = 20 mm, Reference flow stress curve : σ(swift) ) ref
(b) Stress correction by inverse analysis In this section, the results of stress correction by inverse analysis explained in Sect. 4.2 are examined. Fig. 11 shows the history of design variables with the number of iterations. Fig. 12 shows examples of equivalent stress obtained by stress correction (R0 = 3 mm, reference flow stress curve: σ(swift) ). At the end of the eighth iteration, the design variables xI converge ref to a true value. The equivalent stress obtained after eighth iterations is (swift) equal to σ ref . Fig. 13 shows σflow-εeq curves obtained after the eighth
iteration for any case of R0 and σref. The σflow-εeq curves are in good agreement
25
regardless of R0, and σflow is equal to σ(swift) or σ(voce) . Fig. 14 shows numerical ref ref target curves and reproduced P–(a0-a) curves by inverse analysis using the σflow-εeq curve. The numerical target curve can be reproduced with high accuracy by the inverse analysis for specimens with different shapes and σref values. Namely, σref can be highly accurately reproduced independently of the work hardening model when stress correction by inverse analysis using numerical experiment results was carried out under various stress state.
1.0
True value of x1 1.0
0.8 0.7
0.6 0.5
0.9 Design values xI /-
Design variables xI
0.9
x1 系列1 x5 系列3 x9 系列4 x13 系列5
0.8 0.7
0 0.6 1
2
3 4 5 6 Number of iterations
7
8
x5 x9 x13
9
0.5 0 variables 1 2 3 4 on 5 inverse 6 7 8 analysis 9 Fig. 11 History of design (R0 = 3 mm, Reference Number of iterations
flow stress curve : σ(swift) ) ref
26
1400 1200 ( swift ) ref 830( p 0.002)0.22 Initial ( zave )
Equivalent Stress /MPa Equivalent stress /MPa
1200 1000 1000 800 800 600 600 400 400 200 200
00 0.0 0.0
4th cycle
8th cycle
12.9 p
( voce ) ref 602.7 338.0e 1st cycle 2nd cycle
( swift ) ref 830( p 0.002)0.22
1.2 1.0 0.8 0.6 0.4 0.4 0.6 0.8 1.0 1.2 strainεeqεεeq Equivalent Equivalent strain /-/Equivalent strain εeq / or Equivalent plastic , ε p strain ε p / eq 0.2 0.2
Fig. 12 History of stress correction results using inverse analysis (R0 = 3 mm, Reference flow stress curve : σ(swift) ) ref
1200
00
00
1400
00
1200
00
00 0
1000 Stress /MPa
00
800 600
Equivalent Stress /MPa
00
1000 800
( swift ) ref 830( p 0.002)0.22 Swift Voce ( voce) ref 602.7 338.0e
600
12.9 p
Swift Voce method) Corrected flow stress (Proposed
400
200 0
Swift, R0 3mm 系列10 系列9 Swift, R0 6mm 系列8R0 10mm Swift, Swift, 系列7R0 20mm
Voce, R0 3mm Voce, R0 6mm Voce, R0 10mm Voce, R0 20mm
系列10 系列9 0.4 0.6 0.8系列8 Equivalent strain ε ,/-系列7 ε
0.0 0.2 1.0 1.2 400 0.0 0.2 0.4 0.6 0.8 1.0 1.2 p eq 200 Equivalent strain 2ln(a0/a) /Fig. 130 Stress correction results using inverse analysis on numerical tensile 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Equivalent strain 2ln(a0/a) /tests
27
14000
R0 3
14000
Tensile load P /N
12000 10000 8000
Tensile loadP,P F/N/N Tensileload
12000
6
10000
20mm (swift ) ref
8000
R0 3
6000
6 10
4000
6000
2000
4000
0
20mm
Numerical target curves Inverse analysis results
0.0
2000
10
0.2
0.4 0.6 0.8 Change in radius a0 -a /mm
(voce) ref
1.0
1.2
0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 14 Numerical target curves and inverse analysis results Fig. Change in radius a0 -a /mm
5.2 Stress correction using tensile test results of low carbon steel (SS400) and discussion Fig. 15 shows the flowchart of stress correction using tensile test results of SS400. The true σflow-εeq curve of an actual material after the occurrence of necking cannot be not obtained; σflow in two cases of stress corrections were compared with values obtained by extrapolation based on the Swift’s law (Table 2) which parameters were given in the range of uniform elongation.
28
SS400 Tensile tests using image analysis (R0 = 3, 6, 10 and 20mm) σzave-εeq P-(a0-a)
σzave-εeq R, a
Stress correction using inverse analysis
Stress correction using Bridgman’s method
σflow-εeq
σflow-εeq
Compare with a extrapolated Swift’s hardening rule (=Table 2) Fig. 15 Flowchart of validation process of stress corrections for using SS400 tensile test results Fig. 16 shows σzave–εeq curves for different R0 values obtained by tensile tests using SS400 and image analysis. In this experiment of SS400, the same experiment was conducted three times, there was almost no difference in P-(a0-a) curve. To remove the noise originating from image analysis, the change in measured a with time was smoothed using a sixth-order function and then used for inverse analysis. Similarly to numerical experiments, σzave increases with decreasing R0. In addition, with decreasing R0, fracture strain decreases, which is considered to be caused by the increase in stress multiaxiality. A Similar results were reported, for example, by Enami, (2005).
29
Average tensile stress σzave /MPa
/MPa Stress or stress /MPa Equivalent Stress /MPa
1200
Fracture
R0 3 6 10 20mm
1000 800
1200
788( p 0.002)0.19
600
1000
( Table 2)
400 800
Swift
200 600
zave (Filterd) R0=3
zave 系列10
400 0
0 200
0.2
0.4 0.6 0.8 Equivalent Equivalentstrain strainεεeqeq, /ε-p
1
1.2
0
Fig. 16 Average tensile stress curves 0 1 obtained from SS400 tensile tests using Equivalent strain 2ln(a0/a)
image analysis
Fig. 17 shows σflow–εeq curves obtained after stress correction by inverse analysis. It is assumed that uniform strain, εu is 0.1 regardless of R0 and the number of iterations of the optimization algorithm was 8. In the figure, the σflow–εeq curve obtained using the Swift’s law is also shown. σflow–εeq curves obtained after stress correction by inverse analysis are the same regardless of R0 and correspond to the curve using the Swift’s law. For each specimen, σflow is identified until fracture: σflow up to εf = 1.05 was obtained for the specimen with R0 = 20 mm. This strain is more than fivefold the uniform elongation strain. Fig. 18 shows P–(a0−a) curves of experiment and inverse
30
analysis. The experimental results are fairly accurately reproduced.
788( p 0.002)0.19 ( Table 2)
800 1000
Fracture R0 3 6 10 20mm
600
Equivalent stress /MPa
Equivalent stress /MPa
1000
800
Corrected flow stress Smooth (Proposed method) Swift R0 3mm R0=3 R0 6mm R0=6 R0 10mm R0=10 R0 20mm R0=20
400
600
Smooth specimen (ε p 0.19)
200
400
0 200
0
0.2
0 0
0.4 0.6 0.8 Equivalentstrain strainεεeqeq, /ε-p Equivalent
1
1.2
1
Fig. 17 Stress correction results using inverse analysis on SS400 tensile Equivalent strain 2ln(a 0/a) tests
14000
12000
12000
10000
10000 8000
Tensile load P /N
Tensile load P /N
14000
R0 3
8000
10 20mm
6000 4000
Experimental target curves Inverse analysis results
6000 2000
4000 2000
6
0 0.0
0.2
0.4 0.6 0.8 Change in radius a0-a /mm
1.0
1.2
0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Fig. 18 Experimental target curves and inverse analysis results on SS400 Change in radius a0 -a /mm tensile tests
31
Fig. 19 shows σflow–εeq curves obtained after stress correction by Bridgman’s method. σflow–εeq curves obtained after stress correction are almost the same but are slightly higher than the Swift’s law. This finding indicates that the σflow–εeq curves obtained after stress correction by Bridgman’s method are located at upper regions than those obtained after stress correction by inverse analysis. This tendency corresponds to the numerical experiment results described previously. From these findings, σflow–εeq curves obtained after stress correction by inverse analysis represent the actual work hardening behavior more accurately than those obtained by Bridgman’s method.
800
788( p 0.002)0.19
1000
600
( Table 2) Equivalent stress /MPa
Equivalent stress /MPa
1000
400
800
600 Smooth specimen (ε400 p 0.19)
200
CorrectedSmooth flow stress Swift (Bridgman' s method) R0=3 R0 3mm
R0=6 R0 6mm R0=10 R0 10mm R0=20 R0 20mm
200
0 0
0.2
0
0.4 0.6 0.8 Equivalent Equivalent strainεeq ε/-eq, /-ε p Equivalent strainstrain 2ln(a0/a)
0
1
1.2
Fig. 19 Stress correction results using Bridgman’s method on SS400 tensile tests
32
6. Conclusions In this study, we proposed a method of accurately identifying σflow–εeq curves up to fracture for isotropic and homogeneous metallic materials. It was possible to identify the flow stress curve regardless of work hardening model by means of stress correction with the inverse analysis for σzave–εeq curves obtained by notched round bar tensile tests. The evaluation was conducted on numerical experiment and actual material, low carbon steel (SS400). The following conclusions were obtained. (1) In the evaluation using the numerical experiment results, two types of σref (σ(swift) and σ(voce) ) with different work hardening behaviors were accurately ref ref reproduced by the inverse analysis regardless of the stress history. On the other hand, the σflow–εeq curves corrected by Bridgman's method overestimated σref. (2) In the evaluation using the tensile test results of SS400, the σflow–εeq curves corrected by inverse analysis follow Swift’s law determined in the range of uniform elongation. On the other hand, as well as numerical tensile test results, the σflow–εeq curves corrected by Bridgman’s method is higher than that of obtained by the inverse analysis. From these findings, σflow–εeq curves obtained after stress correction by inverse analysis represent the
33
actual work hardening behavior more accurately than those obtained by Bridgman’s method. (3) In the evaluation using the tensile test results of SS400, σflow can be identified in the range of equivalent strain exceeding 1.0.
References Bridgman, P.W., 1952. Studies in Large Plastic Flow and Fracture. McGraw-Hill, New York. Cabezas, E.E., Celentano, D.J., 2004. Experimental and numerical analysis of the tensile test using sheet specimens. Finite Elem. Anal. Des. 40, 555–575. doi:10.1016/S0168-874X(03)00096-9 Coppieters, S., Cooreman, S., Sol, H., Van Houtte, P., Debruyne, D., 2011. Identification of the post-necking hardening behaviour of sheet metal by comparison of the internal and external work in the necking zone. J. Mater. Process. Technol. 211, 545–552. doi:10.1016/j.jmatprotec.2010.11.015 Davidenkov, N.N., Spiridonova, N.I., 1946. Analysis of the State of Stress in the Neck of a Tensile Test Specimen. Proc. ASTM, 46, 1147-1158. Dunand, M., Mohr, D., 2010. Hybrid experimental-numerical analysis of basic ductile fracture experiments for sheet metals. Int. J. Solids Struct. 47,
34
1130–1143. doi:10.1016/j.ijsolstr.2009.12.011 Enami, K., 2005. The effects of compressive and tensile prestrain on ductile fracture initiation in steels. Eng. Fract. Mech. 72, 1089–1105. doi:10.1016/j.engfracmech.2004.07.012 Hasegawa, K., Chen, Z., Nishimura, K., Ikeda, K., 2009. Determination of True Stress–Strain Curves of Sheet Metals in Post-Uniform Elongation Range, Mater. Trans. 50, 138-144. Joun, M., Eom, J.G., Lee, M.C., 2008. A new method for acquiring true stress-strain curves over a large range of strains using a tensile test and finite element method. Mech. Mater. 40, 586–593. doi:10.1016/j.mechmat.2007.11.006 Kajberg, J., Lindkvist, G., 2004. Characterisation of materials subjected to large strains by inverse modelling based on in-plane displacement fields. Int. J. Solids Struct. 41, 3439–3459. doi:10.1016/j.ijsolstr.2004.02.021 Kim, J.H., Serpantié, A., Barlat, F., Pierron, F., Lee, M.G., 2013. Characterization of the post-necking strain hardening behavior using the virtual fields method. Int. J. Solids Struct. 50, 3829–3842. doi:10.1016/j.ijsolstr.2013.07.018 La Rosa, G., Mirone, G., Risitano, A., 2003. Post-necking elastoplastic
35
characterization: degree of approximation in the Bridgman method and properties of the flow-stress/true-stress ratio. Met. Mat. Trans. A 34A (3), 615–624. Mirone, G., 2004. A new model for the elastoplastic characterization and the stress-strain determination on the necking section of a tensile specimen. Int. J. Solids Struct. 41, 3545–3564. doi:10.1016/j.ijsolstr.2004.02.011 Stander, N., Craig, K.J., 2002. On the robustness of a simple domain reduction scheme for simulation based optimization. Eng. Comput. 19, (4), 431-450. Swift, H.W., 1952. Plastic instability under plane stress. J. Mech. Phys. Solids 1, 1–18. Tsuchida, N., Inoue, T., Enami, K., 2012. Estimations of the true stress and true strain until just before fracture by the stepwise tensile test and bridgman equation for various metals and alloys. Mater. Trans. 53, 133-139 Voce, E., 1948. The relationship between stress and strain for homogeneous deformation. J. Inst. Met. 74, 537–562. Yoshida, Y., Yukawa N., Ishikawa T., 2004. Determination of ductile damage parameters by notched round bar tention test using image analysis. Materials Processing and Design: Modeling, Simulation and Applications, NUMIFORM’2004. In: Proceedings of the 8th International Conference on
36
Numerical Methods in Industrial Forming Processes. AIP Conference Proceedings, 712, 1, 1869-1874.
37
S0924-0136(17)30356-4 http://dx.doi.org/doi:10.1016/j.jmatprotec.2017.08.008 PROTEC 15343
To appear in:
Journal of Materials Processing Technology
Received date: Revised date: Accepted date:
16-3-2017 29-7-2017 5-8-2017
Please cite this article as: Murata, Masanobu, Yoshida, Yoshinori, Nishiwaki, Takeshi, STRESS CORRECTION METHOD FOR FLOW STRESS IDENTIFICATION BY TENSILE TEST USING NOTCHED ROUND BAR.Journal of Materials Processing Technology http://dx.doi.org/10.1016/j.jmatprotec.2017.08.008 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
STRESS CORRECTION METHOD FOR FLOW STRESS IDENTIFICATION BY TENSILE TEST USING NOTCHED ROUND BAR
Masanobu Murataa,*, Yoshinori Yoshidab, Takeshi Nishiwakic a
System Technology Dept., Nagoya Municipal Industrial Research Institute,
Nagoya, Aichi 456-0058, Japan b
Center for Advanced Die Engineering and Technology (G-CADET), Gifu
University, Gifu-City, Gifu 501-1193, Japan c
Dept. of Mechanical Engineering, Daido University, Nagoya, Aichi 457-8530,
Japan
Corresponding author. Masanobu Murata E-mail addresses: [email protected] (M. Murata) 3-4-41, Rokuban, Atsuta-ku, Nagoya-shi, Aichi, 456-0058, Japan
1
ABSTRACT In this paper, a stress correction method for flow stress identification using notched round bar tensile test is proposed. Flow stress is evaluated in uniform elongation and local elongation until final fracture in a tensile test with circumference notched round bar tensile test specimens. Tensile load and change in the shape of the notch are measured by image analysis. In order to correct the average tensile stress to the flow stress, inverse analysis is applied to the tensile test. For the validation of the inverse analysis, numerical tensile tests are performed by FEM. As a result of applying the inverse analysis for the numerical tensile tests, the corrected flow stress completely reproduces the two types of reference flow stress curves which are determined by Swift's and Voce's law. On the other hand, the flow stress corrected by Bridgman's method, which is a conventional stress correction method, overestimated these reference flow stress curves. In the case of the actual tensile test of low carbon steel SS400 (in JIS), the flow stress corrected by inverse analysis correspond to Swift’s law determined in uniform elongation. As well as numerical tensile test results, the flow stress corrected by Bridgman’s method is higher than that of obtained by the inverse
2
analysis.
Keywords: tensile test, notched specimen, image analysis, inverse analysis, flow stress, large strain.
1. Introduction In forging and plate forming, plastic working simulations by the finite element method (FEM) have been increasingly applied to shorten the time required for product development and cost reduction. The improvement of the prediction accuracy of the simulation of various factors, including machining force and product shape, is demanded. The improvement of the accuracy of the identification of flow stress curves is required because the prediction accuracy of plastic working simulation is significantly affected by flow stress curves, which represent the work hardening behavior of materials. In general, the flow stress curves of materials are obtained by a tensile test using dumbbell-shaped specimens because of its simple method. Fig. 1 shows the relationship between flow stress and average tensile stress and schematics of round bar specimens used in the tensile test. Once necking occurs in a specimen, the necking area is subjected to multiaxial stress. Therefore, the
3
average stress in the direction of the tensile axis obtained by continuous measurement (hereafter, average tensile stress), σzave = P/A, does not agree with the flow stress of the material, σflow, where P is the tensile load and A is the minimum perpendicular cross-sectional area during continuous measurement. The direct measurement of flow stress after the occurrence of necking is not possible.
Average tensile stress zave P / A
P
P
Necking
A
Stress
A
R
Flow stress
flow
2a P
Equivalent strain Uniform elongation
Post necking
P
zave flow
zave flow
Uniform elongation
Post necking
Fig. 1 Difference between the average tensile stress and the material flow stress in local elongation after necking
4
In the plastic working of actual products, a large strain exceeding the uniform elongation strain is mostly applied to products. Thus, σflow for large strains exceeding the uniform elongation strain is usually predicted by extrapolation on the basis of work hardening models, such as the Swift’s law (Swift, 1952) and Voce’s law (Voce, 1948). However, problems related to the selection of an appropriate hardening model and the prediction accuracy of parameters still remain.
As the method of identifying σflow from σzave after the occurrence of necking, Bridgman’s stress analysis (hereafter, Bridgman’s method) has been proposed. Bridgman (1952) analyzed the stress state at the necking area in a round bar used in the tensile test by elementary analysis assuming that the stress state is an axisymmetric problem. He demonstrated that σzave can be corrected to σflow by continuously measuring the radius of curvature R and the minimum cross-sectional radius a at the bottom of the necking area. A similar analysis was carried out by Davidenkov and Spiridonova (1946). Tsuchida et al. (2012) measured R and a by the stepwise tensile test, flow stress until just before fracture of various metals and alloys was evaluated using Bridgman's method.
5
Yoshida et al. (2004) automated the measurement of the shape of the necking area by image analysis and succeeded in identifying σflow until fracture and the critical damage values. Cabezas and Celentano, (2004) applied Bridgman’s method to uniaxial tensile test in order to obtain σflow after necking occurs. By comparing the tensile test and its FEM analysis, the validity of the obtained σflow was evaluated. Stress correction by Bridgman’s method consists of simple equations; however, many assumptions are made during the derivation of the equations, which may lead to a low accuracy of stress correction. La Rosa et al., (2003) performed FEM analysis using the corrected flow stress curves obtained by the Bridgman’s method. The magnitude of the approximations intrinsic to the Bridgman’s method were quantified from the detailed comparison of the experimental and FEM results. Mirone, (2004) pointed out that the radius of curvature at the necking area is not always necessary for stress correction by comparing experimental results with the FEM analysis results in detail. Mirone, (2004) also developed empirical equations for stress correction with higher accuracy than that of Bridgman’s method. However, there is no guarantee that these equations can be applied to all materials.
Along with the recent significant improvement of computer performance,
6
the number of cases in which inverse analysis by FEM is applied to determine the unknown parameters of materials has been increasing. Hasegawa et al. (2009) identified the strain hardening exponent of the power law hardening model, with the aim of reproducing the relationship between elongation and load of a uniaxial tensile test by FEM. The agreement was improved by assuming the strain hardening exponent as the first and second order functions of the strain. Coppieters et al., (2011) identified the parameters of the Swift’s and Voce’s law by inverse analysis using the strain measurement data of digital image correlation (DIC). Kim et al., (2013) applied virtual fields method (VFM) and inverse analysis to the uniaxial tensile test of the sheet specimen, and identified the parameters of Swift’s and modified Voce’s law. These attempts were premised on work hardening models to express the work hardening behavior of the material after occurrence of necking. However, there is a problem that the accuracy of flow stress curves to be identified depend on the selected work hardening model when the parameters of the work hardening model are the target of identification. On the other hand, other attempts have also been made to predict σflow after occurrence of necking without using the work hardening model. Dunand and Mohr, (2010) calibrated the hardening modulus in each strain section of piecewise
7
linear hardening model divided into three sections, in order to reproduce experimentally-measured force–displacement curve of low ductility aluminum plate. However, parameter calibration are considerable complicated for high ductility materials that many divided sections are required. An attempt using similar piecewise linear hardening model are also performed by Kajberg and Lindkvist, (2004). In this case, the number of section divisions is four. Joun et al., (2008) reported an attempt to increase the number of section divisions. They developed an iterative algorithm to correct the piecewise linear flow stress curves, in order to reproduce experimentally-measured force–displacement curve of uniaxial round bar tensile test. However, since the calibrated data does not include deformation information of the necking, there is no guarantee that deformation of the necking is consistent. In this study, we propose a new stress correction method to identify σflow until fracture including after the necking with high accuracy which does not depend on any work hardening model. We focus on a notched round bar tensile test which is simple in shape of specimen and can easily change stress loading path with change in the initial notch radius. Tensile load and change in the shape of the notch are measured by the tensile tests with image analysis. σflow is then identified by correcting the obtained σzave. For the determination
8
of the amount of stress correction, inverse analysis by FEM is used. We conducted a study which consist of actual experiments and numerical experiments. In the actual experiment, tensile tests were carried out using notched round bar specimens which ware made of low carbon steel SS400 (in JIS) with high ductility and the proposed method was applied to identify σflow until fracture. In addition, to demonstrate the validity of this method, numerical experiments by FEM using notched round bar specimens was carried out to confirm if the reference flow stress curve σref, which is the correct curve
prepared
beforehand,
can
be
appropriately reproduced.
Stress
correction using the conventional method (Bridgman’s method) was also carried out by experiment and by numerical experiments to compare the accuracy of stress correction between the two methods. In this study, in order to confirm the effectiveness of the proposed method, all experiments and numerical calculations are limited to cold and quasi-static conditions.
2. Experimenta method 2.1 Materials and shape of specimens Specimens used in the tensile test were obtained by cutting a low carbon steel SS400 (in JIS) round bar and used in the experiment. Table 1 is a summary
9
of the chemical composition of SS400. As shown in Fig. 2, four notched round bar specimens (a)–(d) and a smooth round bar specimen (e) were used. Table 2 is a summary of the material properties of the smooth round bar specimen obtained by the tensile test and the parameters of the Swift’s law identified in the range of uniform elongation. Here, εp is the equivalent plastic strain. The necking occur at the notch on round bar specimens every time, it is easy to measure the shapes of necking specimens. In addition, by changing the initial notch radius R0, the history of the stress applied to the necking area can be changed during the test. If the obtained flow stress curves are the same as each other, regardless of various stress history, our method will be considered valid.
Table 1 Chemical composition of SS400
C
Si
Mn
P
S
0.07%
0.16%
0.6%
0.024%
0.041%
10
(a)
(b)
(C)
(D)
(E)
Fig. 2 Notched round bar specimens (a)-(d) and a smooth round bar specimen (e) (unit: mm) Table 2 Material properties of SS400
Tensile
Yield
Uniform
strength
strength
elongation
473MPa
357MPa
19%
F*
n*
ε0*
788MPa
0.19
0.002
*Approximated using σ = F(ε0 + εp)n for εp =0.1 - 0.19
2.2 Tensile test method Tensile tests were carried out using the four notched round bar specimens
11
at a tensile speed of 3 mm/min. The process of deformation at the necking area until fracture was recorded using a charge-coupled device (CCD) camera (Point
Grey
Research
Inc.,
Grasshopper
GRAS-20S4M).
The
minimum
cross-sectional radius a and the radius of curvature R at the bottom of the necking area, which is used in Bridgman’s method as explained later, were continuously determined by image analysis. Here, R was determined by least-squares approximation along the circular arc at the bottom of the necking area on the image (Yoshida et al., 2004). The following two types of curve were obtained from the results of the tensile test. One is the average tensile stress (σzave)–equivalent strain (εeq) curve used as the reference for stress correction. The other is the curve of tensile load (P) – change in the cross-sectional radius at the bottom of the necking area (a0−a), which is used as a target curve in the inverse analysis explained later. Assuming that the specimen is a round rod and subjected to isotropic deformation, σzave is given by σzave = P/(π/a2)
(1)
The equivalent strain εeq is given by εeq = 2 ln (a0/a)
(2)
where the average of the strain distributed over the cross-sectional area
12
of the bottom of the necking area and elastic deformation are not taken into consideration.
3. Numerical experiment method To verify the validity of the proposed method, the tensile test was numerically experimented by FEM. Assuming the axial symmetric and mirror symmetric at the bottom of the necking area, half of the longitudinal cross section of a specimen was modeled using axisymmetric elements, as shown in Fig. 3. The size of an average element at the bottom of the necking area is 0.1 mm. It is assumed that the material is homogeneous and isotropic elasto-plastic and follows the von-Mises yield criterion and associated flow rule. LS-DYNA971 (Livermore Software Technology Corporation) was used as the FEM analysis solver. The tensile test was numerically simulated by applying a forced displacement at upper-end nodes.
13
Fig. 3 Mesh in tensile specimen and boundary conditions
Two types of σref, i.e., σ(swift) and σ(voce) , following the Swift’s law given by ref ref Eq. (3) and Voce’s law given by Eq. (4), respectively, were used in the numerical simulation. σ(swift) = 830 (εp + 0.002)0.22 (MPa) ref σ(voce) = 602.7－338.0exp(-12.9εp) ref
(3) (MPa)
(4)
The parameters are adjusted so that σ(swift) and σ(voce) are in good agreement in ref ref the range of uniform elongation and show differences after uniform elongation. Similar to the experiment explained in Section 2.2, the σzave–εeq curve and
14
P–(a0−a) curve were calculated and prepared as the reference curve and numerical target curve, respectively, using the results of tensile tests. The coordinates of nodes in the vicinity of the bottom of the necking area are approximated by a circular arc, and the radius of curvature R was determined for every 0.1 mm of tensile displacement.
4. Stress correction method 4.1 Stress correction by Bridgman’s method (conventional method) According to the stress analysis by Bridgman’s method, the stress components at the bottom of the necking area in the tensile test using round bar specimens are given by a2 2aR r 2 σr σθ σflow ln 2aR
(5)
a2 2aR r 2 σz σflow 1 ln 2aR
(6)
where σr is the radial stress, σθ is the circumferential stress, σz is the stress of tensile direction, and r is the radial distance from the center of the cross section of the necking area. Because the value obtained by integrating σz over the cross section of the bottom of the necking area is equal to P,
15
σflow
1 R a 1 2 ln1 a 2R
σ zave
(7)
is derived. The term before σzave on the right-hand side of Eq. (7) is the restraint coefficient (hereafter, Bridgman’s correction coefficient) used to correct σzave to σflow and is determined from R and a. By multiplying Bridgman’s correction coefficient with the σzave–εeq curves obtained from experimental and numerical results, we obtained σflow–εeq curves.
4.2 Stress correction by inverse analysis (proposed method) (a) Parameter representation of flow stress curves In the proposed method, σzave is corrected by inverse analysis so that the P–(a0−a) curve, the numerical target curve, can be reproduced by FEM analysis. The σflow–εeq curve used for the inverse analysis is defined in Fig. 4 using the σzave–εeq curve obtained from experimental or numerical results as the reference curve. In the range of uniform elongation, σzave values measured using a smooth round bar specimen are assumed to be σflow. For strain exceeding the uniform elongation strain, σflow is expressed by piecewise linear approximation with N sections of the range of strain. Flow stress at each equivalent strain σflow
I
is given by
16
σflow
I
= xIσzave
I
(0≤xI≤1, I = 1, 2,…N)
(8)
σflow I ≤ σflow I+1 (I = 1, 2,…N)
(9)
where the average tensile stress σzaveI is multiplied by x = xI (I = 1, 2, …N). Here, x is the design variable for the optimization calculation and corresponds to Bridgman’s correction coefficient. In this study, the restraint conditions given by Eq. (9) are considered, assuming that the deformations under cold-working and quasistatic conditions, under which work softening rarely occurs, are targeted. In other words, the shaded region in Fig. 4 can be represented by the equation that defines the flow stress curve. The range of equivalent strain is divided with an interval of 0.05 for εu ≤ εeq ≤ 0.5 and 0.1 for 0.5 ≤ εeq ≤ εf. Here, εu is the uniform elongation strain and εf is the average equivalent strain at the bottom of the necking area at fracture. Denoting the radius of the bottom of the necking area at fracture as af, εf = 2ln(a0/af). For instance, Assuming that εu = 0.2 and εf = 1.2 regardless of σref and R0 in the stress correction of numerical simulation results, the number of design variables is 13 (x1, …, x13).
17
Equivalent stress
Corrected flow stress curves Average tensile stress curves zave 2
flow 0 u
zave I
zave 1
flow I
zave N
flow N
u
f
u
Uniform elongation
Equivalent strain
Fig. 4 Corrected flow stress curves for inverse analysis
(b) Method of inverse analysis Tensile test analysis for inverse analysis is performed under the same FEM condition as in section 3, namely it is assumed that the material is homogeneous and isotropic elasto-plastic and follows the von-Mises yield criterion and associated flow rule. The axisymmetric element and the boundary condition of Fig. 3 are used. The parameterized piecewise linear work hardening as shown in Fig. 4 is used as the flow stress curve for inverse analysis. The experimental or numerical target curve of inverse analysis is the P–(a0−a) curve obtained until fracture, as explained previously. The P–(a0−a) curve contains much information on σflow after uniform elongation. Fig. 5 shows the error between a target tensile load Pi and the tensile load
18
calculated by inverse analysis Fi(x). Here, Fi(x) is a calculation result of tensile load, which is also the response surface of a polynomial equation of the first degree with respect to x. The objective function is the minimization of the mean-square error e between Pi and Fi(x), as defined in Eq. (10), and an optimal value of x is determined. 1 e n
n
i 1
Fi(x) Pi Max Pi
2
min
(10)
Here, n is the number of divisions of the target curve. In this study, n = 50 to 100, P–(a0−a) curve obtained from experiments or numerical experiments is divided into 50 to 100. In this study, the successive response surface method (SRSM) developed by Stander and Craig (2002), a type of iterative approximation response surface method, was used to search for the optimal value of x. SRSM is appropriate for local optimization and has shorter computation time than using global optimization such as genetic algorithm (GA). LS-OPT4.2 (Livermore Software Technology Corporation) was used as the optimization software. Fig. 6 shows the flowchart of SRSM. D-optimal design is used to select sampling points for experimental design and tensile test analysis are carried out. Subsequently, a response surface Fi(x) is created for each Pi. Function e,
19
given by Eq. (10), is optimized. To search for the minimum value of e, the adaptive simulated annealing method (ASA), was used. Until e sufficiently decreases, the region of interest for parameter x is moved and reduced during iterative calculation.
Tensile load P, F
Computed curve i 1 i 2
i3
Target curve
Fi ( x )
i
Pi
in
Change in radius a0-a Fig. 5 Error between target curve and computed curve
20
Start Sampling Perform tensile test analysis Creation of metamodels Optimization Move or/and contract the region of interest Evaluation NG
OK
End
Fig. 6 Flowchart of optimization method
5. Results and discussion 5.1 Stress correction using tensile test results of numerical experiment and discussion Fig. 7 shows the flowchart of stress correction for verifying the numerical experiment results. σflow, corrected by Bridgman’s method and inverse analysis, and σref, used as the input of the numerical simulation, were compared to evaluate the reproducibility.
21
Prepare the reference flow stress curves ( swift ) ref 830( p 0.002)0.22 (MPa )
( voce ) ref 602.7 338.0e 12.9p (MPa )
Numerical tensile tests using FEM (R0 = 3, 6, 10 and 20mm) σzave-εeq P-(a0-a)
σzave-εeq R, a
Stress correction using inverse analysis
Stress correction using Bridgman’s method
σflow-εeq
σflow-εeq Compare with each other
Fig. 7 Flowchart of the stress corrections and the validations for using the numerical tensile test results
Fig. 8 shows σzave–εeq curves obtained using Eqs. (1) and (2) and numerical experiment results, as well as a reference flow stress curve using σ(swift) . ref σzave values for any R0 values are always higher than σ(swift) ; this tendency is ref more significant with decreasing R0. This is because, with decreasing R0, the effect of the multiaxial stress state, rather than the uniaxial stress state, becomes significant.
22
Average tensile stress σzave /MPa Stress /MPa stress /MPa or Equivalent
1400 Average tensile stress curves
1200
R0 3
zave
6 10 20mm
1000 800
Reference flow stress curve
600
( swift ) ref 830( p 0.002)0.22
400 200 0 0.0
0.2
0.4 0.6 0.8 Equivalent strain εεeq Equivalent , εp eq /-
1.0
1.2
Fig. 8 Average tensile stress curves obtained from numerical tensile tests (Reference flow stress curve : σ(swift) ) ref
(a) Stress correction by Bridgman’s method Fig. 9 shows the results of the stress correction of σzave–εeq curves by Bridgman’s method in accordance with the procedure explained in Section 4.1. σflow obtained by stress correction using either σ(swift) or σ(voce) almost lies on ref ref a curve regardless of R0. However, with increasing εeq, σflow deviates from σ (swift) ref
and σ(voce) . The flow stress is overestimated by ~12 and ~16% at maximum ref
when the Swift’s law and the Voce’s law are applied respectively. Fig. 10 shows the stress distribution predicted by Bridgman’s method using Eqs. (5)–(7) at the bottom of the necking area at εf = 1.2 (R0 = 20 mm, reference
23
flow stress curve: σ(swift) ). The stress distribution predicted by Bridgman’s ref method cannot sufficiently reproduce the stress of the numerical experiment results. Therefore, the flow stress obtained by Bridgman’s method is considered to be overestimated.
1400
1000
1400
800
1200
600
1000
400
200 0
Stress /MPa
Stress /MPa
1200
800 600
Equivalent stress /MPa
1200 1000 800
( swift ) ref 830( p 0.002)0.22
Swift Voce
600
400 200 0
( voce) ref 602.7Swift 338.0e
12.9 p
Voce s method) Corrected flow stress (Bridgman'
系列10 Swift, R0 3mm 系列9R0 6mm Swift, 系列8R0 10mm Swift, 系列7R0 20mm Swift,
Voce, R0 3mm Voce, R0 6mm Voce, R0 10mm Voce, R0 20mm
系列10 0.4 0.6 0.8 系列9 1.0 系列8 Equivalent strain εεeq , 系列7 εp eq /-
0.0 0.2 1.2 400 0.0 0.2 0.4 0.6 0.8 1.0 1.2 200 Equivalent strain 2ln(a0/a) /0 Fig. 9 Stress correction results using Bridgman’s method about numerical 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Equivalent strain 2ln(a0/a) /-
tensile tests
24
1600
Numerical tensile test σz
1400
Eq. (6) σz
Stress /MPa
1200 Eq. (7) σflow
1000 800
z θ
r
Numerical tensile test σflow
600
Numerical tensile testσr =σθ
400
200
Eq. (5) σr =σθ
0 0.0
0.5
1.0
1.5 2.0 r /mm Distance from center of neck bottom r /mm
2.5
Fig. 10 Distribution of stress components on neck at εf =1.2 (R0 = 20 mm, Reference flow stress curve : σ(swift) ) ref
(b) Stress correction by inverse analysis In this section, the results of stress correction by inverse analysis explained in Sect. 4.2 are examined. Fig. 11 shows the history of design variables with the number of iterations. Fig. 12 shows examples of equivalent stress obtained by stress correction (R0 = 3 mm, reference flow stress curve: σ(swift) ). At the end of the eighth iteration, the design variables xI converge ref to a true value. The equivalent stress obtained after eighth iterations is (swift) equal to σ ref . Fig. 13 shows σflow-εeq curves obtained after the eighth
iteration for any case of R0 and σref. The σflow-εeq curves are in good agreement
25
regardless of R0, and σflow is equal to σ(swift) or σ(voce) . Fig. 14 shows numerical ref ref target curves and reproduced P–(a0-a) curves by inverse analysis using the σflow-εeq curve. The numerical target curve can be reproduced with high accuracy by the inverse analysis for specimens with different shapes and σref values. Namely, σref can be highly accurately reproduced independently of the work hardening model when stress correction by inverse analysis using numerical experiment results was carried out under various stress state.
1.0
True value of x1 1.0
0.8 0.7
0.6 0.5
0.9 Design values xI /-
Design variables xI
0.9
x1 系列1 x5 系列3 x9 系列4 x13 系列5
0.8 0.7
0 0.6 1
2
3 4 5 6 Number of iterations
7
8
x5 x9 x13
9
0.5 0 variables 1 2 3 4 on 5 inverse 6 7 8 analysis 9 Fig. 11 History of design (R0 = 3 mm, Reference Number of iterations
flow stress curve : σ(swift) ) ref
26
1400 1200 ( swift ) ref 830( p 0.002)0.22 Initial ( zave )
Equivalent Stress /MPa Equivalent stress /MPa
1200 1000 1000 800 800 600 600 400 400 200 200
00 0.0 0.0
4th cycle
8th cycle
12.9 p
( voce ) ref 602.7 338.0e 1st cycle 2nd cycle
( swift ) ref 830( p 0.002)0.22
1.2 1.0 0.8 0.6 0.4 0.4 0.6 0.8 1.0 1.2 strainεeqεεeq Equivalent Equivalent strain /-/Equivalent strain εeq / or Equivalent plastic , ε p strain ε p / eq 0.2 0.2
Fig. 12 History of stress correction results using inverse analysis (R0 = 3 mm, Reference flow stress curve : σ(swift) ) ref
1200
00
00
1400
00
1200
00
00 0
1000 Stress /MPa
00
800 600
Equivalent Stress /MPa
00
1000 800
( swift ) ref 830( p 0.002)0.22 Swift Voce ( voce) ref 602.7 338.0e
600
12.9 p
Swift Voce method) Corrected flow stress (Proposed
400
200 0
Swift, R0 3mm 系列10 系列9 Swift, R0 6mm 系列8R0 10mm Swift, Swift, 系列7R0 20mm
Voce, R0 3mm Voce, R0 6mm Voce, R0 10mm Voce, R0 20mm
系列10 系列9 0.4 0.6 0.8系列8 Equivalent strain ε ,/-系列7 ε
0.0 0.2 1.0 1.2 400 0.0 0.2 0.4 0.6 0.8 1.0 1.2 p eq 200 Equivalent strain 2ln(a0/a) /Fig. 130 Stress correction results using inverse analysis on numerical tensile 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Equivalent strain 2ln(a0/a) /tests
27
14000
R0 3
14000
Tensile load P /N
12000 10000 8000
Tensile loadP,P F/N/N Tensileload
12000
6
10000
20mm (swift ) ref
8000
R0 3
6000
6 10
4000
6000
2000
4000
0
20mm
Numerical target curves Inverse analysis results
0.0
2000
10
0.2
0.4 0.6 0.8 Change in radius a0 -a /mm
(voce) ref
1.0
1.2
0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 14 Numerical target curves and inverse analysis results Fig. Change in radius a0 -a /mm
5.2 Stress correction using tensile test results of low carbon steel (SS400) and discussion Fig. 15 shows the flowchart of stress correction using tensile test results of SS400. The true σflow-εeq curve of an actual material after the occurrence of necking cannot be not obtained; σflow in two cases of stress corrections were compared with values obtained by extrapolation based on the Swift’s law (Table 2) which parameters were given in the range of uniform elongation.
28
SS400 Tensile tests using image analysis (R0 = 3, 6, 10 and 20mm) σzave-εeq P-(a0-a)
σzave-εeq R, a
Stress correction using inverse analysis
Stress correction using Bridgman’s method
σflow-εeq
σflow-εeq
Compare with a extrapolated Swift’s hardening rule (=Table 2) Fig. 15 Flowchart of validation process of stress corrections for using SS400 tensile test results Fig. 16 shows σzave–εeq curves for different R0 values obtained by tensile tests using SS400 and image analysis. In this experiment of SS400, the same experiment was conducted three times, there was almost no difference in P-(a0-a) curve. To remove the noise originating from image analysis, the change in measured a with time was smoothed using a sixth-order function and then used for inverse analysis. Similarly to numerical experiments, σzave increases with decreasing R0. In addition, with decreasing R0, fracture strain decreases, which is considered to be caused by the increase in stress multiaxiality. A Similar results were reported, for example, by Enami, (2005).
29
Average tensile stress σzave /MPa
/MPa Stress or stress /MPa Equivalent Stress /MPa
1200
Fracture
R0 3 6 10 20mm
1000 800
1200
788( p 0.002)0.19
600
1000
( Table 2)
400 800
Swift
200 600
zave (Filterd) R0=3
zave 系列10
400 0
0 200
0.2
0.4 0.6 0.8 Equivalent Equivalentstrain strainεεeqeq, /ε-p
1
1.2
0
Fig. 16 Average tensile stress curves 0 1 obtained from SS400 tensile tests using Equivalent strain 2ln(a0/a)
image analysis
Fig. 17 shows σflow–εeq curves obtained after stress correction by inverse analysis. It is assumed that uniform strain, εu is 0.1 regardless of R0 and the number of iterations of the optimization algorithm was 8. In the figure, the σflow–εeq curve obtained using the Swift’s law is also shown. σflow–εeq curves obtained after stress correction by inverse analysis are the same regardless of R0 and correspond to the curve using the Swift’s law. For each specimen, σflow is identified until fracture: σflow up to εf = 1.05 was obtained for the specimen with R0 = 20 mm. This strain is more than fivefold the uniform elongation strain. Fig. 18 shows P–(a0−a) curves of experiment and inverse
30
analysis. The experimental results are fairly accurately reproduced.
788( p 0.002)0.19 ( Table 2)
800 1000
Fracture R0 3 6 10 20mm
600
Equivalent stress /MPa
Equivalent stress /MPa
1000
800
Corrected flow stress Smooth (Proposed method) Swift R0 3mm R0=3 R0 6mm R0=6 R0 10mm R0=10 R0 20mm R0=20
400
600
Smooth specimen (ε p 0.19)
200
400
0 200
0
0.2
0 0
0.4 0.6 0.8 Equivalentstrain strainεεeqeq, /ε-p Equivalent
1
1.2
1
Fig. 17 Stress correction results using inverse analysis on SS400 tensile Equivalent strain 2ln(a 0/a) tests
14000
12000
12000
10000
10000 8000
Tensile load P /N
Tensile load P /N
14000
R0 3
8000
10 20mm
6000 4000
Experimental target curves Inverse analysis results
6000 2000
4000 2000
6
0 0.0
0.2
0.4 0.6 0.8 Change in radius a0-a /mm
1.0
1.2
0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Fig. 18 Experimental target curves and inverse analysis results on SS400 Change in radius a0 -a /mm tensile tests
31
Fig. 19 shows σflow–εeq curves obtained after stress correction by Bridgman’s method. σflow–εeq curves obtained after stress correction are almost the same but are slightly higher than the Swift’s law. This finding indicates that the σflow–εeq curves obtained after stress correction by Bridgman’s method are located at upper regions than those obtained after stress correction by inverse analysis. This tendency corresponds to the numerical experiment results described previously. From these findings, σflow–εeq curves obtained after stress correction by inverse analysis represent the actual work hardening behavior more accurately than those obtained by Bridgman’s method.
800
788( p 0.002)0.19
1000
600
( Table 2) Equivalent stress /MPa
Equivalent stress /MPa
1000
400
800
600 Smooth specimen (ε400 p 0.19)
200
CorrectedSmooth flow stress Swift (Bridgman' s method) R0=3 R0 3mm
R0=6 R0 6mm R0=10 R0 10mm R0=20 R0 20mm
200
0 0
0.2
0
0.4 0.6 0.8 Equivalent Equivalent strainεeq ε/-eq, /-ε p Equivalent strainstrain 2ln(a0/a)
0
1
1.2
Fig. 19 Stress correction results using Bridgman’s method on SS400 tensile tests
32
6. Conclusions In this study, we proposed a method of accurately identifying σflow–εeq curves up to fracture for isotropic and homogeneous metallic materials. It was possible to identify the flow stress curve regardless of work hardening model by means of stress correction with the inverse analysis for σzave–εeq curves obtained by notched round bar tensile tests. The evaluation was conducted on numerical experiment and actual material, low carbon steel (SS400). The following conclusions were obtained. (1) In the evaluation using the numerical experiment results, two types of σref (σ(swift) and σ(voce) ) with different work hardening behaviors were accurately ref ref reproduced by the inverse analysis regardless of the stress history. On the other hand, the σflow–εeq curves corrected by Bridgman's method overestimated σref. (2) In the evaluation using the tensile test results of SS400, the σflow–εeq curves corrected by inverse analysis follow Swift’s law determined in the range of uniform elongation. On the other hand, as well as numerical tensile test results, the σflow–εeq curves corrected by Bridgman’s method is higher than that of obtained by the inverse analysis. From these findings, σflow–εeq curves obtained after stress correction by inverse analysis represent the
33
actual work hardening behavior more accurately than those obtained by Bridgman’s method. (3) In the evaluation using the tensile test results of SS400, σflow can be identified in the range of equivalent strain exceeding 1.0.
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