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A novel method to determine full-range hardening curve for metal bar using hyperbolic shaped compression specimen
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Original Article
A novel method to determine full-range hardening curve for metal bar using hyperbolic shaped compression specimen Junfu Chen a,b , Zhiping Guan a,b,∗ , Jingsheng Xing a,b , Dan Gao a,b , Mingwen Ren a,b a
Key Laboratory of Automobile Materials of Ministry of Education & School of Materials Science and Engineering, Jilin University, 5988 Renmin Street, Changchun 130022, China b International Center of Future Science, Jilin University, Changchun 130012, China
a r t i c l e
i n f o
a b s t r a c t
Article history:
In the determination of full-range hardening curve with compression test, the influence
Received 26 November 2019
of friction and the corresponding stress correction have always been an enormous chal-
Accepted 2 January 2020
lenge. In this study, a hyperbolic shaped specimen is designed to conduct compression test
Available online xxx
for the determination of hardening curve in large range of strains without the effect of
Keywords:
of finite element simulations of compression tests using the proposed specimen with the
Compression test
hardening curves composed with different combinations of initial yield stresses and strain
friction. The associated stress correction method is firstly derived by performing a series
Hyperbolic shaped specimen
hardening exponents. Finally, the hardening curves of mild steel Q420 obtained from the
Hardening curve
compression test using the proposed specimen are implemented in the simulation and
Stress correction
verified with experimental data. The results show that the hyperbolic shaped compres-
Large strain
sion specimen can not only remove the undesirable effect of friction but also determine
Finite element method.
the hardening curve in large range of strains. The established stress correction method for the hyperbolic shaped compression specimen is solely dependent on the strain hardening exponent, which can provide accurate hardening curve merely based on the experimental measurement of load vs. displacement curve and radius of middle cross section. Moreover, the hyperbolic shaped compression specimen provides a feasible approach to determine the strain hardening exponent of material. © 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
∗ Corresponding authors at: Jilin university, School of Materials Science and Engineering, No.5988 Renmin Street, Changchun, Jilin province, 130022, China E-mail: [email protected] (Z. Guan). https://doi.org/10.1016/j.jmrt.2020.01.003 2238-7854/© 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/).
Please cite this article in press as: Chen J, et al. A novel method to determine full-range hardening curve for metal bar using hyperbolic shaped compression specimen. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.01.003
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1.
Introduction
Finite Element Method (FEM) has been widely applied in metal bulk forming to shorten the time required for product development. The precision of simulation results are directly dependent on the accuracy of the inputted constitutive model especially hardening curve. The hardening curve represents the work hardening behavior of metal material, which is normally determined from the uniaxial tension or compression test with cylindrical specimen [1]. As the deformation occurred in bulk forming are generally under compressive stress state, the uniaxial compression test is preferred to obtain the hardening curve. However, due to the inevitable friction at the tool-specimen interface, the hardening curve obtained from compression test with a cylindrical specimen is valid only up to the onset of bulging where the stress state becomes un-uniform, which is corresponding to a rather limited strain range (0–0.4) [2,3]. For the bulk forming in industrial application such as forging, extrusion and rolling, the failure strain of metal part commonly has a large value, even exceeding 1.0, which is much greater than the uniform strain related to the onset of bulging for cylindrical compression specimen [4]. The simulations based on the hardening curve determined from conventional compression test of cylindrical specimen is insufficient to fully reproduce the plastic flow behaviors in metal bulk forming. Thus, to improve the prediction accuracy for metal bulk forming by FEM and further promote the development of precision forming, it is of great significance to obtain the reliable hardening curve in large range of strains from compression test. To obtain the full-range hardening curve subjected to compressive stress state, a number of approaches have been proposed, which can be classified into two main categories. As the main cause for the bulging in cylindrical compression specimen is the friction, the first kind of approaches can be summarized as how to suppress or eliminate the effect of friction. In this field, the conventional ways are to apply lubricants like Vaseline, grease and Teflon membrane in the tool-specimen interface to reduce the friction coefficient, or to machine grooves in the end face of specimen to reduce the contact areas between specimen and tools [5–9]. However, both of the two methods are not sufficient for solving the friction in the case of large deformation for ductile metals, the effect of lubricant will be weakened with the increase of load and the grooves will experience sever plastic deformation, finally leading to the limited reduction of friction [10]. The second method is the so-called extrapolation method. Siebel and Pomp proposed a correction function in terms of the friction coefficient, where the correction factor will approach to unity with the increase of slenderness ratio of specimen, i.e., the effect of friction can be lessened with the increase of slenderness ratio of specimen [11]. In other words, the “friction-free” hardening curve can be obtained by extrapolating the test results to the case of specimen with infinite slenderness ratio [12,13]. The extrapolation method can avoid the analysis of actual friction condition existed in the toolspecimen interface. However, its accuracy and applicability are limited by the available range of slenderness ratio as the specimen with large height and small radius tends to be
unstable in compression test. The third method is to measure the friction coefficient directly and further correct the flow stresses [14–16]. The barrel compression test and ring compression test are the most popular tests in determining the friction coefficients [17–19]. In the application of such two tests, the friction behaviors are assumed to obey linear friction model and the average friction coefficient is considered to characterize the friction properties of the tool-specimen interface. However, in the actual process of compression test, with the increase of load, the individual region of end face will bear different kinds of friction and the transformation of each kind of friction will take place simultaneously [20]. Therefore, it is quite difficult in characterizing the actual friction coefficient accurately in the conventional compression test. Due to the stress state becomes un-uniform after bulging for the cylindrical compression specimen, the second kind of approaches focuses on how to eliminate the bulging phenomenon or correct the flow stresses after bulging. It is very important to understand that such kind of approaches are based on the geometric developments after bulging initiation [21]. From this point of view, a modified compression test was proposed by Bridgman, who suggested performing the compression in stages, each followed by machining of the specimen back to the original proportions with continual decrease in the absolute size [22]. The plastic strains were accumulated and the hardening curve can be extrapolated to rather large strains without considering bulging phenomenon. The disadvantage of such a method is that the stress and strain do not increase monotonically, and disturbance are introduced by the stepwise release and reapplication of stress. The other method is to transfer the average true stress after bulging into equivalent stress [23]. Based on Bridgman’s correction model which are widely used to determine the equivalent stress in the necking region of cylindrical tension specimen, a new stress correction model has been derived from the same arguments that have been used for necking in the tensile test, but in a way suitable for compression of cylindrical specimen [24]. In the application of this stress correction model, the curvature radius of bulging profile and maximum radius of cylindrical compression specimen are necessary. However, this method is derived merely based on geometric changes during deformation and other effective parameters like material properties and friction coefficient are not considered, finally resulting in considerable errors in determining the equivalent stresses in large range of strains. In order to solve the problem that the hardening curve determined from the compression test of cylindrical specimen is limited within a narrow strain range due to friction, a new compression specimen is designed in current study. Inspired from the necking formation of cylindrical tension specimen, the new compression specimen with a hyperbolic gauge section is designed to conduct compression test for the determination of full-range hardening curve without the effect of friction. A series of FE simulations of the compression tests using the proposed specimen will be conducted to derive the corresponding stress correction method. Finally, the hardening curve of mild steel Q420 determined form the compression test using the proposed specimen will be adopted
Please cite this article in press as: Chen J, et al. A novel method to determine full-range hardening curve for metal bar using hyperbolic shaped compression specimen. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.01.003
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in the FE simulation and finally verified with experimental results
2.
Proposed method
2.1. Theory basis of the hyperbolic shaped compression specimen Uniaxial compression test of cylindrical specimen is one of the commonly used tests for obtaining material properties. In the uniform deformation stage, the stress state on each cross section of a cylindrical specimen keeps uniform. The average true stress ave,t and average true strain εave,t can be calculated based on actual measurements of load and cross-sectional area [25]: ave,t =
P a2
εave,t = 2 ∗ ln(
(1) a ) a0
(2)
where P is the current load, a0 and a are the original and instantaneous cross-sectional radius of the specimen, respectively. It is worth noting that the equivalent stress is equal to the average true stress during the uniform deformation stage. Due to the friction between specimen and the two platens, the bulging phenomenon occurs near the equator of specimen after uniform deformation stage, as shown in Fig. 1(a). In the bulged specimen, the element in middle region sub-
3
jects to compressive stress in three major directions. In the bulged section, except for the axial and radial compressive stress, the bulging effect will result in a circumferential tensile stress, i.e., the stress state becomes un-uniform, as shown in Fig. 1(b). Therefore, the average true stress calculated via Eq. (1) can not represent the actual equivalent stress of cylindrical compression specimen after the onset of bulging, which should be corrected. Previous correction methods generally require the identification of friction coefficient and the bulging profile parameters including maximum radius of cross section and curvature radius, which are difficult to execute and time-consuming. In order to conquer these limitations, a hyperbolic shaped compression specimen is design in current study. The sketch of the hyperbolic shaped compression specimen and its compression deformation are depicted in Fig. 1(c). The specimen possesses a standard hyperbolic gauge section with a eccentricity of 2, a minimum diameter of 6 mm in the middle region and two cylindrical ends with a diameter of 16 mm and height of 4 mm. The specific dimensions are shown in Fig. 1(d). Due to the rather weak structure in the gauge section, the deformation will prefer to localize in the middle region. Hence, there are no horizontal deformation occurred in the shoulders, resulting in no friction acting on the end surfaces of the specimen. Unlike the cylindrical compression specimen, the hyperbolic shaped one will experience un-uniform deformation and present un-uniform stress state from the very beginning of compression deformation. As the compression deformation
Fig. 1 – (a) Specimen geometry of cylindrical specimen in the initial and the bulging condition; (b) stress state of cylindrical specimen after bulging; (c) sketch of the hyperbolic shaped compression specimen and its compression deformation;(d) the specific dimensions of hyperbolic shaped specimen (mm). Please cite this article in press as: Chen J, et al. A novel method to determine full-range hardening curve for metal bar using hyperbolic shaped compression specimen. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.01.003
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process of the hyperbolic shaped specimen can be regarded as the reverse process of necking formation in tension deformation of cylindrical specimen, the assumption of uniform distribution of strain on the cross section is also applied to the new compression specimen. When using the hyperbolic shaped compression specimen, the average true strain can be still determined using Eq. (2). However, due to the introduction of the notch in the middle region, stress concentration occurs. As a result, the average true stress calculated via Eq. (1) is not equal to the equivalent stress, which should be corrected by multiplying a correction function as following: eq = ave,t ∗ f (ε)
to the simulation of compression test using proposed specimen firstly. Then, the average true stress vs. strain curves are calculated via Eqs. (1) and (2) based on the simulation results. The ratios between inputted material equivalent stresses and outputted average true stresses corresponding to the same strain are plotted against strain and then adopted to derive the stress correction function. Finally, with the application of the proposed stress correction method, the average true stress vs. strain curve can be converted to material equivalent stress vs. strain curve. The details about the derivation process of stress correction method will be explained in the Section 4.3.
(3)
3.
Numerical experimental procedures
where f (ε) is the stress correction function to be derived.
2.2.
Stress correction based on FEM
A novel stress correction method for the hyperbolic shaped compression specimen will be derived based the simulation results using various material constitutive models. The strategy for derivation of the stress correction method using FEM in the present study is illustrated in Fig. 2. The hardening curves which are generated via various combinations of initial yield stresses and strain hardening exponents are applied
The general FE software Abaqus/Standard (version 6.13) is adopted to simulate the compression test using the hyperbolic shaped specimen. Due to geometrical symmetries, only one half of the specimen which consists of hyperbolic gauge section, arc-transition portion and two cylindrical ends is selected as the geometrical model. The nonlinear geometry option (NLGEOM) is switched on for the simulation of large scale of plastic deformation. Four-node bi-linear axisymmetric quadrilateral element (CAX4I) is used for the discretization. The top and bottom platens are modeled as rigid parts. In
Fig. 2 – Flowchart of the derivation of stress correction method based on FEM: (a) the material equivalent stress vs. strain curve to be inputted into FEM; (b) FE simulation of compression test using the hyperbolic shaped specimen; (c) the average true stress vs. strain curve calculated from the (b). With the application of the proposed stress correction method, the average true stress vs. strain curve in Fig.3 (2) can be transferred to the equivalent one in Fig.2 (a). Please cite this article in press as: Chen J, et al. A novel method to determine full-range hardening curve for metal bar using hyperbolic shaped compression specimen. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.01.003
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hardening model, where the strength coefficient K0 is substin tuted by 0 /(ε0 ) , is adopted here to generate the hardening curves to be inputted into ABAQUS: n
flow = K0 (ε0 + εp ) = 0 (1 +
εp n ) ε0
(4)
where flow , εp are the flow stress and the plastic strain, respectively. The Young modulus E = 200 GPa and the corresponding yield strain ε0 = 0.002 have been used together with the Poisson ratio P = 0.3. In order to explore the influence of strain hardening exponent n and initial yield stress 0 on the stress correction, the hardening curves generated with the combinations of strain hardening exponents varied from 0.075 to 0.225 in an increment of 0.025 and initial yield stresses varied from 200 MPa to 400 MPa in an increment of 100 MPa are investigated in the simulation, covering most of engineering materials. For sake of comparison with the corrected equivalent stress vs. strain curve, the generated material hardening curves should be converted to the material equivalent stressstrain curve by considering the elastic stage. Totally, 21 groups of FE simulations are prepared and analyzed in this study.
Fig. 3 – The global mesh of hyperbolic shaped specimen adopted in simulation of compression test.
order to exclude the influence of mesh size on the simulation results, four different mesh sizes involving 0.1 mm, 0.2 mm, 0.4 mm, and 0.8 mm are utilized to mesh the model. After the sensitive analysis of mesh size, the average mesh size is set as 0.2 mm for reaching an appropriate balance between simulation accuracy and computational cost, as shown in Fig. 3. A downward displacement boundary condition is applied to the top platen along axial direction while the bottom platen is fixed. Symmetric boundary conditions are applied on the middle surface. The surface-to-surface contacts with friction are introduced to model the interaction between the platens and the specimen. In order to investigate the influence of friction on the compression deformation of the proposed specimen, four friction coefficients including 0, 0.1, 0.2 and 0.4 as well as rough friction ( = ∞) are introduced in the model. The material is assumed as homogeneous and isotropic elastic-plastic and follows the von Mises yield criterion. A Swift’s power law
Fig. 4 – (a) The load vs. displacement curves from simulations with different friction conditions; (b) the radius of cross section vs. displacement curves from FE simulations with different friction conditions.
Please cite this article in press as: Chen J, et al. A novel method to determine full-range hardening curve for metal bar using hyperbolic shaped compression specimen. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.01.003
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Fig. 5 – Contours of equivalent plastic strain distributions and radial displacement of the deformed specimen from simulation with different friction condition: (a) and (b) = 0; (c) and (d) rough friction.
4.
Results and discussion
4.1.
Influence of friction on the stress correction
Fig. 4(a) and Fig. 4(b) show the load vs. displacement curves and radius of cross section vs. displacement curves obtained from the simulations of the hyperbolic shaped compression specimen with the friction coefficients = 0, = 0.1, = 0.2 and = 0.4 as well as rough friction. Here, the simulations were conducted based on the hardening curve generated with 0 = 200 MPa and n = 0.1. As can be seen from Figs. 4(a) and 4(b) that the simulations with different friction conditions give similar values of load and radius of cross section corresponding to the same displacement, whose maximum deviations
do not exceed 0.3 %. In other words, for a given material hardening curve, the compression test using the proposed specimen presents the same average true stress vs. strain curve regardless of friction condition between the specimen and platens. Figs. 5(a)-5(d) show the contours of equivalent plastic strain distributions and radial displacement of the deformed specimen from simulation with = 0 and rough friction, respectively. As expected, all of the plastic deformation are localized in the middle region of the specimen while no deformation occur in the shoulders. Hence, there are no horizontal force acting on the end surfaces of the specimen, the friction condition does not affect the stress state of the middle region of specimen. Moreover, the distribution of the radius displacement and equivalent plastic strain for the deformation specimen are consistent corresponding to
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the same displacement for the friction condition of = 0 and rough friction. Namely, the existence of friction or not does not affect the whole compression deformation of the proposed specimen. Therefore, in the case of determination of hardening curve using the hyperbolic shaped compression specimen, the friction condition has none influence on the final results.
4.2. Influence of material properties on the stress correction Fig. 6 shows the material equivalent stress vs. strain curves (black line) as well as the average true stress vs. strain curves calculated by Eqs. (1) and (2) (other color lines), where the material equivalent stress vs. strain curves are generated based on the various combinations of strain hardening exponents varied from 0.075 to 0.225 and initial yield stress varied from 200 MPa to 400 MPa. It can be clearly observed from Fig. 6, for the hyperbolic shaped compression specimen, the average true stress calculated based on the direct application of Eq. (1) results in significant errors compared with the material equivalent stress from the very beginning of deformation due to the un-uniform stress state. For the materials with strain hardening exponents varied from 0.075–0.15, the average true stress vs. strain curves lie above the corresponding material equivalent stress vs. strain curves initially, and then have intersections with material equivalent stress vs. strain curves. As for the remaining materials with strain hardening exponents varied from 0.175–0.225, the values of average true stresses are always larger than those of material equivalent stresses corresponding to the same strain. To reflect the evolution of the deviations directly, the ratios between material equivalent stress eq,m and average true stresses ave,t corresponding to the same strain are plotted against the strain for the materials with various combinations of initial yield stresses and strain hardening exponents, as shown in Fig. 7. As can be seen the eq,m /ave,t vs. strain curves from materials with diverse initial yield stresses are collapsed into one curve, while that from materials with different strain hardening exponents are varied from each other. This phenomenon proves that the stress correction for the hyperbolic shaped compression specimen is solely dependent on strain hardening exponent but has none relationship with the initial yield stress.
4.3. Derivation of the stress correction method based on FEM 4.3.1. Dependencies of stress correction on strain hardening exponent The purpose of this section is to derive a simple correction function to convert the average true stress directly determined from the hyperbolic shaped compression specimen to the material equivalent stress. The stress correction function are composed of a series of stress correction factors, each of them
Fig. 6 – The material equivalent stress–strain curves (black line) as well as the average true stress vs. strain curves (other color lines): (a) 0 =200 MPa; (a) 0 =300 MPa; (a) 0 = 400 MPa.
Please cite this article in press as: Chen J, et al. A novel method to determine full-range hardening curve for metal bar using hyperbolic shaped compression specimen. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.01.003
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Fig. 7 – The ratios between material equivalent stress eq,m and average true stresses ave,t plotted against the strain for the materials with strain hardening exponents varied from 0.075-0.225 and initial yield stress of 200MPa-400 MPa.
Table 1 – The slope K and intercept b determined from the linear fitting of the relationship between the eq,m / ave,t and strain for material with strain hardening exponents varied from 0.075-0.225. Strain hardening exponent n
Slope K
Intercept b
0.075 0.1 0.125 0.15 0.175 0.2 0.225
0.18419 0.15935 0.13523 0.11045 0.08568 0.06037 0.03714
0.88561 0.89405 0.90118 0.9077 0.91357 0.91894 0.92331
is equal to the ratio of the material equivalent stress eq,m and the average true stress ave,t corresponding to the same strain. The evolution of stress correction factor eq,m /ave,t with strain have been previously depicted in Fig. 7. As can be seen the values of eq,m /ave,t varies linearly respect to the strain for all of the selected materials. Therefore, the relationship between stress correction factor and strain can be fitted through linear regression, i.e. the stress correction function, as shown in Fig. 7. The slope K as well as the intercept b in linear regression model are firstly determined for each strain hardening exponent, as listed in Table 1. Obviously, the slope K in the linear regression model increases with the decrease of strain hardening exponent, while the evolution of intercept b shows an opposite tendency. Both of the two parameters are then plotted against strain hardening exponent, respectively. As can be seen from Fig. 8 (a), the slope K presents linear variation respect to the strain hardening exponent. Additionally, the relationship between intercept b and strain hardening exponent can be fitted through a second-order polynomial, as shown in Fig. 8 (b). Therefore, the dependencies of parameters K and b on the strain hardening exponent n can be directly
Fig. 8 – (a) Linear fitting of the relationship between the slope K of the stress correction function and strain hardening exponent; (b) polynomial fitting of the relationship between the intercept b of the stress correction function and strain hardening exponent.
determined through linear regression and polynomial regression, respectively. Thus, the stress correction function can be expressed as following: f (ε) = K ∗ ε + b
K = −0.9838 ∗ n + 0.25791
(5)
b = 0.85718 + 0.42439 ∗ n + 0.58 ∗ n2
4.3.2. Correlating strain hardening exponent with displacement In the derivation of the stress correction function, the strain hardening exponent plays a decisive role. In the numerical experiments, such a parameter is arbitrary to be given. However, in the actual application of the stress correction for the proposed specimen in compression test, the strain hardening exponent of material is unknown. Therefore, we hope to obtain such a material parameter from the compression test using the hyperbolic shaped compression specimen. In the previous section, we have concluded that the compression deformation of the hyperbolic shaped compression specimen is merely affected by strain hardening exponent, showing none relation with friction and initial yield stress.
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Fig. 9 – Contours of axial displacement of the specimen compressed to average true strain of∼0.2 for material with different strain hardening exponents: (a) n = 0.075 (b) n = 0.175; (c) n = 0.225.
That is to say, when the specimen is compressed to the same average true strain, i.e. the same radius of middle cross section, the displacement and geometrical configuration of the specimen for materials with different strain hardening exponent should be diverse and solely dependent on strain hardening exponent. Fig. 9 shows the contours of axial displacements of deformed specimen obtained from the simulation of the hyperbolic shaped compression specimen for the materials with strain hardening exponents of ∼0.075, ∼0.175 and ∼0.225, which were compressed to the same average true strain of ∼0.2, corresponding to the radius of∼3.3155. As can be seen the specimens with different strain hardening exponents finally lead to different axial displacement corresponding to the same average true strain. The displacement is plotted against the average true strain for the material with strain hardening exponents varied from 0.075–0.225, as shown in Fig. 10(a). Similarly, the displacement vs. strain curves related to different strain hardening exponents exhibit different growth trends. For a given strain hardening exponent, the displacement increases with the increase of the strain, showing good linear variation in the strain range of less than 0.4, simultaneously. For an arbitrary strain in the range of 0-0.4, the displacements corresponding the chosen average true strain can be plotted against strain hardening exponents, as shown in Fig. 10(b). Here, the value of average true strain is selected as 0.2. As expected, the strain hardening exponent varies linearly respect to the displacement. The dependencies of strain hardening exponent on the displacement can be finally determined through linear regression as following: n = 0.53752 ∗ L − 0.37375 |ε=0.2
(6)
Therefore, for the hyperbolic shaped compression specimen, the strain hardening exponent can be directly determined by using Eq. (6). Such a method merely requires
Fig. 10 – (a) Displacement vs. average true strain curves for the material with strain hardening exponents varied from 0.075–0.225; (b) linear fitting of the relationship between strain hardening exponent and displacement corresponding to the average true strain of ∼0.2.
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Fig. 11 – The average true stress vs. strain curves, the material equivalent stress vs. strain curves inputted into the FE simulation and the corrected equivalent stress vs. strain curve using the proposed stress correction method for different materials: (a) n = 0.075; (b) n = 0.1; (c) n = 0.125; (d) n = 0.15; (e) n = 0.175;(f) n = 0.2; (g) n = 0.225.
the experimental measurement of the axial displacement and radius evolution of middle cross section of the compression specimen. The additional use of tension test with cylindrical specimen can be avoided effectively.
4.3.3.
The proposed stress correction function
In the above two section, we firstly derived the stress correction function related to strain and determined the dependency of the parameters of the function on strain hardening expo-
Fig. 12 – (a) The photographs of experimental apparatuses of the compression test; (b) the initial snapshot of the hyperbolic shaped compression specimen. Please cite this article in press as: Chen J, et al. A novel method to determine full-range hardening curve for metal bar using hyperbolic shaped compression specimen. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.01.003
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Table 2 – Chemical composition (wt. %) of Q420. Grade
C
Si
Mn
P
S
Fe
Q420
0.067
0.054
0.346
0.021
0.013
Bal.
nent, i.e., Eq.(5). Then, we proposed the measurement method for determining the unknown strain hardening exponent, i.e., Eq. (6). Thus, inserting Eq. (6) into Eq. (5), the final stress correction method for converting the average true stress into equivalent stress can be rewritten as following: eq = ave,t ∗ f (ε)
⎧ f (ε) = K ∗ εave,t + b ⎪ ⎪ ⎪ ⎪ ⎨ K = −0.9838 ∗ n + 0.25791 ⎪ b= ⎪ ⎪ ⎪ ⎩
(7)
0.85718 + 0.42439 ∗ n + 0.58 ∗ n2
n = 0.53752 ∗ L − 0.37375 |ε=0.2
The function proposed in this study is quite easy to apply for two reasons. First of all, the format of the stress correction function is explicitly expressed as Eq. (7). Secondly, the proposed method only requires the measurement of the load vs. displacement curves and the evolution of radius of middle cross section, which are easily obtained from a compression test using the hyperbolic shaped compression specimen.
4.3.4. Simulated verification of the stress correction method In order to verify the proposed stress correction method, it is firstly executed based on the previous simulation results of the hyperbolic shaped compression specimen. The corrected equivalent stress vs. strain curves (red line) transferred from the average true stress vs. strain curves (green line) using Eq. (7) are compared with the material equivalent stress vs. strain curves (black line), as shown in Fig. 11. Very satisfactory agreement can be seen up to strain of ∼0.8 for all the materials except for the material with strain hardening of 0.225. The maximum errors at ε = 0.8 for the material with strain hardening of 0.225 exhibits less than 1 %, which is still in an acceptable level from the view of engineering application. The results indicate that the hyperbolic shaped compression specimen can determine the hardening curve in large range of strains effectively using the associate stress correction method.
5. Experimental verification of the proposed method Since the hyperbolic shaped compression specimen and the corresponding stress correction methods are executed through FEM, which should be further verified with experimental data. The material adopted in current study is mild steel Q420. The chemical composition (weight percent) of Q420 is summarized in Table 2. The compression specimens were machined from metal bar according to the specific dimensions depicted in Fig. 1(d). A universal testing machine (SHIMADZU, AGX) with a load capability of 100 K N was used to conduct
Fig. 13 – (a) The average true strain vs. displacement curve obtained from compression test using the hyperbolic shaped compression specimen; (b) the average true stress vs. strain curve and the equivalent stress vs. strain curves corrected with proposed method; (c) the experimental load vs. displacement curve and the simulated load vs. displacement curves obtained from the simulation based on the average true stress vs. strain curve and the equivalent stress vs. strain curves determined with the proposed method.
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Fig. 14 – (a) The deformed specimen from simulation using the hardening curve determined from the proposed stress correction method; (b) the snapshot of deformed specimen from compression test corresponding to the same displacement (∼4.5 mm) with (a).
the compression test in a constant velocity of 1 mm/min at ambient temperature, as shown in Fig.12 (a). A charge-coupled CCD camera (KOWA-LM16JC1MS) was applied to capture the images of deformed specimen and record the displacement within two gauge marks as well as the radius of middle cross section, simultaneously. The initial snapshot of the specimen is shown in Fig. 12(b). The proposed stress correction method based on FEM was then applied to correct the average true stress to the equivalent stress for Q420. The equivalent stress vs. strain curves determined with the proposed method and the average true stress vs. strain curve were inputted into the FE simulation, outputting the simulated load vs. displacement curves, finally compared with the experimental one. Fig.13 (a) shows the average true strain vs. displacement curve obtained from compression test using the hyperbolic shaped compression specimen. As expected, the displacement varies linear with average true strain in the strain range of 0-0.4, which corresponds to the numerical experiment results described previously. The displacement is identified as 1.01303 corresponding to the average true strain of∼0.2. With the application of Eq. (6), the strain hardening exponent of Q420 is finally identified as 0.17077. Once the strain hardening exponent is known, the average true stress vs. strain curve (green line) can be corrected to equivalent stress vs. strain curve (black line) via Eq. (7), as shown in Fig. 13(b). The simulated load vs. displacement curves, which are outputted from FE simulations based on the hardening curves with and without stress correction, are compared with the experimental load vs. displacement curve (triangle dot), as shown in Fig. 13(c). For the hardening curve without stress correction (green line), there are large deviations between the simulated and experimental load vs. displacement curves from the very beginning of compression deformation, indicating the inadequate of the direct application of Eq. (1) in the determination of hardening curve. While the predicted load vs. displacement curve from the hardening curve corrected with the proposed method (red line) presents good
agreement with the experimental load vs. displacement curve, leading to an exact solution from engineering point of view. Fig. 14(a) shows the deformed specimen from FE simulation using the hardening curve determined from the proposed stress correction method, which is similar to that from the actual compression test corresponding to the same displacement of ∼4.5 mm (Fig.14(b)). Thus, it is confirmed that the hyperbolic shaped compression specimen and the proposed stress correction method are capable of obtaining the hardening in large range of strain precisely without the effect of friction.
6.
Conclusion
In this study, a hyperbolic shaped compression specimen is proposed to obtain the hardening curve in large range of strains without the effect of friction. Through extensive numerical analyses, the associated stress correction method is derived to convert the average true stress obtained directly from compression test of the proposed specimen to equivalent stress. The proposed specimen and stress correction method are verified with experimental data of mild steel Q420. The conclusions can be drawn as follows: 1 The hyperbolic shaped compression specimen can not only remove the undesirable effect of friction but also determine the hardening curve in large range of strains. 2 The established stress correction method for the new compression specimen is solely dependent on the strain hardening exponent, which can provide accurate hardening curve merely based on the experimental measurement of load vs. displacement curve and radius of middle cross section. 3 The hyperbolic shaped compression specimen provides a feasible method to determine the strain hardening exponent of material based on the displacement vs. average true strain curve.
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Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Acknowledgments The National Key Research and Development Program of China (No. 2018YFB1309203) and the Natural Science Foundation of China (Nos. 51575230 and U1810208) are greatly acknowledged. Partial financial support came from the Science and Technology Development Program of Jilin Province (No. 20190302059GX).
references
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Original Article
A novel method to determine full-range hardening curve for metal bar using hyperbolic shaped compression specimen Junfu Chen a,b , Zhiping Guan a,b,∗ , Jingsheng Xing a,b , Dan Gao a,b , Mingwen Ren a,b a
Key Laboratory of Automobile Materials of Ministry of Education & School of Materials Science and Engineering, Jilin University, 5988 Renmin Street, Changchun 130022, China b International Center of Future Science, Jilin University, Changchun 130012, China
a r t i c l e
i n f o
a b s t r a c t
Article history:
In the determination of full-range hardening curve with compression test, the influence
Received 26 November 2019
of friction and the corresponding stress correction have always been an enormous chal-
Accepted 2 January 2020
lenge. In this study, a hyperbolic shaped specimen is designed to conduct compression test
Available online xxx
for the determination of hardening curve in large range of strains without the effect of
Keywords:
of finite element simulations of compression tests using the proposed specimen with the
Compression test
hardening curves composed with different combinations of initial yield stresses and strain
friction. The associated stress correction method is firstly derived by performing a series
Hyperbolic shaped specimen
hardening exponents. Finally, the hardening curves of mild steel Q420 obtained from the
Hardening curve
compression test using the proposed specimen are implemented in the simulation and
Stress correction
verified with experimental data. The results show that the hyperbolic shaped compres-
Large strain
sion specimen can not only remove the undesirable effect of friction but also determine
Finite element method.
the hardening curve in large range of strains. The established stress correction method for the hyperbolic shaped compression specimen is solely dependent on the strain hardening exponent, which can provide accurate hardening curve merely based on the experimental measurement of load vs. displacement curve and radius of middle cross section. Moreover, the hyperbolic shaped compression specimen provides a feasible approach to determine the strain hardening exponent of material. © 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
∗ Corresponding authors at: Jilin university, School of Materials Science and Engineering, No.5988 Renmin Street, Changchun, Jilin province, 130022, China E-mail: [email protected] (Z. Guan). https://doi.org/10.1016/j.jmrt.2020.01.003 2238-7854/© 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/).
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1.
Introduction
Finite Element Method (FEM) has been widely applied in metal bulk forming to shorten the time required for product development. The precision of simulation results are directly dependent on the accuracy of the inputted constitutive model especially hardening curve. The hardening curve represents the work hardening behavior of metal material, which is normally determined from the uniaxial tension or compression test with cylindrical specimen [1]. As the deformation occurred in bulk forming are generally under compressive stress state, the uniaxial compression test is preferred to obtain the hardening curve. However, due to the inevitable friction at the tool-specimen interface, the hardening curve obtained from compression test with a cylindrical specimen is valid only up to the onset of bulging where the stress state becomes un-uniform, which is corresponding to a rather limited strain range (0–0.4) [2,3]. For the bulk forming in industrial application such as forging, extrusion and rolling, the failure strain of metal part commonly has a large value, even exceeding 1.0, which is much greater than the uniform strain related to the onset of bulging for cylindrical compression specimen [4]. The simulations based on the hardening curve determined from conventional compression test of cylindrical specimen is insufficient to fully reproduce the plastic flow behaviors in metal bulk forming. Thus, to improve the prediction accuracy for metal bulk forming by FEM and further promote the development of precision forming, it is of great significance to obtain the reliable hardening curve in large range of strains from compression test. To obtain the full-range hardening curve subjected to compressive stress state, a number of approaches have been proposed, which can be classified into two main categories. As the main cause for the bulging in cylindrical compression specimen is the friction, the first kind of approaches can be summarized as how to suppress or eliminate the effect of friction. In this field, the conventional ways are to apply lubricants like Vaseline, grease and Teflon membrane in the tool-specimen interface to reduce the friction coefficient, or to machine grooves in the end face of specimen to reduce the contact areas between specimen and tools [5–9]. However, both of the two methods are not sufficient for solving the friction in the case of large deformation for ductile metals, the effect of lubricant will be weakened with the increase of load and the grooves will experience sever plastic deformation, finally leading to the limited reduction of friction [10]. The second method is the so-called extrapolation method. Siebel and Pomp proposed a correction function in terms of the friction coefficient, where the correction factor will approach to unity with the increase of slenderness ratio of specimen, i.e., the effect of friction can be lessened with the increase of slenderness ratio of specimen [11]. In other words, the “friction-free” hardening curve can be obtained by extrapolating the test results to the case of specimen with infinite slenderness ratio [12,13]. The extrapolation method can avoid the analysis of actual friction condition existed in the toolspecimen interface. However, its accuracy and applicability are limited by the available range of slenderness ratio as the specimen with large height and small radius tends to be
unstable in compression test. The third method is to measure the friction coefficient directly and further correct the flow stresses [14–16]. The barrel compression test and ring compression test are the most popular tests in determining the friction coefficients [17–19]. In the application of such two tests, the friction behaviors are assumed to obey linear friction model and the average friction coefficient is considered to characterize the friction properties of the tool-specimen interface. However, in the actual process of compression test, with the increase of load, the individual region of end face will bear different kinds of friction and the transformation of each kind of friction will take place simultaneously [20]. Therefore, it is quite difficult in characterizing the actual friction coefficient accurately in the conventional compression test. Due to the stress state becomes un-uniform after bulging for the cylindrical compression specimen, the second kind of approaches focuses on how to eliminate the bulging phenomenon or correct the flow stresses after bulging. It is very important to understand that such kind of approaches are based on the geometric developments after bulging initiation [21]. From this point of view, a modified compression test was proposed by Bridgman, who suggested performing the compression in stages, each followed by machining of the specimen back to the original proportions with continual decrease in the absolute size [22]. The plastic strains were accumulated and the hardening curve can be extrapolated to rather large strains without considering bulging phenomenon. The disadvantage of such a method is that the stress and strain do not increase monotonically, and disturbance are introduced by the stepwise release and reapplication of stress. The other method is to transfer the average true stress after bulging into equivalent stress [23]. Based on Bridgman’s correction model which are widely used to determine the equivalent stress in the necking region of cylindrical tension specimen, a new stress correction model has been derived from the same arguments that have been used for necking in the tensile test, but in a way suitable for compression of cylindrical specimen [24]. In the application of this stress correction model, the curvature radius of bulging profile and maximum radius of cylindrical compression specimen are necessary. However, this method is derived merely based on geometric changes during deformation and other effective parameters like material properties and friction coefficient are not considered, finally resulting in considerable errors in determining the equivalent stresses in large range of strains. In order to solve the problem that the hardening curve determined from the compression test of cylindrical specimen is limited within a narrow strain range due to friction, a new compression specimen is designed in current study. Inspired from the necking formation of cylindrical tension specimen, the new compression specimen with a hyperbolic gauge section is designed to conduct compression test for the determination of full-range hardening curve without the effect of friction. A series of FE simulations of the compression tests using the proposed specimen will be conducted to derive the corresponding stress correction method. Finally, the hardening curve of mild steel Q420 determined form the compression test using the proposed specimen will be adopted
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in the FE simulation and finally verified with experimental results
2.
Proposed method
2.1. Theory basis of the hyperbolic shaped compression specimen Uniaxial compression test of cylindrical specimen is one of the commonly used tests for obtaining material properties. In the uniform deformation stage, the stress state on each cross section of a cylindrical specimen keeps uniform. The average true stress ave,t and average true strain εave,t can be calculated based on actual measurements of load and cross-sectional area [25]: ave,t =
P a2
εave,t = 2 ∗ ln(
(1) a ) a0
(2)
where P is the current load, a0 and a are the original and instantaneous cross-sectional radius of the specimen, respectively. It is worth noting that the equivalent stress is equal to the average true stress during the uniform deformation stage. Due to the friction between specimen and the two platens, the bulging phenomenon occurs near the equator of specimen after uniform deformation stage, as shown in Fig. 1(a). In the bulged specimen, the element in middle region sub-
3
jects to compressive stress in three major directions. In the bulged section, except for the axial and radial compressive stress, the bulging effect will result in a circumferential tensile stress, i.e., the stress state becomes un-uniform, as shown in Fig. 1(b). Therefore, the average true stress calculated via Eq. (1) can not represent the actual equivalent stress of cylindrical compression specimen after the onset of bulging, which should be corrected. Previous correction methods generally require the identification of friction coefficient and the bulging profile parameters including maximum radius of cross section and curvature radius, which are difficult to execute and time-consuming. In order to conquer these limitations, a hyperbolic shaped compression specimen is design in current study. The sketch of the hyperbolic shaped compression specimen and its compression deformation are depicted in Fig. 1(c). The specimen possesses a standard hyperbolic gauge section with a eccentricity of 2, a minimum diameter of 6 mm in the middle region and two cylindrical ends with a diameter of 16 mm and height of 4 mm. The specific dimensions are shown in Fig. 1(d). Due to the rather weak structure in the gauge section, the deformation will prefer to localize in the middle region. Hence, there are no horizontal deformation occurred in the shoulders, resulting in no friction acting on the end surfaces of the specimen. Unlike the cylindrical compression specimen, the hyperbolic shaped one will experience un-uniform deformation and present un-uniform stress state from the very beginning of compression deformation. As the compression deformation
Fig. 1 – (a) Specimen geometry of cylindrical specimen in the initial and the bulging condition; (b) stress state of cylindrical specimen after bulging; (c) sketch of the hyperbolic shaped compression specimen and its compression deformation;(d) the specific dimensions of hyperbolic shaped specimen (mm). Please cite this article in press as: Chen J, et al. A novel method to determine full-range hardening curve for metal bar using hyperbolic shaped compression specimen. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.01.003
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process of the hyperbolic shaped specimen can be regarded as the reverse process of necking formation in tension deformation of cylindrical specimen, the assumption of uniform distribution of strain on the cross section is also applied to the new compression specimen. When using the hyperbolic shaped compression specimen, the average true strain can be still determined using Eq. (2). However, due to the introduction of the notch in the middle region, stress concentration occurs. As a result, the average true stress calculated via Eq. (1) is not equal to the equivalent stress, which should be corrected by multiplying a correction function as following: eq = ave,t ∗ f (ε)
to the simulation of compression test using proposed specimen firstly. Then, the average true stress vs. strain curves are calculated via Eqs. (1) and (2) based on the simulation results. The ratios between inputted material equivalent stresses and outputted average true stresses corresponding to the same strain are plotted against strain and then adopted to derive the stress correction function. Finally, with the application of the proposed stress correction method, the average true stress vs. strain curve can be converted to material equivalent stress vs. strain curve. The details about the derivation process of stress correction method will be explained in the Section 4.3.
(3)
3.
Numerical experimental procedures
where f (ε) is the stress correction function to be derived.
2.2.
Stress correction based on FEM
A novel stress correction method for the hyperbolic shaped compression specimen will be derived based the simulation results using various material constitutive models. The strategy for derivation of the stress correction method using FEM in the present study is illustrated in Fig. 2. The hardening curves which are generated via various combinations of initial yield stresses and strain hardening exponents are applied
The general FE software Abaqus/Standard (version 6.13) is adopted to simulate the compression test using the hyperbolic shaped specimen. Due to geometrical symmetries, only one half of the specimen which consists of hyperbolic gauge section, arc-transition portion and two cylindrical ends is selected as the geometrical model. The nonlinear geometry option (NLGEOM) is switched on for the simulation of large scale of plastic deformation. Four-node bi-linear axisymmetric quadrilateral element (CAX4I) is used for the discretization. The top and bottom platens are modeled as rigid parts. In
Fig. 2 – Flowchart of the derivation of stress correction method based on FEM: (a) the material equivalent stress vs. strain curve to be inputted into FEM; (b) FE simulation of compression test using the hyperbolic shaped specimen; (c) the average true stress vs. strain curve calculated from the (b). With the application of the proposed stress correction method, the average true stress vs. strain curve in Fig.3 (2) can be transferred to the equivalent one in Fig.2 (a). Please cite this article in press as: Chen J, et al. A novel method to determine full-range hardening curve for metal bar using hyperbolic shaped compression specimen. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.01.003
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hardening model, where the strength coefficient K0 is substin tuted by 0 /(ε0 ) , is adopted here to generate the hardening curves to be inputted into ABAQUS: n
flow = K0 (ε0 + εp ) = 0 (1 +
εp n ) ε0
(4)
where flow , εp are the flow stress and the plastic strain, respectively. The Young modulus E = 200 GPa and the corresponding yield strain ε0 = 0.002 have been used together with the Poisson ratio P = 0.3. In order to explore the influence of strain hardening exponent n and initial yield stress 0 on the stress correction, the hardening curves generated with the combinations of strain hardening exponents varied from 0.075 to 0.225 in an increment of 0.025 and initial yield stresses varied from 200 MPa to 400 MPa in an increment of 100 MPa are investigated in the simulation, covering most of engineering materials. For sake of comparison with the corrected equivalent stress vs. strain curve, the generated material hardening curves should be converted to the material equivalent stressstrain curve by considering the elastic stage. Totally, 21 groups of FE simulations are prepared and analyzed in this study.
Fig. 3 – The global mesh of hyperbolic shaped specimen adopted in simulation of compression test.
order to exclude the influence of mesh size on the simulation results, four different mesh sizes involving 0.1 mm, 0.2 mm, 0.4 mm, and 0.8 mm are utilized to mesh the model. After the sensitive analysis of mesh size, the average mesh size is set as 0.2 mm for reaching an appropriate balance between simulation accuracy and computational cost, as shown in Fig. 3. A downward displacement boundary condition is applied to the top platen along axial direction while the bottom platen is fixed. Symmetric boundary conditions are applied on the middle surface. The surface-to-surface contacts with friction are introduced to model the interaction between the platens and the specimen. In order to investigate the influence of friction on the compression deformation of the proposed specimen, four friction coefficients including 0, 0.1, 0.2 and 0.4 as well as rough friction ( = ∞) are introduced in the model. The material is assumed as homogeneous and isotropic elastic-plastic and follows the von Mises yield criterion. A Swift’s power law
Fig. 4 – (a) The load vs. displacement curves from simulations with different friction conditions; (b) the radius of cross section vs. displacement curves from FE simulations with different friction conditions.
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Fig. 5 – Contours of equivalent plastic strain distributions and radial displacement of the deformed specimen from simulation with different friction condition: (a) and (b) = 0; (c) and (d) rough friction.
4.
Results and discussion
4.1.
Influence of friction on the stress correction
Fig. 4(a) and Fig. 4(b) show the load vs. displacement curves and radius of cross section vs. displacement curves obtained from the simulations of the hyperbolic shaped compression specimen with the friction coefficients = 0, = 0.1, = 0.2 and = 0.4 as well as rough friction. Here, the simulations were conducted based on the hardening curve generated with 0 = 200 MPa and n = 0.1. As can be seen from Figs. 4(a) and 4(b) that the simulations with different friction conditions give similar values of load and radius of cross section corresponding to the same displacement, whose maximum deviations
do not exceed 0.3 %. In other words, for a given material hardening curve, the compression test using the proposed specimen presents the same average true stress vs. strain curve regardless of friction condition between the specimen and platens. Figs. 5(a)-5(d) show the contours of equivalent plastic strain distributions and radial displacement of the deformed specimen from simulation with = 0 and rough friction, respectively. As expected, all of the plastic deformation are localized in the middle region of the specimen while no deformation occur in the shoulders. Hence, there are no horizontal force acting on the end surfaces of the specimen, the friction condition does not affect the stress state of the middle region of specimen. Moreover, the distribution of the radius displacement and equivalent plastic strain for the deformation specimen are consistent corresponding to
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the same displacement for the friction condition of = 0 and rough friction. Namely, the existence of friction or not does not affect the whole compression deformation of the proposed specimen. Therefore, in the case of determination of hardening curve using the hyperbolic shaped compression specimen, the friction condition has none influence on the final results.
4.2. Influence of material properties on the stress correction Fig. 6 shows the material equivalent stress vs. strain curves (black line) as well as the average true stress vs. strain curves calculated by Eqs. (1) and (2) (other color lines), where the material equivalent stress vs. strain curves are generated based on the various combinations of strain hardening exponents varied from 0.075 to 0.225 and initial yield stress varied from 200 MPa to 400 MPa. It can be clearly observed from Fig. 6, for the hyperbolic shaped compression specimen, the average true stress calculated based on the direct application of Eq. (1) results in significant errors compared with the material equivalent stress from the very beginning of deformation due to the un-uniform stress state. For the materials with strain hardening exponents varied from 0.075–0.15, the average true stress vs. strain curves lie above the corresponding material equivalent stress vs. strain curves initially, and then have intersections with material equivalent stress vs. strain curves. As for the remaining materials with strain hardening exponents varied from 0.175–0.225, the values of average true stresses are always larger than those of material equivalent stresses corresponding to the same strain. To reflect the evolution of the deviations directly, the ratios between material equivalent stress eq,m and average true stresses ave,t corresponding to the same strain are plotted against the strain for the materials with various combinations of initial yield stresses and strain hardening exponents, as shown in Fig. 7. As can be seen the eq,m /ave,t vs. strain curves from materials with diverse initial yield stresses are collapsed into one curve, while that from materials with different strain hardening exponents are varied from each other. This phenomenon proves that the stress correction for the hyperbolic shaped compression specimen is solely dependent on strain hardening exponent but has none relationship with the initial yield stress.
4.3. Derivation of the stress correction method based on FEM 4.3.1. Dependencies of stress correction on strain hardening exponent The purpose of this section is to derive a simple correction function to convert the average true stress directly determined from the hyperbolic shaped compression specimen to the material equivalent stress. The stress correction function are composed of a series of stress correction factors, each of them
Fig. 6 – The material equivalent stress–strain curves (black line) as well as the average true stress vs. strain curves (other color lines): (a) 0 =200 MPa; (a) 0 =300 MPa; (a) 0 = 400 MPa.
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Fig. 7 – The ratios between material equivalent stress eq,m and average true stresses ave,t plotted against the strain for the materials with strain hardening exponents varied from 0.075-0.225 and initial yield stress of 200MPa-400 MPa.
Table 1 – The slope K and intercept b determined from the linear fitting of the relationship between the eq,m / ave,t and strain for material with strain hardening exponents varied from 0.075-0.225. Strain hardening exponent n
Slope K
Intercept b
0.075 0.1 0.125 0.15 0.175 0.2 0.225
0.18419 0.15935 0.13523 0.11045 0.08568 0.06037 0.03714
0.88561 0.89405 0.90118 0.9077 0.91357 0.91894 0.92331
is equal to the ratio of the material equivalent stress eq,m and the average true stress ave,t corresponding to the same strain. The evolution of stress correction factor eq,m /ave,t with strain have been previously depicted in Fig. 7. As can be seen the values of eq,m /ave,t varies linearly respect to the strain for all of the selected materials. Therefore, the relationship between stress correction factor and strain can be fitted through linear regression, i.e. the stress correction function, as shown in Fig. 7. The slope K as well as the intercept b in linear regression model are firstly determined for each strain hardening exponent, as listed in Table 1. Obviously, the slope K in the linear regression model increases with the decrease of strain hardening exponent, while the evolution of intercept b shows an opposite tendency. Both of the two parameters are then plotted against strain hardening exponent, respectively. As can be seen from Fig. 8 (a), the slope K presents linear variation respect to the strain hardening exponent. Additionally, the relationship between intercept b and strain hardening exponent can be fitted through a second-order polynomial, as shown in Fig. 8 (b). Therefore, the dependencies of parameters K and b on the strain hardening exponent n can be directly
Fig. 8 – (a) Linear fitting of the relationship between the slope K of the stress correction function and strain hardening exponent; (b) polynomial fitting of the relationship between the intercept b of the stress correction function and strain hardening exponent.
determined through linear regression and polynomial regression, respectively. Thus, the stress correction function can be expressed as following: f (ε) = K ∗ ε + b
K = −0.9838 ∗ n + 0.25791
(5)
b = 0.85718 + 0.42439 ∗ n + 0.58 ∗ n2
4.3.2. Correlating strain hardening exponent with displacement In the derivation of the stress correction function, the strain hardening exponent plays a decisive role. In the numerical experiments, such a parameter is arbitrary to be given. However, in the actual application of the stress correction for the proposed specimen in compression test, the strain hardening exponent of material is unknown. Therefore, we hope to obtain such a material parameter from the compression test using the hyperbolic shaped compression specimen. In the previous section, we have concluded that the compression deformation of the hyperbolic shaped compression specimen is merely affected by strain hardening exponent, showing none relation with friction and initial yield stress.
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Fig. 9 – Contours of axial displacement of the specimen compressed to average true strain of∼0.2 for material with different strain hardening exponents: (a) n = 0.075 (b) n = 0.175; (c) n = 0.225.
That is to say, when the specimen is compressed to the same average true strain, i.e. the same radius of middle cross section, the displacement and geometrical configuration of the specimen for materials with different strain hardening exponent should be diverse and solely dependent on strain hardening exponent. Fig. 9 shows the contours of axial displacements of deformed specimen obtained from the simulation of the hyperbolic shaped compression specimen for the materials with strain hardening exponents of ∼0.075, ∼0.175 and ∼0.225, which were compressed to the same average true strain of ∼0.2, corresponding to the radius of∼3.3155. As can be seen the specimens with different strain hardening exponents finally lead to different axial displacement corresponding to the same average true strain. The displacement is plotted against the average true strain for the material with strain hardening exponents varied from 0.075–0.225, as shown in Fig. 10(a). Similarly, the displacement vs. strain curves related to different strain hardening exponents exhibit different growth trends. For a given strain hardening exponent, the displacement increases with the increase of the strain, showing good linear variation in the strain range of less than 0.4, simultaneously. For an arbitrary strain in the range of 0-0.4, the displacements corresponding the chosen average true strain can be plotted against strain hardening exponents, as shown in Fig. 10(b). Here, the value of average true strain is selected as 0.2. As expected, the strain hardening exponent varies linearly respect to the displacement. The dependencies of strain hardening exponent on the displacement can be finally determined through linear regression as following: n = 0.53752 ∗ L − 0.37375 |ε=0.2
(6)
Therefore, for the hyperbolic shaped compression specimen, the strain hardening exponent can be directly determined by using Eq. (6). Such a method merely requires
Fig. 10 – (a) Displacement vs. average true strain curves for the material with strain hardening exponents varied from 0.075–0.225; (b) linear fitting of the relationship between strain hardening exponent and displacement corresponding to the average true strain of ∼0.2.
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Fig. 11 – The average true stress vs. strain curves, the material equivalent stress vs. strain curves inputted into the FE simulation and the corrected equivalent stress vs. strain curve using the proposed stress correction method for different materials: (a) n = 0.075; (b) n = 0.1; (c) n = 0.125; (d) n = 0.15; (e) n = 0.175;(f) n = 0.2; (g) n = 0.225.
the experimental measurement of the axial displacement and radius evolution of middle cross section of the compression specimen. The additional use of tension test with cylindrical specimen can be avoided effectively.
4.3.3.
The proposed stress correction function
In the above two section, we firstly derived the stress correction function related to strain and determined the dependency of the parameters of the function on strain hardening expo-
Fig. 12 – (a) The photographs of experimental apparatuses of the compression test; (b) the initial snapshot of the hyperbolic shaped compression specimen. Please cite this article in press as: Chen J, et al. A novel method to determine full-range hardening curve for metal bar using hyperbolic shaped compression specimen. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.01.003
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Table 2 – Chemical composition (wt. %) of Q420. Grade
C
Si
Mn
P
S
Fe
Q420
0.067
0.054
0.346
0.021
0.013
Bal.
nent, i.e., Eq.(5). Then, we proposed the measurement method for determining the unknown strain hardening exponent, i.e., Eq. (6). Thus, inserting Eq. (6) into Eq. (5), the final stress correction method for converting the average true stress into equivalent stress can be rewritten as following: eq = ave,t ∗ f (ε)
⎧ f (ε) = K ∗ εave,t + b ⎪ ⎪ ⎪ ⎪ ⎨ K = −0.9838 ∗ n + 0.25791 ⎪ b= ⎪ ⎪ ⎪ ⎩
(7)
0.85718 + 0.42439 ∗ n + 0.58 ∗ n2
n = 0.53752 ∗ L − 0.37375 |ε=0.2
The function proposed in this study is quite easy to apply for two reasons. First of all, the format of the stress correction function is explicitly expressed as Eq. (7). Secondly, the proposed method only requires the measurement of the load vs. displacement curves and the evolution of radius of middle cross section, which are easily obtained from a compression test using the hyperbolic shaped compression specimen.
4.3.4. Simulated verification of the stress correction method In order to verify the proposed stress correction method, it is firstly executed based on the previous simulation results of the hyperbolic shaped compression specimen. The corrected equivalent stress vs. strain curves (red line) transferred from the average true stress vs. strain curves (green line) using Eq. (7) are compared with the material equivalent stress vs. strain curves (black line), as shown in Fig. 11. Very satisfactory agreement can be seen up to strain of ∼0.8 for all the materials except for the material with strain hardening of 0.225. The maximum errors at ε = 0.8 for the material with strain hardening of 0.225 exhibits less than 1 %, which is still in an acceptable level from the view of engineering application. The results indicate that the hyperbolic shaped compression specimen can determine the hardening curve in large range of strains effectively using the associate stress correction method.
5. Experimental verification of the proposed method Since the hyperbolic shaped compression specimen and the corresponding stress correction methods are executed through FEM, which should be further verified with experimental data. The material adopted in current study is mild steel Q420. The chemical composition (weight percent) of Q420 is summarized in Table 2. The compression specimens were machined from metal bar according to the specific dimensions depicted in Fig. 1(d). A universal testing machine (SHIMADZU, AGX) with a load capability of 100 K N was used to conduct
Fig. 13 – (a) The average true strain vs. displacement curve obtained from compression test using the hyperbolic shaped compression specimen; (b) the average true stress vs. strain curve and the equivalent stress vs. strain curves corrected with proposed method; (c) the experimental load vs. displacement curve and the simulated load vs. displacement curves obtained from the simulation based on the average true stress vs. strain curve and the equivalent stress vs. strain curves determined with the proposed method.
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Fig. 14 – (a) The deformed specimen from simulation using the hardening curve determined from the proposed stress correction method; (b) the snapshot of deformed specimen from compression test corresponding to the same displacement (∼4.5 mm) with (a).
the compression test in a constant velocity of 1 mm/min at ambient temperature, as shown in Fig.12 (a). A charge-coupled CCD camera (KOWA-LM16JC1MS) was applied to capture the images of deformed specimen and record the displacement within two gauge marks as well as the radius of middle cross section, simultaneously. The initial snapshot of the specimen is shown in Fig. 12(b). The proposed stress correction method based on FEM was then applied to correct the average true stress to the equivalent stress for Q420. The equivalent stress vs. strain curves determined with the proposed method and the average true stress vs. strain curve were inputted into the FE simulation, outputting the simulated load vs. displacement curves, finally compared with the experimental one. Fig.13 (a) shows the average true strain vs. displacement curve obtained from compression test using the hyperbolic shaped compression specimen. As expected, the displacement varies linear with average true strain in the strain range of 0-0.4, which corresponds to the numerical experiment results described previously. The displacement is identified as 1.01303 corresponding to the average true strain of∼0.2. With the application of Eq. (6), the strain hardening exponent of Q420 is finally identified as 0.17077. Once the strain hardening exponent is known, the average true stress vs. strain curve (green line) can be corrected to equivalent stress vs. strain curve (black line) via Eq. (7), as shown in Fig. 13(b). The simulated load vs. displacement curves, which are outputted from FE simulations based on the hardening curves with and without stress correction, are compared with the experimental load vs. displacement curve (triangle dot), as shown in Fig. 13(c). For the hardening curve without stress correction (green line), there are large deviations between the simulated and experimental load vs. displacement curves from the very beginning of compression deformation, indicating the inadequate of the direct application of Eq. (1) in the determination of hardening curve. While the predicted load vs. displacement curve from the hardening curve corrected with the proposed method (red line) presents good
agreement with the experimental load vs. displacement curve, leading to an exact solution from engineering point of view. Fig. 14(a) shows the deformed specimen from FE simulation using the hardening curve determined from the proposed stress correction method, which is similar to that from the actual compression test corresponding to the same displacement of ∼4.5 mm (Fig.14(b)). Thus, it is confirmed that the hyperbolic shaped compression specimen and the proposed stress correction method are capable of obtaining the hardening in large range of strain precisely without the effect of friction.
6.
Conclusion
In this study, a hyperbolic shaped compression specimen is proposed to obtain the hardening curve in large range of strains without the effect of friction. Through extensive numerical analyses, the associated stress correction method is derived to convert the average true stress obtained directly from compression test of the proposed specimen to equivalent stress. The proposed specimen and stress correction method are verified with experimental data of mild steel Q420. The conclusions can be drawn as follows: 1 The hyperbolic shaped compression specimen can not only remove the undesirable effect of friction but also determine the hardening curve in large range of strains. 2 The established stress correction method for the new compression specimen is solely dependent on the strain hardening exponent, which can provide accurate hardening curve merely based on the experimental measurement of load vs. displacement curve and radius of middle cross section. 3 The hyperbolic shaped compression specimen provides a feasible method to determine the strain hardening exponent of material based on the displacement vs. average true strain curve.
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Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Acknowledgments The National Key Research and Development Program of China (No. 2018YFB1309203) and the Natural Science Foundation of China (Nos. 51575230 and U1810208) are greatly acknowledged. Partial financial support came from the Science and Technology Development Program of Jilin Province (No. 20190302059GX).
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Please cite this article in press as: Chen J, et al. A novel method to determine full-range hardening curve for metal bar using hyperbolic shaped compression specimen. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.01.003