Description of fullrange strain hardening behavior of steels
 Tao Li^{1},
 Jinyang Zheng^{1, 3, 4}Email author and
 Zhiwei Chen^{2}
Received: 22 March 2016
Accepted: 3 August 2016
Published: 11 August 2016
The Erratum to this article has been published in SpringerPlus 2016 5:2110
Abstract
Mathematical expression describing plastic behavior of steels allows the execution of parametric studies for many purposes. Various formulas have been developed to characterize stress strain curves of steels. However, most of those formulas failed to describe accurately the strain hardening behavior of steels in the full range which shows various distinct stages. For this purpose, a new formula is developed based on the wellknown Ramberg–Osgood formula to describe the full range strain hardening behavior of steels. Test results of all the six types of steels show a threestage strain hardening behavior. The proposed formula can describe such behavior accurately in the full range using a single expression. The parameters of the formula can be obtained directly and easily through linear regression analysis. Excellent agreements with the test data are observed for all the steels tested. Furthermore, other formulas such as Ludwigson formula, Gardner formula, UGent formula are also applied for comparison. Finally, the proposed formula is considered to have wide suitability and high accuracy for all the steels tested.
Keywords
Background
The description of strain hardening behavior of materials using mathematical expression has been the subject of numerous investigations for many years. Strain hardening response of materials is usually characterized indirectly by the true stress–strain curves obtained from tensile tests. Typically, the strain hardening rate can be calculated numerically from the curves and plotted against strain (or stress). It is now well established that the hardening rate of crystals may be divided into various distinct stages (Nabarro et al. 1964; Asgari et al. 1997; Chinh et al. 2004), typically three stages, labeled Stage I, Stage II and Stage III (KuhlmannWilsdorf 1985). The stages of polycrystalline steels are much less evident than those of the single crystal (Reedhill et al. 1973). Therefore, some forms of analysis are normally to describe the strain hardening behavior of steels. For this purpose, the Ramberg–Osgood formula (Ramberg and Osgood 1943) has been used widely for steels in various engineering fields. However, this formula is inherently deficient to describe the strain hardening behavior of steels in the full range.
Distinct stages strain hardening behavior has been observed in various types of steels (Jha et al. 1987; Nie et al. 2012; Umemoto et al. 2000; Tomita and Okabayashi 1985; Atkinson 1979; Kalidindi 1998; Saha et al. 2007). Many formulas were designed to describe the fullrange hardening and some materialspecific formulas have been proposed for stainless steels (Rasmussen 2003; Gardner and Nethercot 2004a, b; Abdella 2006; Quach et al. 2008; Arrayago et al. 2015), TRIP steels (Tomita and Iwamoto 1995), high strength steels (Gardner and Ashraf 2006) and pipeline steels (Hertelé et al. 2012a, b). Although excellent agreement has been provided for specific materials, the formulas have difficulty being adopted for other materials. Additionally, it should be noted that the strain hardening behavior involves a complex interaction among various factors. At the microscale, this aspect of plastic deformation is intrinsically coupled with all other aspects of plastic deformation such as development of preferred lattice orientations, formation of subgrains, and formation of local shear bands (Wilson 1974). For austenitic steels and TRIP steels, the microstructural phase transformation from austenite to martensite also has a great effect on the plastic deformation. (Leblond et al. 1986a, b; Hallberg et al. 2007; Santacreu et al. 2006; Post et al. 2008; Stringfellow et al. 1992; Bhattacharyya and Weng 1994; Diani et al. 1995; Miller and McDowell 1996; Papatriantafillou et al. 2006; Turteltaub and Suiker 2005; Beese and Mohr 2012; Iwamoto and Tsuta 2000). This has been actively studied for decades. Therefore, it is virtually impossible to develop a complete understanding (Chinh et al. 2004) of the behavior, and no unified theory on the physically based functional description has been found (Cleri 2005). Most of these formulas to describe the strain hardening behavior of steel are purely empirical descriptions.
The purpose of this paper is to present a mathematical description of the fullrange strain hardening behavior for steels with smooth, gradual onset of yielding. Note that many mathematical descriptions have already existed, an overview of existing stress–strain formulas and an expression of the new formula are provided in “Formulas characterizing stress strain curves” section. Test data of various types of steels were referred to in “Test data” section. “Validation and comparison” section validated the proposed formula with test data and comparisons with other formulas were also listed. Then, a limitation of the proposed formula is discussed in “Discussion” section. Finally principal conclusions are drawn in “Conclusion” section.
Formulas characterizing stress strain curves
Overview of existing formulas
The description of the stress–strain curves of metals by mathematical expressions has been a topic of research since the origin of classical mechanics. Numerous formulas have been proposed to describe the stress–strain curves. Osgood (1946) summarized 17 formulas used in the early age of study. Kleemola and Nieminen (1974) discussed the computational method of parameters for some commonly used formulas. Recently, existing common formulas have been reviewed and discussed by Hertelé et al. (2011).
The deficiency of this formula is also very clear: it cannot provide an explicit expression of σ − ε and could have difficulties in describing the smooth, gradual onset of yielding observed in many metallic materials (Hertelé et al. 2011).
The UGent stress–strain model was developed to describe the strain hardening behavior of pipeline steels with two distinct stages. As listed in Eq. (7), for small plastic regions σ ≤ σ _{1}, the UGent model respects a Ramberg–Osgood equation with a true 0.2 % proof stress σ _{0.2} and a first strainhardening exponent n _{1}; for large plastic region σ ≥ σ _{ 2 }, the UGent model respects a Ramberg–Osgood equation with the same 0.2 % proof stress σ _{0.2}, but a possibly different strainhardeing exponent n _{2}; Between these two regions, there is a smooth transition where the curve shape gradually changes.
The deficiency of the UGent formula is that it is too complicated to apply in practice and the parameters are difficult to obtain.
Proposed stress–strain formula

In the small scale yielding plastic area, a Ramberg–Osgood formula with m_{1}, k_{1} is assumed to be followed, defined as ε_{p1}σ line in Fig. 1a. The parameters can be easily obtained through a linear regression analysis as Eq. (9) in the log(ε_{p}) − log(σ) coordinate.$$ m_{1} \cdot \log \varepsilon_{p} + \log K_{1} = \log \sigma $$(11)

In the large scale yielding plastic area, a Ramberg–Osgood formula with m_{2}, k_{2} should be followed, defined as ε_{p2} − σ line in Fig. 1a. The parameters can also be easily obtained through a linear regression analysis in the same way through Eq. (10):$$ m_{2} \cdot \log \varepsilon_{p} + \log K_{2} = \log \sigma $$(12)

In the transition between these two curves mentioned above, the ratio value of ε_{p} − ε_{p1} to ε_{p2} − ε against stress shows a linear relation of Eq. (13), in the coordinate depicted in Fig. 1b. The parameters A, B can be obtained through a linear regression analysis directly.$$ \ln \frac{{\varepsilon_{p}  \varepsilon_{p1} }}{{\varepsilon_{p2}  \varepsilon_{p} }} = A\sigma + B $$(13)
Test data
Tensile characteristics of the steels
Materials  Brand  Elastic modulus (MPa)  Yielding strength R_{p0.2} (MPa)  Tensile strength R_{m} (MPa)  R_{p0.2}/R_{m}  Uniform elongation 

Gun steels  PCrNi3MoVA  215,000  962  1081  0.890  0.066 
G4335V  212,000  972  1160  0.838  0.068  
32CrNi3MoVA  201,000  985  1115  0.883  0.069  
Pipeline steel  API X70  203,700  521  606  0.860  0.085 
TRIP steel  TRIP 690  204,900  493  719  0.686  0.196 
Stainless steel  DIN 1.4662  208,100  490  728  0.673  0.181 
Fitting parameters of other formulas
Formula  Parameters/dimension  G4335V  PCrNi3MoVA  32CrNi3MoVA  Pipeline steel  TRIP steel  Stainless steel 

Ramberg–Osgood  K/MPa  1375  1293  1361  763  1015  1011 
m  0.0433  0.0482  0.0545  0.0642  0.1223  0.1234  
Ludwigson  K_{1}/MPa  1550  1467  1534  832  1180  1260 
m_{1}  0.079  0.085  0.090  0.0915  0.181  0.212  
K_{2}  5.010  5.39  4.540  4.42  5.12  4.80  
m_{2}  −202.4  −296.3  −584.7  −230  −155  −38.1  
Ugent  σ_{0.2}/MPa  –  –  –  521  493  490 
n_{1}  –  –  –  26.5  12.4  5.11  
n_{2}  –  –  –  15.5  8.0  10.7  
σ_{1}/MPa  –  –  –  536  535  490  
σ_{2}/MPa  –  –  –  579  670  460  
Gardner  n  –  –  –  15.1  16.5  4.43 
E_{0.2}/10^{3} MPa  –  –  –  15.9  13.9  44.2  
n _{0.2,1.0} ^{′}  –  –  –  1.55  2.20  3.05 
Validation and comparison
The proposed formula has been applied to the test data of all the six steels. The optimal parameter values for each steel were obtained through the fitting procedure mentioned above (“Proposed stress–strain formula” section). The general Ramberg–Osgood formula (Ramberg and Osgood 1943), Ludwigson formula (Ludwigson 1971), UGent formula (Hertelé et al. 2011) and a materialspecific Gardner formula (Gardner and Nethercot 2004a, b) have also been applied to the data for comparison.
Parameters of the proposed formula
Materials  Parameters of the proposed formula  

A  B  K_{1} (MPa)  m_{1}  K_{2} (MPa)  m_{2}  
G4335V  0.0715  −78.544  1429.39  0.0446  1549.67  0.0789 
PCrNi3MoVA  0.0801  −80.675  1301.3  0.0457  1466.5  0.0845 
32CrNi3MoVA  0.0631  −64.759  1630.76  0.0749  1534.41  0.0899 
Pipeline steel  0.1414  −77.722  700.49  0.0471  815.27  0.0849 
TRIP steel  0.0491  −27.643  897.02  0.0931  1173.82  0.1785 
Stainless steel  0.0391  −24.016  1274.68  0.1558  1247.96  0.2088 
It can be observed from those figures that: First, test data of all the steels show a threestage hardening behavior which can be seen clearly in the strain hardening ratestrain coordinate. Stage I ends at approximately ε = 0.02 for stainless steel and ε = 0.01 for others; Stage II ends at roughly at ε = 0.06 for stainless steel and ε = 0.02 for others.
The difference in the strain hardening rate can be attributed to the operation of different deformation mechanisms (Kocks and Mecking 2003; MontazeriPour and Parsa 2016): Stage I exhibits a distinct decline hardening rate. The sudden drop of hardening rate is associated with crossslip of dislocations bypassing the heads of piled up dislocations (Hockauf and Meyer 2010). After passing the initial Stage I, hardening rate decreases to another region with a constant value defined as Stage II. Stage II exhibits an almost constant hardening rate behavior which is contributed to a steady state for storage and annihilation of dislocations (Zehetbauer and Seumer 1993). After Stage II, the hardening rate decreases continuously into a separate Stage III up to necking point (Kocks and Mecking 2003). Features of Stage III are analogous to Stage I and are considered to be connected with point defect generation and absorption (Zehetbauer and Seumer 1993).
Second, linear relationship assumed in the fitting procedure of the proposed formula is verified for all the test data. The proposed formula provides satisfactory representations of the test data for all the six steels in the full range. It can characterize excellently the threestage strain hardening behavior of steels observed in the test. Six parameters of the formula, all of which are easy to understand and interpret in an intuitive way, can be obtained directly and easily through linear regression.
Third, for other formulas, it can be found that: The Ludwigson formula generally seems to provide accurate description of all curves for large plastic strain, e.g. Stage III, but lacks accuracy at a lower strain, below 0.02 for gun steels and stainless steel. This formula also cannot be utilized directly because there is no explicit expression of strain. The Gardner formula, on the other hand, seems to provide an accurate description of the full range curve for stainless steel and the lower strain parts for pipeline steel up to 0.035 and TRIP steel up to 0.07. The UGent formula provides an accurate description of pipeline steel and TRIP steel up to plastic regions near necking but lack accuracy for stainless steel. The fitting procedure of UGent formula is cumbersome and some parameters are arbitrary.
Discussion
Limitations of the proposed formula are discussed in this section. First, obviously the proposed formulas cannot be utilized to describe the strain hardening behavior of steels with a sharp or specific yielding strength, which can be observed in some carbon steels.
Second, as mentioned in “Test data” section, in this paper the strain hardening response of materials is characterized by the stress–strain curves documented in tensile tests. The parts of the engineering stress–strain curves after necking were ignored due to the local necking effect. However, when extremely large deformation was mentioned, this procedure is not quite enough.
Third, to simplify the loading condition, quasistatic loading mode is considered in this paper. However, it is well known that temperature and strain rate have great effect on the plastic deformation behavior. More works are needed on these issues.
Conclusion
In the present paper, a new formula has been proposed to describe the full range strain hardening behavior of steels. The test results demonstrate that the test data of all the six steels observed have a threestage hardening behavior. The proposed formula, based on two different Ramberg–Osgood formulas, can characterize such behavior in the full range using a single expression. The parameters of the formula can be easily and directly obtained through linear regression analysis. The fitting curves and test results were identified to have excellent agreement for all the six steels.
Notes
Declarations
Authors’ contributions
JZ designed the research. TL performed the analysis and wrote the paper. ZC gave some good suggestions. All authors read and approved the final manuscript.
Acknowledgements
This research was supported by the Special funds for Quality Supervision Research in the Public Interest (“Research on Key Technologies for the Design Standard of UltraHigh Pressure Vessels”, Grant No. 201210242).
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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