- Open Access
Dynamic mechanical response and a constitutive model of Fe-based high temperature alloy at high temperatures and strain rates
© Su et al. 2016
Received: 3 November 2015
Accepted: 14 April 2016
Published: 23 April 2016
The effects of strain rate and temperature on the dynamic behavior of Fe-based high temperature alloy was studied. The strain rates were 0.001–12,000 s−1, at temperatures ranging from room temperature to 800 °C. A phenomenological constitutive model (Power-Law constitutive model) was proposed considering adiabatic temperature rise and accurate material thermal physical properties. During which, the effects of the specific heat capacity on the adiabatic temperature rise was studied. The constitutive model was verified to be accurate by comparison between predicted and experimental results.
Steel alloys are used in a wide range of structural, naval, nuclear, and aerospace applications (Zaera et al. 2012), which have been widely investigated in machining process. Martensitic stainless steels demonstrate a high capability of energy absorption, making it suitable for military applications, where impacts and explosions are often involved (Abed et al. 2014). When it comes to the manufacturing of steam turbine important components, the Fe-based high temperature alloy shows excellent properties under extreme working conditions. The performance of steel alloys under high temperature should be carefully considered during the design and manufacturing process.
High-speed cutting is used for manufacturing steam turbine rotor blades for ultra-supercritical unit. The high cutting temperature generated during the cutting evidently influences tool wear, tool life, surface integrity, and chip formation. The high temperature also leads to the thermal deformation of the cutting tool, which is considered as the major source of error in the machining process (Abukhshim et al. 2006; Takeuchi et al. 1982; List et al. 2012; Özel and Altan 2000).
Power-Law model is controlled by the synergistic effects of temperature, strain and strain rate. During the last decades, many researchers used the experimental stress–strain data from isothermal hot compression tests over a wide range of temperatures (1073–1473 K), but comparatively low strain rates (0.001–1 s−1) (Samantaray et al. 2009). Some researchers conducted quasi-static and high strain rate (up to 4500 s−1) experiments, whereas the high temperature environment was not considered (Pereira et al. 2001). Recently, a few investigations have been done to study the modeling process fully considering a wide range of strain rates and temperatures (Hor et al. 2013a, b; Lee et al. 2011; Hernandez et al. 2011; Wang et al. 2009).
Chemical composition of the Fe-based high temperature alloy (wt%)
Basic mechanical properties of the Fe-based high temperature alloy
Tensile yield stress (MPa)
Tensile strength (MPa)
Reduction of area
Rockwell hardness (HB)
Charpy V impact energy (J)
Quasi-static loading experiments
Dynamic loading experiments
The dynamic compression experiments were performed by the modified Split Hopkinson Pressure Bar with synchronically assembled heating system. The experimental parameters variables were the temperatures and the strain rates: experiments with strain rate of 9000 s−1 at temperatures ranging from 20 to 800 °C, and experiments with strain rates ranging from 4000 to 12,000 s−1 at room temperature.
Each experiment condition is performed at least three times in order to ensure the reliability of the experimental results. Only the results with good repeatability or less scatter can be accepted.
Results and discussion
As shown in Fig. 6, the specific heat capacity increases with temperature ranging from 20 to 700 °C, on the contrary while the temperature is above 700 °C.
As analyzed before, the flow stress decreases with the increase of temperature. However, the flow stress is balanced by work hardening and adiabatic thermal softening, which could be well exhibited in Power-Law model (Ranc et al. 2008).
Determination of the constants
Material constants of constitutive model
−6.0764e − 3
3.4791e − 5
−8.9607e − 8
9.9440e − 11
−3.9930e − 14
Verification of the constitutive model
It can be seen from Fig. 9a that predicted results well correlate with the experimental results. The errors under high strain rates and high true stain are <5 %. The errors between predicted results and experimental results at strain rate 9000 s−1 with temperature ranging from room temperature to 800 °C are larger than that at room temperature with high strain rates, as is shown in Fig. 9b. But the errors are <10 %. So, the Power-Law constitutive model proposed in this study is accurate enough to describe the dynamic mechanical response of the steel under high strain rate and high temperature conditions.
Numerical study of metal cutting process
In order to explain the practicability and accuracy, orthogonal metal cutting process was carried out by both experiments and the finite element method (FEM). The main cutting forces were taken as the research object.
Orthogonal metal cutting experiment
Design of experiments by the orthogonal array
The work length of the insert is 1 mm in keeping with the settings of simulation. Kistler dynamometer typed 9257B was used to collect the cutting force. Nine groups of experiments were carried out, and the parameters are shown in Table 4. By this way, the workload was reduced and experimental effect was guaranteed.
Numerical study of metal cutting process
The simulation of orthogonal metal cutting process was performed using commercial finite element software Third Wave AdvantEdge 6.4, which is widely adopted in the machining industry to facilitate the automation of production. The constitutive model obtained in “Constitutive model” section was input into the software as the costumer materials. The other thermal and mechanical properties of work piece are set as temperature-dependent thermal property. For example, Fig. 6 and Eq. (8) show the specific heat capacity of material. The thermal conductivity and thermal expansion are also temperature-dependent thermal properties, which were tested accurately. The Young’s modulus was set as 211 GPa, and the Poisson’s ratio was 0.33. The properties of cutting tool and coating were offered by the material library in the software. By some trial simulations, the friction coefficient between cutting tool and work piece was set as 0.5.
Result and discussions
The values of simulation results are smaller than the experiment ones. The average error of cutting forces is 7.86 %, which indicates good data reproducibility, and the error is clearly labeled in Fig. 10. The simulated results are consistent with the experimental results, confirming that the model of simulation can provide an accurate prediction of the main cutting force. In other words, the constitutive model obtained in this study is practicable and accurate.
This paper investigated the dynamic mechanical response of the Fe-based high temperature alloy. The work hardening and temperature softening are significant at high strain rates and temperatures, and the results coincided with the actual machining process. The effects of the specific heat capacity on the adiabatic temperature rise was studied, it can be seen that the specific heat capacity affect the adiabatic temperature rise obviously, especially at high temperature (200–800 °C).
A Power-Law constitutive model considering accurate adiabatic temperature rise is obtained through regression analysis of the experimental data gathered from a modified Split Hopkinson Pressure Bar technique with synchronically assembled heating system. The constitutive model was verified to be accurate by comparison between predicted and experimental results. The main cutting forces obtained from the experiments and simulations were compared, of which the average error of cutting forces is 7.86 %, which indicates that the constitutive model obtained in this study is practicable and accurate.
All authors read and approved the final manuscript.
The work has been financially supported by National Science and Technology Major Project of the Ministry of Science and Technology of China (No. 2013ZX04009-022) and Beijing Natural Science Foundation Project (3152013).
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.
- Abed FH (2010) Constitutive modeling of the mechanical behavior of high strength ferritic steels for static and dynamic applications. Mech Time-Depend Mater 14(4):329–345View ArticleGoogle Scholar
- Abed F, Makarem F (2012) Comparisons of constitutive models for steel over a wide range of temperatures and strain rates. J Eng Mater Technol 134(2):021001View ArticleGoogle Scholar
- Abed FH, Ranganathan SI, Serry MA (2014) Constitutive modeling of nitrogen-alloyed austenitic stainless steel at low and high strain rates and temperatures. Mech Mater 77:142–157View ArticleGoogle Scholar
- Abukhshim NA, Mativenga PT, Sheikh MA (2006) Heat generation and temperature prediction in metal cutting: a review and implications for high speed machining. Int J Mach Tools Manuf 46(7–8):782–800View ArticleGoogle Scholar
- Guo W-G, Nemat-Nasser S (2006) Flow stress of Nitronic-50 stainless steel over a wide range of strain rates and temperatures. Mech Mater 38(11):1090–1103View ArticleGoogle Scholar
- Guo QM, Li DF, Guo SL (2012) Microstructural models of dynamic recrystallization in hot-deformed Inconel 625 superalloy. Mater Manuf Process 27(9):990–995View ArticleGoogle Scholar
- Gupta AK, Krishnamurthy HN et al (2013a) Development of constitutive models for dynamic strain aging regime in Austenitic stainless steel 304. Mater Des 45:616–627View ArticleGoogle Scholar
- Gupta AK, Anirudh VK, Singh SK (2013b) Constitutive models to predict flow stress in Austenitic stainless steel 316 at elevated temperatures. Mater Des 43:410–418View ArticleGoogle Scholar
- He A et al (2013) A comparative study on Johnson–Cook, modified Johnson–Cook and Arrhenius-type constitutive models to predict the high temperature flow stress in 20CrMo alloy steel. Mater Des 52:677–685View ArticleGoogle Scholar
- Hernandez C et al (2011) An inverse problem for the characterization of dynamic material model parameters from a single SHPB test. Proc Eng 10:1603–1608View ArticleGoogle Scholar
- Hor A et al (2013a) An experimental investigation of the behaviour of steels over large temperature and strain rate ranges. Int J Mech Sci 67:108–122View ArticleGoogle Scholar
- Hor A et al (2013b) Modelling, identification and application of phenomenological constitutive laws over a large strain rate and temperature range. Mech Mater 64:91–110View ArticleGoogle Scholar
- Hortig C, Svendsen B (2007) Simulation of chip formation during high-speed cutting. J Mater Process Technol 186(1–3):66–76View ArticleGoogle Scholar
- Hou QY, Wang JT (2010) A modified Johnson–Cook constitutive model for Mg–Gd–Y alloy extended to a wide range of temperatures. Comput Mater Sci 50(1):147–152View ArticleGoogle Scholar
- Kajberg J, Wikman B (2007) Viscoplastic parameter estimation by high strain-rate experiments and inverse modelling—speckle measurements and high-speed photography. Int J Solids Struct 44(1):145–164View ArticleGoogle Scholar
- Khan AS, Liu H (2012) Variable strain rate sensitivity in an aluminum alloy: response and constitutive modeling. Int J Plast 36:1–14View ArticleGoogle Scholar
- Khan AS, Sung Suh Y, Kazmi R (2004) Quasi-static and dynamic loading responses and constitutive modeling of titanium alloys. Int J Plast 20(12):2233–2248View ArticleGoogle Scholar
- Lee W-S et al (2011) Dynamic mechanical behaviour and dislocation substructure evolution of Inconel 718 over wide temperature range. Mater Sci Eng, A 528(19–20):6279–6286View ArticleGoogle Scholar
- Liang R, Khan AS (1999) A critical review of experimental results and constitutive models for BCC and FCC metals over a wide range of strain rates and temperatures. Int J Plast 15(9):963–980View ArticleGoogle Scholar
- Lin YC, Chen X-M (2011) A critical review of experimental results and constitutive descriptions for metals and alloys in hot working. Mater Des 32(4):1733–1759View ArticleGoogle Scholar
- Lindholm US (1964) Some experiments with the split hopkinson pressure bar. J Mech Phys Solids 12(5):317–335View ArticleGoogle Scholar
- List G, Sutter G, Bouthiche A (2012) Cutting temperature prediction in high speed machining by numerical modelling of chip formation and its dependence with crater wear. Int J Mach Tools Manuf 54–55:1–9View ArticleGoogle Scholar
- Merchant ME (1945) Mechanics of the metal cutting process. I. Orthogonal cutting and a type 2 chip. J Appl Phys 16(5):267View ArticleGoogle Scholar
- Mirza FA et al (2013) A modified Johnson–Cook constitutive relationship for a rare-earth containing magnesium alloy. J Rare Earths 31(12):1202–1207View ArticleGoogle Scholar
- Özel T, Altan T (2000) Determination of workpiece flow stress and friction at the chip—tool contact for high-speed cutting. Int J Mach Tools Manuf 40(1):133–152View ArticleGoogle Scholar
- Pereira JM, Lerch BA (2001) Effects of heat treatment on the ballistic impact properties of Inconel 718 for jet engine fan containment applications. Int J Impact Eng 25(8):715–733View ArticleGoogle Scholar
- Ranc N et al (2008) Temperature field measurement in titanium alloy during high strain rate loading—adiabatic shear bands phenomenon. Mech Mater 40(4–5):255–270View ArticleGoogle Scholar
- Samantaray D et al (2009) A thermo-viscoplastic constitutive model to predict elevated-temperature flow behaviour in a titanium-modified austenitic stainless steel. Mater Sci Eng, A 526(1–2):1–6View ArticleGoogle Scholar
- Takeuchi Y, Sakamoto M, Sata T (1982) Improvement in the working accuracy of an nc lathe by compensating for thermal expansion. Precis Eng 4(1):19–24View ArticleGoogle Scholar
- Wang QZ, Li W, Xie HP (2009) Dynamic split tensile test of Flattened Brazilian Disc of rock with SHPB setup. Mech Mater 41(3):252–260View ArticleGoogle Scholar
- Wang X et al (2013) Dynamic behavior and a modified Johnson–Cook constitutive model of Inconel 718 at high strain rate and elevated temperature. Mater Sci Eng, A 580:385–390View ArticleGoogle Scholar
- Yu J et al (2014) Numerical study the flow stress in the machining process. Int J Adv Manuf Technol 74(1–4):509–517View ArticleGoogle Scholar
- Zaera R et al (2012) A constitutive model for analyzing martensite formation in austenitic steels deforming at high strain rates. Int J Plast 29:77–101View ArticleGoogle Scholar