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Research Papers

# A Phenomenological Constitutive Equation to Describe Various Flow Stress Behaviors of Materials in Wide Strain Rate and Temperature Regimes

[+] Author and Article Information
Hyunho Shin

Department of Materials Engineering, Gangneung-Wonju National University, Jibyun-dong, Gangneung, Gangwon-do 210-702, Republic of Korea

Jong-Bong Kim1

Department of Automotive Engineering, Seoul National University of Technology, 172 Gongneung2-dong, Nowon-gu, Seoul 139-743, Republic of Koreajbkim@snut.ac.kr

ORIGIN ® software (version 8.0725; OriginLab Corporation; http://www.originlab.com) was used to determine the material parameters. Isothermal assumption during the flow process of the material has been employed throughout the current work. Employing adiabatic assumption at strain rates higher than $1×103$ for the case when flow stress-strain relation is given (Figs.  678910) yielded a change in fitting parameters but no apparent change in the curves of the current model.

1

Corresponding author.

J. Eng. Mater. Technol 132(2), 021009 (Feb 18, 2010) (6 pages) doi:10.1115/1.4000225 History: Received March 17, 2009; Revised September 08, 2009; Published February 18, 2010; Online February 18, 2010

## Abstract

A simple phenomenological constitutive model has been proposed to describe dynamic deformation behavior of various metals in wide strain rate, strain, and temperature regimes. The formulation of the model is, $σ=[A+B{1−exp(−Cε)}][D ln(ε̇/ε̇0)+exp(E⋅ε̇/ε̇0)][1−(T−Tref)/(Tm−Tref)]m$, where $σ$ is the flow stress, $ε$ is the strain, $ε̇$ is the strain rate, $ε̇0$ is the reference strain rate, $T$ is the temperature, $Tref$ is the reference temperature, $Tm$ is the melting temperature, and $A$, $B$, $C$, $D$, $E$, and $m$ are the material parameters. The proposed model successfully describes not only the linear rise of flow stress with logarithmic strain rate for many metals, but also the upturn of the flow stress at strain rate over about $104 s−1$ for the case of copper. It can also describe the exponential increase in the flow stress with logarithmic strain rate for the case of tantalum, and is capable of predicting thermal softening of various metals at high as well as low temperature. The current model can be used for the practical simulation of many high-strain-rate events with improved precision and as a more rigorous comparison standard in the development of a physical model.

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## Figures

Figure 1

Comparison of (a) the Ludwik hardening law (A=100 MPa, B=150 MPa) for varying n with (b) the Voce hardening law (A=100 MPa, B=240 MPa) for varying C

Figure 2

Comparison of (a) the Johnson–Cook strain-rate hardening term with (b) the current model’s counterpart, when (b) D=0.05 and (c) D=0.2

Figure 3

Comparison of (a) the Johnson–Cook thermal softening term with (b) the current model’s thermal softening term. For both models, Tm=1773 K and Tref=298 K.

Figure 4

Comparison of the current model (solid lines) with the experimental flow stress data (symbols) of copper with respect to logarithmic strain rate at 300 K and varying strain, adapted from Follansbee (16) (A=29.8 MPa, B=314.8 MPa, C=6.90, D=0.0124, E=9.44×10−6, and m=2.106, and Tm=1357.8 K for the current model)

Figure 5

Comparison of the current model and the JC model with the initial yield stress data of tantalum with respect to strain rate at 298 K, adapted from Hoge and Mukhergee (17) (A=300 MPa, D=0, E=0.07362 for the current model, and A=300 MPa and C=0.10072 for the JC model)

Figure 6

Comparison of the current model (solid lines) with the shear stress data (symbols) of beryllium with respect to equivalent plastic strain at 573 K and varying strain rate, adapted from Montoya (22). A=116.217 MPa, B=142.139 MPa, C=18.156, D=0.0297, E=2.17×10−4, m=0.0268, and Tm=1551 for shear stress τ, and A=244.655 MPa, B=299.305 MPa, C=18.158, D=0.0297, E=2.17×10−4, m=0.7878 to predict equivalent plastic stress (flow stress, σ=3τ).

Figure 7

Comparison of the current model (solid lines) with the flow stress data (symbols) of uranium with respect to strain at the strain rate of 3500 s−1 and varying temperature, adapted from Armstrong and Wright (23) (A=161.57 MPa, B=859.49 MPa, C=7.8899, D=0.0987, E=0, m=2.106, and Tm=1405.2 K for the current model)

Figure 8

Comparison of the current model (solid lines) with the flow stress data (symbols) of copper with respect to strain at varying strain rate and temperature, adapted from Samanta (24) (A=33.648 MPa, B=97.240 MPa, C=14.464, D=0.1393, E=0, m=0.4643, and Tm=1357.8 K for the current model)

Figure 9

Comparison of the current model (solid lines) with the flow stress data (symbols) of AISI 4340 steel with respect to strain at varying strain rate and temperature, adapted from Lee and Yeh (25). (a) ε̇=500 s−1, (b) ε̇=1500 s−1, and (c) ε̇=2500 s−1 (A=1035.307 MPa, B=338.365 MPa, C=22.318, D=0.022, E=0, m=1.661, and Tm=1793 K for the current model).

Figure 10

Comparison of the current model (solid lines) with the flow stress data (symbols) of tungsten heavy alloy with respect to varying strain rate and temperature, adapted from Lee (26). (a) ε̇=800 s−1, (b) ε̇=1600 s−1, and (c) ε̇=2500 s−1 (A=66.426 MPa, B=24.769 MPa, C=13.080, D=2.1701, E=0, m=0.311, and Tm=1723 K for the current model).

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