Constitutive Equations of Yield Stress Sensitivity to Strain Rate of Metals : A Comparative Study

Several constitutive equations have proposed to model the strain rate sensitivity of metals to strain rate. This paper presents a comparative of six equations reported in the open literature. All equations are used to fit the yield stress of three copper materials and one steel material at two different temperatures. A specific cost function and an optimization problem are defined.The authors recommend the use of the Cowper-Symonds equation or a modified-Eyring equation as both of them fit well the experimental data while using only three material constants. A modified flow stress Johnson-Cook equation is then proposed for metallic materials.


Introduction
Metallic materials are largely used in several industrial fields, for example, aeronautical, naval, automobile, and military industries.In these applications, transportation vehicles have to be designed against impact loads.Thus, the characterization and modeling of metals' sensitivity to strain rate are highly important.The split Hopkinson bar is largely used to characterize materials in the high strain rate range [1,2] while the direct-impact Hopkinson bar is used at the very high strain rate [3,4].In terms of constitutive equations, the Johnson-Cook law [5,6] has been widely used to model the behavior of metallic materials including temperature or (and) strain rate effect(s) [3,[7][8][9].This constitutive equation separates the hardening, temperature, and strain rates.Namely, they are written in a multiplicative form.The strain rate effect is considered as varying linearly in terms of the logarithm of strain rate.
Several studies have showed that the linear variation of yield or flow stress in terms of strain rate is only valid in the quasi-static and intermediate strain rate ranges [10][11][12][13].However, there is a sharp increase in the strain rate sensitivity at high strain rates.This increase cannot be considered by the classical Johnson-Cook equation.Some modified Johnson-Cook equations have then been proposed [14].Huh and Kang [15] proposed a quadratic for the strain rate sensitivity.Tuazon et al. [16] expressed the dependence on the logarithm of strain rate as a power-law.Couque [17] proposed a modified Johnson-Cook equation where the strain rate sensitivity is written in terms of a four-constant equation.El-Qoubaa and Othman [18,19] have proposed a modified-Eyring equation for polymers yield stress sensitivity to strain rates.This model was successfully applied to several metallic materials in [20].
The aim of this work is to compare and discuss the above constitutive equations: the standard Johnson-Cook, Huh-Kang, Tuazon et al., Couque, and modified-Eyring equations.The pioneering Cowper-Symonds equation is also considered [21].In terms of strain rate dependence, it gives similar relation as The Zerilli-Armstrong equation [22].

Experimental Data.
In this work, we are interested in evaluating several constitutive equations.Thus, they are used here to fit the strain rate sensitivity of the yield stress of two metallic materials: steel and copper.These two materials are extensively characterized in the literature.In terms of the copper yield stress, we rely upon the experimental data of Couque [17].For the steel yield stress, we will rely upon the experimental data of Clarke et al. [11].))  and  are two material constants, and ε 0 = 1 s −1 is a reference strain rate Huh-Kang [15]   =  (1 +  1 log ( εε ,  1 ,and  2 are three material constants, and ε 0 = 1 s −1 is a reference strain rate Tuazon et al. [16]   =  (1 + log  ( εε 0 )) , , and  are three material constants, and ε 0 = 1 s −1 is a reference strain rate Couque [17]   =  (1 +  log ( εε , , , and  are four material constants, and ε 0 = 1 s −1 and ε 1 = 10 3 s −1 are two reference strain rates Cowper-Symonds [21] or Zerilli-Armstrong [22]   =  (1 +  ( εε  1.Here, we are only interested in the strain rate sensitivity.We should notice that the equations are sometimes changed from their original form for identification purposed.

Identification Procedure.
In this section, we depict the methodology followed in order to identify the material constants of each of the constitutive equations that are dressed in Table 1.Actually, each constitutive equation depends on a set of two, three, or four material constants.The standard Johnson-Cook model depends on only two material constants:  and .The Huh-Kang, Tuazon et al., Cowper-Symonds, and modified-Eyring equations depend each on three material constants.Finally, the Couque equation depends on four material constants.The main idea of this section is to find for each constitutive equation a set of material constants that reduces the difference between the experimental yield stresses and the yield stresses obtained by that equation.Let Ė = ( ε  ) be a vector that collects the experimental strain rate values which are collected from the literature as explained in Section 2, where ε  denotes the strain rate obtained for a test .Similarly, let Σ = (σ  ) be a vector which collects the yield stresses σ measured at strain rates ε  .Using a constitutive equation  from Table 1, it is possible to build a vector Σ  = (   ) which collects the yield stresses    that are calculated at strain rates ε  .In order to obtain the best material constants for each constitutive equation, we need to optimize a cost function   .This cost function is built in terms of the difference between the experimental and calculated yield stresses.Let ‖ ‖ 2 and ‖ ‖ ∞ be the Euclidean norm and the maximum norm, respectively.It is possible to define an error using the Euclidean norm.
More precisely, Likewise, it is possible to define an error using the maximum norm.Namely, The Euclidean norm-based error gives a measurement of the average difference between the experimental yield stress and the yield stress predicted by the considered constitutive equation.It can be considered as a global error measurement.On the opposite, the max norm-based error focuses on the tests where the maximum difference is encountered.It can then be considered as a local error measurement.
In this study the cost function is defined as the average between the Euclidean norm-based error and the maximum norm-based error: where  (4)     the yield strain at high strain rate.However, it predicts an increase of the yield stress in the quasi-static strain rate range because of the quadratic form of the constitutive equation.This increase at low strain rates was never reported in the open literature, to the best knowledge of the authors.Thus, it is considered here nonphysical.the logarithm of strain rate.More precisely, the expression log( ε / ε 0 ) in the standard Johnson-Cook equation is substituted by the expression log  ( ε / ε 0 ).The Tuazon et al. model fits quite well the experimental data of steel and copper (Figure 3).The error ranges between 3.4% (obtained with copper 26) and 15.2% (obtained with steel 293) (see Table 4).This equation can catch the increase in the strain rate sensitivity at high strain rates.However, it predicts a sharp drop in yield stress at low strain rates.This behavior was never reported in the literature, to the best of the authors knowledge, and thus considered here nonphysical.Hence, we rather recommend using the Tuazon et al. equation for strain rates higher than 10 −2 s −1 .[17] modified the standard Johnson-Cook equation by adding a third term which is written as a power of the strain rate and not the logarithm of strain rate, that is, ( ε / ε 1 )  .This equation fits well the experimental data of copper and steel (Figure 4) over the total strain range, that is, between 10 −3 s −1 and 5 × 10 4 s −1 .The model describes well the sharp increase in the yield stress which is recorded at high strain rate.The error is low and ranges between 2.8% for copper 105 and 6.2% for copper 9 (Table 5).

Cowper-Symonds
Model.The Cowper-Symonds equation uses a power equation of the strain rate.It fits well the experimental data for the copper materials and for the steel at  different temperatures (Figure 5).Particularly, it catches well the sharp increase in the yield stress at high strain rates.The error is quite low (Table 6).It ranges between 4.8% (obtained for copper 105) and 7% (obtained for copper 9).

Modified-Eyring Model.
The modified-Eyring model is developed by El-Qoubaa and Othman [18][19][20] based on the original work of Eyring [23] except that they used an activation volume decreasing with an increasing strain rate.
The model works well here with copper and steel over the studied strain rate range (Figure 6).It fits well the increase in yield stress and the increase in the strain rate sensitivity of the yield stress that is observed at high strain rate.The error is low and ranges between 5.8 obtained with copper 26 and 7.8% obtained with copper 9 (Table 7).

Comparison.
In the previous sections, six equations are used to fit the yield stress of three copper materials and one steel material at two temperatures.The standard Johnson-Cook equation gives the biggest error as mainly it cannot where  0 ,   ,  0 , ε  , , and  are six material constants and   , ,   ,   , ε , ε 0 , and   are the plastic strain, absolute temperature, the room temperature, the melting temperature, the strain rate, a reference strain rate, and the Boltzmann constant, respectively.In this equation, the standard Johnson-Cook flow stress equation is modified using the strain rate sensitivity as predicted for room temperature by the modified-Eyring equation, while keeping the original hardening and temperature effects as first written by the standard equation.

Conclusion
In this work, we have compared six constitutive equations that predict the strain rate sensitivity of metallic materials.They are mainly used to fit the strain rate sensitivity of the yield stress of three copper materials and a steel alloy at two different temperatures.It is recommended to use either the Cowper-Symonds equation or the modified-Eyring equation as both of them give a close fit to the experimental data while using only three material constants each.A modified Johson-Cook equation is then proposed by including the strain rate sensitivity predicted by the modified-Eyring equation in the standard modified Johnson-Cook flow stress equation.

3. 1 .
Standard Johnson-Cook Model.The standard Johnson-Cook equation is used to fit the compression yield stress of three copper materials (Figure 1(a)) and for steel at two temperatures (Figure 2(b)).The material constants, the errors, and the correlation coefficient are calculated and depicted in

Figure 1 :Figure 2 :
Figure 1: Yield stress fitting by the standard Johnson-Cook model: (a) copper and (b) steel.

Figure 4 :
Figure 4: Yield stress fitting by the Couque model: (a) copper and (b) steel.

Table 2
. It is clear that this equation cannot model the increase in strain rate sensitivity at high strain rates.This is observed in the five situations studied here.The error is important.The minimum error is obtained for the case of steel at 273 K.It is equal to 13.4%.It can increase up to 48.2% which is obtained with the case of copper 105.

Table 2 :
Material constants and errors obtained with the standard Johnson-Cook equation.
rate.The fit of this model to the experimental data of copper and steel is shown in Figure2.The material constants and the errors are reported in Table3.The model follows roughly the experimental data.The fit is highly better with steel than copper.The errors are ranging from 5.3% for steel 293 K to 41.4% for copper 105.It can catch the sharp increase in

Table 3 :
Material constants and errors obtained by the Huh-Kang equation.

Table 4 :
Material constants and errors obtained by Tuazon et al. model.

Table 5 :
Material constants and errors obtained by Couque equation.

Table 6 :
Material constants and errors obtained by Cowper-Symonds equation.

Table 7 :
Material constants and errors obtained by the modified-Eyring equation.at high strain rates.The Huh-Kang equation gives only slightly better fit.Moreover, it predicts a sharp increase in the yield stress at quasi-static strain rates and this is a nonphysical behavior.The Tuazon et al. model yield an acceptable fit in the intermediate and high strain rate ranges.However, it predicts a sharp decrease in the yield stress at very low strain rates which is also a nonphysical behavior.Couque, Cowper-Symonds, and the modified-Eyring equations fit well the experimental data.Couque model yields the least value of error, then Cowper-Symonds, and finally the modified-Eyring equation.However, Couque equation uses four material constants while the modified-Eyring and Cowper-Symonds equations use only three each.It is then recommended to use the following modified Johnson-Cook equation for modeling of metallic materials including hardening, strain rate, and temperature effect: ) ,