A Modified Eyring Equation for Modeling Yield and Flow Stresses of Metals at Strain Rates Ranging from 10 − 5 to 5 × 104 s − 1

In several industrial applications, metallic structures are facing impact loads. Therefore, there is an important need for developing constitutive equationswhich take into account the strain rate sensitivity of theirmechanical properties.The Johnson-Cook equation was widely used to model the strain rate sensitivity of metals. However, it implies that the yield and flow stresses are linearly increasing in terms of the logarithm of strain rate. This is only true up to a threshold strain rate. In this work, a three-constant constitutive equation, assuming an apparent activation volume which decreases as the strain rate increases, is applied here for some metals. It is shown that this equation fits well the experimental yield and flow stresses for a very wide range of strain rates, including quasi-static, high, and very high strain rates (from 10 to 5 × 10 s). This is the first time that a constitutive equation is showed to be able to fit the yield stress over a so large strain rate range while using only three material constants.


Introduction
The steel and metallic materials are largely used in several industrial fields, for example, automobile, naval, aeronautical, and military industries.In these applications, transportation vehicles have to be designed against impact loads.Thus, developing constitutive equations that take into account the strain rate sensitivity of metals is of major importance.
The split Hopkinson bar was largely applied to evaluate the strain rate sensitivity of materials at high strain rates; that is, ε ∼< 5000 s −1 [1][2][3].At very high strain rates ( ε >∼ 5000 s −1 ), the direct-impact Hopkinson bar is now increasingly used [4,5].Based on the available experimental data, several constitutive equations were proposed in the open literature [6][7][8][9].Johnson and Cook [10] have proposed a phenomenological equation that includes hardening, temperature, and strain rate effects.This equation has been widely used [11][12][13] and included in several commercial finiteelement software types; this is mainly due to the reduced number of constants that it used and the separation of the hardening, temperature, and strain rate effects.It was then applied to model the strain rate sensitivity mainly for metallic alloys as for titanium [14,15], copper [16], aluminum [17,18], and steel [19][20][21][22] alloys.
The Johnson-Cook equation assumes that the yield and flow stresses are linearly increasing in terms of the logarithm of strain rate.Nevertheless, this assumption is mostly valid up to a threshold strain rate.The strain rate sensitivity highly increases at high strain rates [23,24], indeed.Several modifications have then been proposed to extend the validity of the Johnson-Cook model up to the very high strain rate range [24][25][26][27].
The original Johnson-Cook model uses two constants to take into consideration the strain rate sensitivity.The modified Johnson-Cook equations involve at least four constants in order to catch the stress behavior at very high strain rate.Recently, El-Qoubaa and Othman [28] proposed a modified Eyring equation [29] to model the strain rate sensitivity of the polyetheretherketone's yield stress.Within the framework of the Eyring theory, yielding is a thermally activated process.Ree and Eyring [30] used relaxation processes.Fotheringham et al. [31,32] introduced the idea of the cooperative motion of polymer chain segments.Richeton et al. [33] used the cooperative motion theory and introduced the Arrhenius law for the horizontal and vertical shifts.El-Qoubaa and Othman [28,34] used an apparent activation volume which is decreasing for increasing strain rate.Their equation involves three material constants.
In this work, El-Qoubaa and Othman's equation is examined regarding the yield stress and the flow stress at a given plastic strain, of several metals.Besides, the methodology, to obtain a reasonable first guess of the materials constants, is detailed.

Constitutive Equation
Johnson and Cook [10] have proposed that the flow stress of metals is obtained in the following form: where   , ε , ε 0 , ,   , and   are the plastic strain, strain rate, reference strain rate, temperature, room temperature, and melting temperature, respectively, and , , , , and  are four material constants.The yield stress, at room temperature, is obtained by considering   = 0 and  =   .Thus, With no loss of generality, the reference strain ε 0 can be fixed to 1 s −1 .Consequently, two materials constants, namely,  and , are needed to predict the strain rate sensitivity of metals' yield stress.Similarly, two constants are needed to predict the strain rate sensitivity of the flow stress at a given plastic strain.
Even though the Johnson-Cook equation is a phenomenological model, it can find some foundation in the work of Eyring [29] who argued that yielding is a thermally activation process.In this framework, the following equation for the yield stress is established: where  is the universal gas constant,   and   are two material constants, and   is the activation energy of the transition.The Eyring equation can also be rewritten as where  is the Boltzmann constant.Here also, two constants  0 and  0 are needed to define the strain rate sensitivity of the yield stress. 0 is yield stress at a strain rate ε = ε 0 = 1 s −1 and  0 is the activation volume which is assumed to be constant.
El-Qoubaa and Othman [28] argued for the use of an apparent activation volume that is decreasing with increasing strain rate.More precisely, they suggested that the logarithm of the activation volume is linearly decreasing in terms of strain rate.The apparent activation volume is then written as where ε  is critical strain rate.Substituting the constant activation volume  0 in (4) by the strain-rate-sensitive activation volume of (5) yields a new equation of the yield stress: This modified Eyring equation needs now three constants,  0 ,  0 , and ε  , to predict the strain rate sensitivity of the yield stress.This equation can also be used to model the flow stress at a constant plastic strain.
Equation ( 6) has two main advantages.First, it uses a reduced number of materials constants, namely, 3 constants, whereas the modified Johnson-Cook equations use at least four constants.Second, the three material constants are easily interpretable.Hence,  0 is no more than the yield at a strain rate of 1/s. 0 is the activation volume of the -transition of the Eyring equation.Finally, ε  separates the strain rate range in two parts.For lower strain rates ( ε ≪ ε  ), the yield stress increases linearly in terms of the logarithm of strain rate.At high strain rates, the strain rate sensitivity highly increases with increasing strain rate.These interpretations help in establishing a methodology to obtain a first guess of these constants.
The identification of the materials constants is then divided in two steps.Firstly, a first guess is determined knowing the above interpretations of these constants.Secondly, an optimization procedure is used to better fit the experimental data.More precisely, the first-guess values of  0 and  0 will be determined from two points in the low strain rate range assuming a linear variation of yield stress in terms of the logarithm of strain rate.
Let σ1 and σ2 be the yield stress measured for two strain rates ε 1 and ε 2 , respectively, of the low strain rate range.The first guess of the constants  0 and  0 is respectively.
The third constant ε  will be determined from the threshold strain rate separating the thermally activated regime from the viscous regime [4].Denoting ̂ε  threshold strain rate, the first guess of ε  is defined as These guesses are then considered as initial values of an optimization procedure of which the cost function is an error defined using the relative difference between the experimental yield stress and the yield stress predicted by (6).More precisely, where  is the cost function.

Results and Discussion
In order to check the applicability of the new constitutive equation and the identification procedure to metals, several metallic materials are considered here.First we consider two materials, namely, copper and A304L steel, to show the quality of the first-guess material constants.Couque [4] has already reported experimental values of copper and A304L steel yield stresses for strain rates up to ∼20000/s and ∼10000/ s, respectively.In order to achieve so high strain rates a directimpact Hopkinson bar was used [4,5,28].The two-step identification procedure, detailed above, is used to determine the materials constants of (6) for three copper materials and two AA304L steels.The experimental yield stress data are obtained from Couque [4].The first-guess and optimized material constants are given in Tables 1 and 2. Besides, the experimental yield stress reported by Couque [4] is compared to the yield stress predicted by the new modified Eyring equation ( 6) in Figures 1 and 2.
Using the first-guess material constants, it is easy to fit the quasi-static and intermediate strain rate data (Figures 1(a) and 2(a)).It is less easy to catch the yield stress increase at high strain rates.This can be linked to the difficulty of giving a close first guess of the critical strain rate ε  .It is difficult to determine exactly the transition between thermally activated and viscous regimes, indeed.However, the first guess is very useful to determine initial values for the optimization problem, which is a crucial and important task.In this work, the first-guess parameters are even more important because they give close values of  0 and  0 (Tables 1 and 2).Thus the optimization procedure will mostly concentrate on refining the value of ε  .Using the optimized materials constants, the new modified Eyring equation fits well the experimental data (Figures 1(b) and 2(b)).It catches the behavior in the quasi-static as well as high strain rate ranges.The sharp increase of the yield stress at very high strain rates is well predicted up to ∼20000/s for three copper materials and up to ∼10000/s for two A304L steel materials.
Henceforth, we will concentrate on the results obtained by only the optimized materials constants.The modified Eyring equation is then used to fit the behavior of other metallic materials.Figure 3 compares the flow stress, at 10% of plastic strain, of two aluminum alloys predicted by the new modified Eyring equation to the experimental flow stress reported by Moćko et al. [35].Equation ( 6) fits well the strain rate  sensitivity of the two materials, except for the drop in the flow of AA 6082-T6 at around 1000/s, which should rather be attributed to the change of the testing machine as it is an uncommon behavior.The modified Eyring equation fits well the flow stress at 5% of strain of CVD textured ⟨011⟩ tungsten (Figure 4).The discrepancy obtained at low strain rates could rather be attributed to the unusual important jump of stress between 3 × 10 −4 s −1 and 5 × 10 −3 s −1 .
Dealing with temperature effect on the strain rate sensitivity, Figure 5 compares the yield stress of annealed mild steel as predicted by (6) to the yield stress reported in [38].For the two considered temperatures, that is, 293 and 493 ∘ K, the modified Eyring equation fits well the experimental yield stress over a very wide range of strain rates (from 10 −3 to 5×10 4 s −1 ). Figure 5 shows the capability of the new equation to catch the behavior for strain rates up to 5 × 10 4 s −1 .[37] and Blazynski [38]) and new modified Eyring equation for annealed mild steel.The last example is dealing with tensile yield and ultimate stress of Ti-47Al-2Mn-2Nb (Figure 6).This example shows that the new equation can also fit the tensile behavior.Moreover, it is shown that it works for strain rates down to 10 −5 s −1 .
The material parameters of the different metallic alloys are synthesized in Table 3.They are also compared to the material parameters obtained for PEEK [28].It is worth noting that the activation volume  0 of metallic and polymeric materials is in the same range of [0.25-2.4nm 3 ].This gives a characteristic distance which is in the range of [6-14 Å].Studying copper over a wide temperature range, Suo et al. [40] reported that the activation volume is in the range of 10 3 .
Through multiple examples, the new modified Eyring constitutive equation has been showed here to have a great potential to model the strain rate sensitivity of the yield/flow stress over a very wide range of strain rates (from 10 −5 to 5 × 10 4 s −1 ) using only three material constants.

Conclusion
In this work, a new modified Eyring constitutive equation, predicting the strain rate sensitivity of yield, was validated regarding the experimental yield stress of several metallic materials.This constitutive equation uses only three material constants which were determined using an optimization procedure.A methodology was established in order to obtain a first guess of the material constants, hence simplifying the optimization step.The modified Eyring equation fits well the experimental data on a very wide range of strain rates (over more than 8 decades).This is highly important result.Indeed, it is possible to fit the yield/flow stress on a so large strain rate range by using only three material constants.

Figure 1 :
Figure 1: Comparison between experimental data (Couque [4]) and new modified Eyring equation for copper: (a) first-guess material constants and (b) optimized materials constants.

Figure 2 :
Figure 2: Comparison between experimental data (Couque [4]) and new modified Eyring equation for steel: (a) first-guess material constants and (b) optimized materials constants.

Table 1 :
Material constants for copper.

Table 3 :
Optimized material constants for several metallic materials.