A Modified Eyring Equation for Modeling Yield and Flow Stresses of Metals at Strain Rates Ranging from 10-5 to 5 × 10 4 s-1

Abstract

In several industrial applications, metallic structures are facing impact loads. Therefore, there is an important need for developing constitutive equations which take into account the strain rate sensitivity of their mechanical 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 10-5 to 5 × 104 s-1). 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.

Academic Editor:Gianluca Percoco

Mechanical Engineering Department, Faculty of Engineering, King Abdul-Aziz University, P.O. Box 80248, Jeddah 21589, Saudi Arabia

Received 12 March 2015; Accepted 23 June 2015; 6 July 2015

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. 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, [figure omitted; refer to PDF] s^-1 [1-3]. At very high strain rates ( [figure omitted; refer to PDF] 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 [6-9]. Johnson and Cook [10] have proposed a phenomenological equation that includes hardening, temperature, and strain rate effects. This equation has been widely used [11-13] and included in several commercial finite-element 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 [19-22] 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 [24-27].

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.

2. Constitutive Equation

Johnson and Cook [10] have proposed that the flow stress of metals is obtained in the following form: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are the plastic strain, strain rate, reference strain rate, temperature, room temperature, and melting temperature, respectively, and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are four material constants. The yield stress, at room temperature, is obtained by considering [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Thus, [figure omitted; refer to PDF] With no loss of generality, the reference strain [figure omitted; refer to PDF] can be fixed to 1 s^-1 . Consequently, two materials constants, namely, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , 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: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the universal gas constant, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are two material constants, and [figure omitted; refer to PDF] is the activation energy of the [figure omitted; refer to PDF] -transition. The Eyring equation can also be rewritten as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the Boltzmann constant. Here also, two constants [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are needed to define the strain rate sensitivity of the yield stress. [figure omitted; refer to PDF] is yield stress at a strain rate [figure omitted; refer to PDF] s^-1 and [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is critical strain rate. Substituting the constant activation volume [figure omitted; refer to PDF] in (4) by the strain-rate-sensitive activation volume of (5) yields a new equation of the yield stress: [figure omitted; refer to PDF] This modified Eyring equation needs now three constants, [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , 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, [figure omitted; refer to PDF] is no more than the yield at a strain rate of 1/s. [figure omitted; refer to PDF] is the activation volume of the [figure omitted; refer to PDF] -transition of the Eyring equation. Finally, [figure omitted; refer to PDF] separates the strain rate range in two parts. For lower strain rates ( [figure omitted; refer to PDF] ), 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 [figure omitted; refer to PDF] and [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] and [figure omitted; refer to PDF] be the yield stress measured for two strain rates [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , respectively, of the low strain rate range. The first guess of the constants [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is [figure omitted; refer to PDF] respectively.

The third constant [figure omitted; refer to PDF] will be determined from the threshold strain rate separating the thermally activated regime from the viscous regime [4]. Denoting [figure omitted; refer to PDF] threshold strain rate, the first guess of [figure omitted; refer to PDF] is defined as [figure omitted; refer to PDF] 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, [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the cost function.

3. 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 direct-impact 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.

Table 1: Material constants for copper.

	[figure omitted; refer to PDF] (MPa)	[figure omitted; refer to PDF] (MPa)	[figure omitted; refer to PDF] ( [figure omitted; refer to PDF] )	[figure omitted; refer to PDF] ( [figure omitted; refer to PDF] )	[figure omitted; refer to PDF] ( [figure omitted; refer to PDF] )	[figure omitted; refer to PDF] ( [figure omitted; refer to PDF] )
Copper 105 μ m	103	96.4	2.31	2.35	15000	2037
Copper 26 μ m	172	179.8	1.39	0.737	15000	5919
Copper 9 μ m	278	271.6	0.754	0.612	15000	7163

Table 2: Material constants for A304L steel.

	[figure omitted; refer to PDF] (MPa)	[figure omitted; refer to PDF] (MPa)	[figure omitted; refer to PDF] ( [figure omitted; refer to PDF] )	[figure omitted; refer to PDF] ( [figure omitted; refer to PDF] )	[figure omitted; refer to PDF] ( [figure omitted; refer to PDF] )	[figure omitted; refer to PDF] ( [figure omitted; refer to PDF] )
A304L 100 μ m	336	347.9	0.827	0.376	15000	2946
A304L 10 μ m	714	667.5	0.279	0.737	8000	2113

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.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

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.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

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 [figure omitted; refer to PDF] . 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 [figure omitted; refer to PDF] and [figure omitted; refer to PDF] (Tables 1 and 2). Thus the optimization procedure will mostly concentrate on refining the value of [figure omitted; refer to PDF] .

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 Mocko 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.

Figure 3: Comparison between experimental data (Mocko et al. [35]) and new modified Eyring equation for aluminum.

[figure omitted; refer to PDF]

The modified Eyring equation fits well the flow stress at 5% of strain of CVD textured [figure omitted; refer to PDF] tungsten (Figure 4). The discrepancy obtained at low strain rates could rather be attributed to the unusual important jump of stress between [figure omitted; refer to PDF] s^-1 and [figure omitted; refer to PDF] s^-1 .

Figure 4: Comparison between experimental data (Subhash et al. [36]) and new modified Eyring equation for CVD textured [figure omitted; refer to PDF] tungsten.

[figure omitted; refer to PDF]

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 [figure omitted; refer to PDF] s^-1 ). Figure 5 shows the capability of the new equation to catch the behavior for strain rates up to [figure omitted; refer to PDF] s^-1 .

Figure 5: Comparison between experimental data (Clarke et al. [37] and Blazynski [38]) and new modified Eyring equation for annealed mild steel.

[figure omitted; refer to PDF]

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 .

Figure 6: Comparison between experimental data (Wang et al. [39]) and new modified Eyring equation for Ti-47Al-2Mn-2Nb alloy.

[figure omitted; refer to PDF]

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 [figure omitted; refer to PDF] of metallic and polymeric materials is in the same range of [0.25-2.4 nm³ ]. 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 [figure omitted; refer to PDF] .

Table 3: Optimized material constants for several metallic materials.

	[figure omitted; refer to PDF] (MPa)	[figure omitted; refer to PDF] ( [figure omitted; refer to PDF] )	[figure omitted; refer to PDF] ( [figure omitted; refer to PDF] )
Copper 105 μ m	96.4	2.35	2037
Copper 26 μ m	179.8	0.737	5919
Copper 9 μ m	271.6	0.612	7163
A304L 100 μ m	347.9	0.376	2946
A304L 10 μ m	667.5	0.737	2113
AA 6082-T6	342.7	0.795	9598
AA 7075-T6	718.6	0.907	3970
CVD textured [figure omitted; refer to PDF] tungsten	1176	0.258	3158
Annealed mild steel (293°K)	148.5	0.492	89007
Annealed mild steel (493°K)	105.6	1.370	25711
Ti-47Al-2Mn-2Nb alloy (tensile yield)	451.1	0.497	1462
Ti-47Al-2Mn-2Nb alloy (tensile ultimate strength)	481.8	0.495	590
Polyetheretherketone (PEEK) [28]	159	1.01	12038

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 [figure omitted; refer to PDF] s^-1 ) using only three material constants.

4. 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.

Conflict of Interests

The author declares no conflict of interests.

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