International Journal of Plasticity 90 (2017) 167e176

Contents lists available at ScienceDirect

International Journal of Plasticity journal homepage: www.elsevier.com/locate/ijplas

Development of equations for strain rate sensitivity of UFG aluminum as a function of strain rate Mohammad Sadegh Mohebbi a, *, Abbas Akbarzadeh b a b

Mechanical Engineering Group, Department of Engineering, Qom University of Technology, Khodakaram Blvd., 37195-1519, Qom, Iran Department of Materials Science and Engineering, Sharif University of Technology, Azadi Ave., Tehran, P.O. Box 11155-9466, Iran

a r t i c l e i n f o

a b s t r a c t

Article history: Received 9 October 2016 Received in revised form 5 January 2017 Accepted 16 January 2017 Available online 18 January 2017

Strain rate sensitivity (m-value) of ultraﬁne grain (UFG) AA 1050 and AA 5052 sheets processed by accumulative roll-bonding is investigated versus strain rate by stress relaxation (SR) test at ambient temperature. The results show a weak viscous nature of deformation for AA 5052 specimens as compared to AA 1050 ones. So that much less stress relaxation and negligible strain rate sensitivity are obtained for this material due to dislocation and grain boundary mobility limitation caused by Mg solute atoms. In order to formulate strain rate sensitivity of UFG aluminum as a function of strain rate, three phenomenological and two empirical models are developed and assessed by the experimental results. It is shown that since thermally activated dislocation glide is not the single governing phenomenon, the model developed based on this mechanism of deformation fails to predict the variation of m-value by strain rate. Contribution of grain boundary sliding (GBS) can result in a model ﬁtted on the experimental results at lower strain rates, yet not suitable for higher strain rates. However, the third phenomenological model in which dislocation annihilation was taken into account besides the mentioned phenomena can well predict the trend of m-value at full range of the strain rate. Since this is a parametric model formulated by independent variable of time and with no analytical solution, two empirical equations are presented as more simple and straightforward models. It is shown that these models give useful formulas for estimation and extrapolation of the mvalue at relatively high strain rates of common deformation processes where a monotonic variation by the strain rate is expected. © 2017 Elsevier Ltd. All rights reserved.

Keywords: Ultraﬁne grain aluminum AA 5052 Strain rate sensitivity Stress relaxation Modeling

1. Introduction Among various mechanical aspects of nanocrystalline (NC) and ultraﬁne grain (UFG) metals, strain rate sensitivity exponent (m-value, simply called as strain rate sensitivity) is of particular attention. This parameter is connected to the viscous nature of thermally activated plastic deformation (Miyamoto et al., 2006). Many studies have shown that m-value of UFG metals processed by severe plastic deformation (SPD) is signiﬁcantly enhanced as compared to their coarse grain (CG) counterpart (Khan et al., 2008; Xu et al., 2015). The increased dislocation density which leads to larger contribution to the thermally activated deformation as well as lower activation volume are believed to cause such an enhancement (Miyamoto

* Corresponding author. E-mail address: [email protected] (M.S. Mohebbi). http://dx.doi.org/10.1016/j.ijplas.2017.01.003 0749-6419/© 2017 Elsevier Ltd. All rights reserved.

168

M.S. Mohebbi, A. Akbarzadeh / International Journal of Plasticity 90 (2017) 167e176

et al., 2006; Xu et al., 2015). Moreover, increased contribution of grain diffusion associated deformation and grain boundary sliding (GBS) are thought to improve the strain rate sensitivity (Ivanov and Naydenkin, 2014; Yang et al., 2016). It has been demonstrated that strain rate sensitivity of a particular metal is affected by temperature (Suo et al., 2013). In the same way, it is expected that this parameter is a function of strain rate. Ivanov and Naydenkin (2014) investigated the strain rate sensitivity of UFG pure Al at room temperature in the range of 105 to 8.6 103 s1. Their results showed higher mvalues at lower strain rates. Similar trend has been recognized by Wang and Shan (2008) and Kumar et al. (2009). for UFG AA 1050 at room temperature and UFG Al-3% Mg at 130 C, respectively. In these studies, enhanced contribution of GBS at low strain rates is introduced as the reason for increased strain rate sensitivity. Strain rate jump test is the most widely used method of the m-value evaluation. Calculation from the ﬂow curves at different strain rates is prevalent, as well (Khan and Meredith, 2010; Liu et al., 2015). However, an accurate and reliable evaluation of the strain rate sensitivity needs to be performed at ﬁxed microstructure (Ghosh, 2007). Obviously, the latter procedure, i.e. derivation of ﬂow stresses from the curves at various strain rates, even with the same strain, denies the condition of ﬁxed microstructure, because different deformation histories are behind the considered strain in the curves. Strain rate jump test may satisfy this condition provided a big jump increment is not imposed. Even so, a gradual variation of the ﬂow stress may be detected after the jump, leading to two different m-values as the instantaneous and steady state ones (Khan and Meredith, 2010). Furthermore, since the calculated m-value is linked to the jumps between different strain rates, it is not possible, or at least not accurate, to ﬁnd it as a function of strain rate. Therefore, as can be seen in the mentioned works (Ivanov and Naydenkin, 2014; Wang and Shan, 2008; Kumar et al., 2009), only a discrete estimation of the m-value can be evaluated over the strain rate. With none of the above shortcomings, stress relaxation (SR) test can be considered as the most reliable method for this aim (Ghosh, 2007). In the SR test deformation is interrupted by stopping the cross heads. At this condition a material with viscous nature of deformation continues to be deformed plastically in expense of reduction of the elastic strains in the machine elements, so that the applied stress and the strain rate are reduced over the time (Dotsenko, 1979). Since the kinetics of this relaxation is governed by deformation mechanism, this test is supposed to be helpful in revealing the deformation mechanism and its quantitative parameters (Dotsenko, 1979; Gupta and Li, 1970). Yang et al. (2016). studied strain rate sensitivity of CG, NC and nanotwinned copper by SR test under tensile deformation. However, such a test, particularly in case of UFG materials, should be performed after stabilizing the deformation. This condition can be satisﬁed by imposing some plastic deformation before SR test. Moreover, when m-value is calculated from the tensile data before necking, there is no guarantee that the related values are reliable at larger strains as well. For example, Khan et al. (2008). obtained the m-value of NC aluminum varying by strain. To overcome these problems, Mohebbi et al. (2014a). performed SR test under plane strain compression (PSC) condition after conducting an effective strain of 0.3 on UFG AA 1050 sheets. They showed that this test is a unique approach to assess the strain rate sensitivity of UFG aluminum as a continuous function of the strain rate. In the present study, such a procedure is performed on UFG AA 5052 besides UFG AA 1050 to compare their trends of stress relaxation and strain rate sensitivity. Moreover, the main purpose of this study is to develop a physically meaningful model for variation of the m-value by strain rate. As mentioned above, impact of the strain rate on m-value has already taken some attentions in literature. However, no considerable effort has been allocated to model and formulate such an impact on mvalues of the UFG and NC materials based on their special mechanisms of deformation. In addition to capability of prediction of the mechanical behavior of these materials, such a model can give evidences for the contributing phenomena during deformation of UFG and NC materials.

2. Experimental procedure Sheets of aluminum alloys AA 1050 and AA 5052 with thickness of 1.5 mm were used as the initial material. Before the process, AA 1050 and AA 5052 sheets were annealed at 330 C and solution treated at 550 C, respectively, for 1 h followed by water quenching. Then, they were cut in dimensions of 60 160 mm2 for Accumulative Roll-Bonding (ARB) process up to 8 cycles (AA 1050) and 4 cycles (AA 5052). ARB processing was performed at room temperature by a thickness reduction of 50% by rolls of 150 mm in diameter and speed of 50 rpm with no lubrication. Details of the process have been published in Refs. (Mohebbi et al., 2014a, 2014b). PSC samples with dimensions of 30 20 1.5 mm3 (rolling transverse normal directions) were cut from center of the ARB sheets. The surfaces of samples and indenters were precisely polished and lubricated by a PTFE grease before the test. During the test the indenters with breadth of 3 mm, basal width of 12 mm, height of 10 mm and length of 100 mm were aligned with the transverse direction, in which no spreading was imposed. Initial strain rate of this test was 6.4 103 s1. After a ﬁxed strain of ~0.3 b y PSC, the stress relaxation test was carried out under ﬁxed cross-heads for a period of 600 s. VonMises effective stress and strain were calculated for plane strain condition from the force and displacement, which were registered in 20 equally spaced intervals per second. Displacement was measured by a non-contacting extensometer which was used to track the distance between two marks attached to the main body of the indenters. The reader is referred to Refs (Mohebbi et al., 2014a, 2014b, 2015). for more details on the test as well as the methods for suppression of the elastic displacement of the tools and the effect of friction.

M.S. Mohebbi, A. Akbarzadeh / International Journal of Plasticity 90 (2017) 167e176

169

3. Stress relaxation results Relaxation curves of the specimens are shown in Fig. 1 (a)-(c). Various forms of stress-strain, stress-time and strain-time are depicted and linked in this ﬁgure. The amount of stress relaxation and the relevant plastic strain of AA 5052 specimens are much less than AA 1050 ones. While 25% of the stress value is relaxed in AA 1050-8-cycles specimen, the stress relaxation of AA 5052 specimens is only about 3% after 600 s. Stress relaxation of ultraﬁne grained aluminum is attributed to three phenomena: relaxation of effective (thermal) stress, recovery of internal (athermal) stress and grain boundary sliding (Mohebbi et al., 2015). It can be said that Mg solute atoms affect these contributing phenomena. Thermally activated deformation is hindered by solute atoms. In fact, mobility of dislocations and grain boundaries is much lower in Al-Mg alloys due to the interaction with Mg solute atoms (Soer et al., 2004; Gholinia et al., 2002). Therefore, in compliance with the observed results, very small amount of stress relaxation in UFG AA 5052 is expected. In case of UFG AA 1050, however, the amount of stress relaxation is increased by the number of ARB cycles, because the mentioned effective phenomena are enhanced by the imposed grain reﬁnement (Mohebbi et al., 2015). From the ﬁrst derivation of the strain-time curve, the strain rate is obtained versus time. Having this together with the stress-time data, it is possible to draw stress-strain rate curves. Such curves are represented in Log-Log coordinates in Fig. 1 (d). Since the slope of these curves is equal to the strain rate sensitivity, the m-value can be drawn as a function of strain rate. The results are shown in Fig. 2. Increase of the m-value by the number of ARB cycles in AA 1050 specimens was expected due to enhanced contribution of the thermally activated deformation as well as the reduced activation volume as the results of grain reﬁnement (Mohebbi et al., 2014a). Moreover, increase of the m-value by decrease of the strain rate, as was addressed in the introduction, is mainly explained by enhanced GBS (Ivanov and Naydenkin, 2014; Wang and Shan, 2008; Kumar et al., 2009). Ivanov and Naydenkin (2014) calculated the contribution of GBS to the overall deformation (h) of UFG Al at room temperature. Based on their results, h is increased up to 72% by decrease of the strain rate in the range of 1 105-8.6 103. In addition to GBS, Mohebbi et al. (2014a). have shown that recovery of the internal component of ﬂow stress can be considered as another phenomenon leading to the increase of m-value at lower strain rates.

Fig. 1. (a) Stress versus strain, (b) stress versus time, (c) strain versus time and (d) log-log plot of stress versus strain rate during relaxation test of various ARBed AA 1050 and AA 5052 specimens.

170

M.S. Mohebbi, A. Akbarzadeh / International Journal of Plasticity 90 (2017) 167e176

By comparing the strain rate sensitivity of the two alloys, it is obvious that the m-value of AA 5052 specimens are negligible (note that different scales are used in Fig. 2 for two alloys). This means that plastic deformation of UFG AA 5052 at ambient temperature involves much less viscous nature. Al-Mg alloys are generally known to show negative strain rate sensitivity at a quasi-static loading under ambient temperature which is correlated to dynamic strain aging (Kabirian et al., 2014; Kapoor et al., 2005). Obviously, application of the SR test on a material with negative strain rate sensitivity carries no sense. However, it has been shown that SPD can reduce the susceptibility to dynamic strain aging and enhance the strain rate sensitivity in UFG Al-Mg alloys. High dislocation density in heavily deformed Al-Mg alloys, which reduces the Mg atom atmosphere per dislocation, is thought to be responsible for suppression of dynamic strain aging (Kabirian et al., 2014; Kapoor et al., 2005; Zhao et al., 2014). That is why a positive strain rate sensitivity is measured for the UFG AA 5052 specimens. 4. Modeling Original formulation of the deformation during SR test, which dates back to 1970s (Dotsenko, 1979; Gupta and Li, 1970; de et al., 2011). During this test, total length of Batist and Callens, 1974), is also used in recent studies (Yang et al., 2016; Trojanova the specimen and the machine elements between the cross heads are constant. At this condition, a viscous material deforms plastically in expense of elastic strain of the sample-machine system. Therefore, the plastic and elastic strain rates of the specimen are related as (Dotsenko, 1979; Gupta and Li, 1970):

ε_ p ¼ _εe ¼

s_ 0 E

(1)

0

Where E is the effective elastic modulus of the sample-machine system. In development of original models for SR test, the internal (or athermal) stress component (si) is supposed to be constant. Instead, the effective (or thermal) stress component s*, as the rate dependent component, is believed to be relaxed (s_ ¼ s_ * ) (Dotsenko, 1979). Having Eq. (1) between ε_ (hereafter ε_ represents the plastic strain rate) and s_ , the m-value can be obtained during the SR test as a function of stress and the ﬁrst and second derivatives of that:

m¼

d ln s ε_ ds ε_ s_ s_ 2 ¼ ¼ ¼ d ln_ε s d_ε s €ε s s€

Fig. 2. Strain rate sensitivity of UFG AA 1050 and AA 5052 specimens as a function of strain rate.

(2)

M.S. Mohebbi, A. Akbarzadeh / International Journal of Plasticity 90 (2017) 167e176

171

This equation, in fact, can be considered as a general equation for the m-value during the SR test, so that it is only needed to have s as a function of time to derive the m-value versus time, and therefore, versus strain rate. 4.1. Equation derived from thermally activated dislocation glide If the plastic deformation is undertaken by dislocation movement, the Orowan equation would be used for the plastic strain rate:

ε_ ¼ f rm b v

(3)

Where 4 is a geometric coefﬁcient, rm is the density of mobile dislocations, b is the Burgers vector and v is the average dislocation velocity. This equation implies that the stress dependence of ε_ is related to the stress dependence of v. Dislocation glide is explicitly considered to be a thermally activated process with an activation energy depending on the effective stress. Therefore, v is related to the effective stress in the Arrhenius form:

DG0 v* s* v ¼ v0 exp kT

(4)

Where v0 is the pre-exponential factor, DG0 is the barrier activation energy at zero stress, v* is the activation volume, k is the Boltzmann constant and T is the absolute temperature. It follows from Eqs. (1), (3) and (4) that:

s_ ¼ E0 f rm b v0 exp

DG0 v* s*

(5)

kT

Stress can be obtained from integration of this differential equation (Dotsenko, 1979; de Batist and Callens, 1974):

sðtÞ ¼ sð0Þ a lnðb t þ 1Þ

(6)

In which t is the relaxation time, s(0) is the stress at beginning of the relaxation (t ¼ 0 s) and a and b are: 0 DG0 v* s*ð0Þ kT E f rm b v0 v* exp a ¼ *; b ¼ kT kT v

! (7)

Based on Eq. (6), the following derivatives are obtained:

s_ ¼

ab

ab2

€¼ ; s 2 ðbt þ 1Þ ðbt þ 1Þ

(8)

Which are replaced in Eq. (2) to derive the following equation for the m-value as a function of strain rate:

m¼

a sð0Þ a lnðabÞ þ a lnðE0 ε_ Þ

(9)

Strain rate sensitivity versus strain rate, which is obtained from this equation, is drawn in Fig. 3 for the 1050-8-cycles specimen with a ¼ 10 MPa and b ¼ 0.71 s1. These values are obtained from ﬁtting of Eq. (6) to the relaxation curve of AA 1050-8-cycles specimen (Fig. 1 (b)). E’ is measured from Eq. (1) as 1.25 GPa and s(0) ¼ 248 MPa is taken from relaxation curve of the 1050-8-cycles specimen at t ¼ 0 s (Fig. 1 (a) and (b)). As shown in this ﬁgure (the red dashed line), it is obvious that Eq. (9) can not ﬁt to the experimental results. In fact, the variation of m-value by the strain rate is signiﬁcantly underestimated by this equation. This implies the fact that thermally activated dislocation glide is not the only phenomenon in deformation and SR test of UFG metals. 4.2. Development of the model for UFG aluminum Modeling of the SR behavior based on the thermally activated dislocation movement is a worthy approach with physical meaning. However, it has been shown that the thermally activated dislocation movement is not the single governing event during SR test of UFG metals (Yang et al., 2016; Mohebbi et al., 2015). One operative phenomenon is decrease of the dislocation density. There is a strong driving force for this phenomenon due to high level of stored energy in a severely deformed metal. In addition, the distance between boundaries, which act as sinks for dislocations, is very small in UFG metals. Therefore, strong annihilation of dislocations is expected at beginning of the SR test of these materials. As a general fact, Silbermann et al. (2014). has illustrated that a rapid change of dislocation density is occurred by change of the loading state. Annihilation of dislocations has also been demonstrated by TEM observation of UFG AA 1050 after SR test (Mohebbi et al., 2015). Another possible phenomenon in stress relaxation of UFG metals is the grain boundary sliding (GBS). While ﬁne

172

M.S. Mohebbi, A. Akbarzadeh / International Journal of Plasticity 90 (2017) 167e176

Fig. 3. Strain rate sensitivity of the 1050-8-cycles specimen as a function of strain rate obtained from SR test and various models.

grains and low dislocation density inside the grains conﬁne the common mechanism of dislocation glide in UFG and NC metals, GBS plays an important role in deformation of these materials. In fact, presence of large fraction of high energy nonequilibrium grain boundaries promotes GBS in severely deformed metals (Mishnaevsky and Levashov, 2015). Operation of this phenomenon in deformation of NC and UFG metals has been shown even at room temperature when a low strain rate is applied. Choi et al. (2013). studied the room temperature creep of NC nickel undertaken by grain boundary diffusion and GBS. GBS has also been observed by atomic force microscopy (Chinh et al., 2006) and electron microscopy (Ivanov and Naydenkin, 2012) in UFG aluminum deformed at room temperature. The mentioned phenomena can explain why increase of the m-value by decrease of the strain rate is much faster than what expected from Eq. (9). Hence, in order to develop an appropriate model for prediction of strain rate sensitivity of UFG aluminum versus strain rate, contribution of the mentioned phenomena is added to the original model based on the thermally activated dislocation movement. Similar approach has been successfully implemented to explain why the absolute value of the slope ln ðs_ Þ versus ln (t) plot during the SR test of UFG AA 1050 is smaller than unity (Mohebbi et al., 2015). GBS is taken into account from the following equation (Choi et al., 2013):

ε_ GBS ¼

2 2 105 DGB b4 s* kTG d3

(10)

In which G is the shear modulus, d is the grain diameter, and DGB is the grain boundary diffusion coefﬁcient. DGB can be obtained from the generalized equation for FCC metals (Gust et al., 1985):

d DGB ¼ 9:7 1015 exp

9:07Tm T

m3 s1

(11)

Where Tm is the absolute melting temperature and d is the grain boundary width for which the value of 0.5 nm is always used (Mehrer, 2007). The dislocation glide and the GBS are not mutually exclusive. Assuming that the two mechanisms operate independently, their contributions to the total strain rate is additive. Therefore, ε_ GBS can be added to the ε_ correlated to dislocation glide (Eq. (3)). In this way, Eq. (5) is modiﬁed to:

s_ ¼ E0 f rm b v0 exp

DG0 v* s* kT

E0

2 105 DGB b4 * 2 s k T G d3

(12)

Which can be rewritten using parameters a and b:

s_ ¼ a b exp

s sð0Þ 2 105 DGB b4 E0 ðs si Þ2 a k T G d3

(13)

Note that the equality s sð0Þ ¼ s* s*ð0Þ is used in derivation of this equation. The second derivative of stress is found from Eq. (13) as:

M.S. Mohebbi, A. Akbarzadeh / International Journal of Plasticity 90 (2017) 167e176

s sð0Þ 2 105 DGB b4 E0 2s_ ðs si Þ a k T G d3

s€ ¼ bs_ exp

173

(14)

€, the m-value can be calculated from Eq. In term of s, Eq. (13) should be numerically integrated. Therefore, having s, s_ and s (2). This procedure is accomplished for the 1050-8-cycles specimen with d ¼ 260 nm (obtained from EBSD analysis (Mohebbi et al., 2014a)), T ¼ 298 K, b ¼ 2.86 1010 m, Tm ¼ 925 K and G ¼ 26.1 GPa (Totten, 2003). Regarding the internal stress component, the best ﬁtting is achieved bysi ¼ 156 MPa. This parameter is also experimentally estimated by the relaxation curve saturation method based on which the stress will be saturated to si at very long relaxation time (Dotsenko, 1979). In the present study, this method resulted in a very close value of 163 MPa after relaxation time of 3000 s while it was still decreasing with a very low rate. a, b and E’ are taken with similar values as used for Eq. (9) (the model based on thermally activated dislocation glide). In fact, since the thermally activated dislocation glide is taken from the ﬁrst model with no change, similar values are assigned to a and b. The result of this model is shown by the blue curve in Fig. 3. By comparing this curve with the previous model for dislocation glide, it can be seen that through contribution of GBS, the m-value is enhanced to the experimental results at lower strain rates. However, at higher strain rates, the overestimation by dislocation glide model is still observed. In order to consider the dislocation annihilation, and therefore, a decreasing dislocation density, a ratio of rðtÞ =rð0Þ is introduced. Hence, b is replaced by b(0) which includes rð0Þ . Accordingly, the following equation is obtained by modiﬁcation of Eq. (13):

s_ ¼ a bð0Þ

rðtÞ s sð0Þ 2 105 DGB b4 E0 exp ðs si Þ2 rð0Þ a k T G d3

(15)

From which the second derivative of stress is found as:

s€ ¼ a bð0Þ

!

rðtÞ s sð0Þ 2 105 DGB b4 þ bð0Þ s_ exp 2s_ ðs si Þ E0 rð0Þ a rð0Þ k T G d3 r_

(16)

€ by Eq. (2). Here, Again, s is obtained by numerical integration of Eq. (15). Then, the m-value is calculated from s, s_ and s values of a ¼ 10.6 MPa and b(0) ¼ 1.3 s1 are used for the best ﬁt. It is necessary to note that since the formulation is modiﬁed by considering a varying dislocation density, the proper values of a and b are different from the previous models. In fact, here b(0) includes rð0Þ while, as can be seen in Fig. 4, the ratio of rðtÞ =rð0Þ is considered to decrease by time from the value of 1 at t ¼ 0 s. Regarding the trend of variation, it has been revealed that a concave up decrease of dislocation density in logarithmic scale of time can be taken as a reason for experimentally seen abnormal jSj

Contents lists available at ScienceDirect

International Journal of Plasticity journal homepage: www.elsevier.com/locate/ijplas

Development of equations for strain rate sensitivity of UFG aluminum as a function of strain rate Mohammad Sadegh Mohebbi a, *, Abbas Akbarzadeh b a b

Mechanical Engineering Group, Department of Engineering, Qom University of Technology, Khodakaram Blvd., 37195-1519, Qom, Iran Department of Materials Science and Engineering, Sharif University of Technology, Azadi Ave., Tehran, P.O. Box 11155-9466, Iran

a r t i c l e i n f o

a b s t r a c t

Article history: Received 9 October 2016 Received in revised form 5 January 2017 Accepted 16 January 2017 Available online 18 January 2017

Strain rate sensitivity (m-value) of ultraﬁne grain (UFG) AA 1050 and AA 5052 sheets processed by accumulative roll-bonding is investigated versus strain rate by stress relaxation (SR) test at ambient temperature. The results show a weak viscous nature of deformation for AA 5052 specimens as compared to AA 1050 ones. So that much less stress relaxation and negligible strain rate sensitivity are obtained for this material due to dislocation and grain boundary mobility limitation caused by Mg solute atoms. In order to formulate strain rate sensitivity of UFG aluminum as a function of strain rate, three phenomenological and two empirical models are developed and assessed by the experimental results. It is shown that since thermally activated dislocation glide is not the single governing phenomenon, the model developed based on this mechanism of deformation fails to predict the variation of m-value by strain rate. Contribution of grain boundary sliding (GBS) can result in a model ﬁtted on the experimental results at lower strain rates, yet not suitable for higher strain rates. However, the third phenomenological model in which dislocation annihilation was taken into account besides the mentioned phenomena can well predict the trend of m-value at full range of the strain rate. Since this is a parametric model formulated by independent variable of time and with no analytical solution, two empirical equations are presented as more simple and straightforward models. It is shown that these models give useful formulas for estimation and extrapolation of the mvalue at relatively high strain rates of common deformation processes where a monotonic variation by the strain rate is expected. © 2017 Elsevier Ltd. All rights reserved.

Keywords: Ultraﬁne grain aluminum AA 5052 Strain rate sensitivity Stress relaxation Modeling

1. Introduction Among various mechanical aspects of nanocrystalline (NC) and ultraﬁne grain (UFG) metals, strain rate sensitivity exponent (m-value, simply called as strain rate sensitivity) is of particular attention. This parameter is connected to the viscous nature of thermally activated plastic deformation (Miyamoto et al., 2006). Many studies have shown that m-value of UFG metals processed by severe plastic deformation (SPD) is signiﬁcantly enhanced as compared to their coarse grain (CG) counterpart (Khan et al., 2008; Xu et al., 2015). The increased dislocation density which leads to larger contribution to the thermally activated deformation as well as lower activation volume are believed to cause such an enhancement (Miyamoto

* Corresponding author. E-mail address: [email protected] (M.S. Mohebbi). http://dx.doi.org/10.1016/j.ijplas.2017.01.003 0749-6419/© 2017 Elsevier Ltd. All rights reserved.

168

M.S. Mohebbi, A. Akbarzadeh / International Journal of Plasticity 90 (2017) 167e176

et al., 2006; Xu et al., 2015). Moreover, increased contribution of grain diffusion associated deformation and grain boundary sliding (GBS) are thought to improve the strain rate sensitivity (Ivanov and Naydenkin, 2014; Yang et al., 2016). It has been demonstrated that strain rate sensitivity of a particular metal is affected by temperature (Suo et al., 2013). In the same way, it is expected that this parameter is a function of strain rate. Ivanov and Naydenkin (2014) investigated the strain rate sensitivity of UFG pure Al at room temperature in the range of 105 to 8.6 103 s1. Their results showed higher mvalues at lower strain rates. Similar trend has been recognized by Wang and Shan (2008) and Kumar et al. (2009). for UFG AA 1050 at room temperature and UFG Al-3% Mg at 130 C, respectively. In these studies, enhanced contribution of GBS at low strain rates is introduced as the reason for increased strain rate sensitivity. Strain rate jump test is the most widely used method of the m-value evaluation. Calculation from the ﬂow curves at different strain rates is prevalent, as well (Khan and Meredith, 2010; Liu et al., 2015). However, an accurate and reliable evaluation of the strain rate sensitivity needs to be performed at ﬁxed microstructure (Ghosh, 2007). Obviously, the latter procedure, i.e. derivation of ﬂow stresses from the curves at various strain rates, even with the same strain, denies the condition of ﬁxed microstructure, because different deformation histories are behind the considered strain in the curves. Strain rate jump test may satisfy this condition provided a big jump increment is not imposed. Even so, a gradual variation of the ﬂow stress may be detected after the jump, leading to two different m-values as the instantaneous and steady state ones (Khan and Meredith, 2010). Furthermore, since the calculated m-value is linked to the jumps between different strain rates, it is not possible, or at least not accurate, to ﬁnd it as a function of strain rate. Therefore, as can be seen in the mentioned works (Ivanov and Naydenkin, 2014; Wang and Shan, 2008; Kumar et al., 2009), only a discrete estimation of the m-value can be evaluated over the strain rate. With none of the above shortcomings, stress relaxation (SR) test can be considered as the most reliable method for this aim (Ghosh, 2007). In the SR test deformation is interrupted by stopping the cross heads. At this condition a material with viscous nature of deformation continues to be deformed plastically in expense of reduction of the elastic strains in the machine elements, so that the applied stress and the strain rate are reduced over the time (Dotsenko, 1979). Since the kinetics of this relaxation is governed by deformation mechanism, this test is supposed to be helpful in revealing the deformation mechanism and its quantitative parameters (Dotsenko, 1979; Gupta and Li, 1970). Yang et al. (2016). studied strain rate sensitivity of CG, NC and nanotwinned copper by SR test under tensile deformation. However, such a test, particularly in case of UFG materials, should be performed after stabilizing the deformation. This condition can be satisﬁed by imposing some plastic deformation before SR test. Moreover, when m-value is calculated from the tensile data before necking, there is no guarantee that the related values are reliable at larger strains as well. For example, Khan et al. (2008). obtained the m-value of NC aluminum varying by strain. To overcome these problems, Mohebbi et al. (2014a). performed SR test under plane strain compression (PSC) condition after conducting an effective strain of 0.3 on UFG AA 1050 sheets. They showed that this test is a unique approach to assess the strain rate sensitivity of UFG aluminum as a continuous function of the strain rate. In the present study, such a procedure is performed on UFG AA 5052 besides UFG AA 1050 to compare their trends of stress relaxation and strain rate sensitivity. Moreover, the main purpose of this study is to develop a physically meaningful model for variation of the m-value by strain rate. As mentioned above, impact of the strain rate on m-value has already taken some attentions in literature. However, no considerable effort has been allocated to model and formulate such an impact on mvalues of the UFG and NC materials based on their special mechanisms of deformation. In addition to capability of prediction of the mechanical behavior of these materials, such a model can give evidences for the contributing phenomena during deformation of UFG and NC materials.

2. Experimental procedure Sheets of aluminum alloys AA 1050 and AA 5052 with thickness of 1.5 mm were used as the initial material. Before the process, AA 1050 and AA 5052 sheets were annealed at 330 C and solution treated at 550 C, respectively, for 1 h followed by water quenching. Then, they were cut in dimensions of 60 160 mm2 for Accumulative Roll-Bonding (ARB) process up to 8 cycles (AA 1050) and 4 cycles (AA 5052). ARB processing was performed at room temperature by a thickness reduction of 50% by rolls of 150 mm in diameter and speed of 50 rpm with no lubrication. Details of the process have been published in Refs. (Mohebbi et al., 2014a, 2014b). PSC samples with dimensions of 30 20 1.5 mm3 (rolling transverse normal directions) were cut from center of the ARB sheets. The surfaces of samples and indenters were precisely polished and lubricated by a PTFE grease before the test. During the test the indenters with breadth of 3 mm, basal width of 12 mm, height of 10 mm and length of 100 mm were aligned with the transverse direction, in which no spreading was imposed. Initial strain rate of this test was 6.4 103 s1. After a ﬁxed strain of ~0.3 b y PSC, the stress relaxation test was carried out under ﬁxed cross-heads for a period of 600 s. VonMises effective stress and strain were calculated for plane strain condition from the force and displacement, which were registered in 20 equally spaced intervals per second. Displacement was measured by a non-contacting extensometer which was used to track the distance between two marks attached to the main body of the indenters. The reader is referred to Refs (Mohebbi et al., 2014a, 2014b, 2015). for more details on the test as well as the methods for suppression of the elastic displacement of the tools and the effect of friction.

M.S. Mohebbi, A. Akbarzadeh / International Journal of Plasticity 90 (2017) 167e176

169

3. Stress relaxation results Relaxation curves of the specimens are shown in Fig. 1 (a)-(c). Various forms of stress-strain, stress-time and strain-time are depicted and linked in this ﬁgure. The amount of stress relaxation and the relevant plastic strain of AA 5052 specimens are much less than AA 1050 ones. While 25% of the stress value is relaxed in AA 1050-8-cycles specimen, the stress relaxation of AA 5052 specimens is only about 3% after 600 s. Stress relaxation of ultraﬁne grained aluminum is attributed to three phenomena: relaxation of effective (thermal) stress, recovery of internal (athermal) stress and grain boundary sliding (Mohebbi et al., 2015). It can be said that Mg solute atoms affect these contributing phenomena. Thermally activated deformation is hindered by solute atoms. In fact, mobility of dislocations and grain boundaries is much lower in Al-Mg alloys due to the interaction with Mg solute atoms (Soer et al., 2004; Gholinia et al., 2002). Therefore, in compliance with the observed results, very small amount of stress relaxation in UFG AA 5052 is expected. In case of UFG AA 1050, however, the amount of stress relaxation is increased by the number of ARB cycles, because the mentioned effective phenomena are enhanced by the imposed grain reﬁnement (Mohebbi et al., 2015). From the ﬁrst derivation of the strain-time curve, the strain rate is obtained versus time. Having this together with the stress-time data, it is possible to draw stress-strain rate curves. Such curves are represented in Log-Log coordinates in Fig. 1 (d). Since the slope of these curves is equal to the strain rate sensitivity, the m-value can be drawn as a function of strain rate. The results are shown in Fig. 2. Increase of the m-value by the number of ARB cycles in AA 1050 specimens was expected due to enhanced contribution of the thermally activated deformation as well as the reduced activation volume as the results of grain reﬁnement (Mohebbi et al., 2014a). Moreover, increase of the m-value by decrease of the strain rate, as was addressed in the introduction, is mainly explained by enhanced GBS (Ivanov and Naydenkin, 2014; Wang and Shan, 2008; Kumar et al., 2009). Ivanov and Naydenkin (2014) calculated the contribution of GBS to the overall deformation (h) of UFG Al at room temperature. Based on their results, h is increased up to 72% by decrease of the strain rate in the range of 1 105-8.6 103. In addition to GBS, Mohebbi et al. (2014a). have shown that recovery of the internal component of ﬂow stress can be considered as another phenomenon leading to the increase of m-value at lower strain rates.

Fig. 1. (a) Stress versus strain, (b) stress versus time, (c) strain versus time and (d) log-log plot of stress versus strain rate during relaxation test of various ARBed AA 1050 and AA 5052 specimens.

170

M.S. Mohebbi, A. Akbarzadeh / International Journal of Plasticity 90 (2017) 167e176

By comparing the strain rate sensitivity of the two alloys, it is obvious that the m-value of AA 5052 specimens are negligible (note that different scales are used in Fig. 2 for two alloys). This means that plastic deformation of UFG AA 5052 at ambient temperature involves much less viscous nature. Al-Mg alloys are generally known to show negative strain rate sensitivity at a quasi-static loading under ambient temperature which is correlated to dynamic strain aging (Kabirian et al., 2014; Kapoor et al., 2005). Obviously, application of the SR test on a material with negative strain rate sensitivity carries no sense. However, it has been shown that SPD can reduce the susceptibility to dynamic strain aging and enhance the strain rate sensitivity in UFG Al-Mg alloys. High dislocation density in heavily deformed Al-Mg alloys, which reduces the Mg atom atmosphere per dislocation, is thought to be responsible for suppression of dynamic strain aging (Kabirian et al., 2014; Kapoor et al., 2005; Zhao et al., 2014). That is why a positive strain rate sensitivity is measured for the UFG AA 5052 specimens. 4. Modeling Original formulation of the deformation during SR test, which dates back to 1970s (Dotsenko, 1979; Gupta and Li, 1970; de et al., 2011). During this test, total length of Batist and Callens, 1974), is also used in recent studies (Yang et al., 2016; Trojanova the specimen and the machine elements between the cross heads are constant. At this condition, a viscous material deforms plastically in expense of elastic strain of the sample-machine system. Therefore, the plastic and elastic strain rates of the specimen are related as (Dotsenko, 1979; Gupta and Li, 1970):

ε_ p ¼ _εe ¼

s_ 0 E

(1)

0

Where E is the effective elastic modulus of the sample-machine system. In development of original models for SR test, the internal (or athermal) stress component (si) is supposed to be constant. Instead, the effective (or thermal) stress component s*, as the rate dependent component, is believed to be relaxed (s_ ¼ s_ * ) (Dotsenko, 1979). Having Eq. (1) between ε_ (hereafter ε_ represents the plastic strain rate) and s_ , the m-value can be obtained during the SR test as a function of stress and the ﬁrst and second derivatives of that:

m¼

d ln s ε_ ds ε_ s_ s_ 2 ¼ ¼ ¼ d ln_ε s d_ε s €ε s s€

Fig. 2. Strain rate sensitivity of UFG AA 1050 and AA 5052 specimens as a function of strain rate.

(2)

M.S. Mohebbi, A. Akbarzadeh / International Journal of Plasticity 90 (2017) 167e176

171

This equation, in fact, can be considered as a general equation for the m-value during the SR test, so that it is only needed to have s as a function of time to derive the m-value versus time, and therefore, versus strain rate. 4.1. Equation derived from thermally activated dislocation glide If the plastic deformation is undertaken by dislocation movement, the Orowan equation would be used for the plastic strain rate:

ε_ ¼ f rm b v

(3)

Where 4 is a geometric coefﬁcient, rm is the density of mobile dislocations, b is the Burgers vector and v is the average dislocation velocity. This equation implies that the stress dependence of ε_ is related to the stress dependence of v. Dislocation glide is explicitly considered to be a thermally activated process with an activation energy depending on the effective stress. Therefore, v is related to the effective stress in the Arrhenius form:

DG0 v* s* v ¼ v0 exp kT

(4)

Where v0 is the pre-exponential factor, DG0 is the barrier activation energy at zero stress, v* is the activation volume, k is the Boltzmann constant and T is the absolute temperature. It follows from Eqs. (1), (3) and (4) that:

s_ ¼ E0 f rm b v0 exp

DG0 v* s*

(5)

kT

Stress can be obtained from integration of this differential equation (Dotsenko, 1979; de Batist and Callens, 1974):

sðtÞ ¼ sð0Þ a lnðb t þ 1Þ

(6)

In which t is the relaxation time, s(0) is the stress at beginning of the relaxation (t ¼ 0 s) and a and b are: 0 DG0 v* s*ð0Þ kT E f rm b v0 v* exp a ¼ *; b ¼ kT kT v

! (7)

Based on Eq. (6), the following derivatives are obtained:

s_ ¼

ab

ab2

€¼ ; s 2 ðbt þ 1Þ ðbt þ 1Þ

(8)

Which are replaced in Eq. (2) to derive the following equation for the m-value as a function of strain rate:

m¼

a sð0Þ a lnðabÞ þ a lnðE0 ε_ Þ

(9)

Strain rate sensitivity versus strain rate, which is obtained from this equation, is drawn in Fig. 3 for the 1050-8-cycles specimen with a ¼ 10 MPa and b ¼ 0.71 s1. These values are obtained from ﬁtting of Eq. (6) to the relaxation curve of AA 1050-8-cycles specimen (Fig. 1 (b)). E’ is measured from Eq. (1) as 1.25 GPa and s(0) ¼ 248 MPa is taken from relaxation curve of the 1050-8-cycles specimen at t ¼ 0 s (Fig. 1 (a) and (b)). As shown in this ﬁgure (the red dashed line), it is obvious that Eq. (9) can not ﬁt to the experimental results. In fact, the variation of m-value by the strain rate is signiﬁcantly underestimated by this equation. This implies the fact that thermally activated dislocation glide is not the only phenomenon in deformation and SR test of UFG metals. 4.2. Development of the model for UFG aluminum Modeling of the SR behavior based on the thermally activated dislocation movement is a worthy approach with physical meaning. However, it has been shown that the thermally activated dislocation movement is not the single governing event during SR test of UFG metals (Yang et al., 2016; Mohebbi et al., 2015). One operative phenomenon is decrease of the dislocation density. There is a strong driving force for this phenomenon due to high level of stored energy in a severely deformed metal. In addition, the distance between boundaries, which act as sinks for dislocations, is very small in UFG metals. Therefore, strong annihilation of dislocations is expected at beginning of the SR test of these materials. As a general fact, Silbermann et al. (2014). has illustrated that a rapid change of dislocation density is occurred by change of the loading state. Annihilation of dislocations has also been demonstrated by TEM observation of UFG AA 1050 after SR test (Mohebbi et al., 2015). Another possible phenomenon in stress relaxation of UFG metals is the grain boundary sliding (GBS). While ﬁne

172

M.S. Mohebbi, A. Akbarzadeh / International Journal of Plasticity 90 (2017) 167e176

Fig. 3. Strain rate sensitivity of the 1050-8-cycles specimen as a function of strain rate obtained from SR test and various models.

grains and low dislocation density inside the grains conﬁne the common mechanism of dislocation glide in UFG and NC metals, GBS plays an important role in deformation of these materials. In fact, presence of large fraction of high energy nonequilibrium grain boundaries promotes GBS in severely deformed metals (Mishnaevsky and Levashov, 2015). Operation of this phenomenon in deformation of NC and UFG metals has been shown even at room temperature when a low strain rate is applied. Choi et al. (2013). studied the room temperature creep of NC nickel undertaken by grain boundary diffusion and GBS. GBS has also been observed by atomic force microscopy (Chinh et al., 2006) and electron microscopy (Ivanov and Naydenkin, 2012) in UFG aluminum deformed at room temperature. The mentioned phenomena can explain why increase of the m-value by decrease of the strain rate is much faster than what expected from Eq. (9). Hence, in order to develop an appropriate model for prediction of strain rate sensitivity of UFG aluminum versus strain rate, contribution of the mentioned phenomena is added to the original model based on the thermally activated dislocation movement. Similar approach has been successfully implemented to explain why the absolute value of the slope ln ðs_ Þ versus ln (t) plot during the SR test of UFG AA 1050 is smaller than unity (Mohebbi et al., 2015). GBS is taken into account from the following equation (Choi et al., 2013):

ε_ GBS ¼

2 2 105 DGB b4 s* kTG d3

(10)

In which G is the shear modulus, d is the grain diameter, and DGB is the grain boundary diffusion coefﬁcient. DGB can be obtained from the generalized equation for FCC metals (Gust et al., 1985):

d DGB ¼ 9:7 1015 exp

9:07Tm T

m3 s1

(11)

Where Tm is the absolute melting temperature and d is the grain boundary width for which the value of 0.5 nm is always used (Mehrer, 2007). The dislocation glide and the GBS are not mutually exclusive. Assuming that the two mechanisms operate independently, their contributions to the total strain rate is additive. Therefore, ε_ GBS can be added to the ε_ correlated to dislocation glide (Eq. (3)). In this way, Eq. (5) is modiﬁed to:

s_ ¼ E0 f rm b v0 exp

DG0 v* s* kT

E0

2 105 DGB b4 * 2 s k T G d3

(12)

Which can be rewritten using parameters a and b:

s_ ¼ a b exp

s sð0Þ 2 105 DGB b4 E0 ðs si Þ2 a k T G d3

(13)

Note that the equality s sð0Þ ¼ s* s*ð0Þ is used in derivation of this equation. The second derivative of stress is found from Eq. (13) as:

M.S. Mohebbi, A. Akbarzadeh / International Journal of Plasticity 90 (2017) 167e176

s sð0Þ 2 105 DGB b4 E0 2s_ ðs si Þ a k T G d3

s€ ¼ bs_ exp

173

(14)

€, the m-value can be calculated from Eq. In term of s, Eq. (13) should be numerically integrated. Therefore, having s, s_ and s (2). This procedure is accomplished for the 1050-8-cycles specimen with d ¼ 260 nm (obtained from EBSD analysis (Mohebbi et al., 2014a)), T ¼ 298 K, b ¼ 2.86 1010 m, Tm ¼ 925 K and G ¼ 26.1 GPa (Totten, 2003). Regarding the internal stress component, the best ﬁtting is achieved bysi ¼ 156 MPa. This parameter is also experimentally estimated by the relaxation curve saturation method based on which the stress will be saturated to si at very long relaxation time (Dotsenko, 1979). In the present study, this method resulted in a very close value of 163 MPa after relaxation time of 3000 s while it was still decreasing with a very low rate. a, b and E’ are taken with similar values as used for Eq. (9) (the model based on thermally activated dislocation glide). In fact, since the thermally activated dislocation glide is taken from the ﬁrst model with no change, similar values are assigned to a and b. The result of this model is shown by the blue curve in Fig. 3. By comparing this curve with the previous model for dislocation glide, it can be seen that through contribution of GBS, the m-value is enhanced to the experimental results at lower strain rates. However, at higher strain rates, the overestimation by dislocation glide model is still observed. In order to consider the dislocation annihilation, and therefore, a decreasing dislocation density, a ratio of rðtÞ =rð0Þ is introduced. Hence, b is replaced by b(0) which includes rð0Þ . Accordingly, the following equation is obtained by modiﬁcation of Eq. (13):

s_ ¼ a bð0Þ

rðtÞ s sð0Þ 2 105 DGB b4 E0 exp ðs si Þ2 rð0Þ a k T G d3

(15)

From which the second derivative of stress is found as:

s€ ¼ a bð0Þ

!

rðtÞ s sð0Þ 2 105 DGB b4 þ bð0Þ s_ exp 2s_ ðs si Þ E0 rð0Þ a rð0Þ k T G d3 r_

(16)

€ by Eq. (2). Here, Again, s is obtained by numerical integration of Eq. (15). Then, the m-value is calculated from s, s_ and s values of a ¼ 10.6 MPa and b(0) ¼ 1.3 s1 are used for the best ﬁt. It is necessary to note that since the formulation is modiﬁed by considering a varying dislocation density, the proper values of a and b are different from the previous models. In fact, here b(0) includes rð0Þ while, as can be seen in Fig. 4, the ratio of rðtÞ =rð0Þ is considered to decrease by time from the value of 1 at t ¼ 0 s. Regarding the trend of variation, it has been revealed that a concave up decrease of dislocation density in logarithmic scale of time can be taken as a reason for experimentally seen abnormal jSj