Rate-dependent hardening model for pure titanium considering the effect of deformation twinning

International Journal of Mechanical Sciences 98 (2015) 80–92

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Rate-dependent hardening model for pure titanium considering the effect of deformation twinning Kwanghyun Ahn a, Hoon Huh b,n,1, Jonghun Yoon c a

Central Research Institute, Samsung Heavy Industries, 23 Pangyo-ro 227 beon-gil, Bundang-gu, Seongnam 463-400, Republic of Korea Department of Mechanical Engineering, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 305-701, Republic of Korea c Department of Mechanical Engineering, Hanyang University, 1271 Sa3-dong, Sangrok-gu, Ansan 426-791, Republic of Korea b

ar t ic l e i nf o

a b s t r a c t

Article history: Received 29 September 2014 Received in revised form 11 March 2015 Accepted 9 April 2015 Available online 17 April 2015

This paper is concerned with the strain hardening behavior of commercially pure titanium for a wide range of strain rates. Pure titanium has been described as having three stages of strain hardening behavior during compressive deformation. The stress–strain curve of pure titanium can be divided into three stages according to the strain hardening rate, and the strain hardening rate can be determined by the slope of the stress–strain curve. In the first stage, the strain hardening rate decreases as deformation continues due to dynamic recovery. The strain hardening rate shows a trend similar to general industrial metals. In the second stage, however, the strain hardening rate begins to increase as deformation continues. Following the second stage, the strain hardening rate decreases again in the third stage. Many researchers have determined that a sudden increase in the strain hardening rate in the second stage is caused by the generation of deformation twins, and the strain hardening behavior of pure titanium is influenced by the tendency of twin generation and evolution. In this paper, the effect of deformation twinning on the strain hardening behavior of pure titanium is investigated using OIM (Orientation Imaging Microscopy) analyses. EBSD (Electron Backscatter Diffraction) analyses are conducted to quantitatively observe the generation and evolution of deformation twins. The strain rate effect on the strain hardening is also investigated. Both tensile and compressive tests are conducted at strain rates ranging from 0.001/s to 10/s, and the effect of the strain rate on the three stages of the strain hardening behavior is quantified by observing the micro-structures of deformed specimens at the various strain rates. A novel rate-dependent hardening model is proposed by keeping trace of deformation twins with increase in the compressive strain, which induces the variation of the strain hardening rate. The proposed model is defined with a function of the plastic strain and the strain rate based on OIM results in terms of twin volume fraction and grain size distribution. The three stages of strain hardening behavior of titanium for a wide range of strain rates can be represented by one equation, and this equation can provide useful and simple way that can be applied to the numerical analysis. & 2015 Elsevier Ltd. All rights reserved.

Keywords: TB340H CP titanium Deformation twins EBSD (Electron Backscatter Diffraction) Hardening model Strain rate sensitivity

1. Introduction It is well known that the three stages of strain hardening behavior are observed during the compressive deformation of pure titanium. Kailas et al. [1] suggested that the stress–strain curve of pure titanium can be divided into three stages according to the tendency of the strain hardening rate. The strain hardening rate can be defined as the slope of the stress–strain curve. In the earlier plastic strain region after yielding, the strain hardening rate decreases as deformation continues. As the plastic strain increases, the flow stress increases via the strain hardening mechanism, but

n

Corresponding author. Tel.: þ 82 42 350 3222; fax: þ82 42 350 3210. E-mail address: [email protected] (H. Huh). 1 Present address: 291 Daehak-ro, Yuseong-gu, Daejeon 305-701, Korea.

http://dx.doi.org/10.1016/j.ijmecsci.2015.04.008 0020-7403/& 2015 Elsevier Ltd. All rights reserved.

the slope of the stress–strain curve decreases. This decrease is due to dynamic recovery, and general industrial metals show dynamic recovery with an increase in the plastic strain. Following this region, however, a sudden increase in the strain hardening rate is observed in the second stage. Both the flow stress and its slope increase as the plastic strain increases. In the third stage, the strain hardening rate decreases again as the plastic strain increases. Nemat-Nasser et al. [2] also reported the three stages of the strain hardening behavior of pure titanium observed from compressive tests at room temperature and suggested that the cause of the second stage is dynamic strain aging. According to the research of Doner et al. [3], however, dynamic strain aging in pure titanium is only observed at strain rates ranging from 3
10  5/s to 3
10  2/s and at temperatures ranging from 600 K to 850 K. It can be the clear evidence that the increase in the strain hardening rate in the second stage is not due to dynamic strain aging because the

K. Ahn et al. / International Journal of Mechanical Sciences 98 (2015) 80–92

conditions proposed by Nemat-Nasser et al. do not correspond with the conditions proposed by Doner et al. Salem et al. [4–7] suggested in their series of studies that the second stage for commercially pure titanium is caused by the occurrence of deformation twinning. The authors used high-purity titanium to remove the possibility of dynamic strain aging and demonstrated that three distinct stages of strain hardening behavior were still observed. They also investigated the microstructure of commercially pure titanium during the compression process and demonstrated that the onset of the second stage is correlated with the onset of deformation twinning. The occurrence of deformation twinning in commercially pure titanium has been reported together with its role in strain hardening. Strain hardening of pure titanium is noted to be influenced by the generation and evolution of deformation twins [8–11]. In accordance with Kalidindi et al. [12], the sudden increase in the strain hardening rate in the second stage is due to the combination of several hardening mechanisms caused by deformation twinning. Precise physical connections between deformation twinning and strain hardening have not yet been established. Kalidindi et al. assumed that the strain hardening of pure titanium changes via the Hall–Petch hardening and texture hardening mechanisms. The material is hardened through the Hall–Petch mechanism because the effective slip length is reduced when deformation twins are generated. Additionally, these authors showed, using OIM (Orientation Imaging Microscopy), that the orientation of a twinned region is rotated with respect to the matrix region. The change in the texture orientation induces the change in the yield stress, and it provides clear evidence that the strain hardening of the material changes due to the texture hardening mechanism when deformation twins are generated [13–15]. The complex strain hardening behavior of commercially pure titanium can be understood as a result of the combination of the Hall–Petch hardening and texture hardening mechanisms. Ahn et al. [16] proposed a phenomenological strain hardening model that can represent the three stages of strain hardening behavior of pure titanium based on the investigation of the microstructures during compressive deformation. By measuring the quantitative tendencies of the initiation and evolution of deformation twins using the EBSD analyses, the effect of deformation twinning on the strain hardening of pure titanium was quantified at a quasi-static condition. The role of the initiation and evolution of deformation twins was included into the strain hardening model in terms of the Hall–Petch hardening and texture hardening mechanisms to account for the three stages hardening behavior during the compressive deformation of pure titanium. It has been reported that the generation and evolution of deformation twins are influenced by the strain rate. Chichili et al. [17] investigated the effect of the strain rate on deformation twinning by

81

conducting split Hopkinson pressure bar tests of commercially pure titanium at a strain rate higher than 3
103/s. The results showed that the density of the twins at the same strain condition increases as the strain rate increases. These tests can provide the qualitative results that indicate the twin density increases with an increasing strain rate. In this paper, the compressive tests of commercially pure titanium are conducted at various strain rates ranging from 0.001/s to 10/s to investigate the effect of the strain rate on the three stages of strain hardening behavior induced by the evolution of deformation twinning. The universal testing machine (INSTRON5583) is utilized for the compressive tests at strain rates ranging from 0.001/s to 0.1/s, and the Gleeble3800 system is used for the strain rate ranging from 0.1/s to 10/ s. Tensile tests are also conducted at the same strain rates as the compressive tests to compare the strain hardening behavior with and without deformation twinning because it is reported that very little deformation twinning is generated during tensile deformation [18,19]. The strain rate effect on the three stages of the strain hardening behavior during compressive deformation is quantitatively investigated from the test results at the various strain rates. OIM analyses based on the EBSD are conducted to examine the three stages of strain hardening behavior of titanium. The generation and evolution of deformation twins with increase in the compressive plastic strain are quantitatively investigated. The strain rate effect on the three stages of strain hardening behavior is also investigated by observing the microstructures of the deformed titanium for a wide range of strain rates and by quantifying the strain rate effect on the generation and evolution of deformation twinning. Using the results of the compressive tests and the microscopic investigations, a rate-dependent strain hardening model is proposed. The model is developed based on the investigated effect of deformation twinning on the strain hardening behavior of titanium and its strain rate dependency. The model is capable of representing the three stages of strain hardening for a wide range of strain rates. The purpose of the model is to suggest an applicable form of the strain hardening representation for commercially pure titanium that can be applied to numerical analysis, and the model is shown to accurately represent the strain hardening behavior. The applicability of the model to strain rate conditions higher than thousands/s is also verified by applying the model to the additional SHPB tests.

2. Experiments 2.1. Material CP (commercially pure) titanium is usually divided into four grades according to its chemical composition and strength. A lower grade of CP titanium indicates higher purity and lower strength.

Table 1 Inspection certificate of the tested titanium (Aichi Steel Corporation). Product Applicable Spec. Material Size

Titanium round bars JIS H 4650 (2001) TB340H (MM) 12

Items

Mechanical properties

Min Max Results

Yield strength (MPa) 215 – 304

Elements

Chemical composition (%)

Min Max Results

H – 0.013 0.0008

Date of issue Finish

30-10-2007 Hot rolled, annealed and turned

Tensile strength (MPa) 340 510 435

Elongation (%) 23 – 33

– 56

Hardness (HV) 110 – 166

O – 0.20 0.112

N – 0.03 0.004

Fe – 0.25 0.034

C – 0.08 0.003

Reduction of area (%)

K. Ahn et al. / International Journal of Mechanical Sciences 98 (2015) 80–92

Grade 1 CP titanium, which has the highest purity, has limited applications in industrial structures because of its low strength. Grade 2 CP titanium is the most widely used grade due to its strength and weldability. In this paper, grade 2 CP titanium is selected to investigate the strain hardening behavior of CP titanium. The material investigated is TB340H, which is a rod-type grade 2 CP titanium, and it was obtained from Aichi Steel Corporation, Toyota Group. The material is hot extruded and has a diameter of 12 mm. The material inspection certificate is provided in Table 1. The chemical composition and mechanical properties of the supplied material satisfy the qualifications of grade 2 CP titanium.

1200

True stress [MPa]

82

900

600

10/s 1/s 0.1/s 0.01/s 0.001/s

300

0 0.0

0.1

0.2

0.3

Plastic strain

2.2. Compressive tests

Fig. 1. Specimen orientation and dimensions for the compressive tests.

Hardening rate (dσ /dε/G)

0.06

10/s 1/s 0.1/s 0.01/s 0.001/s

0.04

0.02

0.00 0.0

0.1

0.2

0.3

0.4

Plastic strain Fig. 2. True stress–plastic strain curves and strain hardening rate–plastic strain curves from the compressive tests.

0.020 Hardening rate (dσ/dε/G)

Compressive tests of pure titanium at a wide range of strain rates are conducted to observe the three stages of strain hardening behavior and its strain rate dependency. A universal testing machine, INSTRON5583, is utilized for the compressive tests for the strain rates ranging from 0.001/s to 0.1/s, and the Gleeble3800 system is used for the strain rates ranging from 0.1/s to 10/s. The maximum load and crosshead speed of the INSTRON5583 are 150 kN and 8.3 mm/s, respectively. Due to the speed limitation of the INSTRON5583, compressive tests at a strain rate higher than 0.1/s are conducted using the Gleeble3800. The maximum load and crosshead speed of the Gleeble3800 are 200 kN and 2000 mm/s, respectively. Cross tests using both testing apparatuses are conducted at the strain rate of 0.1/s to verify the deviation caused by using two different machines. The cylindrical specimen shown in Fig. 1 is used for the compressive tests. The compressive test specimens are fabricated from the axial direction of the bulk material because the axial direction is parallel to the extruded direction of the material. The diameter and the length of the specimens are 5 mm and 7.5 mm, respectively. The diameter is determined considering the maximum applicable load of the testing apparatuses, and the length of the specimen is determined to achieve a length/diameter ratio of 1.5. The ASTM E9 standard [20] recommends a length/diameter ratio of 1.5 for determining the general compressive strength properties of metallic materials. The interfaces between the specimen and the compression platens of the testing apparatuses are lubricated to avoid barreling. Fiberflon PTFE glass tape and Dow Corning high vacuum greases are used to minimize the frictional effects between the specimen and the compression platens. For both apparatuses, deformation of the specimen during compression has been measured by LVDT transducer. True strain of the specimen can be calculated based on engineering strain obtained from deformation measurement. Fig. 2 shows the true stress–plastic strain curves and the strain hardening rate–plastic strain curves from the compressive tests. The strain hardening rate is denoted by the slope of the true stress–plastic strain curve calculated by differentiating the true stress with respect to the plastic strain. Since the data has been obtained via numerical differentiation of experimental data,

Original Smoothed (adjacent-averaging)

0.015

0.010

0.005

0.000 0.0

0.2

0.4

0.6

0.8

Plastic strain Fig. 3. Example of the adjacent-averaging method for obtaining smoothed hardening rate.

adjacent-averaging method has been applied for smoothing of original data as shown in Fig. 3. The normalized strain hardening rate is determined by dividing the strain hardening rate by the shear modulus G. The three stages of strain hardening are clearly observed from the strain hardening rate– plastic strain curves for all of the strain rate conditions. Each stage of strain hardening can be classified by its slope: for Stage I, the decreasing strain hardening stage that is similar to dynamic recovery in conventional metals with BCC and FCC crystal structure; for Stage II, there is an increasing strain hardening stage; for Stage III, there is decreasing strain hardening stage. The onset strains of Stage II for five strain rate conditions are indicated using dotted lines. The onset strain of Stage II is reduced as the strain rate increases, and the hardening rate in Stage II shows a more rapid increase at the higher strain rates. From these results, the deformation twins are predicted to be generated earlier and evolve more rapidly as the strain rate increases. This result can be verified using the quantitative microscopic investigation described in the following chapter.

K. Ahn et al. / International Journal of Mechanical Sciences 98 (2015) 80–92

1000

83

Compression Swift fit of tension

True stress [MPa]

800

Fig. 4. Specimen orientation and dimensions for the tensile tests.

600 10/s 1/s 0.1/s 0.01/s 0.001/s

400 200

True stress [MPa]

800 0 0.00

600

400

200

0 0.00

0.10

0.15

0.20

0.25

0.30

Plastic strain Fig. 6. Comparison of the true stress–plastic strain curves from the compressive and the tensile tests.

10/s 1/s 0.1/s 0.01/s 0.001/s 0.05

0.10

0.15

0.20

0.25

0.30

Plastic strain Fig. 5. True stress–plastic strain curves from the tensile tests.

2.3. Tensile tests Tensile tests are also conducted at the same strain rates as the compressive tests. A universal testing machine, INSTRON5583, is utilized for the tensile tests for the strain rates ranging from 0.001/s to 0.1/s, and the HSMTM (High Speed Material Testing Machine) developed by Huh et al. [21] is used for the strain rates ranging from 0.1/s to 10/s. The HSMTM is a servo-hydraulic type tensile testing machine for the dynamic tests at intermediate strain rates. The maximum load and velocity of the apparatus are 30 kN and 7800 mm/s, respectively. The machine is equipped with a gripper fixture specially designed to obtain a constant tensile velocity during the test and to reduce the noise during data acquisition from the load cell. Cross tests using both testing apparatuses are conducted at a strain rate of 0.1/s to verify the deviation caused by using two different machines. The cylindricaltype specimen shown in Fig. 4 is used for the tensile tests. Both ends of the specimen are gripped by the jig via screw tightening to avoid slip during a tensile test. The tensile specimens are also fabricated along the axial direction because the axial direction is parallel to the extruded direction of the material. The diameter and the length of the gage section are determined from a finite element analysis for the gage section to be uniformly elongated. Fig. 5 shows the true stress–plastic strain curves from the tensile tests. Unlike the compression case, abrupt increase in the hardening rate is not observed in the tensile test because the specimen was fractured by necking before pffiffiffi a large deformation was induced. For HCP metals with c/ao 3, such as beryllium, titanium, zirconium, and magnesium, deformation twins are activated by c-axis tension because HCP metals are influenced by the c/a ratio. During compressive deformation, the c-axis in the basal texture tends to be stretched, which induces the tensile twins because the basal textures are initially distributed along the radial direction of the cylindrical specimen. In adverse, deformation twins are hardly observed when the specimen experiences tensile deformation since the c-axis in the basal texture is shortened during the tensile deformation. [18,19]. Fig. 6 shows a comparison of the true stress–plastic strain curves obtained from the compressive and tensile tests. In Fig. 6, the tensile stress–strain curves denote the extrapolated data of the tensile test results using the Swift model σ ¼ Aðεþ ε0 Þn

0.05

ð1Þ

where Aε0 n denotes the initial yield stress of the material. Since the Swift model is rate-independent model, stress–strain curve at each strain rate condition has been fitted using each Swift model. By comparing the compressive curves to the tensile, an increasing behavior in Stage II can be clearly observed. The increasing hardening rate in Stage II can be understood to be a result of deformation twinning. In Stage I, before the occurrence of deformation twinning, the stress–strain curves from the compressive and tensile tests show almost the same behavior, and the increasing hardening due to deformation twinning in Stage II is observed only in the compressive case.

3. Microscopic investigation To quantify the initiation and evolution of deformation twins in titanium during compressive deformation, microstructures and twin formations with a designated applied strain from the initial to nearly 0.4 are investigated with EBSD installed in a field emission scanning electron microscope, Hitachi SU6600. An analysis of the EBSD data is accomplished using the TSL OIM analysis software, and data with a confidence index (CI) value greater than 0.1 are used for the texture and microstructure analysis. Specimens for EBSD analyses are prepared using electropolishing which is performed using a mixture of 500 mL of methanol, 300 mL of butoxyethanol, and 50 mL of perchloric acid after mechanical polishing using 1 μm diamond paste to remove stains and provide a high-quality surface finish. For the investigation of the strain rate effect on the generation and evolution of deformation twins, EBSD analyses are conducted for the specimens compressed with three different strain rate conditions: 0.001/s; 0.1/s; and 10/s. Fig. 7 shows the strain intervals that are investigated: an undeformed specimen and several deformed specimens. From these investigations, the change in the twin volume fraction with increasing plastic strain and its strain rate dependencies can be quantified. For three strain rate conditions, the microstructures observed in Stage I are investigated for denser strain conditions to precisely observe the occurrence of deformation twins. In order to obtain deformed specimens with various strain conditions as shown in Fig. 7, the specimen has been compressed up to designated strain condition – one test should be conducted for one deformed specimen. For the strain rate conditions of 0.001/s and 0.1/s, deformation could be controlled precisely: therefore; deformed specimen with accurate strain condition could be obtained. The specimens could be exactly compressed up to the strain conditions of 0.025, 0.050, 0.075, 0.100, 0.150, 0.200, 0.250, and 0.400. For the strain rate condition of 10/s, however, deformation of the specimen could not be controlled precisely due to high deformation speed even if a stopper is utilized. The specimens could be compressed up to the strain conditions of 0.026, 0.046, 0.065, 0.085, 0.111, 0.147, 0.196, 0.285, and 0.386. Since

K. Ahn et al. / International Journal of Mechanical Sciences 98 (2015) 80–92

1000

0.06

800

0.05 (compression)

600

(d /d )/G

0.04 0.03

400 0.02 200

0.01

0 0.0

0.2

0.4

0.6

Hardening rate (d /d /G)

True stress [MPa]

84

0.00 0.8

Plastic strain

1000

0.05

True stress [MPa]

800 (compression) 600

0.04

(d /d )/G 0.03

400 0.02 200

0.01

0 0.0

0.2

0.4

0.6

Hardening rate (d /d /G)

0.06

0.00 0.8

Plastic strain 1000

0.05

True stress [MPa]

800 (compression) 600

0.04

(d /d )/G 0.03

400 0.02 200

0.01

0 0.0

0.2

0.4

0.6

Hardening rate (d /d /G)

0.06

0.00 0.8

Plastic strain

Fig. 7. Strain intervals of the deformed specimens for the EBSD analyses: (a) 0.001/s; (b) 0.1/s; (c) 10/s.

the purpose of this deformation control test is to investigate changing tendency of micro-structure with increase in the plastic strain, inaccurate deformation control at high strain rate would not be significant. Figs. 8–10 show the ED (Extruded Direction) inverse pole figure maps of the specimens with increase of plastic strain. The f1010g and f21 10g textures are dominantly distributed along the measured area of the undeformed specimen, as shown in Fig. 8(a), because most of the basal plane is parallel to the extrusion direction in the case of an extruded material. Deformation twins are not yet observed in the undeformed specimen in Fig. 8(a). In the case of the deformed specimens, however, deformation twins can be observed as early as a strain condition of approximately 0.025, and these deformation twins evolve as the deformation proceeds. The deformation twin in the EBSD image can be distinguished by its shape and orientation. Fig. 11 shows the deformation twins in the EBSD image. Deformation twins have lenticular shapes, and their orientations are approximately in the {0001} direction while f1010g and f21 10g textures are dominantly distributed in un-twinned regions. In all of the EBSD images in Figs. 8–10, deformation twins can be distinguished by investigating the shape and orientation of the grain. With these results, the twin volume fraction for each strain and strain rate condition can be calculated. Although twin area fraction can be calculated from two-dimensional EBSD image, twin area fraction in the image

has been regarded as twin volume fraction of the whole specimen since the twin area fraction measured on specific cross-sections tends to represent the twin volume fraction generally. The quantitative change in the twin volume fraction with the increase in the compressive strain at the three strain rate conditions is shown in Fig. 12. Deformation twins can be observed during an earlier part of compression right after yielding, and these deformation twins evolve as the deformation continues. The generation and evolution of a deformation twin induces the increase in the strain hardening. Fig. 13 shows a comparison of the strain hardening rate from the tensile and compressive tests at a strain rate of 0.001/s. The strain hardening rate in the compression case clearly begins to show greater values than the values observed in the tension case after deformation twins are generated. Deformation twins are even generated in Stage I, and the strain hardening rate begins to be influenced by the deformation twinning in that stage because the three stages during the compression test are solely classified by the sign of the slope of the strain hardening rate, i.e., the onset of Stage II in this research does not indicate the onset of deformation twinning. In Fig. 12, the deformation twins are initiated during the earlier portion of the plastic strain, evolve as deformation proceeds, and saturate at a specific amount of plastic strain for all of the strain rate conditions. The effect of the strain rate on the change in the twin volume fraction can also be observed. As the strain rate increases, the deformation twins are initiated during the earlier strain, evolve rapidly, and saturate at an earlier strain with a larger twin volume fraction. These tendencies correlate with the effect of the strain rate on the strain hardening rate in Fig. 2(b). As shown in Fig. 2(b), the strain hardening rate begins to increase during the earlier strain, the strain hardening rate increases rapidly, and the increase of the strain hardening rate ends at a lower value of strain as the strain rate increases. These correlations are the clear evidence that the onset of Stage II is caused by the generation and the evolution of deformation twins. In all of the strain rate conditions, there is no additional increase in the strain hardening rate when the twin volume fraction becomes saturated. As shown in Salem et al. [6], a high twin density induces twin–twin and slip– twin interactions, which interfere with the production of new twins in the matrix at high strain levels. Fig. 14 shows the change in the average grain size with the increase in the plastic strain. The average grain size decreases as compression proceeds. The decreasing and saturation tendencies of the average grain size correlate with the increasing and saturation tendencies of the twin volume fraction. By comparing Figs. 12 and 14, the similar tendencies of the twin volume fraction and the average grain size are clearly shown. The decreasing region of the average grain size appears to be the same as the increasing region of the twin volume fraction. The effect of the strain rate on the change in the average grain size also demonstrates the same tendency as the effect of the strain rate on the change in the twin volume fraction. The average grain size begins to decrease with the generation of deformation twins and becomes saturated when the twin volume fraction is saturated. By comparing the EBSD image of the undeformed and the deformed specimens, the average grain size of the deformed specimen is reduced compared with the average grain size of the undeformed specimen because of the generation of deformation twins, although the grain sizes of the un-twinned region in all of the specimens are similar. This finding is the result of the twin boundaries playing a role as the new grain boundaries. For this reason, the occurrence of deformation twins results in a reduction effect of the average grain size. The increase in strain hardening due to the generation of deformation twins can be explained by the Hall–Petch mechanism caused by a reduction in the grain size. A more detailed discussion on the effect of deformation twinning on strain hardening will be presented in the following section.

K. Ahn et al. / International Journal of Mechanical Sciences 98 (2015) 80–92

85

Fig. 8. ED inverse pole figure map of TB340H with an increase in the plastic strain at a strain rate of 0.001/s: (a) initial; (b) ε ¼0.025; (c) ε ¼ 0.050; (d) ε ¼0.075; (e) ε¼ 0.100; (f) ε ¼0.150; (g) ε¼ 0.200; (h) ε ¼0.250; (i) ε¼ 0.400.

86

K. Ahn et al. / International Journal of Mechanical Sciences 98 (2015) 80–92

Fig. 9. ED inverse pole figure map of TB340H with an increase in the plastic strain at a strain rate of 0.1/s: (a) ε¼ 0.025; (b) ε¼ 0.050; (c) ε ¼0.075; (d) ε ¼0.100; (e) ε¼ 0.150; (f) ε ¼0.200; (g) ε ¼0.250; (h) ε ¼0.400.

K. Ahn et al. / International Journal of Mechanical Sciences 98 (2015) 80–92

87

Fig. 10. ED inverse pole figure map of TB340H with an increase in the plastic strain at a strain rate of 10/s: (a) ε¼ 0.026; (b) ε¼ 0.046; (c) ε¼ 0.065; (d) ε¼ 0.085; (e) ε¼ 0.111; (f) ε ¼0.147; (g) ε ¼0.196; (h) ε¼ 0.285; (i) ε ¼0.386.

88

K. Ahn et al. / International Journal of Mechanical Sciences 98 (2015) 80–92

Fig. 11. Occurrence of twinning in the microstructure of the compressed TB340H (ε ¼ 0.150 at a strain rate of 0.001/s).

22

Measured Tanh fit

Average grain size [μ m]

Twin volume fraction

0.15 0.12 0.09 0.06

10/s 0.1/s 0.001/s

0.03 0.00 0.0

0.1

0.2

0.3

0.4

Fig. 12. Effect of the strain rate on the change in the twin volume fraction.

compression tension twin volume fraction

0.06

0.15

0.10 0.04 0.05

0.02 0.00 0.0

0.2

0.4

0.6

Twin volume fraction

Hardening rate (dσ/dε/G)

0.20

0.08

16 14 12

Measured Tanh fit 0.1

0.2

0.3

0.4

0.5

Plasitc strain

Plastic strain

0.10

18

0 0.0

0.5

0.001/s 0.1/s 10/s

20

0.00 0.8

Plastic strain Fig. 13. Effect of the generation and the evolution of the deformation twins on the strain hardening rate (0.001/s).

4. Rate-dependent strain hardening model for pure titanium The dynamic response of a metallic material is indispensable for the numerical analysis of high-speed conditions [21–23]. The

Fig. 14. Effect of the strain rate on the change in the average grain size.

most effective way to apply the dynamic hardening behavior of the material to a numerical simulation is to use the dynamic hardening model. Various rate-dependent hardening models have been suggested by many researchers to represent the effects of the strain and the strain rate simultaneously on the complex hardening characteristics of metallic materials [24]. However, most of the models, such as the Johnson–Cook, the Zerilli–Armstrong, and the Khan–Huang, only describe dynamic recovery with an increase in the plastic strain. Consequently, there is no appropriate hardening model that is capable of representing the three stages strain hardening behavior of pure titanium and its strain rate dependency. The primary purpose of this paper is to propose a ratedependent hardening model for pure titanium that can be simply applied to dynamic numerical simulations. To describe the rate-dependent hardening model for pure titanium that consider the effect of deformation twinning, it is necessary to investigate the quantitative role of deformation twinning on the flow stress of the titanium. Although direct physical connections of deformation twinning to the strain hardening behavior have not yet been established, it is widely accepted that deformation twinning influences the material strength by

K. Ahn et al. / International Journal of Mechanical Sciences 98 (2015) 80–92

changing the strain hardening behavior via several hardening mechanisms. The most widely accepted mechanisms are the Hall–Petch hardening mechanism, which is a result of the grain size reduction, and the texture hardening mechanism, which is a result of lattice re-orientation in the twinned region. As shown in the microstructures of the deformed titanium in the previous section, the occurrence of deformation twinning has an effect of grain refinement, and the reduction in the average grain size induces an increase of the strain hardening rate via the Hall–Petch mechanism. The Hall–Petch mechanism can be explained as the relationship between the grain size and the yield stress by assuming that the grain boundaries act as obstacles of dislocation movement. The number of grain boundaries in the designated area increases as the effective grain size decreases; therefore, a higher driving force is required for plastic deformation to occur according to the Hall–Petch mechanism. The relationship between the grain size and the yield stress can be described mathematically by the Hall–Petch equation σ y ¼ σ 0 þ kd

1 2

ð2Þ

where σ y is the yield stress, σ 0 is a material constant for the initial stress for the dislocation movement, k is the strengthening coefficient, and d is the average grain diameter. The hardening increase of titanium due to deformation twinning can be explained by the Hall–Petch mechanism. Deformation twinning results in a reduction in the average grain size, which was investigated through microscopy. The material is eventually hardened via the Hall–Petch mechanism. In accordance with Kalidindi et al. [12], the hardening increase of titanium cannot be explained solely by the Hall–Petch mechanism. By measuring the average grain size after the onset of deformation twinning, the authors showed that the strain hardening calculated by the Hall–Petch equation is different from that obtained experimentally. After this comparison, they suggested that the material is not hardened by the Hall–Petch mechanism alone but that there should be another hardening mechanism caused by the deformation twinning. Another hardening mechanism resulting from deformation twinning can be explained by texture hardening that is a result of lattice re-orientation in the twinned region. It is well known that the lattice of the twinned region is re-orientated compared to that of the un-twinned region. In accordance with crystal plasticity theory, the yield stress of a material is determined by the combination of the yield stress of each grain, and the yield stress of each grain is influenced by its orientation [25,26]. As investigated through the microscopy in the previous section, the lattice orientation of twinned region is transformed into the {0001} direction while f1010g and f2110g textures are dominantly distributed in un-twined regions, and it is clear that the lattice re-orientation of the twinned region results in the hardening change of material. The change in the yield stress caused by the orientation change can be explained by σ y ¼ Mσ 0

ð3Þ

where σ y is the yield stress of the grain after the orientation change and σ 0 is the yield stress of the grain before the orientation change. M is a parameter that can explain the yield stress change due to the orientation change. Different grains have different values of M because M is a grain-level parameter. According to the microscopic investigation, all of the orientations of the twinned regions are approximately distributed in the {0001} direction. Based on these results, it can be assumed that the M parameter of each grain in the twinned region has the same value. The flow stress change in the entire material due to the flow stress change in the twinned region can be expressed as the following:      σ ¼ σm
1  f t þ σm
M
f t

    ¼ σm
1 þ M  1
f t

89

ð4Þ

where σ m is the stress in the matrix region, f t is the twin volume fraction, and M is a texture hardening parameter that can represent the average yield stress change of the entire twinned region. In Eq. (4), the first bracket indicates the maintenance of the stress in the un-twinned region, and the second bracket indicates the stress change in the twinned region. Although the Hall–Petch and texture hardening mechanisms are not the only mechanisms caused by deformation twinning, it is assumed in this study that the hardening of the material is influenced by these two types of mechanisms alone. When deformation twins are generated, the material is hardened through the combination of the Hall–Petch hardening mechanism induced by grain size reduction and the texture hardening mechanism induced by the lattice re-orientation of the twinned region. The combination of twin hardening mechanisms caused by deformation twinning can be expressed as the combination of Eqs. (2) and (4), which is   1      ð5Þ σ ¼ σ m þ kda 2
1 þ M  1
f t where σ m is the stress in the un-twinned region, da is the average grain size, f t is the twin volume fraction, and k and M are the Hall– Petch hardening parameter and texture hardening parameter, respectively. The first bracket in Eq. (5) indicates the Hall–Petch hardening due to the grain size reduction, which is a result of the deformation twinning, and the second bracket indicates the texture hardening, which is due to the lattice re-orientation of the twinned regions. To represent Eq. (5) by the rate-dependent strain hardening model, each term of σ m , f t , and da in Eq. (5) should be expressed as functions of the strain and strain rate. According to the comparison of the flow stress from the tensile and the compressive tests in Fig. 6, the stress–strain curves from the compressive and the tensile tests demonstrate approximately the same behavior before the occurrence of deformation twinning. Therefore, the flow stress for the un-twinned region, σ m , can be assumed to be the flow stress from the tensile tests. There are many models that are capable of representing the tensile stress–strain curves of titanium because there is no deformation twinning during the tensile deformation. In this research, the Lim–Huh model [27] is selected as the best model for representing the tensile stress–strain curves of the titanium. This selection is based on a comparison of the fitted tensile test results using several dynamic hardening models [24,28]. Fig. 15 shows the fitted tensile stress–strain curves of titanium using the Lim–Huh model and the standard deviation of fitted results [29]. The second term that should be expressed as a function of the strain and strain rate is f t , which is the change in the twin volume fraction with the increase in the strain. As shown in the microscopic investigation in the previous section, the twin volume fraction starts to increase at a specific amount of strain and is then saturated at the end of Stage II. These types of behaviors can be expressed, as shown in Eq. (6), by introducing a hyperbolic tangent function

hε εt i ð6Þ f t ¼ f t ðεÞ ¼ f m
tanh π εs  εt where f m is the saturated value of twin volume fraction and εs is the corresponding saturated plastic strain. εt denotes the onset strain of twinning initiation. The term, hε  εt i, represents following relationship:  hai ¼

a

where a Z 0

0

where a o 0

ð7Þ

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K. Ahn et al. / International Journal of Mechanical Sciences 98 (2015) 80–92

Tensile tests LH fit

800

True stress [MPa]

700 600 500 10/s 1/s 0.1/s 0.01/s 0.001/s

400 300 0 0.00

0.05

0.10 0.15 0.20 Plastic strain

0.25

0.30

Standard deviation [MPa]

50

Fig. 17. Representation of the average grain size as a function of the plastic strain using a hyperbolic tangent function (0.001/s).

twin volume fraction, respectively. Fig. 17 shows the representation of the average grain size using Eq. (8). Eqs. (6) and (8) are the representations of the twin volume fraction and average grain size as functions of the strain, and these equations are independent of the strain rate. The effect of the strain rate on the twin volume fraction and average grain size is already shown in Figs. 12 and 14 and can be summarized as follows.

40 30 20 10 0 1E-3

0.01

0.1

1

10

-1

Strain rate [s ] Fig. 15. Fitting of the tensile test results using the Lim–Huh model: (a) fitted results; (b) standard deviation of fitted results at each strain rate.

(1) Deformation twins are initiated at an earlier strain as the strain rate increases. (2) Deformation twins are saturated at an earlier strain as the strain rate increases. (3) Saturated twin volume fraction increases as the strain rate increases. (4) The average grain size is saturated at a smaller value as the strain rate increases. These four kinds of strain rate dependencies can be expressed by introducing the strain rate-sensitive parameters: C t ; C s ; C f , and C d εt ðε_ Þ ¼ εt ð1  C t ln ε_ Þ

ð9Þ

εs ðε_ Þ ¼ εs ð1 C s ln ε_ Þ 

f m ðε_ Þ ¼ f m 1 þ C f ln ε_

ð10Þ 

d1 ðε_ Þ ¼ d1 ð1  C d ln ε_ Þ Fig. 16. Representation of the twin volume fraction as a function of the plastic strain using a hyperbolic tangent function (0.001/s).

By introducing Eq. (7), the effect of the deformation twinning on the proposed model is activated right after the plastic strain reaches εt . Fig. 16 shows the fitted result for the evolution of the twin volume fraction using Eq. (6). The third term that should be expressed as a function of the strain and strain rate is da , which is the change in the average grain size with an increase in the strain. da can be expressed as a function of the strain using the same form as Eq. (6) because the decreasing and saturation behavior of the average grain size correlate with the increasing and saturation behavior of the twin volume fraction. The reduction of the average grain size due to the increase of twinned region can be expressed as a function of the strain:

hε  εt i þ d0 ð8Þ da ¼ da ðεÞ ¼ ðd1  d0 Þ
tanh π εs  εt where εt and εs in Eq. (8) are the same parameters as in Eq. (6) because the decreasing and saturation regions of the average grain size correlate with the increasing and saturation regions of the

ð11Þ ð12Þ

The construction of the strain rate dependent term, ð1 7 C ln ε_ Þ is adopted from the most widely used rate-dependent model of the Johnson–Cook model. By applying Eqs. (9)–(12) to Eqs. (6) and (8), the effect of the strain rate on the twin volume fraction and the average grain size can be expressed. Figs. 12 and 14 show representations of the twin volume fraction and the average grain size as functions of the strain and the strain rate, respectively. By applying Eqs. (6) and (8)–(12) into Eq. (5), the final form of the rate-dependent strain hardening model, taking into consideration of deformation twinning effect, can be written as follows:  1

 1       1þ M1 f t σ ¼ σ ðε; ε_ Þ ¼ σ m ðε; ε_ Þ þ k da 2  d0 2

ε εt ðε_ Þ þ d0 da ¼ da ðε; ε_ Þ ¼ ðd1 ðε_ Þ  d0 Þ
tanh π εs ðε_ Þ  εt ðε_ Þ where

ε  εt ðε_ Þ f t ¼ f t ðε; ε_ Þ ¼ f m ðε_ Þ
tanh π εs ðε_ Þ εt ðε_ Þ

ð13Þ

Based on the tensile stress–strain relation, σ m ðε; ε_ Þ is the flow stress in the un-twinned region, and the Hall–Petch hardening and texture hardening caused by deformation twinning are applied after the strain εt ðε_ Þ is achieved. σ m ðε; ε_ Þ can be selected for any

K. Ahn et al. / International Journal of Mechanical Sciences 98 (2015) 80–92

True stress [MPa]

1000

800

10/s 1/s 0.1/s 0.01/s 0.001/s

600

400

0 0.0

Proposed model Experiments Base model (Lim-Huh fit of tensile tests)

0.1

0.2

0.3

0.4

Plastic strain Fig. 18. Fitting of the compressive stress–strain curves of TB340H using the proposed model.

Table 2 Model coefficients of the proposed model for TB340H.   k MN=m3=2 d0 ðmÞ d1 ðmÞ fm M

3.79

εt

0.062

20.5
10  6 3.6
10  6 0.574 0.082

εs Ct Cs Cf

0.561 0.107 0.020 0.001

Cd

0.022

Proposed model Experiments

1200

True stress [MPa]

1000 3000/s 2000/s 100/s 10/s 1/s 0.1/s 0.01/s 0.001/s

800 600 400 0 0.0

0.1

0.2 Plastic strain

0.3

0.4

Fig. 19. Fitting of the compressive stress–strain curves of TB340H using the proposed model at a strain rate ranging from 0.001/s to 3000/s.

rate-dependent hardening model that can accurately represent the tensile stress–strain curves. In this research, the Lim–Huh model is selected for σ m ðε; ε_ Þ. k, d0 , and d1 are parameters that represent the hardening effect due to the Hall–Petch mechanism. f m and M are parameters that represent the texture hardening due to the lattice re-orientation. εt and εs are parameters that represent the region affected by twinning, and C t ; C s ; C f , and C d are parameters that represent the strain rate sensitivities of the onset strain of the twins, the saturation strain of the twins, the saturated value of the twin volume fraction, and the saturated value of the average grain 1 2

size, respectively. By subtracting d0 1  da 2

from the effective grain size

in the first bracket of Eq. (13), the Hall–Petch hardening term can express the hardening mechanism due to the reduction of the grain size in the twinned region alone. Fig. 18 shows a comparison of the stress–strain curves obtained from the compression tests and Eq. (13). The proposed model in Eq. (13) presents an accurate representation of the complex strain hardening behavior of the test results at various strain rates. The material constants in Eq. (13) for TB340H CP titanium are shown in Table 2. The proposed model presents a suitable form for the phenomenological representation of the abnormal strain hardening behavior of commercially pure titanium. The material constants of the model can be perceived as the fitting parameters, although each form of the model is constituted based on the effect of deformation

91

twinning on the strain hardening behavior. It is not necessary for the constants to have physically meaningful values because the physical basis of deformation twinning is only introduced to determine the most applicable form of the model. Additionally, the Hall–Petch hardening and texture hardening are assumed to be the all of the hardening mechanisms caused by the deformation twinning in this research, but there are likely other complex mechanisms. The purpose of the model is to suggest an applicable form of ratedependent strain hardening for commercially pure titanium that can be applied to a numerical analysis, and the proposed model can accurately represent the strain hardening behavior, as shown in Fig. 18. Consequently, because the form of the proposed model is based on the phenomenological results, the parameters have to be regarded as the adjustable values in the fitting process. That is, the microscopic investigation was only introduced to determine the reasonable form of the phenomenological model while developing the model, but the model can be fitted without microscopic investigation during the fitting process. The proposed model is developed based on the test results at a strain rate ranging from 0.001/s to 10/s. For this reason, to apply the proposed model to higher strain rates, the applicability of the model to higher strain rates should be verified. For the verification, additional compressive tests at a strain rate of 100/s were conducted using the Gleeble3800. SHPB (Split Hopkinson Pressure Bar) tests are also conducted to obtain the stress–strain curves at strain rates of 2000/s and 3000/s. Fig. 19 shows the stress–strain curves obtained from the compression tests at strain rates ranging from 0.001/s to 3000/s compared with the solution given by Eq. (13). The model is constructed without any microscopically investigated results. The material constants are obtained by fitting only the experimental results. The proposed model presented in Eq. (13) presents an accurate representation of the test results even at strain rates of up to 3000/s.

5. Conclusion Deformation twinning is one of the two principal modes of plastic deformation together with slip, while it becomes prevalent only at high strain rates and low temperatures. However, it is reported that deformation twinning is also observed in quasi-static states and at room temperature for large deformations in several HCP metals because of the lack of a slip system. The three stages of strain hardening behavior of pure titanium investigated in this research are known to be caused by deformation twinning. In this paper, the generation and evolution of deformation twins during the compression of TB340H CP titanium were quantitatively investigated using compressive tests and EBSD analyses. The change in the twin volume fraction and the average grain size were measured quantitatively through a microscopic investigation using EBSD analysis, and the three stages of the strain hardening of CP titanium and the stages' strain rate dependencies were explained from these investigations. A rate-dependent hardening model that is capable of representing the effect of deformation twinning was proposed by assuming that the hardening behavior of the material is influenced by deformation twinning via two mechanisms: Hall–Petch hardening and texture hardening. The effect of deformation twinning was represented by incorporating the combination of two hardening mechanisms. The conclusions of this paper can be summarized in the following. (1) Three stages of the strain hardening behavior are confirmed during the compression of TB340H for all of the strain rate conditions investigated (0.001–10/s). (2) Deformation twins are observed during the compression of TB340H using an EBSD analysis. The three strain hardening

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K. Ahn et al. / International Journal of Mechanical Sciences 98 (2015) 80–92

stages of TB340H can be interpreted as a result of the twinning behavior and, specifically, the initiation, evolution, and saturation of the twins. The effect of the strain rate on the three stages of the strain hardening is a result of the effect of the strain rate on the generation, evolution, and saturation of the deformation twins. (3) The generation of deformation twins produces a reduction effect on the grain size because the twin boundaries play a role as the new grain boundaries. The decreasing and saturation tendencies of the average grain size and their strain rate dependencies correlate with the increasing and saturation tendencies of the twin volume fraction and their strain rate dependencies. (4) Deformation twinning influences the hardening behavior of the material. Two widely accepted mechanisms of Hall–Petch hardening and texture hardening are assumed to be the main effects of deformation twinning on the strain hardening behavior of TB340H. A rate-dependent hardening model has been suggested by incorporating the combination of the two hardening mechanisms of deformation twinning in the tensile model. The twinning effects can be expressed using the Hall– Petch effect through a decrease in the average grain size and the lattice re-orientation of the twinned region through an increase in the twin volume fraction. (5) The proposed model can accurately describe the stress–strain relationship at a higher strain rate condition of up to thousands/s, although the model is developed based on the test results up to 10/s. The applicability of the model to higher strain rate conditions was verified by applying the model to the SHPB test results. The proposed model appears to be helpful and useful for researchers who will conduct dynamic numerical simulations on pure titanium.

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