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Work-hardening stages and deformation mechanism maps during tensile deformation of commercially pure titanium
Computational Materials Science 76 (2013) 52–59
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Work-hardening stages and deformation mechanism maps during tensile deformation of commercially pure titanium Hanka Becker a,b, Wolfgang Pantleon a,⇑ a b
Section of Materials and Surface Engineering, Department of Mechanical Engineering, Technical University of Denmark, Produktionstorvet 425, 2800 Kgs. Lyngby, Denmark Institute of Materials Science, Faculty of Materials Science and Technology, TU Bergakademie Freiberg, 09599 Freiberg, Germany
a r t i c l e
i n f o
Article history: Received 9 November 2012 Received in revised form 12 March 2013 Accepted 20 March 2013 Available online 16 May 2013 Keywords: Tensile deformation Work-hardening Twinning Dislocations Deformation mechanism maps
a b s t r a c t Commercially pure titanium was tensile tested at different strain rates between 2.2 104 s1 and 6.7 101 s1 to characterize the strain rate dependence of plastic deformation and the dominating deformation mechanisms. From true stress-true plastic strain curves, three distinct work-hardening stages are identified. The work-hardening rate decreases linearly with increasing flow stress for all three stages and the work-hardening rate is the controlling factor for the transition between the different stages and mechanisms. During the initial stage (at lowest stresses) plastic deformation is carried mainly 0 1 0i twin by dislocation slip, in the following stage (for moderate stresses), an abundance of 64:6 h1 2gh1 1 2 3i compression twinning. During the last boundaries form indicating the dominance of f1 1 2 0i twin boundaries are detected caused by stage before the onset of necking, additional 84:8 h1 1 2 2gh1 0 1 1i tensile twinning. Based on the microstructural findings and the strain rate sensitivity, f1 0 1 deformation mechanism maps are constructed. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction Nanostructured titanium has become an interesting candidate material for medical implants. Pure titanium is machinable and biocompatible, but its strength is insufficient for load-bearing implants. Nanostructuring of titanium provides a new and promising method for improving its mechanical properties [1–8]. Various methods of severe plastic deformation (SPD) have been successfully applied for grain refinement in pure Ti [1,2]. Commercially pure (CP) titanium has been processed via equal channel angular pressing (ECAP) [3,4], high pressure torsion [5,6] and hydrostatic extrusion [7]. In order to determine the optimal directions of development and design of nanostructured titanium for medical applications, fundamental understanding of the deformation mechanisms and reliable computational models for the virtual, numerical testing of these materials are necessary. The development of such tools is the aim of the EU FP7 Project ViNaT, carried out in collaboration with Russian partners. For validation of the models concise data sets for the mechanical properties are required. As a point of reference, the mechanical behavior during tensile tests in a broad range of strain rates is reported here for conventionally processed commercially pure titanium. Pure titanium has a hexagonal close-packed crystal structure with a c/a ratio of 1.587. Deformation by dislocation slip may occur on a number of different types of slip systems, but activation of ⇑ Corresponding author. Tel.: +45 45252315; fax: +45 45936213. E-mail address: [email protected] (W. Pantleon). 0927-0256/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2013.03.028
only the slip systems with lowest critical resolved shear stresses (basal and prismatic systems) is insufficient to ensure compatible deformation of titanium polycrystals. Therefore, either additional slip systems with higher critical resolved shear stresses as pyramidal slip must be activated or plastic deformation by twinning must occur also. The most common twinning systems are summarized in Table 1. 2. Experimental 2.1. Material Commercially pure titanium of ASTM grade 1 was received asrolled, annealed and pickled sheets of a thickness t = 1 mm. The chemical composition is given in Table 2. Dog-bone specimens for tensile tests with a width w = 12.5 mm and a gauge length l = 75 mm were machined from the as-received sheet material with tensile direction along the original rolling direction (RD). 2.2. Mechanical testing Continuous, strain controlled tensile tests were conducted using two different testing machines: an electromechanical Instron 8874 and a servo hydraulic Instron 88 R 1362. Five different constant cross head velocities v between 1 mm/min and 3000 mm/ min were applied corresponding to nominal strain rates e_ ¼ v =l between 2.2 104 s1 and 6.7 101 s1. Additionally, strain rate jump tests were performed where the cross head velocity was
H. Becker, W. Pantleon / Computational Materials Science 76 (2013) 52–59 Table 1 Common twinning systems for hexagonal crystal structures defined by twinning plane and direction [8–10]. The orientation difference between twin and parent crystal are listed together with the parameter R characterizing the frequency of (almost) coinciding lattice sites between parent crystal and twin [11]. The deformation caused along the c-axis is specified for the case of titanium with c/a ratio less than the ideal value. Twinning system
Rotation angle (°)
Rotation axis
R
Deformation along caxis for Ti
2gh1 0 1 1i f1 0 1 1gh1 1 2 6i f1 1 2 2gh1 1 2 3i f1 1 2 1gh1 0 1 2i f1 0 1
84.8 34.5 64.6 57.4
0i h1 1 2 0 1 0i h1 0 1 0i h1 0i h1 1 2
11 11 7 13
Tensile Tensile Compressive Compressive
Dl l
3.1. Flow curves
ð1Þ
and true stress
F A
rtr ¼ ¼
F Dl 1þ w0 t 0 l
ð2Þ
were calculated from the recorded changes Dl in the cross head position and the required force F. The true plastic strain
epl ¼ etr
rtr
ð3Þ
Eeff
identified by the characteristic disorientation between twin and its parent. The four different types of twins listed in Table 1 were distinguished. According to the Brandon criterion [12],paffiffiffiffimaximal deviation from the ideal orientation difference of 15 = R was accepted determined by the fraction 1/R of (almost) coinciding lattice sites. 3. Results
changed suddenly between two different values several times during the test for determining the instantaneous strain rate sensitivity. Assuming uniform elongation, true strain
etr ¼ ln 1 þ
53
was derived taking into account an effective elastic modulus Eeff obtained from the initial elastic part of true stress versus true strain curves. 2.3. Electron backscatter diffraction The microstructure evolving during tensile testing was investigated by electron backscatter diffraction (EBSD). Pieces of 8 mm 5 mm were cut from the as-received sheet and the tensile tested samples. Free surfaces perpendicular to either the original normal direction (ND) or transversal direction (TD) were grinded with SiC paper, and mechanically polished with diamond paste and OP–S. As a final step, the specimen were electro-polished with Struers A3 solution for 20 s at 40 V (after increasing voltage from 0 V in about 45 s) to achieve an optimal surface quality. For acquiring electron backscattering patterns, two different scanning electron microscopes were utilized: a Zeiss Supra 35 with a Nordlys II EBSD detector using the program Flamenco of HKL Channel 5 for indexing and a JEOL JSM-840 with a phosphor screen as detector using CROMATIC v.2.10 software to index the measured bands. Orientations were gathered at individual positions on regular square grids on the surface of the specimen with step sizes between 0.4 lm and 2 lm. For analyzing the measured data, the programs Tango and Mambo of HKL Channel 5 were applied. The measured orientations are used to calculate pole figures or presented in orientation maps displaying the crystallographic direction along a particular specimen direction. In such orientation maps, twin boundaries were
Fig. 1 summarizes the true stress versus true plastic strain curves of all continuous tensile tests with constant strain rate. Obviously, higher loads are required for deforming with higher stain rates. Some differences between specimens deformed at the same strain rate become obvious, in particular for the largest strain rate. The reason for these deviations could not be identified definitely, but it is most likely caused by a slightly different preparation of the specimens and the use of two different tensile machines with quite different stiffnesses. A comparison of specimens prepared in the same manner and deformed at the same machine only does not show such strong variations. An overview over the mechanical characteristics is given in Table 3. 3.2. Work-hardening behavior The work-hardening behavior is conveniently characterized by the dependence of the work-hardening rate on the flow stress [13]. For obtaining the work-hardening rate
H¼
1 depl drtr drtr ¼ depl e_ pl ;T dt dt
ð4Þ
at any instant during tensile deformation, the evolution of true stress and true plastic strain with time are numerically differentiated by using a five point formula for differentiation. Fig. 2 summarizes the work-hardening behavior for all five different strain rates. The figure displays all recorded data even beyond the onset of necking of the tensile specimen as determined by the Considère criterion [14]
H ¼ rtr :
ð5Þ
Table 2 Chemical composition of as received titanium sheet. Composition
Fe
C
N
O
H
ASTM grade 1 Measured
<0.2 wt.% 0.03 wt.%
<0.08 wt.% 0.006 wt.%
<0.03 wt.% 0.005 wt.%
<0.18 wt.% 0.06 wt.%
<0.015 wt.% 0.002 wt.%
Fig. 1. True stress-true plastic strain curves (calculated assuming homogeneous deformation up to failure) from tensile tests of commercially pure titanium for five different initial strain rates between 2.2 104 s1 and 6.7 101 s1.
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The analysis however is restricted to the initial part where the deformation occurs homogeneously. Before localization of plastic deformation and hence the true tensile strength is reached, three different stages can be clearly distinguished which are designated stages A, B and C here. In each of the three stages the work-hardening rate decreases linearly with flow stress following a Kocks–Mecking behavior [15,16]
H ¼ H0 1
rtr r1
ð6Þ
characterized by only two model parameters, the initial workhardening rate H0 and the saturation stress r1. Eq. (6) represents not only a suitable phenomenological description; the analytical expression can be derived from the evolution of the dislocation density q providing that dislocation slip is the dominating mechanism of plastic deformation [15,16]: mutual accumulation and annihilation of dislocations are described by the evolution equation:
Table 3 Mechanical properties of commercially pure titanium in dependence on strain rate. Nominal strain rate (s1) 2.2 104 1.1 103 1.1 102 1.1 101 6.7 101
Engineering tensile strength (MPa)
True tensile strength (MPa)
True uniform strain
323 ± 1 330 ± 1 340 ± 1 357 ± 1 371 ± 1
409 ± 3 413 ± 2 409 ± 1 424 ± 2 445 ± 1
0.23 ± 0.01 0.21 ± 0.01 0.18 ± 0.01 0.16 ± 0.01 0.17 ± 0.01
dq M ¼ b de
pffiffiffiffi
q
b
2yq
ð7Þ
and the flow stress is determined by the Taylor relation pffiffiffiffi
rtr ¼ Malb q with the Taylor factor M, an interaction coefficient a, the shear modulus l, the Burgers vector b, a factor b characterizing the ratio between the mean free path of dislocation and their pffiffiffiffi mean distance 1= q, and an annihilation length y. The initial work-hardening rate H0 = M2al/2b is solely determined by mutual trapping of dislocations, whereas the saturation stress r1 = Malb/ 2yb results from the balance between accumulation and mutual annihilation of dislocations. Integration of Eq. (6) from an initial flow stress r0 leads to a Voce law [17] for the flow stress
rtr ¼ r1 ðr1 r0 Þ expðH0 e=r1 Þ:
ð8Þ
The values for the model parameters obtained for the three stages are summarized in Table 4. For all stages, both characteristic model parameters depend strongly on the strain rate: For stages A and B the initial work-hardening rate and the saturation stress are higher at larger strain rates – except at the highest strain rates for the initial work-hardening rate of stage B (which might be caused by adiabatic heating). For the saturation stress of stage C, two regimes at lower and higher strain rates must obviously be distinguished. In a similar way, the transition between the stages (A to B and B to C) can be characterized from the intersection points of the linear dependences of the work-hardening rates on the flow stress for the individual stages. The true stresses at which the transition between stages occurs depends strongly on the strain rate – in marked contrast to the work-hardening rate. The work-hardening rate at the transition between two subsequent stages turns out to be apparently independent of the applied strain rate. Comparably, the true plastic strain at the transition is the same for all tested strain rates. Between stages A and B the transition occurs at a work-hardening rate of (1460 ± 20) MPa and a true plastic strain of (0.022 ± 0.001), between stages B and C at a work-hardening rate of (590 ± 10) MPa and (0.098 ± 0.003). This allows construction of the work-hardening stage map shown in Fig. 3. The three work-hardening stages correspond to distinct values of the work-hardening rate irrespective of the applied strain rate, whereas the true stress depends strongly on the strain rate. 3.3. Strain rate jump tests
Fig. 2. Work-hardening behavior of commercially pure titanium for five different nominal strain rates. Three work-hardening stages designated A, B and C are indicated. The thin black line represents the onset of necking according to the Considère criterion.
For quantification of the strain rate dependence of the flow stress, strain rate jump tests are performed where the cross head velocity is suddenly altered between two different cross head velocities at several instants during the test. As an example, the strain jump tests varying the cross head velocity between the two values leading to nominal strain rates of 1.1 103 s1 and 1.1 101 s1 are shown in Fig. 4a together with the curves of continuous tensile tests for these two strain rates. It becomes obvious that the jump tests alternate quite accurately between the two flow curves from continuous tests. After a short transient behavior following the jump in strain rate, the flow curves follow the
Table 4 Model parameters (initial work-hardening rate and saturation stress) obtained by fitting Eq. (6) to the different stages of the work-hardening curves of Fig. 2. Initial strain rate (s1)
2.2 104 1.1 103 1.1 102 1.1 101 6.7 101
Saturation stress r1
Initial work-hardening rate H0 Stage A (GPa)
Stage B (MPa)
Stage C (MPa)
Stage A (MPa)
Stage B (MPa)
Stage C (MPa)
20.7 ± 0.6 21.9 ± 0.8 26.5 ± 1.3 30.7 ± 2.8 31.1 ± 2.8
4240 ± 40 4520 ± 130 5100 ± 130 5500 ± 150 5330 ± 110
1570 ± 50 1480 ± 90 2320 ± 140 2530 ± 100 2210 ± 20
287 ± 1 299 ± 1 311 ± 1 335 ± 3 349 ± 2
400 ± 2 409 ± 3 418 ± 2 436 ± 2 456 ± 1
554 ± 8 582 ± 21 498 ± 10 510 ± 5 557 ± 2
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always show higher strain rate sensitivity than the following jumps for which the strain rate sensitivity remains about constant. For instance, for jumps to higher strain rates, the strain rate sensitivity of the first jump is 0.024, for all later 0.018 independent of the strain rate. For the jumps to lower strain rates a distinction between high and low strain rates has to be made. For jumps from 1.1 102 s1 and higher to lower strain rates the strain rate sensitivities are 0.021 for the first jump and 0.017 for the following, but for strain rates 2.2 103 s1 and 1.1 103 s1 to the lower strain rate 2.2 104 s1 the values are slightly less (0.016 and 0.012).
3.4. Microstructural evolution
Fig. 3. Work-hardening behavior of commercially pure titanium: the transition between the different work-hardening stages (dashed lines) depends mainly on the work-hardening rate. The solid black line represents the onset of necking as determined by the Considère criterion. The colored straight lines display the best descriptions of the experimental work-hardening curves for the lowest and highest strain rate (2.2 104 s1 and 6.7 101 s1) obtained by fitting of Eq. (6) to the individual stages.
behavior at constant strain rate nicely, indicating that the strain rate has only minor effects on the microstructural evolution. The same is seen in the work-hardening curves in Fig. 4b where the strain jump test also oscillates between the two curves obtained from continuous tensile tests. The true stresses at the instant of the jump can be retrieved by extrapolating the different branches of the true stress versus true plastic strain curves to eliminate the effects of possible dynamic strain aging as well as inertia and stiffness of the tensile machine. This allows determination of the instantaneous strain rate sensitivity
m¼
@ ln rtr @ ln e_ pl e;T
ð9Þ
Comparing the strain rate sensitivities determined for the individual jumps, it becomes obvious that the first two jumps (the first to a higher strain rate and the other back to the lower strain rate)
Texture and microstructure were analyzed based on the orientation data gathered by EBSD. The as-received material showed a recrystallized microstructure constituting of (almost) equiaxed grains with a mean chord length of about 9 to 10 lm. In order to elucidate the microstructural evolution during tensile deformation in the different work-hardening stages, additional samples were deformed up to the transition points between the stages. The fraction of twin boundaries (of all high angle boundaries with disorientation angles above 15°) can be used to characterize the occurrence of deformation twins as only a few twin boundaries were observed in the as-received material. Quantitative analysis of the fraction of twin boundaries (summarized in Table 5) revealed that the fraction of twin boundaries even decreases slightly through stage A. In stage B compressive twinning of type 0 1 0i occurs leading to a twin frequency up to 18 % at the 64:6 h1 end of the stage dominating the boundary population entirely. 0 1 0i does not change signifiDuring stage C the ratio of 64:6 h1 0i twin boundaries are formed cantly, but additionally 84:5 h1 1 2 reaching up to 9% at the end of uniform elongation. Hence, it is concluded that during stage A where nearly no new twin boundaries are formed, the plastic strain is carried by dislocation slip, whereas in the following stages, deformation twinning occurs 2gh1 1 2 3i compression twins and dominantly by f1 1 2 2gh1 0 1 1i tensile twins in stage B and C, respectively. f1 0 1 This is illustrated for the strain rate of 1.1 101 s1. Fig. 5a shows an orientation map with inverse pole figure colors after straining to uniform elongation. From the morphology of their parallel boundaries an abundance of twin lamellae can be identified. This is confirmed in Fig. 5b displaying only twin boundaries with
Fig. 4. Strain rate jump tests between 1.1 103 s1 and 1.1 101 s1 and corresponding continuous tensile tests: (a) true stress-true plastic strain curves and (b) workhardening rate versus true stress curves. The thin black line represents the onset of necking according to the Considère criterion.
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Table 5 Twin frequencies observed in orientation maps (summary over several specimens deformed at different strain rates and maps of different step sizes). Rotation angle (°)
Rotation axis
Asreceived (%)
End of stage A (%)
End of stage B (%)
Uniform elongation (%)
84.8 34.5 64.6 57.4
0i h1 1 2 0 1 0i h1 0 1 0i h1 0i h1 1 2
0.7–3.2 <0.2 0.7–1.6 <0.4
0.6–1.3 <0.3 0.3–0.5 <0.3
0.3–2.8 <0.3 5.0–18.3 0.1–1.3
2.5–9.5 <0.3 7.8–17.4 0.6–2.0
(a)
different colors depending on the twin boundary type. The domi 0 1 0i (magenta) and 84:8 h1 1 2 0i (purple) is nance of 64:6 h1 obvious. Stage C is limited by the onset of necking at the ultimate tensile strength and the uniform elongation. The microstructure is charac 0i tensile twinning while the terized by a pronounced 84:8 h1 1 2 fraction of 64:6 h1 0 1 0i compression twins saturates. Both twin types show their highest fraction for a strain rate of 1.1 101 s1, additionally the twins generated at this strain rate are much finer compared to twins generated at the lower stain rates. (Twin lamellae narrower than the step size used for
(b)
Fig. 5. Orientation map obtained by EBSD using a Zeiss Supra 35 with step size 0.6 lm on commercially pure titanium deformed in tension to uniform elongation with a 0 1 0i, purple 84:5 h1 1 2 0i, cyan 57:4 h1 1 2 0i and strain rate 1.1 101 s1. (a) Inverse pole figure colors for RD and (b) delineation of special boundaries (magenta 64:6 h1 0 1 0i twin boundaries). dark blue 34:5 h1
Fig. 6. Pole figures of (a) as-received titanium and (a) titanium deformed to uniform elongation with a strain rate of 2.2 104 s1. RD is vertically and TD horizontally.
H. Becker, W. Pantleon / Computational Materials Science 76 (2013) 52–59
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acquiring the orientation data can obviously not be resolved, but would instead cause non-indexed points which were not observed abundantly). The initial texture of the as-received material is a typical cold-rolling texture. In the pole figures shown in Fig. 6 the maximal pole densities of the basal poles are tilted ±35° from the 0i and h1 21 0i are more or less original ND towards TD, the h0 1 1 evenly distributed along great circles around the basal poles. With increasing strain the basal poles (0 0 0 1) keep their position and the distribution broadens, the most remarkable change however occurs in the two other pole figures where the even distribution around the basal poles is now replaced by an alignment 0g poles with original RD, i.e. the applied tensile of the f1 0 1 direction.
4. Discussion 4.1. Work-hardening stages
Fig. 7. Deformation mechanism map indicating the dominant deformation mechanism in dependence on work-hardening rate and strain rate.
The three different stages identified in the work-hardening behavior are traced to different deformation mechanisms, dislocation slip and deformation twinning of different types. A work-hardening behavior with three distinct stages has been previously reported for compression of pure titanium [18,19], but with a quite different characteristic of the workhardening curves: In this case, an almost linear decrease of the work-hardening rate with increasing stress is reported for stages A and C only, whereas in stage B an increase of the work-hardening rate has been observed. Stage A shows an absence of twinning, but dislocation slip on pyramidal planes in c + a direction takes places additionally to basal and prismatic slip. The workhardening rate is decreasing as expected for dynamic recovery during stage III in a high stacking fault energy metal. The onset of stage B is correlated with emergence of first deformation twins. The twin volume fraction increases during stage B. During stage B the work-hardening rate increases until the twin volume fraction saturates and the strain hardening rate decreases again in stage C. Despite the differences in the work-hardening behavior the designation of the stages as A, B and C is followed in case of tensile deformation as well, as the microstructural investigations indicate no twin activity during stage A, dominant twinning on 0 1 0i systems during stage B, saturation of this twinning 64:6 h1 0i systems during systems and additional twinning on 84:5 h1 1 2 stage C. It should be noted, that the increase in the work-hardening rate in stage B could not be observed during simple shear tests either [19]. The existence of the different stages also explains the differences in the strain rate sensitivity observed for the two first jumps in strain rate experiments. As obvious from Fig. 4b the first jump occurs at the end of stage A where the deformation is dominated by dislocation glide whereas the following jumps occur in stages B and C where pronounced twinning is observed.
Table 6 Summary of instantaneous strain rate sensitivities m (for jumps to higher strain rates) and strain rate exponents n for the saturation stress for the different stages. Stage A
m n n0
0.024 0.025 0.001
Stage B
0.018 0.016 0.002
Stage C Low strain rates
High strain rates
0.018 0.055 0.037
0.018 0.027 0.009
In a similar manner as the instantaneous strain rate sensitivity, a strain rate exponent for the saturation stress
n¼
@ ln r1 @ ln e_ pl T
ð10Þ
is derived from the data in Table 4 leading to small values 0.025 and 0.016 for stages A and B and two different values in stage C: 0.055 and 0.027 for low and high strain rates, respectively. Correcting this apparent strain rate exponent for the trivial instantaneous strain rate effect results in the true strain rate dependence
n0 ¼ n m
ð11Þ
of the saturation stress. If this subtraction is performed individually for the stages (cf. Table 6), the strain rate sensitivity of the flow stress in stages A and B is almost vanishing ðjn0 j 6 0:002Þ, but at the high stresses of stage C the strain rate sensitivity remains 0.037 for low and 0.009 for high strain rates. This indicates that in stage C, a thermal activated process occurs which is more sensitive on strain rate at lower strain rates than at higher strain rates. Such a higher strain rate exponent can be considered as evidence for a contribution of thermal activated dislocation slip. The occurrence of two different strain rate regimes is suggested to be related to the formation of coarser twins at lower and finer twins at higher strain rates. The entire information can be summarized in the deformation mechanism map shown in Fig. 7. For the different deformation modes the controlling parameter seems to be the work-hardening rate. When the work-hardening rate falls below a critical value for a twinning system, twinning on this system occurs, thereby maximizing the work-hardening rate continuously. For work-hardening rates above 1460 MPa dislocation slip is the dominating deformation mechanism independent of the strain rate. If the rate drops be 0 1 0i compression twinning will be low that value 64:6 h1 activated until the work-hardening rate falls below 590 MPa where 0i tensile twinning becomes activated. These transitions 84:8 h1 1 2 are almost independent of the strain rate, the most pronounced effect of the strain rate is the fineness of the tensile twins. 4.2. Effect of nanocrystalline grain size The potential for improving the mechanical behavior of commercially pure titanium by nanostructuring can be assessed by taking into account the effect of a nanocrystalline grain size on the
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Table 7 Material properties used for calculation of flow curves of commercially pure titanium. Shear modulus (l)
Burgers vector (ba)
Interaction coefficient (a)
Taylor factor (M)
Initial flow stress (r0)
44 GPa
0.295 nm
0.5 [23]
5 [23]
190 MPa
evolution of the dislocation density. Grain boundaries restrict the mean free path of mobile dislocation and cause an additional term in the evolution of the dislocation density [20]
pffiffiffiffi q dq M 1 ¼ þ 2yq b D de b
ð12Þ
and hence an additional contribution to the work-hardening rate
H ¼ H0
rD r M 3 a2 l2 b with rD ¼ þ1 2DH0 r1 r
ð13Þ
Integration of Eq. (13) can be performed analytically (cf. [21])
e¼
r1 H0
ðrþ r Þ
r ln
r r r rþ rþ ln r0 r r0 rþ
ð14Þ
with
r ¼
r1 2
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r21 4
þ r1 rD
ð15Þ
Unfortunately, this expression cannot be inverted to gain an analytical equation for the flow stress r(e). Flow curves calculated from Eq. (14) for different grain sizes based on the material parameters summarized in Table 7 are illustrated in Fig. 8: The curve based on a grain size of 10 lm (as in the experimental investigation) shows only slightly larger flow stresses than the theoretical curve calculated without any restriction of the mean free path by grain boundaries, whereas for nanocrystalline titanium with a grain size of 50 nm, a significant enhancement of the flow stress is predicted. For quite small strains, the mutual interaction of dislocations can be ignored and integration of Eq. (13) leads approximately to a Hall Petch like behavior
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M 3 a2 l2 be ke pffiffiffiffi ¼ r0 þ pffiffiffiffi r ¼ r0 þ D D
ð16Þ
With the materials parameters of Table 7 a Hall Petch coefficient k0.2% = 6000 MPa nm1/2 is obtained for the 0.2% proof stress – quite close to the value of ky = 6067 MPa nm1/2 reported for the yield stress of commercially pure titanium grade 2 between 50 nm and 1 lm [7]. Additionally, the behavior of the work-hardening rates according to Eq. (13) is illustrated in Fig. 8 for the two different grain sizes (10 lm and 50 nm). Failure according to the Considère criterion of Eq. (5) occurs when the flow stress reaches the value of the workhardening rate or vice versa the work-hardening rate decreases to the value of the flow stress. Following the predictions of the dislocation density model and Eq. (13), necking of nanocrystalline titanium is expected at smaller plastic strains than in coarse–grained titanium. In case of coarse-grained titanium, such a low workhardening rate required for the onset of necking is not reached during the dislocation–dominated stage A. As soon as the work-hardening rate falls below the critical value for the transition between 0 1 0i comstages A and B HAB = 1460 MPa, deformation by 64:6 h1 pression twinning sets in. Due to the slower decrease of the workhardening rate with stress and strain in both twinning stages B and C, the occurrence of failure is delayed further in the coarse-grained material to even larger strains. In case of nanocrystalline titanium with a grain size of 50 nm, on the other hand, the higher workhardening rate suppresses the occurrence of twinning as with decreasing work-hardening rate the Considère criterion is met before the critical value for the onset of twinning is reached. Such an absence of deformation twinning in nanostructured titanium has been indeed reported [22].
5. Conclusions Three different work-hardening stages have been identified during tensile deformation of commercially pure titanium. All three work-hardening stages show a linear decrease of the workhardening rate with increasing true stress. The transitions between the work-hardening stages occur at constant work-hardening rates – almost independent of the applied strain rate. The occurrence of distinct work-hardening stages is caused by the change in the dominant mechanism of plastic deformation. Initially only dislocation slip takes place, followed by compression twinning by 0 1 0i twins, finally, tensile twinning by 84:8 h1 1 2 0i twins 64:6 h1 occurs simultaneously to dislocation slip as concluded from the microstructural observations and the strain rate sensitivity. Based on this information work-hardening stage and deformation mechanisms maps are constructed. A theoretical analysis of the workhardening behavior shows that an increased work-hardening rate caused by the abundance of grain boundaries in nanostructured material is eventually able to suppress the formation of twinning before the onset of necking in nanocrystalline titanium.
Acknowledgements
Fig. 8. Flow curves (full lines) for commercially pure titanium of different grain sizes (10 lm and 50 nm) calculated from Eq. (14) compared to a flow curve without any grain size effect (denoted 1) obtained from Eq. (8). The corresponding workhardening rates in dependence on strain are also included in the graph (dasheddotted lines). Failure occurs according to the Considère criterion, if flow stress and work-hardening rate become equal; formation of twins starts, when the workhardening rate becomes lower than the critical work-hardening rate for the transition between stage A and B marked as dashed line.
The authors gratefully acknowledge support from the ViNaT project, Contract No. 295322, FP7-NMP-2011-EU-Russia, NMP.2011.1.4-5, and from the Danish-Chinese Center of Nanometals funded by the Danish National Research Foundation. The authors thank Dr. Helmuth Toftegaard for providing the initial material, Frank Adrian for help with performing the mechanical tests and Preben Olesen for support with specimen preparation and EBSD.
H. Becker, W. Pantleon / Computational Materials Science 76 (2013) 52–59
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[11] R. Bonnet, E. Cousineau, D.H. Warrington, Acta Crystallogr. A A37 (1981) 184– 189. [12] D.G. Brandon, Acta Metall. 14 (1966) 1479–1484. [13] U.F. Kocks, H. Mecking, Prog. Mater. Sci. 48 (2003) 171–273. [14] Y. Estrin, L.P. Kubin, Acta Metall. 34 (1986) 2455–2464. [15] U.F. Kocks, J. Eng. Mater. Techn. 98 (1976) 76–85. [16] H. Mecking, U.F. Kocks, Acta Metall. 29 (1981) 1865–1875. [17] E. Voce, J. Inst. Metals 74 (1948) 537–562. [18] A.A. Salem, S.R. Kalidindi, R.D. Doherty, Scripta Mater. 46 (2002) 419–423. [19] A.A. Salem, S.R. Kalidindi, R.D. Doherty, Acta Mater. 51 (2003) 4225–4237. [20] Y. Estrin, H. Mecking, Scripta Metall. 19 (1985) 451–455. [21] D. Breuer, W. Pantleon, U. Martin, U. Mühle, H. Oettel, O. Vöhringer, phys. Stat. Sol. (a) 150 (1995) 281–295. [22] Y.M. Wang, J.Y. Huang, T. Jiao, Y.T. Zhu, A.V. Hamza, J. Mater. Sci. 42 (2007) 1751–1756. [23] T. Tanaka, H. Conrad, Acta Metall. 20 (1972) 1019–1029.
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Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Work-hardening stages and deformation mechanism maps during tensile deformation of commercially pure titanium Hanka Becker a,b, Wolfgang Pantleon a,⇑ a b
Section of Materials and Surface Engineering, Department of Mechanical Engineering, Technical University of Denmark, Produktionstorvet 425, 2800 Kgs. Lyngby, Denmark Institute of Materials Science, Faculty of Materials Science and Technology, TU Bergakademie Freiberg, 09599 Freiberg, Germany
a r t i c l e
i n f o
Article history: Received 9 November 2012 Received in revised form 12 March 2013 Accepted 20 March 2013 Available online 16 May 2013 Keywords: Tensile deformation Work-hardening Twinning Dislocations Deformation mechanism maps
a b s t r a c t Commercially pure titanium was tensile tested at different strain rates between 2.2 104 s1 and 6.7 101 s1 to characterize the strain rate dependence of plastic deformation and the dominating deformation mechanisms. From true stress-true plastic strain curves, three distinct work-hardening stages are identified. The work-hardening rate decreases linearly with increasing flow stress for all three stages and the work-hardening rate is the controlling factor for the transition between the different stages and mechanisms. During the initial stage (at lowest stresses) plastic deformation is carried mainly 0 1 0i twin by dislocation slip, in the following stage (for moderate stresses), an abundance of 64:6 h1 2gh1 1 2 3i compression twinning. During the last boundaries form indicating the dominance of f1 1 2 0i twin boundaries are detected caused by stage before the onset of necking, additional 84:8 h1 1 2 2gh1 0 1 1i tensile twinning. Based on the microstructural findings and the strain rate sensitivity, f1 0 1 deformation mechanism maps are constructed. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction Nanostructured titanium has become an interesting candidate material for medical implants. Pure titanium is machinable and biocompatible, but its strength is insufficient for load-bearing implants. Nanostructuring of titanium provides a new and promising method for improving its mechanical properties [1–8]. Various methods of severe plastic deformation (SPD) have been successfully applied for grain refinement in pure Ti [1,2]. Commercially pure (CP) titanium has been processed via equal channel angular pressing (ECAP) [3,4], high pressure torsion [5,6] and hydrostatic extrusion [7]. In order to determine the optimal directions of development and design of nanostructured titanium for medical applications, fundamental understanding of the deformation mechanisms and reliable computational models for the virtual, numerical testing of these materials are necessary. The development of such tools is the aim of the EU FP7 Project ViNaT, carried out in collaboration with Russian partners. For validation of the models concise data sets for the mechanical properties are required. As a point of reference, the mechanical behavior during tensile tests in a broad range of strain rates is reported here for conventionally processed commercially pure titanium. Pure titanium has a hexagonal close-packed crystal structure with a c/a ratio of 1.587. Deformation by dislocation slip may occur on a number of different types of slip systems, but activation of ⇑ Corresponding author. Tel.: +45 45252315; fax: +45 45936213. E-mail address: [email protected] (W. Pantleon). 0927-0256/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2013.03.028
only the slip systems with lowest critical resolved shear stresses (basal and prismatic systems) is insufficient to ensure compatible deformation of titanium polycrystals. Therefore, either additional slip systems with higher critical resolved shear stresses as pyramidal slip must be activated or plastic deformation by twinning must occur also. The most common twinning systems are summarized in Table 1. 2. Experimental 2.1. Material Commercially pure titanium of ASTM grade 1 was received asrolled, annealed and pickled sheets of a thickness t = 1 mm. The chemical composition is given in Table 2. Dog-bone specimens for tensile tests with a width w = 12.5 mm and a gauge length l = 75 mm were machined from the as-received sheet material with tensile direction along the original rolling direction (RD). 2.2. Mechanical testing Continuous, strain controlled tensile tests were conducted using two different testing machines: an electromechanical Instron 8874 and a servo hydraulic Instron 88 R 1362. Five different constant cross head velocities v between 1 mm/min and 3000 mm/ min were applied corresponding to nominal strain rates e_ ¼ v =l between 2.2 104 s1 and 6.7 101 s1. Additionally, strain rate jump tests were performed where the cross head velocity was
H. Becker, W. Pantleon / Computational Materials Science 76 (2013) 52–59 Table 1 Common twinning systems for hexagonal crystal structures defined by twinning plane and direction [8–10]. The orientation difference between twin and parent crystal are listed together with the parameter R characterizing the frequency of (almost) coinciding lattice sites between parent crystal and twin [11]. The deformation caused along the c-axis is specified for the case of titanium with c/a ratio less than the ideal value. Twinning system
Rotation angle (°)
Rotation axis
R
Deformation along caxis for Ti
2gh1 0 1 1i f1 0 1 1gh1 1 2 6i f1 1 2 2gh1 1 2 3i f1 1 2 1gh1 0 1 2i f1 0 1
84.8 34.5 64.6 57.4
0i h1 1 2 0 1 0i h1 0 1 0i h1 0i h1 1 2
11 11 7 13
Tensile Tensile Compressive Compressive
Dl l
3.1. Flow curves
ð1Þ
and true stress
F A
rtr ¼ ¼
F Dl 1þ w0 t 0 l
ð2Þ
were calculated from the recorded changes Dl in the cross head position and the required force F. The true plastic strain
epl ¼ etr
rtr
ð3Þ
Eeff
identified by the characteristic disorientation between twin and its parent. The four different types of twins listed in Table 1 were distinguished. According to the Brandon criterion [12],paffiffiffiffimaximal deviation from the ideal orientation difference of 15 = R was accepted determined by the fraction 1/R of (almost) coinciding lattice sites. 3. Results
changed suddenly between two different values several times during the test for determining the instantaneous strain rate sensitivity. Assuming uniform elongation, true strain
etr ¼ ln 1 þ
53
was derived taking into account an effective elastic modulus Eeff obtained from the initial elastic part of true stress versus true strain curves. 2.3. Electron backscatter diffraction The microstructure evolving during tensile testing was investigated by electron backscatter diffraction (EBSD). Pieces of 8 mm 5 mm were cut from the as-received sheet and the tensile tested samples. Free surfaces perpendicular to either the original normal direction (ND) or transversal direction (TD) were grinded with SiC paper, and mechanically polished with diamond paste and OP–S. As a final step, the specimen were electro-polished with Struers A3 solution for 20 s at 40 V (after increasing voltage from 0 V in about 45 s) to achieve an optimal surface quality. For acquiring electron backscattering patterns, two different scanning electron microscopes were utilized: a Zeiss Supra 35 with a Nordlys II EBSD detector using the program Flamenco of HKL Channel 5 for indexing and a JEOL JSM-840 with a phosphor screen as detector using CROMATIC v.2.10 software to index the measured bands. Orientations were gathered at individual positions on regular square grids on the surface of the specimen with step sizes between 0.4 lm and 2 lm. For analyzing the measured data, the programs Tango and Mambo of HKL Channel 5 were applied. The measured orientations are used to calculate pole figures or presented in orientation maps displaying the crystallographic direction along a particular specimen direction. In such orientation maps, twin boundaries were
Fig. 1 summarizes the true stress versus true plastic strain curves of all continuous tensile tests with constant strain rate. Obviously, higher loads are required for deforming with higher stain rates. Some differences between specimens deformed at the same strain rate become obvious, in particular for the largest strain rate. The reason for these deviations could not be identified definitely, but it is most likely caused by a slightly different preparation of the specimens and the use of two different tensile machines with quite different stiffnesses. A comparison of specimens prepared in the same manner and deformed at the same machine only does not show such strong variations. An overview over the mechanical characteristics is given in Table 3. 3.2. Work-hardening behavior The work-hardening behavior is conveniently characterized by the dependence of the work-hardening rate on the flow stress [13]. For obtaining the work-hardening rate
H¼
1 depl drtr drtr ¼ depl e_ pl ;T dt dt
ð4Þ
at any instant during tensile deformation, the evolution of true stress and true plastic strain with time are numerically differentiated by using a five point formula for differentiation. Fig. 2 summarizes the work-hardening behavior for all five different strain rates. The figure displays all recorded data even beyond the onset of necking of the tensile specimen as determined by the Considère criterion [14]
H ¼ rtr :
ð5Þ
Table 2 Chemical composition of as received titanium sheet. Composition
Fe
C
N
O
H
ASTM grade 1 Measured
<0.2 wt.% 0.03 wt.%
<0.08 wt.% 0.006 wt.%
<0.03 wt.% 0.005 wt.%
<0.18 wt.% 0.06 wt.%
<0.015 wt.% 0.002 wt.%
Fig. 1. True stress-true plastic strain curves (calculated assuming homogeneous deformation up to failure) from tensile tests of commercially pure titanium for five different initial strain rates between 2.2 104 s1 and 6.7 101 s1.
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H. Becker, W. Pantleon / Computational Materials Science 76 (2013) 52–59
The analysis however is restricted to the initial part where the deformation occurs homogeneously. Before localization of plastic deformation and hence the true tensile strength is reached, three different stages can be clearly distinguished which are designated stages A, B and C here. In each of the three stages the work-hardening rate decreases linearly with flow stress following a Kocks–Mecking behavior [15,16]
H ¼ H0 1
rtr r1
ð6Þ
characterized by only two model parameters, the initial workhardening rate H0 and the saturation stress r1. Eq. (6) represents not only a suitable phenomenological description; the analytical expression can be derived from the evolution of the dislocation density q providing that dislocation slip is the dominating mechanism of plastic deformation [15,16]: mutual accumulation and annihilation of dislocations are described by the evolution equation:
Table 3 Mechanical properties of commercially pure titanium in dependence on strain rate. Nominal strain rate (s1) 2.2 104 1.1 103 1.1 102 1.1 101 6.7 101
Engineering tensile strength (MPa)
True tensile strength (MPa)
True uniform strain
323 ± 1 330 ± 1 340 ± 1 357 ± 1 371 ± 1
409 ± 3 413 ± 2 409 ± 1 424 ± 2 445 ± 1
0.23 ± 0.01 0.21 ± 0.01 0.18 ± 0.01 0.16 ± 0.01 0.17 ± 0.01
dq M ¼ b de
pffiffiffiffi
q
b
2yq
ð7Þ
and the flow stress is determined by the Taylor relation pffiffiffiffi
rtr ¼ Malb q with the Taylor factor M, an interaction coefficient a, the shear modulus l, the Burgers vector b, a factor b characterizing the ratio between the mean free path of dislocation and their pffiffiffiffi mean distance 1= q, and an annihilation length y. The initial work-hardening rate H0 = M2al/2b is solely determined by mutual trapping of dislocations, whereas the saturation stress r1 = Malb/ 2yb results from the balance between accumulation and mutual annihilation of dislocations. Integration of Eq. (6) from an initial flow stress r0 leads to a Voce law [17] for the flow stress
rtr ¼ r1 ðr1 r0 Þ expðH0 e=r1 Þ:
ð8Þ
The values for the model parameters obtained for the three stages are summarized in Table 4. For all stages, both characteristic model parameters depend strongly on the strain rate: For stages A and B the initial work-hardening rate and the saturation stress are higher at larger strain rates – except at the highest strain rates for the initial work-hardening rate of stage B (which might be caused by adiabatic heating). For the saturation stress of stage C, two regimes at lower and higher strain rates must obviously be distinguished. In a similar way, the transition between the stages (A to B and B to C) can be characterized from the intersection points of the linear dependences of the work-hardening rates on the flow stress for the individual stages. The true stresses at which the transition between stages occurs depends strongly on the strain rate – in marked contrast to the work-hardening rate. The work-hardening rate at the transition between two subsequent stages turns out to be apparently independent of the applied strain rate. Comparably, the true plastic strain at the transition is the same for all tested strain rates. Between stages A and B the transition occurs at a work-hardening rate of (1460 ± 20) MPa and a true plastic strain of (0.022 ± 0.001), between stages B and C at a work-hardening rate of (590 ± 10) MPa and (0.098 ± 0.003). This allows construction of the work-hardening stage map shown in Fig. 3. The three work-hardening stages correspond to distinct values of the work-hardening rate irrespective of the applied strain rate, whereas the true stress depends strongly on the strain rate. 3.3. Strain rate jump tests
Fig. 2. Work-hardening behavior of commercially pure titanium for five different nominal strain rates. Three work-hardening stages designated A, B and C are indicated. The thin black line represents the onset of necking according to the Considère criterion.
For quantification of the strain rate dependence of the flow stress, strain rate jump tests are performed where the cross head velocity is suddenly altered between two different cross head velocities at several instants during the test. As an example, the strain jump tests varying the cross head velocity between the two values leading to nominal strain rates of 1.1 103 s1 and 1.1 101 s1 are shown in Fig. 4a together with the curves of continuous tensile tests for these two strain rates. It becomes obvious that the jump tests alternate quite accurately between the two flow curves from continuous tests. After a short transient behavior following the jump in strain rate, the flow curves follow the
Table 4 Model parameters (initial work-hardening rate and saturation stress) obtained by fitting Eq. (6) to the different stages of the work-hardening curves of Fig. 2. Initial strain rate (s1)
2.2 104 1.1 103 1.1 102 1.1 101 6.7 101
Saturation stress r1
Initial work-hardening rate H0 Stage A (GPa)
Stage B (MPa)
Stage C (MPa)
Stage A (MPa)
Stage B (MPa)
Stage C (MPa)
20.7 ± 0.6 21.9 ± 0.8 26.5 ± 1.3 30.7 ± 2.8 31.1 ± 2.8
4240 ± 40 4520 ± 130 5100 ± 130 5500 ± 150 5330 ± 110
1570 ± 50 1480 ± 90 2320 ± 140 2530 ± 100 2210 ± 20
287 ± 1 299 ± 1 311 ± 1 335 ± 3 349 ± 2
400 ± 2 409 ± 3 418 ± 2 436 ± 2 456 ± 1
554 ± 8 582 ± 21 498 ± 10 510 ± 5 557 ± 2
H. Becker, W. Pantleon / Computational Materials Science 76 (2013) 52–59
55
always show higher strain rate sensitivity than the following jumps for which the strain rate sensitivity remains about constant. For instance, for jumps to higher strain rates, the strain rate sensitivity of the first jump is 0.024, for all later 0.018 independent of the strain rate. For the jumps to lower strain rates a distinction between high and low strain rates has to be made. For jumps from 1.1 102 s1 and higher to lower strain rates the strain rate sensitivities are 0.021 for the first jump and 0.017 for the following, but for strain rates 2.2 103 s1 and 1.1 103 s1 to the lower strain rate 2.2 104 s1 the values are slightly less (0.016 and 0.012).
3.4. Microstructural evolution
Fig. 3. Work-hardening behavior of commercially pure titanium: the transition between the different work-hardening stages (dashed lines) depends mainly on the work-hardening rate. The solid black line represents the onset of necking as determined by the Considère criterion. The colored straight lines display the best descriptions of the experimental work-hardening curves for the lowest and highest strain rate (2.2 104 s1 and 6.7 101 s1) obtained by fitting of Eq. (6) to the individual stages.
behavior at constant strain rate nicely, indicating that the strain rate has only minor effects on the microstructural evolution. The same is seen in the work-hardening curves in Fig. 4b where the strain jump test also oscillates between the two curves obtained from continuous tensile tests. The true stresses at the instant of the jump can be retrieved by extrapolating the different branches of the true stress versus true plastic strain curves to eliminate the effects of possible dynamic strain aging as well as inertia and stiffness of the tensile machine. This allows determination of the instantaneous strain rate sensitivity
m¼
@ ln rtr @ ln e_ pl e;T
ð9Þ
Comparing the strain rate sensitivities determined for the individual jumps, it becomes obvious that the first two jumps (the first to a higher strain rate and the other back to the lower strain rate)
Texture and microstructure were analyzed based on the orientation data gathered by EBSD. The as-received material showed a recrystallized microstructure constituting of (almost) equiaxed grains with a mean chord length of about 9 to 10 lm. In order to elucidate the microstructural evolution during tensile deformation in the different work-hardening stages, additional samples were deformed up to the transition points between the stages. The fraction of twin boundaries (of all high angle boundaries with disorientation angles above 15°) can be used to characterize the occurrence of deformation twins as only a few twin boundaries were observed in the as-received material. Quantitative analysis of the fraction of twin boundaries (summarized in Table 5) revealed that the fraction of twin boundaries even decreases slightly through stage A. In stage B compressive twinning of type 0 1 0i occurs leading to a twin frequency up to 18 % at the 64:6 h1 end of the stage dominating the boundary population entirely. 0 1 0i does not change signifiDuring stage C the ratio of 64:6 h1 0i twin boundaries are formed cantly, but additionally 84:5 h1 1 2 reaching up to 9% at the end of uniform elongation. Hence, it is concluded that during stage A where nearly no new twin boundaries are formed, the plastic strain is carried by dislocation slip, whereas in the following stages, deformation twinning occurs 2gh1 1 2 3i compression twins and dominantly by f1 1 2 2gh1 0 1 1i tensile twins in stage B and C, respectively. f1 0 1 This is illustrated for the strain rate of 1.1 101 s1. Fig. 5a shows an orientation map with inverse pole figure colors after straining to uniform elongation. From the morphology of their parallel boundaries an abundance of twin lamellae can be identified. This is confirmed in Fig. 5b displaying only twin boundaries with
Fig. 4. Strain rate jump tests between 1.1 103 s1 and 1.1 101 s1 and corresponding continuous tensile tests: (a) true stress-true plastic strain curves and (b) workhardening rate versus true stress curves. The thin black line represents the onset of necking according to the Considère criterion.
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H. Becker, W. Pantleon / Computational Materials Science 76 (2013) 52–59
Table 5 Twin frequencies observed in orientation maps (summary over several specimens deformed at different strain rates and maps of different step sizes). Rotation angle (°)
Rotation axis
Asreceived (%)
End of stage A (%)
End of stage B (%)
Uniform elongation (%)
84.8 34.5 64.6 57.4
0i h1 1 2 0 1 0i h1 0 1 0i h1 0i h1 1 2
0.7–3.2 <0.2 0.7–1.6 <0.4
0.6–1.3 <0.3 0.3–0.5 <0.3
0.3–2.8 <0.3 5.0–18.3 0.1–1.3
2.5–9.5 <0.3 7.8–17.4 0.6–2.0
(a)
different colors depending on the twin boundary type. The domi 0 1 0i (magenta) and 84:8 h1 1 2 0i (purple) is nance of 64:6 h1 obvious. Stage C is limited by the onset of necking at the ultimate tensile strength and the uniform elongation. The microstructure is charac 0i tensile twinning while the terized by a pronounced 84:8 h1 1 2 fraction of 64:6 h1 0 1 0i compression twins saturates. Both twin types show their highest fraction for a strain rate of 1.1 101 s1, additionally the twins generated at this strain rate are much finer compared to twins generated at the lower stain rates. (Twin lamellae narrower than the step size used for
(b)
Fig. 5. Orientation map obtained by EBSD using a Zeiss Supra 35 with step size 0.6 lm on commercially pure titanium deformed in tension to uniform elongation with a 0 1 0i, purple 84:5 h1 1 2 0i, cyan 57:4 h1 1 2 0i and strain rate 1.1 101 s1. (a) Inverse pole figure colors for RD and (b) delineation of special boundaries (magenta 64:6 h1 0 1 0i twin boundaries). dark blue 34:5 h1
Fig. 6. Pole figures of (a) as-received titanium and (a) titanium deformed to uniform elongation with a strain rate of 2.2 104 s1. RD is vertically and TD horizontally.
H. Becker, W. Pantleon / Computational Materials Science 76 (2013) 52–59
57
acquiring the orientation data can obviously not be resolved, but would instead cause non-indexed points which were not observed abundantly). The initial texture of the as-received material is a typical cold-rolling texture. In the pole figures shown in Fig. 6 the maximal pole densities of the basal poles are tilted ±35° from the 0i and h1 21 0i are more or less original ND towards TD, the h0 1 1 evenly distributed along great circles around the basal poles. With increasing strain the basal poles (0 0 0 1) keep their position and the distribution broadens, the most remarkable change however occurs in the two other pole figures where the even distribution around the basal poles is now replaced by an alignment 0g poles with original RD, i.e. the applied tensile of the f1 0 1 direction.
4. Discussion 4.1. Work-hardening stages
Fig. 7. Deformation mechanism map indicating the dominant deformation mechanism in dependence on work-hardening rate and strain rate.
The three different stages identified in the work-hardening behavior are traced to different deformation mechanisms, dislocation slip and deformation twinning of different types. A work-hardening behavior with three distinct stages has been previously reported for compression of pure titanium [18,19], but with a quite different characteristic of the workhardening curves: In this case, an almost linear decrease of the work-hardening rate with increasing stress is reported for stages A and C only, whereas in stage B an increase of the work-hardening rate has been observed. Stage A shows an absence of twinning, but dislocation slip on pyramidal planes in c + a direction takes places additionally to basal and prismatic slip. The workhardening rate is decreasing as expected for dynamic recovery during stage III in a high stacking fault energy metal. The onset of stage B is correlated with emergence of first deformation twins. The twin volume fraction increases during stage B. During stage B the work-hardening rate increases until the twin volume fraction saturates and the strain hardening rate decreases again in stage C. Despite the differences in the work-hardening behavior the designation of the stages as A, B and C is followed in case of tensile deformation as well, as the microstructural investigations indicate no twin activity during stage A, dominant twinning on 0 1 0i systems during stage B, saturation of this twinning 64:6 h1 0i systems during systems and additional twinning on 84:5 h1 1 2 stage C. It should be noted, that the increase in the work-hardening rate in stage B could not be observed during simple shear tests either [19]. The existence of the different stages also explains the differences in the strain rate sensitivity observed for the two first jumps in strain rate experiments. As obvious from Fig. 4b the first jump occurs at the end of stage A where the deformation is dominated by dislocation glide whereas the following jumps occur in stages B and C where pronounced twinning is observed.
Table 6 Summary of instantaneous strain rate sensitivities m (for jumps to higher strain rates) and strain rate exponents n for the saturation stress for the different stages. Stage A
m n n0
0.024 0.025 0.001
Stage B
0.018 0.016 0.002
Stage C Low strain rates
High strain rates
0.018 0.055 0.037
0.018 0.027 0.009
In a similar manner as the instantaneous strain rate sensitivity, a strain rate exponent for the saturation stress
n¼
@ ln r1 @ ln e_ pl T
ð10Þ
is derived from the data in Table 4 leading to small values 0.025 and 0.016 for stages A and B and two different values in stage C: 0.055 and 0.027 for low and high strain rates, respectively. Correcting this apparent strain rate exponent for the trivial instantaneous strain rate effect results in the true strain rate dependence
n0 ¼ n m
ð11Þ
of the saturation stress. If this subtraction is performed individually for the stages (cf. Table 6), the strain rate sensitivity of the flow stress in stages A and B is almost vanishing ðjn0 j 6 0:002Þ, but at the high stresses of stage C the strain rate sensitivity remains 0.037 for low and 0.009 for high strain rates. This indicates that in stage C, a thermal activated process occurs which is more sensitive on strain rate at lower strain rates than at higher strain rates. Such a higher strain rate exponent can be considered as evidence for a contribution of thermal activated dislocation slip. The occurrence of two different strain rate regimes is suggested to be related to the formation of coarser twins at lower and finer twins at higher strain rates. The entire information can be summarized in the deformation mechanism map shown in Fig. 7. For the different deformation modes the controlling parameter seems to be the work-hardening rate. When the work-hardening rate falls below a critical value for a twinning system, twinning on this system occurs, thereby maximizing the work-hardening rate continuously. For work-hardening rates above 1460 MPa dislocation slip is the dominating deformation mechanism independent of the strain rate. If the rate drops be 0 1 0i compression twinning will be low that value 64:6 h1 activated until the work-hardening rate falls below 590 MPa where 0i tensile twinning becomes activated. These transitions 84:8 h1 1 2 are almost independent of the strain rate, the most pronounced effect of the strain rate is the fineness of the tensile twins. 4.2. Effect of nanocrystalline grain size The potential for improving the mechanical behavior of commercially pure titanium by nanostructuring can be assessed by taking into account the effect of a nanocrystalline grain size on the
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H. Becker, W. Pantleon / Computational Materials Science 76 (2013) 52–59
Table 7 Material properties used for calculation of flow curves of commercially pure titanium. Shear modulus (l)
Burgers vector (ba)
Interaction coefficient (a)
Taylor factor (M)
Initial flow stress (r0)
44 GPa
0.295 nm
0.5 [23]
5 [23]
190 MPa
evolution of the dislocation density. Grain boundaries restrict the mean free path of mobile dislocation and cause an additional term in the evolution of the dislocation density [20]
pffiffiffiffi q dq M 1 ¼ þ 2yq b D de b
ð12Þ
and hence an additional contribution to the work-hardening rate
H ¼ H0
rD r M 3 a2 l2 b with rD ¼ þ1 2DH0 r1 r
ð13Þ
Integration of Eq. (13) can be performed analytically (cf. [21])
e¼
r1 H0
ðrþ r Þ
r ln
r r r rþ rþ ln r0 r r0 rþ
ð14Þ
with
r ¼
r1 2
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r21 4
þ r1 rD
ð15Þ
Unfortunately, this expression cannot be inverted to gain an analytical equation for the flow stress r(e). Flow curves calculated from Eq. (14) for different grain sizes based on the material parameters summarized in Table 7 are illustrated in Fig. 8: The curve based on a grain size of 10 lm (as in the experimental investigation) shows only slightly larger flow stresses than the theoretical curve calculated without any restriction of the mean free path by grain boundaries, whereas for nanocrystalline titanium with a grain size of 50 nm, a significant enhancement of the flow stress is predicted. For quite small strains, the mutual interaction of dislocations can be ignored and integration of Eq. (13) leads approximately to a Hall Petch like behavior
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M 3 a2 l2 be ke pffiffiffiffi ¼ r0 þ pffiffiffiffi r ¼ r0 þ D D
ð16Þ
With the materials parameters of Table 7 a Hall Petch coefficient k0.2% = 6000 MPa nm1/2 is obtained for the 0.2% proof stress – quite close to the value of ky = 6067 MPa nm1/2 reported for the yield stress of commercially pure titanium grade 2 between 50 nm and 1 lm [7]. Additionally, the behavior of the work-hardening rates according to Eq. (13) is illustrated in Fig. 8 for the two different grain sizes (10 lm and 50 nm). Failure according to the Considère criterion of Eq. (5) occurs when the flow stress reaches the value of the workhardening rate or vice versa the work-hardening rate decreases to the value of the flow stress. Following the predictions of the dislocation density model and Eq. (13), necking of nanocrystalline titanium is expected at smaller plastic strains than in coarse–grained titanium. In case of coarse-grained titanium, such a low workhardening rate required for the onset of necking is not reached during the dislocation–dominated stage A. As soon as the work-hardening rate falls below the critical value for the transition between 0 1 0i comstages A and B HAB = 1460 MPa, deformation by 64:6 h1 pression twinning sets in. Due to the slower decrease of the workhardening rate with stress and strain in both twinning stages B and C, the occurrence of failure is delayed further in the coarse-grained material to even larger strains. In case of nanocrystalline titanium with a grain size of 50 nm, on the other hand, the higher workhardening rate suppresses the occurrence of twinning as with decreasing work-hardening rate the Considère criterion is met before the critical value for the onset of twinning is reached. Such an absence of deformation twinning in nanostructured titanium has been indeed reported [22].
5. Conclusions Three different work-hardening stages have been identified during tensile deformation of commercially pure titanium. All three work-hardening stages show a linear decrease of the workhardening rate with increasing true stress. The transitions between the work-hardening stages occur at constant work-hardening rates – almost independent of the applied strain rate. The occurrence of distinct work-hardening stages is caused by the change in the dominant mechanism of plastic deformation. Initially only dislocation slip takes place, followed by compression twinning by 0 1 0i twins, finally, tensile twinning by 84:8 h1 1 2 0i twins 64:6 h1 occurs simultaneously to dislocation slip as concluded from the microstructural observations and the strain rate sensitivity. Based on this information work-hardening stage and deformation mechanisms maps are constructed. A theoretical analysis of the workhardening behavior shows that an increased work-hardening rate caused by the abundance of grain boundaries in nanostructured material is eventually able to suppress the formation of twinning before the onset of necking in nanocrystalline titanium.
Acknowledgements
Fig. 8. Flow curves (full lines) for commercially pure titanium of different grain sizes (10 lm and 50 nm) calculated from Eq. (14) compared to a flow curve without any grain size effect (denoted 1) obtained from Eq. (8). The corresponding workhardening rates in dependence on strain are also included in the graph (dasheddotted lines). Failure occurs according to the Considère criterion, if flow stress and work-hardening rate become equal; formation of twins starts, when the workhardening rate becomes lower than the critical work-hardening rate for the transition between stage A and B marked as dashed line.
The authors gratefully acknowledge support from the ViNaT project, Contract No. 295322, FP7-NMP-2011-EU-Russia, NMP.2011.1.4-5, and from the Danish-Chinese Center of Nanometals funded by the Danish National Research Foundation. The authors thank Dr. Helmuth Toftegaard for providing the initial material, Frank Adrian for help with performing the mechanical tests and Preben Olesen for support with specimen preparation and EBSD.
H. Becker, W. Pantleon / Computational Materials Science 76 (2013) 52–59
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