Influence of the relaxation processes on the structure formation in pure metals and alloys under high-pressure torsion

Acta Materialia 55 (2007) 6039–6050 www.elsevier.com/locate/actamat

Influence of the relaxation processes on the structure formation in pure metals and alloys under high-pressure torsion M.V. Degtyarev, T.I. Chashchukhina, L.M. Voronova *, A.M. Patselov, V.P. Pilyugin Institute of Metal Physics, Ekaterinburg, Russia Received 15 November 2006; received in revised form 6 April 2007; accepted 13 April 2007 Available online 1 October 2007

Abstract A study has been performed on three classes of materials in which different structure-forming processes dominate, namely, cold hardening in iron and structural steels, dynamic recrystallization in copper, and pressure-induced transformation in austenite steel. These processes are shown to affect a staged character of the structure formation upon high-pressure torsion. In the case of cold hardening, the character is controlled by true strain, the growth of which results in an increase in hardness and a refinement of the structure elements. The termination of these processes can be caused by another mechanism of relaxation, e.g. phase transformation induced by high pressure. Upon dynamic recrystallization, the staged character of deformation is determined by the temperature and rate of deformation. Ó 2007 Published by Elsevier Ltd on behalf of Acta Materialia Inc. Keywords: High-pressure torsion; Cold working; Dynamic recrystallization; Phase transformation under pressure; Microstructure

1. Introduction Stability is the most important problem for materials with a submicrocrystalline structure. Upon heating, some such materials exhibit an increase in grain size, in accordance with the kinetics of normal growth [1,2], whereas other cases experience anomalous growth and the fast degradation of an ultradisperse structure [3–6]. One assumes that the type of behavior in the course of heating is controlled by the type of initial structure. Under cold deformation, the structural changes can be traced in stages. Some of the strain–stress dependence and a specific type of structure corresponds to each stage [7,8]. It is assumed that, for each metal, at given temperature and rate of deformation conditions and hydrostatic pressure, a certain stationary microstructure is realized at large monotonous deformations which is retained upon further deformation via constant reproduction [9,10]. *

Corresponding author. E-mail addresses: [email protected], (L.M. Voronova).

[email protected]

The formation of high-angle boundaries is often considered to serve as an indication of the transition to a stationary structure. Based on the size of the structure constituents, this structure is classified as submicrocrystalline (SMC) or nanocrystalline. Such a structure forms, for instance, upon shear under pressure. This method of formation provides the highest true strain of a sample without failure under virtually isothermal conditions. Therefore, it extends the range of true strains in studies of the staged character toward higher values. Much attention has lately been given to studying the deformation mechanisms of submicrocrystalline or nanocrystalline materials on a microlevel (dislocation slip, microtwinning, grain-boundary gliding and grain rotation) [11–13]. However, the relaxation processes (such as phase hardening, dynamic recrystallization, shear phase transformation under pressure) that accompany the deformation at different stages and mainly control the structure being formed have not been taken into consideration. The aim of this work was to study the stages of the structure evolution depending on the relaxation processes that take place under severe plastic deformation.

1359-6454/$30.00 Ó 2007 Published by Elsevier Ltd on behalf of Acta Materialia Inc. doi:10.1016/j.actamat.2007.04.017

6040

M.V. Degtyarev et al. / Acta Materialia 55 (2007) 6039–6050

load made up 0.5 N and for copper, 0.25 N. With such a load chosen, the size of the indentation diagonal did not exceed a quarter of the sample thickness, whereas the depth of hardness indentation was 20–30 times less than the sample thickness after deformation. This enabled us to avoid any effect of sample thickness on material hardness. When investigating the hardness distribution along the sample radius, measurements were carried out for two mutually perpendicular diameters with a step of 0.25 mm and the average value was taken for each Ri. The results of measurements obtained on different samples subjected to the same calculated true strain differed by no more than 7%. While constructing the strain dependence of hardness, its values taken from different samples were averaged over the intervals of true strain, De = 0.2. The distance from the sample center to the area of the structure under study was estimated to within an accuracy of ±0.2 mm. The size of the substructure elements was determined from the electron microscopy patterns by at least 400 measurements, the error being no more than 7%. These results served to construct the size histograms and distribution functions for cells, microcrystallites and recrystallized grains. The following statistic parameters were analyzed: the most probable size (dpr) corresponding to the maximum in the distribution curve, the average size (dav), the minimum (dmin) and the maximum (dmax) size, variation coefficient for the linear dimensions (K) and the half-width of the distribution function (P). The phase content of steel 18-10 was determined by Xray diffraction using Mo Ka radiation in the transmission setting. To adequately allow for the contributions from each phase component, only individual reflections were used that were not superimposed with other diffraction lines: (2 0 0) face-centered cubic, (1 0 1) hexagonal closepacked and (2 0 0) body-centered cubic. The coefficient of linear attenuation l was taken to be equal for all the phases (a, c, e) observed in the steels after deformation. The effect of texture was allowed for via the introduction of a special correction determined by inverse pole figures (ODF). Since the X-ray measurements were performed over the whole sample area, the true strain value was averaged for each sample. To this end, a sample was represented as a set of

Three classes of materials that differ in their structureforming processes proceeding upon deformation at ambient temperature have been examined: (i) iron of different purity and structural steels, which under deformation undergo work hardening; (ii) copper, which undergoes dynamic recrystallization; and (ii) austenite steel 18-10, in which high pressure gives rise to shear (c–e) phase transformation. 2. Experimental The chemical composition of the materials under study is shown in Table 1. The treatment of structural steels prior to deformation consisted of quenching followed by tempering at 920 K for 1 h. Deformation was performed by shear under pressure in Bridgman anvils [14]. The angle of the anvil rotation was varied from 15° to 15 revolutions. All the samples under study were 5 mm in diameter and 0.3 mm thick, and were deformed at a velocity of rotation of x = 1 rev min1. Additionally, copper samples, 12 mm in diameter and 0.6 mm thick, were deformed at a velocity of x = 0.05 rev min1. The velocity of the anvil rotation was constant throughout the course of deformation. The pressure magnitude was chosen to exceed the maximum hardness of the deformed material by a factor of 1.5, thus excluding any sliding between the sample and the anvil surface in the course of deformation (Table 2). The true strain, hardness and size of structural elements were determined from the coordinates of a point under consideration relative to the center of the sample deformed. The threshold deformation by the shear under pressure is limited by the anvil strength and low thickness of the deformed sample. The thickness of the samples under study was measured along two mutually perpendicular diameters (four radii) at a step of 0.5 mm. Each thickness value was taken as the average of four measurements at a corresponding distance from the axis of rotation. For all the samples, hardness was determined by the Vickers method. The load was chosen depending on the sample thickness (for the measurements to be correct, the sample thickness must exceed the indentation diagonal at least threefold). For iron and steels, the Table 1 Chemical composition of the materials studied Material

Pure iron

Content of elements (wt.%) C

Mn

Si

Cr

Ni

Ti

Cu

Bi

Sb

Pb

B

S

P

60.003

60.006

<0.009

<0.008

<0.003

–

–

–

–

–

–

–

–

Armco-iron

0.009

0.18

0.13

–

–

–

0.10

–

–

–

–

0.019

0.010

Steel 13B20

0.20

1.45

0.25

–

–

–

–

–

–

–

0.003

0.012

0.011

Steel 13B30

0.30

1.70

0.28

–

–

–

–

–

–

–

0.003

0.012

0.012

Steel 4330

0.31

1.26

1.08

1.08

2.00

–

–

–

–

–

–

0.004

0.005

Steel 18-10

0.10

Copper

–

–

–

17.75

9.90

0.59

–

–

–

–

–

0.020

0.015

–

–

–

–

–

99.99

0.001

0.001

0.001

–

0.002

–

M.V. Degtyarev et al. / Acta Materialia 55 (2007) 6039–6050 Table 2 The pressure magnitude, maximum deformation degree and corresponding hardness and size of structure constituents for studied materials of different classes Material

Pressure (GPa)

Logarithmic deformations

Hardness (GPa)

Size of structure constituents (lm)

Pure iron

8

9.3

5.7

0.06

Steel 4330

11

7.2

7.5

0.04

Copper

6

11.8 2.4

1.4 1.8

0.18 0.40

Steel 18-10

8

5.1 6.2

4.7 4.7

0.07 0.07

eupsetting ¼ ln

6041

h0 : hiR

ð2Þ

The shear deformation that is equivalent to the true one is calculated, allowing for the von Mises criterion, as was done in Ref. [8], by the equation: pffiffiffi eequiv ¼ c= 3: ð3Þ Calculations by Eqs. (1)–(3) show that, upon shearing under pressure, the upsetting deformation is negligibly small compared with the shear deformation. This fact has served as a ground on which to neglect the upsetting effect in several works [8,10,11,18]. However, it was established in one of the earliest works [18] on the deformation in nickel and chromium–nickel superalloy that a large shear deformation and a comparatively small rolling deformation result in small variations in hardness. This contradiction was successfully eliminated by taking the logarithm of shear deformation [11,18]:

coaxial rings for each of which the true strain was determined by Eq. (1), and the average true strain was determined with allowance for the area of each ring. The results obtained for different samples subjected to the same deformation varied within 5%. The amount of the ferromagnetic a-phase, which formed from the deformation martensite (e) in the course of unloading, was controlled not only by X-ray but also by magnetic measurements of saturation magnetization, with nickel, containing no more that 0.02% impurity, being a reference.

At small angles of anvil rotation /, an equation was proposed in Refs. [11,15] to avoid negative deformation values:

2.1. Calculation of true strain and rate of deformation

etorsion ¼ lnð1 þ c2 Þ

To date, no unified means of estimating the true strain under high-pressure torsion has been developed. Different equations have been employed in the calculations of true strain, and these have led to different results [11]. Otherwise, the deformation of material was estimated simply from the number of anvil revolutions [8,12]. The difficulty in calculation is caused by two peculiar features of the method of ‘‘high-pressure torsion’’: shear deformation is dependent on the distance from the sample center (Ri), and the sample thickness (hiR) decreases upon increasing the angle of the anvil rotation (/) [15,16]. The degree of shear deformation is commonly estimated as in the case of twisting a cylindrical rod when the sample dimensions remain unchanged [10,11,17]: c ¼ uRi =hiR :

ð1Þ

Since the sample after deformation has a thickness on the order of 101 mm and the anvils were rotated by several complete revolutions, the amount of shear deformation, according to Eq. (1), is about 105. Just as in the experiments of Bridgman [14] but unlike Refs. [2,6,8,10], in the work presented here anvils were used in which the sample was not bound to the lateral surface (Fig. 1). Therefore, the pressure imposed prior to the anvil rotation results in a decrease in the sample thickness because of the upsetting and radial pressing-out of the material from under the anvil. The shear deformation upon the anvil rotation is accompanied by sample thinning, i.e. the sample changes its dimensions. In this case, the logarithmic deformation calculated via the changes in the sample thickness is taken as the true deformation:

etorsion ¼ ln c:

ð4Þ

1=2

ð5Þ

:

At large shear deformations c  1, Eq. (5) reduces to Eq. (4). Calculations using Eqs. (4) and (5) show such a remarkable decrease in the shear deformation, in comparison with that calculated by Eq. (3), that a contribution from the upsetting deformation becomes significant. Simply summing up the logarithmic shear deformation (Eq. (5)) and the true deformation upon upsetting (Eq. (2)) yielded Eq. (6): 

e ¼ etorsion þ eupsetting

uRi ¼ ln 1 þ hiR

2 !1=2 þ ln

h0 : hiR

ð6Þ

Since Eq. (6) was derived empirically and is not strictly mathematically grounded, to verify its validity a special experiment was carried out as follows.

P

sample

anvils

P Fig. 1. Sketch of the Bridgman anvils.

M.V. Degtyarev et al. / Acta Materialia 55 (2007) 6039–6050

5

5

4

4

H, GPa

We suggested that in a material the deformation of which results in a continuous accumulation of defects (no dynamic recrystallization or shear phase transformations take place), close hardness values testify to equal accumulated strains. It should be noted that high pressure influences the structure and hardness of the material deformed. Armco-iron and structural steels containing 0.2% and 0.3% carbon were deformed by upsetting in Bridgman anvils under a pressure of 6–10 GPa and crossrolling. For these deformation procedures the true strain is to be correctly calculated from changes in the sample thickness (Eq. (2)). At an equal true strain of e = 0.5, iron acquired an essentially higher hardness upon upsetting under pressure than upon rolling. The difference made up more than 10%. Increasing the carbon content in the alloy resulted in decreasing the differences related to the effect of high pressure. At a carbon concentration of 0.3% no influence of high pressure on hardness was revealed. Therefore, a special investigation was carried out on steel containing 0.3% C, which allowed the substantiation of application of Eq. (6) for the calculation of the true strain upon shear under pressure. Eq. (3) is valid and widely applied when calculating true strains close to e = 1. Steel 4330 was deformed in the range 0.2 < e < 1.4 by tension, drawing under pressure, hydroextrusion with counterpressure and shear under pressure. The hardness values turned out to be equal after equal equivalent deformations independent of the deformation mode. It was supposed that after the deformation with e > 2 calculated by Eq. (3) the steel hardness after shear under pressure and after cross-rolling should change similarly with increasing the true strains. However, it turned out that the steel hardness after cross-rolling is higher than that after the equivalent deformation by shear under pressure and increases with increasing true strain (Fig. 2a). This is connected with the complicated character of the shearunder-pressure deformation when the shear deformation is accompanied by upsetting and radial flaring of the sample, i.e. changes in its dimensions. In this case, the peripheral layers of the material which, in accordance with Eq. (3), are to be the most hardened are continuously extruded out of the anvils and replaced by the material from the central part of the sample, which possesses a lower hardness. The condition for equal hardness and sizes of the structure elements after deformation by shear and cross-rolling to an equal strain is met by Eq. (6), which takes into account the deformation provided by both the anvil rotation and the sample upsetting (Fig. 2b). Further investigation of the structure and radial distribution of hardness of the samples deformed by shear under pressure showed that close hardness values and structure element sizes, corresponding to equal strains, were achieved for different samples independent of the number of anvil revolutions required to gain a given strain when calculated by Eq. (6). Since the structure elements possess

H, GPa

6042

3

2

3

0

2

4

6

8

equivalent deformation, eequ.

2 0

2 true strain, e

4

Fig. 2. Comparison of the hardness values for steel 4330 after deformation by different modes. Calculation of strain upon high-pressure torsion by Eq. (3) (a) and by Eq. (2) (b). s, prior to deformation; h, high-pressure torsion; d, cross-rolling.

submicron sizes, their changes over the sample radius are not apparent enough. To make this fact more obvious, we suggested the use of annealing for recrystallization [15]. Annealing of the deformed Armco-iron sample results in the formation of a rather large grain, which allows any inhomogeneity of the recrystallized structure over the radius to be shown by light metallography. Thus, the true strain calculated by Eq. (6) allows a comparison of the experimental data obtained after the deformation of the samples at different angles of anvil rotation. Eq. (6) was applied when calculating the true strain for all the materials studied. When calculating the true rate of deformation, it was supposed that in each deformed sample the structural changes first were similar to those in the samples deformed with smaller angles of anvil rotation, and the differences arose upon increasing the true strain in the course of subsequent rotation. Therefore, the true rate of deformation was calculated by the equation: e_ ¼ De=Ds;

ð7Þ

where De is the true strain at which the structural changes occur compared to that of the samples deformed with a smaller angle of the anvil rotation, and Ds is the time interval required to gain the corresponding true-strain increment. Note that the true rate of deformation, as the true strain, changes over the sample radius. 3. Results and discussion Table 2 presents the pressure magnitudes, threshold true strain, and related hardness and size of structure elements. For the classes of materials under consideration different relations between the true strain and maximum hardness and minimum size of structure elements were observed. In iron and steels (e.g. 4330) the highest strengthening and refinement of the structure elements are reached at

M.V. Degtyarev et al. / Acta Materialia 55 (2007) 6039–6050

(Fig. 3b and d). Then, the parabolic stages of strengthening will be represented by straight lines with different slopes [19]. The average size of the structure elements also changes in stages and corresponds to the change in hardness (Fig. 3c). In the studied range of true strains, two or three stages can be identified (Fig. 3) [20]. Therefore, it is impossible to judge unambiguously the type of structure at each stage without additional structural studies. The structure elements are dislocation cells with lowangle boundaries and microcrystallites having high-angle boundaries originating from the deformation. The corresponding structure types are: cellular at the first stage; a mixture of cells and microcrystallites at the second stage; and that formed solely by microcrystallites (SMC structure) at the third stage. The electron microscopy method allows the differentiation of these structure constituents by the following signs. Inside the cells, there are observed dislocations. The misorientation upon transition from one cell to another varies gradually, which produces an inhomogeneous dark-field contrast and its progressive movement over the dark-field patterns upon changing the sample tilt in the goniometer. The microdiffraction pattern is characterized by the azimuthally smeared reflections (Fig. 4a and b). In the bulk of microcrystallites, extinction contours are seen which testify to an elastic deformation of the lattice. The dark-field contrast at the microcrystallite boundaries changes abruptly. Upon tilting a sample in the goniometer,

the maximum true strain. This testifies to a continuous accumulation of defects and the absence of any transformations that could violate this regularity. The copper hardness is not related to the true strain and the size of elements that form the structure. The maximal hardness is reached upon a relatively small deformation which does not give rise to the dynamic recrystallization. The subsequent deformation results in a more than twofold decrease in the structure element size, with the copper hardness being lowered due to the dynamic recrystallization. In austenite steels, starting from a certain true strain, the hardness and structure dispersion ceased to change. The c–e transformation developed under pressure essentially decelerates the hardness growth, which results in a lower hardness for the steel 18-10 in comparison with pure iron, having structure elements of similar sizes. Let us now consider the structure evolution upon deformation in the materials of different classes. 3.1. Structure development upon work hardening Fig. 3 shows the dependences of hardness and average size of structure elements on the true strain for iron and structural steels. The changes in the deformation stages match the inflections in the curves. When no distinct inflections are observed (Fig. 3a), the dependence should be reconstructed in the coordinates ‘‘hardness – square root of the true strain’’, in accordance with indications [7]

6

6

2

H, GPa

H, GPa

1 4

2 2

0

4

6

3

4

2

8

1

3

2 0.5

e

e 8

6 3

4

0.9

0.6

2

03

69

e

2 H, GPa

2

d,μm

1

H, GPa

6043

3

6

0.3

4

0

2 2

1

3

0.5

e

Fig. 3. Staged character of deformation in iron and structural steels: (a and b) steel 13B20; (c) pure iron; (d) steel 4330.

6044

M.V. Degtyarev et al. / Acta Materialia 55 (2007) 6039–6050

the microcrystallite becomes extinct as it exits the reflecting position. The microdifraction pattern presents the point reflections (Fig. 4c and d). It is difficult to distinguish between the mixed and SMC structures since both contain microcrystallites. As is seen from Fig. 4, the bright- and dark-field images as well as the microdiffraction patterns corresponding to the mixed (Fig. 4a and b) and SMC (Fig. 4c and d) structures can differ less than the ones corresponding to the SMC structure in different materials (Fig. 4c,d and e,f). Thus, differentiating between these two types of structure is possible only upon the combined analysis of bystage changes in the hardness and electron microscopic data. The deformation of iron and structural steels at all stages results in increasing the hardness and decreasing the size of structure elements (Fig. 5, Table 3) [20]. It should be underlined that at the SMC structure stage, microcrystallites of all sizes become refined since an increase in the true strain leads to a decrease in their maximum, average and the most probable size (Fig. 5). In pure iron at the SMC structure stage there is observed (Table 3) a more intense increase in hardness compared with Armco-iron, and a decrease in the average size of the microcrystallites. We believe that this is related to the fact that in pure iron the acceleration of the cell refinement is favored by the facilitation of the formation of new dislocation walls when further rotation takes place. Table 3 represents the data that show that the stage boundaries depend on the alloy chemical composition. There are no values of hardness, true strain and structure

element size at which cells are replaced by microcrystallites that are common for all materials. At the beginning of each stage the average size of microcrystallites in steels is remarkably less than that in iron. The results obtained allow one to analyze the influence of supersaturation of the solid solution in carbon and of a small volume fraction of second-phase particles on the transition to the rotation deformation modes whose development results in the formation of the high-angle microcrystallite boundaries [21,22]. Alloying of solid solution hinders the dislocation gliding and decreases the true strain corresponding to the formation of both the first microcrystallites, e12, and the homogeneous submicrocrystalline structure, e23. The occurrence of the carbide phase shifts the stage boundaries toward the lower strains still more. The formation of microcrystallites and transition to the SMC structure through the mixed-type structure stage is connected to the change in deformation mechanisms, when the dislocation gliding in hampered to such an extent that it does not provide the relaxation of the external stresses. With the absence of quasi-hydrostatic pressure, this would result in the material fracture. High pressure prevents the material from fracture and enables the development of deformation by a new mechanism that is associated with the movement of partial disclinations [23] and provides high-angle turns of the neighboring structure elements. Dislocations sink from the microcrystallite body at the high-angle boundaries. This is accompanied by the emergence of elastic stresses that result in the appearance of extinction contours in the electron microscopy patterns.

Fig. 4. Microstructure of Armco-iron, e = 4.8 (a and b); pure iron, e = 9.5 (c and d); and quenched steel 13B20, e = 8 (e and f). (a and b) Structure of the mixed type; (c–f) SMC structure.

M.V. Degtyarev et al. / Acta Materialia 55 (2007) 6039–6050

30%

6045

30%

e=3

e=6

10%

10%

0

0.8

1.6

2.4

0

0.4

0.8

50% 30%

e=9.3

e=4 30%

10% 10% 0

0.8

1.6

2.4

0

d, μm

0.4 d, μm

0.8

d, μm

0.20

0.15

themostprobable size average size half-width of distribution function

0.10

0.05

6

7

8

9

e Fig. 5. Variations in the structure parameters upon deformation of pure iron.

Table 3 True strain (e), hardness (H) and the average size of structure elements (d), which are initial for each stage, and their limit values achieved in this paper Material

Mixed

SMC

e1–2

H1–2 (GPa)

d1–2 (lm)

e23

H23 (GPa)

d23 (lm)

e

H (GPa)

d (lm)

Pure iron Armco-iron 13B20 13B30 4330

4.0 2.6 1.6 1.0 1.0

3.0 2.8 2.7 2.8 3.2

0.45 0.35 0.50 0.55 0.40

6.0 5.3 3.7 3.6 3.6

3.2 3.7 3.5 3.6 3.9

0.19 0.17 0.25 0.17 0.15

9.3 9.0 8.2 8.5 7.2

5.7 5.0 6.0 7.6 7.5

0.06 0.10 0.08 0.04 0.04

3.2. Influence of pressure-induced transformation on the structure evolution upon deformation A somewhat different staged character of deformation is observed in the austenite steel 18-10 (Fig. 6). This steel possesses a rather low stacking fault energy (SFE), of just 18 mJ m2 [24]. Therefore, the deformation causes both the development of cellular structure and twinning, and

Limit values

shear band formation [25]. As a result, at the first of the stages under study (1 < e < 5), there form both low- and high-angle boundaries, which correspond to a mixed structure. With this, the hardness increases and the size of the structure elements decreases similarly to the cases of pure iron and structural steels upon work hardening. At the second stage, which, in accordance with the electron microscopy data, is the SMC structure stage, these parameters

6046

M.V. Degtyarev et al. / Acta Materialia 55 (2007) 6039–6050

ment of the structure constituents belonging to the different phases.

5 0.5

3.3. The influence of dynamic recrystallization on changing the stages in structural states 0.3

3

d, μm

H, GPa

4

2 0.1 1 0

2

4

6

phase composition,%

100

γ

80

ε

60 40 20

α

In copper, the size of the structure elements and the hardness change upon deformation nonmonotonously and independently of each other (Fig. 7) [26]. In addition, the shape of the hardness curve changes with the velocity of the anvil rotation. The development of dynamic recrystallization (DR) radically affects the staged character of the structure formation under deformation and the very approach to its study. In this case, the deformation should be considered as a hot process. Consequently, the structure formation will be controlled by the temperature and the rate of deformation rather than the true strain, as is the case of cold deformation. The combined effect of temperature and deformation rate is taken into account by the Zener–Hollomon parameter (Z–H) [27]. Application of this formalism developed to describe the high-temperature creep to active deformation is only possible under condition that the coefficient of deformation hardening tends to zero when the deformation

0 0

2

e

4

ω = 1 rev/min

6 1.9

Fig. 6. Staged character of deformation in steel 18–10: (a) variations of hardness and structure parameters; (b) changes in the phase composition.

1.6

0.4

d, μm

0.5

1.3 0.3 0.2

1

H, GPa

become stable (Fig. 6a). Application of a high pressure induces the c–e shear transformation in this steel. In the course of unloading, the e phase transforms in part into a, and then all three phases are present (Fig. 6b). At the first stage of deformation, the volume fractions of the phases remain almost constant. At the SMC structure stage, the amount of high-pressure phases increases with increasing true strain (Fig. 6b). Hence, the shear transformation can be supposed to provide relaxation at this stage. Therefore, neither the microcrystallite size nor the steel hardness is changed. The stress relaxation connected to this transformation is apparently responsible for a lower hardness value for chromium–nickel steel in comparison with iron at similar sizes of structure elements (Table 2). A combined effect of the deformation and transformation manifests itself in that in steels with a low SFE, very fine elements are formed long before the SMC structure stage and, with increasing the true strain, their amount increases at the expense of the refinement of coarse elements. In such a case, the deformation causes mainly a decrease in the size of the coarsest structure elements. Similar values of interplanar spacing in the crystal lattices (and reflections in the electron-diffraction patterns) do not allow one to discern the microcrystallites of the different phases. This hinders the investigation into the dynamics of refine-

0 1.9

4

8

12

8

12

ω = 0.05 rev/min

1.6

1.3

1 0

4

e Fig. 7. Variations in the hardness and size of the structure elements upon deformation in copper.

M.V. Degtyarev et al. / Acta Materialia 55 (2007) 6039–6050

proceeds in a stationary mode. Fig. 6 shows that upon increasing the true strain for copper e > 2, no hardening takes place, which means that the conditions required are fulfilled. However, the analysis of the Z–H parameter is rarely applied for ascertaining the stage character of deformation [28]. More often it is used to determine the relation between the size of dynamically recrystallized grain and the deformation condition [29,30]. In practice, it is more convenient to represent the Z–H parameter in the form: ln Z ¼ ln e_ þ DH =RT ;

ð8Þ 1

where e_ is the true deformation rate (in s ), DH is the activation energy of the high-temperature diffusion for pure metals (its value is close to that of self-diffusion and in copper is equal to 107 kJ mol1 [2]), R is the gas constant, 8.31 J mol1 K1, and T = 300 K is the deformation temperature. The deformation temperature does not change in the process of high-pressure torsion. The absence of heating is confirmed in a special experiment via measuring temperature using a thermocouple sunk in the sample under deformation [31]. As was shown above, the true deformation rate e_ is changed over the sample radius. In accordance with this, the ln Z values were calculated for different coordinates relative to the sample center (Fig. 8). In the repeated experiments the ln Z values were reproduced to within an error of ±0.3. The involvement of the Z–H parameter allows one to reveal the staged character of the structure formation under DR. Upon the high-temperature deformation of face-centered cubic single crystals [28], DR was found to develop in the range of values of ln Z = 34–42. Moreover,

0

15

45

42

60

with increasing ln Z from 38 to 42, the volume fraction of the DR structure decreases, and above 42 DR does not develop, giving way to work hardening. Basing on the calculation results (Fig. 8), one can suppose that at a velocity of anvil rotation of 1 rev min1, with increasing the angle of rotation, the initially formed structure of hardening should be replaced by the structure that underwent DR first partially and then completely. However, if x is decreased to 0.05, the complete DR stage takes place at a smaller rotation angle. The structural studies showed that in copper deformed by high-pressure torsion at room temperature at ln Z = 41 and e > 2, recrystallization nuclei appeared having an elastically distorted bulk and a striped contrast at the boundaries. At ln Z > 42, no such nuclei are present. At ln Z = 40 and less, postdynamic recrystallization (PDR) develops that results in the appearance of coarse grains (Fig. 9a). Commonly, PDR can be avoided by rapid cooling from the deformation temperature to the ambient one. In our work, PDR is inevitable as these temperatures coincide. After 10 revolutions, when ln Z = 37 (which is the complete DR range), the grain size distribution is narrow; no coarse grains are observed and the course of PDR can be judged from the presence of geometrically regular grains with a low density of dislocations (Fig. 9b). After 15 revolutions of the anvil, ln Z = 38. In the structure, coarse PDR grains appear again (Fig. 9c). The dependence of the structure element size on the Z–H parameter is represented by two straight lines (Fig. 9d) with different slopes, one of which corresponds to the region where the work hardening dominates (dashed line) and

ω = 0.05rev/min

ω = 1 rev/min

0

42 0

1 rev.

40

15

40

0

45

5 rev. lnZ

6047

0

38

38

0

180 1 rev. 5 rev.

15 rev. 10 rev.

36 34

36 34

0

4

8

12

10 rev. 0

4

8

e

e coordinate relative to the center of the sample 0 mm 0.5 mm 1.0 mm 1.5 mm 2.0 mm

3.0 mm 4.0 mm 5.0 mm 6.0 mm

Fig. 8. Variations in the temperature-rate parameters upon deformation in copper.

12

6048

M.V. Degtyarev et al. / Acta Materialia 55 (2007) 6039–6050

40% 40%

20% 20%

0.2 μm

0

2

4

6

0.2 μm

0

4

2

6

d, μm 0.5

40%

0.4

d,μm

60%

20%

0.2 μm

0.3 0.2

0

4

2

6

d, μm

36

38

40

42

lnZ Fig. 9. Variations in the structure parameters upon deformation in copper: (a) ln Z = 40, u = 180°; (b) ln Z = 37, 10 revolutions; (c) ln Z = 38, 15 revolutions; (d) dependence of the average size of the structure elements on ln Z. The solid line corresponds to the regions where DR is dominant; the dashed line, to the regions where the leading part is taken by phase hardening.

the other to the region where the leading part belongs to DR (solid line). With decreasing velocity of anvil rotation, the structure evolution is traceable to the changes in the calculated ln Z values. This means that the work hardening stage is not realized and the stage of complete DR takes place at lower true strains. 3.4. Relation of hardness to the structure element size In this work, the stages of structural evolution were picked out, each of which is featured by a dominant structure-forming process and immanent mechanism of work hardening. Therefore, it is natural to expect changes in the parameters of the hardness dependence on the structure element size upon transition from one stage to another. This dependence is similar to the well-known Hall–Petch relationship for the yield stress [32]: H ¼ H 0 þ kd m ;

ð9Þ

where H0 is the hardness of a given single-crystalline sample, and k and m are the constants for a given structure type which characterize the resistance of deformation boundaries; the m values change from 0.5 for the high-angle boundaries to 1 for the low-angle boundaries [33,34], though in a number of researches some values both lower than 0.5 [35] and higher than 1 [36] were obtained. It was discovered [13,37,38] that when the size of the structure elements are below a certain critical value of some nanometers

the Hall–Petch relationship changes to an inverse one. It is usually assumed that the structure of nanomaterials that is formed under high plastic deformation is similar to a grained structure and, based on this, the m value is taken to be 0.5 [39,40]. In the present work it is shown that, with the size of structure elements of the order of 100 nm in different materials, different types of structure can be realized in which either high-angle or low-angle boundaries are dominant. Therefore, it is by no means always correct to specify m = 0.5, because this might not correspond to the structure type [39], or might result in an overestimated k value and negative value of the yield stress when extrapolating to the range of the coarse-grain state [40]. Three variables entering Eq. (9) cannot be determined unambiguously from experiment without imposing additional conditions. For the recrystallized structure m was specified as 0.5, and the H0 and k values were obtained from the linear dependence H vs. d0.5. For the mixed type and SMC structures, H0 was given and the parameters k and m were determined from the linear dependence ln(H  H0) vs. ln d (Fig. 10) [41]. In our work H and H0 were measured in GPa and k was in MPa m0.5. The approximating functions obtained, based on these prerequisites, for iron and steels are shown in Table 4. In the grain-size scale under study in this work, dependence (9) does not change to an inverse dependence, i.e. the refinement of the structure elements results in an increase in hardness. In pure iron the structure type does not affect the parameters of Eq. (9) (Fig. 10a). The increase

M.V. Degtyarev et al. / Acta Materialia 55 (2007) 6039–6050

of H0 upon transition from the recrystallized to deformed structure (Table 4) may be related to the peculiarities of the experimental method. When the size of the structure elements is much lower that the diagonal of the indentation, there takes place a transition from measuring microhardness to measuring hardness. Alloying of iron with carbon results in the increase of H0 because of the solid-solution hardening (Table 4). The H0 values for the steels containing 0.2% and 0.3% carbon are the same, which purports that a small change in the volume fraction of the carbide phase does not affect the parameters of Eq. (9). At the same time, alloying does not influence the k and m values for the recrystallized structure. For steels with mixed and SMC structures, two equiprobable pairs of the approximating functions were obtained, and our data cannot show a preference for either. In Armco-iron and structural steels, when comparing the recrystallized and mixed structures, the parameters of the equation turn out to be virtually the same. With the SMC structure, there is an increased H0 and a decreased m. These changes are connected with the redistribution of the impurity atoms, first of all carbon, between solid solution, dislocations and microcrystallite boundaries, which results from the carbide particles dissolving and the carbon migrating toward the microcystallite boundaries. The contribution to hardness from different modes of hardening upon changing the type of structure was treated in detail in our previous work [41]. For stainless steel with a mixed type structure the parameter H0 has higher values than for structural steels, which can be traced to stronger solid-solution hardening. In contrast, a lower k value is a consequence of decelerating the hardening because of shear phase transformation, which competes with the work hardening. At the SMC structure stage for this steel, as mentioned above, the val-

2.0

0.05 0.1

d, μm 0.3

1

-1

0.15 0.2

d, μm 0.3

0.6

1.5 ln (H -H 0)

ln (H -H 0)

-2 1.0

-3 0.5

0

6049

Table 4 Approximating functions H = f(d) for iron and steels with structures of different types Structure type

Pure iron

Armco-iron

Structural steels

Austenite steel

Recrystallized 0.6 + 1.2d0.5 0.8 + 1.2d0.5 1.0 + 1.2d0.5 – Mixed

0.8 + 1.2d0.5 0.9 + 1.2d0.5 1.1 + 1.2d0.5 1.1 + 1.0d0.5 1.0 + 1.2d0.55

SMC

0.8 + 1.2d0.5 0.8 + 1.2d0.51 1.0 + 1.1d0.58 – 0.8 + 1.2d0.56

ues of hardness and average size of microcrystallites remain constant. As shown above, in copper the staged changes in the structure are related to the degree of DR evolution. One can pick out two stages of DR: incomplete at 38 < ln Z < 42 and complete at ln Z < 38. When analyzing dependences (9), the experimental data should be grouped in accordance with these values of ln Z. Since the structure contains both high- and low-angle boundaries, the power index in Eq. (9) must not be set a priori but should be determined from the linear dependence of ln(H  H0) on ln d (Fig. 10b). The H0 value was taken as 1.5 GPa, which corresponds to the copper hardness before the DR onset and comprises the contribution from deformation hardening of the initial coarse-grain material [40]. With further deformation, the hardness is increased upon grain refinement and deformation hardening, and, at the same time, is lowered because of the decrease in the density of defects caused by DR, PDR and dynamic recovery. These factors exert the most significant effects on the stages of incomplete DR. From the data of Fig. 10b one can observe the inverse dependence with the positive power index – H = 1.5 + 0.3d0.5. Such dependence can be traced to the effect of several competing processes of hardening and softening. It should be underlined that the dependence observed is not connected with the transition of a material to the nanostructural state; the average size of the structure elements changes from 0.5 to 0.2 lm. At the stage of complete DR, Eq. (9) takes the conventional form – H = 1.5 + 0.04d0.7. In this case, DR is the dominant structure-forming process. The value of power index m falls in the interval from 0.5 to 1, showing the presence in the structure of both grain boundaries and subgrain boundaries [33,34]. The dynamic softening (recrystallization and recovery) results in a decrease in the value of the coefficient k. 4. Conclusion

-3

-2

-1 lnd

0

-4 -2.0

-1.5

-1.0

-0.5

lnd

Fig. 10. Determination of the parameters of the Hall–Petch equation. (a) Pure iron at H0 = 0.8 GPa. h, SMC structure; r, structure of the mixed type; n, cellular structure. (b) Copper at H0 = 1.5 GPa. s, for 38 < ln Z < 42; j, for ln Z < 38.

1. In all the materials studied, the structure development upon high-pressure torsion exhibits a staged character. The changes in microstructure and hardness at each stage are controlled by the dominating structure-forming process – work hardening, dynamic recrystallization or pressure-induced phase transformation. The stage limits were determined for iron and steels by the inflec-

6050

M.V. Degtyarev et al. / Acta Materialia 55 (2007) 6039–6050

tions in the dependences of hardness on the true strain and for copper, by the values of temperature-compensated rate of deformation ln Z. 2. In our experiment the effect of different structure-forming processes results in the final stage of deformation in the structures formed by elements of different types: – in iron and structural steels, microcrystallites produced via work hardening; – in copper, dynamically recrystallized grains; – in stainless steel microcrystallites formed both via the deformation as such and the martensite transformation in the course of high-pressure deformation. Therefore, despite the close sizes of the structure elements, these structures cannot be classified by a single term, as is (unfortunately) common. 3. Under the conditions studied, the stage of a stationary structure was not achieved. With increasing the true strain at the final stage, in iron and steels, there occur a continuous refinement of microcrystallites and a corresponding increase in hardness. In steel 18-10, at constant microcrystallite size and hardness, the phase content changes: the amount of high-pressure phase (e), which remains in the sample after unloading, increases. In copper, the structure depends on the temperature and rate of deformation conditions, which change upon increasing the number of anvil revolutions. Therefore, even at the stage of the complete dynamic recrystallization, the hardness and the size of dynamically recrystallized grains change. 4. The change in the dominating structure-forming process in the material under deformation results in the change in the parameters of the Hall–Petch equation, which relates the material hardness to the average size of its structure elements. Under the action of one structureforming process at different stages of deformation in pure metals the parameters of this equation remain constant upon transition from one stage to another. The change in these parameters under such conditions may take place in the alloys when the change of stages is accompanied by the redistribution of alloying elements. References [1] Degtyarev MV, Voronova LM, Gubernatorov VV, Chashchukhina TI. Docl 2002;47:647. [2] Lian J, Valiev RZ, Baudelet B. Acta Metall 1995;43:4165. [3] Horita Z, Fujinami T, Nemoto M, Longdon TG. Metall Trans A 2000;31:691. [4] Morris DG, Mun˜oz-Morris MA. Acta Mater 2002;50:4047. [5] Torre FD, Swygenhoven HV, Victoria M. Acta Mater 2002;50: 3957.

[6] Zhilyaev AP, Nurislamova GV, Baro MD, Valiev RZ, Longdon TG. Metall Trans A 2002;33:1865. [7] Trefilov VI, Moiseev VF, Pechkovskii EP. In: Deformational strengthening and fracture of metals. Kiev: Naukova Dumka; 1987. p. 242 [in Russian]. [8] Ivanisenko Yu, Lojkowski W, Valiev RZ, Fecht H-J. Acta Mater 2003;51:5555. [9] Beigel‘zimer YaE, Varyukhin VN, Orlov DV, Synkov SG. Screw extrusion as the process of deformation accumulation. Donetsk: TEAN; 2003. p. 87 [in Russian]. [10] Valiev RZ, Ivanisenko YuV, Rauch EF, Baudelet B. Acta Mater 1996;44:4705. [11] Zhilyaev AP, Nurislamova GV, Baro MD, Szpunar JA, Langdon TG. Acta Mater 2003;51:753. [12] Liao XZ, Zhao YH, Zhu YT, Valiev RZ, Gunderov DV. J Appl Phys 2004;96:636. [13] Schiøtz J. Scripta Mater 2004;51:837. [14] Bridgman PW. Studies in large plastic flow and fracture. New York: McGraw-Hill; 1952. [15] Degtyarev MV, Chashchukhina TI, Voronova LM, Davydova LS, Pilyugin VP. Phys Met Metallogr 2000;90(6):83. [16] Vorhauer A, Pippan R. Scripta Mater 2004;51:921. [17] Pugh HLLD, editor. Mechanical behavior of materials under pressure. Amsterdam: Elsevier; 1970. [18] Smirnova NA, Levit VI, Pilyugin VP, et al. Phys Met Metallogr 1986;61(6):1170. [19] van den Beukel A. Scripta Metall 1978;12:809. [20] Degtyarev MV. Phys Met Metallogr 2005;99:595. [21] Rybin VV. Big plastic deformations and fracture of metals. Moscow: Metallurgia; 1986 [in Russian]. [22] Abdulov RZ, Valiev RZ, Krasilnikov NA. Mater Sci Lett 1990;9:1445. [23] Romanov AE, Vladimirov VI. Disclination physics of plastic deformation. In: The structure and crystal defects. Toronto: Elsevier; 1984. p. 418–26. [24] Bampton CC, Jones JP, Lobetto MH. Acta Metall 1978;26:39. [25] Patselov AM, Degtyarev MV, Pilyugin VP, Chashchukhina TI, Voronova LM, Chernyshev EG, et al. Phys Met Metallogr 2004;98:206. [26] Chashchukhina TI, Degtyarev MV, Romanova MYu, Voronova LM. Phys Met Metallogr 2004;98:639. [27] Mc Qeen HJ. Metall Trans A 1977;8:807. [28] Bakhteyeva ND, Levit VI. Phys Met Metallogr 1983;55(4):124. [29] Sakai T, Lonas JJ. Acta Metall 1984;32(2):189. [30] Chang CI, Lee CJ, Huang JC. Scripta Mater 2004;51:509. [31] Teplov VA, Pilyugin VP, Taluts GG. Izv Ross Acad Nauk, Met 1992;2:109. [32] Armstrong RW. Metal Trans 1970;1(5):1169. [33] Langford G, Cohen M. Trans ASM 1969;62:623. [34] Kalish D, Le Fevre BG. Metal Trans A 1975;6:1319. [35] Tompson AW. Acta Metall 1975;23:1337. [36] Abson DJ, Jonas JJ. Met Sci J 1970;4(1):24. [37] Hidaca H, Tsuchiyama T, Takaki S. Scripta Mater 2001;44:1503. [38] Zhao M, Li JC, Jiang Q. J Alloys Compd 2003;361:160. [39] Furukawa M, Iwahashi Y, Horita Z, Nemoto M, Tsenev NK, Valiev RZ, et al. Acta Mater 1997;45:4751. [40] Hansen N. Scripta Mater 2004;51:801. [41] Degtyarev MV, Chashchukhina TI, Voronova LM. Phys Met Metallogr 2004;98(5):98.