Correlation area size of excess dislocations of the same sign as a new mesoscopic characteristic of the dislocation substructure

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Correlation area size of excess dislocations of the same sign as a new mesoscopic characteristic of the dislocation substructure A.I. Dekhtyar G.V. Kurdyumov Institute for Metal Physics of N.A.S. of Ukraine, 36 Vernadski blvd., Kiev 03142, Ukraine

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Article history: Received 6 November 2020 Accepted 8 November 2020 Available online xxxx

The conception of correlation area size (CAS) of excess dislocations of the same sign (EDSS) as a new characteristic of deformation substructure of metals and alloys has been developed. CAS plays important role in elevated temperature creep deformation behavior of metals along with such substructure characteristics as subgrain size and misorientation angle. The main reasons of appearance and some features of CAS of EDSS have been considered. It is shown that CAS can be considered as feature determining the critical grain size. When grain size is less than CAS the creep rate becomes depending on grain size and mechanism changes from high temperature recovery to grain-boundary glide one. Ó 2020 Elsevier B.V. All rights reserved.

Keywords: Metal and alloys Creep Deformation and fracture Microstructure Correlation area size of excess dislocations of the same sign

1. Introduction The dependence of instantaneous creep rate e_ on integral substructure misorientation angle d for single crystalline molybdenum, tungsten and also polycrystalline Ni-1.18%Al was earlier found in [1]. This dependence takes place during creep strain at elevated temperatures

e_ / d2 :

ð1Þ

Angle d had determined by X-ray diffraction method to high statistics as a maximal azimuth broadening of the reciprocal lattice sites dq? corresponding to the active slip planes [4]:

d¼

dq? : q

ð2Þ

Here q is a diffraction vector modulus. On the other hand, angle d immediately depends on the local density of EDSS Dq , which one is connected with the differential quantities describing dislocation substructure [3,4]:

dq? ¼ aL bqdDq ffi aL q

df ðh þ bÞ

l

:

ð3Þ

Here aL is geometrical factor including orientation of reciprocal *

*

lattice site with respect to Burgers vector b , dislocation line l and *

scanning direction L , d is the crystal or grain size, h is the averaged misorientation angle of subboundaries or subgrain boundaries, b is

the averaged misorientation angle within one subgrain or between neighbor subboundaries, l is the averaged subgrain size or distance between neighbor subboundaries, df is the CAS of EDSS. The conception of ‘‘excess dislocations of the same sign”, that we are using over the time of the last ten years as a quantitative response of dislocation substructure, has to be similar to the well-known conception of ‘‘geometrically necessary dislocations” [5–9]. It is used for the description of dislocations necessary for the local crystal bending. Hereinafter we will use the term of ‘‘excess dislocations of the same sign” because it, in our opinion, is some better correlating with the physics of creep process. The CAS of EDSS represents the fundamental feature of creep. The concept of CAS and its basic properties will be presented in this work. 2. Materials and methods Correlation areas of EDSS on X-ray diffractograms at scan of a site of the reciprocal lattice in an azimuthal direction have been exhibited as rather large fragments, between which the deep (till 30–35%) falls of intensity are observed (Fig. 1). If the distribution of intensity from a separate site of the reciprocal lattice places on the edges of X-ray site has the Lorenz shape, it means that structure inside a grain is consisting of subboundaries (or boundaries of subgrains), alternating on the sign [10,11]. It, in turn, means that the size of subgrains is a lower limit of the CAS of EDSS. On the other hand, the CAS of EDSS is limited above by a grain size.

https://doi.org/10.1016/j.matlet.2020.129017 0167-577X/Ó 2020 Elsevier B.V. All rights reserved.

Please cite this article as: A.I. Dekhtyar, Correlation area size of excess dislocations of the same sign as a new mesoscopic characteristic of the dislocation substructure, Materials Letters, https://doi.org/10.1016/j.matlet.2020.129017

A.I. Dekhtyar

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Fig. 1. Topographical picture of X-ray scanning the reciprocal lattice site (0 1 1) in an azimuthal direction of Mo single crystal after creep (T = 1073 K, r = 40 MPa, strain e = 5%), a and b are the apparatus angle coordinates. The vertical lines are indicated the ‘‘borders” between adjacent correlation areas of EDSS.

Fig. 2. On the definition of correlation area size of one sign excess dislocations with the help of light microscopy image. Picture after creep at T = 1073 K, r = 40 MPa, e = 5%. 2df corresponds to the distance between adjacent accommodation kinks Rak on the surface.

However, as the rule, CAS takes intermediate value between the subgrain size and grain size. Each fragment of X-ray diffraction site has an inner structure corresponding to several subgrains, each of which is bordering on others, and in which the excess dislocations have the same sign. Exactly within the ‘‘border” of such adjacent zones the local EDSS have been formed because of intensive annihilation of dislocations of different signs by the way of slip and climb. The large part of these excesses is transformed into subboundaries at following stages of strain. It is necessary to note, that studies on measuring magnitude df for different metals and influencing on it of the different deformation factors are not present in the literature. Everything, which will be explained below, is grounded on rather separate data on analysis of creep of single-crystalline molybdenum and tungsten [13– 16] and of polycrystalline nickel and of Ni-1,18%Al alloy [17,18].

Accommodation kinks are the typical example of visualization of correlation areas of EDSS. In this case EDSS have concentrated along with scalar dislocation density on the one side of deformation kink and excess dislocations of another sign on the other side of kink. Metallographically the size df is measured along direction of slip traces as

df ’ 0:5Rak sinc:

Here Rak is the averaged distance between adjacent accommodation kinks on the surface, c is an angle between position plane of accommodation kink and the surface of observation. In the most cases the values of c are between 45° and 90°. Calculation of df with the help of Fig. 2 data (Rak  160 mm) and with assumption c = 90⁰ leads to the value df = 80 ± 7 mm. We have not bad accordance between X-ray and optical microscopy results. Magnitude df is strongly varied for different metals under the same deformation conditions (homologous temperatures, reduced stresses and so on) (Table 1).

3. Results Relationship (3) shows that substitution of grain size d onto CAS of EDSS df takes place when going from the case of homogeneous bending within grain to the case when df < d. The last means that a set of correlation areas alternated on sign has been fallen into diffraction zone. As the rule, polycrystals investigated here on creep have the grain size about 0.5–0.7 mm whereas correlation area size df is much less. X-ray beam on the specimen surface has the size about 1 mm. The quantity df can be found by scanning X-ray site along azimuthal direction

df ¼

0:5a N

ð5Þ

4. Discussion The noticeable effect on df , and, as the rule, in the direction of its decrease, is caused by preliminary deformation treatments realized by customary ways (active tension at middle temperatures, compression etc.) [1–3]. The usual values of df observed during creep strain for different metals lay within the limits 20–200 lm [14,16,18,19]. When the grain size d becomes less than some distinctive magnitude df the considerable fraction of a strain should be formed by slippage on boundaries of grains. Such situation will arise because of appearance of high component of tangential stresses in the interface area of grains’ boundaries. These stresses, in turn, have been appeared as the result of uncompensated fields of stresses arisen by EDSS at ‘‘cutting” of some part of correlation area of these dislocations. As a result, the reduced applied stresses in great number of grains’ boundaries will be added together with such uncompensated stresses. The last will gives rise to the condition for activation of slip of grain-boundary dislocations. The formation of uncompensated fields of stresses was established and explicitly enough investigated by X-ray diffraction in [16] at analysis of influence of surface layers removal on density and distribution of excess dislocations in plastically bent tungsten single crystals. Therefore, the influence of grain size on creep should be

ð4Þ

Here a is the size of irradiated area by X-ray beam at the specimen surface, N is the quantity of maximums within one X-ray spot. Coefficient 0.5 is appeared due to the presence of altering the areas with the same and opposite signs. Calculation of df with the help of Fig. 1 data (a = 1 mm, N = 6) and expression (4) gives a result df = 73 ± 6 mm. Metallographically the alternation of correlation areas of different signs is exhibited on the crystal surface in model as legible periodic waviness, which is visible as alternation of light and dark bands. Accommodation kinks, which have been as the rule practically perpendicular to the slip traces, are the special case of such bands. They are visible after creep of single crystals especially well [12,13] (Fig. 2). 2

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A.I. Dekhtyar Table 1 Correlation area size of excess dislocations of one sign df for different metals obtained by metallographic and X-ray investigations. Metal Mo W Ni Ni-1.18%Al

Test temperature, K (homologous, T=T m *) 1633 (0.57) 2073 (0.57) 973 (0.57) 973 (0.57)

Applied stress 10 12 25 25

(1.2 (1.2 (1.2 (1.2

   

r, MPa (reduced, r=G*)

4

10 ) 104) 104) 104)

Achieved angle d, mrad

df , mm

20 20 20 20

150 ± 12 130 ± 12 70 ± 6 50 ± 5

*Tm is the melting point; G is the shear modulus.

considerably intensified, when the grain size d becomes less than correlation area size of excess dislocations of one sign (d < df ). Arguments described above have confirmed by early results on high temperature creep of copper [20] and of the most of pure metals with BCC lattice [21]. Particularly as was shown in [20] creep rate of copper does not depend on the grain size until this size is keeping larger than 0.1 mm. When grain size is less than critical one in both cases dependence of creep rate on the grain size is described by the expression:

e_ 

 2 b : d

Acknowledgements Funding: This work was supported by the Science and Technology Center of Ukraine (STCU) [grant No. 50]. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.matlet.2020.129017. References

ð6Þ

[1] A.I. Dekhtyar, L.V. Demchenko, Proc. of the 4th Int. Conf. On Tungsten, Refractory Metals and Alloys, 1997, Princeton, 1998, pp. 309-318.. [2] A.I. Dekhtyar, Metallofiz. Novejš. Tekhnol. 22 (2000) 73–80. [3] A.I. Dekhtyar, Metallofiz. Novejš. Tekhnol. 23 (2001) 123–128. [4] M.A. Krivoglaz, X-ray and Neutron Diffraction in Nonideal Crystals, SpringerVerlag, Berlin, 1996. [5] A. Needleman, J.G. Sevillano, Scr. Mater. 48 (2003) 109–111. [6] H. Gao, Y. Huang, Scr. Mater. 48 (2003) 113–118. [7] L.P. Kubin, A. Mortensen, Scr. Mater. 48 (2003) 119–125. [8] M. Zaizer, E.C. Aifantis, Scr. Mater. 48 (2003) 133–139. [9] D.A. Hughes, N. Hansen, D.J. Bammann, Scr. Mater. 48 (2003) 147–153. [10] K.P. Ryaboshapka, Physics of X-ray Scattering by Deformed Crystals, Naukova Dumka, Kiev, 1993. [11] L.V. Demchenko, A.I. Dekhtyar, V.A. Kononenko, K.P. Ryaboshapka, Met. Phys. 11 (1989) 84–86. [12] A.I. Dekhtyar, O.P. Karasevska, I.V. Moiseyeva, Y.N. Petrov, V.K. Pischak, L.N. Trofimova, Metallofiz. Novejš. Tekhnol. 25 (2003) 205–225. [13] R.I. Barabash, O.P. Karasevska, V.A. Kononenko, Met. Phys. 8 (1986) 43–47. [14] A.I. Dekhtyar, O.P. Karasevska, I.V. Moiseyeva, Y.N. Petrov, V.K. Pischak, L.N. Trofimova, Metallofiz. Novejš. Tekhnol. 24 (2002) 87–111. [15] R.I. Barabash, O.P. Karasevska, V.A. Kononenko, Met. Phys. 10 (1989) 106–108. [16] A.I. Dekhtyar, I.V. Moiseeva, A.P. Starzhinskij, Metallofiz. Novejš. Tekhnol. 17 (1995) 45–50. [17] G.Y. Kozyrski, V.A. Kononenko, Phys. Met. Metallography 22 (1966) 108–111. [18] G.Y. Kozyrski, P.N. Okrainets, V.K. Pischak, Phys. Met. Metallography 34 (1972) 607–611. [19] L.V. Demchenko, V.A. Kononenko, Met. Phys. 47 (1973) 61–64. [20] C.R. Barrett, O.D. Sherby, Trans. AIME 233 (1965) 1116–1119. [21] R.R. Vandervoort, Trans. AIME 242 (1968) 345–347.

This correlates with expression (1) subject to (2) and (3) if instead grain size d to substitute the CAS df . One can suggest that 0.1 mm critical value of grain size is the CAS of EDSS for copper at specified creep test conditions.

5. Conclusions It is quite probable that at grain size larger than critical one the creep rate will depend on CAS of EDSS as well as on grain size accordingly to expression (6) when grain size is the less than critical one. In such cases the function of grain size will be assumed by the CAS of EDSS and creep mechanism is controlled by grainboundary glide.

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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