Critical grain sizes and generalized flow stress grain size dependence. S.A.Firstov, T.G.Rogul, O.A.Shut

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1 Critial grain sizes and generalized flow stress grain size dependene S.A.Firstov, T.G.Rogul, O.A.Shut I.Frantsevih Institute for Problems of Materials Siene, National Aademy of Sienes of Ukraine, Kyiv, Ukraine The generalized equation that desribes the yield stress dependene upon the grain size in the wide range of grain sizes has been obtained. There are two ritial grain sizes (d r, d r ) that orrespond to the hanges in strengthening mehanism. The equation inludes Hall - Peth relation in the range of d> d r. For d r >d> d r, the power in the Hall - Peth relation varies from -/ to -. In a nanometer (nm) range (d r >d) there are possibilities of softening (in ase of weak boundaries) as well as signifiant strengthening (in ase of strong boundaries). Key words: hromium films, ultra-fine grain struture, Hall - Peth relation. There have been enough disussions in the literature about the existene of so-alled inverse Hall Peth (HP) relation [-4 et.], i.e. the derease of yield strength is observed with the redution of the grain size after reahing a ertain grain size d r. For the desription of the deviation from the lassial HP - relation many models are proposed, whih an be ombined into four groups: ) disloation - based models [5, 6]; ) two phase (grain body and grain boundary) - based models (a so-alled mixture rule) [7, 8, 9]; 3) models of grain boundary plastiity mehanisms (diffusion - based models (Coble reep), grain boundary shearing models) [0,, ]; 4) models of wide staking faults (models of twinning mehanisms) [3]; 5) models that take into aount the ompetition of different strengthening mehanisms, beause, as a rule, several mehanisms at simultaneously in real materials [3]. In aordane with those models it s possible to desribe the transition from Hall-Peth strengthening to the inverse HP relation at the some ritial grain size d r. Kumar K.S. [4] proposed a sheme (fig. a) with two ritial grain sizes d r =00 nm and d r =0 nm, in whih the slope of σ(d) dependene hanges. Nevertheless, HP relation () is fulfilled well in the region of large grain sizes in the overwhelming majority of ases. Along with that, it is pointed out in numerous artiles [7, 5, 6] for smaller grains that the strengthening at the grain refinement an be better desribed by the equation ().

2 / d 0 kyd d k d () 0, () where the parameter 0 haraterizes the averaged resistane to disloation motion over the grain body and the oeffiients k y and k haraterize the diffiulty of slip transfer through the grain boundary. The Thompson`s sheme [5], in ontrast to fig a in [4], postulated not the slight slakening of the dependene of the yield strength upon the grain size, but, on the ontrary, the inreasing dependene of the yield strength takes plae due to the transition from the dependene () to the dependene () at d r. Fig. Dependene of flow stress on the grain size а) [4], б) [5] The simplest explanation of suh transition is the following. The oeffiient k y is expressed as / k y r, (3) where Gb / L - the stress of the disloation soure atuation in the neighboring grain ( - oeffiient of an order of, G - shear modulus, b- Burgers vetor, L harateristi length of a Frank Reed soure), r is the distane of the soure from the head of the disloation pile-up. With the grain sizes in a range d r >d >d r it is obvious that the length of disloation soure L and its distane from the grain boundary r an not be larger then d. Supposing that in suh ase L ~ d and r ~ d, it an be shown that the oeffiient is getting dependent upon the grain size as k y = k d / (4) The substitution of (4) into the equation () leads to the equation (). With the further grain size refinement (d r >d), there might be the situation when the disloation soure an not generate disloations at the stress lower than the theoretial shear strength. In this ase flow stress dependene an be desribed as some expression σ 3 (d), whih is determined by one of the onrete deformation mehanisms listed above. It is neessary to underline, that only for the disloation - based models [5, 6], Coble reep [0, ], and also for the so-alled two phase based models [7, 8, 9] the analyti form for suh dependenes existed already.

3 Disloation - based models an be exluded from the examination beause the moving of the disloations inside nanograins is quite diffiult. The realization of the Coble reep at the room temperature for the nanorystalline materials even for the opper needs anomalous high diffusion oeffiient [], whih is unlikely for the metal with higher melting temperature. From this point of view, for hoosing an appropriate analytial dependene for σ 3 (d) (just like in [6]), we will selet the approah based on a mixture rule: d t d d t d d 3 B V. (5) Here t the thikness of grain boundary; σ B the strength of grain boundary (GB strength); σ V the strength of the nanograin volume. Sine the strength of the disloation-free nanograin goes up to the limit (theoretial) strength, we an suggest that σ V is lose to the theoretial strength ( E/30). Earlier, Takeuhi used quite different formula for the yield stress [9]: t y B d, t nm - the effetive thikness of the grain boundary. (6) In our opinion, the expression (5) desribes the omposite behavior better, than the similar formula of Takeuhi (6). Besides, Takeuhi examined only the possible weakening of the dependene beause the strength of the grain volume is taken higher that the GB-strength B, and he did not examine the variant, when the strength of the intergranular material an be equal to the strength of the grain body or even exeed it. To reeive the generalized dependene σ(d) in the wide range of the grain sizes, the method that was used in [7] an be employed, where the distribution of the grains by size an be onsidered. At that, it is needed to onsider the availability of two ritial grain sizes, separating the manifestations of the strength mehanisms, desribed with the equations (), () and (5). Just like in the [7, 8], we will onsider that the grain size log-normal distribution exists in the polyrystalline material: ln( V ) m lnv f ( V ) exp ln s (7) V s V lnv, where s ln V, m ln V - fitting fators. Considering the existene of two ritial grain sizes, the resulting yield stress of the polyrystal an be expressed as follows:

4 d m v v v v Vf ( V ) dv Vf ( V ) dv 3Vf ( V ) dv mv m v v 0, (8) where 3 V dr, 3 dr V The plot orresponding to the equation (8) is built in the fig. for 4 different values of GB-strength σ B using hromium [9] as an example. Following data were used: the strength of the grain volume is losed to the theoretial value V GPa ; / 0 0.GPa ; k y.6*0 GPa* mm ; the thikness of boundaries was t nm ; d r 30 nm and d r 67nm, s lnv. Оne an see, that three ranges of flow stress dependene vs the grain size an be distinguished. The first range extends to d r, where the lassial Hall- Peth`s equation is satisfied. The seond one is up to d r, where the sharp inrease of strength vs the grain size follows ~d -. In the nanometer range of the grain size, the derease of σ(d) with refinement of the grain size as well as it`s inrease take plae depending upon the hange in GB-strength (from E/00 to E/5, fig. ). Fig. Flow stress vs. the grain size for some values of a GB- strength. Experimental data taken from [9] In the fig.3 there are experimental values H/E for the hromium-based materials with different grain sizes, whih are produed with different tehnologies. The data for hromium films with the thikness of 40μm that are produed by magnetron yli sputtering are marked with green spots [9]. The red spots orrespond to the hromium oatings with the thikness of about 6μm, that are produed by magnetron sputtering in the uniform regime and with the ontrolled adding of the oxygen during the sputtering (it was shown that oxygen segregates at the boundaries of grains) []. The blue spots orrespond to the alloy Cr wt.% Ni with different thermal treatment []. It is obvious, that the grain boundaries will be different for all materials. The dependenies shown in fig. 3 have been built aording to the equation (8) onsidering the Marsh formula [0] for two values k y (max and min) and for

5 two values of the GB- strength (for the ase of relatively weak and relatively strong boundaries). The lowest dependene orresponds to the lower values k y =.4*0 - GPa*mm / for the hromium of high purity. The upper dependene orresponds to the higher value k y =.8*0 - GPa*mm /. Aording to the equation (8), it is supposed that at the hange of the size of the grain, the struture and properties of the boundaries and, orrespondingly k y, do not hange. Obviously, though, the state of the grain boundaries hanges ontinuously for suh material during grain size variations with thermal treatment. That is why, the experimental data shown in fig. 3, are loated between two dependenies that reflet different values of k y for eah grain size. Fig.3. The generalized dependene of the normalized hardness (H/E) upon the grain size, whih takes into aount the possible hanges of the boundaries state for two different k y and for two different GB-strength E/00 E/30. Conlusion Summarizing, let us emphasize that, in general, there might be more than two ritial sizes beause strengthening mehanisms mentioned above might ompete in the nano-region. Besides, not only the possible hange in value of k y should be taken into aount in ertain ases but the hange of the parameter σ 0 at the hange of the grain size also. Referenes: [] - E. O. Hall Pro. Phys. So. London B64, 747 (95). [] - N. J. Peth, J. Iron Steel Inst. 74, 5 (953). [3] M.Yu.Gutkin, I.A.Ovid`ko, C.S.Pande Theoretial models of plasti deformation proesses in nanorystalline materials // Rev.Adv.Mater.Si (00) pp [4] M.A. Meyers, A. Mishra, D.J. Benson Mehanial properties of nanorystalline materials // Progress in Materials Siene 5 (006) p [5] - Evans A.G., Hirth J.P. Deformation of nanosale erments // Sr. Met. Mater., 99,v.6,,p [6] - C.S. Pande, R.A. Masumura Proessing and Properties of Nanorystalline Materials. Ed. C.Suryanarayana, J. Singh, F.H. Froes. Warrendale, PA, TMS, 996, p. 387.

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