Plasticity of materials determined by the indentation method

In this review, the development of techniques for determining the plasticity of mate rials by the indentation is considered.

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Date:	2018
Main Authors:	Milman, Yu.V., Chugunova, S.I., Goncharova, I.V., Golubenko, A.A.
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Language:	English
Published:	Інститут металофізики ім. Г.В. Курдюмова НАН України 2018
Series:	Успехи физики металлов
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Cite this:	Plasticity of materials determined by the indentation method / Yu.V. Milman, S.I. Chugunova, I.V. Goncharova, A.A. Golubenko // Progress in Physics of Metals. — 2018. — Vol. 19, No 3. — P. 271-308. — Bibliog.: 80 titles. — eng.

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spelling	nasplib_isofts_kiev_ua-123456789-1679132025-02-09T16:38:14Z Plasticity of materials determined by the indentation method Пластичність матеріялів, що визначається методою індентування Пластичность материалов, определяемая методом индентирования Milman, Yu.V. Chugunova, S.I. Goncharova, I.V. Golubenko, A.A. In this review, the development of techniques for determining the plasticity of mate rials by the indentation is considered. У даному огляді розглянуто розвиток метод визначення пластичности матеріялів індентуванням. В данном обзоре рассмотрено развитие методик определения пластичности материалов индентированием. 2018 Article Plasticity of materials determined by the indentation method / Yu.V. Milman, S.I. Chugunova, I.V. Goncharova, A.A. Golubenko // Progress in Physics of Metals. — 2018. — Vol. 19, No 3. — P. 271-308. — Bibliog.: 80 titles. — eng. 1608-1021 DOI: https://doi.org/10.15407/ufm.19.03.271 PACS numbers: 06.60.Wa, 07.10.-h, 62.20.D-, 62.20.F-, 62.20.fq, 62.20.Qp, 81.40.Jj, 81.40.Lm, 81.70.Bt https://nasplib.isofts.kiev.ua/handle/123456789/167913 en Успехи физики металлов application/pdf Інститут металофізики ім. Г.В. Курдюмова НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	In this review, the development of techniques for determining the plasticity of mate rials by the indentation is considered.
format	Article
author	Milman, Yu.V. Chugunova, S.I. Goncharova, I.V. Golubenko, A.A.
spellingShingle	Milman, Yu.V. Chugunova, S.I. Goncharova, I.V. Golubenko, A.A. Plasticity of materials determined by the indentation method Успехи физики металлов
author_facet	Milman, Yu.V. Chugunova, S.I. Goncharova, I.V. Golubenko, A.A.
author_sort	Milman, Yu.V.
title	Plasticity of materials determined by the indentation method
title_short	Plasticity of materials determined by the indentation method
title_full	Plasticity of materials determined by the indentation method
title_fullStr	Plasticity of materials determined by the indentation method
title_full_unstemmed	Plasticity of materials determined by the indentation method
title_sort	plasticity of materials determined by the indentation method
publisher	Інститут металофізики ім. Г.В. Курдюмова НАН України
publishDate	2018
url	https://nasplib.isofts.kiev.ua/handle/123456789/167913
citation_txt	Plasticity of materials determined by the indentation method / Yu.V. Milman, S.I. Chugunova, I.V. Goncharova, A.A. Golubenko // Progress in Physics of Metals. — 2018. — Vol. 19, No 3. — P. 271-308. — Bibliog.: 80 titles. — eng.
series	Успехи физики металлов
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fulltext	ISSN 1608-1021. Usp. Fiz. Met., 2018, Vol. 19, No. 3 271 https://doi.org/10.15407/ufm.19.03.271 PACS numbers: 06.60.Wa, 07.10.-h, 62.20.D-, 62.20.F-, 62.20.fq, 62.20.Qp, 81.40.Jj, 81.40.Lm, 81.70.Bt Yu.V. MilMan, S.i. ChugunoVa, i.V. gonCharoVa, and А.А. golubenko I.M. Frantsevich Institute for Problems in Materials Science, NAS of Ukraine; 3 Academician Krzhizhanovsky Str.; UA-03142 Kyiv, Ukraine Plasticity of Materials DeterMineD by the inDentation MethoD In this review, the development of techniques for determining the plasticity of mate rials by the indentation is considered. the development of methods for deter mining the plasticity of materials by the indentation is based on the use of funda mental ideas of the physics of strength and plasticity. Significant development of these me thods became possible after the introduction of a new plasticity characteristic δ* = εp/εt, where εр is the plastic deformation, and εt is the total deformation. this plasticity characteristic corresponds to the modern physical definitions of plasticity, in contrast to the widely used elongation to failure δ. the new plasticity characteristic is easily determined by standard determination of hardness by the diamond pyramidal indenters at constant load P (designated as δН) and by instrumental nanoindentation (designated as δА, and δH ≈ δA). A significant advantage of the new plasticity charac teristic is the ability to determine it not only for metals, but for materials, which are brittle at the standard mechanical tests (ceramics, thin layers, coa tings, etc.), as well. In the development of ideas about theoretical strength, concepts of theoretical plasticity under the dislocationfree and dislocation deformation mechanisms are introduced. A number of studies have established a correlation of δН with the elec tronic structure of the material and its physical properties. As shown, the tabor parameter С (C = HM/σS, where HM is the Meyer hardness, and σS is the yield stress) is easily calculated by the δН value. therefore, indentation allows currently determining simply not only the hardness, but also the plasticity and yielding stress of materials. thus, indentation became a simple method for determination of the complex of mechanical properties of materials in a wide temperature range using a sample in the form of a metallographic specimen. Keywords: hardness, plasticity, indentation, yield stress, deformation. introduction Methods of determination of the hardness by indentation with a rigid indenter (ball or pyramid) are simple and extensively used techniques for characterization the mechanical properties of materials. the hard 272 ISSN 1608-1021. Prog. Phys. Met., 2018, Vol. 19, No. 3 Yu.V. Milman, S.I. Chugunova, I.V. Goncharova, and А.А. Golubenko ness, determined by this method, is undoubtedly a strength characteristic connected with the yield strength of the material [1]. however, attempts to determine also a plasticity characteristic by the indentation method were made for many years (see, e.g., refs. [2–6]). An analysis of the indentation process by a ball in determination of the brinell hardness enabled the author of ref. [2] to introduce the no tion of the modulus of plasticity of a material and calculate a certain effective transverse reduction of the area ψ. In ref. [3], the evaluation of two plasticity characteristics, namely, the elongation δ and transverse reduction of the area ψ, by the indentation method was proposed. In this work, some local elongation δ and local transverse reduction ψ at the apex of the pileups around a hardness indent made by a spherical in denter are considered. the values of ψ determined for a number of steels by the indentation method were close to those determined in ten sile tests. the author of ref. [3] used the developed technique of determining the plasticity characteristics to optimize heattreatment regimes of steels. however, these investigations did not find an extensive application. this is evidently due to a substantial difference in the mechanisms of plastic deformation in indention and in uniaxial tension. Moreover, inden tation by a ball cannot be used for ceramics and other highstrength materials, which are brittle in standard mechanical tests. In this connec tion, the possibility to determine approximately δ and ψ by the indentation method did not generate great interest among researchers. In ref. [7], for the first time, the notion of the plasticity index in contact of two surfaces was introduced. the surfaces are assumed to be conventionally plane, but their roughness and the crumpling of asperities on the surface by plastic deformation, which are taken to be spherical with a β radius, are taken into account. the plasticity index was determined as follows: ( ) ,E H′ψ = σ β where E′ = E/(1 – ν2) when the roughness of one surface is taken into account, Е is Young’s modulus, ν is Poisson’s ratio, Н is the hardness determined by a spherical indenter, and σ are standard deviations of the asperityheight distribution. the notion of the region of plastic crumpling of asperities Ар is used, and А is the total contact area. the value of Ар/А was considered in the range of 0.01–0.50, and, in this case, ψ belongs to the range of 0.6–1.0. results of this work are applicable to problems of friction and wear, but they were not used for the study of the plasticity of materials by the method of indentation with a rigid indenter. In refs. [4–11], the term plasticity was introduced to describe the process of plastic deformation in indentation. In these works, the in ISSN 1608-1021. Usp. Fiz. Met., 2018, Vol. 19, No. 3 273 Plasticity of Materials Determined by the Indentation Method stru mented hardness with recording the ‘load on the indenter P– dis pla cement of the indenter h’ curve is predominantly considered. however, in most such works, a plasticity characteristic that enables one to com pa re the plasticity of different materials is not introduced. In refs. [4, 5], the plasticity index determined in instrumented indentation D = Ap /At (where Ар is the work of plastic deformation, and Аt is the work of total deformation) was introduced. this extremely interesting approach is dis cussed in more detail in the section, which deals with plasticity characteristic δA determined in the instrumented indentation. In ref. [12], the inconsistency of the extensively used plasticity cha racteristics (elongation to fracture δ and the transverse reduction of the area to fracture ψ) with the fundamental concepts of physics of strength was noted. In refs. [12, 13], it is noted that two fundamental properties deter mining the mechanical behaviour of materials, namely, their strength and plasticity, can be distinguished in physics of strength. the strength of a material is determined by its capability to resist an applied force. More specifically, the strength of solid bodies can be determined as the resistance to rupture body into two or several parts [14]. the strength is calculated adequately to these definitions in tensile tests as the breaking load divided by the crosssectional area of the spe cimen. In this case, the material is assumed to be perfect, i.e., without cracks and other stress concentrators. however, it should be noted that the hardness correlates with the yield strength rather than with the strength in the sense noted above. the situation with the determination of the plasticity is much more difficult, if it is desired to obtain a characteristic adequate to the phy sical definition of this property. the word ‘plasticity’ comes from the Greek word ‘πλαστικός’, which means suitable for modelling and malleable. Since the present paper is devoted to the fundamental problems of plasticity, we present some definitions of the term ‘plasticity’ in physics of strength. In physics and engineering, plasticity is defined by the susceptibility of a material to undergo residual deformations under load [15]. Plasticity is the property of the materials of solid bodies to deform irreversibly under the action of external forces and internal stresses [16]. Plasticity is the property of solid bodies to retain a part of strain after removal loads that caused it [17]. the plasticity of crystals is the property of crystalline bodies to change irreversibly their sizes and shape under the action of mechanical loads [18]. however, in practice, plasticity is usually characterized by the elon gation δ to fracture in a tensile test or by the transverse reduction in area to fracture ψ. these parameters of the material are of great practical 274 ISSN 1608-1021. Prog. Phys. Met., 2018, Vol. 19, No. 3 Yu.V. Milman, S.I. Chugunova, I.V. Goncharova, and А.А. Golubenko importance but correspond weakly to the definitions of plasticity presented above because they determine the conditions of transition from plastic deformation to fracture and do not always correlate with plasticity in definitions presented above. In refs. [12, 13], it is noted that the parameters δ and ψ are deter mined by not only plasticity in the sense presented above (the malleability of the material and the capability to retain strain after removal of load), but also by at least two conditions: the relationship between the yield stress and the fracture stress as well as the strain hardening value. the parameter δ usually includes the uniform strain and the strain after formation of a stable ‘neck’ and localization of strain in it. the strain εс at which a stable ‘neck’ nucleates is determined by the condition [19]: 1 . 1 m e e ∂σ σ ≥  − ∂   here, е is the true strain, e⋅ is the strain rate and the parameter m characterizes the influence of the strain rate on the yield stress according to the expression σ = const ⋅ e⋅m. therefore, δ depends not only on the compliance of the material to plastic deformation and on the degree of strain, which determines the transition from plastic deformation to fracture, but also on the strain hardening (∂σ/∂e)e⋅ and parameter m, i.e., on the rate sensitivity of the yield stress. It also should be taken into account that the rate of decrease of the crosssection of the ‘neck; after its formation also depends on the parameter m. on the other hand, the strain hardening increases the yield stress and makes the transition to the fracture process more probable. the fracture process, like the process of strain hardening, has a complicated multiform dependence on the test method, structure of the material, temperature and strain rate. In many cases, the dependence of the elongation to fracture δ on a large number of parameters leads to an inadequate estimate of plasticity. the plasticity of a material in its physical definition presented above must increase continuously with increasing temperature because an increase in the temperature facilitates the dislocation motion in solid bodies (except some intermetallics). ho we ver, for most materials, the parameter δ changes nonmonotonously with increasing temperature because the test temperature also influences on the conditions of fracture (transcrystallite or intercrystallite ones; brittle, quasibrittle or ductile fracture [20, 21]) and strain hardening. For instance, in a number of dispersionhardened aluminium alloys, δ decreases as the temperature increases above room temperature as a result of a decrease in the strain hardening and the earlier formation of a stable ‘neck’. At the same time, for these alloys, an increase in the temperature leads to a decrease in the hardness and better deformability ISSN 1608-1021. Usp. Fiz. Met., 2018, Vol. 19, No. 3 275 Plasticity of Materials Determined by the Indentation Method during the metal forming (extrusion, rolling, etc.). In other words, ob jectively, the plasticity of the alloy increases. the transverse reduction of the area ψ characterizes the cross section of a specimen, at which the applied load turns to be sufficient for fracture. the transverse reduction of the area ψ can also consist of two summands: reduction of the area under uniform deformation and reduction of the area under deformation concentrated in the ‘neck’. Since fracture occurs after plastic deformation, which occurs with strain hardening, ψ (like δ) is determined by the condition of transition from plastic deformation to fracture and depends on the strain hardening and the type of fracture of the material. thus, both usually used characteristics δ and ψ often inadequately reflect the physical meaning of the term plasticity, though they are con venient technological tests that characterize the capability of the mate rial to be deformed plastically before fracture during tensile test. It follows from the foregoing that, if the plasticity is characterized by the elongation to fracture δ, then, the notion of plasticity loses its clear physical meaning and, hence, ceases to be a fundamental charac teristic of the material [12, 13]. Mechanical tensile tests of smooth specimens occupied a leading po si tion many decades ago, when plastic materials, namely, steels and metals with f.c.c. and h.c.p. lattices, were the main structural materials. however, in subsequent years, radically new materials, which are low ductile or brittle in mechanical tensile tests at room temperature, were developed. these are ceramics, quasicrystals, metallic glasses, inter metallics, fullerites, and different composites. Alloys based on refractory b.c.c. metals, for which the ductilebrittle temperature is usually higher than room temperature, found extensive application. the cold brittleness phenomenon, which was known for steels, but usually manifested itself below room temperature, turned to be typical for most alloys based on refractory metals with a b.c.c. lattice (Cr, Mo, and W) at room tem perature as well. the efficiency of mechanical tensile tests for materials, which are brittle in tensile tests, is very low, and it is possible to determine only ultimate strength as a fracture stress. As for the plas ticity of these materials in tensile tests, it can be said only that their elongation is δ = 0. Such tests do not give any information on the com parative plasticity (or brittleness) of materials. At the same time, most materials, which are brittle in tensile tests, exhibit some plasticity in other ‘softer’ test methods, in particular, in determination of the hard ness with a rigid indenter [12]. this situation made reasonable the introduction of a new plasticity characteristic as a fundamental property of a material that satisfies the physical definitions presented above. Such plasticity characteristic was 276 ISSN 1608-1021. Prog. Phys. Met., 2018, Vol. 19, No. 3 Yu.V. Milman, S.I. Chugunova, I.V. Goncharova, and А.А. Golubenko proposed in ref. [12] in the form of the dimensionless parameter: * 1 ,p e t t ε ε δ = = − ε ε (1) where εp, εe, and εt are plastic, elastic and total strain, on the contact area specimen–indenter in the direction load P, respectively, and εt = εp + εe. this plasticity characteristic (which, as is seen from eq. (1), is determined by the fraction of the plastic strain in the total elastoplastic strain) corresponds fairly well to the physical definitions of plasticity presented above. Actually, the fraction of plastic strain in the total strain characterizes the malleability of the material, i.e., its capability to change its shape (deform) with preservation of strain after removal of load. the considered plasticity characteristic is universal in the sense that it can be determined by any method of mechanical tests (tension, compression and bending) and, as shown in ref. [12], in indentation. the plastic and elastic components of strain can be determined from the curve of deformation in tension in coordinates stress σ–strain εt (Fig. 1), and the plasticity characteristic δ* can be calculated with the help of these components εp and εe by eq. (1). In uniaxial tension or compression, as it follows from eq. (1) and the hooke law, the plasticity characteristic δ* can be represented in the form [12] as follows: * 1 ,S t E σ δ = − ε (2) where Е is Young’s modulus and σS is the yield stress for the achievement of the strain εt. From expressions (1) and (2), it is seen that δ* depends on the degree of total strain εt. the same can also be said about the value of δ* de termined according to eq. (1) by other methods of mechanical tests. this dependence follows directly from the definitions of plasticity presented in the foregoing. Actually, e.g., in ten sile tests, in the first stages of loading, εt = εe, and plastic strain is absent, even in the ductile metals, i.e., the material does not preserve a part of strain after removal of loads. As the critical shear stress is attained, plastic strain appears, and, subsequently, its fraction rises with increase in the load and εt, i.e., the plastic deformability of a material, and, therefore, δ* increases [12]. It is noted in refs. [12, 13] that the dependence of the plasticity and its characteristic δ* on the degree of strain εt and a comparison of the values of plasticity of different materials should be performed at a certain representative degree of strain εt = const. the condition εt ≈ ≈ const is automatically provided in indentation of materials by an indenter in the form of a pyramid, e.g., a Vickers tetrahedral pyramid or a berkovich trihedral pyramid. At the same time, the small volume ISSN 1608-1021. Usp. Fiz. Met., 2018, Vol. 19, No. 3 277 Plasticity of Materials Determined by the Indentation Method of the deformed material and the specific character of stress fields reduce the susceptibility to macro scopic fracture and decrease abruptly the ductile–brittle transition tempe ra ture Тdb. this makes it possible to determine the hardness and plastici ty characteristics of most materials even at cryogenic temperatures [23–26]. A total degree of strain εt = 8–9% observed in indentation by these indenters is sufficiently rep resentative and convenient for comparison of values of plasticity of different materials. the plasticity characteristic determined according to eq. (1) by the indentation method was denoted by δH in ref. [12]. A theory that makes it possible to determine εр and εе in indentation and calculate δH was developed [12, 13, 24], and experiments on determination of δH for a large number of materials, including materials brittle in standard mechanical tests, were carried out. Values of δH were determined for different materials: f.c.c., b.c.c., and h.c.p. metals, covalent crystals and refractory compounds with a large fraction of the covalent component in the interatomic bond, intermetallics, amorphous metallic alloys and quasicrystals [12, 13, 24]. In view of the locality of the indentation method, it is possible to determine the value of δH for the thin coatings [27]. In the papers [5, 12, 22, 28, 29], the possibility to determine such plasticity characteristic in instrumented indentation was considered. In the recently published work [30], methods for determining of the plasticity characteristics by the indentation method in thin layers and coatings were considered, and the correlation of the plasticity characte ristic with other mechanical properties of these materials was studied. the work [31] reports on the possibility to calculate the tabor parameter С in the relation НМ = СσS (where НM is the Meyer hardness, and σS is the yield strength) from the value of δH and, therefore, to determine the yield strength σS. the review focused on the introduction and development of notions of the plasticity characteristic δH = (plastic strain)/(total strain) and on the application of δH for the determination of the plasticity of materials. Fig. 1. Decomposition of the total strain εt at the point А into the plastic (εр) and the elastic (εе) components for the calculation of the plasticity characteristic δ* in a ten sile test [22] 278 ISSN 1608-1021. Prog. Phys. Met., 2018, Vol. 19, No. 3 Yu.V. Milman, S.I. Chugunova, I.V. Goncharova, and А.А. Golubenko Plasticity characteristic δ H Determined by the indentation Method from the Value of Microhardness, and classification of Materials by the Value of δ H the plasticity characteristic determined by expression (1) was proposed, as has been noted, in ref. [12]. In ref. [12], the mean elastic strain on the indenter–specimen contact area in the direction of the applied load was obtained in the form: 2(1 2 ),e S S S HM E ε = − − ν − ν (3) where НМ is the Meyer hardness, which is considered as the average contact pressure (HM = P/S, where Р is the load on the indenter and S is the projection area of the hardness indent on the surface of the specimen), ЕS is Young’s modulus of the investigated material, and νS is its Poisson’s ratio. expression (3) can be considered as the hooke law for the indentation process. the total strain εt was determined for pyramidal indenters as follows: εt = –ln (sin γ), (4) where γ is the angle between a face and the axis of the pyramid. then, according to eq. (1), for a pyramidal indenter, the plasticity characteristic, determined in indentation, has the form: 21 (1 2 ).H S S S t HM E δ = − − ν − ν ⋅ ε (5) In particular, for the Vickers indenter, with regard for the fact that HV = HM sinγ and γ = 68°, the following relation was obtained: 21 14.3 (1 2 ) .H S S S HV E δ = − − ν − ν (6) For the berkovich hardness, in which a trihedral indenter with an angle γ = 65° is used, the plasticity characteristic is described by the formula: 21 10.2 (1 2 ) .H S S S HM E δ = − − ν − ν (7) the theory of determination of the plasticity characteristics δH was further developed in ref. [13]. In this work, the condition of in comp ressibility of a material under the indenter was used only for the calcu lation of the plastic part of the strain εр, but not for the total strain, as it was done in ref. [12]. this is why the results obtained in ref. [13] can be used to calculate strains and the plasticity characteristic δH for the hard and superhard materials with a large fraction of elastic strain in indentation. ISSN 1608-1021. Usp. Fiz. Met., 2018, Vol. 19, No. 3 279 Plasticity of Materials Determined by the Indentation Method For the plastic strain, the following relation was obtained: 2 * ln 1 ctg .p HM kE  ε = − + γ −    (8) here, k = 0.565 for a trihedral and a tetrahedral pyramid, k = 0.5 for a conic indenter with an apex angle of 2γ, and Е* is the effective Young’s modulus of the indenter–specimen contact pair. 2 2 * 1 11 .S i S i E E E − ν − ν = + (9) the subscripts ‘S’ and ‘i’ correspond to the specimen and indenter, respectively. the plasticity characteristic δH is calculated with the use of relations (8), (3), and (1). In refs. [13, 22], it was shown that, for metals, such calculation gives values of δH coinciding with results obtained by eq. (5). only for the hard and superhard materials, at δH < 0.3, substantial differences are observed, and calculations should be performed with relations (1), (3), and (8). It is seen from eq. (1) that the plasticity characteristic δ* is a dimen sionless parameter and can change from 0 (purely elastic defor mation) up to 1 (for the purely plastic behaviour of the material). In ref. [32], it was shown that the parameter δH correlates to some degree with the elongation to fracture δ determined in a tensile test at a temperature higher than the ductile–brittle transition temperature Тdb. It is natural that, at T < Tdb, the elongation to fracture δ → 0, whereas δH has well defined values charac terizing the plasticity of the mate rial. Figure 2 shows temperature de pen dences of δ (obtained in a ben ding test as the elongation to fracture of edge stret ched fibres) and of the plasticity characteristics δH for a WC–6 mas.% Co hard alloy [32]. In ref. [12], it was experimen tally established that there exists a critical value δHcr. Materials having δH > δHcr are plastic in standard mechanical tensile and bending tests (δ > 0), whereas for δH < δHcr, the elongation in tensile tests δ usually Fig. 2. temperature dependence of the elon gation to fracture δ and of the plastic ity characteristic in microindentation δH for a WC–6 mas.% Co hard alloy with an average grain size d = 1.3 µm [32] 280 ISSN 1608-1021. Prog. Phys. Met., 2018, Vol. 19, No. 3 Yu.V. Milman, S.I. Chugunova, I.V. Goncharova, and А.А. Golubenko approaches to zero. For pure singlephase materials, δHcr ≈ 0.9, whereas for composites and particularly coatings, δHcr can be lower. the introduction of the plasticity characteristic δH made it possible, for the first time, to classify practically all materials by their plasticity. In table 1, which generalizes the results of refs. [12, 24, 31, 33], such a Table 1. Classification of materials by the plasticity characteristic δ H at room temperature [31] Materials НМ, GPa ES, GPa νS δH f.c.c. metals Al 0.173 71 0.350 0.99 Au 0.270 78 0.420 0.99 Cu 0.486 130 0.343 0.98 Ni 0.648 210 0.290 0.98 b.c.c. metals Cr 1.404 298 0.310 0.97 ta 0.972 185 0.342 0.97 V 0.864 127 0.365 0.97 Mo (111) 1.998 324 0.293 0.96 Nb 0.972 104 0.397 0.96 Fe 1.512 211 0.280 0.95 W (001) 4.320 420 0.280 0.92 h.c.p. metals ti 1.112 120 0.360 0.95 zr 1.156 98 0.380 0.95 re 3.024 466 0.260 0.95 Mg 0.324 44.7 0.291 0.95 be 1.620 318 0.024 0.94 Co 1.836 211 0.320 0.94 Intermetallics (IM) Al66Mn11ti23 (IM3) 2.203 168 0.190 0.87 Al61Cr12ti27 (IM2) 3.456 178 0.190 0.81 Al3ti (IM1) 5.335 156 0.300 0.76 Metallic glasses (MG) Fe40Ni38Mo4b18 (MG2) 7.992 152 0.300 0.62 Co50Ni10Fe5Si12b17 (MG3) 9.288 167 0.300 0.60 Fe83b17 (MG1) 10.044 171 0.300 0.58 Quasicrystals (QC) Al70Pd20Mn10 (QC2) 7.560 200 0.280 0.71 Al63Cu25Fe12 (QC1) 8.024 113 0.280 0.48 refractory compounds WC (0001) 18.036 700 0.310 0.81 NbC (100) 25.920 550 0.210 0.54 lab6 (001) 23.220 439 0.200 0.50 tiC (100) 25.920 465 0.191 0.46 zrC (100) 23.760 410 0.196 0.46 Al2o3 (0001) 22.032 323 0.230 0.41 αSiC (0001) 32.400 457 0.220 0.36 Covalent crystals Ge (111) 7.776 130 0.210 0.49 Si (111) 11.340 160 0.220 0.42 ISSN 1608-1021. Usp. Fiz. Met., 2018, Vol. 19, No. 3 281 Plasticity of Materials Determined by the Indentation Method clas sification is presented. As seen from table 1, f.c.c. metals have the highest plasticity, b.c.c. and h.c.p. metals ranking next in order of decrea sing δH. For all metals in a highly pure and possible perfect state, δH > > δHcr, and, therefore, they are ductile in tensile tests. the next group in the table 1 is intermetallics. Plasticity of intermetallics exceeds the plasticity of other materials presented in table 1. however, interme tal lics are brittle at room temperature (δH < δHcr). Metallic glasses, quasicrys tals, refractory compounds, and purely covalent crystals usually have an even smaller plasticity. Plasticity characteristic δ A Determined in the instrumented indentation Note also that the plasticity characteristic δH can also be determined in instrumented indentation (which is more often used in determination of nanohardness), during which loading and unloading curves are recorded in coordinates ‘load Р–displacement of an indenter h’ (Fig. 3) [22]. In this case, the plasticity characteristic has the form: 1 ,p e A t t A A A A δ = = − (10) where Ар, Ае and Аt are the works expended on plastic, elastic and total deformations, respectively, during penetration of the indenter; А t = А p + + А e. the ratio Ар/А t can be determined from the ratio of areas under the unloading and the loading curve. the authors of refs. [4, 5, 29, 34–37], etc., in measurement of the nanohardness, also use the ratio Ар/А t for the charac teri zation of the plastic behaviour of materials and denote it by PI (the socalled plasticity index). In ref. [28], it was shown that δH ≈ δA if both δA and δH are de termined by identical indenters under equal loads on the indenter. In Figure 4, results obtained in ref. [28] (with the use of a berko vich indenter) in the form of a dependence of δA and δH on the ratio HM (1 – νS – – ν2 S)/ES (see eq. (7)) are shown. As seen, for all metals and most ceramic materials investigated in ref. [28], at Fig. 3. Diagram of penetration of a pyra midal indenter in coordinates ‘load P− displacement of the indenter h’; he and hp are the elastic and plastic displacement approaching the indenter and the speci men; Ae and Ap are the elastic and plastic components of the work of deformation in instrumented indentation [22] 282 ISSN 1608-1021. Prog. Phys. Met., 2018, Vol. 19, No. 3 Yu.V. Milman, S.I. Chugunova, I.V. Goncharova, and А.А. Golubenko δA > 0.5, the values of the characteristics δA and δH practically coincide (with considering the error in calculations of the values of НМ and ES). In recent years, the plasticity parameter δA is often used for the determination of the plasticity characteristic by the indentation method [4, 5, 30, 36, 38]. In this case, different notation of the parameter δA is used. In ref. [4], it is presented as the ductility index D = Ар /А t, and, in ref. [5], the same parameter is denoted by PI (plasticity index). In some works, the parameter δH = εp/εt is also denoted by PI. For instance, formula (10) was used for the calculation of the plas icity in refs. [4, 5]. In ref. [5], the following values of δA were obtained: 0.91 for W, 0.95 for Cu, 0.98 for Al, and 0.52 for Al2o3, which are close to the results presented in Fig. 4 and table 1. It should be noted that table 1 presents data obtained for singlecrystal Al2o3, whereas ref. [5] reports on data for polycrystalline Al2o3. An advantage of the characteristic δA (or PI) is that, in its calculation, the preliminary determination of the hardness, Young’s modulus and Poisson’s ratio is not required that increases the accuracy of determination of this characteristic. It is important that δA correlates well with the plasticity parameter δH, which, as shown above, has a clear physical meaning coinciding with the physical definition of plasticity. Fig. 4. Dependence of the plasticity characteristics δA and δH on the ratio 2(1 2 )/ S S S HM E− ν − ν . Values of δA and δH were obtained with the use of a berkovich indenter under the same load for each material [28] ISSN 1608-1021. Usp. Fiz. Met., 2018, Vol. 19, No. 3 283 Plasticity of Materials Determined by the Indentation Method size Dependence (ise) of the Plasticity characteristic δ H the plasticity characteristic δH is calculated by formulas (6) and (7) or with the use of eqs. (3) and (8), into which the hardness HM enters. Since in determination of the microhardness and particularly nano hard ness, hM depends on the size of the hardness indent (and, hence, on the load on the indenter, P), δH also must depend on P and the depth of plastic penetration of the indenter h. the size dependence of the hard ness (indentation size effect, abbreviated as ISe) is widely discussed in the literature: see, e.g., refs. [38–41]. For the explanation of the ISe, dislocation notions [39, 40], which lead to a dependence of the type H2 = K (1 + K1/h), where K and K1 are constants, were developed [38]. however, in many cases, the difference in mechanisms of plastic deformation in indentation of different crystalline materials does not make it possible to use theoretical values of K and K1. Moreover, in some cases, the dependence H2 ∝ 1/h has a bilinear character. In ref. [38], it was shown that a large ISe in measurement of the hardness of crystalline materials is caused by the fact that, in the case of using pyramidal indenters, the relation εp + εe ≈ const holds. With decrease in the size of the hardness indent, plastic deformation is impeded because of the hindrance of the work of dislocation sources and Fig. 5. elastic εe, plastic εp, and total εt strains vs. the applied load P in indentation of a copper single crystal (111) [38] Fig. 6. Influence of the load on an indenter P on the nanohardness H and plasticity characteristic δH on a copper single crys tal (111) [38] 284 ISSN 1608-1021. Prog. Phys. Met., 2018, Vol. 19, No. 3 Yu.V. Milman, S.I. Chugunova, I.V. Goncharova, and А.А. Golubenko a decrease in the length of mean free path of dislocations [38]. As a result, with decrease in the size of a hardness indent or in the load on the indenter, P, the fraction of the plastic strain εp diminishes, whereas the fraction of the elastic strain εe increases (Fig. 5). the decrease in εp leads to a decrease in the plasticity cha rac teristic δH accor ding to eq. (1). the increase in εe leads to an increase in the hardness H according to the hooke law for indentation (3). In Figure 6, a decrease in δH and a rise in the hardness H with dec reasing load on an indenter P are illustrated for the case of nanoinden ta tion of copper. In ref. [38], for the calculation of the ISe in crystalline materials, it was proposed to use the empirical Meyer relation in the form: 0 , m h P N h   =     (11) where N and m are constants and h0 is a unit of length in the used system of units. For nanohardness, it is assumed that h0 = 1 nm. Table 2. Values of the modulus of elasticity (Young’s modulus) E S , nanohardness H and constants m (see No. 11), n (see No. 12) and i (see No. 13) [38] No. Material Pmax, mN ES, GPa hmax, nm H, GPa M n i Hf, at hf = = 100 nm Hf at hf = = 1000 nm 1 beo * 10 400 181.5 12.8 1.58 −0.27 −0.42 16.5 6.2 2 tiN * 50 440 394.3 24.6 1.72 −0.16 −0.28 36.2 18.9 3 Si3N4 50 324 415.3 24.3 1.67 −0.20 −0.33 39.0 18.2 4 NbC * 50 550 404.8 25.2 1.82 −0.10 −0.18 32.5 21.4 5 NbC * 50 550 359.3 31.3 1.65 −0.21 −0.35 48.9 21.9 6 zrN * 50 400 400.7 24.3 1.65 −0.21 −0.35 39.7 17.6 7 tib2 ** 50 540 308.2 44.1 1.63 −0.22 −0.37 66.7 28.6 8 WC * 50 700 310.6 39.8 1.59 −0.26 −0.41 63.6 24.5 9 lab6 * 50 439 336.6 38.7 1.53 −0.30 −0.46 68.0 23.3 10 βSiC * 50 460 323.2 44.3 1.70 −0.17 −0.30 62.8 31.6 11 zrC 50 480 386.0 26.4 1.63 −0.22 −0.37 43.3 18.6 12 b4C * 10 500 123.3 48.9 1.64 −0.22 −0.36 52.8 22.8 13 Al2o3 * 10 409 144.9 33.3 1.64 −0.22 −0.36 38.0 16.6 14 Mgo * 50 310 584.0 9.46 1.74 −0.15 −0.26 15.1 8.2 15 W * 10 420 301.3 6.10 1.85 −0.08 −0.15 7.2 5.1 16 Mo * 50 324 931.2 3.21 1.71 −0.17 −0.29 6.1 3.1 17 Cr * 50 279 1025.3 2.63 1.66 −0.20 −0.34 5.7 2.6 18 Nb 50 104 1460.2 1.26 1.84 −0.08 −0.16 1.9 1.3 19 ta 50 185 1259.2 1.74 1.75 −0.14 −0.24 3.2 1.8 20 Cu (111) 62.5 170 2100.8 0.66 1.72 −0.16 −0.28 1.6 0.8 21 Al * 120 70 3148.0 0.66 1.73 −0.16 −0.27 1.7 0.9 here, * — denotes single crystal, — polycrystalline, and * — individual grain. ISSN 1608-1021. Usp. Fiz. Met., 2018, Vol. 19, No. 3 285 Plasticity of Materials Determined by the Indentation Method Using eq. (11), the authors of ref. [38] proposed the following equation: H = N1P n, (12) where n = 1 − 2/m, and H = N2h i, (13) where i = m – 2, N1 and N2 are constants. equations (12) and (13) enable us to recalculate the hardness H1 from the load P1 and the depth of the indent h = h1 to the hardness H2 under the load P2 and at h = h2 according to the expressions: 2 2 1 1 , n P H H P   =     (14) 2 2 1 1 . i h H H h   =     (15) the value of the parameter m can be experimentally determined by the relation P (h). For some materials, values of m, n, and i are presented in table 2 according to ref. [38]. In ref. [38], to prevent the influence of the ISe on the value of the hardness, it is proposed to determine the instrumented hardness at a con stant value h = hf, rather than at a constant value of the load Р. If this is impossible, it is proposed to recalculate the values of H for the fixed value of the depth of plastic penetration, namely, hf = 1000 nm for metals and hf = 100 nm for ceramics, refractory compounds, and other highstrength materials, which are brittle in standard mechanical tests of materials. influence of structural factors on the Plasticity characteristic δ H the complex physical meaning of the elongation to fracture δ in a tensile test made it impossible to develop a theory of dependence of δ on struc tural factors, temperature, and strain rate for many years of using this quantity. the theory of structural sensitivity of δH was developed in ref. [12, 22, 42] with the use of the notion of the structural sensitivity of the yield strength. Since the new plasticity characteristic δH is pro portional to HM = CσS (where С is the tabor parameter [1], and δS is the yield strength), these prob lems are easily solved for δH. In refs. [12, 22], for the case where the dependence of σS on the grain size d is described well by the known hall–Petch equation σH = = σ0 + Kyd –1/2, the following equation was obtained using eq. (5): δH = δ H0 – K1d –1/2, (16) where δ H0 denotes the plasticity of a single crystal, and K1 = = CKy/ ES εt (1 – νS – 2ν2 S). 286 ISSN 1608-1021. Prog. Phys. Met., 2018, Vol. 19, No. 3 Yu.V. Milman, S.I. Chugunova, I.V. Goncharova, and А.А. Golubenko For example, a dependence of δH on the grain size for nano structured copper (based on data of the hardness taken from ref. [43]) and a dependence of δA on the grain size for nanostructured iron (calculated with eq. (6)) [42] are shown in Fig. 7. It is seen from Fig. 7 that eq. (16) is well satisfied if d >> 25 nm for Cu and if d > 200 nm for Fe. For a smaller value of d, the hall–Petch equation is not satisfied due to slippage along the grain boundaries. In ref. [44], eq. (16) was confirmed in a study of the influence of the grain size in a Ni–48.4 at.%Al alloy on the plasticity characteristic δH. If the dependence of the yield strength on the density of chao tically distributed dislocations ρ is described by the reliably established relation 0S G bσ = σ + α ρ , where σ0 is the yield strength of a dislocationfree crystal, G is the modulus of elasticity, b is the modulus of the burgers vector, and α is a constant, then, according to ref. [12]: 1 2 ,H H Kδ = δ − ρ (17) where δ H1 is the plasticity of the crystal at ρ = 0, and 2 2 (1 2 ).S S S t C G b K E α = − ν − ν ε It is seen from the presented relations that the plasticity characteristic δH decreases with increasing dislocation density and decreasing grain size. It follows from eq. (5) that δH decreases also in the case of any other hardening that leads to a rise in НМ. influence of temperature on the Plasticity characteristic δ H An experimental determination of the temperature dependence of the hardness of materials enables one to calculate and analyze the change in the plasticity characteristic δH with temperature [12, 22–24]. Since the hardness measured by pyramidal indenters corresponds to the yield stress at a certain fixed degree of strain, for the description of the temperature dependence of the hardness, it is reasonable to use the Fig. 7. Dependence of the plasticity characteristic δH on the grain size for copper and δA for iron according to ref. [42] ISSN 1608-1021. Usp. Fiz. Met., 2018, Vol. 19, No. 3 287 Plasticity of Materials Determined by the Indentation Method theory that describes the conventional yield strength rather than the critical shear stress or the lower yield point. In refs. [23, 24], the theory of conventional yield strength developed in ref. [45, 46] was used for description the dependence δH (Т). In particular, for the interval of cold deformation, at a temperature T < T * (where Т * is the characteristic temperature of deformation, below which an intensive rise in the yield strength is observed, and the cold brittleness is possible [47]), the following relation was obtained [22]: (0) ln ,H H AT M VE δ = δ + ε (18) where δH (0) is the plasticity at 0 K, 2 2(0)(0) (0) 1 (1 2 ) 1 (1 2 ),S H S S S S S t S t CHM E E σ δ = − − ν − ν = − − ν − ν ε ε (18а) 2(1 2 ),S S t Ck A = − ν − ν ε V is the activation volume, ε⋅ is the strain rate, k is the boltzmann constant, and М is a material constant. temperature dependences of δH for different materials are shown in Figs. 8–11. As is seen in these figures, a linear dependence δH (T) is observed for most materials at low temperatures according to formula (18). the strong dependence of the yield stress, hardness and plasticity characteristic δH on temperature is observed for crystals with a high Peierls–Nabarro stress (covalent crystals, b.c.c. metals, etc.). It is seen Fig. 8. temperature dependence of the plasticity characteristic δH for Cr, Mo and Nb. For Mo and Cr, the values of the characteristic temperature Т * and recrystal lization temperature Тr are marked [23] 288 ISSN 1608-1021. Prog. Phys. Met., 2018, Vol. 19, No. 3 Yu.V. Milman, S.I. Chugunova, I.V. Goncharova, and А.А. Golubenko in Figs. 8–11 that, in contrast to elongation to fracture δ, δH always rises with increasing temperature. In covalent and partially covalent crystals, the linear dependence δH (Т), which follows from eq. (18) and, as is seen in Fig. 11, can be broken in the temperature range adjacent to 0 K due to the semiconductor– metal phase transition under the pressure of an indenter (Si, Ge) or due to the fact that the formation of a hardness indent occurs as a result of brittle fracture rather than through plastic deformation [48]. on the curves of the dependence δH (T), three temperature regions with different character of the dependence δH (T) can be distinguished. these are regions of cold, warm, and hot deformation [52, 53]. on the δH (T) curves, these three regions manifest themselves most clearly for refractory metals with a b.c.c. lattice (Fig. 9). the characteristic tempe rature of deformation Т * is the boundary between the temperature in tervals of cold and warm deformation, and the recrystallization tempe rature Тr is the boundary between the tem perature intervals of warm and hot deformation. Characteristic features of the behaviour of δH (T) in different temperature intervals were discussed in refs. [23, 24, 33]. the behaviour of the plasticity in the interval of cold deformation (T < < T ), studied for the first time, turned to be particularly important because, in this temperature interval, the elon gation to fracture tends to zero (δ → 0) for many materials, whereas δH has welldefined values, which characterize the plasticity of the material. As seen in Figs. 8–11, for all materials, except covalent crystals shown in Fig. 12, in this temperature interval, an abrupt decrease in δH is Fig. 9. temperature dependence of the plasticity characteristic for aluminium and copper [23]. Data for the calculation of δH were taken from ref. [49] for aluminium and from refs. [50, 51] for copper Fig. 10. temperature dependence of the plasticity characteristic δH for refractory compounds WС, NbC, zrC and tiC [23] ISSN 1608-1021. Usp. Fiz. Met., 2018, Vol. 19, No. 3 289 Plasticity of Materials Determined by the Indentation Method Fig. 11. temperature dependence of the plas ticity characteristic δH for Si (the plane (111)), Ge (the plane (111)), and SiC (the plane (0001)) single crystals [23]. the extrapola tion of δH (Т) from the region of the disloca tion mechanism of deformation was per for med to determine temperatures, at which δH = 0 in the absence of a phase transition in Si and Ge in the indentation process. For SiC, the lowtemperature region, on which fracture is a leading mechanism of forma tion of an indent, is not taken into account observed as the temperature decrea ses. Note that, in the tempe ra ture region adjacent to 0 K, the de pen dence δH (T) has a linear character. According to refs. [52, 54], in the interval of cold deformation, a chaotic distribution of dislocations is typical. here, dislocation stoppers are unstable due to a high level of external stresses, and dislocation boundaries practically do not form. the strain hardening is caused by the interaction of mobile dislocations with forest dislocations and is the strongest in this interval. the mobility of dislocation in crystals with a substantial fraction of the covalent component in the interatomic bond is predominantly determined by the thermally activated overcoming of Peierls barriers and diminishes with decreasing temperature. this is why, Fig. 12. Scheme of the temperature dependen ce of the plasticity δH in the regions of cold, warm, and hot deforma tion. the change in the granular and disloca tion structure under de formation is shown sche matically [55] 290 ISSN 1608-1021. Prog. Phys. Met., 2018, Vol. 19, No. 3 Yu.V. Milman, S.I. Chugunova, I.V. Goncharova, and А.А. Golubenko with decrease in the temperature, plastic deformation decreases, the hardness НМ increases, and the plasticity characteristic δH decreases. From equation (18), it follows that, near low temperatures, the characteristic δH must rise linearly with increasing temperature, which is observed in practice. In this case, dδH/dT ∝ A/VE, i.e., Young’s modulus Е and the activation volume V are the most important para meters, which cause the rise in δH with increasing temperature. At higher tem pe ra tures, near the cha racteristic tem perature of defor ma tion Т , σS decreases exponentially as the temperature increases, whereas δH exponentially rises with increasing temperature Т [45, 55]. In the interval of warm deformation, a cellular dislocation structure forms, and the dependence δH (T) is very weak in this interval. For most crystalline singlephase materials, δH attains the value 0.9 and somewhat exceeds it in this temperature interval [55]. In the temperature interval of hot deformation, δH increases, attaining the maximum value δH = 1, but, even for most plastic metals, remains slightly smaller than 1 [55]. theoretical Plasticity of Materials the notion of theoretical plasticity was introduced into the development of the knowledge of theoretical strength [55]. theoretical plasticity is considered as the plasticity of an ideal crystal upon attainment of the theoretical shear strength, and the dislocationfree mechanism of shear deformation is assumed. the introduction of the notion of theoretical plasticity seems to be rational because both characteristics, namely, strength and plasticity, can adequately characterize the mechanical behaviour of materials. the insufficient development of the physical knowledge of plasticity at cryogenic temperatures and the absence of the notion of theoretical plasticity until the publication of ref. [55] are explained by the imperfection of the extensively used plasticity characteristics δ and ψ. In ref. [55], theoretical plasticity was considered with regard for the fact that a solid body deforms purely elastically until the attainment of the theoretical strength σtheor, and then plastic deformation develops without strain hardening. In ref. [56], for the calculation of the theoretical plasticity δHtheor, the following expression was obtained: 2 theor 22.8 (1 2 ) 1 .t S S H S C E τ − ν − ν δ = − (19) here, С is the tabor parameter: C = HM/δS. the value of С can be calculated according to the Johnson theory (see section about relationship between the plasticity characteristic δH and tabor parameter C). Values of the theoretical shear strength τt for different materials were taken from the book [57]. ISSN 1608-1021. Usp. Fiz. Met., 2018, Vol. 19, No. 3 291 Plasticity of Materials Determined by the Indentation Method Values of the theoretical plasticity δHtheor are presented in table 3 in comparison with values of δH at 20 °С and δH at 0 K at the dislocation mechanism of deformation (see section on plasticity at 0 K) if δH is determined with the use of a Vickers indenter. As seen from eq. (19), the theoretical plasticity is determined by three parameters: the value of τt, Young’s modulus ES and Poisson’s ratio νS. In this case, a reduction in τt and an increase in ES and νS lead to a rise in the theoretical plasticity. the results presented in table 3 show that, for all studied crystals δH theor < δH (0), which is natural, because, in the case of the dislocation mechanism of deformation even without the help of thermal oscillations of atoms, the plasticity must be larger than that in the case of the dislocationfree mechanism (see the physical understanding of δH (0) in the section about plasticity at 0 K). It is seen from table 3 that f.c.c. metals have the largest theoreti cal plasticity. however, even for these metals, δHtheor < δHcr (except gold), i.e., in tensile tests, in the case of the dislocationfree deforma tion mechanism, their elongation to fracture δ will be equal to zero at 0 K. the theoretical strength of crystals decreases as the temperature of deformation rises, and, at room temperature, it can be less by a factor of two than the presented estimates [57]. this is why, in ref. [55], it was suggested that f.c.c. crystals presented in table 3 would have plas ticity in tensile tests at room temperature in the case of the disloca tion free mechanism of deformation. For b.c.c. metals (Fe and W), δHtheor is substantially lower than δHcr and, in the case of the dislo cationfree deformation mechanism in tensile tests, δ for them will be equal to zero. All the more, this situation extends to covalent crystals. It is seen from the analysis of expression (19) that an increase in the theoretical strength and τt leads to a decrease in the theoretical plasticity Table 3. Hardness НМ, Young’s modulus Е S , Poisson’s ratio ν S , plasticity characteristic δ H (at temperatures of 20 °С and 0 K), theoretical strength τ t and theoretical plasticity δ Htheor Material HM at 20 °С, GPa ES, GPa νS δH at 20 °С δH at 0 K τt, GPa δHtheor Al 0.16 70 0.35 0.988 0.976 0.90 0.78 Cu 0.90 130 0.343 0.961 0.935 1.20 0.85 Ag 0.03 29.5 0.38 0.995 – 0.77 0.76 Au 0.05 78 0.42 0.998 – 0.74 0.91 zn 0.06 110 0.231 0.995 – 2.30 0.44 W 5.63 420 0.28 0.900 0.850 16.50 0.50 Fe 1.34 200 0.28 0.950 0.880 6.60 0.58 Al2o3 22.04 325 0.23 0.400 0.250 16.90 0.21 292 ISSN 1608-1021. Prog. Phys. Met., 2018, Vol. 19, No. 3 Yu.V. Milman, S.I. Chugunova, I.V. Goncharova, and А.А. Golubenko δHtheor. this is why all formulated conditions of choosing highstrength materials with a high theoretical strength (see, e.g., ref. [57]) should be discussed with regard for a decrease in the theoretical plasticity δHtheor with increase in τt. Plasticity at 0 K, δ H (0) In ref. [23], it was shown that the plasticity characteristic at 0 K δH (0) is also a fundamental characteristic of materials. the δH (0) is determined under the same conditions of plastic deformation as the Peierls–Nabarro stress σP–N, which is required for a dislocation to overcome potential barriers of the crystal lattice without the help of thermal oscillations of atoms, i.e., at 0 K. σP–N is practically the theoretical strength in the case of the dislocation mechanism of deformation, and δH (0) can be considered as the theoretical plasticity in the case of the dislocation mechanism of deformation. As has been noted, for most real materials, the value of δH (0) can be determined by extrapolation of the temperature dependence of δH to 0 K, because the linear character of the dependence δH (Т) [23] was shown theoretically (eq. (18)) and experimentally (Figs. 8–11). It is very important to note that Young’s modulus ЕS, which is the most important parameter that determines the theoretical strength of crystals [57], enters also into expression (18a), and an increase in ЕS leads not only to an increase in the theoretical strength, but also to a rise in the plasticity δH (0). It is seen from table 1 and in Fig. 9 that, in f.c.c. metals (Al and Cu), the value of δH at room temperature exceed substantially the critical value of δHcr, and, as is known, these metals have a high plasticity to fracture δ not only at room temperature, but also at cryogenic tempe ratures, including the temperature of liquid hydrogen and even the temperature of liquid helium [49, 50]. It is interesting that the values of δ in these metals usually even increases as the temperature decreases below room temperature. this can be explained by an increase in the strain hardening with decreasing temperature. the increase in the strain hardening, as has been noted, extends the stage of uniform deformation before formation of a stable ‘neck’, but does not cause the transition to brittle fracture in these metals because the yield strength in them in creases fairly weakly with decreasing temperature and remains lower than the fracture stress. estimates show that, for these metals, δH (0) > δHcr also at 0 K. the plasticity reserve in these metals is so substantial that, even in the case of grain refinement down to 1 µm, δH (0) > δHcr not only at room temperature, but also at 0 K. however, in nanostructured copper, at a grain size d ≈ 0.25 nm, already at room temperature, δH < δHcr (see Fig. 7), and the plasticity to fracture δ is only 1–2% [50]. ISSN 1608-1021. Usp. Fiz. Met., 2018, Vol. 19, No. 3 293 Plasticity of Materials Determined by the Indentation Method In highly pure single crystals of b.c.c. metals, the inequality δH > δHcr also holds at room temperature, i.e., they have some plasticity also in tensile tests. however, with decrease in the temperature, due to the substantial fraction of the covalent component in the interatomic bond and a high Peierls–Nabarro stress, in these metals, the yield strength and hardness increase abruptly [25], whereas the plasticity characteristic δH decreases sharply (Fig. 8), and at 0 K, δH (0) < δHcr, i.e., in these metals, as the temperature decreases below room temperature, the ductile–brittle transition occurs. In commercially pure b.c.c. metals and alloys based on them, the ductile–brittle transition temperature can also be higher than room temperature [25] (particularly, in the group VIA metals: Cr, Mo and W) and, hence, at room temperature, δH < δHcr. In covalent silicon and germanium crystals, already at room tempe rature, δH < δHcr. In these crystals, the sharpest increase in the yield strength is observed as the temperature decreases. however, in a wide temperature range adjacent to 0 K, in these crystals, indentation is accompanied by the semiconductor–metal phase transition, and the hardness has a nearly constant value and does not reflect the yield stress any longer [58–60]. this is why, to evaluate δH in these crystals in the case of the dislocation mechanism of deformation, the extrapola tion of the dependence δH (Т) can be performed only from the more high temperature region, where the mechanism of deformation has a dis location character. Such extrapolation showed that, in these crystals, the value δH = 0 is attained at a certain temperature ТС that is much higher than 0 K (see Fig. 11). It follows from Fig. 11 that ТC is 200 °С for germanium and 400 °C for silicon. In the considered covalent crystals, δH (0) will be equal to 0 in the case of the dislocation mechanism of deformation, but, actually, because of the phase transition, δH (0) is substantially higher. In refractory compounds with a substantial frac tion of the covalent component of the interatomic bond (Al2o3, tiC, zrC, NbC and WC), the inequality δH < δHcr holds already at 20 °С, and a further reduction in δH is observed with decrease in the temperature (Fig. 10), so that δH (0) is very low for most of these crystals. Note that tungsten carbide WC has a higher value of δH than the other considered carbides both at room temperature and at 0 K that can be explained by a large value of Young’s modulus ЕS and a smaller ratio НМ/ЕS. In silicon carbide SiC, the lowtemperature athermal segment on the dependence Н (Т) and δH (T) is caused by a change in the deformation mechanism, namely, at low temperatures, fracture rather than plastic deformation becomes the leading mechanism of formation of an indent in indentation [48, 61]. As is seen from the presented results, the plasticity δH (0) differs substantially for materials with different types of the interatomic bond and different atomic structures: from extremely high values for f.c.c. 294 ISSN 1608-1021. Prog. Phys. Met., 2018, Vol. 19, No. 3 Yu.V. Milman, S.I. Chugunova, I.V. Goncharova, and А.А. Golubenko metals to zero value for covalent crystals. the δH (0) characterizes the plasticity in the case of the dislocation mechanism of deformation, but, in the absence of thermal oscillations of atoms, it can be considered, as has been noted, as a fundamental characteristic of materials [23, 55]. In ref. [23], for the consideration of the dependence of the plasticity at 0 K on the parameters of the thermally activated dislocation motion, melting point, and Young’s modulus, on the basis of eq. (18), Fig. 12, and results obtained in ref. [46], the following relation was obtained: 1/2 1/2 ln( / ) (0) ,m H HW S U T M B VE ε δ ≈ δ −  (20) where U is the activation energy of dislocation motion, Тm is the melting point, δHW is the plasticity in the temperature interval of warm defor mation (see Fig. 12), δHW ≈ const, M is a material constant (see eq. (18) and ref. [46]), and 2 1/2 (1 2 ) . 21 S S t Ck B k − ν − ν = ε expression (20) relates δH (0) to the parameters of thermoactivated dis location motion U and V. From this expression, it follows that an increase in height of potential barriers U and a reduction in their width V leads to a decrease in δH (0). An increase in the melting point Тm also decreases δH (0). A rise in the modulus of elasticity ЕS leads to an increase in δH (0). relationship between the Plasticity characteristic δ H and tabor Parameter c At present, the physical relationship between the hardness HM and the yield stress σS rather than the relationship between the hardness HM and the strength of the material can be thought to be justified [1]. this relationship is usually investigated in the form of the simple relation: HM = C σS, (21) where C is the tabor parameter. For steel and a number of other structural alloys, the tabor parameter lies in the rather narrow interval C = 2.8–3.1. however, for pure f.c.c. metals, C can be much higher, whereas for ceramics, C approaches to unit. the physical meaning of the parameter C was revealed in ref. [31]. In this work, the improved Johnson inclusion core model of indentation was used [62, 63]. the scheme of interaction of the indenter under a load P with the surface of a specimen in the improved inclusion core model is shown in Fig. 13 [31]. As we can see in Fig. 13, in the inclusion core model, a core of de formation with a radius с, in which purely plastic deformation occurs, and a zone of elastoplastic deformation with a radius bS are considered. ISSN 1608-1021. Usp. Fiz. Met., 2018, Vol. 19, No. 3 295 Plasticity of Materials Determined by the Indentation Method In this model, three transcendental equations with three unknowns were obtained. these are the yield stress σS, the relative size of the elastoplastic zone х = bS/c and z ≈ ctgψ, where 2ψ is the angle at the apex of the conic indenter under the load Р. * 3 cot cot 2 / , (1 ) ( ) / , (2/3 2ln ) / 0. i i S S S S S S z HM E x z x HM  = ψ = γ −  − θ σ − α = β σ  + − σ = (22a) (22b) (22c) In these equations, 2γi is the apex angle of the conic indenter without load, * 2 2(1 2 ) 2 (1 2 ) , and , . 3 (1 ) 6 (1 ) 1 S S S i S S S i S S S i E E E E − ν − ν α = β = θ = = − ν − ν − ν the equations of system (22) enable to calculate all three indicated un knowns from the value of the hardness HM and the elastic charac te ris tics ES and νS. In ref. [31], using the system of eqs. (22) and eq. (6) for the deter mination of the plasticity characteristic δH, the authors obtained the equations, which relate the size of the zone of elastoplastic deformation x and tabor parameter C to the plasticity characteristic δH (formulas were obtained under the condition θSσS << 1, which is satisfied well for metals): 3 2.21 (2 3 2ln ) 1 ,H S S z x x + δ = − λ − α (23) where 2 2 1,5 1 1 2 2.21 1 2 and 1 . 1 1 S S S S S H C S S S z C e − − ν − ν ν λ λ = = − δ = − − ν − ν − α (24) In Figs. 14 and 15, experimental data obtained in ref. [31] and theoretical dependences δH = f (x) and δH = f (C) calculated by formulas (23) and (24) for z = 0.38 and νS = 0.27 are presented. For these values Fig. 13. Scheme of inter action of an indenter and a specimen under a load Р in a spherical coordi nate system {0rθψ}, HM = = P/(πc2) [31] 296 ISSN 1608-1021. Prog. Phys. Met., 2018, Vol. 19, No. 3 Yu.V. Milman, S.I. Chugunova, I.V. Goncharova, and А.А. Golubenko of the parameters νS and z, the standard deviations of experimental results from theoretical curves have a minimum value. As we can see from eqs. (23) and (24), along with Figs. 14 and 15, there exists a clear correlation between the plasticity characte ristic δH, relative size of the zone of elastoplastic deformation x, and tabor parameter С. the larger the plasticity δH, the greater the values of x and С. thus, in ref. [31], it was shown that the values of x and С are determined by the value of the plasticity characteristic δH accor ding to universal regulari ties (23) and (24), which hold for all homogeneous materials with different types of crystal lattices and different character of interatomic bonds. the presented results of ref. [31] enabled to explain for the first time the physical nature of the tabor parameter С. the results of ref. [31] show that as the plasticity of a material rises, the size of the zone of elastoplastic deformation х in crea ses, and, therefore, the pres sure on the indenter p = HM must provide plastic deformation in the increasing zone of elas top lastic deformation with a ra dius bS. Fig. 15. relation between the tabor parameter C = HM/σS and the plasticity char acteristic σН. experimental results and theoretical curve based on eq. (24) [31] Fig. 14. relation between the tabor parameter C = HM/σS and the plasticity char acteristic δН. experimental results and theoretical curve based on eq. (24) [31] ISSN 1608-1021. Usp. Fiz. Met., 2018, Vol. 19, No. 3 297 Plasticity of Materials Determined by the Indentation Method In this connection, the excess of the pressure p above the yield strength σS must rise with increasing х, i.e., with increasing plasti city δH . thus, the plasticity characteristic δH acquires universal meaning. the value of the parameter δH enables one to explain not only the plastic characteristics of a material, but also the relationship between the hardness and the yield strength. Since the parameter С is unambiguously related to the plasticity characteristic δH, its value, as is seen from ref. [31], can be used as plasticity characteristic. In this case, the greater parameter C, the higher plasticity of the material. In Figure 15, it is seen that a critical value Сcr = 2.5 (which corresponds to δHcr = 0.9) can be introduced, and, therefore, materials with С < Ccr are brittle in standard mechanical tensile tests. thus, it is possible to formulate clearly the dual nature of the hardness НМ, which was assumed in a number of works: [2, 3, 5], etc. the НМ is proportional to the yield strength σS, which is a strength characteristic of the material. however, the proportionality coefficient, namely, the tabor parameter С = НМ/σS, is determined by the plasticity of the material and can even be used as a plasticity characteristic. Note also that, in practical terms, if the plasticity characteristic δH has been determined by formula (6), then it is possible to determine the tabor parameter C by expression (24) or from Fig. 15, and calculate the yield strength σS by formula (21) [31]. the knowledge presented in ref. [31] makes it possible to raise the efficiency of study of the hardness of materials. the indentation method enable one not only to determine the hardness of the material (which is an important strength characteristic of the material), but also to deter mine easily the plasticity characteristic δН, tabor parameter С (from Fig. 15 or eq. (24)) and yield stress of the material σS. examples of Using the Plasticity characteristic obtained by indentation In refs. [12, 64], it is noted that it is reasonable to perform the cal culation of the characteristic δH in any investigations of the influence of the chemical composition, heat treatment, metal forming and struc tural state on the hardness and mechanical properties of materials. the use of the combination of the strength characteristic Н and plasticity characteristic δH makes it possible to characterize more completely the mechanical behaviour of the material than the use of only the hardness Н. Moreover, as is seen from the section deals with plasticity at 0 K, it turns out that, from the known value of δН, it is easy to calculate the yield stress σS. It is also important that modern techniques of measuring 298 ISSN 1608-1021. Prog. Phys. Met., 2018, Vol. 19, No. 3 Yu.V. Milman, S.I. Chugunova, I.V. Goncharova, and А.А. Golubenko the hardness make it possible to determine the microhardness and plasticity characteristic δН in a wide temperature range, from cryogenic temperatures close to 0 K to a temperature of 1000 °С with the use of a diamond indenter and to 1350 °С with the use of an Al2o3 indenter. the use of the characteristic δH allows one to explain the phenomenal properties of WC–Co hard alloys because, as has been noted, WC is characterized by hardness sufficiently high for tool materials in com bi nation with a plasticity δH very high for refractory compounds. repeated attempts to replace WC by the harder carbides zrC and tiC were not successful because these carbides have a substantially lower plasticity δH. An analysis of δH values (see table 1) allows classifying groups of materials (including new lowplasticity and brittle materials) by plasticity δH. the value of δH decreases in the order as follows: metals–intermetallics– metallic glasses–ceramics–quasicrystals–covalent crystals. As shown in ref. [65], for superplastic materials, the determination of the temperature dependence of δH enables one to determine the optimal temperature of superplastic deformation. It is well known that the deformed (workhardening) metals usually have a smaller plasticity than annealed metals (except for the case where deformation reduces the ductile–brittle transition temperature Тdb). In contrast to metals, for quasicrystals, it was shown (with the use of the plasticity characteristic) that plastic deformation reduces the hardness Н and increases the plasticity characteristic δH, whereas annea ling increases Н and decreases δH [66]. Fig. 16. Dependence of the plasticity of metals and carbides (δH) on their energy of surface tension (eSt) [68] ISSN 1608-1021. Usp. Fiz. Met., 2018, Vol. 19, No. 3 299 Plasticity of Materials Determined by the Indentation Method After publication of ref. [12] and introduction of the plasticity index PI = δA [4, 5], the plasticity characteristics δH or δA = PI were extensively used in works of many researchers. For instance, in ref. [67], the character of plastic flow in two slip sys tems of a MoSi2 single crystal ({110} 〈001〉 and {101} 〈010〉) was studied. the plasticity characteristic δH was determined, and it was shown that δH = 0.72 and 0.80 and that the larger value δH = 0.80 corresponds to the ‘soft’ ori entation {101} 〈010〉, which agrees well with data obtained in a study of the slip systems by the transmission electron microscopy (teM) method. It is interesting to note that, in ref. [68], a correlation dependence of the surface energy of metals and carbides on their plasticity δH was established. With increase in the plasticity of materials δH, an increase in the energy of surface tension is observed (Fig. 16). this regularity is obser ved for metals and refractory compounds. In ref. [30], it was shown that the plasticity characteristic δH correlates with the wear rate of magnetic steels with different sizes of cementite grains (Fig. 17). According to ref. [30], the value of δH correlates with the fracture probability of Al–ti–N intermetallic coatings. the fracture probability decreases with increase in δH. It is known that, in metals with a b.c.c. lattice, a correlation between the electronic structure and plasticity of the materials is observed: a decrease in the covalent component in the interatomic bond leads to an increase in the plasticity [25, 26, 69]. For covalent crystals with the diamond lattice, it was impossible to perform an experimental investigation of the relationship between the electronic structure and plasticity δ because, for all these crystals, the plasticity δ determined in tension is equal to zero. the introduction of the plasticity characteristic δH, determined by the indentation method, made it possible to perform such investigations. In ref. [70], it was shown that the plasticity δH rises for these crystals with increase in the concentration of free electrons (Fig. 18). In ref. [71], it was also shown that the increase in the value of the pseu dopotential W111 for covalent crystals with the diamond lattice correlates with the plasticity characte ristic δH: the greater the va lue of W111, the Fig. 17. Variation in relative wear rate in highperformance stamping of mag netic steels with plasticity index (δH) for different cemented carbide grades [30] 300 ISSN 1608-1021. Prog. Phys. Met., 2018, Vol. 19, No. 3 Yu.V. Milman, S.I. Chugunova, I.V. Goncharova, and А.А. Golubenko less is the va lue of the plas ticity (Fig. 19). these results agree well with the modern know ledge of the influence of the electronic structure on the mobility of dislocations in these crystals [72–74]. the plasticity characteristic δH turned to be sensitive and informative in the investigation of the mechanical properties of microlaminates [75]. It was shown that, for Nb5Si3/Nb microlaminates, depending on the method of their preparation, the values of δH vary in the wide range from 0.373 to 0.824. the authors of refs. [50, 51, 76] used extensively the plasticity characteristic δH in the in vesti gation of the micro me cha nical properties of nano crystalline materials at cryogenic tempe ra tures. For instance, for nano crys tal line titanium, δH = 0.76 and δH = 0.83 at 77 K and 300 K, respectively [50]. the value of δH for nanocrystalline ti tanium is much smaller than that for ti in the poly crys talline state (δH = = 0.973). In investigation of the effect of the microstructure on the plastic deformation of copper at a temperature of 295 and 77 K, the authors of ref. [50] calculated the plasticity characteristic δH and showed that, indepen dently of the grain size and temperature, δH = 0.93–0.96 that testifies to the high plasticity of copper under an indenter at low tempe ratures. these values of δH are typical of metals with an h.c.p. lattice [23]. In ref. [77], the tempe rature dependence of the hard ness and plasticity characte ris tic δH of the Fe–28Al–3Cr interme tal lic in the tempe rature range 300–1273 K was investigated. At 300 K, δH = 0.85, δH rises with increasing temperature, and only at a temperature above 800 K, δH > 0.9, i.e., the material becomes ductile. Fig. 19. Plasticity index (δH) vs. pseudopoten tial, W111, of covalent crystals [71] Fig. 18. Plasticity index (δH) vs. roomtemper ature concentration of free electrons for cova lent crystals [70] ISSN 1608-1021. Usp. Fiz. Met., 2018, Vol. 19, No. 3 301 Plasticity of Materials Determined by the Indentation Method In refs. [78, 79], by the example of a group of ceramic coatings, the efficiency of using the developed technique of determining the plasti city characteristic δH of coatings for application in engineering prac tice was shown. It was shown that the investigated coatings could be arranged in the order of increasing plasticity characteristic δH as follows: carbides of group IVA metals, iron borides, carbides of the group VA and VIA metals, titanium nitride tiN, and silicides of refractory metals. on carbide coatings, the influence of the temperature on the plasticity characteristic δH was studied. In refs. [78, 79], differences in the change of the plasticity characteristic δH for coatings with increasing temperature were associated with different mobility of dis locations in carbides of the group VА and VIА metals. It was shown that, for ceramic coatings with a high hardness HV ≈ 20 GPa, which are used in cutting and upsetting tools, the plasticity characteristic must be at a level δH ≥ 0.45. thus, thin coating can be efficiently used at values of the plasticity characteristic much lesser than the critical value δH = 0.9 for massive materials. this is explained by the small thickness of coatings, at which bending deformation has a purely elastic character that reduces the risk of brittle fracture of coatings. In all works where the values of δH or PI = δA were determined, their values agreed well with the values of δH presented in table 1 and refs. [12, 22, 31]. In the development and application of a wide class of functional materials (thin coatings, films, gradient materials, etc.), the necessity of determining the mechanical properties of these new materials arose. these problems can be successfully solved with the use of the nano in dentation method, which enables one not only to determine the hardness, Young’s modulus, etc., but also to calculate the plasticity characteristic δА (plasticity index) by the technique proposed in refs. [5, 22, 29] (see section covers plasticity characteristic δA determined in the instrumen ted indentation). Note also that the operation and the control of hardnesstesting in struments for instrumented indentation can be performed remotely, which makes it possible to determine the plasticity characteristic, hardness, and yield stress for the purposes of nuclear power enginee ring with the use of simple specimens in the form of metallographic specimens practically without damaging them in the process of measu rement [64]. 302 ISSN 1608-1021. Prog. Phys. Met., 2018, Vol. 19, No. 3 Yu.V. Milman, S.I. Chugunova, I.V. Goncharova, and А.А. Golubenko conclusion the development of the technique of determination of plasticity by the indentation method for two last decades [5, 12, 13, 22–24, 28, 29, 31, 55, 64] extended substantially the possibilities to characterize the mechanical properties of materials by the simple highly efficient indentation method. A new technique of determination of plasticity by the indentation method became possible due to the successful solution of the more general problem, namely, the introduction of the universal dimensionless plasticity characteristic of materials δ* = εp/εt, i.e., the ratio of the plas tic part of strain εр to the total strain εt. this plasticity characteristic corresponds to the physical understanding of the plasticity of materials in the physics of strength. In the present review, the technique of determination of the plasticity δ* by the indentation method, which has been developed in recent years, is considered (in this case, the notation of the new plasticity characteristic by δН was adopted). Undoubtedly, in the future, the development of a technique of determination of δ* with the use of other methods of determination of mechanical properties, in particular, in tensile tests, will be possible. the following developments may be the main results of the intro duction of the plasticity characteristic δН or plasticity index PI = δA. the characterization of all materials (including metallic glasses, quasicrystals, and other materials brittle in standard mechanical tests) by their plasticity is possible in a wide temperature range, including cryogenic and elevated temperatures (up to 1000 °С with the use of a diamond indenter and up to 1350 °С with the use of an Al2o3 indenter). there is the possibility to characterize the plasticity of coatings and thin layers of different materials, including ceramic and other brittle coatings. the theory of the influence of structural factors (grain size and dislocation density), the temperature and strain rate on the plasticity δН has been developed. the notion of theoretical plasticity has been introduced, and a technique of its determination has been developed. In this case, the theoretical plasticity, like the theoretical strength, is considered for the case of dislocationfree deformation of a perfect crystal in the absence of thermal oscillations of atoms, i.e., at 0 K. the notion of theoretical plasticity in the case of the dislocation mechanism of deformation, but without the help of thermal oscillations of atoms has been introduced. this plasticity characteristic has been defined as the plasticity δН at 0 K. the δН (0) is determined under the same conditions as the Peierls–Nabarro stress (stress required for the motion of dislocations through potential barriers of a crystal lattice ISSN 1608-1021. Usp. Fiz. Met., 2018, Vol. 19, No. 3 303 Plasticity of Materials Determined by the Indentation Method without the help of thermal oscillations, that is practically the theoretical strength in the dislocation mechanism of deformation) and, hence, gives additional necessary data on the dislocation mechanism of deformation under the indicated conditions. In choosing the basic material for operation under specific conditions, the possibility to take into account not only its high theoretical strength, but also the necessity of a combination of a high theoretical strength with a sufficient theoretical plasticity appears. the choice of a basic element with a high Peierls–Nabarro stress for highstrength materials is more practiceoriented; in this case, the combination of a high Peierls– Nabarro stress with a sufficient plasticity characteristic δН (0) is necessary. the use of the characteristic δН in development of real high strength alloys and coatings, which must combine a high strength with a sufficient plasticity in a wide temperature range, is even more practice oriented. It has been shown that the plasticity characteristic δA (Plasticity Index) that can be determined in instrumented indentation is approximately equal to δН (δA ≈ δН), if indentation is carried out by identical indenters and under equal loads on an indenter. In this case, for the determination of δA, only areas under loading and unloading curves of an indenter in coordinates ‘load on an indenter Р–displacement of the indenter h’ are used, and the necessity of determining Young’s modulus and Poisson’s ratio νS disappears. It seems reasonable to introduce the definition of the plasticity characteristic in nanoindentation into the standard of nanohardness testing method [80]. It has turned to be possible to understand the dual nature of the hardness НМ, which depends both on the strength characteristic, namely, the yield strength σS, and on the plasticity of the material. the hardness НМ is proportional to the yield strength σS, but the pro portionality coefficient, namely, the tabor parameter С, is determined by the plasticity characteristic δН of the material. For the first time, the possibility to establish the correlation between the plasticity of materials, which fracture in a brittle manner in standard mechanical tests, with their electronic structure and different physical properties has appeared. the introduction of the plasticity characteristic δН made it possible for the first time to determine easily the tabor parameter С for single phase materials and calculate the yield strength σS from the simple expres sion σS = HM/C. As a result, the possibilities and efficiency of the indentation method, which now includes the determination of the hardness, plasticity, and yield strength, have been substantially extended. remotely controlled instrumented hardness with the determination of the hardness Н, plasticity, Young’s modulus Е and yield strength σS can be used in the nuclear power industry and space research. 304 ISSN 1608-1021. Prog. Phys. Met., 2018, Vol. 19, No. 3 Yu.V. Milman, S.I. Chugunova, I.V. Goncharova, and А.А. 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Академіка Кржижановського, 3; 03142 Київ, Україна ПЛАСТИЧНІСТЬ МАТЕРІЯЛІВ, щО ВИЗНАЧАЄТЬСЯ МЕТОДОЮ ІНДЕНТУВАННЯ У даному огляді розглянуто розвиток метод визначення пластичности матеріялів індентуванням. Розроблення способів визначення пластичности матеріялів за допомогою методи індентування засновано на використанні фундаментальних уявлень фізики міцности та пластичности. Істотний розвиток цих способів став можливим після введення нової характеристики пластичности δ* = εp/εt, де εр — пластична деформація, а εt — загальна деформація. Ця характеристика плас тич ности відповідає сучасним фізичним визначенням пластичности на відміну від подовження до руйнування δ, яке широко використовується. Нова характеристика пластичности легко визначається при стандартному визначенні твердости алмазними пірамідальними інденторами за сталого навантаження Р (позначаєть ся δН) і при інструментальному наноіндентуванні (позначається δА); при цьому δH ≈ δA. Істотною перевагою нової характеристики пластичности є можливість визначення її як для металів, так і для крихких при стандартних механічних випробуваннях матеріялів, включаючи кераміку, тонкі шари та покриття. У роз виток уявлень про теоретичну міцність введено уявлення про теоретичну плас тичність при бездислокаційному та дислокаційному механізмах деформації. У ряді робіт встановлено кореляцію δН з електронною будовою матеріялу та його фізичними властивостями. Показано, що параметер Тейбора С (C = HM/σS, де НМ — твердість за Мейєром, σS — межа плинности) легко розраховується за δН. Тому індентування уможливлює нині достатньо просто визначити не тільки твер дість, а й пластичність і межу плинности матеріялів. Таким чином, інден тування стало простою методою визначення комплексу ме ха нічних властивостей матеріялів у широкому температурному інтервалі з використанням зразка у вигляді металографічного шліфа. Ключові слова: твердість, пластичність, індентування, межа плинности, де фор мація. 308 ISSN 1608-1021. Prog. Phys. Met., 2018, Vol. 19, No. 3 Yu.V. Milman, S.I. Chugunova, I.V. Goncharova, and А.А. Golubenko Ю. В. Мильман, С. И. Чугунова, И. В. Гончарова, А. А. Голубенко Институт проблем материаловедения им. И. Н. Францевича НАН Украины; ул. Академика Кржижановского, 3; 03142 Киев, Украина ПЛАСТИЧНОСТЬ МАТЕРИАЛОВ, ОПРЕДЕЛЯЕМАЯ МЕТОДОМ ИНДЕНТИРОВАНИЯ В данном обзоре рассмотрено развитие методик определения пластичности мате риалов индентированием. Разработка способов определения пластичности мате риалов методом индентирования основана на использовании фундаментальных представлений физики прочности и пластичности. Существенное развитие этих способов стало возможным после введения новой характеристики пластичности δ* = εp/εt, где εр — пластическая деформация, а εt — общая деформация. Эта ха рактеристика пластичности соответствует современным физическим определени ям пластичности, в отличие от широко используемого удлинения до разрушения δ. Новая характеристика пластичности легко определяется при стандартном определении твёрдости алмазными пирамидальными инденторами при постоян ной нагрузке P (получила обозначение δН) и при инструментальном наноинден тировании (обозначение δА); при этом δH ≈ δA. Существенным преимуществом новой характеристики пластичности является возможность её определения как для металлов, так и для хрупких при стандартных механических испытаниях материалов, включая керамику, тонкие слои и покрытия. В развитие представ лений о теоретической прочности введены представления о теоретической пла стичности при бездислокационном и дислокационном механизмах деформации. В ряде работ установлена корреляция δН с электронным строением материала и его физическими свойствами. Показано, что параметр Тейбора C (C = HM/σS, где НМ — твёрдость по Мейеру, а σS — предел текучести) легко рассчитывается по δH. Поэтому индентирование позволяет в настоящее время достаточно просто определить не только твёрдость, но и пластичность, и предел текучести материа лов. Таким образом, индентирование стало простым методом определения ком плекса механических свойств материалов в широком температурном интервале с использованием образца в виде металлографического шлифа. Ключевые слова: твёрдость, пластичность, индентирование, предел текучести, деформация.

Plasticity of materials determined by the indentation method

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