- Email: [email protected]
Modeling of the plasticity of microstructured and nanostructured materials
Available online at www.sciencedirect.com
St. Petersburg Polytechnical University Journal: Physics and Mathematics 1 (2015) 405–409 www.elsevier.com/locate/spjpm
Modeling of the plasticity of microstructured and nanostructured materials✩ Natalia R. Kudinova a,∗, Vladimir A. Polyanskiy a, Anatoly M. Polyanskiy a, Yuri A. Yakovlev b a Institute b Research
of Problems of Mechanical Engineering RAS, 61 Bolshoi Pr. V.O., St. Petersburg 199178, Russian Federation & Development Company Electronic & Beam Technologies, 6 Bronevaya, St. Petersburg 198188, Russian Federation Available online 9 February 2016
Abstract A new approach to modeling of the plasticity of materials with nanostructure and ultrafine one has been proposed. Its main advantage is the minimum number of physical parameters in use. In the context of the proposed model, we calculated the volumetric density of the energy of surface tension of the material grains. This energy is a significant part of the internal energy during deformation. The size dependence of the melting temperature of nanoparticles was compared with experimental data. We obtained size dependence of the yield point on its basis. Yield point was interpreted as the result of changes of grains surface energy during the deformation. The obtained yield point dependence on the grain size was related to the Hall–Petch law, and this resulted in confirmation of the hypothesis on the crucial role of surface tension forces in the initial stage of plastic deformation of ultrafine materials. Copyright © 2016, St. Petersburg Polytechnic University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Ultrafine structure; Nanostructure; Yield point; Melting temperature.
1. Introduction Materials with nano- and ultrafine structures have extreme mechanical characteristics, which presents a challenge in choosing the appropriate materials when designing and computing machines and constructions. The internal structure of a material can be modified during the element construction process. For example, ✩ Peer review under responsibility of St. Petersburg Polytechnic University. ∗ Corresponding author. E-mail addresses: [email protected] (N.R. Kudinova), [email protected] (V.A. Polyanskiy), [email protected] (A.M. Polyanskiy), [email protected] (Y.A. Yakovlev).
an element can be rendered the required properties in its various parts through pressing. These opportunities necessitate having to predict the mechanical characteristics of a material at the machine design stage. This prediction is even more important for designing new materials. In order to make such a prediction, a model for modifiying the mechanical properties of a material depending on the size and the shape of structural elements must be developed. The most important mechanical parameter of a construction material is its yield point. On the one hand, the linear character of the stress–strain curve in the absence of fluidity allows to carry out fast and precise strength calculations; on the other hand, the plastic flow of the material allows
http://dx.doi.org/10.1016/j.spjpm.2016.02.001 2405-7223/Copyright © 2016, St. Petersburg Polytechnic University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). (Peer review under responsibility of St. Petersburg Polytechnic University).
406
N.R. Kudinova et al. / St. Petersburg Polytechnical University Journal: Physics and Mathematics 1 (2015) 405–409
to smooth out the stress field singularities. An additional strength margin emerges due to the hardening phenomenon occurring in plastic zones. Plastic flow is the main type of strain that occurs under pressing and stamping. In this case, knowing the mechanical characteristics of the material, such as the yield point, is necessary for calculating the parameters of the technological process. Plastic straining is accompanied by inelastic energy losses. The yield point value indicates the elastic energy required for starting the plastic strain process in the material. Even though the plastic flow mode closely resembles the liquid-phase material straining, the specific energies required for heating the material up to the melting point and to complete melting down have significantly higher (about ten times) values than those of the elastic strain and the maximum possible (until material failure) plastic strain energies. Due to this difference, it is generally considered that the plastic strain mechanism is connected either to metal grain sliding [1,2], or to the generation and the motion of the dislocations leading to microcrack growth [3–6]. Petch attributed the experimental dependence of the yield point on the mean grain size of the material to microcrack growth [7]. The relation of this type, i.e., the Hall–Petch relationship, is violated only for grain sizes less than 1 μm. In a way, it can be regarded as a method for testing various plasticity theories taking into account the grain size [1–3]. At the same time, microcrack formation does not always occur when the yield point is reached, especially in materials with an ultrafine structure. According to the dislocation plasticity theory [8], all mechanical characteristics strongly depend on the dislocation density on the surface of the grains. This quantity cannot be measured directly, so the predictive value of the dislocation approach is not particularly high. Mesomechanics [9] describes the plasticity as the result of combined dislocation effects, i.e., from a modeling standpoint, it includes the dislocation theory with all of its advantages and disadvantages. Therefore, the known plasticity mechanisms do not allow constructing a predictive calculation model for a material that would make it possible to predict the specific yield point values. Aside from yield point change, structural element refinement can also affect the melting temperature. The latter is determined experimentally for micro- and nanodisperse materials, but not as definitively as the yield point. For example, aggregate and powder metallic nanomaterials may have different melting temper-
atures, whereas metallic nanopowders of pure metals in an inert-gas blanket are characterized by a decrease in the melting point with a decrease in nanoparticle sizes from 20 to 1–2 nm [10]. The change in the melting point is, apparently, connected to a reduction of the surface tension coefficient, since the probability of isolated atoms and molecules dislodging from the crystallite increases due to thermal motion. Additionally, the experimental dependence of the melting temperature on the surface tension coefficient is known to be linear for various substances [11]. This experimental behavior indicates that interfacial interaction in crystallites does not have a strong effect on the surface tension forces and the melting process for an average crystallite size of about 100 μm. Consequently, the surface energy of crystallites can be regarded as a separate component of the internal energy of the material. The influence of surface tension forces in crystallites on the yield point of the material has not yet been investigated. However, the very experimental procedure for measuring the surface tension forces based on the zero-creep mechanism [12] proves that such an influence exists. Therefore, if the dependence of the surface energy on the crystallite size is known, it is possible to predict the yield point variations while the structure of the metal is being refined. This is a prerequisite for constructing a model that would allow predicting yield point values from the bulk density of the surface tension energy.
2. An estimate for the relationship between the surface tension coefficient of the particles and their size The Gibbs–Tolman–König–Buff equation [13,14] for a spherical particle has the following form: δ 1 δ2 1 + + R 3 R2 ∂ ln σ , = 2 2δ δ ∂ ln R 1 + R 1 + R + 13 Rδ 2 2δ R
(1)
where σ is the surface tension coefficient; R is the radius of a particle; δ is the Tolman constant equal (in order of magnitude) to the surface layer thickness of a particle. In the general case, this equation is unsolvable in the explicit form, since the constant δ depends on the radius. To solve it, the condition R δ is accepted, which allows to eliminate the summands of the order O( 2δ ). Integrating the Eq. (1) in this case leads to the R
N.R. Kudinova et al. / St. Petersburg Polytechnical University Journal: Physics and Mathematics 1 (2015) 405–409
Fig. 1. Plots of the surface tension coefficient normalized to the constant σ0 versus the particle radius normalized by the constant δ. Curve 1 is the exact solution (4), curves 2 and 3 are the approximations (2) and (3), respectively.
following well-known formula [15]: σ =
σ0 , 1 + 2δ R
(2)
where σ0 is the surface tension coefficient for a flat surface. Taking into account the condition R δ, it is possible to obtain another asymptotic expression for the surface tension coefficient [15]: 2δ (3) σ = σ0 exp − R The solutions (2) and (3) are typically regarded as the closest to the exact one. It is these solutions that are commonly used for comparisons to the experimental data. However, there is a general solution of Eq. (1), assuming that δ does not depend on R. It can be represented in an analytical form [12]: 3 xk2 ln ( R − xk ) σ0 R δ exp − σ = (4) δ 3xk2 + 4xk + 2 k=1 where x1 = −0.558, x2,3 = −0.721 ± 0.822i are the roots of a cubic equation 3x 3 + 6x 2 + 6x + 2 = 0. Fig. 1 shows the comparison of the exact solution (4) to the most frequently used approximations (2) and (3).
407
Fig. 2. The experimental (points) and the calculated (lines 1–3) dependences of the gold melting point versus the average diameter d of its particles. The calculated values were obtained from the approximate Eq. (6) and (7) (curves 2 and 3, respectively) and the exact solution (5) of the Eq. (1) (curve 1).
Using the known linear relationship between the melting temperature of the material and the surface tension coefficient of the particles, and the solutions (2)–(4), we constructed a model dependence of the melting temperature on the particle size. We obtained the following expressions for the temperatures: 3 xk2 ln ( R − xk ) T0 R δ exp − Tl = , (5) δ 3xk2 + 4xk + 2 k=1 T0
, Tl = 1 + 2δ R
(6)
2δ . Tl = T0 exp − R
(7)
Fig. 2 shows these dependences for pure gold (experimental points and calculated curves). Comparing the plots shows that the exact solution (5) is good for describing the experimental data in the particle size range from 3 to 20 nm. The approximate dependences, obtained by the Formulae (6) and (7), produce a substantial error only in the range from 1 to 5 nm. Therefore, the exact solution (5) allows to satisfactorily describe the changes in the melting point. This result makes it possible to apply the obtained dependence of the surface tension coefficient on the particle sizes to our further calculations with complete confidence.
408
N.R. Kudinova et al. / St. Petersburg Polytechnical University Journal: Physics and Mathematics 1 (2015) 405–409
3. The connection between the yield point of a material and the surface tension energy of its particles As follows from Fig. 1, the surface tension coefficient rapidly decreases with a decrease in the particle size R/δ of the material, for particles with sizes less than 20δ (about 20 nm). Its values for most metals with larger-sized structural elements are about σ 0 =1– 2 J/m2 . The elastic energy stored in a metal through uniaxial tension up to the yield point is about 90 kJ/m3 for steels. Let us compare this energy with the bulk density of the surface tension energy ρ σ . Let us assume, to simplify the estimates, that the particles have a spherical form, then the energy is going to equal ρ σ =3σ 0 /R. Already for a typical crystallite size of about 2R = 120 – 200 μm, the ρ σ value becomes equal to the maximum elastic strain energy. Consequently, the surface energy needs to be taken into account when constructing plastic deformation models for ultrafine materials. We can speculate that it is this quantity that determines the plastic deformation mechanisms in this case. To test this hypothesis, let us use the Hall–Petch law [16,7] which was discovered empirically. To explain this law, Petch suggested examining the formation and the growth of a microcrack in a grain with a characteristic dimension d (this diameter d = 2R for our assumptions). Currently, the law itself, and the deviations from it for nanometer-sized crystallites are interpreted solely on the basis of the dislocation mechanism of plastic deformation. According to the Hall–Petch law, the yield point τ depends on the grain dimension d = 2R of a polycrystalline material and is expressed as τ = τ0 + K (2R )− 2 , 1
where τ 0 is some friction stress required for the dislocation glide; K is the material constant, often referred to as the Hall–Petch coefficient (both values are assessed from the experimental data). Let us consider the beginning of the plastic straining process. For most metals, the strain grows essentially without any additional increase in stress. In the simplest case of uniaxial tension, this stress can be taken as constant. Therefore, the plastic strain energy should be proportional to the yield point. Let us assume that plastic strains lead to a growth of the internal crystallite surface without substantially changing their volume; then, when the plastic flow starts, the mechanical stresses should compensate both the elastic strains of the crystallites and the surface ten-
Fig. 3. The normalized bulk density of the surface energy ρσ δ/σ0 versus the (R/δ)−1/2 value, and the linear approximations of the curve parts by the Hall–Petch dependence. Eq. (8) was used.
sion forces. It follows, then, that the yield point is linearly related to the bulk density of the surface energy. The behavior of this relationship makes it possible to approximate the Hall–Petch law using the dependence of the specific surface energy of the material on the particle size. With the exact solution of the Gibbs–Tolman–König–Buff Eq. (4), we can obtain an expression for the bulk density of the surface tension energy in a material containing spherical particles. The obtained relationship is expressed in the following way: 3 xk2 ln ( R − xk ) 3σ0 δ exp − ρσ = (8) δ 3xk2 + 4xk + 2 k=1 The normalized bulk density of the surface energy ρσ δ/σ0 is shown in Fig. 3 versus the (R/δ)−1/2 value. The curve has several quasilinear parts. The figure shows linear approximations of the curve parts corresponding to the Hall–Petch dependence. We should note that for grain sizes (2R) ranging from 4δ to 2δ, there is a strong change in the approximating dependence parameters. The obtained dependence can be approximated using the Hall–Petch law; however, when the grain size is less than 2δ, the bulk density of the surface energy reaches its limit value (about 0.913 σ0 /δ). The plots shown in Fig. 3 give reason for interpreting the plasticity as a variation in the surface energy. In other words, if the material started flowing, it means that the allowed value of the surface energy has been exceeded. The obtained dependences prove that surface energy plays a key role in the transition from the elastic to the plastic strain. The proposed analysis of such
N.R. Kudinova et al. / St. Petersburg Polytechnical University Journal: Physics and Mathematics 1 (2015) 405–409
dependences allows predicting the yield point values from the known specific surface energies, without involving any dislocation models. The estimates we have presented show that the exact solutions of the Gibbs–Tolman–König–Buff equation allow to calculate the melting point with an accuracy that is acceptable for practical use. The curve of the normalized bulk density of the surface tension versus the particle diameter is in agreement with the Hall–Petch law in the d > 4δ range and differs from it for lower diameter values, which is consistent with the experimental data. In the second case the exponent at the particle size 2R (that was equal to –1/2) changes in the Hall–Petch law. For many metals this size is just (4–8) δ.
4. Conclusions Thus, the surface tension energy plays a crucial role in the plastic straining of ultrafine materials. This mechanism allows constructing new calculation models for plastic straining without having to use dislocation theory. The exact solution of the Gibbs–Tolman–König– Buff equation makes it possible to describe the relationship between the melting point and the particle sizes ranging from 3 to 20 nm more adequately, which goes to further prove that surface tension forces play the decisive role in the initial phase of plastic straining of ultrafine materials. The available experimental data on the plastic-flow mesoprocesses depending on the curvature radius of nanostructural elements [17] confirm this point of view.
Acknowledgment The study was supported by the Russian Foundation for Basic Research grants no. 12-08-00386 and no. 1408-00646.
409
References [1] U.F. Kocks, The relation between polycrystal deformation and single crystal deformation, Metal. Trans. 1 (5) (1970) 1121– 1143. [2] H.S. Kim, A composite model for mechanical properties of nanocrystalline materials, Scripta Mater. 39 (8) (1998) 1057– 1061. [3] M.Yu. Gutkin, I.A. Ovidko, Physical Mechanics of Deformed Nanostructures: Nanocrystalline Materials, Vol. 1, Yanus, St. Petersburg, 2003. [4] H.K. Birnbaum, P. Sofronis, Hydrogen-enhanced localized plasticity – a mechanism for hydrogen-related fracture, Mater. Sci. Eng 176 (1–2) (1994) 191–202. [5] P. Sofronis, Y. Liang, N. Aravas, Hydrogen induced shear localization of the plastic flow in metals and alloys, Europ. J. Mech. A. Solids. 20 (6) (2001) 857–872. [6] D. Delafosse, T. Magnin, Interfaces in stress corrosion cracking: a case study in duplex stainless steels, Solid State Phenomena 59-60 (1998) 221–250. [7] N.J. Petch, The cleavage strength of polycrystals, J. Iron Steel Inst. 174 (1953) 25–28. [8] I.A. Ovid’ko, N.V. Skiba, Enhanced dislocation emission from grain boundaries in nanocrystalline materials, Scripta Materialia 67 (1) (2012) 13–16. [9] V.E. Panin, Overview on mesomechanics of plastic deformation and fracture of solids, Theor. Appl. Fracture Mech 30 (1) (1998) 1–11. [10] Ph. Buffat, J.-P. Borel, Size effect on the melting temperature of gold particles, Phys. Rev. A 13 (6) (1976) 2287–2298. [11] H. Udin, A.J. Shaler, J. Wulff, The surface tension of solid copper, Trans. AIME. 185 (1949) 186–190. [12] S.Sh. Rekhviashvili, E.V. Kishtikova, On the temperature of melting of nanoparticles and nanocrystalline substances, Tech. Phys. Lett. 32 (5) (2006) 439–441. [13] P.F. Buff, Spherical interface. ii. molecular theory, J. Chem. Phys. 23 (3) (1955) 419–427. [14] F.O. Koenig, On the thermodynamic relation between surface tension and curvature, J. Chem. Phys. 18 (4) (1950) 449–459. [15] V.M. Yurov, V.S. Portnov, N.Kh. Ibrayev, S.A. Guchenko, Poverhnostnoe natyazhenie tverdyh tel, malyh chastic i plenok [The surface tension of solid bodies, small particles and thin membrans], Uspehi sovremennogo estestvoznania 11 (2011) 55–58. [16] E.O. Hall, Deformation and ageing of mild steel, Proc. Phys. Soc. Sec. B. 64 (6) (1951) 495–502. [17] E. V., A.V. Panin, T.F. Panin, Yu.F. Elsukova, Popkova, Fundamentalnaya rol’ krivizny kristallicheskoy struktury v plastichnosti i prochnosti tverdyh tel [Fundamental role of curvature of the crystal structure in plasticity and strength of solids], Fizicheskaya mezomehanika 17 (6) (2014) 7–8.
St. Petersburg Polytechnical University Journal: Physics and Mathematics 1 (2015) 405–409 www.elsevier.com/locate/spjpm
Modeling of the plasticity of microstructured and nanostructured materials✩ Natalia R. Kudinova a,∗, Vladimir A. Polyanskiy a, Anatoly M. Polyanskiy a, Yuri A. Yakovlev b a Institute b Research
of Problems of Mechanical Engineering RAS, 61 Bolshoi Pr. V.O., St. Petersburg 199178, Russian Federation & Development Company Electronic & Beam Technologies, 6 Bronevaya, St. Petersburg 198188, Russian Federation Available online 9 February 2016
Abstract A new approach to modeling of the plasticity of materials with nanostructure and ultrafine one has been proposed. Its main advantage is the minimum number of physical parameters in use. In the context of the proposed model, we calculated the volumetric density of the energy of surface tension of the material grains. This energy is a significant part of the internal energy during deformation. The size dependence of the melting temperature of nanoparticles was compared with experimental data. We obtained size dependence of the yield point on its basis. Yield point was interpreted as the result of changes of grains surface energy during the deformation. The obtained yield point dependence on the grain size was related to the Hall–Petch law, and this resulted in confirmation of the hypothesis on the crucial role of surface tension forces in the initial stage of plastic deformation of ultrafine materials. Copyright © 2016, St. Petersburg Polytechnic University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Ultrafine structure; Nanostructure; Yield point; Melting temperature.
1. Introduction Materials with nano- and ultrafine structures have extreme mechanical characteristics, which presents a challenge in choosing the appropriate materials when designing and computing machines and constructions. The internal structure of a material can be modified during the element construction process. For example, ✩ Peer review under responsibility of St. Petersburg Polytechnic University. ∗ Corresponding author. E-mail addresses: [email protected] (N.R. Kudinova), [email protected] (V.A. Polyanskiy), [email protected] (A.M. Polyanskiy), [email protected] (Y.A. Yakovlev).
an element can be rendered the required properties in its various parts through pressing. These opportunities necessitate having to predict the mechanical characteristics of a material at the machine design stage. This prediction is even more important for designing new materials. In order to make such a prediction, a model for modifiying the mechanical properties of a material depending on the size and the shape of structural elements must be developed. The most important mechanical parameter of a construction material is its yield point. On the one hand, the linear character of the stress–strain curve in the absence of fluidity allows to carry out fast and precise strength calculations; on the other hand, the plastic flow of the material allows
http://dx.doi.org/10.1016/j.spjpm.2016.02.001 2405-7223/Copyright © 2016, St. Petersburg Polytechnic University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). (Peer review under responsibility of St. Petersburg Polytechnic University).
406
N.R. Kudinova et al. / St. Petersburg Polytechnical University Journal: Physics and Mathematics 1 (2015) 405–409
to smooth out the stress field singularities. An additional strength margin emerges due to the hardening phenomenon occurring in plastic zones. Plastic flow is the main type of strain that occurs under pressing and stamping. In this case, knowing the mechanical characteristics of the material, such as the yield point, is necessary for calculating the parameters of the technological process. Plastic straining is accompanied by inelastic energy losses. The yield point value indicates the elastic energy required for starting the plastic strain process in the material. Even though the plastic flow mode closely resembles the liquid-phase material straining, the specific energies required for heating the material up to the melting point and to complete melting down have significantly higher (about ten times) values than those of the elastic strain and the maximum possible (until material failure) plastic strain energies. Due to this difference, it is generally considered that the plastic strain mechanism is connected either to metal grain sliding [1,2], or to the generation and the motion of the dislocations leading to microcrack growth [3–6]. Petch attributed the experimental dependence of the yield point on the mean grain size of the material to microcrack growth [7]. The relation of this type, i.e., the Hall–Petch relationship, is violated only for grain sizes less than 1 μm. In a way, it can be regarded as a method for testing various plasticity theories taking into account the grain size [1–3]. At the same time, microcrack formation does not always occur when the yield point is reached, especially in materials with an ultrafine structure. According to the dislocation plasticity theory [8], all mechanical characteristics strongly depend on the dislocation density on the surface of the grains. This quantity cannot be measured directly, so the predictive value of the dislocation approach is not particularly high. Mesomechanics [9] describes the plasticity as the result of combined dislocation effects, i.e., from a modeling standpoint, it includes the dislocation theory with all of its advantages and disadvantages. Therefore, the known plasticity mechanisms do not allow constructing a predictive calculation model for a material that would make it possible to predict the specific yield point values. Aside from yield point change, structural element refinement can also affect the melting temperature. The latter is determined experimentally for micro- and nanodisperse materials, but not as definitively as the yield point. For example, aggregate and powder metallic nanomaterials may have different melting temper-
atures, whereas metallic nanopowders of pure metals in an inert-gas blanket are characterized by a decrease in the melting point with a decrease in nanoparticle sizes from 20 to 1–2 nm [10]. The change in the melting point is, apparently, connected to a reduction of the surface tension coefficient, since the probability of isolated atoms and molecules dislodging from the crystallite increases due to thermal motion. Additionally, the experimental dependence of the melting temperature on the surface tension coefficient is known to be linear for various substances [11]. This experimental behavior indicates that interfacial interaction in crystallites does not have a strong effect on the surface tension forces and the melting process for an average crystallite size of about 100 μm. Consequently, the surface energy of crystallites can be regarded as a separate component of the internal energy of the material. The influence of surface tension forces in crystallites on the yield point of the material has not yet been investigated. However, the very experimental procedure for measuring the surface tension forces based on the zero-creep mechanism [12] proves that such an influence exists. Therefore, if the dependence of the surface energy on the crystallite size is known, it is possible to predict the yield point variations while the structure of the metal is being refined. This is a prerequisite for constructing a model that would allow predicting yield point values from the bulk density of the surface tension energy.
2. An estimate for the relationship between the surface tension coefficient of the particles and their size The Gibbs–Tolman–König–Buff equation [13,14] for a spherical particle has the following form: δ 1 δ2 1 + + R 3 R2 ∂ ln σ , = 2 2δ δ ∂ ln R 1 + R 1 + R + 13 Rδ 2 2δ R
(1)
where σ is the surface tension coefficient; R is the radius of a particle; δ is the Tolman constant equal (in order of magnitude) to the surface layer thickness of a particle. In the general case, this equation is unsolvable in the explicit form, since the constant δ depends on the radius. To solve it, the condition R δ is accepted, which allows to eliminate the summands of the order O( 2δ ). Integrating the Eq. (1) in this case leads to the R
N.R. Kudinova et al. / St. Petersburg Polytechnical University Journal: Physics and Mathematics 1 (2015) 405–409
Fig. 1. Plots of the surface tension coefficient normalized to the constant σ0 versus the particle radius normalized by the constant δ. Curve 1 is the exact solution (4), curves 2 and 3 are the approximations (2) and (3), respectively.
following well-known formula [15]: σ =
σ0 , 1 + 2δ R
(2)
where σ0 is the surface tension coefficient for a flat surface. Taking into account the condition R δ, it is possible to obtain another asymptotic expression for the surface tension coefficient [15]: 2δ (3) σ = σ0 exp − R The solutions (2) and (3) are typically regarded as the closest to the exact one. It is these solutions that are commonly used for comparisons to the experimental data. However, there is a general solution of Eq. (1), assuming that δ does not depend on R. It can be represented in an analytical form [12]: 3 xk2 ln ( R − xk ) σ0 R δ exp − σ = (4) δ 3xk2 + 4xk + 2 k=1 where x1 = −0.558, x2,3 = −0.721 ± 0.822i are the roots of a cubic equation 3x 3 + 6x 2 + 6x + 2 = 0. Fig. 1 shows the comparison of the exact solution (4) to the most frequently used approximations (2) and (3).
407
Fig. 2. The experimental (points) and the calculated (lines 1–3) dependences of the gold melting point versus the average diameter d of its particles. The calculated values were obtained from the approximate Eq. (6) and (7) (curves 2 and 3, respectively) and the exact solution (5) of the Eq. (1) (curve 1).
Using the known linear relationship between the melting temperature of the material and the surface tension coefficient of the particles, and the solutions (2)–(4), we constructed a model dependence of the melting temperature on the particle size. We obtained the following expressions for the temperatures: 3 xk2 ln ( R − xk ) T0 R δ exp − Tl = , (5) δ 3xk2 + 4xk + 2 k=1 T0
, Tl = 1 + 2δ R
(6)
2δ . Tl = T0 exp − R
(7)
Fig. 2 shows these dependences for pure gold (experimental points and calculated curves). Comparing the plots shows that the exact solution (5) is good for describing the experimental data in the particle size range from 3 to 20 nm. The approximate dependences, obtained by the Formulae (6) and (7), produce a substantial error only in the range from 1 to 5 nm. Therefore, the exact solution (5) allows to satisfactorily describe the changes in the melting point. This result makes it possible to apply the obtained dependence of the surface tension coefficient on the particle sizes to our further calculations with complete confidence.
408
N.R. Kudinova et al. / St. Petersburg Polytechnical University Journal: Physics and Mathematics 1 (2015) 405–409
3. The connection between the yield point of a material and the surface tension energy of its particles As follows from Fig. 1, the surface tension coefficient rapidly decreases with a decrease in the particle size R/δ of the material, for particles with sizes less than 20δ (about 20 nm). Its values for most metals with larger-sized structural elements are about σ 0 =1– 2 J/m2 . The elastic energy stored in a metal through uniaxial tension up to the yield point is about 90 kJ/m3 for steels. Let us compare this energy with the bulk density of the surface tension energy ρ σ . Let us assume, to simplify the estimates, that the particles have a spherical form, then the energy is going to equal ρ σ =3σ 0 /R. Already for a typical crystallite size of about 2R = 120 – 200 μm, the ρ σ value becomes equal to the maximum elastic strain energy. Consequently, the surface energy needs to be taken into account when constructing plastic deformation models for ultrafine materials. We can speculate that it is this quantity that determines the plastic deformation mechanisms in this case. To test this hypothesis, let us use the Hall–Petch law [16,7] which was discovered empirically. To explain this law, Petch suggested examining the formation and the growth of a microcrack in a grain with a characteristic dimension d (this diameter d = 2R for our assumptions). Currently, the law itself, and the deviations from it for nanometer-sized crystallites are interpreted solely on the basis of the dislocation mechanism of plastic deformation. According to the Hall–Petch law, the yield point τ depends on the grain dimension d = 2R of a polycrystalline material and is expressed as τ = τ0 + K (2R )− 2 , 1
where τ 0 is some friction stress required for the dislocation glide; K is the material constant, often referred to as the Hall–Petch coefficient (both values are assessed from the experimental data). Let us consider the beginning of the plastic straining process. For most metals, the strain grows essentially without any additional increase in stress. In the simplest case of uniaxial tension, this stress can be taken as constant. Therefore, the plastic strain energy should be proportional to the yield point. Let us assume that plastic strains lead to a growth of the internal crystallite surface without substantially changing their volume; then, when the plastic flow starts, the mechanical stresses should compensate both the elastic strains of the crystallites and the surface ten-
Fig. 3. The normalized bulk density of the surface energy ρσ δ/σ0 versus the (R/δ)−1/2 value, and the linear approximations of the curve parts by the Hall–Petch dependence. Eq. (8) was used.
sion forces. It follows, then, that the yield point is linearly related to the bulk density of the surface energy. The behavior of this relationship makes it possible to approximate the Hall–Petch law using the dependence of the specific surface energy of the material on the particle size. With the exact solution of the Gibbs–Tolman–König–Buff Eq. (4), we can obtain an expression for the bulk density of the surface tension energy in a material containing spherical particles. The obtained relationship is expressed in the following way: 3 xk2 ln ( R − xk ) 3σ0 δ exp − ρσ = (8) δ 3xk2 + 4xk + 2 k=1 The normalized bulk density of the surface energy ρσ δ/σ0 is shown in Fig. 3 versus the (R/δ)−1/2 value. The curve has several quasilinear parts. The figure shows linear approximations of the curve parts corresponding to the Hall–Petch dependence. We should note that for grain sizes (2R) ranging from 4δ to 2δ, there is a strong change in the approximating dependence parameters. The obtained dependence can be approximated using the Hall–Petch law; however, when the grain size is less than 2δ, the bulk density of the surface energy reaches its limit value (about 0.913 σ0 /δ). The plots shown in Fig. 3 give reason for interpreting the plasticity as a variation in the surface energy. In other words, if the material started flowing, it means that the allowed value of the surface energy has been exceeded. The obtained dependences prove that surface energy plays a key role in the transition from the elastic to the plastic strain. The proposed analysis of such
N.R. Kudinova et al. / St. Petersburg Polytechnical University Journal: Physics and Mathematics 1 (2015) 405–409
dependences allows predicting the yield point values from the known specific surface energies, without involving any dislocation models. The estimates we have presented show that the exact solutions of the Gibbs–Tolman–König–Buff equation allow to calculate the melting point with an accuracy that is acceptable for practical use. The curve of the normalized bulk density of the surface tension versus the particle diameter is in agreement with the Hall–Petch law in the d > 4δ range and differs from it for lower diameter values, which is consistent with the experimental data. In the second case the exponent at the particle size 2R (that was equal to –1/2) changes in the Hall–Petch law. For many metals this size is just (4–8) δ.
4. Conclusions Thus, the surface tension energy plays a crucial role in the plastic straining of ultrafine materials. This mechanism allows constructing new calculation models for plastic straining without having to use dislocation theory. The exact solution of the Gibbs–Tolman–König– Buff equation makes it possible to describe the relationship between the melting point and the particle sizes ranging from 3 to 20 nm more adequately, which goes to further prove that surface tension forces play the decisive role in the initial phase of plastic straining of ultrafine materials. The available experimental data on the plastic-flow mesoprocesses depending on the curvature radius of nanostructural elements [17] confirm this point of view.
Acknowledgment The study was supported by the Russian Foundation for Basic Research grants no. 12-08-00386 and no. 1408-00646.
409
References [1] U.F. Kocks, The relation between polycrystal deformation and single crystal deformation, Metal. Trans. 1 (5) (1970) 1121– 1143. [2] H.S. Kim, A composite model for mechanical properties of nanocrystalline materials, Scripta Mater. 39 (8) (1998) 1057– 1061. [3] M.Yu. Gutkin, I.A. Ovidko, Physical Mechanics of Deformed Nanostructures: Nanocrystalline Materials, Vol. 1, Yanus, St. Petersburg, 2003. [4] H.K. Birnbaum, P. Sofronis, Hydrogen-enhanced localized plasticity – a mechanism for hydrogen-related fracture, Mater. Sci. Eng 176 (1–2) (1994) 191–202. [5] P. Sofronis, Y. Liang, N. Aravas, Hydrogen induced shear localization of the plastic flow in metals and alloys, Europ. J. Mech. A. Solids. 20 (6) (2001) 857–872. [6] D. Delafosse, T. Magnin, Interfaces in stress corrosion cracking: a case study in duplex stainless steels, Solid State Phenomena 59-60 (1998) 221–250. [7] N.J. Petch, The cleavage strength of polycrystals, J. Iron Steel Inst. 174 (1953) 25–28. [8] I.A. Ovid’ko, N.V. Skiba, Enhanced dislocation emission from grain boundaries in nanocrystalline materials, Scripta Materialia 67 (1) (2012) 13–16. [9] V.E. Panin, Overview on mesomechanics of plastic deformation and fracture of solids, Theor. Appl. Fracture Mech 30 (1) (1998) 1–11. [10] Ph. Buffat, J.-P. Borel, Size effect on the melting temperature of gold particles, Phys. Rev. A 13 (6) (1976) 2287–2298. [11] H. Udin, A.J. Shaler, J. Wulff, The surface tension of solid copper, Trans. AIME. 185 (1949) 186–190. [12] S.Sh. Rekhviashvili, E.V. Kishtikova, On the temperature of melting of nanoparticles and nanocrystalline substances, Tech. Phys. Lett. 32 (5) (2006) 439–441. [13] P.F. Buff, Spherical interface. ii. molecular theory, J. Chem. Phys. 23 (3) (1955) 419–427. [14] F.O. Koenig, On the thermodynamic relation between surface tension and curvature, J. Chem. Phys. 18 (4) (1950) 449–459. [15] V.M. Yurov, V.S. Portnov, N.Kh. Ibrayev, S.A. Guchenko, Poverhnostnoe natyazhenie tverdyh tel, malyh chastic i plenok [The surface tension of solid bodies, small particles and thin membrans], Uspehi sovremennogo estestvoznania 11 (2011) 55–58. [16] E.O. Hall, Deformation and ageing of mild steel, Proc. Phys. Soc. Sec. B. 64 (6) (1951) 495–502. [17] E. V., A.V. Panin, T.F. Panin, Yu.F. Elsukova, Popkova, Fundamentalnaya rol’ krivizny kristallicheskoy struktury v plastichnosti i prochnosti tverdyh tel [Fundamental role of curvature of the crystal structure in plasticity and strength of solids], Fizicheskaya mezomehanika 17 (6) (2014) 7–8.