Nghiên cứu biến dạng đàn hồi – phi tuyến của kim loại, hợp kim xen kẽ hai và ba thành phần.Nghiên cứu biến dạng đàn hồi – phi tuyến của kim loại, hợp kim xen kẽ hai và ba thành phần.Nghiên cứu biến dạng đàn hồi – phi tuyến của kim loại, hợp kim xen kẽ hai và ba thành phần.Nghiên cứu biến dạng đàn hồi – phi tuyến của kim loại, hợp kim xen kẽ hai và ba thành phần.Nghiên cứu biến dạng đàn hồi – phi tuyến của kim loại, hợp kim xen kẽ hai và ba thành phần.Nghiên cứu biến dạng đàn hồi – phi tuyến của kim loại, hợp kim xen kẽ hai và ba thành phần.Nghiên cứu biến dạng đàn hồi – phi tuyến của kim loại, hợp kim xen kẽ hai và ba thành phần.Nghiên cứu biến dạng đàn hồi – phi tuyến của kim loại, hợp kim xen kẽ hai và ba thành phần.Nghiên cứu biến dạng đàn hồi – phi tuyến của kim loại, hợp kim xen kẽ hai và ba thành phần.Nghiên cứu biến dạng đàn hồi – phi tuyến của kim loại, hợp kim xen kẽ hai và ba thành phần.Nghiên cứu biến dạng đàn hồi – phi tuyến của kim loại, hợp kim xen kẽ hai và ba thành phần.Nghiên cứu biến dạng đàn hồi – phi tuyến của kim loại, hợp kim xen kẽ hai và ba thành phần.Nghiên cứu biến dạng đàn hồi – phi tuyến của kim loại, hợp kim xen kẽ hai và ba thành phần.Nghiên cứu biến dạng đàn hồi – phi tuyến của kim loại, hợp kim xen kẽ hai và ba thành phần.Nghiên cứu biến dạng đàn hồi – phi tuyến của kim loại, hợp kim xen kẽ hai và ba thành phần.
MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION NGUYEN DUC HIEN STUDY ON ELASTIC AND NONLINEAR DEFORMATION OF METALS, BINARY AND TERNARY INTERSTITIAL ALLOYS Speciality: Mathematical and Theoretical Physics Code: 9.44.01.03 SUMMARY OF PhD THESIS HANOI – 2023 This work is completed at Hanoi National University of Education Scientific supervisor 1: Assoc Prof PhD Nguyen Quang Hoc Scientific supervisor 2: Assoc Prof PhD Hoang Van Tich Reviewer 1: Prof PhD Bach Thanh Cong – University of Science, Vietnam National University, Hanoi Reviewer 2: Prof PhD Vu Van Hung – University of Education, Vietnam National University, Hanoi Reviewer 3: Assoc Prof PhD Nguyen Hong Quang – Institute of Physics - Vietnam Academy of Science and Technology The thesis will be defended at the University-level thesis evaluation Council at Hanoi National University of Education at the hour date month year The thesis is available at - Hanoi National Library - Library of Hanoi National University of Education INTRODUCTION Reasons for choosing the topic Metals and alloys are materials with a long history of development and are widely used in industries and practical daily life There are two types of alloys as substitutional alloys and interstitial alloys Steel is a typical form of exportation with special importance in construction, transportation, machine manufacturing An interstitial alloy Fe is FeC It is called carbon steel and accounts for a large proportion in the steel industry Fe and its interstitial alloys such as FeSi, FeC, FeH make up most of the Earth's core and celestial bodies Their thermodynamic, elastic and melting properties provide us with information about the composition, structure, evolution of the Earth, celestial bodies Information about metals such as Au, Cu, Fe, Cr and some of their alloys are especially important for designing and manufacturing equipments to serve the needs of human life The study of mechanical properties including the deformation process of materials such as metals and alloys, including interstitial alloys, has attracted the attention of many researchers, both theoretically and experimentally These research results play a very important role in science, technology and human life Statistical moment method (SMM) has been successfully applied to study the deformation of metals and binary substitutional alloys However, the study of elastic and nonlinear properties of cubic ternary and binary interstitial alloys by SMM is still an open problem For the above reasons, we choose the topic “Study on elastic and nonlinear deformation of metals, binary and ternary interstitial alloys” Purpose, object and scope of research The purpose of the thesis is to develop SMM to study elasticnonlinear deformation, effect of strain on diffusion for ternary and binary interstitial alloy taking into account the influence of temperature, pressure, concentrations of substitutional and interstitial atoms The object of the study is the elastic-nonlinear deformation, the effect of the strain on the diffusion of cubic ternary and binary interstitial alloy Research scope: Studied temperature range from zero to the melting temperature of the main metal; the studied substitutional atomic concentration range from zero to 10%; the studied interstitial atomic concentration range is from zero to 5% and the studied pressure range is from zero to 100 GPa Research methods The main research method is the statistical moment methold (SMM) In addition, in numerical calculations we use the Maple sofware and approximate methods Scientific and practical significance The results from the thesis provide the information about the nonlinear-elastic deformation and the influence of the deformation on the diffusion for metals and alloys The thesis contributes to the development of SMM in studying properties of materials in general and in studying the deformation and the influence of the deformation on the diffusion for interstitial alloy materials in particular Some numerical results of the thesis can be references for forecasting and orienting experimental studies New contributions of thesis Build analytic expressions for nonlinear and elastic deformation quanties of cubic ternary and binary interstitial alloy using SMM The thesis contributes to supplementing and perfecting the deformation theory and cubic interstitial alloy Apply the obtained theory to calculate numerically for some metals and alloys The obtained numerical results are compared with the experimental data and the results calculated by other theoretical methods Some numerical results can be predictive and guide future experiments Thesis outline In addition to the Introduction, the Conclusion, the References and the Appendix the thesis content is presented in chapters The content of the thesis has been reported at national and international specialized scientific conferences and published in articles in domestic and international scientific journals CHAPTER 1: OVERVIEW ON DEFORMATION OF METALS AND ALLOYS 1.1 Interstitial alloys 1.2 Deformation theory 1.2.1 Elastic deformation Under the action of an external force, a solid is deformed, that is, it changes shape and size When deformed, the points or atoms of the solid move Elastic deformation is the deformation of a solid object under the action of an external force that, after unloading, the solid body returns to its original shape and size 1.2.2 Nonlinear deformation Nonlinear strain (inelastic strain, residual strain) is the deformation of a solid body under the action of an external force, but after unloading, the deformation is not lost and the solid body does not return to shape, size and shape original size That happens when the external force (load) must be large enough Nonlinear strain does not change the deformation volume 1.2.3 Elastic wave in solids The longitudinal and transverse wave velocities have the form Vd 2C44 C12 C , Vn 44 (1.28) 1.2.4 Influence of deformation on diffusion The dependence of the diffusion coefficient D on the biaxial tensile stress has the form 2 r m m V V V/ / , Dx ( ) Dx ( 0) exp kBT (1.31) where Dx ( ) is the diffusion coefficient in the x direction of the r system subjected to biaxial stress , V is the volume of recovery m and V/ / is the volume component displaced in the direction parallel to the direction of diffusion 1.3 Some major methods of research There are many different theoretical methods in studying nonlinear and elastic deformation of metals and alloys such as the harmonic theory, the pseudoharmonic theory, the molecular dynamics method, the finite element method, the calculations from first principles (ab initio), the tight-binding Hamiltonian method, the density functional theory, the machine learning method, the lattice Green function method, the calculation of phase diagram, the modified embedded atomic method, etc Although there are many theoretical methods used to study the properties of objects during elastic and nonlinear deformation process, these methods are still limited in one aspect or the other Most of the methods not take into account the anharmonicity effect in the lattice vibrations, not to mention the effect of pressure on the deformation processes of the object, not to mention the dependence of the deformation quantities on the concentrations of substitutional and interstitial atoms, the results of deformation studies are mostly for metals and limited for alloys, not taking into account the effect of stress or strain on the diffusion in the alloy 1.4 Statistical moment method Kˆ n 1 a Kˆ n a Qˆ n 1 a Kˆ n an 1 a B2 m i 2m ! m 0 2m Kˆ n (2 m ) an 1 1.57 a Conclusion of Chapter In chapter 1, we present an overview of strain theory and methods of studying elastic - nonlinear deformation of metals and alloys, which refers to the theory of elastic deformation, nonlinear strain, and elastic deformation linear, elastic wave velocity and effect of strain on diffusion in the material; Basic content and advantages and disadvantages of popular deformation research methods Introduce a traceability formula that relates higher moments to lower moments and use it to determine the free energy of the system CHAPTER 2: STUDY ON NOLINEAR AND ELASTIC DEFORMATION OF INTERSTITIAL ALLOY AC 2.1 Alloy model and Helmholtz free energy Fig 2.1 Model of interstitial alloy AC with BCC structure (a) and FCC structure(b) c AC X X TScAC , (2.1) X 2 Y X U X X 2 X YX2 1X X k X 2 k X4 4 YX YX 2 X YX 1X 21 X 2 X YX , xX X 3 x X ln e , YX x coth x, X A, C , A1 , A2 , (2.2) c A 1 7cC , c A1 2cC , c A2 4cC c A 1 15cC , c A1 6cC , for BCC lattice, c A2 8cC for FCC lattice 2.2 Cohesive energy, alloy parameters and mean nearest neighbor distance between two atoms For BCC lattice, u0 A 4 AA r1 A 3 AA r2 A , r2 A kA r1 A , (2.4) 2 d AA r1 A d AA r1A d AA r2 A 3r1A dr1 A dr12A dr22A d AA r2 A , r2 A dr2 A (2.5) 1 A d AA r1A d AA r1A d AA r1A d AA r1A + 2 54 9r1 A dr1 A dr1 A dr1 A 9r1 A dr1 A 9r1 A + d AA r2 A d AA r2 A d AA r2 A - , 2 24 dr2 A dr2 A 4r2 A dr2 A 4r2 A 2 A (2.6) d AA r1A d AA r1 A d AA r1A d AA r2 A - + 2.7 dr1A 2r2 A dr1 A 3r1 A dr1 A 3r1 A dr23A u0C AC r1C 2 AC r2C , r2C 2r1C , (2.8) kC d AC r1C d AC r2C d AC r2C , r1C dr1C r2C dr2C dr2C C 4 1C 2C , 1C (2.10) d AC (r1C ) d AC (r1C ) d AC (r2C ) - dr1C 48 8r12C dr12C 8r1C dr24C d 3 AC (r2C ) d 2 AC (r2C ) d AC ( r2C ) , 2 8r2C dr2C dr2C 16r2C dr2C 16r23C 2C + (2.9) (2.11) d 3 AC (r1C ) d 2 AC (r1C ) d AC (r2C ) - + 4r1C dr1C dr13C 2r1C dr12C 2r1C d 3 AC (r2C ) d 2 AC (r2C ) d AC (r2C ) , 2 4r2C dr2C dr2C 4r2C dr2C 4r2C u0 A1 u0 A 3 A1C r1A1 , k A1 k A d 2 A1C r1 A1 dr12A1 (2.13) d A1C r1A1 , r1A1 dr1 A1 A1 4 1 A1 2 A1 , 1 A1 1 A (2.12) (2.14) (2.15) d A1C d A1C 24 dr1 A r1A1 dr13A 1 d A1C d A1C , 4r1 A1 dr1 A1 4r1 A1 dr1 A1 (2.16) γ 2A1 2 A d A1C ( r1A1 ) , 4r1 A1 dr13A1 (2.17) u0 A2 u0 A 6 A2 C r1A2 , d 2 A2 C r1 A2 k A2 k A dr12A2 (2.18) d A2 C r1A2 , r1 A2 dr1 A2 (2.19) A2 4 1 A2 2 A2 , 1 A2 1A (2.20) d A2 C r1A2 24 dr14A 2 A2 2 A 15 d A2 C (r1A2 ) 15 d A2 C (r1 A2 ) , dr1A2 dr12A2 4r12A2 4r13A2 (2.21) d A2 C (r1A2 ) d A2 C (r1A2 ) 4r1 A2 dr14A2 dr13A2 d A2 C (r1A2 ) d A2C (r1A2 ) - , dr1 A2 dr12A2 8r12A2 8r1A2 (2.22) For BCC lattice, u0 A 6 AA r1 A 3 AA r2 A , r2 A kA 1 A d AA r1 A dr12A 2r1 A , (2.23) d AA r1 A d AA r2 A d AA r2 A , r1A dr1 A r dr2 A dr2 A 2A (2.24 ) d AA r1A d AA r1A d AA r1A d AA r1A + 2 24 4r1A dr1 A dr1 A dr1A 8r1 A dr1 A 8r1 A + d AA r2 A d AA r2 A d AA r2 A - , 2 24 dr2 A dr2 A 4r2 A dr2 A 4r2 A 2 A (2.25) d AA r1 A d AA r1A d AA r1A d AA r1 A + 2 4r1A dr1 A dr1A dr1 A 8r1A dr1A 8r1A according to formulas (2.52) to (2.59) 2.4 Nonlinear deformation r1FX P, r1 X P, (2.61) r1FX P, T r1 X P, T r1X P, (2.62) F AC F X c X TScACF , (2.63) X AC AC f AC ( ) X 2 F 2v AC c X X 1 F v v AC AC (2.64) AC , 1 2 r01F X F v AC f AC ( F ) f AC max C AC AC max F ACmax 1 AC max T 2r01X , T 2 F F F F X2 2r01 FX X r r1 X T 1X f AC ( ) C AC AC , XF F r1X f ACmax , C AC F (2.70) (2.72) (2.73) (2.74) AC max f AC max 1F C AC F (1 F ) (2.75) F AC , AC (2.76) lAC max AC C AC F 1F f AC ( 0,2 ) , (2.77) , (2.78) e AC , e (2.80) AC 0,2 0,2 ACe AC AC EYAC e AC e AC e (2.81) 2.5 Numerical results and discussion for metals and alloys Using the Mie-Lennard-Jones n-m, the Finnis – Sinclair and the Morse potentials 10 The numerical results for Fe, FeSi, FeH, FeC, Au, Cu, and CuSi are summarized in Table 2.2 to 2.10, Table 2.13 to 2.20, illustrated from Figure 2.2 to 2.19 7000 6800 6600 Vd (ms-1) 6400 6200 6000 5800 SMM Experiments of Shibazaki at al (2016) Experiments of Antonangeli and Ohtani (2015) Experiments of Decremps at al (2014) Experiments of Liu at al (2014) 5600 5400 5200 5000 10 P (GPa) Figure 2.3 Vd(P) for Fe at T = 300 K calculated by SMM and from experiments of Antonangeli (2015)[11], Decremps (2014)[25], Liu (2014) [80] Shibazaki (2016)[111] According to Table 2.3 for FeSi at P = and the same temperature when increasing Si concentration, the elastic modulus, elastic constant and elastic wave velocity all decrease sharply For example, for FeSi at P = and T = 1000K with increasing Si concentration from to 5%, E decreases 65.48% For FeSi at P = and the same Si concentration, with increasing temperature, the elastic modulus, elastic constant and elastic wave velocity all decrease For example, for FeSi at P = and cSi = 5% with increasing temperature from 100 to 1000K, E decreases 61.88% According to Table 2.4 for FeSi at T = 300K and the same Si concentration with increasing pressure, the elastic modulus, elastic constant and elastic wave velocity all increase For example, for FeSi at T = 300K and cSi = 5% with increasing pressure from to 10GPa, E increases 35.85% Table 2.7 EY(cC) (GPa) for FeC at P = and T = 300 K calculated by SMM and from experiments [115] E c (10 cC (%) 1,4 2,3 10 Pa/%) 11 SMM Experiments[115] δ (%) 208.2 208.2 198.5 204.8 3.1 194.6 201.5 3.4 188.8 197.9 4.6 185.9 193.8 4.1 -0.6309 -0.6180 2.1 Bảng 2.8 EY(cH) (GPa) for FeH at P = and T = K calculated by SMM and ab initio of Psiachos at al (2011)[104] cH (%) PPTKMM ab initio[104] (%) 222.8 229.2 2.79 216.9 225.9 3.98 211.1 222.6 5.17 205.4 219.4 6.38 199.8 216.1 5.01 194.2 212.8 8.74 180 PPTKMM Experiments of Santra at al (2014) Experiments of Ledbetter and Naimon (1974) 160 EY (GPa) 140 120 100 80 cSi(%) Figure 2.11 EY (cSi ) for CuSi at T = 300K and P = calculated by SMM and from experiments of Ledbetter and Naimon (1974)[75], Santra at al (2014)[109] 18 1200 16 cSi = cSi = 2% cSi = 5% 14 800 1(MPa) f(GPa) 12 1000 10 400 Fe, SMM Fe, XEXPT of Smith et al.(2020) Fe, DEXPT of Smith et al.(2020) FeSi2%, SMM FeSi5%, SMM 200 (a) 600 0 (%) (%) 12 Figure 2.13 (a) f ( , cSi ) and (b) ( , cSi ) for FeSi at T = 300K, P = calculated by SMM and from experiments of Smith at al (2020)[113] Conclusion of Chapter In chapter 2, we build the theory of elastic and nonlinear strain of interstitial alloy AC with cubic structure in which general analytic expressions of free energy, displacement of particles from lattice are drawn, mean nearest neighbor distance between two atoms, elastic deformation quantities such as isothermal and adiabatic elastic moduli, Young's modulus, mass compression modulus, shear modulus, elastic constants, longitudinal wave velocity, transverse wave velocity, nonlinear strain quantities such as strain energy density, maximum real stress, elastic strain limit to determine stress strain curve Analytical results of the resulting strain-specific quantities depend on the temperature, pressure and the interstitial atom concentration The obtained theoretical results are numerically calculated for metals Fe, Au, Cu and interstitial alloy FeSi, FeH, FeC, AuSi, CuSi The law of deformation depends on temperature and pressure of interstitial alloy is consistent with the law of deformation depending on temperature and pressure of the main metal in interstitial alloy Young E modules, mass compression modulus K, shear modulus G, elastic constants C11, C12, C44 and elastic wave propagation velocities Vd, Vn of metal and interstitial alloy decrease with increasing temperature and increase with increasing temperature pressure increase The elastic strain quantities of interstitial alloy all decrease with the increase of interstitial atom concentration The maximum real stress and elastic strain limit of metal and interstitial alloy decrease with the increase of temperature and increase with the increase of pressure The maximum real stress and elastic strain limit of interstitial alloy decrease with the increase of interstitial atom concentration Many numerical results obtained by SMM, especially elastic and nonlinear deformation of metals, have been compared with numerical results by other theoretical methods and experimental data and have some very consistent results good Many numerical results obtained by SMM on elastic and nonlinear strain of interstitial alloy are new, predictive, experimentally oriented 13 CHAPTER 3: STUDY ON NOLINEAR AND ELASTIC DEFORMATION OF INTERSTITIAL ALLOY ABC 3.1 Alloy model Fig 3.1 Model of substitutional and interstitial alloy ABC with BCC structure (a) and FCC structure(b) 3.2 Helmholtz free energy ABC AC cB B A TScAC TScABC , (3.1) 3.3 Mean nearest neighbor distance a ABC c AC a AC BTAC B cB aB TB , BT c AC BTAC cB BTB BT BT (3.2) 3.4 Elastic deformation EYABC EYAC cB EYA EYB , (3.9) Determining Young's modulus, we can also determine other elastic modules, elastic constants and elastic wave propagation speed according to formulas (3.12) to (3.19) 3.5 Nonlinear deformation (3.19) r1FX P, r1 X P, , r1FX P, T r1X P, T r1 X P , , F F ABC AC cB BF AF TScACF TScABCF , ABC ABC ABC , 1 14 (3.20) (3.21) (3.22) f ABC ( ) X 2 F 2v ABC c X X 2 F F X2 r1X 1 F v v ABC ABC F 2r01 X T cB A F v v ABC ABC 2 F 2v ABC 2 r01F X F v ABC F FX r 1X 2 r01F A F v ABC AF F r1A T 2r01A , T f ABC ( F ) f ABC max C ABC ABC max F (3.26) (3.27) (3.28) f ABCmax , C ABC F (3.29) ABC max f ABC max 1F C ABC F (1 F ) (3.30) ABCmax 1 ABC max 2r01 X T 2 F F F A2 2r01F A FA r r1 A T 1A f ABC ( ) C ABC ABC , XF F r1X F ABC , ABC F lABC max ABC C ABC AC 1F f ABC ( 0,2 ) , (3.32) , (3.33) e ABC , 1 e (3.35) ABC 0,2 0,2 ABCe ABC (3.31) EYABC e ABC e ABC e (3.36) 3.6 Numerical results and discussion for alloys The numerical results for FeCrSi, AuCuSi are summarized in Table 3.1 to Table 3.18 and illustrated from Figure 3.3 to Figure 3.10 15 340 240 320 220 300 280 200 K (GPa) 240 220 200 SMM Ab initio of Zhang (2010) Ab initio of Olsson (2003) Experiments of Heintze (2009) Experiments of Speich (1972) 160 140 (a) 120 100 180 160 140 180 SMM Ab initio of Zhang (2010) Ab initio of Olsson (2003) Experiments of Heintze (2009) Experiments of Specich (1972) 120 (b) 100 80 10 cCr (%) cCr (%) E c K c Figure 3.2 (a) Cr and (b) Cr for FeCr at T = 298K, P = calculated by SMM, ab initio of Olsson [100], ab initio of Zhang [131] and from experiments of Heintze [40] and Specich [115] 35 cSi = cSi = 1% cSi = 3% cSi = 5% 30 EY (1010 Pa) EY (GPa) 260 25 20 15 10 10 20 30 40 50 60 70 P (GPa) Figure 3.3 EY(P, cSi) of AuCuSi at cCu = 10% and T = 300K 16 10 18 750 16 cSi=0 cSi=2% cSi=5% 14 700 f(GPa) f(GPa) 12 10 650 600 cSi=0 cSi=2% cSi=5% 550 (a) 0 (b) 500 Figure 3.4 (a) (%) (%) f ( , cSi ) and (b) ( , cSi ) for FeCrSi at cCr =10%, T = 300K and P = 25 800 P=0 P = 2,55 GPa P = 4,88 GPa P = 9,47 GPa P = 18,78 GPa 20 600 500 1(MPa) f(GPa) 15 P=0 P = 2,55 GPa P = 4,88 GPa P = 9,47 GPa P = 18,78 GPa 700 10 400 300 200 100 (a) 0 10 12 (b) 10 12 (%) (%) ( , P) for AuCuSi at cCu =10%, Figure 3.9 (a) f ( , P) and (b) cSi = 1% T = 300K Conclusion of Chapter In chapter 3, we build the theory of elastic-nonlinear deformation of substitutional and interstitial ternary alloy ABC with the cubic structure, which derives the general analytic expressions of free energy, magnitude displacement of the particle from the lattice, average nearest neighbor distance between two atoms, elastic deformation quantities such as isothermal and adiabatic elastic moduli, Young's modulus, mass compression modulus, shear modulus, constants elastic, longitudinal wave velocity, transverse 17