上海交通大学：《金属材料强韧化与组织调控》课程教学资源（课堂练习）exam_example4

Al:Texture(Chapter 2) 7.5 points a)Define the concept of the crystallographic texture.Please describe shortly how textures can be measured?(2.5 points) b) Please sketch the {200)pole figure of a cube,which has been clockwise rotated by 45 around the transverse direction (TD).Please determine the Miller indices and the rotation matrix of this orientation.(5 points) w s 1 A2:Crystal Defects (Chapter 3) 10 points a)Name the most important types of crystal defects.From a physical metallurgy point of view,each of these defects has some profound influence on certain specific processes happening in metals.Name these processes.(3 points) b) During the formation of n vacancies in a crystal the Gibbs free energy changes according to AG=nH-T(nS,+S). n=number of vacancies H=vacancy formation enthalpy St=vibration entropy per vacancy Sa=configuration entropy The variety of arrangements of n vacancies on N different lattice sites can be N! expressed as @ Derive by application of Stirling's formula (N-n)!n! In(x!)=x.Inx-x an expression for the atomic equilibrium concentration of vacancies ca.(7 points) Estimate the formation enthalpy of a vacancy in tungsten.The equilibrium concentration of vacancies at the melting point (3410C)is given as 10,the vibration entropy can be assumed as 2-k.Express the result with the dimension of eV. (1.5 points)

A1: Texture (Chapter 2) 7.5 points a) Define the concept of the crystallographic texture. Please describe shortly how textures can be measured? (2.5 points) b) Please sketch the {200} pole figure of a cube, which has been clockwise rotated by 45° around the transverse direction (TD). Please determine the Miller indices and the rotation matrix            w s l v r k u q h g of this orientation. (5 points) A2: Crystal Defects (Chapter 3) 10 points a) Name the most important types of crystal defects. From a physical metallurgy point of view, each of these defects has some profound influence on certain specific processes happening in metals. Name these processes. (3 points) b) During the formation of n vacancies in a crystal the Gibbs free energy changes according to   L L    G nH T nS S B k  . n = number of vacancies L HB = vacancy formation enthalpy L S = vibration entropy per vacancy Sk = configuration entropy The variety of arrangements of n vacancies on N different lattice sites can be expressed as   ! ! ! N N N nn    . Derive by application of Stirling’s formula ( ln ! ln   x   x xx ) an expression for the atomic equilibrium concentration of vacancies a L c . (7 points) c) Estimate the formation enthalpy of a vacancy in tungsten. The equilibrium concentration of vacancies at the melting point (3410°C) is given as 10–4, the vibration entropy can be assumed as 2·k. Express the result with the dimension of eV. (1.5 points)

A3:Intermetallic Phases(Chapter 4) 9 points a)A requirement for a distinct solubility in binary alloys is for example,that both alloying partners must crystallize in the same crystall structure.Name additional requirements for a distinct solubility(Hume-Rothery-rules).(3 points) b) How is the long range order parameter defined (equation)?Give a relation and explain the different quantities appearing in it.Which range of values does this parameter show in case of an AB alloy?And for an AB3 alloy?Support your answer using calculations.(5 points) c) The occurrence of long range order phases can be confirmed by the appearance of so- called superlattice reflexes.Explain shortly,what is the meaning of superlattice reflexes and why do they appear.(2 points) A4:Diffusion (Chapter 5) 11 points a) Name the differences between vacancy diffusion and self diffusion.And explain what are their activation energies composed of.(2 points) b) Demonstrate that the diffusion coefficient for interstitial diffusion through octahedral sites in the fcc lattice can be determined by the relation: 6t (D-Diffusion coefficient,-Jump time,-Jump difference of the diffusing atom) Sketch all the necessary figures and explain all the involved quantities.(Derive the equation step by step)(11 points) A5 Dislocations a)Sketch a FCC unit cell with the proper atom positions with a standard cartisen co- ordinate system. b)You berve that n ede disocation with aBurgers vectorin thes plane splits into two Shockley-Partial dislocations: b,=52刘umdb,=8h1可

A3: Intermetallic Phases (Chapter 4) 9 points a) A requirement for a distinct solubility in binary alloys is for example, that both alloying partners must crystallize in the same crystall structure. Name additional requirements for a distinct solubility (Hume-Rothery-rules). (3 points) b) How is the long range order parameter defined (equation)? Give a relation and explain the different quantities appearing in it. Which range of values does this parameter show in case of an AB alloy? And for an AB3 alloy? Support your answer using calculations. (5 points) c) The occurrence of long range order phases can be confirmed by the appearance of so￾called superlattice reflexes. Explain shortly, what is the meaning of superlattice reflexes and why do they appear. (2 points) A4: Diffusion (Chapter 5) 11 points a) Name the differences between vacancy diffusion and self diffusion. And explain what are their activation energies composed of. (2 points) b) Demonstrate that the diffusion coefficient for interstitial diffusion through octahedral sites in the fcc lattice can be determined by the relation:   6 2 D  (D – Diffusion coefficient,  – Jump time,  - Jump difference of the diffusing atom) Sketch all the necessary figures and explain all the involved quantities. (Derive the equation step by step) (11 points) A5 Dislocations a) Sketch a FCC unit cell with the proper atom positions with a standard cartisen co￾ordinate system. b) You observe that an edge dislocation with a Burgers vector   110 2 a b  in the slip plane splits into two Shockley-Partial dislocations:   121 6 a b1  und 211 6 a b2 

Sketch in the unit cell of the task a)the burgers verctors of the partial dislocations in such a way that the splitting of the dislocation into partials is seen clearly. c)Compute the Line element s of the edge dislocation in the standard form. Line normal n of the glideplane,in which the edge dislocation lies:n= d)Calculate the dissociation width Xo of this edge dislocation in Cu. The dissociation width is given as: 02 G.1 2 YSFE 色e,色×e- Assume a physically acceptable value for the Poisson's ratio. e)Explain why (energetics)as given in part b)a perfect dislocation dissociates into Shockley partials. f Compute with the help of the Peach-Koehler equation the force between two parallel edge dislocations.Show the dislocation positions in an orthogonal coordinate system with proper notations. O Ox 0 Stress tensor for the edge dislocation in z-direction:= 0 (0 0 A6: Time-dependent deformation(Chapter 6) 5.5 points Draw a schematic creep curve at)and creep rate curve &(t)as a function of time for two different temperatures each(TI<T2).Mark unambiguously which temperature corresponds to which curve.Indicate and name the three characteristic parts of the creep curve

Sketch in the unit cell of the task a) the burgers verctors of the partial dislocations in such a way that the splitting of the dislocation into partials is seen clearly. c) Compute the Line element s of the edge dislocation in the standard form. Line normal n of the glideplane, in which the edge dislocation lies: 2 1 a n 1 12 1              d) Calculate the dissociation width x0 of this edge dislocation in Cu. The dissociation width is given as:                       1  1 b s b s b s b s 1 2 G x 1 2 1 2 SFE 0 Assume a physically acceptable value for the Poisson´s ratio. e) Explain why (energetics) as given in part b) a perfect dislocation dissociates into Shockley partials. f) Compute with the help of the Peach-Koehler equation the force between two parallel edge dislocations. Show the dislocation positions in an orthogonal coordinate system with proper notations. Stress tensor for the edge dislocation in z-direction:       xyz xx xy xy yy zz            0 0 0 0 A6: Time-dependent deformation (Chapter 6) 5.5 points Draw a schematic creep curve (t) and creep rate curve (t) as a function of time for two different temperatures each (T1 < T2). Mark unambiguously which temperature corresponds to which curve. Indicate and name the three characteristic parts of the creep curve

A7: Grain boundary motion(Chapter 7) 13.5 points a) The velocities of a moving grain boundary in aluminium have been measured at different temperatures(see following table).The driving force during the experiment was kept constant.Determine from this data the temperature dependence of the grain boundary velocity v=f(T).Please represent the analysis in a graph and show your calculation steps with table.Calculate the activation energy (in eV).(7 points) v [um/s] 87 17,0 32,4 T [C] 400 415 430 b) Deduce the time law for discontinuous grain growth under the assumption that the average velocity of the grain boundary is proportional to the change of discontinuously growing grains'diameter over time.Explain the occurring variables. (4.5 points) c) The grain boundary motion can be hindered by foreign atoms.Sketch the dependency of velocity on the driving force in the case of a free moving boundary and in the case of a loaded boundary.Name a way,how to transfer the motion of a loaded boundary into a free moving boundary.(2 points)

A7: Grain boundary motion (Chapter 7) 13.5 points a) The velocities of a moving grain boundary in aluminium have been measured at different temperatures (see following table). The driving force during the experiment was kept constant. Determine from this data the temperature dependence of the grain boundary velocity v = f(T). Please represent the analysis in a graph and show your calculation steps with table. Calculate the activation energy (in eV). (7 points) v [µm/s] 8,7 17,0 32,4 T [°C] 400 415 430 b) Deduce the time law for discontinuous grain growth under the assumption that the average velocity of the grain boundary is proportional to the change of discontinuously growing grains’ diameter over time. Explain the occurring variables. (4.5 points) c) The grain boundary motion can be hindered by foreign atoms. Sketch the dependency of velocity on the driving force in the case of a free moving boundary and in the case of a loaded boundary. Name a way, how to transfer the motion of a loaded boundary into a free moving boundary. (2 points)

A8:Phase transformations(Chapter 8&9) 16 points a Calculate the critical work of nucleation and the critical edge length of a cubic nucleus during homogeneous nucleation from the melt.Indicate all used variables.(6 points) b) Draw the curve of the free energy AG as a function of nucleus size (edge length D)for T>Tm and T Tm.Mark also the critical edge length De.Explain the curve progressions(what happens to the"nucleus",if DDe?).(5 points) c)Explain,which additional significant influence needs to be considered in the case of homogeneous nucleation for a transformation in the solid state.Write down an expression for the free energy AG as a function of the edge length D.(2 points) d Martensitic Transformation is of great technical importance,due to its great strengthening effect.How is a martensitic phase transformation defined?Name two of the mechanisms,which lead to the high strength of martensite (with short explanations).(3 points) A9: Electron Therory (Chapter 10) 5 points Electrons in matter follow the Pauli's exclusion principle and fill,according to the Band theory,successive finite energy levels.By absorption of thermal energy (k T),an electron can reach a state of higher energy.The temperature dependence of the probability of occupancy for a state of energy E is described by the Fermi distribution: f(E)dE = -de 1+exp E-EF a) Draw the distribution function fE)against the energy E,first for T =OK (absolute zero)and then for a finite temperature T>OK.Indicate on the diagram the Fermi energy EF.(3 points) Calculate the Fermi temperature TF,i.e.the temperature corresponding to the Fermi energy &r for Silver (F(Ag)=5.48 eV).(2 points) R=8.3144J/molK,k=8.6210-5eV/K=1.38-10-23J/K,=6.023-1023

A8: Phase transformations (Chapter 8&9) 16 points a) Calculate the critical work of nucleation and the critical edge length of a cubic nucleus during homogeneous nucleation from the melt. Indicate all used variables. (6 points) b) Draw the curve of the free energy ΔG as a function of nucleus size (edge length D) for T > Tm and T Dc?). (5 points) c) Explain, which additional significant influence needs to be considered in the case of homogeneous nucleation for a transformation in the solid state. Write down an expression for the free energy ΔG as a function of the edge length D. (2 points) d) Martensitic Transformation is of great technical importance, due to its great strengthening effect. How is a martensitic phase transformation defined? Name two of the mechanisms, which lead to the high strength of martensite (with short explanations). (3 points) A9: Electron Therory (Chapter 10) 5 points Electrons in matter follow the Pauli's exclusion principle and fill, according to the Band theory, successive finite energy levels. By absorption of thermal energy (k·T), an electron can reach a state of higher energy. The temperature dependence of the probability of occupancy for a state of energy E is described by the Fermi distribution: dE kT E f E dE F           1 exp 1 ( ) a) Draw the distribution function f(E) against the energy E, first for T = 0K (absolute zero) and then for a finite temperature T > 0K. Indicate on the diagram the Fermi energy F. (3 points) b) Calculate the Fermi temperature TF, i.e. the temperature corresponding to the Fermi energy F for Silver (F(Ag) = 5.48 eV). (2 points) R=8.3144 J/molK, k=8.6210-5eV/K = 1.3810-23J/K, NL=6.0231023