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_example3

A1:Hexagonal Crystal Structures(Chapter 2) 7.5 points a) Draw in the following unit cells of a hexagonal close packed crystal structure (respectively three unit cells are displayed for visualization of the crystal symmetry) the respectively given directions and planes,and please indicate them clearly.Please keep in mind,that in each case specific directions and planes are addressed,whose position in the unit cell is defined unambiguously.(4 Points) [1120] [2i3] 木C a3 a3 a2 a2 K a1 k a1 ī00) 122) 个C a a3 a2 a2 a1 a b) Which slip system is the most important slip system and the most important twinning system in the hexagonal dense packed crystal lattice?(2 points) 9 Discuss the twinning of hexagonal materials in dependence of the c/a-ration in case of compression perpendicular to the c-axis.(1.5 points)

1 A1: Hexagonal Crystal Structures (Chapter 2) 7.5 points a) Draw in the following unit cells of a hexagonal close packed crystal structure (respectively three unit cells are displayed for visualization of the crystal symmetry) the respectively given directions and planes, and please indicate them clearly. Please keep in mind, that in each case specific directions and planes are addressed, whose position in the unit cell is defined unambiguously. (4 Points) [11 20] [1 2 1 3] (1 100) (11 22) b) Which slip system is the most important slip system and the most important twinning system in the hexagonal dense packed crystal lattice? (2 points) c) Discuss the twinning of hexagonal materials in dependence of the c/a-ration in case of compression perpendicular to the c-axis. (1.5 points)

A2: Structure-and Orientation Determination(Chapter 2) 9.5 points a Derive an expression for the diffraction of X-rays with wave length A in a crystal lattice as function of the incidence angle 0.Explain the derivation with a diagram. Indicate all symbols.(3 points) b) What is the extinction rule for fcc-lattice?(1 point) c Calculate the theoretical diffraction angle 0 for the first three lattice planes,which you can obtain from an X-ray diffractogram (see below),for the Nickel (aN=3,524 A). The wave length of the X-radiation is Aco=1.7902 A.(4 points) d) Assign the corresponding diffracting crystallographic planes to the peaks in an X-ray diffractogram of Ni powder according to your results from part c).Consider that the 0- angles obtained from an X-ray diffractogram can vary from the theoretically calculated ones by up to 0.1.(1.5 points) 3000 2000- 1000- 0- 20 30 40 50 Theta [ 2

2 A2: Structure- and Orientation Determination (Chapter 2) 9.5 points a) Derive an expression for the diffraction of X-rays with wave length λ in a crystal lattice as function of the incidence angle θ . Explain the derivation with a diagram. Indicate all symbols. (3 points) b) What is the extinction rule for fcc-lattice? (1 point) c) Calculate the theoretical diffraction angle θ for the first three lattice planes, which you can obtain from an X-ray diffractogram (see below), for the Nickel (aΝι = 3,524 Å). The wave length of the X-radiation is λCo = 1.7902 Å. (4 points) d) Assign the corresponding diffracting crystallographic planes to the peaks in an X-ray diffractogram of Ni powder according to your results from part c). Consider that the θ- angles obtained from an X-ray diffractogram can vary from the theoretically calculated ones by up to 0.1°. (1.5 points)

A3:Coincidence Site Boundaries(Chapter 3) 7 points a)A special form of appearance of a high angle grain boundary is a coincidence site boundary,which can be described by the value2.What is meant by a coincidence site lattice,what is the definition for E and which values can E assume in the cubic crystal structures?(3 points) b) The below displayed drawing shows a 22.6(1 00)grain boundary.Highlight all coincidence sites and draw the coincidence site lattice(CSL).Calculate the value E for this boundary.Sketch additionally in a unit cell of the coincidence site lattice the DSC-lattice.(4 points) 0 0=22.62° 3

3 A3: Coincidence Site Boundaries (Chapter 3) 7 points a) A special form of appearance of a high angle grain boundary is a coincidence site boundary, which can be described by the valueΣ. What is meant by a coincidence site lattice, what is the definition for Σ and which values can Σ assume in the cubic crystal structures? (3 points) b) The below displayed drawing shows a 22.6° (1 0 0) grain boundary. Highlight all coincidence sites and draw the coincidence site lattice (CSL). Calculate the value Σ for this boundary. Sketch additionally in a unit cell of the coincidence site lattice the DSC-lattice. (4 points) θ θ = 22.62° a

A4:Alloying(Chapter4) 16points a Give the Gibbs phase rule.Identify all variables in this relation.(2.5 points) b) Construct a qualitative phase diagram for the three phases S,a and B as function of the solute content cB from the Gibbs free energy curves given at the four temperatures TI to T4.Indicate all phase fields.(5 points) c)How much is the solubility limit at T=OK?Explain your answer by drawing curves of the enthalpy H(c),the entropy S(c)and the Gibbs free energy G(c)as a function of the solute concentration c in a graph.(2.5 points) 少 Draw the unit cell of the ordered alloy compounds CuAu and Ni3Al.(2 points) e) How is the long range order parameter s defined(equation)?Give a relation and explain the different quantities appearing in it.Which range of values does this parameter take in case of an AB alloy?And for an AB3 alloy?Account for your answer using simple calculations.(4 points)

4 A4: Alloying (Chapter4) 16points a) Give the Gibbs phase rule. Identify all variables in this relation. (2.5 points) b) Construct a qualitative phase diagram for the three phases S, α and β as function of the solute content cB from the Gibbs free energy curves given at the four temperatures T1 to T4. Indicate all phase fields. (5 points) c) How much is the solubility limit at T= 0K? Explain your answer by drawing curves of the enthalpy H(c), the entropy S(c) and the Gibbs free energy G(c) as a function of the solute concentration c in a graph. (2.5 points) d) Draw the unit cell of the ordered alloy compounds CuAu and Ni3Al. (2 points) e) How is the long range order parameter s defined (equation)? Give a relation and explain the different quantities appearing in it. Which range of values does this parameter take in case of an AB alloy? And for an AB3 alloy? Account for your answer using simple calculations. (4 points)

G T=konst. G T,=konst. B B S 0 2040 60 80 100 0 20 40 60 80 100 A Cal%l B A Cpl% B T=konst. G G T=konst. S B B 0 20 4060 80 100 0 20 40 6080100 A c%, B A c%, B Ts(A T Ts(B) T2 T 0 20 40 60 80 100 A CB [% B 5

5 G T3=konst. G T4=konst. S α β β S α 0 20 40 60 80 100 A B cB [%] 0 20 40 60 80 100 A B cB [%] G T1=konst. G T2=konst. S α β β α S 0 20 40 60 80 100 A B cB [%] 0 20 40 60 80 100 A B cB [%] T4 T3 T2 T1 TS(A) TS(B) 0 20 40 60 80 100 A B cB [%]

A5:Diffusion(Chapter 5) 6 points a) Due to charge neutrality no single vacancies can exist in an ion crystal.Name and explain two different defects,from which the charge neutrality in a crystal is preserved.(2 points) b) Sketch the ion conductivity o as a function of the temperature T and name the two occurring ranges.(4 points) A6:Crystal Plasticity (Chapter 6) 13 points a Draw a technical diagram of the nominal stress(schematic)over strain with continuous transition between elastic and plastic range.(1.5 points) b) Indicate and specify the five most important characteristics in this diagram.(2.5 points) c) Further,draw a qualitative diagram of the true stress over strain in the same diagram. (1 point) S To determine the uniform strain the Considere-criterion can be consulted.Derive the formula and depict it graphically.Explain the used formula signs.(6 points) e On which material parameters depend the critical resolved shear stress in bcc metals and in fcc metals?At which crystal structure(fcc or bcc)exist a clear temperature dependency of the critical shear stress and explain the reason briefly?(2 points) A7:Time-dependent Deformation(Chapter 6) 5.5 points Draw a schematic creep curve s(t)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.(5.5 points) 6

6 A5: Diffusion (Chapter 5) 6 points a) Due to charge neutrality no single vacancies can exist in an ion crystal. Name and explain two different defects, from which the charge neutrality in a crystal is preserved. (2 points) b) Sketch the ion conductivity σ as a function of the temperature T and name the two occurring ranges. (4 points) A6: Crystal Plasticity (Chapter 6) 13 points a) Draw a technical diagram of the nominal stress (schematic) over strain with continuous transition between elastic and plastic range. (1.5 points) b) Indicate and specify the five most important characteristics in this diagram. (2.5 points) c) Further, draw a qualitative diagram of the true stress over strain in the same diagram. (1 point) d) To determine the uniform strain the Considère-criterion can be consulted. Derive the formula and depict it graphically. Explain the used formula signs. (6 points) e) On which material parameters depend the critical resolved shear stress in bcc metals and in fcc metals? At which crystal structure (fcc or bcc) exist a clear temperature dependency of the critical shear stress and explain the reason briefly? (2 points) A7: Time-dependent Deformation (Chapter 6) 5.5 points Draw a schematic creep curve ε(t) and creep rate curve ε&( )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. (5.5 points)

A8:Softening(Chapter 7) 13 points a Explain the difference between "continuous grain growth"and "discontinuous grain growth".(2 points) b) Derive an expression for the driving force of continuous grain growth.Explain the basic ideas with the help of a sketch.(3 points) c) Derive an expression for the driving force of discontinuous grain growth,assuming grain having a cuboidal shape.Describe the basic ideas with the help of a sketch.(3 points) d In an AlMg alloy exist spherical precipitates with radius r=0.1 um.Calculate the effective driving force for continuous grain growth,assuming that the volume fraction of the precipitates is 3%and curvature radius of the grain boundary is ten times as large as the average grain size D(D=5 um).(3 points) e) At which grain size would continuous grain growth be stopped?(2 points) (y=1 Jm2) A9: Transformations:solid-liquid solid-solid (Chapter 8&9) 7.5 points a) Draw schematically an eutectoid phase diagram and indicate all phase regions.(1 point) b) Draw the Gibbs free energy curve for this transition temperature.Indicate precisely in which concentration range precipitates form by spinodal decomposition and where via regular nucleation.(2.5 points) What are the characteristic features of spinodal decomposition and what are the differences compared to precipitation via regular nucleation?(4 points) A10: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.When the Temperature of the matter is raised,corresponding to an energy increase of(k.T),an electron can leave its energy state to reach a state of higher energy.The probability that a state of energy E is >

7 A8: Softening (Chapter 7) 13 points a) Explain the difference between "continuous grain growth" and "discontinuous grain growth". (2 points) b) Derive an expression for the driving force of continuous grain growth. Explain the basic ideas with the help of a sketch. (3 points) c) Derive an expression for the driving force of discontinuous grain growth, assuming grain having a cuboidal shape. Describe the basic ideas with the help of a sketch. (3 points) d) In an AlMg alloy exist spherical precipitates with radius r = 0.1 µm. Calculate the effective driving force for continuous grain growth, assuming that the volume fraction of the precipitates is 3% and curvature radius of the grain boundary is ten times as large as the average grain size D (D = 5 µm). (3 points) e) At which grain size would continuous grain growth be stopped? (2 points) (γ = 1 Jm-2) A9: Transformations: solid – liquid & solid – solid (Chapter 8 & 9) 7.5 points a) Draw schematically an eutectoid phase diagram and indicate all phase regions. (1 point) b) Draw the Gibbs free energy curve for this transition temperature. Indicate precisely in which concentration range precipitates form by spinodal decomposition and where via regular nucleation. (2.5 points) c) What are the characteristic features of spinodal decomposition and what are the differences compared to precipitation via regular nucleation? (4 points) A10: 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. When the Temperature of the matter is raised, corresponding to an energy increase of (k·T), an electron can leave its energy state to reach a state of higher energy. The probability that a state of energy E is

occupied by an electron is then given by the Fermi-Dirac distribution.This Fermi- Dirac distribution is a function of the temperature T and the Fermi energy 8F: 1 f(E)dE=- dE 1+exp (E-F a Draw the distribution function f(E)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 sr. (3 points) b) Calculate the Fermi temperature Tr,i.e.the temperature corresponding to the Fermi energy sF for Silver (F(Ag)=5.48 ev).(2 points) R=8.3144 J/molK,k=8.6210-5eV/K=1.38-10-23J/K,NL=6.023-1023 Material Constants for Aluminium: G=27x10N/m2,a=4.04A,Tm=660℃，p=2.7g/cm3,A=27.0g/mol,=0.34 YSFE-18x10-2J/m2.YGB-0.6J/m2 Material Constants for Copper: G=48x109NWm2,a=3.61A,Tm=1083℃，p=87gcm3,A=63.5g/mol,V=0.35 YSFE-5x10-2J/m2,YGB-0.5J/m2.YSurface-1J/m2 8

8 occupied by an electron is then given by the Fermi-Dirac distribution. This Fermi￾Dirac distribution is a function of the temperature T and the Fermi energy εF: dE kT E f E dE F ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + = ε 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 εF. (3 points) b) Calculate the Fermi temperature TF, i.e. the temperature corresponding to the Fermi energy εF for Silver (εF(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 Material Constants for Aluminium: G=27×109N/m2, a=4.04Å, Tm=660°C, ρ=2.7g/cm3, A=27.0g/mol, ν=0.34 γSFE=18×10-2J/m2, γGB=0.6J/m2 Material Constants for Copper: G=48×109 N/m2, a=3.61Å, Tm=1083°C, ρ=8.7g/cm3, A=63.5g/mol, ν=0.35 γSFE=5×10-2J/m2, γGB=0.5J/m2, γSurface=1J/m2