1.Intergranular fracture of steels can occur under a variety of metallurgical conditions.Discuss in detail three distinct causes of Intergranular fracture. Describe the characteristics of the various modes by which a structural transformation that involves no compositional change can take place in binary metallic alloys. 3. (a)What is the mechanism by which dislocations surmount the Peierls barrier in a crystal?What assumptions are involved in describing the process?What are the reasons for the Peierls barrier being effective at successively higher values of T/Tm,where Tm is the melting temperature,for fcc metals,bcc metals,and diamond-cubic semi-metals,respectively? (b)Why does the effective line tension differ for a bowing-out,initially screw-oriented dislocation and screw dislocation decreasing in length?Which line tension is larger? 4. a)Discuss in detail the operation of Frank-Read source for dislocation segment pinned by(a)super jogs and(b)5 um second phase particles.Is this model applicable in real systems?Has it been observed experimentally?What approximations are involved in the quantitative description of the process? b)For what types of metals and material and under what conditions of system variables is the Peierls barrier important in metal deformation?What is the specific mechanism for its influence on mechanical properties? 5.(a)Discuss the process of intersection of two perpendicular screw dislocations. In what temperature and strain regime would you expect this process to be important?Why? (b)What is the Peach-Koehler equation for the force on a dislocation?Illustrate its use dor the case of glide interaction of two edge dislocation with the same slip system but whose glide planes are separated by a normal distance h. 6. Discuss critically,proposed mechanisms for the initiation of the cellular microstructure in discontinuous precipitation reactions. 7.The individual particles of atomized aluminum and titanium alloys are coated with a thin oxide film.Discuss the effects of the oxide film on the processing (cold compaction,hot pressing,extrusion,welding)and mechanical properties of the consolidated powder. 8.Discuss in detail the process of cross-slip in a brass.For an exact calculation of the energy barrier,describe how you would proceed.Illustrate your best guess of the critical configuration.How would the process depend on dislocation character?
1. Intergranular fracture of steels can occur under a variety of metallurgical conditions. Discuss in detail three distinct causes of Intergranular fracture. 2. Describe the characteristics of the various modes by which a structural transformation that involves no compositional change can take place in binary metallic alloys. 3. (a) What is the mechanism by which dislocations surmount the Peierls barrier in a crystal? What assumptions are involved in describing the process? What are the reasons for the Peierls barrier being effective at successively higher values of T/Tm, where Tm is the melting temperature, for fcc metals, bcc metals, and diamond-cubic semi-metals, respectively? (b) Why does the effective line tension differ for a bowing-out, initially screw-oriented dislocation and screw dislocation decreasing in length? Which line tension is larger? 4. a) Discuss in detail the operation of Frank-Read source for dislocation segment pinned by (a) super jogs and (b) 5 μm second phase particles. Is this model applicable in real systems? Has it been observed experimentally? What approximations are involved in the quantitative description of the process? b) For what types of metals and material and under what conditions of system variables is the Peierls barrier important in metal deformation? What is the specific mechanism for its influence on mechanical properties? 5. (a) Discuss the process of intersection of two perpendicular screw dislocations. In what temperature and strain regime would you expect this process to be important? Why? (b) What is the Peach-Koehler equation for the force on a dislocation? Illustrate its use dor the case of glide interaction of two edge dislocation with the same slip system but whose glide planes are separated by a normal distance h. 6. Discuss critically, proposed mechanisms for the initiation of the cellular microstructure in discontinuous precipitation reactions. 7. The individual particles of atomized aluminum and titanium alloys are coated with a thin oxide film. Discuss the effects of the oxide film on the processing (cold compaction, hot pressing, extrusion, welding) and mechanical properties of the consolidated powder. 8. Discuss in detail the process of cross-slip inα brass. For an exact calculation of the energy barrier, describe how you would proceed. Illustrate your best guess of the critical configuration. How would the process depend on dislocation character?
9 (a)The introduction of discrete defects into an otherwise elastic solid containing a large sharp crack can either increase or decrease the apparent fracture toughness of the solid.Describe the physical basis common to both effects. (b)What effect will a dislocation withb=10=001]have on the Mode I stress intensity factor of a crack on the (010)plane with crack front parallel to [0011.This dislocation is located directly ahead (=0)of the crack.Given: cos -sin e sin 301 √2πr 2 2 2 e. 0.38 022=- -cos-[l+sin。sin 2πr 2 2 62 K,-sin号cos2co .0030 2πr2 2 2 (c)Ignoring the areal fraction contribution,will an array of small internally pressurized bubbles,uniformly distributed in the matrix,affect the fracture toughness of steel below the NDT?Explain. 10.Discuss the motion of the dislocations shown in each of cases below.Be as quantitative as possible,discussing the particular path of motion the dislocation will take,showing detailed expressions where possible, explaining your steps,and interpreting your results.Assume that the direction of dislocation motion coincides with direction of dislocation motion coincides with the direction of force acting on the dislocation. Hint:The Peach-Koehler equation defining the force exerted on a unit length of dislocation line with direction s and Burgers vector b,due to an external stress fieldo,is: R =(o·b)×5 L Case(a) A specimen of uniform thickness with a dislocation located at position (xl=a, x2=b)from a corner,and defined by b=bes 5=e3
9. (a) The introduction of discrete defects into an otherwise elastic solid containing a large sharp crack can either increase or decrease the apparent fracture toughness of the solid. Describe the physical basis common to both effects. (b) What effect will a dislocation with , have on the Mode I stress intensity factor of a crack on the (010) plane with crack front parallel to [001]. This dislocation is located directly ahead ( b b [ ] 100 → = ξ [001] → = θ = 0) of the crack. Given: 11 22 33 3 cos [1 sin sin ] 2 2 2 3 cos [1 sin sin ] 2 2 2 3 sin cos cos 2 22 2 I I I K r K r K r 2 2 θ θ θ σ π θ θ θ σ π θθ θ σ π = − = + = (c) Ignoring the areal fraction contribution, will an array of small internally pressurized bubbles, uniformly distributed in the matrix, affect the fracture toughness of steel below the NDT? Explain. 10. Discuss the motion of the dislocations shown in each of cases below. Be as quantitative as possible, discussing the particular path of motion the dislocation will take, showing detailed expressions where possible, explaining your steps, and interpreting your results. Assume that the direction of dislocation motion coincides with direction of dislocation motion coincides with the direction of force acting on the dislocation. Hint: The Peach-Koehler equation defining the force exerted on a unit length of dislocation line with direction ξ and Burgers vector b, due to an external stress fieldσ , is: ( ) F b L = σ ⋅ ×ξ Case (a) A specimen of uniform thickness with a dislocation located at position (x1=a, x2=b) from a corner, and defined by 3 3 ˆ ˆ b be ξ e = =
。(a,b) e1 Free Surfaces Case(b) A specimen thinned down in the e,~direction,under tensile loading o.The dislocation is very far from all surfaces,except the upper and lower surfaces into which the dislocation runs.The dislocation and stress field are defined by: b=be 5=e =e(ie.,is the only nonzero component of stress) 3-D view applied stress 2 front view showing important diΠiensi0ns 11. Describe the role of stress(static and cyclic)in metallic corrosion phenomena. Discuss the use of fracture mechanics. 12. There is a long-standing interest in adding ceramic "particles"to metal matrices.Some examples of such commercial materials are WC-Co cermets, Al-SiC composites,and TD-nickel(thoria dispersed nickel).Describe in detail the various strengthening mechanisms that are possible in such systems(at both low and high temperature).Clearly delineate the ranges of particle size and volume fraction at which each mechanism operates.Also describe how the material behavior changes when such particles are added. 13. With the aid of suitable diagrams,deduce the types of monoclinic Bravais
Case (b) A specimen thinned down in the ~direction, under tensile loading 3 eˆ σ . The dislocation is very far from all surfaces, except the upper and lower surfaces into which the dislocation runs. The dislocation and stress field are defined by: 1 2 ˆ ˆ b be ξ e = = 1 1 σ = σe eˆ ˆ (i.e., σ11 is the only nonzero component of stress) 11. Describe the role of stress (static and cyclic) in metallic corrosion phenomena. Discuss the use of fracture mechanics. 12. There is a long-standing interest in adding ceramic "particles" to metal matrices. Some examples of such commercial materials are WC-Co cermets, Al-SiC composites, and TD-nickel (thoria dispersed nickel). Describe in detail the various strengthening mechanisms that are possible in such systems (at both low and high temperature). Clearly delineate the ranges of particle size and volume fraction at which each mechanism operates. Also describe how the material behavior changes when such particles are added. 13. With the aid of suitable diagrams, deduce the types of monoclinic Bravais
lattices which may be obtained from the stacking of two-dimensional lattices. 14.Write an essay which describes and explains the effects due to grain boundaries during precipitation from solid solution. 15.Consider a straight edge dislocation in a crystal.Discuss the types of forces that can operate on the dislocation.For one type of force,give an explicit answer in terms of the Peach-Koehler formula. 16.Discuss in detail the classification of types of kinks on dislocations,the physical basis for their presence on dislocations,and macroscopic properties that indicate the presence of kinks.Give an expression for the velocity of a dislocation moving under a rate-controlling process of kink formation. 17. Consider cleavage fracture of a single crystal of LiF.The crack plane Is the (001)and the crack front-lies parallel to the [100]direction.The surface energy of LiF Is .35 j/m and the slip systems are a/2 {110).Assume cross slip does not occur. (a)Assuming Jc >2y,predict the characteristics(size,shape,dislocation makeup) of the plastic zone as the applied J Increases from zero to above 2Y.Also, describe any distinguishing differences between the surface zone and the midthickness zone. (b)If the material is altered in some way such that J>2Y,what happens to the plastic zone? (c)Assuming that J=2y (exactly),sketch and describe the loading/unloading load-displacement curve for a cracked single crystal DCB specimen with an Initial slot of length a(and"a"can't be less than a )In constructing your sketch assume that crack extension occurs. 18. Discuss the microstructural basis for the Orowan-Friedel relation.Consider the validity of the relation in view of the approximations involved In its derivation.For what types of microstructural features should the relation apply? 15.(a)Symmetry axes are sometimes described in terms of rotation axes and rotation-reflection axes.rather than rotation axes and inversion axes.The operation of a rotation-reflection axis Sn consists of a rotation through 27/n followed by reflection in a plane normal to the rotation-reflection axis.Sketch stereograms showing the operation of 1,2,3,4,and 6-fold reflection axes on a single pole in a general position.Deduce the inversion axis to which each is equivalent. (b)Pure Au and pure Cu are both cubic with atoms at 0,0,0;0,1/2,1/2;1/2,0,1/2; 1/2,1/2,0.The compound Cu-Au is cubic with an Au atom at 0,0,0 and Cu atoms at 0,1/2,1/2;1/2,0,1/2;1/2,1/2,0.Determine the lattice type of Au,Cu,and Cu3Au,and in each case state the number of atoms associated with each lattice
lattices which may be obtained from the stacking of two-dimensional lattices. 14. Write an essay which describes and explains the effects due to grain boundaries during precipitation from solid solution. 15. Consider a straight edge dislocation in a crystal. Discuss the types of forces that can operate on the dislocation. For one type of force, give an explicit answer in terms of the Peach-Koehler formula. 16. Discuss in detail the classification of types of kinks on dislocations, the physical basis for their presence on dislocations, and macroscopic properties that indicate the presence of kinks. Give an expression for the velocity of a dislocation moving under a rate-controlling process of kink formation. 17. Consider cleavage fracture of a single crystal of LiF. The crack plane Is the (001) and the crack front-lies parallel to the [100] direction. The surface energy of LiF Is .35 j/m and the slip systems are a/2 {110}. Assume cross slip does not occur. (a) Assuming Jc > 2y , predict the characteristics (size, shape, dislocation makeup) of the plastic zone as the applied J Increases from zero to above 2Y. Also, describe any distinguishing differences between the surface zone and the midthickness zone. (b)If the material is altered in some way such that J→2Y, what happens to the plastic zone? (c) Assuming that J = 2y (exactly) , sketch and describe the loading/unloading load-displacement curve for a cracked single crystal DCB specimen with an Initial slot of length a (and “a” can't be less than a ). In constructing your sketch assume that crack extension occurs. 18. Discuss the microstructural basis for the Orowan-Friedel relation. Consider the validity of the relation in view of the approximations involved 1n its derivation. For what types of microstructural features should the relation apply? 15. (a) Symmetry axes are sometimes described in terms of rotation axes and rotation-reflection axes, rather than rotation axes and inversion axes. The operation of a rotation-reflection axis Sn consists of a rotation through 2π /n followed by reflection in a plane normal to the rotation-reflection axis. Sketch stereograms showing the operation of 1, 2, 3, 4, and 6-fold reflection axes on a single pole in a general position. Deduce the inversion axis to which each is equivalent. (b) Pure Au and pure Cu are both cubic with atoms at 0,0,0; 0,1/2,1/2; 1/2,0,1/2; 1/2,1/2,0. The compound Cu-Au is cubic with an Au atom at 0,0,0 and Cu atoms at 0,1/2,1/2; 1/2,0,1/2; 1/2,1/2,0. Determine the lattice type of Au, Cu, and Cu3Au, and in each case state the number of atoms associated with each lattice
point. 16. Discuss the influence of kinks and jogs on dislocation motion.Qualitatively compare the regimes of temperature where these defects could be important in metals and in cubic (Ge and Si)materials 17. Consider the two dislocations shown in the figure below.Dislocation A is at the origin of global Cartesian coordinate system (x),and dislocation B is positioned at x1=2,x2=1.A second Cartesian coordinate system (x')has origin at dislocation B,and is oriented relative to the global coordinate system by a counterclockwise rotation of 45 about die x3-axis.The two dislocations have Burgers vector b and dislocation line g;described by: b4=2[100] Referred to the global (x) 54=-[00则 coordinate system. b=[100] Referred to the local (x') B=1[001] coordinate system. Use me pictorial,mathematical,and textual information to answer the following questions: (a)What are the components of the stress field caused by dislocation A at the dislocation B position,as referred to global(x)coordinate system? (b)What are the components of me stress field caused by dislocation A at the dislocation B position,as referred to me local(x')coordinate system? (c)What are the glide and climb directions of dislocation B,as referred to the local (x')coordinate system? (d)What arc the glide and climb forces,as referred to the local (x')coordinate system,that act on dislocation B due to the presence of dislocation A?Show your basic equations and steps carefully. 18. A thin,linear-elastic film of thickness h is deposited on a thick,linear-elastic substrate.A two-dimensional representation of the assembly is shown below. free surface film interface substrate The stress-free lattice spacing af of the film in the e^-direction is less than the corresponding spacing as in the substrate,and this produces a uniform tensile coherency strain(as-af)/af in the film when the interface is coherent.The thickness h is small enough to assume uniform stresses and strains through the thickness of the film.Further,the whole sample is very thick in thein-
point. 16. Discuss the influence of kinks and jogs on dislocation motion. Qualitatively compare the regimes of temperature where these defects could be important in metals and in cubic (Ge and Si) materials. 17. Consider the two dislocations shown in the figure below. Dislocation A is at the origin of global Cartesian coordinate system (x), and dislocation B is positioned at x1=2, x2=l. A second Cartesian coordinate system (x') has origin at dislocation B, and is oriented relative to the global coordinate system by a counterclockwise rotation of 45° about die x3-axis. The two dislocations have Burgers vector b and dislocation line ξ ; described by: 2[100] -l[00l] [100] 1[001] A A B B b b ξ ξ = = = = Referred to the global (x) coordinate system. Referred to the local (x') coordinate system. ⎧ ⎨ ⎩ ⎧ ⎨ ⎩ Use me pictorial, mathematical, and textual information to answer the following questions: (a) What are the components of the stress field caused by dislocation A at the dislocation B position, as referred to global (x) coordinate system? (b) What are the components of me stress field caused by dislocation A at the dislocation B position, as referred to me local (x') coordinate system? (c) What are the glide and climb directions of dislocation B, as referred to the local (x') coordinate system? (d) What arc the glide and climb forces, as referred to the local (x') coordinate system, that act on dislocation B due to the presence of dislocation A? Show your basic equations and steps carefully. 18. A thin, linear-elastic film of thickness h is deposited on a thick, linear-elastic substrate. A two-dimensional representation of the assembly is shown below. The stress-free lattice spacing af of the film in the e^-direction is less than the corresponding spacing as in the substrate, and this produces a uniform tensile coherency strain 11 ε = (as-af)/af in the film when the interface is coherent. The thickness h is small enough to assume uniform stresses and strains through the thickness of the film. Further, the whole sample is very thick in the 3e -in-
direction(pointed out of the page). (a)Estimate per unit area of interface the total elastic energy contained in the thin film as a function of af,as,h,and Young's modulus E and Poisson's ratio v, both of which are assumed to be the same for the film and substrate. (b)Edge disocationswith linedirectionpnted ou ofthe paper in the in direction and Burgers vectorpointed in the-direction are observed to form along the interface as shown below.Using additional information given below,provide a relation between the equilibrium distance B separating adjacent dislocations,the film thickness h,Burgers vector magnitude b,stress-free lattice spacings af and as,cutoff Ra,and elastic parameters E and Y. Make a reasonable assumption as to include only the most important interaction energies between dislocations,and to neglect less important interaction terms.Outline and explain your steps clearly along the way. free surface B subs女ate Additional information to solve the problem is given below: (i)Self energy per unit length of a straight edge dislocation with Burgers vector magnitude b in an infinite body described by elastic shear modulus 4=E/21+y): (ii)Interaction energy per unit length between two parallel dislocation lines described by individual Burgers vectors b and b,and common line direction. The vector R denotes the perpendicular distance between the two dislocation lines and Ra is taken to be a constant:: 座_4渔望R-,”×3色xR2 L2π R21-y） R21-R×9R飞×9风 19.(a)What are tilt and twist boundaries,and how are they formed from a single crystal? (b)Show that the interfacial dislocation network formed between the same two misoriented crystals may be either tilt,twist,or mixed,without any
direction (pointed out of the page). (a) Estimate per unit area of interface the total elastic energy contained in the thin film as a function of af, as, h, and Young's modulus E and Poisson's ratio v, both of which are assumed to be the same for the film and substrate. (b) Edge dislocations with line direction ξ pointed out of the paper in the indirection and Burgers vector b pointed in the 3e -direction are observed to form along the interface as shown below. Using additional information given below, provide a relation between the equilibrium distance B separating adjacent dislocations, the film thickness h, Burgers vector magnitude b,stress-free lattice spacings af and as,cutoff Ra, and elastic parameters E and γ . Make a reasonable assumption as to include only the most important interaction energies between dislocations, and to neglect less important interaction terms. Outline and explain your steps clearly along the way. Additional information to solve the problem is given below: (i) Self energy per unit length of a straight edge dislocation with Burgers vector magnitude b in an infinite body described by elastic shear modulus μ = + E / 2(1 ) γ : 1 2 2 Ws b L = μ (ii) Interaction energy per unit length between two parallel dislocation lines described by individual Burgers vectors 1b and 2 b and common line directionξ . The vector R denotes the perpendicular distance between the two dislocation lines and Ra is taken to be a constant:: 12 1 2 1 2 2 1 2 ( )( )ln [( )( )]ln [( ) ][( ) ] 2 2 (1 ) 2 (1 ) a a W b b R R b b b Rb R L R RR μ ξ ξ μ μ ξ ξ ξ ξ π πγ πγ ⋅ ⋅ = −× ×− × ⋅ × ⋅ − − 19. (a) What are tilt and twist boundaries, and how are they formed from a single crystal? (b) Show that the interfacial dislocation network formed between the same two misoriented crystals may be either tilt, twist, or mixed, without any
change in misorientation. (c)A dislocation network between two misoriented crystals is made up of three families of dislocations having non-coplanar Burgers vectors,b1,b2, and b3,and line directions rl,r2,and r3,respectively.If Y is a unit vector normal to the boundary plane,and u is a unit vector along the rotation axis about which the misorientation occurs,show that if n2b2+nb3=r×1w (1) V=0 (2) (where ni is the number of dislocations of set bi,and w=u)then the line direction of the dislocation set 1 is: 片=[u×(b2×b]×y with similar expressions for r2 and r3. 20. (a)Prove that the reciprocal lattice for FCC is BCC,and vice versa. (b)Derive a condition for defining the boundary faces for the first Brillouin zone(BZ)for a given crystal structure. (c)Describe the shape of the first BZ for an FCC crystal.Show that it has the same shape as the Wigner-Seitz zone for a BCC crystal.What is the significance of each? 21. What is the physical basis for the stress concentrations of a pileup?Outline (do not derive in detail)the derivation of an expression for the stress concentration.Compare the magnitude of the pileup stress at the tip for three cases: (a)a grain boundary in pure aluminum, (b)a grain boundary in an Al-4%Ge alloy (low stacking fault energy),and (c)the aluminum phase in a laminar composite of Al and Al203. In all three cases the local resolved applied stress on the glide plane is the same. Discuss your answer. 22. (a)A monocrystalline slab of bcc material is oriented so that the x,y,and z axes are parallel to [100],[010]and [001],respectively.If the slab is subjected to balanced biaxial deformation(ex=ey =e),then the elastic strain energy per unit volume We may be expressed as follows: wel=Ye2. Express the biaxial modulus Y in terms of the elastic constants Cu and C12. (b)Consider a two-dimensional crystal with point group symmetry 4mm.Let
change in misorientation. (c) A dislocation network between two misoriented crystals is made up of three families of dislocations having non-coplanar Burgers vectors, b1, b2, and b3, and line directions r1, r2, and r3, respectively. If γ is a unit vector normal to the boundary plane, and u is a unit vector along the rotation axis about which the misorientation θ occurs, show that if 22 33 1 nb nb r w + = × (1) 0 ir ⋅ν = (2) (where ni is the number of dislocations of set bi, and w = uθ ) then the line direction of the dislocation set 1 is: 1 23 r ubb = [ ( )] ×× ×ν with similar expressions for r2 and r3. 20. (a) Prove that the reciprocal lattice for FCC is BCC, and vice versa. (b) Derive a condition for defining the boundary faces for the first Brillouin zone (BZ) for a given crystal structure. (c) Describe the shape of the first BZ for an FCC crystal. Show that it has the same shape as the Wigner-Seitz zone for a BCC crystal. What is the significance of each? 21. What is the physical basis for the stress concentrations of a pileup? Outline (do not derive in detail) the derivation of an expression for the stress concentration. Compare the magnitude of the pileup stress at the tip for three cases: (a) a grain boundary in pure aluminum, (b) a grain boundary in an Al-4% Ge alloy (low stacking fault energy), and (c) the aluminum phase in a laminar composite of Al and Al2O3. In all three cases the local resolved applied stress on the glide plane is the same. Discuss your answer. 22. (a) A monocrystalline slab of bcc material is oriented so that the x, y, and z axes are parallel to [100], [010] and [001], respectively. If the slab is subjected to balanced biaxial deformation (ex = ey = e), then the elastic strain energy per unit volume Wel may be expressed as follows: Wel = Ye2 . Express the biaxial modulus Y in terms of the elastic constants C11 and C12. (b) Consider a two-dimensional crystal with point group symmetry 4mm. Let
an axis system be chosen so that X1//[10]and x2//[01].The elastic constants of the crystal may be represented by a fourth-rank tensor Cik;i,j, k,1=1,2.Because of symmetry the elastic constants may be represented by a symmetric matrix Cmn Cnm,m,n=1,2,3,where a pair of consecutive indices in Ciikl *s reduced to a single index in Cmn according to the following rules: 11->122->2120r21->3 The resulting matrix Cmn has the form shown below: 0 C21 0 00C3 Suppose that a new axis system(xI',x2)is used,which is related to the old one by a45°rotation. (1)Express C in the new axis system in terms of Cu,Ci2 and C33 (2)From the expression in (1)obtain a relation which must hold if the crystal is elastically isotropic. 23.a)A metallographic specimen of pure lead is mechanically polished and then etched .If this procedure is repeated several times,a different microstructure is observed after each etch.Explain this behavior and specify a metallographic procedure which would yield the actual microstructure of lead. b)An austenitic 304 stainless steel contains a small quantity of body-centered cubic martensite which is difficult to recognize in the microstructure if one employs the standard metallographic procedures.Assuming that one is restricted to optical microscopy specify a metallographic technique which would delineate the martensite in the microstructure. 24.Two FCC crystal are rotated 1/2 about [112],and have a lattice parameter of 4A.If the boundary is symmetric tilt boundary made up of edge dislocations, and the tilt angle is 1/2. i)What is the boundary plane? ii)What is the length of dislocation line per unit area of boundary? iii)Calculate the angle at which the individual dislocation making up the boundary become indistinguishable. iv)Show that these dislocation are stable with respect to displacement of any one of them on its slip plane. 25. a)Discuss and compare slip and twinning in alpha iron. b)Assuming that the twin plane is(112)and that the twinning direction is [111], draw a two-dimensional diagram which shows the atom movements required to
an axis system be chosen so that X1 // [10] and x2 // [01]. The elastic constants of the crystal may be represented by a fourth-rank tensor Cijkl; i, j, k, 1=1,2. Because of symmetry the elastic constants may be represented by a symmetric matrix Cmn = Cnm , m, n=l, 2, 3, where a pair of consecutive indices in Cijkl *s reduced to a single index in Cmn according to the following rules: 11->1 22->2 12 or 21-> 3 The resulting matrix Cmn has the form shown below: 11 12 21 22 33 0 0 0 0 C C C C C ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ Suppose that a new axis system (x1',x2') is used, which is related to the old one by a 45° rotation. (1) Express in the new axis system in terms of C '' C11 11, C12 and C33. (2) From the expression in (1) obtain a relation which must hold if the crystal is elastically isotropic. 23. a) A metallographic specimen of pure lead is mechanically polished and then etched .If this procedure is repeated several times, a different microstructure is observed after each etch. Explain this behavior and specify a metallographic procedure which would yield the actual microstructure of lead. b) An austenitic 304 stainless steel contains a small quantity of body-centered cubic martensite which is difficult to recognize in the microstructure if one employs the standard metallographic procedures. Assuming that one is restricted to optical microscopy specify a metallographic technique which would delineate the martensite in the microstructure. 24. Two FCC crystal are rotated 1/2 about [112], and have a lattice parameter of 4A. If the boundary is symmetric tilt boundary made up of edge dislocations, and the tilt angle is 1/2. ⅰ) What is the boundary plane? ⅱ) What is the length of dislocation line per unit area of boundary? ⅲ) Calculate the angle at which the individual dislocation making up the boundary become indistinguishable. ⅳ) Show that these dislocation are stable with respect to displacement of any one of them on its slip plane. 25. a) Discuss and compare slip and twinning in alpha iron. b) Assuming that the twin plane is (112) and that the twinning direction is [111− ], draw a two-dimensional diagram which shows the atom movements required to
form a twin in alpha iron. c)How does one distinguish slip bands and mechanical twins in the microstructure of alpha iron? 26 a)In term of the lattice parameters a,b,c,what is the interplanar spacing d for the following cases: (i)(121)planes,cubic (ii)(211)planes,cubic (iii)(12 1)planes,orthorhombic (iv)(211)planes,orthorhombic b)What is the linear density of atoms (i.e.,number of atoms per unit length along a line )along one of the close-packed direction in a bcc crystal? c)What is the planar density of atoms (i.e.,number of atoms per unit area of plane)for the most nearly close packed plane in a bcc crystal? 27.a)Discuss in detail the process of intersection of a gliding dislocation in alpha-brass with two (forest)dislocations lying on an intersecting glide plane and whose burgers vectors are also inclined to the glide plane of the moving dislocation. b)What is the importance of the Peierls barrier in controlling mechanical properties?What experimental evidence is there for the existence of such a barrier? 28.Suppose a binary alloy of aluminum in nickel was solution treated,quenched into the Ni3Al-Ni(Al)phase field and aged at a temperature adequate for the solid solution to decompose rapidly.Given that Ni(Al)is a random fee solid solution,Ni3Al is an ordered fee phase,and that the lattice parameters of Ni3Al and Ni(Al)are very similar yet not the same,answer the following questions: (a)What microstructure would you expect to see shortly after the beginning of the aging reaction and why would you expect to see it? (b)What changes (in detail,paying attention to the nature of interfaces) would you expect to occur within the microstructure as aging continues indefinitely.Give explanations. 29. Describe,with particular attention to the contrasts among,the various modes by which transformation of structure but not of composition can be affected in solid alloys You are given a series of micrographs labeled in order,A through F. a)Which of the micrographs shows evidence of or influence of (1)Twining (2)Plastic Deformation
form a twin in alpha iron. c) How does one distinguish slip bands and mechanical twins in the microstructure of alpha iron? 26 a) In term of the lattice parameters a, b, c, what is the interplanar spacing d for the following cases: ( ) (1 ⅰ 2 − 1) planes, cubic ( ) (211) planes, cubic ⅱ ( ) (1 ⅲ 2 − 1) planes, orthorhombic ( ) (211) planes, orthorhombic ⅳ b) What is the linear density of atoms (i.e., number of atoms per unit length along a line ) along one of the close-packed direction in a bcc crystal? c) What is the planar density of atoms (i.e., number of atoms per unit area of plane) for the most nearly close packed plane in a bcc crystal? 27. a) Discuss in detail the process of intersection of a gliding dislocation in alpha-brass with two (forest) dislocations lying on an intersecting glide plane and whose burgers vectors are also inclined to the glide plane of the moving dislocation. b) What is the importance of the Peierls barrier in controlling mechanical properties? What experimental evidence is there for the existence of such a barrier? 28. Suppose a binary alloy of aluminum in nickel was solution treated, quenched into the Ni3Al-Ni(Al) phase field and aged at a temperature adequate for the solid solution to decompose rapidly. Given that Ni(Al) is a random fee solid solution, Ni3Al is an ordered fee phase, and that the lattice parameters of Ni3Al and Ni(Al) are very similar yet not the same, answer the following questions: (a) What microstructure would you expect to see shortly after the beginning of the aging reaction and why would you expect to see it? (b) What changes (in detail, paying attention to the nature of interfaces) would you expect to occur within the microstructure as aging continues indefinitely. Give explanations. 29. Describe, with particular attention to the contrasts among, the various modes by which transformation of structure but not of composition can be affected in solid alloys You are given a series of micrographs labeled in order, A through F. a) Which of the micrographs shows evidence of or influence of (1) Twining (2) Plastic Deformation
(3)Transformation form a liquid (4)Interfacial energy (5)Texture (6)Heterogeneous nucleation (7)Solid state diffusion Note:You may use A,B,C,etc.,more than once,if appropriate. b)Briefly state what each material could be,and briefly describe a plausible history for each(i.e.,how it could have got that way)
(3) Transformation form a liquid (4) Interfacial energy (5) Texture (6) Heterogeneous nucleation (7) Solid state diffusion Note: You may use A, B, C, etc., more than once, if appropriate. b) Briefly state what each material could be, and briefly describe a plausible history for each (i.e., how it could have got that way)