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 toan 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