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.Letchange 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