(1)

Determine the index for the direction EA and the Miller index for the plane ABD shown on the right.

$E = 0a, \ 1b, \ 1c$ $A = 1a, \ 0b, \ \frac{1}{2}c$ $A - E = 1a, \ -1b, \ -\frac{1}{2}c$ $AE = [2\bar{2}\bar{1}]$
$A,B,C = -\frac{3}{4}, \ -1, \ -\frac{1}{2}$ $\frac{na}{A},\frac{nb}{B},\frac{nc}{C} = -n\frac{4}{3}, \ -n, \ -2n$ $n = 3$ $(\bar{4}\bar{3}\bar{6})$

(2)

Please list the indices of all planes that belong to the $\{120\}$ family in a tetragonal crystal with lattice parameters $a = b \ne c$ and $\alpha = \beta = \gamma = 90^{\circ}$.

$[120]$ $[1\bar{2}0]$ $[210]$ $[2\bar{1}0]$

(3)

Please provide the definition of the following terms: isotropy, anisotropy, polymorphism, and allotropy.

Isotropy

Uniform in all directions. An isotropic material's mechanical properties are not dependant on it's orientation.

Anisotropy

An anisotropic material's mechanical properties are dependant on it's orientation.

Polymorphism

The existance of multiple crystalline structures for the same compound.

Allotropy

The existance of multiple crystalline structures for the same element.

(4)

Please list all the crystalline defects that you learned in Chapter 4.

Point Defects

Linear Defects

Interfacial/Planar defects

Volume defects

(5)

Iron (Fe) can have either a face-centered cubic (FCC) structure or a body-centered cubic (BCC) structure. The lattice parameter for FCC Fe and BCC Fe is 0.3571 nm and 0.2866 nm, respectively. Please calculate the radius of Fe in the two structures by assuming the hard sphere model in which the nearest atoms touch each other. Please also calculate the linear density along the close-packed directions in the FCC and BCC Fe.

Fe (FCC)

$a = 0.3571nm$ $a = 2R\sqrt{2}$ (for FCC) $R = \cfrac{a}{2\sqrt{2}}$ $R = 0.1263nm$ $LD = \cfrac{no. \ atoms}{direction \ length}$ Close packed direction for FCC [110] $no. \ atoms = 2$ $Length = a\sqrt{2}$ $LD = \cfrac{\sqrt{2}}{a}$ $LD =3.96 nm^{-1}$

Fe (BCC)

$a = 0.2866 nm$ $a = \cfrac{4R}{\sqrt{3}}$ (for BCC) $R = \cfrac{a\sqrt{3}}{4}$ $R = 0.1241nm$ Close packed direction for BCC [111] $no. \ atoms = 2$ $Length = a\sqrt{3}$ $LD = \cfrac{2}{a\sqrt{3}}$ $LD =4.029 nm^{-1}$

(6)

Please list four strategies for strengthening metallic materials. Briefly describe how these strategies can strengthen materials and provide empirical strengthening relationships. Please also briefly explain why metals with FCC structures are in general more ductile and present better formability than metals with HCP structures.

Reduce Grain Size

Grain boundries are barriers to slip. By reducing the grain size, the number of barriers to slip increases strengthening the material. $$\sigma_{Y} = \sigma_0 + k_y \sqrt{d}$$

Solid Solution

Impurity atoms distort the lattice and create stresses opposing dislocation motion. Increasing the concentration of impurities can strengthen the material. $$\sigma_Y \sim \sqrt{C}$$

Precipitation Strengthening

Hard percipitants can make shearing difficult. As the percipitants may require more stress in order to sheer and or may have a misorientated slip planes. Introducing percipitants can strengthen a material. Percipitants act as "pinning" sites with spacing S. $$\sigma_Y \sim \frac{1}{S}$$

Cold Work

Cold work creates dislocations within the sample. The higher the dislocation density within the material, the harder it is for further dislocations to occur. $$\tau_{CRSS} = \tau_0 + A\sqrt{\rho_d}$$

Ductility of FCC vs HCP

The FCC crystalline structure generallly has more slip systems than HCP structure. More slip sytems increases the propensity for dislocation motion. Ductility is dependant on plastic derformation (i.e. dislocation motion). In general metals with FCC will be more ductile and formable than metals with HCP as FCC metals have more slip systems.

(7)

Annealing of cold worked metals can lead to recovery, recrystallisation and grain growth. Please describe the three phenomena and explain the driving force of the phenomena. Also briefly mention how these phenomena affect the mechanical properties of the metals.

Recovery

The process in which dislocation density is reduced by dislocation annihilation. The driving force for recovery is reducing the energy of the system. Decreasing dislocation density will weaken the material and increase its ductility.

Recrystallisation

The process in which new crystal grains are formed from the existing grains. Complete once only new crystal grains remain. The driving force for recrystallisation is atomic diffusion, which requires high temperature. Although dislocation density will be significantly reduced the number of grains will increase causing more slip boundries.

Grain Growth

The crystals formed in the recrystallisation proccess will begin to grow, consuming smaller crystals. Grain boundary area and the energy of the system is reduced, making dislocation motion easier. The driving force for grain growth is atomic diffusion, which requires high temperature. With increased grain size the meterial will weaken and become more ductile.