(a) Determine the principal stresses. (b) Determine the unit vectors defining the principal directions. (c) Determine whether yielding is predicted according to the Tresca criterion. (d) Determine whether yielding is predicted according to the von Mises criterion.

P.1.5. A shear stress ty acts in a two-dimensional field in which the maximum allowable shear stress is denoted by Tmax and the major principal stress by o1. Derive, using the geometry of Mohr's circle of stress, expressions for the maximum values of direct stress which may be applied to the x and y planes in terms of these three parameters. \text { Answer: } \quad \sigma_{x}=\sigma_{I}-\tau_{\max }+\sqrt{\tau_{\max }^{2}-\tau_{x y}^{2}} \sigma_{y}=\sigma_{I}-\tau_{\max }-\sqrt{\tau_{\max }^{2}-\tau_{x y}^{2}}

In homogeneous nucleation, below the melting temperature the free energy of the solid phase (Gsolid) is:greater than, less than, or equal to (circle one) the free energy of the liquid phase (Griquia). [1 pt)

7. The d311 inter planar spacing in an FCC metal is 0.1218 nm. (a) What is its lattice constant a? (b) What is the atomic radius of the metal? (c) What could this metal be? (6 points)

10. (15 pts) Utilizing the information we have discussed all semester, as well as the information in your text,choose the most appropriate material for the following application. You need to design an economical internal combustion (IC) engine exhaust valve that must withstand temperatures greater than 500 °F, has a yield strength greater than 300 MPa, has a hardness of at least 200 HB, and has a moderate wear resistance. Narrow it down as-far you reasonably can (be very specific in heat treatment, alloy details, etc., listing a generie material like "steel"will earn you no points) and justify all decisions based on referencing specific figures/data in your text. Place your answer in the box below to receive full credit.

5. Sketch the following planes: (001), (100), (111), (100), (342).

1. In an extrusion process, the effect of cross-sectionalshape on pressure can be assessed by means of the dieshape factor K,, please calculate the shape factor of theright part. Assume the cross-sectional area is 314 cm? andits perimeter is 200cm, respectively. (6')

9. (15 pts total) Here you are going to put your knowledge learned in class to the test. You have been given a mystery material that you must identify. Just like in lab you have the capability to run a tensile test, hardness test,charpy impact test, or a jominy bar test, as well as subjecting the material to a series of heat-treating steps. ) Briefly describe each test and what the results are (i.e. what the test tells you). b) (2 pts) If you were restricted to only two tests, which would you choose and why (be specific on what you are looking for)? c) (3 pts) For the sake of argument say you chose a tensile test and calculated your Young's modulus as 193 GPa,what does this tell you about the material type (be specific)? What other piece of information from this test would be needed to narrow down your options? d) (3 pts) If you ran a hardness test on the same sample from part (c) and found a hardness of 201 HB, could you narrow down your material choices even more? (If you can – you must explain how to receive credit, if you can't please explain why.) e) (4 pts) Continuing with the sample from part (d) if a tensile test showed a yield strength of 515 MPa and a tensile strength of 860 MPa can you identify your material? (If you can - be as specific as you can, must list material along with its heat-treating. If you cannot - be very clear as to what else would be needed and how you would find it.)

8. For the solidification of iron, calculate the critical radius r* and the activation free energy AG* if nucleation is homogeneous. Values for the latent heat of fusion and surface free energy are -1.85 × 10ºJ/m³ and 0.204 J/m², respectively. Use the supercooling value found in Table 10.1. (4pt)

4. (a) List the three primary classifications of solid materials,and then cite the distinctive chemical feature of each.(b) Note the four types of advanced materials and, for each, its distinctive feature(s).