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J. B. Puthoff Cal Poly Pomona MTE 207

Problem #1. [10 pts.] You are hardening a steel surface by carburization at 1,000 °C using a hydrocarbon gas. (a) Suppose you know that the carbon concentration drops from 5 at.% at 1 mm depth to 4 at.% at 2 mm depth. Estimate the flux of C atoms in this region. (b) Next, suppose that you are given that the hydrocarbon gas establishes a surface concentration of Cs = 1.0 wt.% C and that the initial, uniform composition of your metal is C0 = 0.2 wt.% C. How long will it take your steel to reach a composition 0.6 wt.% C at a depth of 1 mm? Data: ρFe = 7.63 g/cm3 at 1,000 °C, D0 = 2.0 × 10–5 m2/s, Q = 142 kJ/mol. Hint: You will need a way to calculate erf–1(z), i.e., a table or calculator.

J. B. Puthoff Cal Poly Pomona MTE 207

Problem #2. [15 pts.] You have the following materials available for manufacturing a spherical pressure vessel: Alloy σy [MPa] ρ [g/cm3] price [$/kg] 1040 plain-carbon steel 600 7.80 0.63 304 stainless steel 205 7.80 3.70 3003-H14 aluminum 145 2.73 3.00 Ti-5Al-2.5Sn 827 4.46 15.00 If the stress σ in the (thin) wall of spherical vessel pressurized to pressure p is given by

σ = 𝑝𝑅 2𝑡

, where t is the thickness of the vessel wall and R is its radius. Calculate the mass of a vessel constructed so that the stress in the wall is less than the yield stress, including a safety factor of 3. Which material gives the lightest vessel? Which material gives the cheapest? Hint: for a thin- walled pressure vessel in which t is small, t2 and t3 are ≈ 0.

J. B. Puthoff Cal Poly Pomona MTE 207

Problem #3. [5 pts.] The function

𝑐A(𝑥, 𝑡) = 𝑏

√π𝐷A𝑡 exp [−

𝑥2

4𝐷A𝑡 ]

describes the concentration c (in atom fraction) after diffusion of A in a cylinder of material B which has had a disk of material A welded to one of its ends (see diagram). At what depth x in the cylinder of B does the concentration of A reach 0.1% after time t = 1,000 s at 1,000 K? Data: D0 = 10–10 m2/s, b = 10,000 m, and Q = 20,000 J/mol.

x = 0

t > 0 t = 0

B A cA(x,t)

J. B. Puthoff Cal Poly Pomona MTE 207

Problem #4. [10 pts.] The critical shear stress for dislocation motion τc is a physical parameter that depends on a number of different structural variables, such as the dislocation density ρ. This relationship is sometimes summarized as

τc(ρ) = τ0 + 𝐴√ρ , where τ0 and A are constants. From experiments performed on copper, τc = 0.69 MPa for ρ = 104 mm–2 and τc = 2.0 MPa for ρ = 105 mm–2. What is the yield stress σy of single crystal of copper when ρ = 106 mm–2 and the crystal is loaded along an axis that makes an angle of φ = 65° with the slip plane? (Take the slip direction angle to be λ = 45°.)

J. B. Puthoff Cal Poly Pomona MTE 207

Problem #5. [10 pts.] Label the various phases in this phase diagram (L, α, β, etc.).

Weight percent B → A B

J. B. Puthoff Cal Poly Pomona MTE 207

Problem #6. [10 pts.] A sample of brass was deformed in a tensile test, producing the following data. The initial length of the specimen was ℓ0 = 35 mm and the initial area was A0 = 10.5 mm2. What are the (a) elastic modulus, (b) the 0.2% offset yield stress, and (c) the ultimate tensile stress. What are the true stress and true strain at the point of highest load? Making a plot of these data could help.

Load [N] Extension [mm] 0 0.0000

66 0.0112 177 0.0157 327 0.0199 462 0.0240 797 1.72

1350 5.55 1720 8.15 2220 13.07 2690 22.77 2410 25.25

J. B. Puthoff Cal Poly Pomona MTE 207

Problem #7. [5 pts. Extra Credit] Suppose that you are designing a component that has a volume of 50 cm3. If the total mass of the component is to be no more than 60 g and the elastic modulus must be at least 2.0 GPa, what polymers can be used to fabricate the part?