ENGINEERING 86 – HOMEWORK #1

Problem 6.21

(a) A cylindrical metal specimen 15 mm in diameter and 150 mm long is to be subjected to a tensile stress of 50 MPa; at this stress level, the resulting deformation will be totally elastic.  If the elongation must be less than 0.072 mm, which of the metals in Table 6.1 of the textbook are suitable candidates? Why?

[WRITE YOUR SOLUTION ON THIS SIDE OF THE PAGE. ANYTHING WRITTEN ON THE BACK WILL NOT BE GRADED.]

Problem 6.21

(b) If, in addition, the maximum permissible diameter decrease is 0.0023 mm when the tensile stress of 50 MPa is applied, which of the metals that satisfy the criterion in part (a) are suitable candidates? Why?

Problem 6.31

A specimen of magnesium having a rectangular cross section of dimensions 3.2 mm X 19.1 mm is deformed in tension. Using the load–elongation data shown in the following table, complete parts (a) through (f).

(a)  USING GRAPHING SOFTWARE (e.g. Excel, Google Sheets), plot the data as engineering stress versus engineering strain.  Replace page 5 of this document with a page containing this plot. Answer parts (b) through (f) on page 6.

(b) Compute the modulus of elasticity.

(c)  Determine the yield strength at a strain offset of 0.002.

(d)  Determine the tensile strength of this alloy. (e)  Compute the modulus of resilience.

(f)  What is the ductility, in percent elongation?

Problem 6.31

Write your solutions to parts (b) - (f) on this page.

Problem 6.42

Show that Equations 6.18a and 6.18b in the textbook are valid when there is no volume change during deformation.

Problem 6.48

For a brass alloy, the following engineering stresses produce the corresponding plastic engineering strains prior to necking:

Based on this information, compute the engineering stress necessary to produce an engineering strain of 0.28.

Problem 8.14

The following tabulated data were gathered from a series of Charpy impact tests on a commercial low-carbon steel alloy.

Problem 8.34

A cylindrical component 50 mm long constructed from an S-590 alloy is to be exposed to a tensile   load of 70,000 N.  What minimum diameter is required for it to experience an elongation of no more than 8.2 mm after an exposure for 1,500 h at 650。C? Assume that the sum of instantaneous and primary creep elongations is 0.6 mm. Please use the graph shown to the right.

Problem 2.19

For a Na+–Cl–  ion pair, attractive and repulsive energies EA and ER, respectively, depend on the distance between the ions r, according to

For these expressions, energies are expressed in electron volts per Na+–Cl– pair, and r is the distance in nanometers. The net energy EN  is just the sum of the preceding two expressions.

(a)  USING GRAPHING SOFTWARE (e.g. Excel, Google Sheets), superimpose on a single plot EN, ER, and EA versus r up to 1.0 nm.

(b) On the basis of this plot, determine (i) the equilibrium spacing r0 between the Na+  and Cl–  ions, and (ii) the magnitude of the bonding energy E0  between the two ions.

(c)  Mathematically determine the r0  and E0 values using the solutions to Problem 2.18 (POSTED ON SAKAI IN HOMEWORK 01 FOLDER), and compare these with the graphical results from part (b).

Problem 2.19 (cont.)

Write your answers to parts (b) and (c) here.

Problem 2.21

(a) The net potential energy EN between two adjacent ions is sometimes represented by the expression

in which r is the interionic separation and C, D, and ρ are constants whose values depend on the specific material. Derive an expression for the bonding energy E0 in terms of the equilibrium interionic separation r0 and the constants D and ρ using the following procedure:

(i) Differentiate EN with respect to r, and set the resulting expression equal to zero. (ii) Solve for C in terms of D, ρ, and r0.

(iii) Determine the expression for E0  by substitution for C in the above equation.

Problem 2.21

(b) Derive another expression for E0 in terms of r0, C, and ρ using a procedure analogous to the one outlined in part (a).

Problem 3.4

For the HCP crystal structure, show that the ideal c/a ratio is 1.633.

NOTE: For this problem, you cannot assume that the APF = 0.74 or c = 2a2,3.  These are a result of the c/a ratio being 1.633.

Problem 3.15

Niobium (Nb) has an atomic radius of 0.1430 nm and a density of 8.57 g/cm3. Determine whether it has an FCC or a BCC crystal structure.

Problem 3.27

List the point coordinates of both the zinc (Zn) and sulfur (S) atoms for a unit cell of the zinc blende (ZnS) crystal structure (see below).

Problem 3.36

(a) Determine the indices for the direction shown in the following cubic unit cell.

Problem 3.41

(a) Determine the indices for the direction shown in the following hexagonal unit cell.

Problem 3.41

(b) Determine the indices for the direction shown in the following hexagonal unit cell.

Problem 3.50

(a) Cite the indices of the direction that results from the intersection of the (110) and (111) planes in a cubic crystal system.

Problem 3.50

(b) Cite the indices of the direction that results from the intersection of the (110) and (110) planes in a cubic crystal system.

Non-Callister Problem 1

Cite the indices of the direction that results from the intersection of the (212) and (210) planes.  Hint : You may need to review the cross product and dot product of two vectors (see Lecture 8 slides 32-33).

Problem 3.54

The figure below shows three different crystallographic planes for a unit cell of a hypothetical metal. The circles represent atoms.

(a)  Draw the unit cell.  Show atoms in their correct positions, and label the dimensions of each side of the cell. (b) To what crystal system does the unit cell belong?

(c)  What would this crystal structure be called?

Problem 3.57

Sketch the (0111) and (2110) planes in a hexagonal unit cell.

Problem 3.60

(a) Derive planar density expressions for FCC (100) and (111) planes in terms of the atomic radius R. (b) Compute and compare planar density values for these same two planes for aluminum (Al).

Problem 3.60 (cont.)

Non-Callister Problem 2

(a) It has been discovered that a dislocation in an FCC crystal structure has a Burgers vector b = 2 [110] and a

dislocation line vector [112].  Is this an edge or screw dislocation?  Explain.

Non-Callister Problem 2

(b) As a follow-up to part (a), clearly draw the FCC crystal structure, the slip plane, the dislocation line, and the Burgers vector with correct magnitude and direction.  Label your origin, axes, planes, and directions.

Problem 4.8

(a) Compute the radius r of an impurity atom that will just fit into an FCC octahedral site in terms of the atomic radius R of the host atom (without introducing lattice strains).