(a) Show, for a tensile test, that %CW = a P P + 1 b * 100

To provide some perspective on the dimensions of atomic defects, consider a metal specimen with a dislocation density of 105 mm2 . Suppose that all the dislocations in 1000 mm3 (1 cm3 ) were somehow removed and linked end to end. How far (in miles) would this chain extend? Now suppose that the density is increased to 109 mm2 by cold working. What would be the chain length of dislocations in 1000 mm3 of material?

Consider two edge dislocations of opposite sign and having slip planes separated by several atomic distances, as indicated in the following diagram. Briefly describe the defect that results when these two dislocations become aligned with each other.

Is it possible for two screw dislocations of opposite sign to annihilate each other? Explain your answer.

For each of edge, screw, and mixed dislocations, cite the relationship between the direction of the applied shear stress and the direction of dislocation line motion.

(a) Define a slip system. (b) Do all metals have the same slip system? Why or why not?

(a) Compare planar densities (Section 3.11 and Problem 3.60) for the (100), (110), and (111) planes for FCC. (b) Compare planar densities (Problem 3.61) for the (100), (110), and (111) planes for BCC.

One slip system for the BCC crystal structure is 5110681119. In a manner similar to Figure 7.6b, sketch a 51106-type plane for the BCC structure, representing atom positions with circles. Now, using arrows, indicate two different 81119 slip directions within this plane.

One slip system for the HCP crystal structure is 500016811209. In a manner similar to Figure 7.6b, sketch a 501116-type plane for the HCP structure and, using arrows, indicate three different 811209 slip directions within this plane. You may find Figure 3.9 helpful.

Equations 7.1a and 7.1b, expressions for Burgers vectors for FCC and BCC crystal structures, are of the form b = a 2 8uyw9 where a is the unit cell edge length. The magnitudes of these Burgers vectors may be determined from the following equation: b = a 2 (u2 + y2 + w2 ) 1>2 (7.11) determine the values of b for copper and iron. You may want to consult Table 3.1. 7.

(a) In the manner of Equations 7.1a to 7.1c, specify the Burgers vector for the simple cubic crystal structure whose unit cell is shown in Figure 3.3. Also, simple cubic is the crystal structure for the edge dislocation of Figure 4.4, and for its motion as presented in Figure 7.1. You may also want to consult the answer to Concept Check 7.1. (b) On the basis of Equation 7.11, formulate an expression for the magnitude of the Burgers vector, b , for the simple cubic crystal structure.

Sometimes cosf cosl in Equation 7.2 is termed the Schmid factor. Determine the magnitude of the Schmid factor for an FCC single crystal oriented with its [120] direction parallel to the loading axis.

Consider a metal single crystal oriented such that the normal to the slip plane and the slip direction are at angles of 60 and 35, respectively, with the tensile axis. If the critical resolved shear stress is 6.2 MPa (900 psi), will an applied stress of 12 MPa (1750 psi) cause the single crystal to yield? If not, what stress will be necessary?

A single crystal of zinc is oriented for a tensile test such that its slip plane normal makes an angle of 65 with the tensile axis. Three possible slip directions make angles of 30, 48, and 78 with the same tensile axis. (a) Which of these three slip directions is most favored? (b) If plastic deformation begins at a tensile stress of 2.5 MPa (355 psi), determine the critical resolved shear stress for zinc.

Consider a single crystal of nickel oriented such that a tensile stress is applied along a [001] direction. If slip occurs on a (111) plane and in a [101] direction and is initiated at an applied tensile stress of 13.9 MPa (2020 psi), compute the critical resolved shear stress.

A single crystal of a metal that has the FCC crystal structure is oriented such that a tensile stress is applied parallel to the [100] direction. If the critical resolved shear stress for this material is 0.5 MPa, calculate the magnitude(s) of applied stress(es) necessary to cause slip to occur on the (111) plane in each of the [101], [101], and [011] directions

(a) A single crystal of a metal that has the BCC crystal structure is oriented such that a tensile stress is applied in the [100] direction. If the magnitude of this stress is 4.0 MPa, compute the resolved shear stress in the [111] direction on each of the (110), (011), and (101) planes. (b) On the basis of these resolved shear stress values, which slip system(s) is (are) most favorably oriented?

Consider a single crystal of some hypothetical metal that has the BCC crystal structure and is oriented such that a tensile stress is applied along a [121] direction. If slip occurs on a (101) plane and in a [111] direction, compute the stress at which the crystal yields if its critical resolved shear stress is 2.4 MPa.

Consider a single crystal of some hypothetical metal that has the FCC crystal structure and is oriented such that a tensile stress is applied along a [112] direction. If slip occurs on a (111) plane and in a [011] direction, and the crystal yields at a stress of 5.12 MPa, compute the critical resolved shear stress.

The critical resolved shear stress for copper (Cu) is 0.48 MPa (70 psi). Determine the maximum possible yield strength for a single crystal of Cu pulled in tension.

List four major differences between deformation by twinning and deformation by slip relative to mechanism, conditions of occurrence, and final result.

Briefly explain why small-angle grain boundaries are not as effective in interfering with the slip process as are high-angle grain boundaries.

Briefly explain why HCP metals are typically more brittle than FCC and BCC metals.

Describe in your own words the three strengthening mechanisms discussed in this chapter (i.e., grain size reduction, solid-solution strengthening, and strain hardening). Explain how dislocations are involved in each of the strengthening techniques.

(a) From the plot of yield strength versus (grain diameter)1/2 for a 70 Cu30 Zn cartridge brass in Figure 7.15, determine values for the constants s0 and ky in Equation 7.7. (b) Now predict the yield strength of this alloy when the average grain diameter is 2.0 103 mm.

The lower yield point for an iron that has an average grain diameter of 1 102 mm is 230 MPa (33,000 psi). At a grain diameter of 6 103 mm, the yield point increases to 275 MPa (40,000 psi). At what grain diameter will the lower yield point be 310 MPa (45,000 psi)? 7.

If it is assumed that the plot in Figure 7.15 is for non-cold-worked brass, determine the grain size of the alloy in Figure 7.19; assume its composition is the same as the alloy in Figure 7.15.

In the manner of Figures 7.17b and 7.18b, indicate the location in the vicinity of an edge dislocation at which an interstitial impurity atom would be expected to be situated. Now, briefly explain in terms of lattice strains why it would be situated at this position.

(a) Show, for a tensile test, that %CW = a P P + 1 b * 100 if there is no change in specimen volume during the deformation process (i.e., A0l0  Ad ld). (b) Using the result of part (a), compute the percent cold work experienced by naval brass (for which the stressstrain behavior is shown in Figure 6.12) when a stress of 415 MPa (60,000 psi) is applied.

Two previously undeformed cylindrical specimens of an alloy are to be strain hardened by reducing their cross-sectional areas (while maintaining their circular cross sections). For one specimen, the initial and deformed radii are 15 and 12 mm, respectively. The second specimen, with an initial radius of 11 mm, must have the same deformed hardness as the first specimen; compute the second specimens radius after deformation

Two previously undeformed specimens of the same metal are to be plastically deformed by reducing their cross-sectional areas. One has a circular cross section, and the other is rectangular; during deformation, the circular cross section is to remain circular, and the rectangular is to remain rectangular. Their original and deformed dimensions are as follows: Circular Rectangular (diameter, mm) (mm) Original dimensions 18.0 20 50 Deformed dimensions 15.9 13.7 55.1 Which of these specimens will be the hardest after plastic deformation, and why?

A cylindrical specimen of cold-worked copper has a ductility (%EL) of 15%. If its cold-worked radius is 6.4 mm (0.25 in.), what was its radius before deformation?

2 (a) What is the approximate ductility (%EL) of a brass that has a yield strength of 345 MPa (50,000 psi)? (b) What is the approximate Brinell hardness of a 1040 steel having a yield strength of 620 MPa (90,000 psi)?

Experimentally, it has been observed for single crystals of a number of metals that the critical resolved shear stress tcrss is a function of the dislocation density rD as tcrss = t0 + A1rD (7.12) where t0 and A are constants. For copper, the critical resolved shear stress is 0.69 MPa (100 psi) at a dislocation density of 104 mm2 . If it is known that the value of t0 for copper is 0.069 MPa (10 psi), compute tcrss at a dislocation density of 106 mm2 .

Briefly cite the differences between the recovery and recrystallization processes.

Estimate the fraction of recrystallization from the photomicrograph in Figure 7.21c

Explain the differences in grain structure for a metal that has been cold worked and one that has been cold worked and then recrystallized.

(a) What is the driving force for recrystallization? (b) What is the driving force for grain growth?

(a) From Figure 7.25, compute the length of time required for the average grain diameter to increase from 0.03 to 0.3 mm at 600C for this brass material. (b) Repeat the calculation, this time using 700C.

Consider a hypothetical material that has a grain diameter of 2.1 102 mm. After a heat treatment at 600C for 3 h, the grain diameter has increased to 7.2 102 mm. Compute the grain diameter when a specimen of this same original material (i.e., d0  2.1 102 mm) is heated for 1.7 h at 600C. Assume the n grain diameter exponent has a value of 2. 7.4

A hypothetical metal alloy has a grain diameter of 1.7 102 mm. After a heat treatment at 450C for 250 min, the grain diameter has increased to 4.5 102 mm. Compute the time rerequired for a specimen of this same material (i.e., d0  1.7 102 mm) to achieve a grain diameter of 8.7 102 mm while being heated at 450C. Assume the n grain diameter exponent has a value of 2.1.

The average grain diameter for a brass material was measured as a function of time at 650C, which is shown in the following table at two different times: Time (min) Grain Diameter (mm) 40 5.6 102 100 8.0 102 (a) What was the original grain diameter? (b) What grain diameter would you predict after 200 min at 650C? 7.

An undeformed specimen of some alloy has an average grain diameter of 0.050 mm. You are asked to reduce its average grain diameter to 0.020 mm. Is this possible? If so, explain the procedures you would use and name the processes involved. If it is not possible, explain why.

Grain growth is strongly dependent on temperature (i.e., rate of grain growth increases with increasing temperature), yet temperature is not explicitly included in Equation 7.9. (a) Into which of the parameters in this expression would you expect temperature to be included? (b) On the basis of your intuition, cite an explicit expression for this temperature dependence.

A non-cold-worked brass specimen of average grain size 0.01 mm has a yield strength of 150 MPa (21,750 psi). Estimate the yield strength of this alloy after it has been heated to 500C for 1000 s, if it is known that the value of s0 is 25 MPa (3625 psi).

The following yield strength, grain diameter, and heat treatment time (for grain growth) data were gathered for an iron specimen that was heat treated at 800C. Using these data, compute the yield strength of a specimen that was heated at 800C for 3 h. Assume a value of 2 for n, the grain diameter exponent. Grain Yield Heat diameter Strength Treating (mm) (MPa) Time (h) 0.028 300 10 0.010 385 1

For crystals having cubic symmetry, generate a spreadsheet that allows the user to determine the angle between two crystallographic directions, given their directional indices.

Determine whether it is possible to cold work steel so as to give a minimum Brinell hardness of 240 and at the same time have a ductility of at least 15%EL. Justify your answer.

Determine whether it is possible to cold work brass so as to give a minimum Brinell hardness of 150 and at the same time have a ductility of at least 20%EL. Justify your answer.

A cylindrical specimen of cold-worked steel has a Brinell hardness of 240. (a) Estimate its ductility in percent elongation. (b) If the specimen remained cylindrical during deformation and its original radius was 10 mm (0.40 in.), determine its radius after deformation.

It is necessary to select a metal alloy for an application that requires a yield strength of at least 310 MPa (45,000 psi) while maintaining a minimum ductility (%EL) of 27%. If the metal may be cold worked, decide which of the following are candidates: copper, brass, or a 1040 steel. Why?

A cylindrical rod of 1040 steel originally 11.4 mm (0.45 in.) in diameter is to be cold worked by drawing; the circular cross section will be maintained during deformation. A coldworked tensile strength in excess of 825 MPa (120,000 psi) and a ductility of at least 12%EL are desired. Furthermore, the final diameter must be 8.9 mm (0.35 in.). Explain how this may be accomplished.

A cylindrical rod of brass originally 10.2 mm (0.40 in.) in diameter is to be cold worked by drawing; the circular cross section will be maintained during deformation. A cold-worked yield strength in excess of 380 MPa (55,000 psi) and a ductility of at least 15%EL are desired. Furthermore, the final diameter must be 7.6 mm (0.30 in.). Explain how this may be accomplished.

A cylindrical brass rod having a minimum tensile strength of 450 MPa (65,000 psi), a ductility of at least 13%EL, and a final diameter of 12.7 mm (0.50 in.) is desired. Some brass stock of diameter 19.0 mm (0.75 in.) that has been cold worked 35% is available. Describe the procedure you would follow to obtain this material. Assume that brass experiences cracking at 65%CW.

Consider the brass alloy discussed in Problem 7.41. Given the following yield strengths for the two specimens, compute the heat treatment time required at 650C to give a yield strength of 90 MPa. Assume a value of 2 for n, the grain diameter exponent. Time (min) Yield Strength (MPa) 40 80 100 70

Plastically deforming a metal specimen near room temperature generally leads to which of the following property changes? (A) An increased tensile strength and a decreased ductility (B) A decreased tensile strength and an increased ductility (C) An increased tensile strength and an increased ductility (D) A decreased tensile strength and a decreased ductility

A dislocation formed by adding an extra halfplane of atoms to a crystal is referred to as a (an) (A) screw dislocation (B) vacancy dislocation (C) interstitial dislocation (D) edge dislocation

The atoms surrounding a screw dislocation experience which kinds of strains? (A) Tensile strains (B) Shear strains (C) Compressive strains (D) Both B and C